Introduction to Quantum Noise, Measurement, and Amplification

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REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL–JUNE 2010

Introduction to quantum noise, measurement, and ampliﬁcation

A. A. Clerk* Department of Physics, McGill University, 3600 rue University Montréal, Quebec, Canada H3A 2T8

M. H. Devoret Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA

S. M. Girvin Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA

Florian Marquardt Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, D-80333 München, Germany

R. J. Schoelkopf Department of Applied Physics, Yale University, P.O. Box 208284, New Haven, Connecticut 06520-8284, USA ͑Published 15 April 2010͒

The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, optical–quantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linear-response framework. This approach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.

DOI: 10.1103/RevModPhys.82.1155 PACS number͑s͒: 72.70.ϩm

CONTENTS 1. Heuristic weak-measurement noise constraints 1173 2. Generic linear-response detector 1174 I. Introduction 1156 3. Quantum constraint on noise 1175 II. Quantum Noise Spectra 1159 4. Evading the detector quantum noise A. Introduction to quantum noise 1159 inequality 1176 B. Quantum spectrum analyzers 1161 B. Quantum limit on QND detection of a qubit 1177 III. Quantum Measurements 1162 V. Quantum Limit on Linear Amplifiers and Position A. Weak continuous measurements 1164 Detectors 1178 B. Measurement with a parametrically coupled A. Preliminaries on amplification 1178 resonant cavity 1164 B. Standard Haus-Caves derivation of the quantum 1. QND measurement of the state of a qubit limit on a bosonic amplifier 1179 using a resonant cavity 1167 C. Nondegenerate parametric amplifier 1181 2. Quantum limit relation for QND qubit state 1. Gain and added noise 1181 detection 1168 2. Bandwidth-gain trade-off 1182 3. Measurement of oscillator position using a 3. Effective temperature 1182 resonant cavity 1169 D. Scattering versus op-amp modes of operation 1183 IV. General Linear-Response Theory 1173 E. Linear-response description of a position detector 1185 A. Quantum constraints on noise 1173 1. Detector back-action 1185 2. Total output noise 1186 3. Detector power gain 1186 *[email protected] 4. Simplifications for a quantum-ideal detector 1188

5. Quantum limit on added noise and noise tons off an electron, thereby detecting its position to an temperature 1188 accuracy set by the wavelength of photons used. The F. Quantum limit on the noise temperature of a fact that its momentum will suffer a large uncontrolled voltage amplifier 1190 perturbation, affecting its future motion, is of no con- 1. Classical description of a voltage amplifier 1191 cern here. Only as one tries to further increase the res- 2. Linear-response description 1192 olution will one finally encounter relativistic effects ͑pair 3. Role of noise cross correlations 1193 production͒ that set a limit given by the Compton wave- G. Near quantum-limited mesoscopic detectors 1194 length of the electron. The situation is obviously very 1. dc superconducting quantum interference different if one attempts to observe the whole trajectory device amplifiers 1194 of the particle. As this effectively amounts to measure- 2. Quantum point contact detectors 1194 ments of both position and momentum, there has to be a 3. Single-electron transistors and trade-off between the accuracies of both, set by the resonant-level detectors 1194 Heisenberg uncertainty relation. This is enforced in H. Back-action evasion and noise-free amplification 1195 practice by the uncontrolled perturbation of the momen- 1. Degenerate parametric amplifier 1196 tum during one position measurement adding to the 2. Double-sideband cavity detector 1196 noise in later measurements, a phenomenon known as 3. Stroboscopic measurements 1197 “measurement back-action.” VI. Bosonic Scattering Description of a Two-Port Just such a situation is encountered in “weak mea- Amplifier 1198 surements” ͑Braginsky and Khalili, 1992͒, where one in- A. Scattering versus op-amp representations 1198 tegrates the signal over time, gradually learning more 1. Scattering representation 1198 about the system being measured; this review will focus 2. Op-amp representation 1199 on such measurements. There are many good reasons 3. Converting between representations 1200 why one may be interested in doing a weak measure- B. Minimal two-port scattering amplifier 1201 ment, rather than an instantaneous, strong, projective 1. Scattering versus op-amp quantum limit 1201 2. Why is the op-amp quantum limit not measurement. On a practical level, there may be limita- achieved? 1203 tions to the strength of the coupling between the system VII. Reaching the Quantum Limit in Practice 1203 and the detector, which have to be compensated by in- A. Importance of QND measurements 1203 tegrating the signal over time. One may also deliberately B. Power matching versus noise matching 1204 opt not to disturb the system too strongly, e.g., to be able VIII. Conclusions 1204 to apply quantum feedback techniques for state control. Acknowledgments 1205 Moreover, as one reads out an oscillatory signal over ͑ References 1205 time, one effectively filters away noise e.g., of a technical nature͒ at other frequencies. Finally, consider an example like detection of the collective coordinate of mo- I. INTRODUCTION tion of a micromechanical beam. Its zero-point uncertainty ͑ground-state position fluctuation͒ is typi- Recently several advances have led to a renewed interest in the quantum-mechanical aspects of noise in me- cally on the order of the diameter of a proton. It is out soscopic electrical circuits, detectors, and amplifiers. of the question to reach this accuracy in an instanta- One motivation is that such systems can operate simul- neous measurement by scattering photons of such a taneously at high frequencies and at low temperatures, small wavelength off the structure, since they would in- entering the regime where ប␻Ͼk T. As such, quantum stead resolve the much larger position fluctuations of the B ͑ zero-point fluctuations will play a more dominant role in individual atoms comprising the beam and induce all ͒ determining their behavior than the more familiar ther- kinds of unwanted damage , instead of reading out the mal fluctuations. A second motivation comes from the center-of-mass coordinate. The same holds true for relation between quantum noise and quantum measure- other collective degrees of freedom. ment. There exists an ever-increasing number of experi- The prototypical example we discuss is that of a weak ments in mesoscopic electronics where one is forced to measurement detecting the motion of a harmonic oscil- think about the quantum mechanics of the detection lator ͑such as a mechanical beam͒. The measurement process, and about fundamental quantum limits which then actually follows the slow evolution of amplitude constrain the performance of the detector or amplifier and phase of the oscillations ͑or, equivalently, the two used. quadrature components͒, and the SQL derives from the Given the above, we will focus in this review on dis- fact that these two observables do not commute. It es- cussing what is known as the “standard quantum limit” sentially says that the measurement accuracy will be lim- ͑SQL͒ on both displacement detection and amplifica- ited to resolving both quadratures down to the scale of tion. To preclude any possible confusion, it is worthwhile the ground-state position fluctuations, within one me- to state explicitly from the start that there is no limit to chanical damping time. Note that, in special applica- how well one may resolve the position of a particle in an tions, one might be interested only in one particular instantaneous measurement. Indeed, in the typical quadrature of motion. Then the Heisenberg uncertainty Heisenberg microscope setup, one would scatter pho- relation does not enforce any SQL and one may again

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1157 obtain unlimited accuracy, at the expense of renouncing principle, and realizing in practice, amplifiers whose all knowledge of the other quadrature. noise reaches this minimum quantum limit. Achieving Position detection by weak measurement essentially the quantum limit on continuous position detection has amounts to amplification of the quantum signal up to a been one of the goals of many recent experiments on classically accessible level. Therefore, the theory of quantum electromechanical ͑Cleland et al., 2002; Knobel quantum limits on displacement detection is intimately and Cleland, 2003; LaHaye et al., 2004; Naik et al., 2006; connected to limits on how well an amplifier can work. If Flowers-Jacobs et al., 2007; Etaki et al., 2008; Poggio et an amplifier does not have any preference for any par- al., 2008; Regal et al., 2008͒ and optomechanical systems ticular phase of the oscillatory signal, it is called “phase ͑Arcizet et al., 2006; Gigan et al., 2006; Schliesser et al., preserving,” which is the case relevant for amplifying 2008; Thompson et al., 2008; Marquardt and Girvin, and thereby detecting both quadratures equally well.1 2009͒. As we will show, having a near-quantum-limited We derive and discuss the SQL for phase-preserving lin- detector would allow one to continuously monitor the ear amplifiers ͑Haus and Mullen, 1962; Caves, 1982͒. quantum zero-point fluctuations of a mechanical resona- Quantum mechanics demands that such an amplifier tor. It is also necessary to have a quantum-limited detec- adds noise that corresponds to half a photon added to tor is for such tasks as single-spin NMR detection ͑Ru- each mode of the input signal, in the limit of high gar et al., 2004͒, as well as gravitational wave detection photon-number gain G. In contrast, for small gain, the ͑Abramovici et al., 1992͒. The topic of quantum-limited minimum number of added noise quanta, ͑1−1/G͒/2, detection is also directly relevant to recent activity ex- can become arbitrarily small as the gain is reduced down ploring feedback control of quantum systems ͑Wiseman to 1 ͑no amplification͒. One might ask, therefore, and Milburn, 1993, 1994; Doherty et al., 2000; Korotkov, whether it should not be possible to evade the SQL by 2001b; Ruskov and Korotkov, 2002͒; such schemes re- being content with small gains. The answer is no, since quire need a close-to-quantum-limited detector. high gains Gӷ1 are needed to amplify the signal to a This review is organized as follows. We start in Sec. II level where it can be read out ͑or further amplified͒ us- by providing a review of the basic statistical properties ing classical devices without their noise having any fur- of quantum noise, including its detection. In Sec. III we ther appreciable effect, converting 1 input photon into turn to quantum measurements and give a basic intro- Gӷ1 output photons. According to Caves, it is neces- duction to weak continuous measurements. To make sary to generate an output that “we can lay our grubby, things concrete, we discuss heuristically measurements classical hands on” ͑Caves, 1982͒. It is a simple exercise of both a qubit and an oscillator using a simple resonant to show that feeding the input of a first, potentially low- cavity detector, giving an idea of the origin of the quan- gain amplifier into a second amplifier results in an over- tum limit in each case. Section IV is devoted to a more all bound on the added noise that is just the one ex- rigorous treatment of quantum constraints on noise aris- pected for the product of their respective gains. ing from general quantum linear-response theory. The Therefore, as one approaches the classical level, i.e., heart of the review is contained in Sec. V, where we give large overall gains, the SQL always applies in its simpli- a thorough discussion of quantum limits on amplification fied form of half a photon added. and continuous position detection. We also discuss vari- Unlike traditional discussions of the amplifier SQL, ous methods for beating the usual quantum limits on here we devote considerable attention to a general added noise using back-action evasion techniques. We linear-response approach based on the quantum relation are careful to distinguish two very distinct modes of am- ͑ between susceptibilities and noise. This approach treats plifier operation the “scattering” versus “op-amp” ͒ the amplifier or detector as a black box with an input modes ; we expand on this in Sec. VI, where we discuss port coupling to the signal source and an output port to both modes of operation in a simple two-port bosonic access the amplified signal. It is more suited for mesos- amplifier. Importantly, we show that an amplifier can be copic systems than the quantum optics scattering-type quantum limited in one mode of operation, but fail to be approach, and it leads us to the quantum noise inequal- quantum limited in the other mode of operation. Finally, ity: a relation between the noise added to the output and in Sec. VII we highlight a number of practical consider- the back-action noise feeding back to the signal source. ations that one must keep in mind when trying to per- In the ideal case ͑what we term a “quantum-limited de- form a quantum-limited measurement. Table I provides tector”͒, the product of these two contributions reaches a synopsis of the main results discussed in the text as the minimum value allowed by quantum mechanics. We well as definitions of symbols used. show that optimizing this inequality on noise is a neces- In addition to the above, we have supplemented the sary prerequisite for having a detector achieve the quan- main text with several pedagogical appendixes that tum limit in a specific measurement task, such as linear cover basic background topics. Particular attention is amplification. given to the quantum mechanics of transmission lines There are several motivations for understanding in and driven electromagnetic cavities, topics that are especially relevant given recent experiments making use of microwave stripline resonators. These appendixes are 1In the literature this is often referred to as a “phase insensi- contained in a separate online-only supplementary tive” amplifier. We prefer the term “phase preserving” to avoid document ͑Clerk et al., 2009͒͑see also http://arxiv.org/ any ambiguity. abs/0810.4729͒. In Table II, we list the contents of these

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1158 Clerk et al.: Introduction to quantum noise, measurement, …

TABLE I. Table of symbols and main results.

Symbol Deﬁnition or result

General definitions ͓␻͔ ͑ ͒ ͓␻͔ ͐ϱ ͑ ͒ i␻t f Fourier transform of the function or operator f t , defined via f = −ϱdtf t e ˆ†͓␻͔ ͐ϱ ˆ†͑ ͒ i␻t ˆ†͓␻͔ ͑ˆ͓ ␻͔͒† „Note that for operators, we use the convention f = −ϱdtf t e , implying f = f − … S ͓␻͔ S ͓␻͔ ͐+ϱ i␻t͗ ͑ ͒ ͑ ͒͘ FF Classical noise spectral density or power spectrum: FF = −ϱdte F t F 0 S ͓␻͔ ͓␻͔ ͐+ϱ i␻t͗ ˆ ͑ ͒ ˆ ͑ ͒͘ FF Quantum noise spectral density: SFF = −ϱdte F t F 0 ¯ ͓␻͔ ¯ ͓␻͔ 1 ͑ ͓␻͔ ͓ ␻͔͒ 1 ͐+ϱ i␻t͕͗ ˆ ͑ ͒ ˆ ͑ ͖͒͘ SFF Symmetrized quantum noise spectral density SFF = 2 SFF +SFF − = 2 −ϱdte F t ,F 0 ␹ ͑ ͒ AB t General linear-response susceptibility describing the response of A to a perturbation that couples to B;inthe ␹ ͑ ͒ ͑ ប͒␪͑ ͓͒͗ ˆ ͑ ͒ ˆ ͑ ͔͒͘ ͓ ͑ ͔͒ quantum case, given by the Kubo formula AB t =− i/ t A t ,B 0 Eq. 2.14 A ͑ ͒ ˆ ͑ ͒␴ ˆ ˆ Coupling constant dimensionless between measured system and detector or amplifier,e.g., V=AF t ˆ x, V=AxˆF, ˆ ប␻ ␴ † or V=A c ˆ zaˆ aˆ M,⍀ Mass and angular frequency of a mechanical harmonic oscillator ͱប ⍀ xZPF Zero-point uncertainty of a mechanical oscillator, xZPF= /2M ␥ ˆ ˆ ␥ ͑ 2 ប⍀͒͑ ͓⍀͔ 0 Intrinsic damping rate of a mechanical oscillator due to coupling to a bath via V=AxˆF: 0 = A /2M SFF ͓ ⍀͔͒ ͓ ͑ ͔͒ −SFF − Eq. 2.12 ␻ c Resonant frequency of a cavity ␬ ␻ ␬ ,Qc Damping, quality factor of a cavity: Qc = c / Sec. II ͓␻͔ ␻ ͓␻͔ ͓ ␻͔ Teff Effective temperature at a frequency for a given quantum noise spectrum, defined via SFF /SFF − ͑ប␻ ͓␻͔͒ ͓ ͑ ͔͒ =exp /kBTeff Eq. 2.8 Fluctuation-dissipation theorem relating the symmetrized noise spectrum to the dissipative part for an ¯ ͓␻͔ 1 ͑ប␻ ͒͑ ͓␻͔ ͓ ␻͔͒ ͓ ͑ ͔͒ equilibrium bath: SFF = 2 coth /2kBT SFF −SFF − Eq. 2.16 Sec. III ⌬ ⌬␪ജ 1 ͓ ͑ ͒ ͑ ͔͒ Number-phase uncertainty relation for a coherent state: N 2 Eqs. 3.6 and G12 N˙ Photon-number flux of a coherent beam ␦␪ Imprecision noise in the measurement of the phase of a coherent beam 1 ˙ ˙ ͓ ͑ ͒ ͑ ͔͒ Fundamental noise constraint for an ideal coherent beam: SNNS␪␪= 4 Eqs. 3.8 and G21 ¯ 0 ͑␻͒ Symmetrized spectral density of zero-point position fluctuations of a damped harmonic oscillator Sxx ¯ ͑␻͒ Total output noise spectral density ͑symmetrized͒ of a linear position detector, referred back to the oscillator Sxx,tot ¯ ͑␻͒ Added noise spectral density ͑symmetrized͒ of a linear position detector, referred back to the oscillator Sxx,add Sec. IV xˆ Input signal Fˆ Fluctuating force from the detector, coupling to xˆ via Vˆ =AxˆFˆ Iˆ Detector output signal ¯ ͓␻͔ ¯ ͓␻͔ ͉ ¯ ͓␻͔͉2 General quantum constraint on the detector output noise, back-action noise, and gain: SII SFF − SIF ജ͉ប␹ ͓␻͔ ͉2͑ ⌬ ¯ ͓␻͔ ប␭˜ ͓␻͔ ͓͒ ͑ ͔͒ ␹ ͓␻͔ϵ␹ ͓␻͔ ␹ ͓␻͔ ⌬͓ ͔ ͓͉ 2͉ ͑ ͉ ͉2͔͒ ˜ IF /2 1+ †SIF / /2‡ Eq. 4.11 , where ˜ IF IF −† FI ‡* and z = 1+z − 1+ z /2 ⌬͓ ͔ജ ⌬ ͑ ͒ „Note that 1+ z 0 and =0 in most cases of relevance; see discussion around Eq. 4.17 … ␣ ͉␣͉2 ¯ ¯ ͑ ␣͓␻͔͒ Complex proportionality constant characterizing a quantum-ideal detector: =SII/SFF and sin arg ប͉␭͓␻͔͉ ͱ¯ ͓␻͔ ¯ ͓␻͔͓ ͑ ͒ ͑ ͔͒ = /2/ SII SFF Eqs. 4.18 and I17 ⌫ ͓ ͑ ͒ ͔͓ ͑ ͔͒ meas Measurement rate for a quantum nondemolition QND qubit measurement Eq. 4.24 ⌫␸ Dephasing rate ͑due to measurement back-action͓͒Eqs. ͑3.27͒ and ͑4.19͔͒ ␩ ⌫ ⌫ ഛ ͓ ͑ ͔͒ Constraint on weak, continuous QND qubit state detection: = meas/ ␸ 1 Eq. 4.25 Sec. V G Photon-number ͑power͒ gain, e.g., in Eq. ͑5.7͒ Input-output relation for a bosonic scattering amplifier: bˆ † =ͱGaˆ † +Fˆ † ͓Eq. ͑5.7͔͒

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1159

TABLE I. ͑Continued.͒

Symbol Deﬁnition or result

͑⌬a͒2 ͑⌬ ͒2 ϵ 1 ͕͗ †͖͘ Symmetrized field operator uncertainty for the scattering description of a bosonic amplifier: a 2 aˆ ,aˆ −͉͗a͉͘2 Standard quantum limit for the noise added by a phase-preserving bosonic scattering amplifier in the high-gain ӷ ͗͑⌬ ͒2͘ 1 ͑⌬ ͒2 ജ͑⌬ ͒2 1 ͓ ͑ ͔͒ limit, G 1, where a ZPF= 2 : b /G a + 2 Eq. 5.10 ͓␻͔ ͑ GP Dimensionless power gain of a linear position detector or voltage amplifier maximum ratio of the power delivered by the detector output to a load, vs the power fed into signal source͒: ͓␻͔ ͉␹ ͓␻͔͉2 ͓͑ ␹ ͓␻͔ ␹ ͓␻͔͒ ͓ ͑ ͔͒ GP = IF / 4Im FF Im II Eq. 5.52 Ӎ͓͑ ␣ ͉␣͉͒͑ ប␻͔͒2 ͓ ͑ ͔͒ For a quantum-ideal detector, in the high-gain limit: GP Im / 4 kBTeff/ Eq. 5.57 ¯ ͓␻ ͔ ¯ ͓␻ ͔ ប ͑ប␻ ͒͑ ␹ ͓␻͔͒ ͓ ͑ ͔͒ Sxx,eq ,T Intrinsic equilibrium noise Sxx,eq ,T = coth /2kBT −Im xx Eq. 5.59 ␻ 4 ͓␻͔ Aopt Optimal coupling strength of a linear position detector which minimizes the added noise at frequency : Aopt ¯ ͓␻͔ ͉͑␭͓␻͔␹ ͓␻͔͉2 ¯ ͓␻͔͒ ͓ ͑ ͔͒ =SII / xx SFF Eq. 5.64 ␥͓ ͔ ␥͓ ͔ ␥ Aopt Detector-induced damping of a quantum-limited linear position detector at optimal coupling; satisfies Aopt / 0 ␥͓ ͔ ͉ ␣ ␣͉͑ ͱ ͓⍀͔͒ ប⍀ Ӷ ͓ ͑ ͔͒ + Aopt = Im / 1/ GP = /4kBTeff 1 Eq. 5.69 Standard quantum limit for the added noise spectral density of a linear position detector ͑valid at each frequency ␻͒ ͓␻͔ജ ͓ ͔͓ ͑ ͔͒ : Sxx,add limT→0 Sxx,eq w,T Eq. 5.62 Effective increase in oscillator temperature due to coupling to the detector back-action, for an ideal detector, ប⍀ Ӷ Ӷ ϵ͑␥ ␥ ͒ ͑␥ ␥ ͒→ប⍀ ͓ ͑ ͔͒ with /kB Tbath Teff: Tosc Teff+ 0Tbath / + 0 /4kB +Tbath Eq. 5.70 Zin,Zout Input and output impedances of a linear voltage amplifier Zs Impedance of signal source attached to input of a voltage amplifier ␭ V Voltage gain of a linear voltage amplifier V˜ ͑t͒ Voltage noise of a linear voltage amplifier „Proportional to the intrinsic output noise of the generic ͓ ͑ ͔͒ linear-response detector Eq. 5.81 … I˜͑t͒ Current noise of a linear voltage amplifier „Related to the back-action force noise of the generic linear-response ͓ ͑ ͔͒ detector Eqs. 5.80 … ͓ ͑ ͔͒ TN Noise temperature of an amplifier defined in Eq. 5.74 ͓ ͑ ͔͒ ZN Noise impedance of a linear voltage amplifier Eq. 5.77 ͓␻͔ജប␻ ͓ ͑ ͔͒ Standard quantum limit on the noise temperature of a linear voltage amplifier: kBTN /2 Eq. 5.89 Sec. VI ˆ ͑ ˆ ͒ Voltage at the input ͑output͒ of the amplifier Va Vb Relation to bosonic mode operators: Eq. ͑6.2a͒ ˆ ͑ˆ ͒ Current drawn at the input ͑leaving the output͒ of the amplifier Ia Ib Relation to bosonic mode operators: Eq. ͑6.2b͒ ␭Ј I Reverse current gain of the amplifier s͓␻͔ Input-output 2ϫ2 scattering matrix of the amplifier ͓Eq. ͑6.3͔͒ ␭ ␭Ј ͑ ͒ Relation to op-amp parameters V , I ,Zin,Zout: Eqs. 6.7 ˆ ˆ ͑ ͒ V˜ ͑I˜͒ Voltage current noise operators of the amplifier Fˆ ͓␻͔ Fˆ ͓␻͔ Input ͑output͒ port noise operators in the scattering description ͓Eq. ͑6.3͔͒ a , b ˆ ˆ Relation to op-amp noise operators V˜ ,I˜:Eq.͑6.9͒

Appendixes. Note that, while some aspects of the topics material presented, especially in our discussion of quan- discussed in this review have been studied in the quantum ampliﬁcation ͑see Secs. V. D and VI͒. tum optics and quantum dissipative systems communities and are the subject of several comprehensive books II. QUANTUM NOISE SPECTRA ͑Braginsky and Khalili, 1992; Weiss, 1999; Gardiner and ͒ Zoller, 2000; Haus, 2000 , they are somewhat newer to A. Introduction to quantum noise the condensed matter physics community; moreover, some of the technical machinery developed in these In classical physics, the study of a noisy time- ﬁelds is not directly applicable to the study of quantum dependent quantity invariably involves its spectral den- noise in quantum electronic systems. Finally, note that sity S͓␻͔. The spectral density tells us the intensity of the while this article is a review, there is considerable new noise at a given frequency and is directly related to the

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TABLE II. Contents of online appendix material. Page num- Note, however, that in the quantum case the canonical bers refer to the supplementary material. commutation relation between position and momentum implies there must be some correlations between the Section Page two, namely, ͗xˆ͑0͒pˆ ͑0͒͘−͗pˆ ͑0͒xˆ͑0͒͘=iប. These correla- ˆ ˆ A. Basics of classical and quantum noise 1 tions are easily evaluated by writing x and p in terms of harmonic oscillator ladder operators. We find that in B. Quantum spectrum analyzers: further details 4 thermal equilibrium ͗pˆ ͑0͒xˆ͑0͒͘=−iប/2 and ͗xˆ͑0͒pˆ ͑0͒͘ C. Modes, transmission lines and classical 8 ប input-output theory =+i /2. Not only are the position and momentum correlated, but their correlator is imaginary:3 This means D. Quantum modes and noise of a transmission line 15 that, despite the fact that the position is Hermitian ob- E. Back-action and input-output theory for driven 18 servable with real eigenvalues, its autocorrelation func- damped cavities tion is complex and given from Eq. ͑2.2͒ by F. Information theory and measurement rate 29 G. Number phase uncertainty 30 G ͑t͒ = x2 ͕n ͑ប⍀͒e+i⍀t + ͓n ͑ប⍀͒ +1͔e−i⍀t͖, ͑2.3͒ H. Using feedback to reach the quantum limit 31 xx ZPF B B I. Additional technical details 34 2 ϵប ⍀ where xZPF /2M is the RMS zero-point uncertainty of x in the quantum ground state and nB is the Bose- Einstein occupation factor. The complex nature of the autocorrelation function of the noise.2 In a similar fash- autocorrelation function follows from the fact that the ion, the study of quantum noise involves quantum noise operator xˆ does not commute with itself at different spectral densities. These are defined in a manner that times. mimics the classical case Because the correlator is complex it follows that the ϱ + spectral density is not symmetric in frequency, ͓␻͔ ͵ i␻t͗ ͑ ͒ ͑ ͒͘ ͑ ͒ Sxx = dte xˆ t xˆ 0 . 2.1 −ϱ S ͓␻͔ =2␲x2 ͕n ͑ប⍀͒␦͑␻ + ⍀͒ Here xˆ is a quantum operator ͑in the Heisenberg repre- xx ZPF B ͓ ͑ប⍀͒ ͔␦͑␻ ⍀͖͒ ͑ ͒ sentation͒ whose noise we are interested in, and the an- + nB +1 − . 2.4 gular brackets indicate the quantum statistical average evaluated using the quantum density matrix. Note that In contrast, a classical autocorrelation function is always we use S͓␻͔ throughout this review to denote the spec- real, and hence a classical noise spectral density is al- tral density of a classical noise, while S͓␻͔ will denote a ways symmetric in frequency. Note that in the high- ӷប⍀ ͑ប⍀͒ϳ ͑ប⍀͒ quantum noise spectral density. temperature limit kBT we have nB nB As a simple introductory example illustrating impor- ϳ ប⍀ ͓␻͔ +1 kBT/ . Thus, in this limit Sxx becomes sym- tant differences from the classical limit, consider the po- metric in frequency as expected classically, and coincides sition noise of a simple harmonic oscillator having mass with the classical expression for the position spectral M and frequency ⍀. The oscillator is maintained in equi- density ͓cf. Eq. ͑A12͔͒. librium with a large heat bath at temperature T via some The Bose-Einstein factors suggest a way to under- infinitesimal coupling, which we ignore in considering stand the frequency asymmetry of Eq. ͑2.4͒: the positive- the dynamics. The solutions of the Heisenberg equations frequency part of the spectral density has to do with of motion are the same as for the classical case but with stimulated emission of energy into the oscillator and the the initial position x and momentum p replaced by the negative-frequency part of the spectral density has to do corresponding quantum operators. It follows that the with emission of energy by the oscillator. That is, the position autocorrelation function is positive-frequency part of the spectral density is a mea- ͑ ͒ ͗ ͑ ͒ ͑ ͒͘ ͗ ͑ ͒ ͑ ͒͘ ͑⍀ ͒ sure of the ability of the oscillator to absorb energy, Gxx t = xˆ t xˆ 0 = xˆ 0 xˆ 0 cos t while the negative-frequency part is a measure of the 1 ability of the oscillator to emit energy. As we will see, + ͗pˆ ͑0͒xˆ͑0͒͘ sin͑⍀t͒. ͑2.2͒ M⍀ this is generally true, even for nonthermal states. Figure 1 shows this idea for the case of the voltage noise spec- Classically the second term on the right-hand side tral density of a resistor ͑see Appendix D.3 for more ͑RHS͒ vanishes because in thermal equilibrium x and p details͒. Note that the result Eq. ͑2.4͒ can be extended to are uncorrelated random variables. As we will see the case of a bath of many harmonic oscillators. As de- shortly for the quantum case, the symmetrized ͑some- scribed in Appendix D a resistor can be modeled as an times called the “classical”͒ correlator vanishes in ther- infinite set of harmonic oscillators and from this model mal equilibrium, just as it does classically: ͗xˆpˆ +pˆ xˆ͘=0. the Johnson or Nyquist noise of a resistor can be derived.

2For readers unfamiliar with the basics of classical noise, a compact review is given in Appendix A ͑supplementary 3Notice that this occurs because the product of two noncom- material͒. muting Hermitian operators is not itself a Hermitian operator.

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10 tailed balance relation holds. However, if we are con- 8 cerned only with a single particular frequency, then it is 6 S [ ]/2Rk T always possible to define an “effective temperature” Teff VV B 4 for the noise using Eq. ͑2.7͒, i.e., 2 0 ប␻ k T ͓␻͔ϵ . ͑2.8͒ -10 -5 0 5 10 B eff ͓␻͔ ͓ ␻͔ log SFF /SFF − h/kT † ‡ <0 B >0 emission absorption Note that for a nonequilibrium system Teff will in gen- by reservoir by reservoir eral be frequency dependent. In NMR language, Teff will simply be the “spin temperature” of our TLS spectrom- FIG. 1. Quantum noise spectral density of voltage fluctuations eter once it reaches steady state after being coupled to across a resistor ͑resistance R͒ as a function of frequency at zero temperature ͑dashed line͒ and finite temperature ͑solid the noise source. line͒. Another simple quantum noise spectrometer is a harmonic oscillator ͑frequency ⍀, mass M, and position x͒ coupled to a noise source ͓see, e.g., Schwinger ͑1961͒ B. Quantum spectrum analyzers and Dykman ͑1978͔͒. The coupling Hamiltonian is now The qualitative picture described previously can be Vˆ = AxˆFˆ = A͓x ͑aˆ + aˆ †͔͒Fˆ , ͑2.9͒ confirmed by considering simple systems which act as ZPF effective spectrum analyzers of quantum noise. The sim- ˆ ˆ plest such example is a quantum two-level system ͑TLS͒ where a is the oscillator annihilation operator, F is the coupled to a quantum noise source ͑Aguado and Kou- operator describing the fluctuating noise, and A is again wenhoven, 2000; Gavish et al., 2000; Schoelkopf et al., a coupling constant. We see that −AFˆ plays the role of a 2003͒. With the TLS described as a fictitious spin-1/2 fluctuating force acting on the oscillator. In complete particle with spin down ͑spin up͒ representing the analogy to the previous section, noise in Fˆ at the oscil- ͑ ͒ ˆ ⍀ ground state excited state , its Hamiltonian is H0 lator frequency can cause transitions between the os- ͑ប␻ ͒␴ ប␻ = 01/2 ˆ z, where 01 is the energy splitting between cillator energy eigenstates. The corresponding Fermi the two states. The TLS is then coupled to an external golden rule transition rates are again simply related to ͓␻͔ noise source via an additional term in the Hamiltonian, the noise spectrum SFF . Incorporating these rates into a simple master equation describing the probability ˆ ˆ ␴ ͑ ͒ V = AF ˆ x, 2.5 to find the oscillator in a particular energy state, one finds that the stationary state of the oscillator is a Bose- where A is a coupling constant and the operator Fˆ rep- Einstein distribution evaluated at the effective tempera- ͓⍀͔ ͑ ͒ resents the external noise source. The coupling Hamil- ture Teff defined in Eq. 2.8 . Further, one can use the tonian Vˆ can lead to the exchange of energy between the master equation to derive a classical-looking equation two-level system and noise source and hence transitions for the average energy ͗E͘ of the oscillator ͑see Appen- between its two eigenstates. The corresponding Fermi dix B.2͒, golden rule transition rates can be compactly expressed d͗E͘/dt = P − ␥͗E͘, ͑2.10͒ in terms of the quantum noise spectral density of Fˆ , ͓␻͔ SFF , where 2 2 ⌫↑ = ͑A /ប ͒S ͓− ␻ ͔, ͑2.6a͒ FF 01 ͑ 2 ͒͑ ͓⍀͔ ͓ ⍀͔͒ ϵ 2 ¯ ͓⍀͔ P = A /4M SFF + SFF − A SFF /2M, ⌫ ͑ 2 ប2͒ ͓ ␻ ͔ ͑ ͒ ↓ = A / SFF + 01 . 2.6b ͑2.11͒

Here ⌫↑ is the rate at which the qubit is excited from its ␥ ͑ 2 2 ប2͒͑ ͓⍀͔ ͓ ⍀͔͒ ͑ ͒ ground to excited state and ⌫↓ is the corresponding rate = A xZPF/ SFF − SFF − . 2.12 for the opposite relaxation process. As expected, ͑ ͒ positive- ͑negative-͒ frequency noise corresponds to ab- The two terms in Eq. 2.10 describe, heating and damp- sorption ͑emission͒ of energy by the noise source. Note ing of the oscillator by the noise source, respectively. that if the noise source is in thermal equilibrium at tem- The heating effect of the noise is completely analogous perature T, the transition rates of the TLS must satisfy to what happens classically: a random force causes the −␤ប␻ oscillator’s momentum to diffuse, which in turn causes the detailed balance relation ⌫↑ /⌫↓ =e 01, where ␤ ͗E͘ to grow linearly in time at rate proportional to the =1/kBT. This in turn implies that in thermal equilibrium the quantum noise spectral density must satisfy force noise spectral density. In the quantum case, Eq. ͑2.11͒ indicates that it is the symmetric-in-frequency part ␤ប␻ S ͓+ ␻ ͔ = e 01S ͓− ␻ ͔. ͑2.7͒ ¯ ͓⍀͔ FF 01 FF 01 of the noise spectrum, SFF , which is responsible for The more general situation is where the noise source is this effect, and which thus plays the role of a classical ¯ ͓␻͔ not in thermal equilibrium; in this case, no general de- noise source. This is another reason why SFF is often

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referred to as the “classical” part of the noise.4 In con- A2S¯ ͓⍀͔ =2k TM␥. ͑2.17͒ trast, we see that the asymmetric-in-frequency part of FF B the noise spectrum is responsible for the damping. This Further insight into the ﬂuctuation-dissipation theorem also has a simple heuristic interpretation: damping is is provided in Appendix C.3, where we discuss it in the caused by the net tendency of the noise source to ab- simple but instructive context of a transmission line ter- sorb, rather than emit, energy from the oscillator. minated by an impedance Z͓␻͔. The damping induced by the noise source may equiva- We have thus considered two simple examples of lently be attributed to the oscillator’s motion inducing an methods of measure quantum noise spectral densities. ͗ ͘ average value to F which is out of phase with x, i.e., Further details, as well as examples of other quantum ␦͗AF͑t͒͘=−M␥x˙͑t͒. Standard quantum linear-response noise spectrum analyzers, are given in Appendix B. theory yields

␦͗AFˆ ͑t͒͘ = A2 ͵ dtЈ␹ ͑t − tЈ͒͗xˆ͑tЈ͒͘, ͑2.13͒ FF III. QUANTUM MEASUREMENTS

where we have introduced the susceptibility Having introduced both quantum noise and quantum spectrum analyzers, we are now in a position to intro- ␹ ͑ ͒ ͑ ប͒␪͑ ͓͒͗ ˆ ͑ ͒ ˆ ͑ ͔͒͘ ͑ ͒ FF t = − i/ t F t ,F 0 . 2.14 duce the general topic of quantum measurements. All practical measurements are affected by noise. Certain Using the fact that the oscillator’s motion involves only quantum measurements remain limited by quantum the frequency ⍀, we thus have noise even though they use completely ideal apparatus. As we will see, the limiting noise here is associated with ␥ ͑ 2 2 ប͒ ␹ ͓⍀͔ ͑ ͒ = 2A xZPF/ †−Im FF ‡. 2.15 the fact that canonically conjugate variables are incom- patible observables in quantum mechanics. ͑ ͒ ␹ A straightforward manipulation of Eq. 2.14 for FF The simplest idealized description of a quantum mea- shows that this expression for ␥ is exactly equivalent to surement, introduced by von Neumann ͑von Neumann, our previous expression, Eq. ͑2.12͒. 1932; Wheeler and Zurek, 1984; Bohm, 1989; Haroche In addition to giving insight on the meaning of the and Raimond, 2006͒, postulates that the measurement symmetric and asymmetric parts of a quantum noise process instantaneously collapses the system’s quantum spectral density, the above example also directly yields state onto one of the eigenstates of the observable to be the quantum version of the fluctuation-dissipation theo- measured. As a consequence, any initial superposition of rem ͑Callen and Welton, 1951͒. As we saw earlier, if our these eigenstates is destroyed and the values of observ- noise source is in thermal equilibrium, the positive- and ables conjugate to the measured observable are per- negative-frequency parts of the noise spectrum are turbed. This perturbation is an intrinsic feature of quan- strictly related to one another by the condition of de- tum mechanics and cannot be avoided in any tailed balance ͓cf. Eq. ͑2.7͔͒. This in turn lets us link the measurement scheme, be it of the projection type de- classical symmetric-in-frequency part of the noise to the scribed by von Neumann or a rather weak continuous damping ͑i.e., the asymmetric-in-frequency part of the measurement to be analyzed further below. ͒ ␤ ͑ ͒ noise . Setting =1/kBT and making use of Eq. 2.7 ,we To form a more concrete picture of quantum measure- have ment, we begin by noting that every quantum measurement apparatus consists of a macroscopic “pointer” S ͓⍀͔ + S ͓− ⍀͔ coupled to the microscopic system to be measured. ͓A S¯ ͓⍀͔ϵ FF FF FF 2 specific model is discussed by Allahverdyan et al. ͑2001͒.͔ This pointer is sufficiently macroscopic that its 1 ͑␤ប⍀ ͒͑ ͓⍀͔ ͓ ⍀͔͒ = 2 coth /2 SFF − SFF − position can be read out “classically.” The interaction ប⍀M between the microscopic system and the pointer is ar- = coth͑␤ប⍀/2͒ ␥͓⍀͔. ͑2.16͒ ranged so that the two become strongly correlated. One A2 of the simplest possible examples of a quantum mea- Thus, in equilibrium, the condition that noise-induced surement is that of the Stern-Gerlach apparatus, which transitions obey detailed balance immediately implies measures the projection of the spin of an S=1/2 atom that noise and damping are related to one another via along some chosen direction. What is really measured in the temperature. For Tӷប⍀, we recover the more fa- the experiment is the final position of the atom on the miliar classical version of the fluctuation-dissipation detector plate. However, the magnetic field gradient in theorem the magnet causes this position to be perfectly correlated ͑“entangled”͒ with the spin projection so that the latter can be inferred from the former. Suppose, for ex- ϱ 4Note that with our definition ͗Fˆ 2͘=͐ ͑d␻/2␲͒S¯ ͓␻͔.Itis ample that the initial state of the atom is the product of −ϱ FF ␰ ͑ជ͒ common in engineering contexts to define so-called one-sided a spatial wave function 0 r centered on the entrance to classical spectral densities, which are equal to twice our the magnet and a spin state that is the superposition of ␴ definition. up and down spins corresponding to the eigenstate of ˆ x,

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+d providing a way to confirm the accuracy of the measurement scheme. These optimal kinds of repeatable mea- z z surements are called quantum nondemolition ͑QND͒ measurements ͑Braginsky et al., 1980; Braginsky and FIG. 2. ͑Color online͒ Schematic of position distributions of an Khalili, 1992, 1996; Peres, 1993͒. A simple example atom in the detector plane of a Stern-Gerlach apparatus whose field gradient is in the z direction. For small values of the would be a sequential pair of Stern-Gerlach devices ori- displacement d ͑described in the text͒, there is significant over- ented in the same direction. In the absence of stray mag- lap of the distributions and the spin cannot be unambiguously netic perturbations, the second apparatus would always inferred from the position. For large values of d, the spin is yield the same answer as the first. In practice, one terms perfectly entangled with position and can be inferred from the a measurement QND if the observable being measured position. This is the limit of strong projective measurement. is an eigenstate of the ideal Hamiltonian of the measured system ͑i.e., one ignores any couplings between this system and sources of dissipation͒. This is reason- ͉⌿ ͘ ͑ ͱ ͕͉͒↑͘ ͉↓͖͉͘␰ ͘ ͑ ͒ 0 = 1/ 2 + 0 . 3.1 able if such couplings give rise to dynamics on time After passing through a magnet with field gradient in the scales longer than needed to complete the measurement. z direction, an atom with spin up is deflected upward This point is elaborated in Sec. VII, where we discuss and an atom with spin down is deflected downward. By practical considerations related to the quantum limit. the linearity of quantum mechanics, an atom in a spin We also discuss in that section the fact that the repeat- superposition state thus ends up in a superposition of ability of QND measurements is of fundamental practi- the form cal importance in overcoming detector inefficiencies ͑Gambetta et al., 2007͒. ͉⌿ ͘ ͑ ͱ ͕͉͒↑͉͘␰ ͘ ͉↓͉͘␰ ͖͘ ͑ ͒ 1 = 1/ 2 + + − , 3.2 A common confusion is to think that a QND measurement has no effect on the state of the system being mea- where ͗rជ͉␰ ͘=␺ ͑rជ±dzˆ͒ are spatial orbitals peaked in the ± 1 sured. While this is true if the initial state is an eigen- plane of the detector. The deflection d is determined by state of the observable, it is not true in general. Consider the device geometry and the magnetic field gradient. again our example of a spin oriented in the x direction. The z-direction position distribution of the particle for The result of the first ␴ˆ measurement will be that the each spin component is shown in Fig. 2.Ifd is suffi- z state randomly and completely unpredictably collapses ciently large compared to the wave packet spread then, onto one of the two ␴ˆ eigenstates: the state is indeed given the position of the particle, one can unambigu- z altered by the measurement. However, all subsequent ously determine the distribution from which it came and measurements using the same orientation for the detec- hence the value of the spin projection of the atom. This tors will always agree with the result of the first mea- is the limit of a strong “projective” measurement. surement. Thus QND measurements may affect the In the initial state one has ͗⌿ ͉␴ˆ ͉⌿ ͘=+1, but in the 0 x 0 state of the system, but never the value of the observ- final state one has able ͑once it is determined͒. Other examples of QND ͗⌿ ͉␴ ͉⌿ ͘ 1 ͕͗␰ ͉␰ ͘ ͗␰ ͉␰ ͖͘ ͑ ͒ measurements include ͑i͒ measurement of the electro- 1 ˆ x 1 = 2 − + + + − . 3.3 magnetic field energy stored inside a cavity by determin- ␰ For sufficiently large d the states ± are orthogonal and ing the radiation pressure exerted on a moving piston ␴ thus the act of ˆ z measurement destroys the spin coher- ͑Braginsky and Khalili, 1992͒, ͑ii͒ detection of the presence ence of a photon in a cavity by its effect on the phase of ͑ ͗⌿ ͉␴ˆ ͉⌿ ͘ → 0. ͑3.4͒ an atom’s superposition state Nogues et al., 1999; 1 x 1 Haroche and Raimond, 2006͒, and ͑iii͒ the “dispersive” This is what we mean by projection or wave function measurement of a qubit state by its effect on the fre- “collapse.” The result of measurement of the atom po- quency of a driven microwave resonator ͑Blais et al., 1 ͒ sition will yield a random and unpredictable value of ± 2 2004; Wallraff et al., 2004; Lupas¸cu et al., 2007 , which is for the z projection of the spin. This destruction of the the first canonical example described below. coherence in the transverse spin components by a strong In contrast to the above, in non-QND measurements measurement of the longitudinal spin component is the the back-action of the measurement will affect the ob- first of many examples we will see of the Heisenberg servable being studied. The canonical example we con- uncertainty principle in action. Measurement of one sider is the position measurement of a harmonic oscilla- variable destroys information about its conjugate vari- tor. Since the position operator does not commute with able. We study several examples in which we understand the Hamiltonian, the QND criterion is not satisfied. microscopically how it is that the coupling to the mea- Other examples of non-QND measurements include ͑i͒ surement apparatus causes the back-action quantum photon counting via photodetectors that absorb the pho- noise which destroys our knowledge of the conjugate tons, ͑ii͒ continuous measurements where the observ- variable. able does not commute with the Hamiltonian, thus in- In the special case where the eigenstates of the ob- ducing a time dependence of the measurement result, servable we are measuring are also stationary states ͑i.e., and ͑iii͒ measurements that can be repeated only after a energy eigenstates͒, a second measurement of the ob- time longer than the energy relaxation time of the sys- ͑ ͒ servable would reproduce exactly the same result, thus tem e.g., for a qubit, T1 .

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A. Weak continuous measurements saw that a crucial feature of quantum noise is the asymmetry between positive and negative frequencies; we In discussing “real” quantum measurements, another further saw that this corresponds to the difference be- key notion to introduce is that of weak continuous mea- tween absorption and emission events. For measure- surements ͑Braginsky and Khalili, 1992͒. Many measurements, another key aspect of quantum noise will be im- ments in practice take an extended time interval to com- portant: as will be discussed extensively, quantum plete, which is much longer than the “microscopic” time mechanics places constraints on the noise of any system scales ͑oscillation periods, etc.͒ of the system. The rea- capable of acting as a detector or amplifier. These con- son may be quite simply that the coupling strength be- straints in turn place restrictions on any weak continu- tween the detector and the system cannot be made arbi- ous measurement, and lead directly to quantum limits trarily large, and one has to wait for the effect of the on how well one can make such a measurement. system on the detector to accumulate. For example, in In the remainder of this section, we give an introduc- our Stern-Gerlach measurement, suppose that we are tion to how one describes a weak continuous quantum only able to achieve small magnetic field gradients and measurement, considering the specific examples of using that, consequently, the displacement d cannot be made parametric coupling to a resonant cavity for QND detec- large compared to the wave packet spread ͑see Fig. 2͒. tion of the state of a qubit and the ͑necessarily non- ͒ In this case the states ␰ would have nonzero overlap QND detection of the position of a harmonic oscillator. ± ͑ ͒ and it would not be possible to reliably distinguish them: In the following section Sec. IV , we give a derivation we thus would have only a weak measurement. How- of a very general quantum-mechanical constraint on the ever, by cascading together a series of such measure- noise of any system capable of acting as a detector, and ments and taking advantage of the fact that they are show how this constraint directly leads to the quantum QND, we can eventually achieve an unambiguous strong limit on qubit detection. Finally, in Sec. V, we turn to the projective measurement: at the end of the cascade, we important but slightly more involved case of a quantum ␴ linear amplifier or position detector. We show that the are certain of which ˆ z eigenstate the spin is in. During ␰ basic quantum noise constraint derived Sec. IV again this process, the overlap of ± would gradually fall to zero corresponding to a smooth continuous loss of phase leads to a quantum limit; here this limit is on how small coherence in the transverse spin components. At the end one can make the added noise of a linear amplifier. of the process, the QND nature of the measurement en- Before leaving this section, it is worth pointing out ␴ ↑ ↓ that the theory of weak continuous measurements is sures that the probability of measuring z = or will accurately reflect the initial wave function. Note that it is sometimes described in terms of some set of auxiliary only in this case of weak continuous measurements that systems which are sequentially and momentarily weakly coupled to the system being measured ͑see Appendix it makes sense to define a measurement rate in terms of ͒ a rate of gain of information about the variable being E . One then envisions a sequence of projective von measured, and a corresponding dephasing rate, the rate Neumann measurements on the auxiliary variables. The at which information about the conjugate variable is be- weak entanglement between the system of interest and ing lost. We see that these rates are intimately related one of the auxiliary variables leads to a kind of partial collapse of the system wave function ͑more precisely, the via the Heisenberg uncertainty principle. ͒ While strong projective measurements may seem to density matrix , which is described in mathematical terms not by projection operators, but rather by positive be the ideal, there are many cases where one may inten- ͑ ͒ tionally desire a weak continuous measurement; as dis- operator valued measures POVMs . We will not use cussed in the Introduction. There are many practical ex- this and the related “quantum trajectory” language here, but direct the interested reader to the literature for amples of weak continuous measurement schemes. ͑ These include ͑i͒ charge measurements, where the cur- more information on this important approach Peres, ͓ 1993; Brun, 2002; Haroche and Raimond, 2006; Jordan rent through a device e.g., quantum point contact ͒ ͑QPC͒ or single-electron transistor ͑SET͔͒ is modulated and Korotkov, 2006 . by the presence or absence of a nearby charge, and where it is necessary to wait for a sufficiently long time to overcome the shot noise and distinguish between the B. Measurement with a parametrically coupled resonant two current values, ͑ii͒ the weak dispersive qubit mea- cavity surement discussed below, and ͑iii͒ displacement detection of a nanomechanical beam ͑e.g., optically or by ca- A simple yet experimentally practical example of a pacitive coupling to a charge sensor͒, where one looks at quantum detector consists of a resonant optical or rf the two quadrature amplitudes of the signal produced at cavity parametrically coupled to the system being mea- the beam’s resonance frequency. sured. Changes in the variable being measured ͑e.g., the Not surprisingly, quantum noise plays a crucial role in state of a qubit or the position of an oscillator͒ shift the determining the properties of a weak continuous quan- cavity frequency and produce a varying phase shift in tum measurement. For such measurements, noise both the carrier signal reflected from the cavity. This changing determines the back-action effect of the measurement phase shift can be converted ͑via homodyne interferom- on the measured system and how quickly information is etry͒ into a changing intensity; this can then be detected acquired in the measurement process. Previously, we using diodes or photomultipliers.

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In this section, we analyze weak continuous measure- ther, coherent states are overcomplete and states of dif- ments made using such a parametric cavity detector; this ferent phase are not orthogonal to each other; this di- will serve as a good introduction to the more general rectly implies ͑see Appendix G͒ that there is an approaches presented later. We show that this detector is uncertainty in any measurement of the phase. For large capable of reaching the “quantum limit” meaning that it N¯ , this is given by can be used to make a weak continuous measurement, as optimally as is allowed by quantum mechanics. This is ͑⌬␪͒2 = 1/4N¯ . ͑3.5͒ true for both the ͑QND͒ measurement of the state of a qubit and the ͑non-QND͒ measurement of the position Thus, large-N¯ coherent states obey the number-phase of a harmonic oscillator. Complementary analyses of uncertainty relation weak continuous qubit measurement are given by Ma- ͑ ͒ ͑ ⌬ ⌬␪ 1 ͑ ͒ khlin et al. 2000, 2001 using a single-electron transis- N = 2 , 3.6 tor͒ and by Gurvitz ͑1997͒, Korotkov ͑2001b͒, Korotkov and Averin ͑2001͒, Pilgram and Büttiker ͑2002͒, and analogous to the position-momentum uncertainty rela- Clerk et al. ͑2003͒ ͑using a quantum point contact͒.We tion. Equation ͑3.6͒ can also be usefully formulated in focus here on a high-Q cavity detector; weak qubit mea- terms of noise spectral densities associated with the surement with a low-Q cavity was studied by Johansson measurement. Consider a continuous photon beam car- et al. ͑2006͒. ¯ It is worth noting the widespread usage of cavity de- rying an average photon flux N˙ . The variance in the tectors in experiment. One important current realization number of photons detected grows linearly in time and ͑⌬ ͒2 is a microwave cavity used to read out the state of a can be represented as N =SN˙ N˙ t, where SN˙ N˙ is the superconducting qubit ͑Il’ichev et al., 2003; Blais et al., white-noise spectral density of photon flux fluctuations. 2004; Izmalkov et al., 2004; Lupas¸cu et al., 2004, 2005; On a physical level, it describes photon shot noise, and is Wallraff et al., 2004; Duty et al., 2005; Schuster et al., ¯ given by S ˙ ˙ =N˙ . 2005; Sillanpää et al., 2005͒. Another class of examples NN Consider now the phase of the beam. Any homodyne are optical cavities used to measure mechanical degree measurement of this phase will be subject to the same of freedom. Examples of such systems include those photon shot noise fluctuations discussed above ͑see Ap- where one of the cavity mirrors is mounted on a canti- pendix G for more details͒. Thus, if the phase of the lever ͑Arcizet et al., 2006; Gigan et al., 2006; Kleckner beam has some nominal small value ␪ , the output signal and Bouwmeester, 2006͒. Related systems involve a 0 freely suspended mass ͑Abramovici et al., 1992; Corbitt from the homodyne detector integrated up to time t will ␪ ͐t ␶␦␪͑␶͒ ␦␪ et al., 2007͒, an optical cavity with a thin transparent be of the form I= 0t+ 0d , where is a noise ␪ membrane in the middle ͑Thompson et al., 2008͒, and, representing the imprecision in our measurement of 0 more generally, an elastically deformable whispering gal- due to the photon shot noise in the output of the homo- lery mode resonator ͑Schliesser et al., 2006͒. Systems dyne detector. An unbiased estimate of the phase ob- ␪ ͗␪͘ ␪ where a microwave cavity is coupled to a mechanical tained from I is =I/t, which obeys = 0. Further, one 2 element are also under active study ͑Blencowe and has ͑⌬␪͒ =S␪␪/t, where S␪␪ is the spectral density of the Buks, 2007; Regal et al., 2008; Teufel et al., 2008͒. ␦␪ white noise. Comparison with Eq. ͑3.5͒ yields We start our discussion with a general observation. ¯ The cavity uses interference and the wave nature of light S␪␪ = 1/4N˙ . ͑3.7͒ to convert the input signal to a phase-shifted wave. For small phase shifts we have a weak continuous measure- The above results lead us to the fundamental wave- ment. Interestingly, it is the complementary particle na- particle relation for ideal coherent beams, ture of light that turns out to limit the measurement. As ͱ 1 S ˙ ˙ S␪␪ = . ͑3.8͒ we will see, it both limits the rate at which we can make NN 2 a measurement ͑via photon shot noise in the output Before we study the role that these uncertainty rela- beam͒ and also controls the back-action disturbance of tions play in measurements with high-Q cavities, con- the system being measured ͑due to photon shot noise sider the simplest case of reflection of light from a mir- inside the cavity acting on the system being measured͒. ror without a cavity. The phase shift of the beam ͑having These dual aspects are an important part of any weak wave vector k͒ when the mirror moves a distance x is continuous quantum measurement; hence, an under- 2kx. Thus, the uncertainty in the phase measurement standing of both the output noise ͑i.e., the measurement corresponds to a position imprecision which can again imprecision͒ and back-action noise of detectors will be I be represented in terms of a noise spectral density Sxx 2 crucial. =S␪␪/4k . Here the superscript I refers to the fact that All of our discussion of noise in the cavity system will this is noise representing imprecision in the measure- be framed in terms of the number-phase uncertainty re- ment, not actual fluctuations in the position. We also lation for coherent states. A coherent photon state con- need to worry about back-action: each photon hitting tains a Poisson distribution of the number of photons, the mirror transfers a momentum 2បk to the mirror, so implying that the fluctuations in photon number obey photon shot noise corresponds to a random back-action ͑⌬ ͒2 ¯ ¯ ប2 2 N =N, where N is the mean number of photons. Fur- force noise spectral density SFF=4 k SN˙ N˙ . Multiplying

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these together, we have the central result for the product being perfectly reflecting. In this case, a wave incident of the back-action force noise and the imprecision, on the cavity ͓say, in a one-dimensional ͑1D͒ waveguide͔ ␪ I 2 2 will be perfectly reflected, but with a phase shift deter- S S = ប S ˙ ˙ S␪␪ = ប /4, ͑3.9͒ FF xx NN mined by the cavity and the value of zˆ. The reflection ␻ or in analogy with Eq. ͑3.6͒ coefficient at the bare cavity frequency c is simply given by ͑Walls and Milburn, 1994͒ ͱS SI = ប/2. ͑3.10͒ FF xx ͑ ͒ ͑ ͒ ͑ ͒ r =− 1+2iAQczˆ / 1−2iAQczˆ . 3.12 Not surprisingly, the situation considered here is as ideal as possible. Thus, the RHS above is actually a lower Note that r has unit magnitude because all incident pho- bound on the product of imprecision and back-action tons are reflected if the cavity is lossless. For weak cou- noise for any detector capable of significant amplifica- pling we can write the reflection phase shift as r=−ei␪, tion; we will prove this rigorously in Sec. IV.A Equation where ͑3.10͒ thus represents the quantum limit on the noise of ␪ Ϸ ͑ ␻ ͒ ͑ ͒ our detector. As we will see shortly, having a detector 4QcAzˆ = A czˆ tWD. 3.13 with quantum-limited noise is a prerequisite for reaching We see that the scattering phase shift is simply the fre- the quantum limit on various different measurement quency shift caused by the parametric coupling multi- tasks ͑e.g., continuous position detection of an oscillator plied by the Wigner delay time ͑Wigner, 1955͒ and QND qubit state detection͒. Note that in general, a ͒ ͑ ␻ ␬ץ ץ -given detector will have more noise than the quantum tWD =Im ln r/ =4/ . 3.14 limited value; we devote considerable effort later to determining the conditions needed to achieve the lower Thus the measurement-imprecision noise power for a bound of Eq. ͑3.10͒. ¯ given photon flux N˙ incident on the cavity is given by We now turn to the story of measurement using a high-Q cavity; it will be similar to the above discussion, I ͑ ␻ ͒2 ͑ ͒ Szz = S␪␪/ A ctWD . 3.15 except that we have to account for the filtering of the noise by the cavity response. We relegate relevant calcu- The random part of the generalized back-action force lational details to Appendix E. The cavity is simply de- conjugate to zˆ is, from Eq. ͑3.11͒, scribed as a single bosonic mode coupled weakly to elec- ͒ ͑ ␦ ប␻ ץ ˆ ץ ϵ ˆ tromagnetic modes outside the cavity. The Hamiltonian Fz − H/ zˆ =−A c nˆ , 3.16 of the system is given by ˆ where, since zˆ is dimensionless, Fz has units of energy. ˆ ប␻ ͑ ͒ † ˆ ͑ ͒ H = H0 + c 1+Azˆ aˆ aˆ + Henvt. 3.11 Here ␦nˆ =nˆ −n¯ =aˆ †aˆ −͗aˆ †aˆ͘ represents the photon- number fluctuations around the mean n¯ inside the cavity. Here H0 is the unperturbed Hamiltonian of the system whose variable zˆ ͑which is not necessarily a position͒ is The back-action force noise spectral density is thus being measured, aˆ is the annihilation operator for the S = ͑Aប␻ ͒2S . ͑3.17͒ ␻ FzFz c nn cavity mode, and c is the cavity resonance frequency in the absence of the coupling A. We will take both A and As shown in Appendix E, the cavity filters the photon ˆ ␻Ӷ␬ zˆ to be dimensionless. The term Henvt describes the elec- shot noise so that at low frequencies , the number tromagnetic modes outside the cavity, and their coupling fluctuation spectral density is simply to the cavity; it is responsible for both driving and damp- ͑ ͒ ing the cavity mode. The damping is parametrized by Snn = n¯ tWD. 3.18 ␬ rate , which tells us how quickly energy leaks out of the The mean photon number in the cavity is found to be cavity; we consider the case of a high-quality factor cav- ¯ ¯ ϵ␻ ␬ӷ n¯ =N˙ t , where again N˙ is the mean photon flux inci- ity, where Qc c / 1. WD Turning to the interaction term in Eq. ͑3.11͒,wesee dent on the cavity. From this it follows that that the parametric coupling strength A determines the S = ͑Aប␻ t ͒2S ˙ ˙ . ͑3.19͒ change in frequency of the cavity as the system variable FzFz c WD NN zˆ changes. We assume for simplicity that the dynamics of Combining this with Eq. ͑3.15͒ again yields the same zˆ is slow compared to ␬. In this limit the reflected phase result as Eq. ͑3.10͒ obtained without the cavity. The shift simply varies slowly in time adiabatically following parametric cavity detector ͑used in this way͒ is thus a the instantaneous value of zˆ. We also assume that the quantum-limited detector, meaning that the product of coupling A is small enough that the phase shifts are al- its noise spectral densities achieves the ideal minimum ways very small and hence the measurement is weak. value. Many photons will have to pass through the cavity be- We now examine how the quantum limit on the noise fore much information is gained about the value of the of our detector directly leads to quantum limits on dif- phase shift and hence the value of zˆ. ferent measurement tasks. In particular, we consider the We first consider the case of a one-sided cavity where cases of continuous position detection and QND qubit only one of the mirrors is semitransparent, the other state measurement.

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I(t) While Eq. ͑3.20͒ makes it clear that the state of the ΔI qubit modulates the cavity frequency, we can easily re- write this equation to show that this same interaction term is also responsible for the back-action of the measurement ͑i.e., the disturbance of the qubit state by the t measurement process͒

ˆ ͑ប ͒͑␻ ␻ † ͒␴ ប␻ † ˆ ͑ ͒ H = /2 01 +2A caˆ aˆ ˆ z + caˆ aˆ + Henvt. 3.24 We now see that the interaction can also be viewed as ͑ ͒ ͑ ͒ providing a “light shift” i.e., ac Stark shift of the qubit FIG. 3. Color online Distribution of the integrated output for ͑ the cavity detector I͑t͒ for the two different qubit states. The splitting frequency Blais et al., 2004; Schuster et al., ͒ ␻ separation of the means of the distributions grows linearly in 2005 which contains a constant part 2An¯ A c plus a ran- ͱ ⌬␻ ˆ ប time, while the widths of the distributions grow only as t. domly fluctuating part 01=2Fz / , that depends on nˆ =aˆ †aˆ, the number of photons in the cavity. During a 1. QND measurement of the state of a qubit using a resonant measurement, nˆ will fluctuate around its mean and act as cavity a fluctuating back-action “force” on the qubit. In the present QND case, noise in nˆ =aˆ †aˆ cannot cause transi- Here we specialize to the case where the system op- tions between the two qubit eigenstates. This is the op- ␴ erator zˆ = ˆ z represents the state of a spin-1/2 quantum posite of the situation considered in Sec. II.B, where we ͑ ͒ bit. Equation 3.11 becomes wanted to use the qubit as a spectrometer. Despite the lack of any noise-induced transitions, there still is a ˆ 1 ប␻ ␴ˆ ប␻ ͑ ␴ˆ ͒ˆ † ˆ ˆ ͑ ͒ H = 2 01 z + c 1+A z a a + Henvt. 3.20 back-action here, as noise in nˆ causes the effective splitting frequency of the qubit to fluctuate in time. For weak We see that ␴ˆ commutes with all terms in the Hamil- z coupling, the resulting phase diffusion leads to tonian and is thus a constant of the motion ͑assuming measurement-induced dephasing of superpositions in ˆ ϱ͒ that Henvt contains no qubit decay terms so that T1 = the qubit ͑Blais et al., 2004; Schuster et al., 2005͒ accord- and hence the measurement will be QND. From Eq. ing to ͑3.13͒ we see that the two states of the qubit produce t phase shifts ±␪ where ␸ 0 ͗e−i ͘ = ͳexpͩ− i͵ d␶⌬␻ ͑␶͒ͪʹ. ͑3.25͒ 01 ␪ ␻ ͑ ͒ 0 0 = A ctWD. 3.21 For weak coupling the dephasing rate is slow and thus ␪ Ӷ As 0 1, we need many reflected photons before we we are interested in long times t. In this limit the integral are able to determine the state of the qubit. This is a is a sum of a large number of statistically independent direct consequence of the unavoidable photon shot terms and thus we can take the accumulated phase to be noise in the output of the detector, and is a basic feature Gaussian distributed. Using the cumulant expansion we of weak measurements—information on the input is ac- then obtain quired only gradually in time. t 2 Let I͑t͒ be the homodyne signal for the wave reflected ␸ 1 ͗e−i ͘ = expͩ− ͳͫ͵ d␶⌬␻ ͑␶͒ͬ ʹͪ 01 from the cavity integrated up to time t. Depending on 2 0 the state of the qubit the mean value of I will be ͗I͘ ␪ 2 =± 0t, and the rms Gaussian fluctuations about the = expͩ− S tͪ. ͑3.26͒ ͱ 2 FzFz mean will be ⌬I= S␪␪t. As shown in Fig. 3 and discussed ប ͑ ͒ by in Makhlin et al. 2001 , the integrated signal is drawn Note also that the noise correlator above is naturally from one of two Gaussian distributions, which are better symmetrized—the quantum asymmetry of the noise ͑ and better resolved with increasing time as long as the plays no role for this type of coupling. Equation ͑3.26͒ ͒ measurement is QND . The state of the qubit thus be- yields the dephasing rate comes ever more reliably determined. The signal energy 2 2 to noise energy ratio becomes ⌫␸ = ͑2/ប ͒S =2␪ S ˙ ˙ . ͑3.27͒ FzFz 0 NN ͗ ͘2 ͑⌬ ͒2 ␪2 ͑ ͒ Using Eqs. ͑3.23͒ and ͑3.27͒, we find the interesting SNR = I / I = 0/S␪␪t, 3.22 conclusion that the dephasing rate and measurement which can be used to define the measurement rate via rates coincide, 2 I 2 I ⌫ ϵ SNR/2 = ␪ /2S␪␪ = 1/2S . ͑3.23͒ ⌫␸/⌫ = ͑4/ប ͒S S =4S ˙ ˙ S␪␪ =1. ͑3.28͒ meas 0 zz meas zz FzFz NN There is a certain arbitrariness in the scale factor of 2 As we will see and prove rigorously, this represents the appearing in the definition of the measurement rate; this ideal quantum-limited case for QND qubit detection: particular choice is motivated by precise information- the best one can do is measure as quickly as one ͑ ⌫ theoretic grounds as defined, meas is the rate at which dephases. In keeping with our earlier discussions, it rep- the “accessible information” grows, Appendix F͒. resents the enforcement of the Heisenberg uncertainty

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principle. The faster you gain information about one quantum mechanics enforces the constraint that in a variable, the faster you lose information about the con- QND qubit measurement the best you can possibly do is ⌫ ⌫ ͑ jugate variable. Note that, in general, the ratio ␸ / meas measure as quickly as you dephase Devoret and Schoel- will be larger than 1, as an arbitrary detector will not kopf, 2000; Korotkov and Averin, 2001; Makhlin et al., reach the quantum limit on its noise spectral densities. 2001; Averin, 2000b, 2003; Clerk et al., 2003͒, Such a nonideal detector produces excess back-action beyond what is required quantum mechanically. In addition to the quantum noise point of view pre- ⌫ ഛ⌫ ͑ ͒ meas ␸. 3.35 sented above, there is a second complementary way in which to understand the origin of measurement-induced ͑ ͒ dephasing Stern et al., 1990 , which is analogous to our While a detector with quantum-limited noise has an description of loss of transverse spin coherence in the equality above, most detectors will be very far from this ͑ ͒ Stern-Gerlach experiment in Eq. 3.3 . The measure- ideal limit, and will dephase the qubit faster than they ment takes the incident wave, described by a coherent acquire information about its state. We provide a proof ͉␣͘ ͑ state , to a reflected wave described by a phase of Eq. ͑3.35͒ in Sec. IV.B; for now, we note that its heu- ͒ ͉ ␣͘ ͉ ␣͘ shifted coherent state r↑ · or r↓ · , where r↑/↓ is the ristic origin rests on the fact that both measurement and ͑ ͒ qubit-dependent reflection amplitude given in Eq. 3.12 . dephasing rely on the qubit becoming entangled with Considering now the full state of the qubit-plus-detector, the detector. Consider again Eq. ͑3.29͒, describing the the measurement results in a state change: evolution of the qubit-detector system when the qubit is initially in a superposition of ↑ and ↓. To say that we 1 1 +i␻ t/2 ͉͑↑͘ + ͉↓͒͘ ͉␣͘ → ͑e 01 ͉↑͘ ͉r↑ · ␣͘ have truly measured the qubit, the two detector states ͱ2 ͱ2 ͉r↑␣͘ and ͉r↓␣͘ need to correspond to different values of −i␻ t/2 + e 01 ͉↓͘ ͉r↓ · ␣͒͘. ͑3.29͒ the detector output ͑i.e., phase shift ␪ in our example͒; this necessarily implies they are orthogonal. This in turn As ͉r↑ ·␣͉͘r↓ ·␣͘, the qubit has become entangled with implies that the qubit is completely dephased: ␳↑↓=0, the detector: the above state cannot be written as a ͑ ͒ product of a qubit state times a detector state. To assess just as we saw in Eq. 3.4 in the Stern-Gerlach example. the coherence of the final qubit state ͑i.e., the relative Thus, measurement implies dephasing. The opposite is ͉ ␣͘ ͉ ␣͘ phase between ↑ and ↓͒, one looks at the off-diagonal not true. The two states r↑ and r↓ could in principle matrix element of the qubit’s reduced density matrix, be orthogonal without them corresponding to different values of the detector output ͑i.e., ␪͒. For example, the ␳ ϵ ͗↓͉␺͗͘␺͉↑͑͒͘ ↓↑ Trdetector 3.30 qubit may have become entangled with extraneous mi-

␻ croscopic degrees of freedom in the detector. Thus, on a ͑ +i 01t ͒͗ ␣͉ ␣͑͒͘ = e /2 r↑ · r↓ · 3.31 heuristic level, the origin of Eq. ͑3.35͒ is clear. Returning to our one-sided cavity system, we see from ␻ ͑ +i 01t ͒ ͓ ͉␣͉2͑ * ͔͒ ͑ ͒ = e /2 exp − 1−r↑r↓ . 3.32 Eq. ͑3.28͒ that the one-sided cavity detector reaches the In Eq. ͑3.31͒ we have used the usual expression for the quantum limit. It is natural to now ask why this is the overlap of two coherent states. We see that the measure- case: Is there a general principle in action here that allows the one-sided cavity to reach the quantum limit? ment reduces the magnitude of ␳↑↓: this is dephasing. The amount of dephasing is directly related to the over- The answer is yes: reaching the quantum limit requires that there is no “wasted” information in the detector lap between the different detector states that result ͑ ͒ when the qubit is up or down; this overlap can be Clerk et al., 2003 . There should not exist any unmeasured quantity in the detector which could have been ͑ ͒ ͉␣͉2 ¯ straightforwardly found using Eq. 3.32 and =N probed to learn more about the state of the qubit. In the ¯ =N˙ t, where N¯ is the mean number of photons that have single-sided cavity detector, information on the state of reflected from the cavity after time t. We have the qubit is only available in ͑that is, is entirely encoded in͒ the phase shift of the reflected beam; thus, there is no ͦ ͓ ͉␣͉2͑ * ͔͒ͦ ͓ ¯ ␪2͔ϵ ͓ ⌫ ͔ exp − 1−r↑r↓ = exp −2N 0 exp − ␸t , “wasted” information, and the detector does indeed ͑3.33͒ reach the quantum limit. To make this idea of “no wasted information” more with the dephasing rate ⌫␸ given by concrete, we now consider a simple detector that fails to reach the quantum limit precisely due to wasted infor- ⌫ ␪2 ˙¯ ͑ ͒ ␸ =2 0N 3.34 mation. Consider again a 1D cavity system where now in complete agreement with the previous result in Eq. both mirrors are slightly transparent. Now, a wave inci- ␻ ͑3.27͒. dent at frequency R on one end of the cavity will be partially reflected and partially transmitted. If the initial incident wave is described by a coherent state ͉␣͘, the 2. Quantum limit relation for QND qubit state detection scattered state can be described by a tensor product of We now return to the ideal quantum limit relation of the reflected wave’s state and the transmitted wave’s Eq. ͑3.28͒. As stated previously, this is a lower bound: state,

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1169

͉␣͘ → ͉r␴ · ␣͉͘t␴ · ␣͘, ͑3.36͒ Why does the two-sided cavity fail to reach the quantum limit? The answer is clear from Eq. ͑3.46͒: even where the qubit-dependent reflection and transmission though we are not monitoring it, there is information on ͑ amplitudes r␴ and t␴ are given by Walls and Milburn, the state of the qubit available in the phase of the re- ͒ 1994 flected wave. Note from Eq. ͑3.38͒ that the magnitude of ͑ ͒ ͑ ͒ the reflected wave is weak ͑ϰA2͒, but ͑unlike the trans- t↓ =1/ 1+2iAQc , 3.37 mitted wave͒ the difference in the reflection phase asso- ͑ ͒ ͑ ͒ ciated with the two qubit states is large ͑±␲/2͒. The r↓ =2iQcA/ 1+2iAQc , 3.38 “missing information” in the reflected beam makes a di- ͑ ͒ ͑ ͒ with t↑ = t↓ * and r↑ = r↓ *. Note that the incident beam is rect contribution to the dephasing rate ͓i.e., the second ͉ ͉2 ͑ ͒2 almost perfectly transmitted: t␴ =1−O AQc . term in Eq. ͑3.46͔͒, making it larger than the measure- Similar to the one-sided case, the two-sided cavity ment rate associated with measurement of the transmis- could be used to make a measurement by monitoring sion phase shift. In fact, there is an equal amount of the phase of the transmitted wave. Using the expression information in the reflected beam as in the transmitted for t␴ above, we find that the qubit-dependent transmis- beam, so the dephasing rate is doubled. We thus have a sion phase shift is given by concrete example of the general principle connecting a failure to reach the quantum limit to the presence of ␪˜ ␪˜ ͑ ͒ ↑/↓ =± 0 =±2AQc, 3.39 wasted information. Note that the application of this where again the two signs correspond to the two differ- principle to generalized quantum point contact detectors ent qubit eigenstates. The phase shift for transmission is is found in Clerk et al. ͑2003͒. only half as large as in reflection so the Wigner delay Returning to our cavity detector, we note in closing time associated with transmission is that it is often technically easier to work with the transmission of light through a two-sided cavity, rather than ␬ ͑ ͒ ˜tWD =2/ . 3.40 reflection from a one-sided cavity. One can still reach the quantum limit in the two-sided cavity case if one Upon making the substitution of ˜tWD for tWD, the one- sided cavity Eqs. ͑3.15͒ and ͑3.17͒ remain valid. How- uses an asymmetric cavity in which the input mirror has ever, the internal cavity photon-number shot noise re- much less transmission than the output mirror. Most mains fixed so that Eq. ͑3.18͒ becomes photons are reflected at the input, but those that enter the cavity will almost certainly be transmitted. The price ͑ ͒ Snn =2n¯˜tWD. 3.41 to be paid is that the input carrier power must be increased. which means that

˙¯ 2 ˙ ˙ 2 ͑ ͒ Snn =2N˜tWD =2SNN˜tWD 3.42 3. Measurement of oscillator position using a resonant cavity and The qubit measurement discussed previously was an 2 example of a QND measurement: the back-action did S =2ប2A2␻ ˜t2 S ˙ ˙ . ͑3.43͒ FzFz c WD NN not affect the observable being measured. We now con- As a result the back-action dephasing doubles relative to sider the simplest example of a non-QND measurement, the measurement rate and we have namely, the weak continuous measurement of the position of a harmonic oscillator. The detector will again be 1 ⌫ ⌫ ˙ ˙ ͑ ͒ meas/ ␸ =2SNNS␪␪ = 2 . 3.44 a parametrically coupled resonant cavity, where the position of the oscillator x changes the frequency of the Thus the two-sided cavity fails to reach the quantum ͑ ͓͒ ͑ ͔͒ limit by a factor of 2. cavity as per Eq. 3.11 see, e.g., Tittonen et al. 1999 . Using the entanglement picture, we may again alter- Similarly to the qubit case, for a sufficiently weak cou- natively calculate the amount of dephasing from the pling the phase shift of the reflected beam from the cavity will depend linearly on the position x of the oscillator overlap between the detector states corresponding to ͓ ͑ ͔͒ the qubit states ↑ and ↓ ͓cf. Eq. ͑3.31͔͒.Wefind cf. Eq. 3.13 ; by reading out this phase, we may thus measure x. The origin of back-action noise is the same as −⌫␸t e = ͉͗t↑␣͉t↓␣͗͘r↑␣͉r↓␣͉͘ ͑3.45͒ before, namely, photon shot noise in the cavity. Now, however, this represents a random force which changes 2 =exp͓− ͉␣͉ ͑1−͑t↑͒*t↓ − ͑r↑͒*r↓͔͒. ͑3.46͒ the momentum of the oscillator. During the subsequent time evolution these random force perturbations will re- Note that both the changes in the transmission and re- appear as random fluctuations in the position. Thus the flection amplitudes contribute to the dephasing of the measurement is not QND. This will mean that the mini- qubit. Using the above expressions, we find mum uncertainty of even an ideal measurement is larger 2 2 2 2 ¯ ͑by exactly a factor of 2͒ than the “true” quantum un- ⌫␸t =4͑␪˜ ͒ ͉␣͉ =4͑␪˜ ͒ N¯ =4͑␪˜ ͒ N˙ t =2⌫ t. ͑3.47͒ 0 0 0 meas certainty of the position ͑i.e., the ground-state uncer- Thus, in agreement with the quantum noise result, the tainty͒. This is known as the standard quantum limit on two-sided cavity misses the quantum limit by a factor of weak continuous position detection. It is also an ex- 2. ample of a general principle that a linear phase-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1170 Clerk et al.: Introduction to quantum noise, measurement, … ) preserving ampliﬁer necessarily adds noise, and that the 12 minimum added noise exactly doubles the output noise for the case where the input is vacuum ͑i.e., zero-point͒ 10

noise. A more general discussion of the quantum limit (arb. units 8 area prop. to T on ampliﬁers and position detectors will be presented in Sec. V. (ω) 6 measurement

S imprecision We start by emphasizing that we are speaking here of 4 a weak continuous measurement of the oscillator position. The measurement is sufficiently weak that the po- 2 sition undergoes many cycles of oscillation before sig- 0 nificant information is acquired. Thus we are not talking Output Noise 0.0 0.5 1.0 1.5 2.0 about the instantaneous position but rather the overall Frequency ω/Ω amplitude and phase, or more precisely the two quadrature amplitudes describing the smooth envelope of the FIG. 4. ͑Color online͒ Spectral density of the symmetrized motion, output noise S¯ ␪␪͓␻͔ of a linear position detector. The oscillator’s noise appears as a Lorentzian on top of a noise floor ͑i.e., ˆ͑ ͒ ˆ ͑ ͒ ͑⍀ ͒ ˆ ͑ ͒ ͑⍀ ͒ ͑ ͒ x t = X t cos t + Y t sin t . 3.48 the measurement imprecision͒. As discussed in the text, the One can easily show that, for an oscillator, the two width of the peak is proportional to the oscillator damping rate ␥ ˆ ˆ 0, while the area under the peak is proportional to tempera- quadrature amplitudes X and Y are canonically conju- ture. This latter fact can be used to calibrate the response of gate and hence do not commute with each other, the detector. ͓ ˆ ˆ ͔ ប ⍀ 2 ͑ ͒ X,Y = i /M =2ixZPF. 3.49 ditional fluctuations of the phase ␪, over and above the As the measurement is both weak and continuous, it will intrinsic shot noise-induced phase fluctuations S␪␪.We yield information on both Xˆ and Yˆ . As such, one is ef- consider the usual case where the noise spectrometer fectively trying to simultaneously measure two incom- being used to measure the noise in ␪ ͑i.e., the noise in patible observables. This basic fact is intimately related the homodyne current͒ measures the symmetric-in- to the property mentioned above, that even a com- frequency noise spectral density; as such, it is the pletely ideal weak continuous position measurement will symmetric-in-frequency position noise that we detect. In have a total uncertainty that is twice the zero-point un- ӷប⍀ the classical limit kBT , this is given by certainty. We are now ready to start our heuristic analysis of 1 S¯ ͓␻͔ϵ ͑S ͓␻͔ + S ͓− ␻͔͒ position detection using a cavity detector; relevant cal- xx 2 xx xx culational details presented in Appendix E.3. Consider k T ␥ first the mechanical oscillator we wish to measure. We Ϸ B 0 . ͑3.51͒ ⍀2 ͉͑␻͉ ⍀͒2 ͑␥ ͒2 take it to be a simple harmonic oscillator of natural fre- 2M − + 0/2 ⍀ ␥ quency and mechanical damping rate 0. For weak damping, and at zero coupling to the detector, the spec- If we ignore back-action effects, we expect to see this tral density of the oscillator’s position fluctuations is Lorentzian profile riding on top of the background im- given by Eq. ͑2.4͒ with the delta function replaced by a precision noise floor; this is shown in Fig. 4. Lorentzian5 Note that additional stages of amplification would also add noise, and would thus further augment this back- ␥ ground noise floor. If we subtract off this noise floor, the S ͓␻͔ = x2 ͭn ͑ប⍀͒ 0 xx ZPF B ͑␻ ⍀͒2 ͑␥ ͒2 full width at half maximum of the curve will give the + + 0/2 damping parameter ␥ , and the area under the experi- ␥ 0 + ͓n ͑ប⍀͒ +1͔ 0 ͮ. ͑3.50͒ mental curve, B ͑␻ ⍀͒2 ͑␥ ͒2 − + 0/2 ϱ d␻ k T When we now weakly couple the oscillator to the cav- ͵ S¯ ͓␻͔ = B , ͑3.52͒ ␲ xx ⍀2 ͓ ͑ ͒ ͔ −ϱ 2 M ity as per Eq. 3.11 , with zˆ =xˆ /xZPF and drive the cavity on resonance, the phase shift ␪ of the reflected beam ␦␪͑ ͒ ͓ ␪ ͔ ͑ ͒ measures the temperature. What the experimentalist ac- will be proportional to x „i.e., t = d /dx x t ….As tually plots in making such a curve is the output of the such, the oscillator’s position fluctuations will cause ad- entire detector-plus-following-amplifier chain. Impor- tantly, if the temperature is known, then the area of the measured curve can be used to calibrate the coupling of 5This form is valid only for weak damping because we are the detector and the gain of the total overall amplifier assuming that the oscillator frequency is still sharply defined. ͑ We have evaluated the Bose-Einstein factor exactly at fre- chain see, e.g., LaHaye et al., 2004; Flowers-Jacobs et quency ⍀ and have assumed that the Lorentzian centered at al., 2007͒. One can thus make a calibrated plot where the positive ͑negative͒ frequency has negligible weight at negative measured output noise is referred back to the oscillator ͑positive͒ frequencies. position.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1171

Consider now the case where the oscillator is at zero 10 ]

͑ ͒ 0 Total added position noise

temperature. Equation 3.50 then yields for the symme- γ Measurement imprecision trized noise spectral density / 8 Contribution from backaction 2 ) ␥ 6 0 2 0/2 ZPT

¯ ͓␻͔ ͑ ͒ x Sxx = xZPF 2 2 . 3.53 ( ͉͑␻͉ ⍀͒ ͑␥ ͒ [ − + 0/2 4

One might expect that one could see this Lorentzian add ͑ ␪ xx, 2 directly in the output noise of the detector i.e., the S noise͒, above the measurement-imprecision noise floor. However, this neglects the effects of measurement back- 0.1 1 10 action. From the classical equation of motion we expect Coupling SFF / SFF,opt the response of the oscillator to the back-action force ͓ ͑ ͔͒ ␻ F=Fz /xZPF cf. Eq. 3.16 at frequency to produce an FIG. 5. ͑Color online͒ Noise power of the added position noise ␦ ͓␻͔ ␹ ͓␻͔ ͓␻͔ additional displacement x = xx F , where of a linear position detector, evaluated at the oscillator’s reso- ␹ ͓␻͔ ͑ ¯ ͓⍀͔͒ xx is the mechanical susceptibility nance frequency Sxx,add as a function of the magnitude of the back-action noise spectral density S . S is proportional 1 1 FF FF ␹ ͓␻͔ϵ . ͑3.54͒ to the oscillator-detector coupling, and in the case of the cavity xx ⍀2 ␻2 ␥ ␻ M − − i 0 detector is also proportional to the power incident on the cavity. The optimal value of S is given by S =បM⍀␥ /2 ͓cf. These extra oscillator fluctuations will show up as addi- FF FF,opt 0 Eq. ͑3.59͔͒. We have assumed that there are no correlations tional fluctuations in the output of the detector. For sim- between measurement-imprecision noise and back-action plicity, we focus on this noise at the oscillator’s reso- noise, as is appropriate for the cavity detector. nance frequency ⍀. As a result of the detector’s back- action, the total measured position noise ͑i.e., inferred spectral density͒ at the frequency ⍀ is given by in Appendix E; the more general case with nonzero noise correlations and back-action damping is discussed ͉␹ ͓⍀͔͉2 S¯ ͓⍀͔ = S¯ 0 ͓⍀͔ + xx S ͓+ ⍀͔ + S ͓− ⍀͔ in Sec. V. E . xx,tot xx 2 † FF FF ‡ Assuming we have a quantum-limited detector that ͑ ͒͑ I ប2 ͒ 1 obeys Eq. 3.9 i.e., SxxSFF= /4 and that the shot I ͓ ⍀͔ I ͓ ⍀͔ ͑ ͒ + †Sxx + + Sxx − ‡ 3.55 noise is symmetric in frequency, the added position noise 2 spectral density at resonance ͓i.e., second term in Eq. ͑3.56͔͒ becomes =S¯ 0 ͓⍀͔ + S¯ ͓⍀͔. ͑3.56͒ xx xx,add ប2 1 ¯ ͓⍀͔ ͉ͫ␹͓⍀͔͉2 ͬ ͑ ͒ The first term here is just the intrinsic zero-point noise Sxx,add = SFF + . 3.58 4 SFF of the oscillator, Recall from Eq. ͑3.19͒ that the back-action noise is pro- ¯ 0 ͓⍀͔ 2 ␥ ប͉␹ ͓⍀͔͉ ͑ ͒ Sxx =2xZPF/ 0 = xx . 3.57 portional to the coupling of the oscillator to the detector and to the intensity of the drive on the cavity. The added ¯ The second term Sxx,add is the total noise added by the position-uncertainty noise is plotted in Fig. 5 as a func- measurement, and includes both the measurement im- tion of SFF. We see that for high drive intensity the back- I ϵ I 2 precision Sxx SzzxZPF and the extra fluctuations caused action noise dominates the position uncertainty, while ͑ by the back-action. We stress that S¯ corresponds to a for low drive intensity the output shot noise the last xx,tot ͒ position noise spectral density inferred from the output term in the equation above dominates. ͑ ¯ ͓⍀͔͒ of the detector: one simply scales the spectral density of The added noise and hence the total noise Sxx,tot ¯ ͓⍀͔ ͑ ␪ ͒2 is minimized when the drive intensity is tuned so that total output fluctuations S␪␪,tot by d /dx . Implicit in Eq. ͑3.57͒ is the assumption that the back- SFF is equal to SFF,opt, with action noise and the imprecision noise are uncorrelated S = ប/2͉␹ ͓⍀͔ = ͑ប/2͒M⍀␥ . ͑3.59͒ and thus add in quadrature. It is not obvious that this is FF,opt xx 0 correct, since in the cavity detector the back-action noise The more heavily damped is the oscillator, the less sus- and output shot noise are both caused by the vacuum ceptible it is to back-action noise and hence the higher is noise in the beam incident on the cavity. It turns out that the optimal coupling. At the optimal coupling strength, there are indeed correlations, however, the symmetrized the measurement-imprecision noise and back-action ͑ ͒ ¯ noise each make equal contributions to the added noise, i.e., classical correlator S␪F does vanish for our choice of a resonant cavity drive. Further, Eq. ͑3.55͒ assumes yielding that the measurement does not change the damping rate S¯ ͓⍀͔ = ប/M⍀␥ = S¯ 0 ͓⍀͔. ͑3.60͒ of the oscillator. Again, while this will not be true for an xx,add 0 xx arbitrary detector, it is the case here for the cavity de- Thus, the spectral density of the added position noise is tector when ͑as we have assumed͒ it is driven on reso- exactly equal to the noise power associated with the os- nance. Details justifying both these statements are given cillator’s zero-point fluctuations. This represents a mini-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1172 Clerk et al.: Introduction to quantum noise, measurement, … mum value for the added noise of any linear position 2.0 detector, and is referred to as the standard quantum total noise limit on position detection. Note that this limit only involves the added noise of the detector, and thus has 1.5 ] eq. + back Ω nothing to do with the initial temperature of the oscilla- [ action noise xx 0

tor. S /

We emphasize that to reach the above quantum limit ] 1.0

ω equilibrium on weak continuous position detection one needs the [ noise (T=0) detector itself to be quantum limited, i.e., the product tot I xx, SFFSxx must be as small as is allowed by quantum me- S 0.5 chanics, namely, ប2 /4. Having a quantum-limited detector, however, is not enough: in addition, one must be able to achieve sufficiently strong coupling to reach the 0.0 optimum given in Eq. ͑3.59͒. Further, the measured out- 0.0 0.4 0.8 1.2 put noise must be dominated by the output noise of the Frequency ω/Ω cavity, not by the added noise of following amplifier stages. FIG. 6. ͑Color online͒ Spectral density of measured position A related, stronger quantum limit refers to the total fluctuations of a harmonic oscillator S¯ ͓␻͔ as a function of ¯ ͓␻͔ xx,tot inferred position noise from the measurement Sxx,tot . frequency ␻, for a detector which reaches the quantum limit at It follows from Eqs. ͑3.60͒ and ͑3.56͒ that at resonance the oscillator frequency ⍀. We have assumed that without the the smallest this can be is twice the oscillator’s zero- coupling to the detector the oscillator would be in its ground point noise state. The y axis has been normalized by the zero-point position noise spectral density S¯ 0 ͓␻͔, evaluated at ␻=⍀.One ¯ ͓⍀͔ ¯ 0 ͓⍀͔ ͑ ͒ xx Sxx,tot =2Sxx . 3.61 clearly sees that the total noise at ⍀ is twice the zero-point value, and that the peak of the Lorentzian rises a factor of 3 Half the noise here is from the oscillator itself, half is above the background. This background represents the mea- from the added noise of the detector. It is even more surement imprecision and is equal to 1/2 of S¯ 0 ͑⍀͒. challenging to reach this quantum limit: one needs to xx both reach the quantum limit on the added noise and cool the oscillator to its ground state. optimal coupling at the quantum limit: ͑i͒ the total Finally, we emphasize that the optimal value of the added noise at resonance ͑back-action plus measure- coupling derived above was specific to the choice of ment imprecision͒ is equal to the zero-point noise and minimizing the total position noise power at the reso- ͑ii͒ back-action and measurement imprecision make nance frequency. If a different frequency had been cho- equal contributions to the total added noise. Somewhat sen, the optimal coupling would have been different; surprisingly, the same maximum peak-to-floor ratio is one again finds that the minimum possible added noise obtained when one tries to continuously monitor coher- corresponds to the ground-state noise at that frequency. ent qubit oscillations with a linear detector which is It is interesting to ask what the total position noise transversely coupled to the qubit ͑Korotkov and Averin, would be as a function of frequency, assuming that the 2001͒; this is also a non-QND situation. Finally, if one coupling has been optimized to minimize the noise at only wants to detect the noise peak ͑as opposed to mak- the resonance frequency, and that the oscillator is ini- ing a continuous quantum-limited measurement͒, one tially in the ground state. From our results above we could use two independent detectors coupled to the os- have cillator and look at the cross correlation between the ␥ /2 two output noises: in this case, there need not be any S¯ ͓␻͔ = x2 0 noise floor ͑Jordan and Büttiker, 2005a; Doiron et al., xx,tot ZPF͉͑␻͉ ⍀͒2 ͑␥ ͒2 − + 0/2 2007͒. ប ͉␹ ͓␻͔͉2 In Table III, we give a summary of recent experiments ͫ xx ͉␹ ͓⍀͔͉ͬ + + xx which approach the quantum limit on weak continuous 2 ͉␹ ͓⍀͔͉ xx position detection of a mechanical resonator. Note that x2 ͑␥ /2͒2 in many of these experiments the effects of detector Ϸ ZPFͭ1+3 0 ͮ, ͑3.62͒ ␥ ͉͑␻͉ − ⍀͒2 + ͑␥ /2͒2 back-action were not seen. This could either be the re- 0 0 sult of too low a detector-oscillator coupling or due to which is plotted in Fig. 6. Assuming that the detector is the presence of excessive thermal noise. As shown, the quantum limited, one sees that the Lorentzian peak rises back-action force noise serves to slightly heat the oscil- above the constant background by a factor of 3 when the lator. If it is already at an elevated temperature due to coupling is optimized to minimize the total noise power thermal noise, this additional heating can be hard to re- at resonance. This represents the best one can do when solve. continuously monitoring zero-point position fluctua- In closing, we stress that this section has given only a tions. Note that the value of this peak-to-floor ratio is a rudimentary introduction to the quantum limit on posi- direct consequence of two simple facts which hold for an tion detection. A complete discussion which treats the

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1173

TABLE III. Synopsis of recent experiments approaching the quantum limit on continuous position detection of a mechanical ¯ I resonator. The second column corresponds to the best measurement-imprecision noise spectral density Sxx achieved in the experi- ¯ 0 ment. This value is compared against the zero-point position noise spectral density Sxx, calculated using the total measured resonator damping ͑which may include a back-action contribution͒. All spectral densities are at the oscillator’s resonance fre- ⍀ ¯ I quency . As discussed in the text, there is no quantum limit on how small one can make Sxx; for an ideal detector, one needs to ¯ I ¯ 0 tune the detector-resonator coupling so that Sxx=Sxx/2 in order to reach the quantum limit on position detection. The third column presents the product of the measured imprecision noise ͑unless otherwise noted͒ and measured back-action noises, divided by ប/2; this quantity must be one to achieve the quantum limit on the added noise.

Imprecision noise Detector noise Mechanical a frequency ͓Hz͔ zero-point noise product ⍀ ͑ ␲͒ ͱ¯ I ¯ 0 ͓⍀͔ ͱ¯ I ¯ ͑ប ͒ Experiment / 2 Sxx/Sxx SxxSFF/ /2 Cleland et al. ͑2002͒ ͑quantum point contact͒ 1.5ϫ106 4.2ϫ104 Knobel and Cleland ͑2003͒ ͑single-electron transistor͒ 1.2ϫ108 1.8ϫ102 LaHaye et al. ͑2004͒ ͑single-electron transistor͒ 2.0ϫ107 5.4 Naik et al. ͑2006͒ ͑single-electron transistor͒ 2.2ϫ107 5.3b 8.1ϫ102 ͑ ¯ I ͒ ϫ 1 if Sxx had been limited by SET shot noise 1.5 10 Arcizet et al. ͑2006͒ ͑optical cavity͒ 8.1ϫ105 0.87 Flowers-Jacobs et al. ͑2007͒ ͑atomic point-contact͒ 4.3ϫ107 29 1.7ϫ103 Regal et al. ͑2008͒ ͑microwave cavity͒ 2.4ϫ105 21 Schliesser et al. ͑2008͒ ͑optical cavity͒ 4.1ϫ107 0.50 Poggio et al. ͑2008͒ ͑quantum point contact͒ 5.2ϫ103 63 Etaki et al. ͑2008͒ ͑dc SQUID͒ 2.0ϫ106 47 Groblacher et al. ͑2009͒ ͑optical cavity͒ 9.5ϫ102 0.57 Schliesser et al. ͑2009͒ ͑optical cavity͒ 6.5ϫ107 5.5 1.0ϫ102 Teufel et al. ͑2009͒ ͑microwave cavity͒ 1.0ϫ106 0.63

aA blank value in this column indicates that back-action was not measured in the experiment. bNote that back-action effects dominated the mechanical Q in this measurement, lowering it from 1.2ϫ105 to ϳ4.2ϫ102. If one ͱ¯ I ¯ 0 ͓⍀͔ϳ compares the imprecision against the zero-point noise of the uncoupled mechanical resonator, one ﬁnds Sxx/Sxx 0.33. important topics of back-action damping, effective tem- eral quantum constraint on noise. We introduce both the perature, noise cross correlation, and power gain is notion of a generic linear-response detector and the ba- given in Sec. V. E . sic quantum constraint on detector noise. Next, we discuss how this noise constraint leads to the quantum limit on QND state detection of a qubit. The quantum limit IV. GENERAL LINEAR-RESPONSE THEORY on a linear ampliﬁer ͑or a position detector͒ is discussed in the next section. A. Quantum constraints on noise

In this section, we further develop the connection between quantum limits and noise discussed previously, fo- 1. Heuristic weak-measurement noise constraints cusing now on a more general approach. As before, we emphasize the idea that reaching the quantum limit re- As stressed in the introduction, there is no fundamen- quires a detector having quantum-ideal noise properties. tal quantum limit on the accuracy with which a given The approach here is different from typical treatments observable can be measured, at least not within the in the quantum optics literature ͑Gardiner and Zoller, framework of nonrelativistic quantum mechanics. For 2000; Haus, 2000͒, and uses nothing more than features example, one can, in principle, measure the position of a of quantum linear response. Our discussion here will ex- particle to arbitrary accuracy in the course of a projec- pand upon Clerk et al. ͑2003͒ and Clerk ͑2004͒; some- tion measurement. However, the situation is different what similar approaches to quantum measurement are when we specialize to continuous non-QND measure- also discussed by Braginsky and Khalili ͑1992͒ and ments. Such a measurement can be envisaged as a series Averin ͑2003͒. of instantaneous measurements, in the limit where the In this section, we start by heuristically sketching how spacing between the measurements ␦t is taken to zero. constraints on noise ͓similar to Eq. ͑3.9͒ for the cavity Each measurement in the series has a limited resolution detector͔ can emerge directly from the Heisenberg un- and perturbs the conjugate variables, thereby affecting certainty principle. We then present a rigorous and gen- the subsequent dynamics and measurement results. We

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had Fˆ =nˆ , the cavity photon number. As we are interested in weak couplings, we assume a simple bilinear form for the detector-signal interaction Hamiltonian ˆ ˆ ͑ ͒ Hint = AxˆF. 4.1 signal input output "load" ͑ source operator Here the operator xˆ which is not necessarily a position operator͒ carries the input signal. Note that because xˆ detector belongs to the signal source, it necessarily commutes ˆ ˆ FIG. 7. ͑Color online͒ Schematic of a generic linear-response with the detector variables I,F. detector. We always assume the coupling strength A to be small enough that we can accurately describe the output of the detector using linear response.6 We thus have discuss this for the example of a series of position measurements of a free particle. ͗ˆ͑ ͒͘ ͗ˆ͘ ͵ Ј␹ ͑ Ј͒͗ ͑ Ј͒͘ ͑ ͒ After initially measuring the position with an accuracy I t = I 0 + A dt IF t − t xˆ t , 4.2 ⌬x, the momentum suffers a random perturbation of size ⌬pജប/͑2⌬x͒. Consequently, a second position mea- ͗ˆ͘ where I 0 is the input-independent value of the detector surement taking place a time ␦t later will have an addi- ␹ ͑ ͒ output at zero coupling and IF t is the linear-response tional uncertainty of size ␦t͑⌬p/m͒ϳប␦t/m⌬x. Thus, susceptibility or gain of our detector. Note that Clerk et when one is trying to obtain a good estimate of the po- al. ͑2003͒ and Clerk ͑2004͒ denoted this gain coefficient sition by averaging several such measurements, it is not ␭. Using standard time-dependent perturbation theory optimal to make ⌬x too small, because otherwise this in the coupling Hˆ , one can easily derive Eq. ͑4.2͒, with additional perturbation, called the back-action of the int ␹ ͑t͒ given by a Kubo-like formula, measurement device, will become large. The back-action IF can be described as a random force ⌬F=⌬p/␦t. A mean- ␹ ͑ ͒ ͑ ប͒␪͑ ͓͒͗ˆ͑ ͒ ˆ ͑ ͔͒͘ ͑ ͒ IF t =− i/ t I t ,F 0 0. 4.3 ingful limit ␦t→0 is obtained by keeping both ⌬x2␦t ϵ ¯ ⌬ 2 ␦ ϵ ¯ Here and in what follows the operators Iˆ and Fˆ are Sxx and p / t SFF fixed. In this limit, the deviations ␦x͑t͒ describing the finite measurement accuracy and the Heisenberg operators with respect to the detector fluctuations of the back-action force F can be described Hamiltonian, and the subscript 0 indicates an expecta- ͗␦ ͑ ͒␦ ͑ ͒͘ ¯ ␦͑ ͒ tion value with respect to the density matrix of the un- as white noise processes, x t x 0 =Sxx t and coupled detector. ͗ ͑ ͒ ͑ ͒͘ ¯ ␦͑ ͒ F t F 0 =SFF t . The Heisenberg uncertainty relation As discussed, there will be unavoidable noise in both ⌬ ⌬ ജប ¯ ¯ ജប2 ͑ the input and output ports of our detector. This noise is p x /2 then implies SxxSFF /4 Braginsky and Khalili, 1992͒. Note that this is completely analogous to subject to quantum-mechanical constraints, and its pres- the relation ͑3.9͒ we derived for the resonant cavity de- ence is what limits our ability to make a measurement or tector using the fundamental number-phase uncertainty amplify a signal. We thus need to quantitatively charac- relation. In this section, we derive rigorously more gen- terize the noise in both these ports. Recall from the dis- eral quantum limit relations on noise spectral densities cussion in Sec. II.B that it is the symmetric-in-frequency of this form. part of a quantum noise spectral density which plays a role akin to classical noise. We thus want to characterize the symmetrized noise correlators of our detector ͑de- 2. Generic linear-response detector noted as always with an overbar͒. Redefining these op- To rigorously discuss the quantum limit, we start with erators so that their average value is zero at zero cou- a description of a detector that is as general as possible. ͑ ˆ → ˆ ͗ ˆ ͘ ˆ → ˆ ͗ˆ͘ ͒ pling i.e., F F− F 0, I I− I 0 , we have To that end, we think of a detector as some physical ϱ ͑ ˆ 1 ␻ system described by some unspecified Hamiltonian Hdet S¯ ͓␻͔ϵ ͵ dt ei t͕͗Fˆ ͑t͒,Fˆ ͑0͖͒͘ , ͑4.4a͒ ␳ ͒ FF 2 0 and some unspecified density matrix ˆ 0 which is time −ϱ independent in the absence of coupling to the signal ϱ source. The detector has both an input port, character- 1 ¯ ͓␻͔ϵ i␻t͕͗ˆ͑ ͒ ˆ͑ ͖͒͘ ͑ ͒ ˆ SII ͵ dt e I t ,I 0 0, 4.4b ized by an operator F, and an output port, characterized 2 −ϱ by an operator Iˆ ͑see Fig. 7͒. The output operator Iˆ is simply the quantity that is read out at the output of the 6 detector ͑e.g., the current in a single-electron transistor, The precise condition for the breakdown of linear response or the phase shift in the cavity detector of the previous will depend on specific details of the detector. For example, in the cavity detector discussed in Sec. III.B, one would need the section͒. The input operator Fˆ is the detector quantity Ӷ ͗ ͘ dimensionless coupling A to satisfy A 1/Qc zˆ to ensure that that directly couples to the input signal ͑e.g., the qubit͒ the nonlinear dependence of the phase shift ␪ on the signal ͗zˆ͘ and that causes a back-action disturbance of the signal is negligible. This translates to the signal modulating the cavity source; in the cavity example of the previous section, we frequency by an amount much smaller than its linewidth ␬.

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ϱ ͓␻͔ 1 the unsymmetrized I-F quantum noise correlator, SIF . ¯ ͓␻͔ϵ ͵ i␻t͕͗ˆ͑ ͒ ˆ ͑ ͖͒͘ ͑ ͒ SIF dt e I t ,F 0 0, 4.4c This spectral density, which need not be symmetric in 2 −ϱ frequency, is defined as ϱ where ͕,͖ indicates the anticommutator, S¯ represents II S ͓␻͔ = ͵ dt ei␻t͗Iˆ͑t͒Fˆ ͑0͒͘ . ͑4.7͒ ¯ IF 0 the intrinsic noise in the output of the detector, and SFF −ϱ describes the back-action noise seen by the source of the Using the definitions, one can easily show that input signal. In general, there will be some correlation between these two kinds of noise; this is described by S¯ ͓␻͔ = 1 S ͓␻͔ + S ͓− ␻͔* , ͑4.8a͒ ¯ IF 2 † IF IF ‡ the cross correlator SIF. Finally, we must also allow for the possibility that our ␹ ͓␻͔ ␹ ͓␻͔ ͑ ប͒ ͓␻͔ ͓ ␻͔ ͑ ͒ IF − FI * =− i/ †SIF − SIF − *‡. 4.8b detector could operate in reverse ͑i.e., with input and ¯ output ports playing opposite roles͒. We thus introduce Thus, while SIF represents the classical part of the I-F ␹ ␹ ␹ the reverse gain FI of our detector. This is the response quantum noise spectral density, the gains IF, FI are de- coefficient describing an experiment where we couple termined by the quantum part of this spectral density. our input signal to the output port of the detector ͑i.e., This also demonstrates that though the gains have an ˆ ˆ͒ explicit factor of 1/ប in their definitions, they have a Hint=AxÎ , and attempt to observe something at the in- well-defined ប→0 limit, as the asymmetric-in-frequency put port ͓i.e., in ͗Fˆ ͑t͔͒͘. In complete analogy to Eq. ͑4.2͒, part of S ͓␻͔ vanishes in this limit. one would then have IF

͗ ˆ ͑ ͒͘ ͗ ˆ ͘ ͵ Ј␹ ͑ Ј͒͗ ͑ Ј͒͘ ͑ ͒ 3. Quantum constraint on noise F t = F 0 + A dt FI t − t xˆ t 4.5 Despite having said nothing about the detector’s ͑ with Hamiltonian or state except that it is time indepen- dent͒, we can nonetheless derive a general quantum con- ␹ ͑t͒ =−͑i/ប͒␪͑t͓͒͗Fˆ ͑t͒,Iˆ͑0͔͒͘ . ͑4.6͒ straint on its noise properties. Note first that for purely FI 0 classical noise spectral densities one always has the in- Note that if our detector is in a time-reversal symmetric, equality thermal equilibrium state, then Onsager reciprocity re- ¯ ͓␻͔ ¯ ͓␻͔ ͉ ¯ ͓␻͔͉2 ജ ͑ ͒ ␹ ␹* ͑ SII SFF − SIF 0. 4.9 lations would imply either IF= FI if I and F have the ͒ ␹ ␹* ͑ same parity under time reversal or IF=− FI if I and F This simply expresses the fact that the correlation be- have the opposite parity under time reversal͓͒see, e.g., tween two different noisy quantities cannot be arbi- Pathria ͑1996͔͒. Thus, if the detector is in equilibrium, trarily large; it follows immediately from the Schwartz the presence of gain necessarily implies the presence of inequality. In the quantum case, this simple constraint reverse gain. Nonzero reverse gain is also found in many becomes modified whenever there is an asymmetry be- standard classical electrical amplifiers such as op-amps tween the detector’s gain and reverse gain. This asym- ͑Boylestad and Nashelsky, 2006͒. ␹ ͓␻͔ metry is parametrized by the quantity ˜ IF , The reverse gain is something that we must worry ␹ ͓␻͔ϵ␹ ͓␻͔ ͑␹ ͓␻͔͒ ͑ ͒ about even if we are not interested in operating our de- ˜ IF IF − FI *. 4.10 tector in reverse. To see why, note that to make a mea- We show below that the following quantum noise in- surement of the output operator Iˆ, we must necessarily equality ͑involving symmetrized noise correlators͒ is al- ␹ ͓ ͑ ͒ couple to it in some manner. If FI 0, the noise associ- ways valid see also Eq. 6.36 in Braginsky and Khalili ated with this coupling could in turn lead to additional ͑1996͔͒, back-action noise in the operator Fˆ . Even if the reverse S¯ ͓␻͔S¯ ͓␻͔ − ͉S¯ ͓␻͔͉2 gain did nothing but amplify vacuum noise entering the II FF IF output port, this would heat up the system being mea- ប␹˜ ͓␻͔ 2 S¯ ͓␻͔ ജ ͯ IF ͯ ͩ1+⌬ͫ IF ͬͪ, ͑4.11͒ sured at the input port and hence produce excess back- ប␹ ͓␻͔ ␹ 2 ˜ IF /2 action. Thus, the ideal situation is to have FI=0, implying a high asymmetry between the input and output of where the detector, and requiring the detector to be in a state ⌬͓z͔ = ͓͉1+z2͉ − ͑1+͉z͉2͔͒/2. ͑4.12͒ far from thermodynamic equilibrium. We note that almost all mesoscopic detectors that have been studied in To interpret the quantum noise inequality Eq. ͑4.11͒, detail ͑e.g., single-electron transistors and generalized note that 1+⌬͓z͔ജ0. Equation ͑4.11͒ thus implies that if quantum point contacts͒ have been found to have a van- our detector has gain and does not have a perfect sym- ishing reverse gain: ␹ =0 ͑Clerk et al., 2003͒. For this ͑ ␹ ␹* ͒ FI metry between input and output i.e., IF FI , then it reason, we often focus on the ideal ͑but experimentally must in general have a minimum amount of back-action ͒ ␹ relevant situation where FI=0 in what follows. and output noise; moreover, these two noises cannot be Before proceeding, it is worth emphasizing that there perfectly anticorrelated. As shown in the following sec- ␹ ␹ is a relation between the detector gains IF and FI and tions, this constraint on the noise of a detector directly

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1176 Clerk et al.: Introduction to quantum noise, measurement, … leads to quantum limits on various different measure- Iˆ and Fˆ is bounded by the value of their commutator at ment tasks. Note that in the zero-frequency limit ␹˜ and IF that frequency. The fact that Iˆ and Fˆ do not commute is ¯ ⌬ SIF are both real, implying that the term involving in necessary for the existence of linear response ͑gain͒ Eq. ͑4.11͒ vanishes. The result is a simpler-looking in- from the detector, but also means that the noise in both equality found elsewhere ͑Averin, 2003; Clerk et al., Iˆ and Fˆ cannot be arbitrarily small. A more detailed 2003; Clerk, 2004͒. derivation, yielding additional important insights, is de- While Eq. ͑4.11͒ may appear reminiscent of the stan- scribed in Appendix I.1. dard ﬂuctuation-dissipation theorem, its origin is quite Given the quantum noise constraint of Eq. ͑4.11͒,we different: in particular, the quantum noise constraint ap- can now very naturally deﬁne a quantum-ideal detector plies irrespective of whether the detector is in equilib- ͑ ␻͒ rium. Equation ͑4.11͒ instead follows directly from at a given frequency as one which minimizes the left-hand side ͑LHS͒ of Eq. ͑4.11͒—a quantum-ideal de- Heisenberg’s uncertainty relation applied to the fre- ␻ ˆ ˆ tector has a minimal amount of noise at frequency .We quency representation of the operators I and F.Inits are often interested in the ideal case where there is no most general form, the Heisenberg uncertainty relation ͑ ˆ gives a lower bound for the uncertainties of two observ- reverse gain i.e., measuring I does not result in addi- ables in terms of their commutator and their noise cor- tional back-action noise in Fˆ ͒; the condition to have a relator ͑Gottfried, 1966͒, quantum-limited detector thus becomes

͑⌬ ͒2͑⌬ ͒2 ജ 1 ͕͗ ˆ ˆ ͖͘2 1 ͉͓͗ ˆ ˆ ͔͉͘2 ͑ ͒ ¯ ͓␻͔ ¯ ͓␻͔ ͉ ¯ ͓␻͔͉2 A B 4 A,B + 4 A,B . 4.13 SII SFF − SIF ͗ ˆ ͘ ͗ ˆ ͘ ប␹ ͓␻͔ 2 ¯ ͓␻͔ Here we have assumed A = B =0. We now choose the ͯ IF ͯ ͩ ⌬ͫ SIF ͬͪ ͑ ͒ = 1+ ប␹ ͓␻͔ , 4.17 Hermitian operators Aˆ and Bˆ to be given by the cosine 2 IF /2 transforms of Iˆ and Fˆ , respectively, over a finite time where ⌬͓z͔ is given in Eq. ͑4.12͒. Again, as discussed interval T, below, in most cases of interest ͑e.g., zero frequency and/or large amplifier power gain͒ the last term on the 2 T/2 Aˆ ϵ ͱ ͵ dt cos͑␻t + ␦͒Iˆ͑t͒, ͑4.14a͒ RHS will vanish. In the following, we demonstrate that T −T/2 the ideal noise requirement of Eq. ͑4.17͒ is necessary in order to achieve the quantum limit on QND detection of 2 T/2 a qubit, or on the added noise of a linear amplifier. Bˆ ϵ ͱ ͵ dt cos͑␻t͒Fˆ ͑t͒. ͑4.14b͒ Before leaving our general discussion of the quantum T −T/2 noise constraint, it is worth emphasizing that achieving Eq. ͑4.17͒ places a strong constraint on the properties of Note that we have phase shifted the transform of Iˆ rela- the detector. In particular, there must exist a tight con- ˆ tive to that of F by a phase ␦. In the limit T→ϱ we find, nection between the input and output ports of the at any finite frequency ␻0 detector—in a certain restricted sense, the operators Iˆ ͑⌬ ͒2 ¯ ͓␻͔ ͑⌬ ͒2 ¯ ͓␻͔ ͑ ͒ ˆ ͓ ͑ ͒ A = SII , B = SFF , 4.15a and F must be proportional to one another see Eq. I13 in Appendix I.1͔. As is discussed in Appendix I.1, this ͕͗Aˆ ,Bˆ ͖͘ =2Reei␦S¯ ͓␻͔, ͑4.15b͒ proportionality immediately tells us that a quantum- IF ideal detector cannot be in equilibrium. The proportion- +ϱ ality exhibited by a quantum-ideal detector is param- ͓͗Aˆ ,Bˆ ͔͘ = ͵ dt cos͑␻t + ␦͓͒͗Iˆ͑t͒,Fˆ ͑0͔͒͘ etrized by a single complex-valued number ␣͓␻͔, whose −ϱ magnitude is given by ប i␦ ␹ ͓␻͔ ͑␹ ͓␻͔͒ ͑ ͒ = i Re†e „ IF − FI *…‡. 4.15c ͉␣͓␻͔͉2 ¯ ͓␻͔ ¯ ͓␻͔ ͑ ͒ = SII /SFF . 4.18 In the last equality, we have simply made use of the While this proportionality requirement may seem purely Kubo formula definitions of the gain and reverse gain formal, it does have a simple heuristic interpretation; as ͓cf. Eqs. ͑4.3͒ and ͑4.6͔͒. As a consequence of Eqs. discussed by Clerk et al. ͑2003͒, it may be viewed as a ͑4.15a͒, ͑4.15b͒, and ͑4.15c͒, the Heisenberg uncertainty formal expression of the principle that a quantum- relation ͑4.13͒ directly yields limited detector must not contain any wasted informa- ប2 tion ͑cf. Sec. III.B.2͒. S¯ ͓␻͔S¯ ͓␻͔ ജ ͕Re͑ei␦S¯ ͓␻͔͖͒2 + Re ei␦ ␹ ͓␻͔ II FF IF 4 † „ IF ͑␹ ͓␻͔͒ 2 ͑ ͒ − FI *…‡ . 4.16 4. Evading the detector quantum noise inequality Maximizing the RHS of this inequality over ␦ then yields We now turn to situations where the RHS of Eq. the general quantum noise constraint of Eq. ͑4.11͒. ͑4.11͒ vanishes, implying that there is no additional With this derivation, we can now interpret the quan- quantum constraint on the noise of our detector beyond tum noise constraint Eq. ͑4.11͒ as stating that the noise what exists classically. In such situations, one could have at a given frequency given frequency in two observables a detector with perfectly correlated back-action and out-

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͑ ¯ ¯ ͉ ¯ ͉2͒ imaginary, and larger in magnitude than ប/2. In this put noises i.e., SFFSII= SIF , or even with a vanishing ¯ case, it would again be possible to have the LHS of the back-action SFF=0. Perhaps not surprisingly, these situ- noise constraint of Eq. ͑4.11͒ equal to 0. However, one ations are not of much utility. As we now show, in cases ͑ ͒ again finds that in such a case the dimensionless power where the RHS of Eq. 4.11 vanishes, the detector may gain of the detector is at most equal to 1; it thus does not be low noise, but will necessarily be deficient in another amplify. This is shown explicitly in Appendix I.2. An important regard: it will not be good enough that we can important related statement is that a quantum-limited ignore the extra noise associated with the measurement detector with a large power gain must have the quantity of the detector output Iˆ. As discussed, the reading out of ¯ ␹ SIF/ IF be real. Thus, at the quantum limit, correlations Iˆ will invariably involve coupling the detector output to between the back-action force and the intrinsic output some other physical system. In the ideal case, this cou- noise fluctuations must have the same phase as the gain ␹ pling will not generate any additional back-action on the IF. As discussed in Sec. V. F , this requirement can be system coupled to the detector’s input port. In addition, interpreted in terms of the principle of no wasted infor- the signal at the detector output should be large enough mation introduced in Sec. III.B.2. that any noise introduced in measuring Iˆ is negligible; we already came across this idea in our discussion of the resonant cavity detector ͓see discussion following Eq. B. Quantum limit on QND detection of a qubit ͑3.60͔͒. This means that we need our detector to truly In Sec. III.B, we discussed the quantum limit on QND amplify the input signal, not simply reproduce it at the qubit detection in the specific context of a resonant cav- output with no gain in energy. As now shown, a detector ity detector. We now show how the full quantum noise that evades the quantum constraint of Eq. ͑4.11͒ by constraint of Eq. ͑4.11͒ directly leads to this quantum making the RHS of the inequality zero will necessarily limit for an arbitrary weakly coupled detector. Similar to fail in one or both of the above requirements. Sec. III.B, we couple the input operator of our generic The most obvious case where the quantum noise con- linear-response detector to the ␴ˆ operator of the qubit straint vanishes is for a detector which has equal forward z we wish to measure ͓i.e., we take xˆ =␴ˆ in Eq. ͑4.1͔͒;we and reverse gains, ␹ =␹* . As mentioned, this relation z FI IF ␴ˆ will necessarily hold if the detector is time-reversal sym- also consider the QND regime, where z commutes with the qubit Hamiltonian. As we saw in Sec. III.B, the metric and in equilibrium, and Iˆ and Fˆ have the same ⌫ quantum limit in this case involves the inequality meas parity under time reversal. In this case, the relatively ഛ⌫ ⌫ ⌫ ␸, where meas is the measurement rate and ␸ is the large reverse gain implies that in the analysis of a given back-action dephasing rate. For the latter quantity, we measurement task, it is not sufficient to just consider the can directly use the results of our calculation for the noise of the detector: one must necessarily also consider cavity system, where we found the dephasing rate was the noise associated with whatever system is coupled to set by the zero-frequency noise in the cavity photon Iˆ to read out the detector output, as this noise will be fed number ͓see Eq. ͑3.27͔͒. In complete analogy, the back- back to the detector input port, causing additional back- action dephasing rate will be determined here by the action; we give an explicit example of this in the next zero-frequency noise in the input operator Fˆ of our de- subsection, where we discuss QND qubit detection. tector, ␹ ␹* Even more problematically, when FI= , there is IF ⌫ ͑ 2 ប2͒ ¯ ͑ ͒ never any amplification by the detector. As discussed in ␸ = 2A / SFF. 4.19 Sec. V.E.3, the proper metric of the detector’s ability to We omit frequency arguments in this section, as it is amplify is its dimensionless power gain: What is the always the zero-frequency susceptibilities and spectral power supplied at the output of the detector versus the densities that appear. amount of power drawn at the input from the signal The measurement rate ͑the rate at which information source? When ␹ =␹* , one has negative feedback, with FI IF on the state of qubit is acquired͒ is also defined in com- the result that the power gain cannot be larger than 1 plete analogy to what was done for the cavity detector. ͓ ͑ ͔͒ see Eq. 5.53 . There is thus no amplification when We imagine we turn the measurement on at t=0 and ␹ =␹* . Further, if one also insists that the noise con- FI IF start to integrate up the output I͑t͒ of our detector, straint of Eq. ͑4.11͒ is optimized, then one finds the power gain must be exactly 1; this is explicitly demon- t ˆ ͑ ͒ ͵ Јˆ͑ Ј͒ ͑ ͒ strated in Appendix I.2. The detector thus will simply m t = dt I t . 4.20 0 act as a transducer, reproducing the input signal at the output without any increase in energy. We have here a The probability distribution of the integrated output specific example of a more general idea that will be dis- mˆ ͑t͒ will depend on the state of the qubit; for long times, cussed in Sec. V: if a detector acts only as a transducer, it we may approximate the distribution corresponding to need not add any noise. each qubit state as being Gaussian. Noting that we have At finite frequencies, there is a second way to make chosen Iˆ so that its expectation vanishes at zero cou- ͑ ͒ the RHS of the quantum noise constraint of Eq. 4.11 pling, the average value of ͗mˆ ͑t͒͘ corresponding to each ¯ ͓␻͔ ␹ ͑ ͒ vanish: one needs the quantity SIF / ˜ IF to be purely qubit state is in the long-time limit of interest

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͗ ͑ ͒͘ ␹ ͗ ͑ ͒͘ ␹ ͑ ͒ the two detectors in series as ␹ ␹ , while the back- mˆ t ↑ = A IFt, mˆ t ↓ =−A IFt. 4.21 I2F2 IF action driving the qubit dephasing is described by the The variance of both distributions is, to leading order, spectral density ͑␹ ͒2S . Using the fact that our sec- FI F2F2 independent of the qubit state, ond detector must itself satisfy the quantum noise inequality, we have ͗ 2͑ ͒͘ ͗ ͑ ͒͘2 ϵ͗͗ 2͑ ͒͘͘ ¯ ͑ ͒ mˆ t ↑/↓ − mˆ t ↑/↓ mˆ t ↑/↓ = SIIt. 4.22 ͓͑␹ ͒2S¯ ͔S¯ ജ ͑ប2/4͒͑␹ ␹ ͒2. ͑4.26͒ For the last equality above, we have taken the long-time FI F2F2 I2I2 I2F2 IF limit, which results in the variance of mˆ being deter- Thus, the overall chain of detectors satisfies the usual mined completely by the zero-frequency output noise zero-reverse-gain quantum noise inequality, implying ¯ ͓␻ ͔ ␩ഛ SII =0 of the detector. The assumption here is that, that we still have 1. due to the weakness of the measurement, the measure- ⌫ ment time 1/ meas will be much longer than the autocor- V. QUANTUM LIMIT ON LINEAR AMPLIFIERS AND relation time of the detector’s noise. POSITION DETECTORS We can now define the measurement rate, in complete analogy to the cavity detector of the previous section ͓cf. In the previous section we established the fundamen- Eq. ͑3.23͔͒, by how quickly the resolving power of the tal quantum constraint on the noise of any system ca- 7 measurement grows, pable of acting as a linear detector; we further showed 1 2 2 2 that this quantum noise constraint directly leads to the ͓͗mˆ ͑t͒͘↑ − ͗mˆ ͑t͒͘↓͔ /͓͗͗mˆ ͑t͒͘͘↑ + ͗͗mˆ ͑t͒͘͘↓͔ϵ⌫ t. 4 meas quantum limit on nondemolition qubit detection using a ͑4.23͒ weakly coupled detector. In this section, we turn to the more general situation where our detector is a phase- This yields preserving quantum linear amplifier: the input to the detector is described by some time-dependent operator ⌫ = A2͑␹ ͒2/2S¯ . ͑4.24͒ meas IF II xˆ͑t͒, which we wish to have amplified at the output of Putting this all together, we find that the “efficiency” our detector. As we see, the quantum limit in this case is ␩ ⌫ ⌫ ratio = meas/ ␸ is given by a limit on how small one can make the noise added by the amplifier to the signal. The discussion in this section ␩ ϵ ⌫ ⌫ ប2͑␹ ͒2 ¯ ¯ ͑ ͒ meas/ ␸ = IF /4SIISFF. 4.25 both furthers and generalizes the heuristic discussion of position detection using a cavity detector presented in In the case where our detector has a vanishing reverse Sec. III.B. gain ͑i.e., ␹ =0͒, the quantum-limit bound ␩ഛ1 follows FI In this section, we start by presenting a heuristic dis- immediately from the quantum noise constraint of Eq. cussion of quantum constraints on amplification. We ͑4.11͒. We thus see that achieving the quantum limit for then demonstrate explicitly how the previously dis- QND qubit detection requires both a detector with cussed quantum noise constraint leads directly to the quantum-ideal noise properties, as defined by Eq. ͑4.17͒, quantum limit on the added noise of a phase-preserving and a detector with a vanishing noise cross correlator: linear amplifier; we examine the cases of both a generic ¯ SIF=0. linear position detector and a generic voltage amplifier, ␹ If in contrast FI 0, it would seem that it is possible following the approach outlined by Clerk ͑2004͒.We to have ␩ജ1. This is of course an invalid inference: as also spend time explicitly connecting the linear-response ␹ discussed, FI 0 implies that we must necessarily con- approach used here to the bosonic scattering formula- sider the effects of extra noise injected into the detection of the quantum limit favored by the quantum optics ͑ tor’s output port when one measures Iˆ, as the reverse community Haus and Mullen, 1962; Caves, 1982; Gras- ͒ gain will bring this noise back to the qubit, causing extra sia, 1998; Courty et al., 1999 , paying particular attention dephasing. The result is that one can do no better than to the case of a two-port scattering amplifier. We will see ␩=1. To see this explicitly, consider the extreme case that there are some important subtleties involved in con- ␹ ␹ ¯ ¯ verting between the two approaches. In particular, there IF= FI and SII=SFF=0, and suppose we use a second exists a crucial difference between the case where the detector to read out the output Iˆ of the first detector. input signal is tightly coupled to the input of the ampli- ˆ fier ͑the case usually considered in the quantum optics This second detector has input and output operators F2, community͒, versus the case where, as in an ideal op- Iˆ ; we also take it to have a vanishing reverse gain, so 2 amp, the input signal is only weakly coupled to the input that we do not have to also worry about how its output of the amplifier ͑the case usually considered in the solid is read out. Coupling of the detectors linearly in the state community͒. ͑ ˆ ˆ ͒ standard way i.e., Hint,2 =IF2 , gives the overall gain of

A. Preliminaries on amplification 7The factor of 1/4 here is purely chosen for convenience; we are defining the measurement rate based on the information- What exactly does one mean by amplification? As we theoretic definition given in Appendix F. This factor of 4 is will see ͑Sec. V.E.3͒, a precise definition requires that the consistent with the definition used in the cavity system. energy provided at the output of the amplifier be much

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larger than the energy drawn at the input of the to equal the sum of the oscillators’ frequencies. The re- amplifier—the power gain of the amplifier must be sulting scheme is called a phase-preserving nondegener- larger than 1. For the moment, however, we work with ate parametric amplifier ͑see Sec. V. C ͒. the cruder definition that amplification involves making Crucially, there is a price to pay for the introduction of some time-dependent signal larger. To set the stage, we an extra degree of freedom: there will be noise associ- first consider an extremely simple classical analog of a ated with the idler oscillator, and this noise will contrib- linear amplifier. Imagine that the “signal” we wish to ute to the noise in the output of the amplifier. Classi- amplify is the coordinate x͑t͒ of a harmonic oscillator; cally, one could make the noise associated with the idler we can write this signal as oscillator arbitrarily small by simply cooling it to zero temperature. This is not possible quantum mechanically; ͑ ͒ ͑ ͒ ͑␻ ͒ ͓ ͑ ͒ ␻ ͔ ͑␻ ͒ ͑ ͒ x t = x 0 cos St + p 0 /M S sin St . 5.1 there are always zero-point fluctuations of the idler oscillator to contend with. It is this noise which sets a fun- Our signal has two quadrature amplitudes, i.e., the am- damental quantum limit for the operation of the ampli- plitude of the cosine and sine components of x͑t͒.To fier. We thus have a heuristic accounting for the amplify this signal, we start at t=0 to parametrically existence of a quantum limit on the added noise of a ␻ drive the oscillator by changing its frequency S periodi- phase-preserving linear amplifier: one needs extra de- ␻ ͑ ͒ ␻ ␦␻ ͑␻ ͒ cally in time: S t = 0 + sin Pt , where we assume grees of freedom to amplify both signal quadratures, and ␦␻Ӷ␻ 0. The well-known physical example is a swing such extra degrees of freedom invariably have noise as- whose motion is being excited by effectively changing sociated with them. the length of the pendulum at the right frequency and ␻ phase. For a “pump frequency” P equalling twice the B. Standard Haus-Caves derivation of the quantum limit on a ␻ ␻ “signal frequency,” P =2 S, the resulting dynamics will bosonic amplifier lead to an amplification of the initial oscillator position, with the energy provided by the external driving, We now make the ideas of the previous section more precise by sketching the standard derivation of the ͑ ͒ ͑ ͒ ␭t ͑␻ ͒ ͓ ͑ ͒ ␻ ͔ −␭t ͑␻ ͒ x t = x 0 e cos St + p 0 /M S e sin St . quantum limit on the noise added by a phase-preserving ͑5.2͒ amplifier. This derivation is originally due to Haus and Mullen ͑1962͒, and was both clarified and extended by Thus, one of the quadratures is amplified exponentially, Caves ͑1982͒; the amplifier quantum limit was also mo- 8 at a rate ␭=␦␻/2, while the other one decays. In a tivated in a slightly different manner by Heffner ͑1962͒. quantum-mechanical description, this produces a While extremely compact, the Haus-Caves derivation squeezed state out of an initial coherent state. Such a can lead to confusion when improperly applied; we dis- system is called a “degenerate parametric amplifier,” cuss this in Sec. V. D , as well as in Sec. VI, where we and we discuss its quantum dynamics in Sec. V. H .We apply this argument carefully to the important case of a see that such an amplifier, which amplifies only a single two-port quantum voltage amplifier and discuss the con- quadrature, is not required quantum mechanically to nection to the general linear-response formulation of add any noise ͑Caves et al., 1980; Caves, 1982; Braginsky Sec. IV. and Khalili, 1992͒. The starting assumption of this derivation is that both Can we now change this parametric amplification the input and output ports of the amplifier can be de- scheme slightly in order to make both signal quadratures scribed by sets of bosonic modes. If we focus on a nar- grow with time? It turns out that this is impossible as row bandwidth centered on frequency ␻, we can de- long as we restrict ourselves to a driven system with a scribe a classical signal E͑t͒ in terms of a complex single degree of freedom. The reason in classical me- number a defining the amplitude and phase of the signal chanics is that Liouville’s theorem requires phase-space ͑or equivalently the two quadrature amplitudes͒͑Haus volume to be conserved during motion. More formally, and Mullen, 1962; Haus, 2000͒ this is related to the conservation of Poisson brackets, or ␻ ␻ ͑ ͒ ϰ ͓ −i t * +i t͔ ͑ ͒ in quantum mechanics to the conservation of commuta- E t i ae − a e . 5.3 tion relations. Nevertheless, it is certainly desirable to In the quantum case, the two signal quadratures of E͑t͒ have an amplifier that acts equally on both quadratures ͓i.e., the real and imaginary parts of a͑t͔͒ cannot be mea- ͑a so-called phase-preserving or phase-insensitive ampli- sured simultaneously because they are canonically con- fier͒, since the signal’s phase is often not known before- jugate; this is in complete analogy to a harmonic oscilla- hand. The way around the restriction created by Liou- tor ͓see Eq. ͑3.48͔͒. As a result a,a* must be elevated to ville’s theorem is to add more degrees of freedom, such the status of photon ladder operators: a→aˆ, a* →a†. that the phase-space volume can expand in both quadra- Consider the simplest case where there is only a single ͑ ͒ tures i.e., position and momentum of the interesting mode at both the input and output, with corresponding signal degree of freedom, while being compressed in other directions. This is achieved most easily by coupling the signal oscillator to another oscillator, the “idler 8Note that Caves ͑1982͒ provided a discussion of why the mode.” The external driving now modulates the cou- derivation of the amplifier quantum limit given by Heffner pling between these oscillators, at a frequency that has ͑1962͒ is not rigorously correct.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1180 Clerk et al.: Introduction to quantum noise, measurement, … operators aˆ and bˆ .9 It follows that the input signal into amplified noise of the input, while the second term rep- the amplifier is described by the expectation value ͗aˆ͘, resents the noise added by the amplifier. Note that if there is no amplification ͑i.e., G=1͒, there need not be ͗ ˆ ͘ while the output signal is described by b . Correspond- any added noise. However, in the more relevant case of ingly, the symmetrized noise in both these quantities is large amplification ͑Gӷ1͒, the added noise cannot van- described by ish. It is useful to express the noise at the output as an ͑⌬a͒2 ϵ 1 ͕͗aˆ,aˆ †͖͘ − ͉͗aˆ͉͘2, ͑5.4͒ equivalent noise at the input by simply dividing out the 2 photon gain G. Taking the large-G limit, we have with an analogous definition for ͑⌬b͒2. ͑⌬ ͒2 ജ ͑⌬ ͒2 1 ͑ ͒ To derive a quantum limit on the added noise of the b /G a + 2 . 5.10 amplifier, one uses two simple facts. First, both the input Thus, we have a simple demonstration that an amplifier and the output operators must satisfy the usual commu- with a large photon gain must add at least half a quan- tation relations tum of noise to the input signal. Equivalently, the minimum value of the added noise is equal to the zero-point ͓ˆ ˆ †͔ ͓ ˆ ˆ †͔ ͑ ͒ a,a =1, b,b =1. 5.5 noise associated with the input mode; the total output Second, the linearity of the amplifier and the fact that it noise ͑referred to the input͒ is at least twice the zero- is phase preserving ͑i.e., both signal quadratures are am- point input noise. Note that both these conclusions are plified the same way͒ implies a simple relation between identical to what we found in our analysis of the resonant cavity position detector in Sec. III.B.3. We discuss the output operator bˆ and the input operator aˆ, later how this conclusion can also be reached using the ͑ bˆ = ͱGaˆ, bˆ † = ͱGaˆ †, ͑5.6͒ general linear-response language of Sec. IV cf. Secs. V. E and V. F ͒. where G is the dimensionless photon-number gain of the As discussed, the added noise operator F is associated amplifier. It is clear, however, that this expression cannot with additional degrees of freedom ͑beyond input and possibly be correct as written because it violates the fun- output modes͒ necessary for phase-preserving amplifica- damental bosonic commutation relation ͓bˆ ,bˆ †͔=1. We tion. To see this more concretely, note that every linear are therefore forced to write amplifier is inevitably a nonlinear system consisting of an energy source and a “spigot” controlled by the input bˆ = ͱGaˆ + Fˆ , bˆ † = ͱGaˆ † + Fˆ †, ͑5.7͒ signal, which redirects the energy source partly to the output channel and partly to some other channel͑s͒. where Fˆ is an operator representing additional noise Hence there are inevitably other degrees of freedom in- added by the amplifier. Based on the discussion of the volved in the amplification process beyond the input and previous section, we can anticipate what Fˆ represents: it output channels. An explicit example is the quantum is noise associated with the additional degrees of free- parametric amplifier, discussed in the next subsection. dom that must invariably be present in a phase- Further insights into amplifier-added noise and its con- preserving amplifier. nection to the fluctuation-dissipation theorem can be ob- Fˆ tained by considering a simple model where a transmis- As represents noise, it has a vanishing expectation sion line is terminated by an effective negative value; in addition, one also assumes that this noise is impedance; we discuss this model in Appendix C.4. uncorrelated with the input signal, implying ͓Fˆ ,aˆ͔ To see explicitly the role of the additional degrees of =͓Fˆ ,aˆ †͔=0 and ͗Fˆ aˆ͘=͗Fˆ aˆ †͘=0. Insisting that ͓bˆ ,bˆ †͔=1 freedom, note first that for GϾ1 the RHS of Eq. ͑5.8͒ is thus yields negative. Hence the simplest possible form for the added noise is ͓Fˆ ,Fˆ †͔ =1−G. ͑5.8͒ Fˆ = ͱG −1dˆ †, Fˆ † = ͱG −1dˆ , ͑5.11͒ The question now becomes how small can we make ˆ ˆ † the noise described by Fˆ ? From Eqs. ͑5.7͒, the noise at where d and d represent a single additional mode of the the amplifier output ⌬b is given by system. This is the minimum number of additional degrees of freedom that must inevitably be involved in the ͑⌬ ͒2 ͑⌬ ͒2 1 ͕͗Fˆ Fˆ †͖͘ ജ ͑⌬ ͒2 1 ͉͓͗Fˆ Fˆ †͔͉͘ amplification process. Note that for this case the in- b = G a + 2 , G a + 2 , equality in Eq. ͑5.9͒ is satisfied as an equality, and the ജ ͑⌬ ͒2 ͉ ͉ ͑ ͒ G a + G −1/2. 5.9 added noise takes on its minimum possible value. If in- ͑ We have used here a standard inequality to bound the stead we have, say, two additional modes coupled in- equivalently͒ expectation of ͕Fˆ ,Fˆ †͖. The first term here is simply the Fˆ ͱ ͑ ␪ ˆ † ␪ ˆ ͒ ͑ ͒ = G −1 cosh d1 + sinh d2 , 5.12 9To relate this to the linear-response detector of Sec. IV.A, it is straightforward to show that the added noise is in- one could naively write xˆ, the operator carrying the input sig- evitably larger than the minimum. This again can be in- nal as, xˆ =aˆ +aˆ †, and the output operator Iˆ as, Iˆ =bˆ +bˆ † ͑we terpreted in terms of wasted information, as the extra discuss how to make this correspondence in Sec. VI͒. degrees of freedom are not being monitored as part of

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this approximation were not valid, then our amplifier ω would in any case not be the linear amplifier we seek. I With this approximation we can hereafter ignore the dynamics of the pump degree of freedom and deal with the ω reduced system Hamiltonian P ͑␻ ␻ ͒ ˆ ប͑␻ ˆ † ˆ ␻ ˆ † ˆ ͒ ប␭͑ˆ † ˆ † −i I+ S t ω Hsys = IaI aI + SaSaS + i aSaI e S ͑␻ ␻ ͒ +i I+ S t͒ ͑ ͒ − aˆ Saˆ Ie , 5.15 ␭ϵ␩␺ where P. Transforming to the interaction representation we are left with the following time- ͑ ͒ FIG. 8. Color online Energy level scheme of the nondegen- independent quadratic Hamiltonian for the system erate ͑phase-preserving͒ parametric oscillator. ˆ ប␭͑ † † ͒ ͑ ͒ Vsys = i aˆ Saˆ I − aˆ Saˆ I . 5.16 the measurement process and so information is being To get some intuitive understanding of the physics, we lost. temporarily ignore the damping of the cavity modes that would result from their coupling to modes outside the cavity. We now have a pair of coupled EOMs for the two C. Nondegenerate parametric amplifier modes

1. Gain and added noise ˙ ␭ † ˙ † ␭ ͑ ͒ aˆ S =+ aˆ I , aˆ I =+ aˆ S, 5.17 Before we start our discussion of the connection of the Haus-Caves formulation of the quantum limit to the for which the solutions are general linear-response approach of Sec. IV, it is useful ͑ ͒ ͑␭ ͒ ͑ ͒ ͑␭ ͒ †͑ ͒ aˆ S t = cosh t aˆ S 0 + sinh t aˆ I 0 , to consider a specific example. To that end, we analyze ͑ ͒ here a nondegenerate parametric amplifier, a linear 5.18 †͑ ͒ ͑␭ ͒ ͑ ͒ ͑␭ ͒ †͑ ͒ phase-preserving amplifier which reaches the quantum aˆ I t = sinh t aˆ S 0 + cosh t aˆ I 0 . ͑ limit on its added noise Louisell et al., 1961; Gordon et We see that the amplitude in the signal channel grows ͒ al., 1963; Mollow and Glauber, 1967a, 1967b and di- exponentially in time and that the effect of the time evo- rectly realizes the ideas of the previous subsection. One lution is to perform a simple unitary transformation ͑ ͒ possible realization Yurke et al., 1989 is a cavity with which mixes aˆ with aˆ † in such a way as to preserve the three internal resonances that are coupled together by a S I ͑ ͒ commutation relations. Note the close connection with nonlinear element such as a Josephson junction whose the form found from general arguments in Eqs. symmetry permits three-wave mixing. The three modes ͑5.7͒–͑5.11͒. are called the pump, idler, and signal and their energy We may now tackle the full system which includes the ͑ ͒ ␻ ␻ ␻ level structure shown in Fig. 8 obeys P = I + S. The coupling between the cavity modes and modes external system Hamiltonian is then to the cavity. Such a coupling is of course necessary in order to feed the input signal into the cavity, as well as Hˆ = ប͑␻ aˆ † aˆ + ␻ aˆ †aˆ + ␻ aˆ †aˆ ͒ + iប␩͑aˆ †aˆ †aˆ sys P P P I I I S S S S I P extract the amplified output signal. It will also result in † ͒ ͑ ͒ − aˆ Saˆ Iaˆ P . 5.13 the damping of the cavity modes, which will cut off the exponential growth found above and yield a fixed ampli- We have made the rotating wave approximation in the tude gain. We present the main results in this section, three-wave mixing term, and without loss of generality relegating details to how one treats the bath modes ͓so- we take the nonlinear susceptibility ␩ to be real and called input-output theory ͑Walls and Milburn, 1994͔͒ to positive. The system is driven at the pump frequency Appendix E. Working in the standard Markovian limit, and the three-wave mixing term permits a single pump we obtain the following EOMs in the interaction repre- photon to split into an idler photon and a signal photon. sentation: This process is stimulated by signal photons already present and leads to gain. A typical mode of operation aˆ˙ =−͑␬ /2͒aˆ + ␭aˆ † − ͱ␬ bˆ , would be the negative-resistance reflection mode in S S S I S S,in ͑5.19͒ which the input signal is reflected from a nonlinear cav- ˙ † ͑␬ ͒ † ␭ ͱ␬ ˆ † ity and the reflected beam extracted using a circulator aˆ I =− I/2 aˆ I + aˆ S − IbI,in. ͑Yurke et al., 1989; Bergeal et al., 2008͒. Here ␬ and ␬ are the respective damping rates of the The nonlinear equations of motion ͑EOMs͒ become S I cavity signal mode and the idler mode. The coupling to tractable if we assume the pump has large amplitude and extra-cavity modes also lets signals and noise enter the can be treated classically by making the substitution cavity from the baths: this is described by the bosonic ␻ ͑␻ ␻ ͒ aˆ = ␺ e−i Pt = ␺ e−i I+ S t, ͑5.14͒ ˆ ˆ P P P operators bS,in and bI,in which drive the signal and idler ␺ ˆ where without loss of generality we take P to be real modes, respectively. bS,in describes both the input signal and positive. We note here the important point that if to be amplified and vacuum noise entering from the bath

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ˆ frequency ␻ ; for a finite bandwidth, we also need to coupled to the signal mode, whereas bI,in simply de- S scribes vacuum noise.10 understand how the power gain varies as a function of Let us fix our attention on signals inside a frequency frequency over the entire signal bandwidth. As we see, a ␦␻ ␻ ͑ parametric amplifier suffers from the fact that as one window centered on S hence zero frequency in the interaction representation͒. For simplicity, we first con- increases the overall magnitude of the gain at the center ␻ ͑ ͒ sider the case where the signal bandwidth ␦␻ is almost frequency S e.g., by increasing the pump amplitude , infinitely narrow ͑i.e., much smaller than the damping one simultaneously narrows the frequency range over rate of the cavity modes͒. It then suffices to find the which the gain is appreciable. Heuristically, this is be- steady state solution of these EOMs, cause parametric amplification involves using the pump energy to decrease the damping and hence increase the ͑ ␭ ␬ ͒ † ͑ ͱ␬ ͒ ˆ ͑ ͒ aˆ S = 2 / S aˆ I − 2/ S bS,in, 5.20 quality factor of the signal mode resonance. This increase in quality factor leads to amplification, but it also † ͑ ␭ ␬ ͒ ͑ ͱ␬ ͒ ˆ † ͑ ͒ reduces the bandwidth over which aˆ S can respond to the aˆ I = 2 / I aˆ S − 2/ I bI,in. 5.21 input signal bˆ . The output signal of the nondegenerate paramp is the S,in To deal with a finite signal bandwidth, one simply signal leaving the cavity signal mode and entering the Fourier transforms Eqs. ͑5.19͒. The resulting equations ˆ external bath modes; it is described by an operator bS,out. are easily solved and substituted into Eq. ͑5.22͒, result- The standard input-output theory treatment of the ing in a frequency-dependent generalization of the extra-cavity modes ͑Walls and Milburn, 1994͒, presented input-output relation given in Eq. ͑5.25͒, in Appendix E, yields the simple relation ͓cf. Eq. ͑E37͔͒ bˆ ͓␻͔ =−g͓␻͔bˆ ͓␻͔ − gЈ͓␻͔bˆ † ͓␻͔. ͑5.26͒ ˆ ˆ ͱ␬ ͑ ͒ S,out S,in I,in bS,out = bS,in + Saˆ S. 5.22 ͓␻͔ The first term corresponds to the reflection of the signal Here g is the frequency-dependent gain of the ampli- Ј͓␻͔ ͉ Ј͓␻͔͉2 ͉ ͓␻͔͉2 and noise incident on the cavity from the bath, while the fier, and g satisfies g = g −1. In the rel- ͉ ͓ ͔͉2 ӷ ͑ second term corresponds to radiation from the cavity evant limit where G0 = g 0 1 i.e., large gain at the ͒ mode into the bath. Using this, we find that the output signal frequency , one has to a good approximation signal from the cavity is given by ͱ ͓͑␬ ␬ ͒ ͑␬ ␬ ͔͒͑␻ ͒ G0 − i S − I / I + S /D 2 g͓␻͔ = , ͑5.27͒ Q +1 2Q † 1−i͑␻/D͒ bˆ = bˆ + bˆ , ͑5.23͒ S,out Q2 −1 S,in Q2 −1 I,in with ϵ ␭ ͱ␬ ␬ where Q 2 / I S is proportional to the pump ampli- 1 ␬ ␬ tude and inversely proportional to the cavity decay D = S I . ͑5.28͒ 2 Ͻ ͱ ␬ ␬ rates. We have to require Q 1 to make sure that the G0 S + I parametric amplifier does not settle into self-sustained oscillations, i.e., it works below threshold. Under that As always, we work in an interaction picture where the condition, we can define the photon-number gain G via signal frequency has been shifted to zero. D represents 0 the effective operating bandwidth of the amplifier. Com- ͱ ͑ 2 ͒ ͑ 2 ͒ ͑ ͒ − G0 = Q +1 / Q −1 , 5.24 ponents of the signal with frequencies ͑in the rotating ͉͒␻͉ Ӷ such that frame D are strongly amplified, while components with frequencies ͉␻͉ ӷD are not amplified at all, but can ˆ ͱ ˆ ͱ ˆ † ͑ ͒ in fact be slightly attenuated. As already anticipated, the bS,out =− G0bS,in − G0 −1bI,in. 5.25 amplification bandwidth D becomes progressively ˆ ˆ In the ideal case, the noise associated with bI,in,bS,in is smaller as the pump power and G0 are increased, with simply vacuum noise. As a result, the input-output rela- ͱ the product G0D remaining constant. In a parametric tion Eq. ͑5.25͒ is precisely of the Haus-Caves form ͑5.11͒ amplifier increasing the gain via increasing the pump for an ideal quantum-limited amplifier. It demonstrates strength comes with a price: the effective operating that the nondegenerate parametric amplifier reaches the bandwidth is reduced. quantum limit for minimum added noise. In the limit of large gain the output noise ͑referred to the input͒ for a vacuum input signal is precisely doubled. 3. Effective temperature Recall that in Sec. II.B we introduced the concept of 2. Bandwidth-gain trade-off an effective temperature of a nonequilibrium system, The above results neglected the finite bandwidth ␦␻ Eq. ͑2.8͒. As we will discuss, this concept plays an im- of the input signal to the amplifier. The gain G0 given in portant role in quantum-limited amplifiers; the degener- Eq. ͑5.24͒ is only the gain at precisely the mean signal ate paramp gives us a first example of this. Returning to the behavior of the paramp at the signal frequency, we note that Eq. ͑5.25͒ implies that, even for vacuum input 10Note that the bˆ operators are not dimensionless, as bˆ †bˆ to both the signal and idler ports, the output will contain represents a photon flux ͑see Appendix E͒. a real photon flux. To quantify this in a simple way, it is

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1183 useful to introduce temporal modes which describe the ͑b†͒n͉0,0͘ = ͱn!͉n,0͑͘5.34͒ input and output fields during a particular time interval S ͓j⌬t,͑j+1͒⌬t͔͑where j is an integer͒, to obtain ͑ ͒⌬ ϱ 1 j+1 t ˆ ͵ ␶ ˆ ͑␶͒ ͑ ͒ ͉⌿ ͘ = Z−1/2͚ ␣n͉n,n͑͘5.35͒ BS,in,j = ͱ d bS,in , 5.29 out ⌬t j⌬t n=0 ˆ ˆ and hence with the temporal modes BS,out,j and BI,in,j defined analogously. These temporal modes are discussed in Appen- Z =1/͑1−͉␣͉2͒͑5.36͒ dix D.2, where we discuss the windowed Fourier trans- ͉␣͉2 Ͻ form ͓see Eq. ͑D18͔͒. so the state is normalizable only for 1. With the above definition, we find that the output Because this output is obtained by unitary evolution mode will have a real occupancy even if the input mode from the vacuum input state, the output state is a pure is empty, state with zero entropy. In light of this, it is interesting to consider the reduced density matrix obtain by tracing ͗ ͉ ˆ † ˆ ͉ ͘ n¯ S,out = 0 BS,out,jBS,out,j 0 over the idler mode. The pure-state density matrix is ϱ mء␣͘ ͑ ͒͗ ͉ ˆ ˆ † ͉ ͘ ␣n ͉ ˆ † ˆ ͉ ͗ = G0 0 BS,in,jBS,in,j 0 + G0 −1 0 BI,in,jBI,in,j 0 ␳ ͉⌿ ͗͘⌿ ͉ ͉ ͘ ͗ ͉ ͑ ͒ = out out = ͚ n,n m,m . 5.37 ͑ ͒ m,n=0 Z = G0 −1. 5.30 If we now trace over the idler mode we are left with the The dimensionless mode occupancy n¯ S,out is best thought of as a photon flux per unit bandwidth ͓see Eq. ͑D26͔͒. reduced density matrix for the signal channel ϱ This photon flux is equivalent to the photon flux that ͉␣͉2nS 1 ␤ប␻ † ␳ ͕␳͖ ͉ ͘ ͗ ͉ϵ − Sa aS would appear in equilibrium at the very high effective ˜ S =TrIdler = ͚ nS nS e S , ͑ ͒ Z Z temperature assuming large gain G0 nS=0 Ϸ ប␻ ͑ ͒ ͑5.38͒ Teff SG0. 5.31 This is an example of a more general principle, discussed which is a pure thermal equilibrium distribution with in Sec. V.E.4: a high-gain amplifier must have associated effective Boltzmann factor ␤ប␻ with it a large effective temperature scale. Referring this e− S = ͉␣͉2 Ͻ 1. ͑5.39͒ total output noise back to the input, we have ͑in the limit G ӷ1͒ The effective temperature can be obtained from the re- 0 quirement that the signal mode occupancy is G −1, ប␻ 0 S ␤ប␻ T /G = ប␻ /2 + ប␻ /2 = + T . ͑5.32͒ 1/͑e S −1͒ = G −1, ͑5.40͒ eff 0 S S 2 N 0 which in the limit of large gain reduces to Eq. ͑5.31͒. This corresponds to the half photon of vacuum noise This appearance of finite entropy in a subsystem even associated with the signal source, plus the added noise of ͑ when the full system is in a pure state is a purely quan- a half photon of our phase-preserving amplifier i.e., the tum effect. Classically the entropy of a composite system noise temperature TN is equal to its quantum-limited is at least as large as the entropy of any of its compo- ͒ value . Here the added noise is simply the vacuum noise nents. Entanglement among the components allows this associated with the idler port. lower bound on the entropy to be violated in a quantum The above argument is merely suggestive that the out- system.11 In this case the two-mode squeezed state has put noise looks like an effective temperature. In fact, it strong entanglement between the signal and idler chan- is possible to show that the photon-number distribution nels ͑since their photon numbers are fluctuating identi- of the output is precisely that of a Bose-Einstein distri- cally͒. ͑ ͒ bution at temperature Teff. From Eq. 5.13 we see that the action of the paramp is to destroy a pump photon and create a pair of new photons, one in the signal chan- D. Scattering versus op-amp modes of operation nel and one in the idler channel. Using the SU͑1,1͒ symmetry of the quadratic hamiltonian in Eq. ͑5.16͒ it is We now begin to address the question of how the possible to show that, for vacuum input, the output of standard Haus-Caves derivation of the amplifier quan- the paramp is a so-called “two-mode squeezed state” of tum limit presented in Sec. V. B relates to the general the form ͑Caves and Schumaker, 1985; Gerry, 1985; linear-response approach of Sec. IV. Recall that in Sec. Knight and Buzek, 2004͒ III.B.3 we already used this latter approach to discuss position detection with a cavity detector, reaching simi- ␣ † † ͉⌿ ͘ −1/2 b bI ͉ ͘ ͑ ͒ out = Z e S 0 , 5.33 ␣ where is a constant related to the gain and, to simplify 11This paradox has prompted Charles Bennett to remark that the notation, we have dropped the “out” labels on the a classical house is at least as dirty as its dirtiest room, but a operators. The normalization constant Z can be worked quantum house can be dirty in every room and still perfectly out by expanding the exponential and using clean over all.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1184 Clerk et al.: Introduction to quantum noise, measurement, …

aout bin TABLE IV. Two different ampliﬁer modes of operation. amplifier

ain bout Input Signal Output Signal Mode s͑t͒ o͑t͒ input line output line ͑ ͒ ͑ ͒ ͑ ͑ ͒ ͑ ͒͒ Scattering s t =ain t o t =bout t FIG. 9. ͑Color online͒ Schematic of a two-port bosonic ampli- ͑ ͒ ͑ ͒ ain indep. of aout bout indep. of bin fier. Both the inputs and outputs of the amplifier are attached Op-amp s͑t͒=V ͑t͒ o͑t͒=V ͑t͒ to transmission lines. The incoming and outgoing wave ampli- a b ͑a depends on a ͒ ͑b depends on b ͒ tudes in the input ͑output͒ transmission line are labeled in out out in ͑ ˆ ˆ ͒ aˆ in,aˆ out bin,bout , respectively. The voltages at the end of the ͑ ˆ ˆ ͒ two lines Va ,Vb are linear combinations of incoming and outgoing wave amplitudes. Further, the output signal is taken to be the amplitude of the outgoing wave exiting the output of the amplifier ͑ ͒ i.e., bout , again irrespective of whatever might be enter- lar conclusions ͑i.e., at best, the detector adds noise ing the output port ͑see Table IV͒. In this situation, the equal to the zero-point noise͒. In that linear-response- Haus-Caves description of the quantum limit in the pre- based discussion, we saw that a crucial aspect of the vious section is almost directly applicable; we make this quantum limit was the trade-off between back-action precise in Sec. VI. Back-action is indeed irrelevant, as noise and measurement-imprecision noise. We saw that the prescribed experimental conditions mean that it reaching the quantum limit required both a detector plays no role. We call this mode of operation the scat- with ideal noise, as well as an optimization of the tering mode, as it is most relevant to time-dependent detector-oscillator coupling strength. Somewhat disturb- experiments where the experimentalist launches a signal ingly, none of these ideas appeared explicitly in the pulse at the input of the amplifier and looks at what exits Haus-Caves derivation; this can give the misleading im- the output port. One is usually only interested in the pression that the quantum limit never has anything to do scattering mode of operation in cases where the source with back-action. A further confusion comes from the producing the input signal is matched to the input of the fact that many detectors have input and outputs that amplifier: only in this case is the input wave ain perfectly cannot be described by a set of bosonic modes. How transmitted into the amplifier. As we see in Sec. VI, such does one apply the above arguments to such systems? a perfect matching requires a relatively strong coupling The first step in resolving these seeming inconsisten- between the signal source and the input of the amplifier; cies is to realize that there are really two different ways as such, the amplifier will strongly enhance the damping in which one can use a given amplifier or detector. In of the signal source. deciding how to couple the input signal ͑i.e., the signal to The second mode of linear amplifier operation is what be amplified͒ to the amplifier, and in choosing what we call the op-amp mode; this is the mode one usually quantity to measure, the experimentalist essentially en- has in mind when thinking of an amplifier which is forces boundary conditions; as now shown, there are in weakly coupled to the signal source, and will be the next general two distinct ways in which to do this. For con- focus. The key difference from the scattering mode is creteness, consider the situation shown in Fig. 9: a two- that here the input signal is not simply the amplitude of port voltage amplifier where the input and output ports a wave incident on the input port of the amplifier; simi- of the amplifier are attached to one-dimensional trans- larly, the output signal is not the amplitude of a wave mission lines ͑see Appendix C for a quick review of exiting the output port. As such, the Haus-Caves deriva- quantum transmission lines͒. As in the previous section, tion of the quantum limit does not directly apply. For the we focus on a narrow bandwidth signal centered about a bosonic amplifier discussed here the op-amp mode frequency ␻. At this frequency, there exists both a right- would correspond to using the amplifier as a voltage op- moving and a left-moving wave in each transmission amp. The input signal would thus be the voltage at the line. We label the corresponding amplitudes in the input end of the input transmission line. Recall that the volt- ͑ ͒ ͑ ͒ output line with ain,aout bin,bout , as per Fig. 9. Quan- age at the end of a transmission line involves the ampli- ͑ ͒ tum mechanically these amplitudes become operators, tude of both left- and right-moving waves, i.e., Va t ϰ ͓ ͑ ͒ ͑ ͔͒ much in the same way that we treated the mode ampli- Re ain t +aout t . At first this might seem quite con- ͑ ͒ tude a as an operator in the previous section. We ana- fusing: If the signal source determines Va t , does this ͑ ͒ ͑ ͒ lyze this two-port bosonic amplifier in Sec. VI; here we mean it sets the values of both ain t and aout t ? Does only sketch its operation to introduce the two different not this violate causality? The signal source enforces the ͑ ͒ ͑ ͒ amplifier operation modes. This will then allow us to value of Va t by simply changing ain t in response to ͑ ͒ understand the subtleties of the Haus-Caves quantum the value of aout t . While there is no violation of causal- limit derivation. ity, the fact that the signal source is dynamically re- In the first kind of setup, the experimentalist arranges sponding to what comes out of the amplifier’s input port things so that ain, the amplitude of the wave incident on implies that back-action is indeed relevant. the amplifier’s input port, is precisely equal to the signal The op-amp mode of operation is relevant to the typi- to be amplified ͑i.e., the input signal͒, irrespective of the cal situation of weak coupling between the signal source ͑ ͒ amplitude of the wave leaving the input port i.e., aout . and amplifier input. By weak coupling, we mean here

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1185

cold properties of a generic linear-response detector in Sec. load IV, including the fundamental quantum noise constraint vacuum noise uin uout of Eq. ͑4.11͒. For simplicity, we start with the problem of

aout bin continuous position detection of a harmonic oscillator. amplifier Our discussion will thus generalize the discussion of po- input line output line sition detection using a cavity detector given in Sec. ain bout circulator III.B.3. We start with a generic detector ͑as introduced in Sec. IV.A͒ coupled at its input to the position xˆ of a 13 FIG. 10. Illustration of a bosonic two-port amplifier used in harmonic oscillator ͓see Eq. ͑4.1͔͒. We want to under- the scattering mode of operation. The signal is an incoming stand the total output noise of our amplifier in the pres- wave in the input port of the amplifier, and does not depend on ence of the oscillator, and more importantly how small what is coming out of the amplifier. This is achieved by con- we can make the amplifier’s contribution to this noise. necting the input line to a circulator and a “cold load” ͑i.e., a The resulting lower bound is known as the standard zero-temperature resistor͒: all that goes back toward the quantum limit ͑SQL͒ on position detection, and is analo- source of the input signal is vacuum noise. gous to the quantum limit on the added noise of a voltage amplifier ͑discussed in Sec. V. F ͒. something stronger than just requiring that the amplifier be linear: we require additionally that the amplifier does 1. Detector back-action not appreciably change the dissipation of the signal We first consider the consequence of noise in the de- source. This is analogous to the situation in an ideal volt- tector input port. As seen in Sec. II.B, the fluctuating age op-amp, where the amplifier input impedance is ˆ much larger than the impedance of the signal source. We back-action force F acting on our oscillator will lead to stress that the op-amp mode and this limit of weak cou- both damping and heating of the oscillator. To model the ͑ ͒ pling is the relevant situation in most electrical measure- intrinsic i.e., detector-independent heating and damp- ments. ing of the oscillator, we also assume that our oscillator is Thus, we see that the Haus-Caves formulation of the coupled to an equilibrium heat bath. In the weak- quantum limit is not directly relevant to amplifiers or coupling limit that we are interested in, one can use detectors operated in the usual op-amp mode of opera- lowest-order perturbation theory in the coupling A to tion. We clearly need some other way to describe quan- describe the effects of the back-action force Fˆ on the tum amplifiers used in this regime. As we demonstrated oscillator. A full quantum treatment ͑see Appendix I.4͒ in the remainder of this section, the general linear- shows that the oscillator is described by an effective clas- response approach of Sec. IV is exactly what is needed. sical Langevin equation,14 To see this, expand the discussion of Sec. IV to include ¨͑ ͒ ⍀2 ͑ ͒ ␥ ˙͑ ͒ ͑ ͒ the concepts of input and output impedance as well as Mx t =−M x t − M 0x t + F0 t power gain. The linear-response approach will allow us 2 to see ͑similar to Sec. III.B.3͒ that reaching the quantum − MA ͵ dtЈ␥͑t − tЈ͒x˙͑tЈ͒ − AF͑t͒. ͑5.41͒ limit in the op-amp mode does indeed require a trade- off between back-action and measurement imprecision, The position x͑t͒ in the above equation is not an opera- and requires use of an amplifier with ideal quantum tor, but is simply a classical variable whose fluctuations ͑ ͒ ͑ ͒ noise properties ͓see Eq. ͑4.11͔͒. This approach also has are driven by the fluctuating forces F t and F0 t . None- the added benefit of being directly applicable to systems theless, the noise in x calculated from Eq. ͑5.41͒ corre- where the input and output of the amplifier are not de- ¯ ͓␻͔ 12 sponds precisely to Sxx , the symmetrized quantum- scribed by bosonic modes. In Sec. VI, we return to the mechanical noise in the operator xˆ. The fluctuating force scattering description of a two-port voltage amplifier exerted by the detector ͑which represents the heating ͑Fig. 10͒, and show explicitly how an amplifier can be part of the back-action͒ is described by AF͑t͒ in Eq. quantum limited when used in the scattering mode of ͑5.41͒; it has zero mean, and a spectral density given by operation, but miss the quantum limit when used in the 2 ¯ ͓␻͔ ͑ ͒ ␥͑ ͒ op-amp mode of operation. A SFF in Eq. 4.4a . The kernel t describes the damping effect of the detector. It is given by the asym-

E. Linear-response description of a position detector 13For consistency with previous sections, our coupling Hamil- In this section, we examine the amplifier quantum tonian does not have a minus sign. This is different from the limit for a two-port linear amplifier in the usual weak- convention of Clerk ͑2004͒, where the coupling Hamiltonian is ˆ coupling, op-amp regime of operation. Our discussion written Hint=−Axˆ ·F. here will make use of the results obtained for the noise 14Note that we have omitted a back-action term in this equation which leads to small renormalizations of the oscillator frequency and mass. These terms are not important for the fol- 12Note that the Haus-Caves derivation for the quantum limit lowing discussion, so we have omitted them for clarity; one can of a scattering amplifier has been generalized to the case of consider M and ⍀ in this equation to be renormalized quanti- fermionic operators ͑Gavish et al., 2004͒. ties. See Appendix I.4 for more details.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1186 Clerk et al.: Introduction to quantum noise, measurement, … metric part of the detector’s quantum noise, as was de- have the measurement-imprecision noise discussed in rived in Sec. II.B ͓see Eq. ͑2.12͔͒. Sec. III.B.3. The second term corresponds to the ampli- Equation ͑5.41͒ also describes the effects of an equi- fied fluctuations of the oscillator, which are in turn given ͑ ͒ librium heat bath at temperature T0 which models the by solving Eq. 5.41 , intrinsic ͑i.e., detector-independent͒ damping and heat- 1/M ing of the oscillator. The parameter ␥ is the damping ␦ ͓␻͔ ͫ ͬ͑ ͓␻͔ ͓␻͔͒ 0 x =− F − AF ͑␻2 − ⍀2͒ + i␻⍀/Q͓␻͔ 0 arising from this bath and F0 is the corresponding fluc- ϵ ␹ ͓␻͔͑ ͓␻͔ ͓␻͔͒ ͑ ͒ tuating force. The spectral density of the F0 noise is de- xx F0 − AF , 5.45 termined by ␥ and T via the fluctuation-dissipation 0 0 ͓␻͔ ⍀ ͑␥ ␥͓␻͔͒ ͓ ͑ ͔͒ ␥ where Q = / 0 + is the oscillator quality fac- theorem see Eq. 2.16 . T0 and 0 have a simple physical significance: they are the temperature and damping tor. It follows that the spectral density of the total noise in the detector output is given classically by of the oscillator when the coupling to the detector A is S ͓␻͔ S ͓␻͔ ͉␹ ͓␻͔␹ ͓␻͔͉2͑ 4S ͓␻͔ set to zero. II,tot = II + xx IF A FF To make further progress, we recall from Sec. II.B + A2S ͓␻͔͒ that even though our detector will in general not be in F0F0 equilibrium, we may nonetheless assign it an effective +2A2 Re ␹ ͓␻͔␹ ͓␻͔S ͓␻͔ . ͑5.46͒ temperature T ͓␻͔ at each frequency ͓see Eq. ͑2.8͔͒. † xx IF IF ‡ eff S S S ͑ ͒ The effective temperature of an out-of-equilibrium de- Here II, FF, and IF are the classical detector noise tector is simply a measure of the asymmetry of the de- correlators calculated in the absence of any coupling to tector’s quantum noise. We are often interested in the the oscillator. Note importantly that we have included limit where the internal detector time scales are much the fact that the two kinds of detector noise ͑in Iˆ and in faster than the time scales relevant to the oscillator ͑i.e., Fˆ ͒ may be correlated with one another. ⍀−1 ␥−1 ␥−1͒ ␻→ , , 0 . We may then take the 0 limit in the To apply the classically derived Eq. ͑5.46͒ to our quan- expression for Teff, yielding tum detector-plus-oscillator system, we recall from Sec. II.B that symmetrized quantum noise spectral densities 2k T ϵ S¯ ͑0͒/M␥͑0͒. ͑5.42͒ B eff FF play the role of classical noise. The LHS of Eq. ͑5.46͒ In this limit, the oscillator position noise calculated from thus becomes SII,tot, the total symmetrized quantum- Eq. ͑5.41͒ is given by mechanical output noise of the detector, while the RHS will now contain the symmetrized quantum-mechanical 1 2͑␥ + ␥͒k ␥ T + ␥T S¯ ͓␻͔ = 0 B 0 eff . ¯ ¯ ¯ xx M ͑␻2 − ⍀2͒2 + ␻2͑␥ + ␥ ͒2 ␥ + ␥ detector noise correlators SFF, SII, and SIF, defined as in 0 0 Eq. ͑4.4a͒. Though this may seem rather ad hoc, one can ͑ ͒ 5.43 easily demonstrate that Eq. ͑5.46͒ thus interpreted This is exactly what would be expected if the oscillator would be quantum mechanically rigorous if the detector were only attached to an equilibrium Ohmic bath with a correlation functions obeyed Wick’s theorem. Thus, ͑ ͒ ␥ ␥ ␥ ¯ quantum corrections to Eq. 5.46 will arise solely from damping coefficient ⌺ = 0 + and temperature T ͑␥ ␥ ͒ ␥ the non-Gaussian nature of the detector noise correla- = 0T+ Teff / ⌺. tors. We expect from the central limit, theorem that such corrections will be small in the relevant limit, where ␻ is 2. Total output noise much smaller that the typical detector frequency ϳ ប kBTeff/ , and neglect these corrections in what fol- The next step in our analysis is to link fluctuations in lows. Note that the validity of Eq. ͑5.46͒ for a specific ͓ the position of the oscillator as determined from Eq. model of a tunnel junction position detector has been ͑ ͔͒ 5.41 to noise in the output of the detector. As dis- explicitly verified by Clerk and Girvin ͑2004͒. cussed in Sec. III.B.3, the output noise consists of the intrinsic output noise of the detector ͑i.e., “measurement-imprecision noise”͒ plus the amplified 3. Detector power gain position fluctuations in the position of the oscillator. The Before proceeding, we need to consider our detector latter contains both an intrinsic part and a term due to once again in isolation, and return to the fundamental the response of the oscillator to the back-action. question of what we mean by amplification. To be able To start, imagine that we can treat both the oscillator to say that our detector truly amplifies the motion of the ͑ ͒ ͑ ͒ position x t and the detector output I t as classically oscillator, it is not sufficient to simply say that the re- fluctuating quantities. Using the linearity of the detec- sponse function ␹ must be large ͑note that ␹ is not ␦ IF IF tor’s response, we can then write Itotal, the fluctuating dimensionless͒. Instead, true amplification requires that part of the detector’s output, as the power delivered by the detector to a following am- ␦ ͓␻͔ ␦ ͓␻͔ ␹ ͓␻͔␦ ͓␻͔ ͑ ͒ plifier be much larger than the power drawn by the de- Itotal = I0 + A IF x . 5.44 tector at its input—i.e., the detector must have a dimen- ͑␦ ͒ ͑ The first term I0 describes the intrinsic oscillator- sionless power gain G ͓␻͔ much larger than 1. As ͒ P independent fluctuations in the detector output, and discussed, if the power gain was not large, we would ¯ ͓␻͔ ͉␹ ͉2 has a spectral density SII . If we scale this by IF ,we need to worry about the next stage in the amplification

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Aλ Bg ␥ ͓␻͔ ͑ 2 ␻͓͒ ␹ ͓␻͔͔ g y out = B /M −Im II S ͓␻͔ − S ͓− ␻͔ = ͑B2/Mប␻͒ͫ II II ͬ, ͑5.48͒ FD x F I y 2 A B ␹ γ0 + γ Ag γout + γld where II is the linear-response susceptibility which determines how ͗Iˆ͘ responds to a perturbation coupling to ͑ ͒ FIG. 11. Color online Schematic of a generic linear-response ˆ position detector, where an auxiliary oscillator y is driven by I the detector output. ϱ i ␹ ͓␻͔ ͵ ͓͗ˆ͑ ͒ ˆ͑ ͔͒͘ i␻t ͑ ͒ II =−ប dt I t ,I 0 e . 5.49 0 of our signal, and how much noise is added in that process. Having a large power gain means that by the time As the oscillator y is being driven on resonance, the ͓␻͔ ␹ ͓␻͔ ͓␻͔ our signal reaches the following amplifier, it is so large relation between y and I is given by y = yy I ␹ ͓␻͔ ␻ ␥ ͓␻͔ −1 that the added noise of this following amplifier is unim- with yy =−i† M out ‡ . From conservation of en- portant. The power gain is analogous to the dimension- ergy, we have that the net power flow into the output less photon-number gain G that appears in the standard oscillator from the detector is equal to the power dissi- Haus-Caves description of a bosonic linear amplifier pated out of the oscillator through the intrinsic damping ␥ ͓see Eq. ͑5.7͔͒. ld. We thus have To make the above more precise, we start with the ϵ ␥ 2 ␹ Pout M ldy˙ idea case of no reverse gain FI=0. We define the power ͓␻͔ ␥ ␻2͉␹ ͓␻͔͉2͉ ␹ ␹ ͓␻͔ ͉2 gain GP of our generic position detector in a way that = M ld yy BA IF xx FD is analogous to the power gain of a voltage amplifier. 1 ␥ Imagine we drive the oscillator we are trying to measure = ld ͉BA␹ ␹ ͓␻͔F ͉2. ͑5.50͒ ␻ ͑␥ ␥ ͓␻͔͒2 IF xx D with a force 2FD cos t; this will cause the output of our M ld + out detector ͗Iˆ͑t͒͘ to also oscillate at frequency ␻. To opti- Using the above definitions, we find that the ratio be- ␥ mally detect this signal in the detector output, we fur- tween Pout and Pin is independent of 0, but depends on ␥ ther couple the detector output I to a second oscillator ld, with natural frequency ␻, mass M, and position y: there P 1 A2B2͉␹ ͓␻͔͉2 ␥ /␥ ͓␻͔ is a new coupling term in our Hamiltonian HЈ =BIˆ ·yˆ, out = IF ld out . ͑5.51͒ int 2␻2 ␥ ͓␻͔␥͓␻͔ ͑ ␥ ␥ ͓␻͔͒2 where B is a coupling strength. The oscillations in ͗I͑t͒͘ Pin M out 1+ ld/ out will now act as a driving force on the auxiliary oscillator We now define the detector power gain G ͓␻͔ as the ͑ ͒ P y see Fig 11 . We can consider the auxiliary oscillator y value of this ratio maximized over the choice of ␥ . The as a “load” we are trying to drive with the output of our ld maximum occurs for ␥ =␥ ͓␻͔͑i.e., the load oscillator detector. ld out is “matched” to the output of the detector͒, resulting in To find the power gain, we need to consider both Pout, the power supplied to the output oscillator y from the P 1 A2B2͉␹ ͉2 G ͓␻͔ϵmaxͫ outͬ = IF detector, and Pin, the power fed into the input of the P 2␻2 ␥ ␥ Pin 4M out amplifier. Consider first P . This is simply the time- in ͉␹ ͓␻͔͉2 averaged power dissipation of the input oscillator x IF ͑ ͒ ␥͓␻͔ = ␹ ͓␻͔ ␹ ͓␻͔ . 5.52 caused by the back-action damping . Using an over- 4Im FF Im II bar to denote a time average, we have In the last equality, we have used the relation between ␥͓␻͔ ␥ ͓␻͔ the damping rates and out and the linear- P ϵ M␥͓␻͔x˙ 2 = M␥͓␻͔␻2͉␹ ͓␻͔͉2F2 . ͑5.47͒ ␹ ͓␻͔ ␹ ͓␻͔͓ in xx D response susceptibilities FF and II see Eqs. ͑2.15͒ and ͑5.48͔͒. We thus find that the power gain is a ␹ ͓␻͔ simple dimensionless ratio formed by the three different Note that the oscillator susceptibility xx depends on both the back-action damping ␥͓␻͔ and the intrinsic os- response coefficients characterizing the detector, and is ␥ ͓ ͑ ͔͒ independent of the coupling constants A and B.Aswe cillator damping 0 see Eq. 5.45 . Next, we need to consider the power supplied to the see in Sec. V. F , it is completely analogous to the power load oscillator y at the detector output. This oscillator gain of a voltage amplifier, which is also determined by will have some intrinsic detector-independent damping three parameters: the voltage gain, the input impedance, ␥ ␥ and the output impedance. Note that there are other ld, as well as a back-action damping out. In the same way that the back-action damping ␥ of the input oscilla- important measures of power gain commonly in use in the engineering community: we comment on these in ˆ ͓ tor x is determined by the quantum noise in F see Eqs. Sec. VII.B. ͑ ͒ ͑ ͒ ͑ ͔͒ 2.14 , 2.13 , and 2.12 , the back-action damping of the Finally, the above results are easily generalized to the load oscillator y is determined by the quantum noise in ␹ case where the detector’s reverse gain FI is nonvanish- the output operator Iˆ, ing. For simplicity, we present results for the case where

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␤ ͑␹ ␹ ͒ ͉␹ ͉2 ജ 2 2 2 =Re IF FI / IF 0, implying that there is no posi- ͑Im ␣͒ coth ͑ប␻/2k T ͒ + ͑Re ␣͒ G ͓␻͔ = B eff , ͑5.56͒ tive feedback. Maximizing the ratio of Pout/Pin over P ͉␣͉2 ␥ choices of ld now yields ͑ ␤ ͱ ␤ ͒ ഛ ␤ ͑ ͒ ␣͓␻͔ GP,rev =2GP/ 1+2 GP + 1+4 GP 1/ . 5.53 where is the parameter characterizing a quantum- limited detector in Eq. ͑4.18͒; recall that ͉␣͓␻͔͉2 deter- Here G is the power gain in the presence of reverse P,rev mines the ratio of S and S . It follows immediately gain, while G is the zero-reverse gain power gain given II FF P that for a detector with ideal noise to also have a large by Eq. ͑5.52͒. One can confirm that G is a monotonic P,rev power gain ͑G ӷ1͒, one absolutely needs k T ӷប␻:a increasing function of G , and is bounded by 1/␤.As P B eff P large power gain implies a large effective detector tem- noted in Sec. IV.A.4,if␹ =␹* , there is no additional FI IF perature. In the large-G limit, we have quantum noise constraint on our detector beyond what P exists classically ͓i.e., the RHS of Eq. ͑4.11͒ vanishes͔. 2 ␹ ␹* Im ␣ k T We now see explicitly that when FI= , the power gain Ӎ ͫ B effͬ ͑ ͒ IF GP . 5.57 of our detector can be at most 1, as ␤=1. Thus, while ͉␣͉ ប␻/2 there is no minimum back-action noise required by quantum mechanics in this case, there is also no ampli- Thus, the effective temperature of a quantum-ideal de- fication: at best, our detector would act as a transducer. tector does more than just characterize the detector ␹ ␹* Note further that if the detector has FI= IF and opti- back-action—it also determines the power gain. ͑ ͒ ͓␻͔ mizes the inequality of Eq. 4.11 , then one can show Finally, an additional consequence of the large-GP , ͑ ͒ ␹ GP,rev must be 1 see Appendix I.2 : the detector is sim- large Teff limit is that the gain IF and noise cross cor- ply a transducer. This is in keeping with the results ob- relator S¯ are in phase: S¯ /␹ is purely real, up to tained using the Haus-Caves approach, which also yields IF IF IF corrections that are as small as ␻/T . This is shown the conclusion that a noiseless detector is a transducer. eff explicitly in Appendix I.3. Thus, we find that a large power gain detector with ideal quantum noise cannot 4. Simplifications for a quantum-ideal detector have significant out-of-phase correlations between its output and input noises. This last point may be under- We now consider the important case where our detec- stood in terms of the idea of wasted information: if there tor has no reverse gain ͑allowing it to have a large power were significant out-of-phase correlations between Iˆ and gain͒, and also has ideal quantum noise ͓i.e., it satisfies the ideal noise condition of Eq. ͑4.17͔͒. Fulfilling this Fˆ , it would be possible to improve the performance of condition immediately places some powerful constraints the amplifier by using feedback. We discuss this point in ¯ ␹ on our detector. Sec. VI. Note that because SIF/ IF is real the last term in First, note that we have defined in Eq. ͑5.42͒ the ef- the quantum noise constraint of Eq. ͑4.11͒ vanishes. fective temperature of our detector based on what happens at the input port; this is the effective temperature seen by the oscillator we are trying to measure. We 5. Quantum limit on added noise and noise could also consider the effective temperature of the de- temperature tector as seen at the output ͑i.e., by the oscillator y used in defining the power gain͒. This output effective tem- We now turn to calculating the noise added to our ͓ ͗ ͑ ͔͒͘ perature is determined by the quantum noise in the out- signal i.e., xˆ t by our generic position detector. To ˆ characterize this added noise, it is useful to take the total put operator I, ͑symmetrized͒ noise in the output of the detector and ͓␻͔ϵប␻ ͑ ͓ ␻͔ ͓ ␻͔͒ ͑ ͒ refer it back to the input by dividing out the gain of the kBTeff,out /log SII + /SII − . 5.54 detector, For a general out-of-equilibrium amplifier, Teff,out does not have to be equal to the input effective temperature ͑ ͒ ¯ ͓␻͔ϵ¯ ͓␻͔ 2͉␹ ͓␻͔͉2 ͑ ͒ Teff defined by Eq. 2.8 . However, for a quantum-ideal Sxx,tot SII,tot /A IF . 5.58 detector, the effective proportionality between input and output operators ͓see Eq. ͑I13͔͒ immediately yields ¯ ͓␻͔ Sxx,tot is simply the frequency-dependent spectral ͓␻͔ ͓␻͔ ͑ ͒ density of position fluctuations inferred from the output Teff,out = Teff . 5.55 of the detector. It is this quantity that will directly deter- Thus, a detector with quantum-ideal noise necessarily mine the sensitivity of the detector–given a certain de- has the same effective temperature at its input and its tection bandwidth, what is the smallest variation of x output. This is all the more remarkable given that a ¯ ͓␻͔ quantum-ideal detector cannot be in equilibrium, and that can be resolved? The quantity Sxx,tot will have contributions from the intrinsic fluctuations of the input thus Teff cannot represent a real physical temperature. Another important simplification for a quantum-ideal signal as well as a contribution due to the detector. We ¯ ͓␻ ͔ detector is the expression for the power gain. Using the first define Sxx,eq ,T to be the symmetrized equilib- proportionality between input and output operators ͓cf. rium position noise of our damped oscillator ͑whose ͑ ͔͒ ␥ ␥͒ Eq. I13 , one finds damping is 0 + at temperature T,

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tion is the same as the cavity position detector of Sec. S¯ ͓␻,T͔ = ប coth͑ប␻/2k T͒͑−Im␹ ͓␻͔͒, ͑5.59͒ xx,eq B xx III.B.3: there is an optimal value of the coupling ␹ ͓␻͔ where the oscillator susceptibility xx is defined in strength A which corresponds to a trade-off between im- Eq. ͑5.45͒. The total inferred position noise may then be precision noise and back-action ͓i.e., the first and second written as ͑ ͔͒ ¯ ͓␻͔ terms in Eq. 5.61 . We thus expect Sxx,add to attain a ¯ ͓␻͔ϵ͓␥ ͑␥ ␥͔͒ ¯ ͓␻ ͔ ¯ ͓␻͔ minimum value at an optimal choice of coupling A Sxx,tot 0/ 0 + · Sxx,eq ,T0 + Sxx,add . =Aopt where both these terms make equal contributions ͑5.60͒ ͑ ͒ ␾͓␻͔ ␹ ͓␻͔ see Fig. 5 . Defining =arg xx , we thus have the In the usual case where the detector noise can be ap- bound proximated as being white, this spectral density will con- ¯ ¯ sist of a Lorentzian sitting atop a constant noise floor ¯ ͫͱSIISFF S ͓␻͔ ജ 2͉␹ ͓␻͔͉ ͑ ͒ ͑ ͒ xx,add xx ͉␹ ͉2 see Fig. 6 . The first term in Eq. 5.60 represents posi- IF ␦ ͑ ͒ tion noise arising from the fluctuating force F0 t asso- ͑ ͒ Re͓␹* e−i␾͓␻͔S¯ ͔ ciated with the intrinsic detector-independent dissipa- IF IF ͬ + , ͑5.63͒ tion of the oscillator ͓see Eq. ͑5.41͔͒. The prefactor of ͉␹ ͉2 IF this term arises because the strength of the intrinsic Langevin force acting on the oscillator is proportional to where the minimum value at frequency ␻ is achieved ␥ ␥ ␥ 0, not to 0 + . when The second term in Eq. ͑5.60͒ represents the added 2 ͱ¯ ͓␻͔ ͉␹ ͓␻͔␹ ͓␻͔͉2 ¯ ͓␻͔ ͑ ͒ position noise due to the detector. It has contributions Aopt = SII / IF xx SFF . 5.64 ¯ both from the detector’s intrinsic output noise SII from Using the inequality X2 +Y2 ജ2͉XY͉ we see that this the detector’s back-action noise S¯ , and may be written ¯ FF value serves as a lower bound on Sxx,add even in the as presence of detector-dependent damping. In the case where the detector-dependent damping is negligible, the S¯ S¯ ͓␻͔ = II + A2͉␹ ͉2S¯ RHS of Eq. ͑5.63͒ is independent of A, and thus Eq. xx,add ͉␹ ͉2 2 xx FF IF A ͑5.64͒ can be satisfied by simply tuning the coupling ͓␹* ͑␹ ͒ ¯ ͔ strength A; in the more general case where there is 2Re xx *SIF + IF . ͑5.61͒ detector-dependent damping, the RHS is also a function ͉␹ ͉2 ͑ ␹ ͓␻͔͒ IF of A through the response function xx , and it may no longer be possible to achieve Eq. ͑5.64͒ by simply For clarity, we have omitted writing the explicit fre- tuning A.15 quency dependence of the gain ␹ , susceptibility ␹ , IF xx While Eq. ͑5.63͒ is certainly a bound on the added and noise correlators; they should all be evaluated at the ¯ ͓␻͔ frequency ␻. Note that the first term on the RHS corre- displacement noise Sxx,add , it does not in itself repre- sponds to the measurement-imprecision noise of our de- sent the quantum limit. Reaching the quantum limit requires more than simply balancing the detector back- tector, S¯ I ͑␻͒. xx action and intrinsic output noises ͓i.e., the first two terms We can now finally address the quantum limit on the in Eq. ͑5.61͔͒; one also needs a detector with quantum- added noise in this setup. As discussed in Sec. V. D , the ideal noise properties, that is a detector which satisfies Eq. Haus-Caves derivation of the quantum limit ͑cf. Sec. (4.17). Using the quantum noise constraint of Eq. ͑4.11͒ V. B ͒ is not directly applicable to the position detector ¯ ͓␻͔ we are describing here; nonetheless, we may use its re- to further bound Sxx,add , we obtain sult to guess what form the quantum limit will take here. ␹ ͓␻͔ The Haus-Caves argument told us that the added noise ¯ ͓␻͔ ജ ͯ xx ͯ Sxx,add 2 ␹ of a phase-preserving linear amplifier must be at least as IF large as the zero-point noise. We thus anticipate that, if ប͉␹ ͉ 2 2S¯ our detector has a large power gain, the spectral density ϫ ͫͱͩ IF ͪ ͩ ⌬ͫ IF ͬͪ ͉ ¯ ͉2 1+ ប␹ + SIF ͑ ¯ ͓␻͔͒ 2 IF of the noise added by the detector i.e., Sxx,add must be at least as large as the zero-point noise of our Re͓␹* e−i␾͓␻͔S¯ ͔ damped oscillator, IF IF ͬ ͑ ͒ + ͉␹ ͉ , 5.65 IF ¯ ͓␻͔ ജ ¯ ͓ ͔ ͉ប ␹ ͓␻͔͉ ͑ ͒ Sxx,add lim Sxx,eq w,T = Im xx . 5.62 T→0 where the function ⌬͓z͔ is defined in Eq. ͑4.12͒. The ¯ ͓␻͔ ͑ ͒ We now show that the bound above is rigorously correct minimum value of Sxx,add in Eq. 5.65 is now ͓ at each frequency ␻. achieved when one has both an optimal coupling i.e., The first step is to examine the dependence of the ¯ ͓␻͔͓ ͑ ͔͒ 15 added noise Sxx,add as given by Eq. 5.61 on the Note that, in the heuristic discussion of position detection coupling strength A. If we ignore for a moment the using a resonant cavity detector in Sec. III.B.3, these concerns detector-dependent damping of the oscillator, the situa- did not arise as there was no back-action damping.

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Eq. ͑5.64͔͒ and a quantum-limited detector, that is one across the entire oscillator resonance. A more reason- which satisfies Eq. ͑4.11͒ as an equality. ¯ ͓␻͔ ␻ ⍀ able goal is to optimize Sxx,add at resonance, = .As Next, we consider the relevant case where our detec- ␹ ͓⍀͔ ͑ ͒ ¯ ͓␻͔ӷ xx is imaginary, Eq. 5.67 tells us that SIF should be tor is a good amplifier and has a power gain GP 1 over the width of the oscillator resonance. As discussed, zero. Assuming we have a quantum-limited detector with a large power gain ͑k T ӷប⍀͒, the remaining ¯ ␹ B eff this implies that the ratio SIF/ IF is purely real, up to condition on the coupling A ͓Eq. ͑5.64͔͒ may be written ប␻ ͑ small /kBTeff corrections see Sec. IV.A.4 and Appen- as dix I.3 for more details͒. This in turn implies that ␥͓A ͔ Im ␣ 1 ប⍀ ⌬͓2S¯ /ប␹ ͔=0; we thus have opt ͯ ͯ ͑ ͒ IF IF ␥ ␥͓ ͔ = ␣ ͱ = . 5.69 0 + Aopt 2 G ͓⍀͔ 4kBTeff 2 P ប 2 S¯ ¯ ͓␻͔ ജ ͉␹ ͓␻͔͉ͫͱͩ ͪ ͩ IFͪ As ␥͓A͔ϰA2 is the detector-dependent damping of Sxx,add 2 xx + ␹ 2 IF the oscillator, we thus have that to achieve the quantum- ¯ ͓⍀͔ cos ␾͓␻͔ S¯ limited value of Sxx,add with a large power gain, one † ‡ IFͬ ͑ ͒ + ␹ . 5.66 needs the intrinsic damping of the oscillator to be much IF larger than the detector-dependent damping. The ¯ ␹ ͑ detector-dependent damping must be small enough to Finally, as there is no further constraint on SIF/ IF beyond the fact that it is real͒, we can minimize the expres- compensate the large effective temperature of the detector; if the bath temperature satisfies ប⍀/k ӶT ¯ ͓␻͔ B bath sion over its value. The minimum Sxx,add is achieved Ӷ ͑ ͒ Teff, Eq. 5.69 implies that at the quantum limit the for a detector whose cross correlator satisfies temperature of the oscillator will be given by ͉ ¯ ͓␻͔ ␹ ͉ ͑ប ͒ ␾͓␻͔ ͑ ͒ ϵ͑␥ ␥ ͒ ͑␥ ␥ ͒ → ប⍀ SIF / IF optimal =− /2 cot , 5.67 Tosc · Teff + 0 · Tbath / + 0 /4kB + Tbath. with the minimum value given by ͑5.70͒

S¯ ͉͓␻͔͉ = ប͉Im ␹ ͓␻͔͉ = lim S¯ ͓␻,T͔, ͑5.68͒ Thus, at the quantum limit and for large Teff, the detec- xx,add min xx xx,eq ប⍀ 16 T→0 tor raises the oscillator’s temperature by /4kB. As expected, this additional heating is only half the zero- ¯ ͓␻ ͔ where Sxx,eq ,T is the equilibrium contribution to point energy; in contrast, the quantum-limited value of S¯ ͓␻͔ defined in Eq. ͑5.59͒. Thus, in the limit of a ¯ ͓␻͔ xx,tot Sxx,add corresponds to the full zero-point result, as it large power gain, we have that at each frequency the also includes the contribution of the intrinsic output minimum displacement noise added by the detector is noise of the detector. precisely equal to the noise arising from a zero- Finally, we return to Eq. ͑5.65͒; this is the constraint temperature bath. This conclusion is irrespective of the on the added noise S¯ ͓␻͔ before we assumed our strength of the intrinsic ͑detector-independent͒ oscillator xx,add detector to have a large power gain, and consequently a damping. large T . Note crucially that if we did not require a We have thus derived the amplifier quantum limit ͑in eff large power gain, then there need not be any added the context of position detection͒ for a two-port ampli- noise. Without the assumption of a large power gain, the fier used in the op-amp mode of operation. Though we ¯ ␹ reached a conclusion similar to that given by the Haus- ratio SIF/ IF can be made imaginary with a large magni- ⌬͓ ¯ ␹ ͔→ Caves approach, the linear-response, quantum noise ap- tude. In this limit, 1+ 2SIF/ IF 0: the quantum con- proach used is quite different. This approach makes ex- straint on the amplifier noises ͓e.g., the RHS of Eq. plicitly clear what is needed to reach the quantum limit. ͑4.11͔͒ vanishes. One can then easily use Eq. ͑5.65͒ to We find that to reach the quantum limit on the added show that the added noise S¯ ͓␻͔ can be zero. This ¯ ͓␻͔ xx,add displacement noise Sxx,add with a large power gain, confirms a general conclusion that we have seen several one needs ͑1͒ a quantum-limited detector, that is, a de- times now ͑see Secs. IV.A.4 and V. B ͒: if a detector does tector which satisfies the ideal noise condition of Eq. not amplify ͑i.e., the power gain is unity͒, it need not ͑4.17͒, and hence the proportionality condition of Eq. produce any added noise. ͑I13͒; ͑2͒ a coupling A which satisfies Eq. ͑5.64͒; and ͑3͒ ¯ ͑ ͒ a detector cross correlator SIF that satisfies Eq. 5.67 . F. Quantum limit on the noise temperature of a voltage ͑ ͒ Recall that condition 1 is identical to what is re- amplifier quired for quantum-limited detection of a qubit; it is rather demanding, and requires that there is no wasted We now turn our attention to the quantum limit on information about the input signal in the detector which the added noise of a generic linear voltage amplifier is not revealed in the output ͑Clerk et al., 2003͒. Also note that cot ␾ changes quickly as a function of fre- 16 ¯ If in contrast our oscillator was initially at zero temperature quency across the oscillator resonance, whereas SIF will ͑ ͒ ͑ ͑ ͒ i.e., Tbath=0 , one finds that the effect of the back-action at be roughly constant; condition 2 thus implies that it ӷ ͒ the quantum limit and for GP 1 is to heat the oscillator to a ¯ ͓␻͔ ប⍀ will not be possible to achieve a minimal Sxx,add temperature /kB ln 5.

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~ V age amplifier represents back-action: this fluctuating λV current ͑spectral density S˜˜͓␻͔͒ flows back across the Zin II vin Zout input output parallel combination of the source impedance and am- ~ λI IL Vout I ZS Z plifier input impedance, producing an additional fluctu- L ating voltage at its input. The current noise is thus analo- S gous to the back-action noise FF of our generic linear- FIG. 12. Schematic of a linear voltage amplifier, including a response detector. ␭ ˜ ˜ reverse gain I. V and I represent the standard voltage and Putting the above together, the total voltage at the current noises of the amplifier, as discussed in the text. The input terminals of the amplifier is case with reverse gain is discussed in Sec. VI. ͑ ͒ Zin ͓ ͑ ͒ ˜ ͑ ͔͒ ZsZin ˜͑ ͒ vin,tot t = vin t + V t − I t used in the op-amp mode of operation ͓see, e.g., De- Zin + Zs Zs + Zin ͑ ͔͒ voret and Schoelkopf 2000 . For such amplifiers, the Ӎ v ͑t͒ + V˜ ͑t͒ − Z I˜͑t͒. ͑5.71͒ added noise is usually expressed in terms of the “noise in s temperature” of the amplifier; we define this concept In the second line, we have taken the usual limit of an and demonstrate that, when appropriately defined, this ideal voltage amplifier which has an infinite input im- ប␻ ͑ ͒ ͑ ͒ noise temperature must be bigger than / 2kB , where pedance i.e., the amplifier draws zero current . The ␻ is the signal frequency. Though the voltage amplifier is spectral density of the total input voltage fluctuations is closely analogous to the position detector treated previ- thus ously, its importance makes it worthy of a separate dis- S ͓␻͔ = S ͓␻͔ + S ͓␻͔. ͑5.72͒ cussion. As in the previous section, our discussion here VV,tot vinvin VV,add S will use the general linear-response approach. In con- Here v v is the spectral density of the voltage fluctua- trast, in Sec. VI, we present the bosonic scattering de- in in ͑ ͒ tions of the input signal vin t and SVV,add is the amplifi- scription of a two-port voltage amplifier, a description er’s contribution to the total noise at the input similar to that used in formulating the Haus-Caves proof S ͓␻͔ S ˜ ˜ ͉ ͉2S˜˜ ͓ *S ˜ ˜͔ ͑ ͒ of the amplifier quantum limit. We will then be in a good VV,add = VV + Zs II −2Re Zs VI . 5.73 position to contrast the linear-response and scattering For clarity, we have dropped the frequency index for the approaches and will see there that the scattering and the spectral densities appearing on the RHS of this equa- op-amp modes of operation discussed here are not tion. equivalent. We stress that the general treatment pre- It is useful now to consider a narrow bandwidth input sented here can also be applied directly to the system signal at a frequency ␻, and ask the following question: discussed in Sec. VI. if the signal source was simply an equilibrium resistor at a temperature T0, how much hotter would it have to be 1. Classical description of a voltage amplifier S ͓␻͔ to produce a voltage noise equal to VV,tot ? The re- We begin by recalling the standard schematic descrip- sulting increase in the source temperature is defined as ͓␻͔ tion of a voltage amplifier ͑see Fig. 12͒. The input volt- the noise temperature TN of the amplifier and is a ͑ ͒ convenient measure of the amplifier’s added noise. It is age to be amplified vin t is produced by a circuit which standard among engineers to define the noise tempera- has a Thevenin-equivalent impedance Zs, the source impedance. We stress that we are considering the op-amp ture with the assumption that the initial temperature of ӷប␻ mode of amplifier operation, and thus the input signal the resistor T0 . One may then use the classical ex- does not correspond to the amplitude of a wave incident pression for the thermal noise of a resistor, which yields upon the amplifier ͑see Sec. V. D ͒. The amplifier itself the definition has an input impedance Zin and an output impedance 2ReZ k T ͓␻͔ϵS ͓␻͔. ͑5.74͒ ␭ s B N VV,tot Zout, as well as a voltage gain coefficient V: assuming no ͑ →ϱ ͒ ͉ ͉ i␾ current is drawn at the output i.e., Zload in Fig. 12 , Writing Zs = Zs e , we have the output voltage V ͑t͒ is simply ␭ times the voltage out V S ˜ ˜ 1 VV ␾ across the input terminals of the amplifier. ͫ ͉ ͉S˜˜ ͑ −i S ˜ ˜͒ͬ 2kBTN = ␾ ͉ ͉ + Zs II −2Re e VI . The added noise of the amplifier is usually repre- cos Zs sented by two noise sources placed at the amplifier in- ͑5.75͒ put. There is both a voltage noise source V˜ ͑t͒ in series It is clear from this expression that TN will have a mini- with the input voltage source and a current noise source ͉ ͉ ͉ ͉ mum as a function of Zs . For Zs too large, the back- I˜͑t͒ in parallel with the input voltage source ͑Fig. 12͒. action current noise of the amplifier will dominate TN, ˜ ͑ ͒ ͉ ͉ The voltage noise produces a fluctuating voltage V t while for Zs too small, the voltage noise of the amplifier ͑ S ͓␻͔͒ ͑ ͒ spectral density V˜ V˜ , which simply adds to the sig- i.e., its intrinsic output noise will dominate. The situa- nal voltage at the amplifier input, and is amplified at the tion is completely analogous to that of the position de- output; as such, it is completely analogous to the intrin- tector of the last section; there we needed to optimize S sic detector output noise II of our linear-response de- the coupling strength A to balance back-action and in- tector. In contrast, the current noise source of the volt- trinsic output noise contributions and thus minimize the

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1192 Clerk et al.: Introduction to quantum noise, measurement, …

total added noise. Optimization of the source impedance S˜˜͓␻͔ ↔ ␻2S¯ ͓␻͔. ͑5.80͒ thus yields a completely classical minimum bound on II QQ TN, ˆ Similarly, the ﬂuctuations in the operator Vout, when referred back to the ampliﬁer input, will correspond to the k T ജ ͱS ˜ ˜ S˜˜ − ͓Im S ˜ ˜͔2 −ReS ˜ ˜ , ͑5.76͒ B N VV II VI VI voltage noise V˜ ͑t͒ discussed above,

where the minimum is achieved for an optimal source S ͓␻͔ ↔ ¯ ͓␻͔ ͉␭ ͉2 ͑ ͒ V˜ V˜ SV V / V . 5.81 impedance that satisﬁes out out A similar correspondence holds for the cross correlator ͉ ͓␻͔͉ ͱS ͓␻͔ S ͓␻͔ϵ ͑ ͒ of these noise sources, Zs opt = V˜ V˜ / I˜I˜ ZN, 5.77

S ˜ ˜͓␻͔ ↔ + i␻S¯ ͓␻͔/␭ . ͑5.82͒ VI VoutQ V sin ␾͉͓␻͔͉ =−ImS ˜ ˜͓␻͔/ͱS ˜ ˜ ͓␻͔S˜˜͓␻͔. ͑5.78͒ opt VI VV II To proceed, we need to identify the input and output impedances of the amplifier, and then define its power The above equations define the so-called noise imped- gain. The first step in this direction is to assume that the ance Z . We stress again that the discussion so far in this N ͑ ˆ ͒ section has been completely classical. output of the amplifier Vout is connected to an external circuit via a term ˆ Ј ͑ ͒ ˆ ͑ ͒ 2. Linear-response description Hint = qout t Vout, 5.83 ˜ It is easy to connect the classical description of a volt- where iout=dqout/dt is the current in the external circuit. age amplifier to the quantum-mechanical description of We may now identify the input and output impedances a generic linear-response detector; in fact, all that is of the amplifier in terms of the damping at the input and needed is a “relabeling” of the concepts and quantities output. Use of the Kubo formulas for conductance and introduced in Sec. V. E when discussing a linear position resistance yields ͓cf. Eq. ͑2.12͒ and Eq. ͑5.48͒, with the detector. Thus, the quantum voltage amplifier will be substitutions Fˆ →Qˆ and Iˆ →Vˆ ͔ characterized by both an input operator Qˆ and an out- 1/Z ͓␻͔ = i␻␹ ͓␻͔, ͑5.84͒ ˆ ˆ ˆ in QQ put operator Vout; these play the roles of F and I in the ˆ ͓␻͔ ␹ ͓␻͔ ͑ ␻͒ ͑ ͒ position detector, respectively. Vout represents the out- Zout = VV / − i , 5.85 ˆ put voltage of the amplifier, while Q is the operator ͗˜ ͘ ͑ ͓␻͔͒ ͓␻͔ ͗ ͘ ͓␻͔˜ ͓␻͔ ͑ ͒ i.e., Iin ␻ = 1/Zin vin and V ␻ =Zout iout , which couples to the input signal vin t via a coupling ␻ Hamiltonian where the subscript indicates the Fourier transform of a time-dependent expectation value. We consider throughout this section the case of no ˆ ͑ ͒ ˆ ͑ ͒ Hint = vin t Q. 5.79 reverse gain, ␹ =0. We can define the power gain G QVout P exactly as in Sec. V.E.3 for a linear position detector. GP ˜ˆ ˆ In more familiar terms, Iin−dQ/dt represents the cur- is defined as the ratio of the power delivered to a load rent flowing into the amplifier.17 The voltage gain of our attached to the amplifier output divided by the power ␭ drawn by the amplifier, maximized over the impedance amplifier V will again be given by the Kubo formula of ͑ ͒ ˆ → ˆ ˆ → ˆ ͑ of the load. One finds Eq. 4.3 , with the substitutions F Q,I Vout we as- ͒ ͉␭ ͉2 ͑ ͒ ͑ ͒ ͑ ͒ sume these substitutions throughout this section . GP = V /4 Re Zout Re 1/Zin . 5.86 We can now easily relate the fluctuations of the input and output operators to the noise sources used to de- Expressing this in terms of the linear-response coeffi- ␭ ␹ scribe the classical voltage amplifier. As usual, symme- cients VV and QQ, we obtain an expression that is com- ͑ ͒ ¯ ͓␻͔ pletely analogous to Eq. 5.52 for the power gain for a trized quantum noise spectral densities S will play the position detector role of the classical spectral densities S͓␻͔ appearing in ͉␭ ͉2 ␹ ␹ ͑ ͒ the classical description. First, as the operator Qˆ repre- GP = V /4 Im QQ Im VV. 5.87 sents a back-action force, its fluctuations correspond to ͓␻͔ Finally, we define again the effective temperature Teff the amplifier’s current noise I˜͑t͒, of the amplifier via Eq. ͑2.8͒, and define a quantum- limited voltage amplifier as one that satisfies the ideal- noise condition of Eq. ͑4.17͒. For such an amplifier, the 17 Note that one could have instead written the coupling power gain will be determined by the effective tempera- ˆ ͑ ͒ ␾͑ ͒˜ˆ ͑ ͒ Hamiltonian in the more traditional form Hint t = t Iin, ture via Eq. 5.56 . ␾ ͐ ͑ ͒ where = dtЈvin tЈ is the flux associated with the input volt- Turning to the noise, we calculate the total symme- age. The linear-response results we obtain are exactly the trized noise at the output port of the amplifier following same. We prefer to work with the charge Qˆ in order to be the same argument used to get the output noise of the consistent with the rest of the text. position detector ͓cf. Eq. ͑5.46͔͒. As we did in the clas-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1193 sical approach, we assume that the input impedance of Z ͓␻͔ ␣ ប␻ 1 ͯ N ͯ = ͯ ͯ = Ӷ 1. the amplifier is much larger than the source impedance: ͓␻͔ ␣ ͱ ͓␻͔ ӷ Re Zin Im 4kBTeff 2 GP Zin Zs; we test this assumption for consistency at the end of the calculation. Focusing only on the amplifier ͑5.90͒ contribution to this noise ͑as opposed to the intrinsic ͉ ͉Ӷ͉ ͉ ͒ It follows that ZN Zin in the large-power-gain, large- noise of the input signal , and referring this noise back effective-temperature regime of interest, thus justifying to the amplifier input, we find that the symmetrized the form of Eq. ͑5.73͒. Equation ͑5.90͒ is analogous to quantum noise spectral density describing the added the case of the displacement detector, where we found ¯ ͓␻͔ noise of the amplifier SVV,add satisfies the same equa- that reaching the quantum limit on resonance required tion we found for a classical voltage amplifier, Eq. ͑5.73͒, the detector-dependent damping to be much weaker with each classical spectral density S͓␻͔ replaced by the than the intrinsic damping of the oscillator ͓cf. Eq. corresponding symmetrized quantum spectral density ͑5.69͔͒. S¯ ͓␻͔ as per Eqs. ͑5.80͒–͑5.82͒. Thus, similarly to the situation of the displacement It follows that the amplifier noise temperature will detector, the linear-response approach allows us both to again be given by Eq. ͑5.75͒, and that the optimal noise derive rigorously the quantum limit on the noise tem- temperature ͑after optimizing over the source imped- perature TN of an amplifier and to state conditions that ance͒ will be given by Eq. ͑5.76͒. Whereas classically must be met to reach this limit. To reach the quantum- nothing more could be said, quantum mechanically we limited value of TN with a large power gain, one needs now get a further bound from the quantum noise con- both a tuned source impedance Zs and an amplifier that straint of Eq. ͑4.11͒ and the requirement of a large possesses ideal noise properties ͓cf. Eqs. ͑4.17͒ and Eq. power gain. The latter requirement tells us that the volt- ͑I13͔͒. age gain ␭ ͓␻͔ and the cross correlator S¯ must be V Vout Q in phase ͑cf. Sec. IV.A.4 and Appendix I.3͒. This in turn 3. Role of noise cross correlations ¯ means that SV˜ I˜ must be purely imaginary. In this case, Before leaving the topic of a linear voltage amplifier, the quantum noise constraint may be rewritten as we pause to note the role of cross correlations in current and voltage noise in reaching the quantum limit. First, ͑ ͒ ¯ ͓␻͔ ¯ ͓␻͔ ͓ ¯ ͔2 ജ ͑ប␻ ͒2 ͑ ͒ note from Eq. 5.78 that in both the classical and quan- SV˜ V˜ SI˜I˜ − Im SV˜ I˜ /2 . 5.88 tum treatments the noise impedance ZN of the amplifier ͑ ͒ will have a reactive part i.e., Im ZN 0 if there are Using these results in Eq. ͑5.76͒, we find the ultimate out-of-phase correlations between the amplifier’s current 18 ͑ ͒ quantum limit on the noise temperature, and voltage noises i.e., if Im SVI 0 . Thus, if such correlations exist, it will not be possible to minimize the noise temperature ͑and hence reach the quantum limit͒, k T ͓␻͔ ജប␻/2 ͑5.89͒ B N if one uses a purely real source impedance Zs. More significantly, note that the final classical expres- As for the position detector, reaching the quantum limit sion for the noise temperature TN explicitly involves the ͓ ͑ ͔͒ here is not simply a matter of tuning the coupling ͓i.e., real part of the SVI correlator cf. Eq. 5.76 . In contrast, ¯ tuning the source impedance Zs to match the noise im- we have shown that in the quantum case Re SVI must be pedance, cf. Eq. ͑5.77͒ and ͑5.78͔͒; one also needs to zero if one wishes to reach the quantum limit while hav- have an amplifier with ideal quantum noise, that is, an ing a large power gain ͑see Appendix I.3͒; as such, this amplifier satisfying Eq. ͑4.17͒. quantity does not appear in the final expression for the Finally, we need to test our initial assumption that minimal TN. It also follows that to reach the quantum ͉ ͉Ӷ͉ ͉ ͉ ͉ Zs Zin , taking Zs to be equal to its optimal value limit while having a large power gain, an amplifier can- ͑ ͒ ZN. Using the proportionality condition of Eq. I13 and not have significant in-phase correlations between its the fact that we are in the large-power-gain limit current and voltage noise. ͑ ͓␻͔ӷ ͒ GP 1 ,wefind This last statement can be given a heuristic explanation. If there are out-of-phase correlations between current and voltage noise, we can easily make use of these 18 Note that our definition of the noise temperature TN con- by appropriately choosing our source impedance. How- forms with that of Devoret and Schoelkopf ͑2000͒ and most ever, if there are in-phase correlations between current electrical engineering texts, but is slightly different from that of and voltage noise, we cannot use these simply by tuning ͑ ͒ Caves 1982 . Caves assumed the source is initially at zero tem- the source impedance. We could, however, have used perature ͑i.e., T =0͒, and consequently used the full quantum 0 them by implementing feedback in our amplifier. The expression for its equilibrium noise. In contrast, we have assumed that k T ӷប␻. The different definition of the noise fact that we have not done this means that these corre- B 0 lations represent a kind of missing information; as a re- temperature used by Caves leads to the result kBTN ജប␻/͑ln 3͒ as opposed to our Eq. ͑5.89͒. We stress that the sult, we must necessarily miss the quantum limit. In Sec. difference between these results has nothing to do with phys- VI.B, we explicitly give an example of a voltage ampli- ics, but only with how one defines the noise temperature. fier which misses the quantum limit due to the presence

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1194 Clerk et al.: Introduction to quantum noise, measurement, … of in-phase current and voltage fluctuations; we show charges, and thus the QPC acts as a charge-to-current how this amplifier can be made to reach the quantum amplifier. It has been shown theoretically that the QPC limit by adding feedback in Appendix H. can achieve the amplifier quantum limit, both in the regime where transport is due to tunneling ͑Gurvitz, 1997͒, G. Near quantum-limited mesoscopic detectors and in regimes where the transmission is not small ͑Aleiner et al., 1997; Levinson, 1997; Korotkov and Having discussed the origin and precise definition of Averin, 2001; Pilgram and Büttiker, 2002; Clerk et al., the quantum limit on the added noise of a linear, phase- 2003͒. Experimentally, QPCs are in widespread use as preserving amplifier, we now provide a review of work detectors of quantum dot qubits. The back-action examining whether particular detectors are able in prin- dephasing of QPC detectors was studied by Buks et al. ciple to achieve this ideal limit. We focus on the op-amp ͑1998͒ and Sprinzak et al. ͑2000͒; good agreement was mode of operation discussed in Sec. V. D , where the de- found with the theoretical prediction, confirming that tector is only weakly coupled to the system producing the QPC has quantum-limited back-action noise. the signal to be amplified. As repeatedly stressed, reaching the quantum limit in this case requires the detector to have quantum-ideal noise, as defined by Eq. ͑4.17͒. 3. Single-electron transistors and resonant-level detectors Heuristically, this corresponds to the general require- A metallic single-electron transistor ͑SET͒ consists of ment of no wasted information: there should be no other a small metallic island attached via tunnel junctions to quantity besides the detector output that could be moni- larger source and drain electrodes. Because of ͑ tored to provide information on the input signal Clerk Coulomb-blockade effects, the conductance of a SET is ͒ et al., 2003 . We have already given one simple but rel- sensitive to nearby charges, and hence it acts as a sensi- evant example of a detector which reaches the amplifier tive charge-to-current amplifier. Considerable work has quantum limit: the parametric cavity detector, discussed investigated whether metallic SETs can approach the in Sec. III.B. Here we turn to other more complex de- quantum limit in various different operating regimes. tectors. Theoretically, the performance of a normal-metal SET in the sequential tunneling regime was studied by Shnir- 1. dc superconducting quantum interference device amplifiers man and Schön ͑1998͒, Devoret and Schoelkopf ͑2000͒, The dc superconducting quantum interference device Makhlin et al. ͑2000͒, Aassime et al. ͑2001͒, and Johans- ͑SQUID͒ is a detector based on a superconducting ring son et al. ͑2002, 2003͒. In this regime, where transport is having two Josephson junctions. It can in principle be via a sequence of energy-conserving tunnel events, one used as a near quantum-limited voltage amplifier or is far from optimizing the quantum noise constraint of flux-to-voltage amplifier. Theoretically, this was investi- Eq. ͑4.17͒, and hence one cannot reach the quantum gated using a quantum Langevin approach ͑Koch et al., limit ͑Shnirman and Schön, 1998; Korotkov, 2001b͒.If 1981; Danilov et al., 1983͒, as well as more rigorously by one instead chooses to work with a normal-metal SET in using perturbative techniques ͑Averin, 2000b͒ and map- the cotunneling regime ͑a higher-order tunneling pro- pings to quantum impurity problems ͑Clerk, 2006͒. Ex- cess involving a virtual transition͒, then one can indeed periments on SQUIDS have also confirmed their poten- approach the quantum limit ͑Averin, 2000a; van den tial for near-quantum-limited operation. Mück et al. Brink, 2002͒. However, by virtue of being a higher-order ͑ ͒ process, the related currents and gain factors are small, 2001 were able to achieve a noise temperature TN approximately 1.9 times the quantum-limited value at an impinging on the practical utility of this regime of opera- operating frequency of ␻=2␲ϫ519 MHz. Working at tion. It is worth noting that while most theory on SETs lower frequencies appropriate to gravitational wave de- assume a dc voltage bias, to enhance bandwidth, experi- tection applications, Vinante et al. ͑2001͒ were able to ments are usually conducted using the rf-SET configura- tion ͑Schoelkopf et al., 1998͒, where the SET changes achieve a TN approximately 200 times the quantum- ␻ ␲ϫ the damping of a resonant LC circuit. Korotkov and limited value at a frequency =2 1.6 kHz; more re- ͑ ͒ cently, the same group achieved a T approximately ten Paalanen 1999 showed that this mode of operation for N a sequential tunneling SET increases the measurement- times the quantum limit at a frequency ␻=2␲ imprecision noise by approximately a factor of 2. The ϫ1.6 kHz ͑Falferi et al., 2008͒. In practice, it can be dif- measurement properties of a normal-metal, sequential- ficult to achieve the theoretically predicted quantum- tunneling rf-SET ͑including back-action͒ were studied limited performance due to spurious heating caused by experimentally by Turek et al. ͑2005͒. the dissipation in the shunt resistances used in the Measurement using superconducting SETs has also SQUID. This effect can be significantly ameliorated, been studied. Clerk et al. ͑2002͒ showed that so-called however, by adding cooling fins to the shunts ͑Wellstood incoherent Cooper-pair tunneling processes in a super- et al., 1994͒. conducting SET can have a noise temperature which is approximately a factor of 2 larger than the quantum- 2. Quantum point contact detectors limited value. The measurement properties of supercon- A quantum point contact ͑QPC͒ is a narrow conduct- ducting SETs biased at a point of incoherent Cooper- ing channel formed in a two-dimensional gas. The cur- pair tunneling have been probed recently in experiment rent through the constriction is sensitive to nearby ͑Thalakulam et al., 2004; Naik et al., 2006͒.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1195

The quantum measurement properties of phase- However, due to the dynamics of a harmonic oscillator, coherent noninteracting resonant level detectors have this disturbance will not affect the measured quadrature also been studied theoretically ͑Averin, 2000b; Clerk at later times. One can already see this from the classical and Stone, 2004; Mozyrsky et al., 2004; Gavish et al., equations of motion. Suppose our oscillator is driven by 2006͒. These systems are similar to metallic SET, except a time-dependent force F͑t͒ which has appreciable band- that the central island only has a single level ͑as opposed width only near ⍀. We may write this as to a continuous density of states͒, and Coulomb- blockade effects are typically neglected. These detectors ͑ ͒ ͑ ͒ ͑⍀ ␦͒ ͑ ͒ ͑⍀ ␦͒ ͑ ͒ F t = FX t cos t + + FY t sin t + , 5.93 can reach the quantum limit in the regime where the voltage and temperature are smaller than the intrinsic where F ͑t͒, F ͑t͒ are slowly varying compared to ⍀. energy broadening of the level due to tunneling. They X Y Using the fact that the oscillator has a high-quality fac- can also reach the quantum limit in a large-voltage re- tor Q=⍀/␥, one can easily find the equations of motion, gime that is analogous to the cotunneling regime in a metallic SET ͑Averin, 2000b; Clerk and Stone, 2004͒. ͑ ͒ ͑␥ ͒ ͑ ͒ ͑ ͒ ⍀ ͑ ͒ The influence of dephasing processes on such a detector dX␦ t /dt =− /2 X␦ t − FY t /2m , 5.94a was studied by Clerk and Stone ͑2004͒. ͑ ͒ ͑␥ ͒ ͑ ͒ ͑ ͒ ⍀ ͑ ͒ dY␦ t /dt =− /2 Y␦ t + FX t /2m . 5.94b H. Back-action evasion and noise-free amplification ͑ ͒ ͑ ͒ Thus, as long as FY t and FX t are uncorrelated and sufficiently slow, the dynamics of the two quadratures Having discussed in detail quantum limits on phase- are completely independent; in particular, if Y␦ is subject preserving linear amplifiers ͑i.e., amplifiers which mea- to a narrow-bandwidth, noisy force, it is of no conse- sure both quadratures of a signal equally well͒, we now quence to the evolution of X␦. An ideal measurement of return to the situation discussed in Sec. V. A : imagine we X␦ will result in a back-action force having the form of wish only to amplify a single quadrature of some time- ͑ ͒ ͑ ͒ ͑ ͒ dependent signal. For this case, there need not be any Eq. 5.93 with FY t =0, implying that X␦ t will be com- added noise from the measurement. Unlike the case of pletely unaffected by the measurement. amplifying both quadratures, Liouville’s theorem does Not surprisingly, if one can measure and amplify X␦ not require the existence of any additional degrees of without any back-action, there need not be any added freedom when amplifying a single quadrature: phase noise due to the amplification. In such a setup, the only space volume can be conserved during amplification added noise is the measurement-imprecision noise asso- simply by contracting the unmeasured quadrature ͓see ciated with intrinsic fluctuations of the amplifier output. Eq. ͑5.2͔͒. As no extra degrees of freedom are needed, These may be reduced in principle to an arbitrarily small ͑ there need not be any extra noise associated with the value by simply increasing the amplifier gain e.g., by ͒ amplification process. increasing the detector-system coupling : in an ideal Alternatively, single-quadrature detection can take a setup, there is no back-action penalty on the measured form similar to a QND measurement, where the back- quadrature associated with this increase. action does not affect the dynamics of the quantity being The above conclusion can lead to what seems like a measured ͑Thorne et al., 1978; Braginsky et al., 1980; contradiction. Imagine we use a back-action evading ͑ Caves et al., 1980; Caves, 1982; Braginsky and Khalili, amplifier to make a “perfect” measurement of X␦ i.e., ͒ 1992; Bocko and Onofrio, 1996͒. For concreteness, con- negligible added noise . We would then have no uncer- sider a high-Q harmonic oscillator with position x͑t͒ and tainty as to the value of this quadrature. Consequently, resonant frequency ⍀. Its motion may be written in we would expect the quantum state of our oscillator to terms of quadrature operators defined as in Eq. ͑3.48͒, be a squeezed state, where the uncertainty in X␦ is much smaller than xZPF. However, if there is no back-action xˆ͑t͒ = Xˆ ␦͑t͒cos͑⍀t + ␦͒ + Yˆ ␦͑t͒sin͑⍀t + ␦͒. ͑5.91͒ acting on X␦, how is the amplifier able to reduce its uncertainty? This seeming paradox can be fully resolved by ͑ ͒ Here xˆ t is the Heisenberg-picture position operator of considering the conditional aspects of an ideal single- the oscillator. The quadrature operators can be written quadrature measurement, where one considers the state ͑ ͒ in terms of the Schrödinger-picture oscillator creation of the oscillator given a particular measurement history and destruction operators as ͑Ruskov et al., 2005; Clerk et al., 2008͒. i͑⍀t+␦͒ † −i͑⍀t+␦͒ It is worth stressing that the possibility of amplifying a Xˆ ␦͑t͒ = x ͑cê + cˆ e ͒, ͑5.92a͒ ZPF single quadrature without back-action ͑and hence without added noise͒ relies crucially on our oscillator resem- ˆ ͑ ͒ ͑ i͑⍀t+␦͒ † −i͑⍀t+␦͒͒ ͑ ͒ Y␦ t =−ixZPF cê − cˆ e . 5.92b bling a perfect harmonic oscillator: the oscillator Q must As previously discussed, the two quadrature ampli- be large, and nonlinearities ͑which could couple the two quadratures͒ must be small. In addition, the envelope of tude operators Xˆ ␦ and Yˆ ␦ are canonically conjugate ͓cf. the nonvanishing back-action force F ͑t͒ must have a Eq. ͑3.49͔͒. Making a measurement of one quadrature X narrow bandwidth. One should further note that a very ˆ amplitude, say X␦, will thus invariably lead to back- high-precision measurement of X␦ will produce a very ˆ action disturbance of the other conjugate quadrature Y␦. large back-action force FX. If the system is not nearly

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1196 Clerk et al.: Introduction to quantum noise, measurement, … perfectly harmonic, then the large amplitude imparted The dimensionless quadrature operators correspond- to the conjugate quadrature Y␦ will inevitably leak back ing to the signal mode are into X␦. xˆ = ͑1/ͱ2͒͑aˆ † + aˆ ͒, yˆ = ͑i/ͱ2͒͑aˆ † − aˆ ͒, ͑5.98͒ Amplifiers or detectors that treat the two signal S S S S S S ͓ ͔ quadratures differently are known in the quantum optics which obey xˆ S ,yˆ S =i. We can define quadrature opera- literature as phase sensitive; we prefer the designation tors Xˆ , Yˆ corresponding to the input and out- phase nonpreserving since they do not preserve the S,in/out S,in/out put fields in a completely analogous manner. phase of the original signal. Such amplifiers invariably The steady state solution of Eq. ͑5.97͒ for the output rely on some internal clock ͑i.e., an oscillator with a fields becomes well-defined phase͒ which breaks time-translation invariance and picks out the phase of the quadrature that ˆ ͱ ˆ ˆ ˆ ͱ ͑ ͒ XS,out = GXS,in, YS,out = YS,in/ G, 5.99 will be amplified ͓i.e., the choice of ␦ used to define the two quadratures in Eq. ͑5.91͔͒; we see this explicitly in where the number gain G is given by what follows. This leads to an important caveat: even in ͓͑␭ ␬ ͒ ͑␭ ␬ ͔͒2 ͑ ͒ G = + S/2 / − S/2 . 5.100 a situation where the interesting information is in a single signal quadrature, to benefit from using a phase- We thus see clearly that the amplifier treats the two nonpreserving amplifier, we must know in advance the quadratures differently. One quadrature is amplified and precise phase of this quadrature. If we do not know this the other attenuated, with the result that the commuta- phase, we either have to revert to a phase-preserving tion relation can be preserved without the necessity of amplification scheme ͑and thus be susceptible to added extra degrees of freedom and added noise. Note that the noise͒ or we would have to develop a sophisticated and large-amplitude pump mode has played the role of a high speed quantum feedback scheme to dynamically clock in the degenerate paramp: it is the phase of the adapt the measurement to the correct quadrature in real pump that picks out which quadrature of the signal will time ͑Armen et al., 2002͒. In what follows, we make the be amplified. above ideas concrete by considering a few examples of Before ending our discussion here, it is important to quantum phase-nonpreserving amplifiers.19 stress that while the degenerate parametric amplifier is phase sensitive and has no added noise, it is not an example of back-action evasion ͓see Caves et al. ͑1980͒, 1. Degenerate parametric amplifier footnote on p. 342͔. This amplifier is operated in the Perhaps the simplest example of a phase nonpreserv- scattering mode of amplifier operation, a mode where ing amplifier is the degenerate parametric amplifier; the ͑as discussed in Sec. V. D ͒ back-action is not relevant at classical version of this system was described in Sec. V. A all. Recall that in this mode of operation the amplifier ͓see Eq. ͑5.2͔͒. The setup is similar to the nondegenerate input is perfectly impedance matched to the signal parametric amplifier discussed in Sec. V. C , except that source, and the input signal is simply the amplitude of an the idler mode is eliminated, and the nonlinearity con- incident wave on the amplifier input. This mode of op- verts a single pump photon into two signal photons at eration necessarily requires a strong coupling between frequency ␻ =␻ /2. As we now show, the resulting dy- ͑ ͗ ˆ ͒͘ S P the input signal and the amplifier input i.e., bS,in .If namics causes one signal quadrature to be amplified one instead tried to weakly couple the degenerate para- while the other is attenuated, in such a way that it is not metric amplifier to a signal source, and operate it in the necessary to add extra noise to preserve the canonical op-amp mode of operation ͑cf. Sec. V. D ͒, one finds that commutation relations. there is indeed a back-action disturbance of the mea- The system Hamiltonian is sured quadrature. We have yet another example which demonstrates that one must be careful to distinguish the ˆ ប͑␻ ˆ † ˆ ␻ ˆ † ˆ ͒ ប␩͑ˆ † ˆ † ˆ ˆ ˆ ˆ † ͒ Hsys = PaPaP + SaSaS + i aSaSaP − aSaSaP . op-amp and scattering modes of amplifier operation. ͑5.95͒

If the pump is treated classically as before, the analog of 2. Double-sideband cavity detector Eq. ͑5.16͒ is We now turn to a simple but experimentally relevant ˆ ប͑␭ ͒͑ † † ͒ ͑ ͒ detector that is truly back-action evading. We take as Vsys = i /2 aˆ Saˆ S − aˆ Saˆ S , 5.96 our input signal the position xˆ of a mechanical oscillator. ␭ ␩␺ ͑ ͒ where /2= P, and the analog of Eq. 5.19 is The ampliﬁer setup we consider is almost identical to the cavity position detector discussed in Sec. III.B.3:we aˆ˙ =−͑␬ /2͒aˆ + ␭aˆ † − ͱ␬ bˆ . ͑5.97͒ S S S S S S,in again have a single-sided resonant cavity whose frequency depends linearly on the oscillator’s position, with the Hamiltonian given by Eq. ͑3.11͒͑with zˆ =xˆ /x ͒. 19One could in principle generalize the linear-response ap- ZPF proach of Sec. V to deal with phase-nonpreserving detectors. We showed in Sec. III.B.3 and Appendix E.3 that, by However, as such detectors are not time-translational invari- driving the cavity on resonance, it could be used to make ant, such a description becomes rather cumbersome and is not a quantum-limited position measurement: one can oper- particularly helpful. We prefer instead to present concrete ate it as a phase-preserving ampliﬁer of the mechanical examples. oscillator’s position and achieve the minimum possible

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amount of added noise. To use the same system to make ͱ † ϰ N˙ ͑aˆ + aˆ ͒Xˆ ␦. ͑5.102b͒ a back-action free measurement of one oscillator quadrature only, one simply uses a different cavity drive. Instead of driving at the cavity resonance frequency ␻ , c This is exactly what is found in a full calculation ͑see one drives at the two sidebands associated with the me- ͒ ͑ ␻ ⍀ ⍀ Appendix E.4 . Note that we have written the interac- chanical motion i.e., at frequencies c ± , where is as always the frequency of the mechanical resonator͒.As tion in an interaction picture in which the fast oscilla- we see, such a drive results in an effective interaction tions of the cavity and oscillator operators have been that couples the cavity to only one quadrature of the removed. In the second line, we have made use of Eqs. ͑ ͒ oscillator’s motion. This setup was first proposed as a 5.92 to show that the effective interaction involves only means of back-action evasion by Braginsky et al. ͑1980͒; the Xˆ ␦ oscillator quadrature. further discussion can be found in Caves et al. ͑1980͒ and We thus see from Eq. ͑5.102b͒ that the cavity is Braginsky and Khalili ͑1992͒, as well as in Clerk et al. coupled only to the oscillator X␦ quadrature. As shown ͑2008͒, which gives a fully quantum treatment and con- rigorously in Appendix E.3, the result is that the system siders conditional aspects of the measurement. In what only measures and amplifies this quadrature: the light follows, we sketch the operation of this system following leaving the cavity has a signature of Xˆ ␦ but not of Yˆ ␦. Clerk et al. ͑2008͒; details are provided in Appendix E.4. Further, Eq. ͑5.102b͒ implies that the cavity operator We start by requiring that our system be in the “good- ␻ ӷ⍀ӷ␬ ͑␬ ͱ ˙ ͑ †͒ cavity” limit, where c is the damping of the N aˆ +aˆ will act as a noisy force on the Y␦ quadrature. cavity mode͒; we also require the mechanical oscillator While this will cause a back-action heating of Y␦, it will to have a high-Q factor. In this regime, the two side- not affect the measured quadrature X␦. We thus have a ␻ ⍀ bands associated with the mechanical motion at c ± true back-action-evading amplifier: the cavity output ␻ are well separated from the main cavity resonance at c. light lets one measure X␦ free from any back-action ef- Making a single-quadrature measurement requires that fect. Note that, in deriving Eq. ͑5.102a͒, we have used one drives the cavity equally at the two sideband fre- the fact that the cavity operators have fluctuations in a ¯ ϳ␬Ӷ⍀ quencies. The amplitude of the driving field bin entering narrow bandwidth : the back-action force noise is the cavity will be chosen to have the form slow compared to the oscillator frequency. If this were not the case, we could still have a back-action heating of the measured X␦ quadrature. Such effects, arising from a ͱ ˙ nonzero ratio ␬/⍀, are treated in Clerk et al. ͑2008͒. ¯ i N i␦ −i͑␻ −⍀͒t −i␦ −i͑␻ +⍀͒t b ͑t͒ =− ͑e e c − e e c ͒ Finally, as there is no back-action on the measured X␦ in 4 quadrature, the only added noise of the amplification ͱ ˙ scheme is measurement-imprecision noise ͑e.g., shot N ␻ = sin͑⍀t + ␦͒e−i ct. ͑5.101͒ noise in the light leaving the cavity͒. This added noise 2 can be made arbitrarily small by increasing the gain of the detector by, for example, increasing the strength of the cavity drive N˙ . In a real system where ␬/␻ is non- Here N˙ is the photon-number flux associated with the M zero, the finite bandwidth of the cavity number fluctua- cavity drive ͑see Appendix E for more details on how to tions leads to a small back-action on the X␦. As a result, properly include a drive using input-output theory͒. one cannot make the added noise arbitrarily small, as Such a drive could be produced by taking a signal at the cavity resonance frequency and amplitude-modulating it too large a cavity drive will heat the measured quadra- ␬ ␻ at the mechanical frequency. ture. Nonetheless, for a sufficiently small ratio / M, To understand the effect of this drive, note that it one can still beat the standard quantum limit on the ␻ ⍀ added noise ͑Clerk et al., 2008͒. sends the cavity both photons with frequency c − and ␻ ⍀ photons with frequency c + . The first kind of drive photon can be converted to a cavity photon if a quantum is absorbed from the mechanical oscillator; the second 3. Stroboscopic measurements kind of drive photon can be converted to a cavity photon if a quantum is emitted to the mechanical oscillator. With sufficiently high bandwidth it should be possible The result is that we can create a cavity photon by either to do stroboscopic measurements synchronized with the absorbing or emitting a mechanical oscillator quantum. oscillator motion, which could allow one to go below the i2␦ If there is a well-defined relative phase of e between standard quantum limit in one quadrature of motion the two kinds of drive photons, we would expect the ͑Caves et al., 1980; Braginsky and Khalili, 1992͒.Toun- double-sideband drive to yield an effective cavity- derstand this idea, imagine an extreme form of phase- oscillator interaction of the form sensitive detection in which a Heisenberg microscope makes a strong high-resolution measurement which projects the oscillator onto a state of well-defined posi- ϰ ͱ ˙ ͓ †͑ i␦ −i␦ †͒ ͔͑͒ Veff N aˆ e cˆ + e cˆ + H.c. 5.102a tion X0 at time t=0,

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ϱ mission line can be described as a set of noninteracting ⌿ ͑t͒ = ͚ a e−i͑n+1/2͒⍀t͉n͘, ͑5.103͒ bosonic modes ͑see Appendix D for a quick review͒. X0 n n=0 Denoting the input transmission line with an a and the ͗ ͉ ͘ output transmission line with a b, the current and volt- where the coefficients obey an = n X0 . Because the position is well defined the momentum is extremely uncer- age operators in these lines may be written as tain. ͑Equivalently, the momentum kick delivered by the ϱ d␻ ␻ back-action of the microscope makes the oscillator mo- Vˆ ͑t͒ = ͵ ͑Vˆ ͓␻͔e−i t + H.c.͒, ͑6.1a͒ q 2␲ q mentum uncertain.͒ Thus the wave packet quickly 0 spreads out and the position uncertainty becomes large. ϱ d␻ However, because of the special feature that the har- ˆ ͑ ͒ ␴ ͵ ͑ˆ ͓␻͔ −i␻t ͒ ͑ ͒ Iq t = q ␲ Iq e + H.c. , 6.1b monic oscillator levels are evenly spaced, we can see 0 2 from Eq. ͑5.103͒ that the wave packet re-assembles itself precisely once each period of oscillation because ein⍀t with =1 for every integer n. ͑At half periods, the packet re- ប␻ ˆ ͓␻͔ ͱ ͑ ͓␻͔ ͓␻͔͒ ͑ ͒ assembles at position −X .͒ Hence stroboscopic mea- Vq = Zq qˆ in + qˆ out , 6.2a 0 2 surements made once ͑or twice͒ per period will be back- action evading and can go below the standard quantum ប␻ limit. The only limitations will be the finite anharmonic- ˆ ͓␻͔ ͱ ͑ ͓␻͔ ͓␻͔͒ ͑ ͒ Iq = qˆ in − qˆ out . 6.2b ity and damping of the oscillator. Note that the possibil- 2Zq ity of using mesoscopic electron detectors to perform ␴ Here q can be equal to a or b, and we have a =1, stroboscopic measurements has recently received atten- ␴ ͓␻͔ ͓␻͔ ͑ ͒ b =−1. The operators aˆ in , aˆ out are bosonic annihi- tion Jordan and Büttiker, 2005b; Ruskov et al., 2005 . ͓␻͔ lation operators; aˆ in describes an incoming wave in the input transmission line ͑i.e., incident on the ampli- ͒ ␻ ͓␻͔ VI. BOSONIC SCATTERING DESCRIPTION OF A fier having frequency , while aˆ out describes an out- TWO-PORT AMPLIFIER ␻ ˆ ͓␻͔ going wave with frequency . The operators bin and ˆ ͓␻͔ In this section, we return again to the topic of Sec. bout describe analogous waves in the output transmis- ˆ V. F , quantum limits on a quantum voltage amplifier. We sion line. We can think of Va as the input voltage to our now discuss the physics in terms of the bosonic voltage amplifier and Vˆ as the output voltage. Similarly, Iˆ is the amplifier first introduced in Sec. V. D . Recall that in that b a ˆ section we demonstrated that the standard Haus-Caves current drawn by the amplifier at the input and Ib is the derivation of the quantum limit was not directly relevant current drawn at the output of the amplifier. Finally, Za ͑ ͒ ͑ ͒ to the usual weak-coupling op-amp mode of amplifier Zb is the characteristic impedance of the input output operation, a mode where the input signal is not simply transmission line. the amplitude of a wave incident on the amplifier. In this As we have seen, amplification invariably requires ad- section, we expand upon that discussion, giving an ex- ditional degrees of freedom. Thus, to amplify a signal at plicit discussion of the differences between the op-amp a particular frequency ␻, there will be 2N bosonic description of an amplifier presented in Sec. V. E , and modes involved, where the integer N is necessarily the scattering description often used in the quantum op- larger than 2. Four of these modes are simply the tics literature ͑Grassia, 1998; Courty et al., 1999͒.Wesee frequency-␻ modes in the input and output lines ͑i.e., that what one means by back-action and added noise are ͓␻͔ ͓␻͔ ˆ ͓␻͔ ˆ ͓␻͔͒ ͑ aˆ in , aˆ out , bin , and bout . The remaining 2 N not the same in the two descriptions! Further, even −2͒ modes describe auxiliary degrees of freedom in- though an amplifier may reach the quantum limit when volved in the amplification process; these additional ͑ used in the scattering mode i.e., its added noise is as modes could correspond to frequencies different from ͒ small as allowed by commutation relations , it can none- the signal frequency ␻. The auxiliary modes can also be theless fail to achieve the quantum limit when used in divided into incoming and outgoing modes. It is thus the op-amp mode. Finally, the discussion here will also convenient to represent them as additional transmission allow us to highlight important aspects of the quantum lines attached to the amplifier; these additional lines limit not easily discussed in the more general context of could be semi-infinite or could be terminated by active Sec. IV. elements.

A. Scattering versus op-amp representations 1. Scattering representation In the bosonic scattering approach, a generic linear In general, our generic two-port bosonic amplifier will amplifier is modeled as a set of coupled bosonic modes. be described by an NϫN scattering matrix, which deter- To make matters concrete, we consider the specific case mines the relation between the outgoing and incoming of a voltage amplifier with distinct input and output mode operators. The form of this matrix is constrained ports, where each port is a semi-infinite transmission line by the requirement that the output modes obey the ͑see Fig. 9͒. We start by recalling that a quantum trans- usual canonical bosonic commutation relations. It is con-

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venient to express the scattering matrix in a form which 2. Op-amp representation only involves the input and output lines explicitly, In the usual op-amp amplifier mode of operation ͑de- scribed in Sec. IV͒, the input and output signals are not simply incoming and outgoing wave amplitudes; thus, aˆ ͓␻͔ ͓␻͔ ͓␻͔ aˆ ͓␻͔ Fˆ ͓␻͔ ͩ out ͪ ͩs11 s12 ͪͩ in ͪ ͩ a ͪ the scattering representation is not an optimal descrip- = ͓␻͔ ͓␻͔ + . tion of our amplifier. The system we are describing here bˆ ͓␻͔ s21 s22 bˆ ͓␻͔ Fˆ ͓␻͔ out in b is a voltage amplifier: thus, in the op-amp mode, the ͑6.3͒ experimentalist would ensure that the voltage at the end ͑ ˆ ͒ of the input line Va is equal to the signal to be amplified, and would read out the voltage at the end of the Here Fˆ ͓␻͔ and Fˆ ͓␻͔ are each an unspecified linear ͑ ˆ ͒ a b output transmission line Vb as the output of the ampli- combination of the incoming auxiliary modes introduced fier. From Eq. ͑6.1a͒, we see that this implies that the above. They thus describe noise in the outgoing modes amplitude of the wave going into the amplifier, ain, will of the input and output transmission lines which arises depend on the amplitude of the wave exiting the ampli- from the auxiliary modes involved in the amplification fier, aout. process. Note the similarity between Eqs. ͑6.3͒ and ͑5.7͒ Thus, if we want to use our amplifier as a voltage for the simple one-port bosonic amplifier considered in amplifier, we want to find a description that is more tai- Sec. V. B . lored to our needs than the scattering representation of In the quantum optics literature, one typically views Eq. ͑6.3͒. This can be found by simply re-expressing the Eq. ͑6.3͒ as the defining equation of the amplifier; we scattering matrix relation of Eq. ͑6.3͒ in terms of volt- call this the scattering representation of our amplifier. ages and currents. The result will be what we term the The representation is best suited to the scattering mode op amp representation of our amplifier, a representation of amplifier operation described in Sec. V. D . In this which is standard in the discussion of classical amplifiers mode of operation, the experimentalist ensures that ͓see, e.g., Boylestad and Nashelsky ͑2006͔͒. In this rep- ͗aˆ ͓␻͔͘ is precisely equal to the signal to be amplified, ˆ ˆ in resentation, one views Va and Ib as inputs to the ampli- irrespective of what is coming out of the amplifier. Simi- ˆ larly, the output signal from the amplifier is the amplifier: Va is set by whatever we connect to the amplifier input, while Iˆ is set by whatever we connect to the am- tude of the outgoing wave in the output line, ͗bˆ ͓␻͔͘.If b out plifier output. In contrast, the outputs of our amplifier we focus on bˆ , we have precisely the same situation as out are the voltage in the output line, Vˆ , and the current described in Sec. 5.10, where we presented the Haus- b ˆ Caves derivation of the quantum limit ͓see Eq. ͑5.7͔͒.It drawn by the amplifier at the input, Ia. Note that this follows that in the scattering mode of operation the ma- interpretation of voltages and currents is identical to the ͓␻͔ way we viewed the voltage amplifier in the linear- trix element s21 represents the gain of our amplifier at frequency ␻, ͉s ͓␻͔͉2 the corresponding “photon- response and quantum noise treatment of Sec. V. F . 21 Using Eqs. ͑6.1a͒ and ͑6.1b͒ and suppressing fre- Fˆ number gain,” and b the added noise operator of the quency labels for clarity, Eq. ͑6.3͒ may be written explic- Fˆ amplifier. The operator a represents the back-action itly in terms of the voltages and current in the input noise in the scattering mode of operation; this back- ͑ ˆ ˆ ͒ ͑ ˆ ˆ ͒ Va ,Ia and output Vb ,Ib transmission lines, action has no effect on the added noise of the amplifier ␭ in the scattering mode. V − Zout ˆ Vˆ Vˆ ␭ · V˜ Similar to Sec. V. B , one can now apply the standard ͩ b ͪ = 1 ͩ a ͪ + V . ͑6.4͒ ΂ ␭Ј ΃ ΂ ΃ argument of Haus and Mullen ͑1962͒ and Caves ͑1982͒ ˆ ˆ Ia Z I Ib ˜ˆ to our amplifier. This argument tells us that since the in I “out” operators must have the same commutation rela- The coefficients in the above matrix are completely de- tions as the “in” operators, the added noise Fˆ cannot termined by the scattering matrix of Eq. ͑6.3͓͒see Eqs. b ͑ ͒ ͔ be arbitrarily small in the large-gain limit ͑i.e., ͉s ͉ӷ1͒. 6.7 below ; moreover, they are familiar from the dis- 21 ␭ ͓␻͔ Note that this version of the quantum limit on the added cussion of a voltage amplifier in Sec. V. F . V is the ␭Ј͓␻͔ noise has nothing to do with back-action. As discussed, voltage gain of the amplifier, I is the reverse current this is perfectly appropriate for the scattering mode of gain of the amplifier, Zout is the output impedance, and operation, as in this mode the experimentalist ensures Zin is the input impedance. The last term on the RHS of that the signal going into the amplifier is completely in- Eq. ͑6.4͒ describes the two familiar kinds of amplifier ˆ dependent of whatever is coming out of the amplifier. noise. V˜ is the usual voltage noise of the amplifier ͑re- This mode of operation could be realized in time- ͒ ˜ˆ dependent experiments, where a pulse is launched at the ferred back to the amplifier input , while I is the usual amplifier. This mode is not realized in most weak- current noise of the amplifier. Recall that in this stan- coupling amplification experiments, where the signal to dard description of a voltage amplifier ͑see Sec. V. F ͒ I˜ be amplified is not identical to an incident wave ampli- represents the back-action of the amplifier: the system tude. producing the input signal responds to these current

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fluctuations, resulting in an additional fluctuation in the ͑ ͒͑ ͒ ͑ ͒ D = 1+s11 1−s22 + s12s21. 6.8 ␭ ˜ input signal going into the amplifier. Similarly, VV represents the intrinsic output noise of the amplifier: this Further, the voltage and current noises in the op-amp contribution to the total output noise does not depend representation are simple linear combinations of the Fˆ Fˆ on properties of the input signal. Note that we are using noises a and b appearing in the scattering representa- ͗ˆ ͘ a sign convention where a positive Ia indicates a cur- tion, rent flowing into the amplifier at its input, while a positive ͗Iˆ ͘ indicates a current flowing out of the amplifier 1 1+s11 b ˜ˆ − Fˆ ˆ ˆ V ͱ 2 2s21 a at its output. Also note that the operators Va and Ib on ប␻ ͩ ͪ ͑ ͒ ΂ ΃ = 2 Za . 6.9 the RHS of Eq. ͑6.4͒ will have noise; this noise is en- ΂ s22 −1 s12 ΃ Fˆ ˆ Z · I˜ − b tirely due to the systems attached to the input and out- a D D put of the amplifier, and as such should not be included in what we call the added noise of the amplifier. Again, all quantities above are evaluated at frequency ␻. Additional important properties of our amplifier fol- Equation ͑6.9͒ immediately leads to an important con- low immediately from quantities in the op-amp repre- clusion and caveat: what one calls the ‘‘back-action’’ and ͑ F sentation. As discussed in Sec. V. E , the most important ‘‘added noise’’ in the scattering representation i.e., a F ͒ measure of gain in our amplifier is the dimensionless and b are not the same as the ‘‘back-action’’ and power gain. This is the ratio between power dissipated ‘‘added noise’’ defined in the usual op-amp representa- at the output to that dissipated at the input, taking the ˆ tion. For example, the op-amp back-action I˜ does not in output current I to be V /Z , B B out Fˆ general coincide with a, the back-action in the scatter- ͑␭ ͒2 ␭ ␭Ј −1 ϵ V Zin ͩ V I Zin ͪ ͑ ͒ ing picture. If we are indeed interested in using our am- GP 1+ . 6.5 plifier as a voltage amplifier, we are interested in the 4 Zout 2 Zout total added noise of our amplifier as defined in the op- Another important quantity is the loaded input im- amp representation. As we saw in Sec. V. F ͓cf. Eq. pedance: what is the input impedance of the amplifier in ˜ˆ ˜ˆ the presence of a load attached to the output? In the ͑5.71͔͒, this quantity involves both the noises I and V. ␭Ј We thus see explicitly something discussed in Sec. V. D :it presence of reverse current gain I 0, the input impedance will depend on the output load. Taking the load is dangerous to make conclusions about how an ampli- impedance to be Z , some simple algebra yields fier behaves in the op-amp mode of operation based on load its properties in the scattering mode of operation. As we 1/Z =1/Z + ␭Ј␭ /͑Z + Z ͒. ͑6.6͒ see, even though an amplifier is ideal in the scattering in,loaded in I V load out ͑ F ͒ mode i.e., a as small as possible , it can nonetheless It is of course undesirable to have an input impedance fail to reach the quantum limit in the op-amp mode of which depends on the load. Thus, we see yet again that it operation. is undesirable to have appreciable reverse gain in our ˜ˆ amplifier ͑cf. Sec. IV.A.2͒. In what follows, we calculate the op-amp noises V and ˆ I˜ in a minimal bosonic voltage amplifier, and show explicitly how this description is connected to the more 3. Converting between representations general linear-response treatment of Sec. V. F . However, ͑ ͒ After some straightforward algebra we now express before proceeding, it is worth noting that Eqs. 6.7a , ͑ ͒ ͑ ͒ ͑ ͒ the op-amp parameters appearing in Eq. ͑6.4͒ in terms 6.7b , 6.7c , and 6.7d are themselves completely con- of the scattering matrix appearing in Eq. ͑6.3͒, sistent with linear-response theory. Using linear re- ␭ sponse, one would calculate the op-amp parameters V, s ␭Ј, Z , and Z using Kubo formulas ͓cf. Eqs. ͑5.84͒ and ␭ ͱZb 21 ͑ ͒ I in out V =2 , 6.7a ͑5.85͒ and the discussion following Eq. ͑5.79͔͒. These in Za D ˆ ˆ turn would involve correlation functions of Ia and Vb s evaluated at zero coupling to the amplifier input and ␭Ј ͱZb 12 ͑ ͒ I =2 , 6.7b output. Zero coupling means that there is no input volt- Za D age to the amplifier ͑i.e., a short circuit at the amplifier input, Vˆ =0͒ and there is nothing at the amplifier output ͑1+s ͒͑1+s ͒ − s s a 11 22 12 21 ͑ ͒ drawing current ͑i.e., an open circuit at the amplifier Zout = Zb , 6.7c D ˆ ͒ ͑ ͒ ˆ output, Ib =0 . Equation 6.4 tells us that in this case Vb ˆ ␭ ˜ˆ ˜ˆ 1 1 ͑1−s ͒͑1−s ͒ − s s and Ia reduce to the noise operators VV and I, respec- 11 22 12 21 ͑ ͒ = , 6.7d tively. Using the fact that the commutators of Fˆ and Fˆ Zin Za D a b are completely determined by the scattering matrix ͓cf. where all quantities are evaluated at the same frequency Eq. ͑6.3͔͒, we verify explicitly in Appendix I.5 that the ␻, and D is defined as Kubo formulas yield the same results for the op-amp

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1201 gains and impedances as Eqs. ͑6.7a͒, ͑6.7b͒, ͑6.7c͒, and cold load ͑6.7d͒ above. vacuum noise uin uout

aout bin B. Minimal two-port scattering amplifier input line output line ain bout 1. Scattering versus op-amp quantum limit 1/2 1/2 † cin cout=G cin+(G-1) v in In this section we demonstrate that an amplifier which ideal is ideal and minimally complex when used in the scatter- 1-port ing operation mode fails, when used as a voltage op- amp. amp, to have a quantum-limited noise temperature. The vout vin vacuum noise system we look at is similar to the amplifier considered by Grassia ͑1998͒, though our conclusions are somewhat different than those found there. FIG. 13. Schematic of a minimal two-port amplifier which In the scattering representation, one might guess that reaches the quantum limit in the scattering mode of operation, but misses the quantum limit when used as a weakly coupled an ideal amplifier would be one where there are no re- op-amp. See text for further description. flections of signals at the input and output, and no way for incident signals at the output port to reach the input. In this case, Eq. ͑6.3͒ takes the form ˆ ˆ some correlations between I˜ and V˜ if there is to be gain Fˆ ͓i.e., ␭ is given by a Kubo formula involving these op- aˆ out 00 aˆ in a V ͩ ͪ = ͩ ͪͩ ͪ + ͩ ͪ, ͑6.10͒ erators, see Eq. ͑4.3͔͒. ˆ ͱG 0 ˆ Fˆ bout bin b To make further progress, we note again that commu- ͱ Fˆ Fˆ where we have defined Gϵs . All quantities above tators of the noise operators a and b are completely 21 ͑ ͒ should be evaluated at the same frequency ␻; for clarity, determined by Eq. 6.10 and the requirement that the we omit the explicit ␻ dependence of quantities output operators obey canonical commutation relations. throughout this section. We thus have Turning to the op-amp representation, the above ͓Fˆ Fˆ †͔ ͑ ͒ equation implies that our amplifier has no reverse gain, a, a =1, 6.13a and that the input and output impedances are simply given by the impedances of the input and output trans- ͑ ͒ ͓Fˆ Fˆ †͔ ͉ ͉ ͑ ͒ mission lines. From Eqs. 6.7 , we have b, b =1− G , 6.13b ␭ ͱ ͑ ͒ V =2 ZbG/Za, 6.11a ͓Fˆ ,Fˆ ͔ = ͓Fˆ ,Fˆ †͔ =0. ͑6.13c͒ ␭Ј ͑ ͒ a b a b I =0, 6.11b We will be interested in the limit of a large power ͑ ͒ gain, which requires ͉G͉ӷ1. A minimal solution to the Zout = Zb, 6.11c above equations would be to have the noise operators ͑ ͒ determined by two independent ͑i.e., mutually commut- 1/Zin =1/Za. 6.11d ing͒ auxiliary input mode operators u and v† , We immediately see that our amplifier looks less ideal as in in an op-amp. The input and output impedances are the Fˆ ͑ ͒ same as those of the input and output transmission line. a = uˆ in, 6.14 However, for an ideal op-amp, we would have liked Z →ϱ and Z →0. in out Fˆ ͱ͉ ͉ † ͑ ͒ Also of interest are the expressions for the amplifier b = G −1vˆ in. 6.15 noises in the op-amp representation Further, to minimize the noise of the amplifier, we take 1 ˜ˆ 1 Fˆ the operating state of the amplifier to be the vacuum for V − a ͱ ប␻ 2 ͱ ͩ ͪ ͑ ͒ both these modes. With these choices, our amplifier is in ΂ ΃ =− 2 Za΂ 2 G ΃ . 6.12 ˆ Fˆ ͑ ͒ Z · I˜ 1 0 b exactly the minimal form described by Grassia 1998 :an a input and output line coupled to a negative resistance As s12=0, the back-action noise is the same in both the box and an auxiliary “cold load” via a four-port circula- op-amp and scattering representations: it is determined tor ͑see Fig. 13͒. The negative resistance box is nothing Fˆ but the single-mode bosonic amplifier discussed in Sec. completely by the noise operator a. However, the voltage noise ͑i.e., the intrinsic output noise͒ involves both V. B ; an explicit realization of this element would be the parametric amplifier discussed in Sec. V. C . The ‘‘cold Fˆ and Fˆ . We thus have the unavoidable consequence a b load’’ is a semi-infinite transmission line which models ˆ ˆ that there will be correlations in I˜ and V˜ . Note that from dissipation due to a resistor at zero temperature ͑i.e., its basic linear-response theory we know that there must be noise is vacuum noise, cf. Appendix D͒.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1202 Clerk et al.: Introduction to quantum noise, measurement, …

Note that within the scattering picture one would con- S¯ ͓␻͔S¯ ͓␻͔ − ͓Im S¯ ͔2 =2ϫ ͑ប␻/2͒2. ͑6.20͒ clude that our amplifier is ideal: in the large-gain limit, VV II VI ˆ ¯ ¯ the noise added by the amplifier to bout corresponds to a The noise product SVVSII is precisely twice the quantum- single quantum at the input limited value. As a result, the general theory of Sec. V. F tells us that if one couples an input signal weakly to this ͕͗Fˆ Fˆ †͖͘ ͉ ͉ ͓͉͑ ͉ ͒ ͉ ͉͔͕͗ † ͖͘ → ͑ ͒ ͑ Ӷ ͒ b, b / G = G −1 / G vˆ in,vˆ in 1. 6.16 amplifier i.e., Zs Zin , it is impossible to reach the quantum limit on the added noise. Thus, while our am- This, however, is not the quantity which interests us: as plifier is ideal in the scattering mode of operation ͓cf. we want to use this system as a voltage op-amp, we want Eq. ͑6.16͔͒, it fails to reach the quantum limit when used to know if the noise temperature defined in the op-amp in the weak-coupling, op-amp mode of operation. Our picture is as small as possible. We are also usually inter- amplifier’s failure to have ideal quantum noise also ested in the case of a signal that is weakly coupled to our means that if we tried to use it to do QND qubit detec- amplifier; here weak coupling means that the input im- tion, the resulting back-action dephasing would be twice pedance of the amplifier is much larger than the imped- as large as the minimum required by quantum mechan- ͑ ӷ ͒ ance of the signal source i.e., Zin Zs . In this limit, the ics ͑cf. Sec. IV.B͒. amplifier only slightly increases the total damping of the One might object to the above conclusions based on signal source. the classical expression for the minimal noise tempera- To address whether we can reach the op-amp quan- ture, Eq. ͑5.76͒. Unlike the quantum noise constraint of tum limit in the weak-coupling regime, we can make use Eq. ͑5.88͒, this equation also involves the real part of of the results of the general theory presented in Sec. V. F . S¯ , and is optimized by our minimal amplifier. However, In particular, we need to check whether the quantum VI ͑ ͒ this does not mean that one can achieve a noise tem- noise constraint of Eq. 5.88 is satisfied, as this is a pre- ប␻ requisite for reaching the ͑weak-coupling͒ quantum perature of /2 at weak coupling. Recall from Sec. V. F limit. Thus, we need to calculate the symmetrized spec- that in the usual process of optimizing the noise tem- tral densities of the current and voltage noises, and their perature one starts by assuming the weak-coupling con- cross correlation. It is easy to confirm from the defini- dition that the source impedance Zs is much smaller tions of Eqs. ͑6.1a͒ and ͑6.1b͒ that these quantities take than the amplifier input impedance Zin. One then finds ͉ ͉ the form that to minimize the noise temperature, Zs should be tunedͱ to match the noise impedance of the amplifier ¯ ͓␻͔ ͕͗ ˜ˆ ͓␻͔ ˜ˆ †͑␻ ͖͒͘ ␲␦͑␻ ␻ ͒ ͑ ͒ ϵ ¯ ¯ SVV = V ,V Ј /4 − Ј , 6.17a ZN SVV/SII. However, in our minimal bosonic ampli- ͑ ͒ ͱ ϳ fier, it follows from Eqs. 6.18 that ZN =Zin/ 2 Zin: the noise impedance is on the order of the input impedance. ¯ ͓␻͔ ͕͗˜ˆ͓␻͔ ˜ˆ†͑␻Ј͖͒͘ ␲␦͑␻ ␻Ј͒ ͑ ͒ SII = I ,I /4 − , 6.17b Thus, it is impossible to match the source impedance to the noise impedance while at the same time satisfying ˆ ˆ Ӷ S¯ ͓␻͔ = ͕͗V˜ ͓␻͔,I˜†͑␻Ј͖͒͘/4␲␦͑␻ − ␻Ј͒. ͑6.17c͒ the weak-coupling condition Zs Zin. VI Despite its failings, our amplifier can indeed yield a The expectation values here are over the operating state quantum-limited noise temperature in the op-amp mode of the amplifier; we have chosen this state to be the of operation if we no longer insist on a weak coupling to the input signal. To see this explicitly, imagine we con- vacuum for the auxiliary mode operators uˆ in and vˆ in to minimize the noise. nected our amplifier to a signal source with source im- ͉ ͉ӷ ͑ ͒ ͑ ͒ pedance Z . The total output noise of the amplifier, re- Taking s21 1, and using Eqs. 6.14 and 6.15 ,we s have ferred back to the signal source, will now have the form

¯ ͓␻͔ ប␻ ͑␴ ␴ ͒ ប␻ ͑ ͒ ˜ˆ ͓ ͑ ͔͒˜ˆ ˜ˆ ͑ ͒ SVV = Za uu + vv /4 = Za/2, 6.18a Vtot =− ZsZa/ Zs + Za I + V. 6.21 Note that this classical-looking equation can be rigor- ¯ ͓␻͔ ប␻␴ ប␻ ͑ ͒ SII = uu/Za = /Za, 6.18b ously justified within the full quantum theory if one starts with a full description of the amplifier and the ͑ ¯ ͓␻͔ ប␻␴ ប␻ ͑ ͒ signal source e.g., a parallel LC oscillator attached in SVI = uu/2 = /2, 6.18c parallel to the amplifier input͒. Plugging in the expres- ˆ ˆ where we have defined sions for I˜ and V˜ ,wefind

␴ ϵ͗aˆbˆ † + bˆ †aˆ͑͘6.19͒ ប␻ Z Z 2 ab ˜ˆ ͱ ͫͩ s a ͪͩ ˆ ͪ ͱ ͑ ˆ ˆ † ͒ͬ Vtot = uin − Za uin − vin 2 Z + Z ͱZ and have used the fact that there cannot be any correla- s a a tions between the operators u and v in the vacuum state ប␻Z Z − Z ͱ a ͫͩ s a ͪ † ͬ ͑ ͒ ͑i.e., ͗uˆ vˆ †͘=0͒. = uˆ in − vˆ in . 6.22 2 Zs + Za It follows immediately from the above equations that our minimal ampliﬁer does not optimize the quantum Thus, if one tunes Zs to Za =Zin, the mode uˆ in does not noise constraint of Eq. ͑5.88͒, contribute to the total added noise, and one reaches the

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1203 quantum limit. Physically speaking, by matching the sig- ͉␭ ͉2S¯ /Z = Z S¯ ϵ 2k T . ͑6.23͒ nal source to the input line the back-action noise de- V VV out in II B eff scribed by Fˆ =uˆ does not feed back into the input of In contrast, our minimal bosonic amplifier has very dif- a in ferent input and output effective temperatures the amplifier. Note that to achieve this matching explicitly requires one to be far from weak coupling. Having ¯ ប␻ ͑ ͒ 2kBTeff,in = ZinSII = /2, 6.24 Zs =Za means that when we attach the amplifier to the signal source, we dramatically increase the damping of 2k T = ͉␭ ͉2 · S¯ /Z =2͉G͉ប␻. ͑6.25͒ the signal source. B eff,out V VV out This large difference in effective temperatures means that it is impossible for the system to possess ideal quantum noise, and thus it cannot reach the weak-coupling 2. Why is the op-amp quantum limit not achieved? quantum limit. While it implies that one is not at the quantum limit, Returning to the more interesting case of a weak the fact that T ӶT can nonetheless be viewed as amplifier-signal coupling, one might still be puzzled as to eff,in eff,out an asset. From a practical point of view, a large Teff,in can why our seemingly ideal amplifier misses the quantum be dangerous. Even though the direct effect of the large limit. While the mathematics behind Eq. ͑6.20͒ is fairly Teff,in is offset by an appropriately weak coupling to the transparent, it is also possible to understand this result amplifier ͓see Eq. ͑5.70͒ and following discussion͔, this heuristically. To that end, note again that the amplifier large Teff,in can also heat up other degrees of freedom if ¯ noise cross correlation SIV does not vanish in the large- they couple strongly to the back-action noise of the am- gain limit ͓cf. Eq. ͑6.18c͔͒. Correlations between the two plifier. This can in turn lead to unwanted heating of the amplifier noises represent a kind of information, as by input system. As Teff,in is usually constant over a broad making use of them, we can improve the performance of range of frequencies, this unwanted heating effect can the amplifier. It is easy to take advantage of out-of-phase be quite bad. In the minimal amplifier discussed here, correlations between I˜ and V˜ ͑i.e., Im S¯ ͒ by simply tun- this problem is circumvented by having a small Teff,in. VI The only price that is paid is that the added noise will be ing the phase of the source impedance ͓cf. Eq. ͑5.75͔͒. ͱ However, one cannot take advantage of in-phase noise 2 the quantum limit value. We discuss this issue further in Sec. VII. ͑ ¯ ͒ correlations i.e., Re SVI as easily. To take advantage of the information here, one needs to modify the amplifier itself. By feeding back some of the output voltage to the VII. REACHING THE QUANTUM LIMIT IN PRACTICE input, one could effectively cancel out some of the back- A. Importance of QND measurements action current noise I˜ and thus reduce the overall magnitude of S¯ . Hence, the unused information in the cross The fact that QND measurements are repeatable is of II fundamental practical importance in overcoming detec- ¯ correlator Re SVI represents a kind of wasted informa- tor inefficiencies ͑Gambetta et al., 2007͒. A prototypical tion: had we made use of these correlations via a feed- example is the electron shelving technique ͑Nagourney back loop, we could have reduced the noise temperature et al., 1986; Sauter et al., 1986͒ used to measure trapped and increased the information provided by our amplifier. ions. A related technique is used in present implemen- ¯ The presence of a nonzero Re SVI thus corresponds to tations of ion-trap-based quantum computation. Here wasted information, implying that we cannot reach the the ͑extremely long-lived͒ hyperfine state of an ion is quantum limit. Recall that within the linear-response ap- read out via state-dependent optical fluorescence. With proach we were able to prove rigorously that a large- properly chosen circular polarization of the exciting la- gain amplifier with ideal quantum noise must have ser, only one hyperfine state fluoresces and the transition is cycling; that is, after fluorescence the ion almost al- ¯ ͓ ͑ ͔͒ Re SVI=0 cf. the discussion following Eq. 5.57 ; thus, a ways returns to the same state it was in prior to absorb- ¯ nonvanishing Re SVI rigorously implies that one cannot ing the exciting photon. Hence the measurement is be at the quantum limit. In Appendix H, we give an QND. Typical experimental parameters ͑Wineland et al., explicit demonstration of how feedback may be used to 1998͒ allow the cycling transition to produce Nϳ106 utilize these cross correlations to reach the quantum fluorescence photons. Given the photomultiplier quan- limit. tum efficiency and typically small solid angle coverage, Finally, yet another way of seeing that our amplifier only a small number n¯ d will be detected on average. The does not reach the quantum limit ͑in the weak-coupling probability of getting zero detections ͑ignoring dark regime͒ is to realize that this system does not have a counts for simplicity͒ and hence misidentifying the hy- well-defined effective temperature. Recall from Sec. V. F perfine state is P͑0͒=e−n¯ d. Even for a very poor overall ͓ −5 that a system with ideal quantum noise i.e., one that detection efficiency of only 10 , we still have n¯ d =10 and satisfies Eq. ͑5.88͒ as an equality͔ necessarily has the nearly perfect fidelity F=1−P͑0͒Ϸ0.999 955. It is impor- same effective temperature at its input and output ports tant to note that the total time available for measure- ͓ ͑ ͔͒ ͑ ͒ cf. Eq. 5.55 . Here that implies the requirement ment is not limited by the phase coherence time T2 of

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 1204 Clerk et al.: Introduction to quantum noise, measurement, … the qubit or by the measurement-induced dephasing ␭2 2 2 vin Zin Zload/Zout ͑Korotkov, 2001a; Makhlin et al., 2001; Schuster et al., = ͩ ͪ . Z Z + Z ͑1+Z /Z ͒2 2005; Gambetta et al., 2006͒, but rather only by the rate out in s load out at which the qubit makes real transitions between mea- As a function of Zload, Pout is maximized when Zload ͑␴ ͒ surement ˆ z eigenstates. In a perfect QND measure- =Zout, ment there is no measurement-induced state mixing ␭2v2 Z 2 ͑Makhlin et al., 2001͒ and the relaxation rate 1/T is in ͩ in ͪ ͑ ͒ 1 Pout,max = . 7.3 unaffected by the measurement process. 4Zout Zin + Zs The available power gain is now defined as ␭2 2 B. Power matching versus noise matching ϵ Pout,max Zs ͩ Zin ͪ GP,avail = Pin,avail Zout Zin + Zs In Sec. V, we saw that an important part of reaching 4Z /Z the quantum limit on the added noise of an amplifier = G s in . ͑7.4͒ P ͑ ͒2 ͑when used in the op-amp mode of operation͒ is to op- 1+Zs/Zin timize the coupling strength to the amplifier. For a posi- We see that GP,avail is strictly less than or equal to the tion detector, this condition corresponds to tuning the power gain G ; equality is achieved only when Z =Z ␥ P s in strength of the back-action damping to be much ͑i.e., when the source impedance is “power matched” to ͓ smaller than the intrinsic oscillator damping cf. Eq. the input of the amplifier͒. The general situation where ͑5.69͔͒. For a voltage amplifier, this condition corre- Ͻ GP,avail GP indicates that we are not drawing as much sponds to tuning the impedance of the signal source to power from the source as we could, and hence the actual ͓ ͑ ͔͒ be equal to the noise impedance cf. Eq. 5.77 ,anim- power supplied to the load is not as large as it could be. pedance that is much smaller than the amplifier’s input Consider now a situation where we have achieved the ͓ ͑ ͔͒ impedance cf. Eq. 5.90 . quantum limit on the added noise. This necessarily In this section, we make the simple point that optimiz- means that we have “noise matched,” i.e., taken Z to be ing the coupling ͑i.e., source impedance͒ to reach the s equal to the noise impedance Z . The available power quantum limit is not the same as what one would do to N gain in this case is optimize the power gain. To understand this, we need to Ӎ ␭2 Ӎ ͱ Ӷ ͑ ͒ introduce another measure of power gain commonly GP,avail ZN/Zout 2 GP GP. 7.5 used in the engineering community, the available power We have used Eq. ͑5.90͒, which tells us that the noise gain G . For simplicity, we discuss this quantity in P,avail impedance is smaller than the input impedance by a the context of a linear voltage amplifier, using the nota- ͑ ͱ ͒ tions of Sec. V. F ; it can be analogously defined for the large factor of 1/ 2 GP . Thus, as reaching the quantum limit requires the use of a source impedance much position detector of Sec. V. E . GP,avail tells us how much power we are providing to an optimally matched output smaller than Zin, it results in a dramatic drop in the load relative to the maximum power we could in prin- available power gain compared to the case where we ͑ ͒ ciple have extracted from the source. This is in marked power match i.e., take Zs =Zin . In practice, one must decide whether it is more important to minimize the contrast to the power gain GP, which was calculated using the actual power drawn at the amplifier input. added noise or maximize the power provided at the output of the amplifier: one cannot do both at the same For the available power gain, we first consider Pin,avail. This is the maximum possible power that could be deliv- time. ered to the input of the amplifier, assuming we optimized both the value of the input impedance Zin and the VIII. CONCLUSIONS load impedance Zload while keeping Zs fixed. For simplicity, we take all impedances to be real in our discus- In this review, we have given an introduction to quan- sion. In general, the power drawn at the input of the tum limits for position detection and amplification, lim- 2 ͑ ͒2 its that are tied to fundamental constraints on quantum amplifier is Pin=vinZin/ Zs +Zin . Maximizing this over Zin, we obtain the available input power Pin,avail, noise correlators. We end by emphasizing notable current developments and pointing out future perspectives 2 ͑ ͒ in the field. Pin,avail = vin/4Zs. 7.1 As emphasized, much of our discussion has been di-

The maximum occurs for Zin=Zs. rectly relevant to the measurement of mechanical nan- The output power supplied to the load Pout oresonators, a topic attracting considerable recent atten- =v2 /Z is calculated as before, keeping Z and Z tion. These nanoresonators are typically studied by load load in s ͑ distinct. One has coupling them either to electrical often superconduct- ing͒ circuits or to optical cavities. A key goal is to 2 2 achieve quantum-limited continuous position detection vout Zload P = ͩ ͪ ͑cf. Sec. V. E ͒; current experiments are coming tantaliz- out Z Z + Z load out load ingly close to this limit ͑cf. Table III͒. Although the abil- ͑ ͒ 7.2 ity to follow the nanoresonator’s motion with a precision

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Clerk et al.: Introduction to quantum noise, measurement, … 1205 set by the quantum limit is in principle independent cussed in this review, it can be turned into a linear am- from being at low temperatures, it becomes interesting plifier for a sufficiently weak input signal. An interesting only when the systems are near their ground state; one and important open question is whether such a setup can could then, e.g., monitor the oscillator’s zero-point fluc- reach the quantum limit on linear amplification. tuations ͑cf. Sec. III.B.3͒. Given the comparatively small Both qubit detection and mechanical measurements values of mechanical frequencies ͑mostly less than 1 in electrical circuits would benefit from quantum-limited GHz͒, this often calls for the application of nonequilib- on-chip amplifiers. Such amplifiers are now being devel- rium cooling techniques which exploit back-action to re- oped using the tools of circuit quantum electrodynamics, duce the effective temperature of the mechanical device, employing Josephson junctions or SQUIDs coupled to a technique that has been demonstrated recently in both microwave transmission line cavities ͑Bergeal et al., superconducting circuits and optomechanical setups ͓see 2008; Castellanos-Beltran et al., 2008͒. Such an amplifier Marquardt and Girvin ͑2009͒ for a review͔. has already been used to perform continuous position The ability to perform quantum-limited position de- detection with a measurement imprecision below the tection will in turn open up many new interesting av- SQL level ͑Teufel et al., 2009͒. enues of research. Among the most significant is the ͑ possibility of quantum feedback control Wiseman and ACKNOWLEDGMENTS Milburn, 1993, 1994͒, where one uses the continuously obtained measurement output to tailor the state of the This work was supported in part by NSERC, the Ca- mechanical resonator. The relevant theoretical frame- nadian Institute for Advanced Research ͑CIFAR͒, NSA work is that of quantum conditional evolution and quan- under ARO Contract No. W911NF-05-1-0365, the NSF tum trajectories ͑see, e.g., Brun, 2002; Jacobs and Steck, under Grant No. DMR-0653377 and No. DMR-0603369, 2006͒, where one tracks the state of a measured quan- the David and Lucile Packard Foundation, the W. M. tum system in a particular run of the experiment. Appli- Keck Foundation, and the Alfred P. Sloan Foundation. cation of these ideas has only recently been explored in We acknowledge the support and hospitality of the As- condensed matter contexts ͑Korotkov, 1999, 2001b; pen Center for Physics where part of this work was car- Goan and Milburn, 2001; Goan et al., 2001; Oxtoby et al., ried out. F.M. acknowledges support via DIP, GIF, and 2006, 2008; Bernád et al., 2008͒. Fully understanding the through SFB 631, SFB/TR 12, the Nanosystems Initia- potential of these techniques, as well as differences that tive Munich, and the Emmy-Noether program. S.M.G. occur in condensed matter versus atomic physics con- acknowledges the support of the Centre for Advanced texts, remains an active area of research. 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Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010 Introduction to Quantum Noise, Measurement and Ampliﬁcation: Online Appendices

A.A. Clerk,1 M.H. Devoret,2 S.M. Girvin,3 Florian Marquardt,4 and R.J. Schoelkopf2 1Department of Physics, McGill University, 3600 rue University Montréal,QC Canada H3A 2T8∗ 2Department of Applied Physics, Yale University PO Box 208284, New Haven, CT 06520-8284 3Department of Physics, Yale University PO Box 208120, New Haven, CT 06520-8120 4Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-UniversitätMünchen Theresienstr. 37, D-80333 München,Germany (Dated: July 2, 2009)

Contents 2. Explicit examples 32 3. Op-amp with negative voltage feedback 33 A. Basics of Classical and Quantum Noise 1 1. Classical noise correlators 1 I. Additional Technical Details 34 2. The Wiener-Khinchin Theorem 2 1. Proof of quantum noise constraint 34 3. Square law detectors and classical spectrum analyzers 3 2. Proof that a noiseless detector does not amplify 36 3. Simplifications for a quantum-limited detector 36 B. Quantum Spectrum Analyzers: Further Details 4 4. Derivation of non-equilibrium Langevin equation 37 1. Two-level system as a spectrum analyzer 4 5. Linear-response formulas for a two-port bosonic amplifier37 2. Harmonic oscillator as a spectrum analyzer 5 a. Input and output impedances 38 3. Practical quantum spectrum analyzers 6 b. Voltage gain and reverse current gain 38 a. Filter plus diode 6 6. Details for the two-port bosonic voltage amplifier with b. Filter plus photomultiplier 7 feedback 39 c. Double sideband heterodyne power spectrum 7 References 40 C. Modes, Transmission Lines and Classical Input/Output Theory 8 1. Transmission lines and classical input-output theory 8 Appendix A: Basics of Classical and Quantum Noise 2. Lagrangian, Hamiltonian, and wave modes for a transmission line 10 3. Classical statistical mechanics of a transmission line 11 1. Classical noise correlators 4. Amplification with a transmission line and a negative resistance 13 Consider a classical random voltage signal V (t). The signal is characterized by zero mean hV (t)i = 0, and D. Quantum Modes and Noise of a Transmission Line 15 1. Quantization of a transmission line 15 autocorrelation function 2. Modes and the windowed Fourier transform 16 0 0 3. Quantum noise from a resistor 17 GVV (t − t ) = hV (t)V (t )i (A1)

E. Back Action and Input-Output Theory for Driven whose sign and magnitude tells us whether the voltage Damped Cavities 18 fluctuations at time t and time t0 are correlated, anti- 1. Photon shot noise inside a cavity and back action 19 2. Input-output theory for a driven cavity 20 correlated or statistically independent. We assume that 3. Quantum limited position measurement using a cavity the noise process is stationary (i.e., the statistical proper- detector 24 ties are time translation invariant) so that GVV depends 4. Back-action free single-quadrature detection 28 only on the time difference. If V (t) is Gaussian distributed, then the mean and autocorrelation completely F. Information Theory and Measurement Rate 29 1. Method I 29 specify the statistical properties and the probability dis- 2. Method II 30 tribution. We will assume here that the noise is due to the sum of a very large number of fluctuating charges G. Number Phase Uncertainty 30 so that by the central limit theorem, it is Gaussian dis- H. Using feedback to reach the quantum limit 31 tributed. We also assume that GVV decays (sufficiently 1. Feedback using mirrors 31 rapidly) to zero on some characteristic correlation time scale τc which is finite. The spectral density of the noise as measured by a spectrum analyzer is a measure of the intensity of the ∗Electronic address: [email protected] signal at different frequencies. In order to understand 2 the spectral density of a random signal, it is useful to with a large heat bath at temperature T via some in- define its ‘windowed’ Fourier transform as follows: finitesimal coupling which we will ignore in considering the dynamics. The solution of Hamilton’s equations of Z +T/2 1 iωt motion are VT [ω] = √ dt e V (t), (A2) T −T/2 1 x(t) = x(0) cos(Ωt) + p(0) sin(Ωt) where T is the sampling time. In the limit T τc the MΩ integral is a sum of a large number N ≈ T of random p(t) = p(0) cos(Ωt) − x(0)MΩ sin(Ωt), (A9) τc uncorrelated terms. We can think of the value of the integral as the end point of a random walk in the complex where x(0) and p(0) are the (random) values of the po- plane which starts at the√ origin. Because the distance sition and momentum at time t = 0. It follows that the traveled will scale with T , our choice of normalization position autocorrelation function is makes the statistical properties of V [ω] independent of the sampling time T (for sufficiently large T ). Notice Gxx(t) = hx(t)x(0)i (A10) √ 1 that VT [ω] has the peculiar√ units of volts secs which is = hx(0)x(0)i cos(Ωt) + hp(0)x(0)i sin(Ωt). usually denoted volts/ Hz. MΩ The spectral density (or ‘power spectrum’) of the noise Classically in equilibrium there are no correlations be- is defined to be the ensemble averaged quantity tween position and momentum. Hence the second term 1 2 2 2 vanishes. Using the equipartition theorem MΩ hx i = SVV [ω] ≡ lim h|VT [ω]| i = lim hVT [ω]VT [−ω]i (A3) 2 T →∞ T →∞ 1 2 kBT , we arrive at The second equality follows from the fact that V (t) is k T real valued. The Wiener-Khinchin theorem (derived in B Gxx(t) = 2 cos(Ωt) (A11) Appendix A.2) tells us that the spectral density is equal MΩ to the Fourier transform of the autocorrelation function which leads to the spectral density Z +∞ S [ω] = dt eiωtG (t). (A4) kBT VV VV Sxx[ω] = π [δ(ω − Ω) + δ(ω + Ω)] (A12) −∞ MΩ2 The inverse transform relates the autocorrelation func- which is indeed symmetric in frequency. tion to the power spectrum Z +∞ dω −iωt 2. The Wiener-Khinchin Theorem GVV (t) = e SVV [ω]. (A5) −∞ 2π From the definition of the spectral density in Eqs.(A2- We thus see that a short auto-correlation time implies A3) we have a spectral density which is non-zero over a wide range of frequencies. In the limit of ‘white noise’ Z T Z T 1 0 iω(t−t0) 0 SVV [ω] = dt dt e hV (t)V (t )i 2 T 0 0 GVV (t) = σ δ(t) (A6) Z T Z +2B(t) 1 iωτ the spectrum is flat (independent of frequency) = dt dτ e hV (t + τ/2)V (t − τ/2)i T 0 −2B(t) 2 SVV [ω] = σ (A7) (A13) In the opposite limit of a long autocorrelation time, the where signal is changing slowly so it can only be made up out of a narrow range of frequencies (not necessarily centered B (t) = t if t < T/2 on zero). = T − t if t > T/2. Because V (t) is a real-valued classical variable, it naturally follows that GVV (t) is always real. Since V (t) is not If T greatly exceeds the noise autocorrelation time τc a quantum operator, it commutes with its value at other then it is a good approximation to extend the bound B(t) times and thus, hV (t)V (t0)i = hV (t0)V (t)i. From this it in the second integral to infinity, since the dominant con- follows that GVV (t) is always symmetric in time and the tribution is from small τ. Using time translation invari- power spectrum is always symmetric in frequency ance gives

T +∞ SVV [ω] = SVV [−ω]. (A8) 1 Z Z S [ω] = dt dτ eiωτ hV (τ)V (0)i VV T As a prototypical example of these ideas, let us con- 0 −∞ sider a simple harmonic oscillator of mass M and fre- Z +∞ = dτ eiωτ hV (τ)V (0)i . (A14) quency Ω. The oscillator is maintained in equilibrium −∞ 3

This proves the Wiener-Khinchin theorem stated in 3. Square law detectors and classical spectrum analyzers Eq. (A4). A useful application of these ideas is the following. Now that we understand the basics of classical noise, Suppose that we have a noisy signal V (t) = V¯ + η(t) we can consider how one experimentally measures a clas- which we begin monitoring at time t = 0. The integrated sical noise spectral density. With modern high speed signal up to time t is given by digital sampling techniques it is perfectly feasible to directly measure the random noise signal as a function of Z T I(T ) = dt V (t) (A15) time and then directly compute the autocorrelation func- 0 tion in Eq. (A1). This is typically done by first per- forming an analog-to-digital conversion of the noise sig- and has mean nal, and then numerically computing the autocorrelation hI(T )i = VT.¯ (A16) function. One can then use Eq. (A4) to calculate the noise spectral density via a numerical Fourier transform. Provided that the integration time greatly exceeds the Note that while Eq. (A4) seems to require an ensemble autocorrelation time of the noise, I(T ) is a sum of a large average, in practice this is not explicitly done. Instead, number of uncorrelated random variables. The central one uses a sufficiently long averaging time T (i.e. much limit theorem tells us in this case that I(t) is gaussian longer than the correlation time of the noise) such that distributed even if the signal itself is not. Hence the a single time-average is equivalent to an ensemble aver- probability distribution for I is fully specified by its mean age. This approach of measuring a noise spectral density and its variance directly from its autocorrelation function is most appro- Z T priate for signals at RF frequencies well below 1 MHz. h(∆I)2i = dtdt0 hη(t)η(t0)i. (A17) For microwave signals with frequencies well above 1 0 GHz, a very different approach is usually taken. Here, the From the definition of spectral density above we have the standard route to obtain a noise spectral density involves simple result that the variance of the integrated signal first shifting the signal to a lower intermediate frequency grows linearly in time with proportionality constant given via a technique known as heterodyning (we discuss this by the noise spectral density at zero frequency more in Sec. B.3.c). This intermediate-frequency signal

2 is then sent to a filter which selects a narrow frequency h(∆I) i = SVV [0] T. (A18) range of interest, the so-called ‘resolution bandwidth’. As a simple application, consider the photon shot noise Finally, this filtered signal is sent to a square-law detector of a coherent laser beam. The total number of photons (e.g. a diode), and the resulting output is averaged over detected in time T is a certain time-interval (the inverse of the so-called ‘video bandwidth’). It is this final output which is then taken Z T to be a measure of the noise spectral density. N(T ) = dt N˙ (t). (A19) 0 It helps to put the above into equations. Ignoring for simplicity the initial heterodyning step, let The photo-detection signal N˙ (t) is not gaussian, but rather is a point process, that is, a sequence of delta func- Vf [ω] = f[ω]V [ω] (A24) tions with random Poisson distributed arrival times and be the voltage at the output of the filter and the input ˙ mean photon arrival rate N. Nevertheless at long times of the square law detector. Here, f[ω] is the (ampli- the mean number of detected photons tude) transmission coefficient of the filter and V [ω] is the Fourier transform of the noisy signal we are measuring. hN(T )i = NT˙ (A20) From Eq. (A5) it follows that the output of the square will be large and the photon number distribution will be law detector is proportional to gaussian with variance Z +∞ dω hIi = |f[ω]|2S [ω]. (A25) 2 2π VV h(∆N) i = SN˙ N˙ T. (A21) −∞ Since we know that for a Poisson process the variance is Approximating the narrow band filter centered on fre- 1 equal to the mean quency ±ω0 as 2 h(∆N)2i = hN(T )i, (A22) |f[ω]| = δ(ω − ω0) + δ(ω + ω0) (A26) it follows that the shot noise power spectral density is

˙ 1 SN˙ N˙ (0) = N. (A23) A linear passive filter performs a convolution Vout(t) = R +∞ 0 0 0 −∞ dt F (t − t )Vin(t ) where F is a real-valued (and causal) Since the noise is white this result happens to be valid at function. Hence it follows that f[ω], which is the Fourier trans- all frequencies, but the noise is gaussian distributed only form of F , obeys f[−ω] = f ∗[ω] and hence |f[ω]|2 is symmetric at low frequencies. in frequency. 4 we obtain noise). As a result, for times exceeding the autocorrelation time τc of the noise, the integral will not grow hIi = SVV (−ω0) + SVV (ω0) (A27) linearly with time but rather only as the square root of time, as expected for a random walk. We can now com- showing as expected that the classical square law detector pute the probability measures the symmetrized noise power. We thus have two very different basic approaches for A2 Z t Z t 2 −iω01(τ1−τ2) the measurement of classical noise spectral densities: for pe(t) ≡ |αe| = 2 dτ1dτ2 e F (τ1)F (τ2) ~ 0 0 low RF frequencies, one can directly measure the noise (B4) autocorrelation, whereas for high microwave frequencies, which we expect to grow quadratically for short times one uses a filter and a square law detector. For noise t < τc, but linearly for long times t > τc. Ensemble signals in intermediate frequency ranges, a combination averaging the probability over the random noise yields of different methods is generally used. The whole story becomes even more complicated, as at very high frequen- A2 Z t Z t p¯ (t) = dτ dτ e−iω01(τ1−τ2) hF (τ )F (τ )i cies (e.g. in the far infrared), devices such as the so- e 2 1 2 1 2 ~ 0 0 called ‘Fourier Transform spectrometer’ are in fact based (B5) on a direct measurement of the equivalent of an auto- Introducing the noise spectral density correlation function of the signal. In the infrared, visible Z +∞ and ultraviolet, noise spectrometers use gratings followed iωτ SFF (ω) = dτ e hF (τ)F (0)i, (B6) by a slit acting as a filter. −∞ and utilizing the Fourier transform defined in Eq. (A2) Appendix B: Quantum Spectrum Analyzers: Further Details and the Wiener-Khinchin theorem from Appendix A.2, we find that the probability to be in the excited state 1. Two-level system as a spectrum analyzer indeed increases linearly with time at long times,2

2 In this sub-appendix, we derive the Golden Rule tran- A p¯e(t) = t SFF (−ω01) (B7) sition rates Eqs. (2.6) describing a quantum two-level sys- ~2 tem coupled to a noise source (cf. Sec. II.B). Our deriva- The time derivative of the probability gives the transition tion is somewhat unusual, in that the role of the contin- rate from ground to excited states uum as a noise source is emphasized from the outset. We start by treating the noise F (t) in Eq. (2.5) as being a A2 Γ↑ = SFF (−ω01) (B8) classically noisy variable. We assume that the coupling ~2 A is under our control and can be made small enough Note that we are taking in this last expression the spec- that the noise can be treated in lowest order perturba- tral density on the negative frequency side. If F were a tion theory. We take the state of the two-level system to strictly classical noise source, hF (τ)F (0)i would be real, be and SFF (−ω01) = SFF (+ω01). However, because as we α (t) discuss below F is actually an operator acting on the en- |ψ(t)i = g . (B1) h i αe(t) vironmental degrees of freedom, Fˆ(τ), Fˆ(0) 6= 0 and In the interaction representation, first-order time- SFF (−ω01) 6= SFF (+ω01). dependent perturbation theory gives Another possible experiment is to prepare the two-level system in its excited state and look at the rate of decay Z t into the ground state. The algebra is identical to that i ˆ |ψI(t)i = |ψ(0)i − dτ V (τ)|ψ(0)i. (B2) above except that the sign of the frequency is reversed: ~ 0 2 If we initially prepare the two-level system in its ground A Γ↓ = SFF (+ω01). (B9) state, the amplitude to find it in its excited state at time ~2 t is from Eq. (B2) We now see that our two-level system does indeed act as a quantum spectrum analyzer for the noise. Operationally, iA Z t αe = − dτ he|σˆx(τ)|giF (τ), ~ 0 iA Z t iω01τ = − dτ e F (τ). (B3) 2 Note that for very long times, where there is a significant de- 0 ~ pletion of the probability of being in the initial state, first-order perturbation theory becomes invalid. However, for sufficiently Since the integrand in Eq. (B3) is random, αe is a sum small A, there is a wide range of times τc t 1/Γ for which of a large number of random terms; i.e. its value is the Eq. B7 is valid. Eqs. (2.6a) and (2.6b) then yield well-defined endpoint of a random walk in the complex plane (as dis- rates which can be used in a master equation to describe the full cussed above in defining the spectral density of classical dynamics including long times. 5 we prepare the system either in its ground state or in its It it is important to keep in mind that our expressions excited state, weakly couple it to the noise source, and for the transition rates are only valid if the autocorrela- after an appropriate interval of time (satisfying the above tion time of our noise is much shorter that the typical inequalities) simply measure whether the system is now time we are interested in; this typical time is simply the in its excited state or ground state. Repeating this pro- inverse of the transition rate. The requirement of a short tocol over and over again, we can find the probability autocorrelation time in turn implies that our noise source of making a transition, and thereby infer the rate and must have a large bandwidth (i.e. there must be large hence the noise spectral density at positive and nega- number of available photon frequencies in the vacuum) tive frequencies. Naively one imagines that a spectrom- and must not be coupled too strongly to our system. This eters measures the noise spectrum by extracting a small is true despite the fact that our final expressions for the amount of the signal energy from the noise source and transition rates only depend on the spectral density at analyzes it. This is not the case however. There must the transition frequency (a consequence of energy con- be energy flowing in both directions if the noise is to be servation). fully characterized. One standard model for the continuum is an infinite We now rigorously treat the quantity Fˆ(τ) as a quan- collection of harmonic oscillators. The electromagnetic tum Heisenberg operator which acts in the Hilbert space continuum in the hydrogen atom case mentioned above is of the noise source. The previous derivation is unchanged a prototypical example. The vacuum electric field noise (the ordering of Fˆ(τ1)Fˆ(τ2) having been chosen cor- coupling to the hydrogen atom has an extremely short rectly in anticipation of the quantum treatment), and autocorrelation time because the range of mode frequen- Eqs. (2.6a,2.6b) are still valid provided that we interpret cies ωα (over which the dipole matrix element coupling the angular brackets in Eq. (B5,B6) as representing a the atom to the mode electric field E~α is significant) is quantum expectation value (evaluated in the absence of extremely large, ranging from many times smaller than the coupling to the spectrometer): the transition frequency to many times larger. Thus, the autocorrelation time of the vacuum electric field noise is Z +∞ −15 iωτ X ˆ ˆ considerably less than 10 s, whereas the decay time of SFF (ω) = dτ e ραα hα|F (τ)|γihγ|F (0)|αi. −9 −∞ the hydrogen 2p state is about 10 s. Hence the inequal- α,γ ities needed for the validity of our expressions are very (B10) easily satisfied. Here, we have assumed a stationary situation, where the density matrix ρ of the noise source is diagonal in the energy eigenbasis (in the absence of the coupling to 2. Harmonic oscillator as a spectrum analyzer the spectrometer). However, we do not necessarily assume that it is given by the equilibrium expression. This We now provide more details on the system described yields the standard quantum mechanical expression for in Sec. II.B, where a harmonic oscillator acts as a spec- the spectral density: trometer of quantum noise. We start with the coupling +∞ Hamiltonian givein in Eq. (2.9). In analogy to the TLS Z i iωτ X (α−γ )τ ˆ 2 SFF (ω) = dτ e ραα e ~ |hα|F |γi| spectrometer, noise in Fˆ at the oscillator frequency Ω −∞ α,γ can cause transitions between its eigenstates. We as- X sume both that A is small, and that our noise source = 2π ρ |hα|Fˆ|γi|2δ( − − ω)(B11). ~ αα γ α ~ has a short autocorrelation time, so we may again use α,γ perturbation theory to derive rates for these transitions. Substituting this expression into Eqs. (2.6a,2.6b), we de- There is a rate for increasing the number of quanta in rive the familiar Fermi Golden Rule expressions for the the oscillator by one, taking a state |ni to |n + 1i: two transition rates. 2 A 2 In standard courses, one is not normally taught that Γn→n+1 = (n + 1)xZPF SFF [−Ω] ≡ (n + 1)Γ↑ ~2 the transition rate of a discrete state into a continuum (B12) as described by Fermi’s Golden Rule can (and indeed As expected, this rate involves the noise at −Ω, as en- should!) be viewed as resulting from the continuum act- ergy is being absorbed from the noise source. Similarly, ing as a quantum noise source which causes the am- there is a rate for decreasing the number of quanta in the plitudes of the different components of the wave func- oscillator by one: tion to undergo random walks. The derivation presented here hopefully provides a motivation for this interpreta- 2 A 2 tion. In particular, thinking of the perturbation (i.e. the Γn→n−1 = 2 nxZPF SFF [Ω] ≡ nΓ↓ (B13) coupling to the continuum) as quantum noise with a ~ small but finite autocorrelation time (inversely related This rate involves the noise at +Ω, as energy is being to the bandwidth of the continuum) neatly explains why emitted to the noise source. the transition probability increases quadratically for very Given these transition rates, we may immediately write short times, but linearly for very long times. a simple master equation for the probability pn(t) that 6 there are n quanta in the oscillator: a harmonic oscillator) in which we can separately measure the up and down transition rates between states d pn = [nΓ↑pn−1 + (n + 1)Γ↓pn+1] differing by some precise energy ~ω > 0 given by the dt frequency of interest. The down transition rate tells us − [nΓ↓ + (n + 1)Γ↑] pn (B14) the noise spectral density at frequency +ω and the up The first two terms describe transitions into the state |ni transition rate tells us the noise spectral density at −ω. from the states |n + 1i and |n − 1i, and hence increase While we have already discussed experimental implemen- pn. In contrast, the last two terms describe transitions tation of these ideas using two-level systems and oscilla- out of the state |ni to the states |n + 1i and |n − 1i, and tors, similar schemes have been implemented in other sys- hence decrease pn. The stationary state of the oscillator tems. A number of recent experiments have made use of d is given by solving Eq. (B14) for pn = 0, yielding: superconductor-insulator-superconductor junctions (Bil- dt langeon et al., 2006; Deblock et al., 2003; Onac et al., −n Ω/(kBTeff ) − Ω/(kBTeff ) pn = e ~ 1 − e ~ (B15) 2006) to measure quantum noise, as the current-voltage characteristics of such junctions are very sensitive to where the effective temperature Teff [Ω] is defined in the absorption or emission of energy (so-called photon- Eq. (2.8). Eq. (B15) describes a thermal equilibrium assisted transport processes). It has also been suggested distribution of the oscillator, with an effective oscillator that tunneling of flux in a SQUID can be used to measure temperature Teff [Ω] determined by the quantum noise quantum noise (Amin and Averin, 2008). spectrum of Fˆ. This is the same effective temperature In this subsection, we discuss additional methods for that emerged in our discussion of the TLS spectrum an- the detection of quantum noise. Recall from Sec. A.3 that alyzer. As we have seen, if the noise source is in thermal one of the most basic classical noise spectrum analyzers equilibrium at a temperature Teq, then Teff [Ω] = Teq. consists of a linear narrow band filter and a square law In the more general case where the noise source is not detector such as a diode. In what follows, we will consider in thermal equilibrium, Teff only serves to characterize a simplified quantum treatment of such a device where we the asymmetry of the quantum noise, and will vary with do not explicitly model a diode, but instead focus on the frequency 3. energy of the filter circuit. We then turn to various noise We can learn more about the quantum noise spectrum detection schemes making use of a photomultiplier. We of Fˆ by also looking at the dynamics of the oscillator. will show that depending on the detection scheme used, In particular, as the average energy hEi of the oscillator one can measure either the symmetrized quantum noise P∞ 1 spectral density S¯[ω], or the non-symmetrized spectral is just given by hE(t)i = n=0 ~Ω n + 2 pn(t), we can use the master equation Eq. (B14) to derive an equa- density S[ω]. tion for its time dependence. One thus finds Eq. (2.10). By demanding dhEi/dt = 0 in this equation, we find that the combination of damping and heating effects a. Filter plus diode causes the energy to reach a steady state mean value of hEi = P/γ. This implies that the finite ground state Using the simple treatment we gave of a harmonic os- energy hEi = ~Ω/2 of the oscillator is determined via the cillator as a quantum spectrum analyzer in Sec. B.2, one balance between the ‘heating’ by the zero-point fluctua- can attempt to provide a quantum treatment of the clas- tions of the environment (described by the symmetrized sical ‘filter plus diode’ spectrum analyzer discussed in correlator at T = 0) and the dissipation. It is possible to Sec. A.3. This approach is due to Lesovik and Loosen take an alternative but equally correct viewpoint, where (1997) and Gavish et al. (2000). The analysis starts by only the deviation hδEi = hEi − ~Ω/2 from the ground modeling the spectrum analyzer’s resonant filter circuit state energy is considered. Its evolution equation as a harmonic oscillator of frequency Ω weakly coupled d to some equilibrium dissipative bath. The oscillator thus hδEi = hδEi(Γ − Γ ) + Γ Ω (B16) dt ↑ ↓ ↑~ has an intrinsic damping rate γ0 Ω, and is initially at a finite temperature Teq. One then drives this damped only contains a decay term at T = 0, leading to hδEi → 0. oscillator (i.e. the filter circuit) with the noisy quantum force Fˆ(t) whose spectrum at frequency Ω is to be measured. 3. Practical quantum spectrum analyzers In the classical ‘filter plus diode’ spectrum analyzer, As we have seen, a ‘quantum spectrum analyzer’ can the output of the filter circuit was sent to a square law in principle be constructed from a two level system (or detector, whose time-averaged output was then taken as the measured spectral density. To simplify the analysis, we can instead consider how the noise changes the average energy of the resonant filter circuit, taking this 3 Note that the effective temperature can become negative if the quantity as a proxy for the output of the diode. Sure noise source prefers emitting energy versus absorbing it; in the enough, if we subject the filter circuit to purely classical present case, that would lead to an instability. noise, it would cause the average energy of the circuit 7 hEi to increase an amount directly proportional to the neff → 0), the above condition can never be satisfied, and classical spectrum SFF [Ω]. We now consider hEi in the thus it is impossible to ignore the damping effect of the case of a quantum noise source, and ask how it relates to noise source on the filter circuit. In the zero point limit, the quantum noise spectral density SFF [Ω]. this damping effect (i.e. second term in Eq. (B20)) will The quantum case is straightforward to analyze using always be greater than or equal to the expected heating the approach of Sec. B.2. Unlike the classical case, the effect of the noise (i.e. first term in Eq. (B20)). noise will both lead to additional fluctuations of the filter circuit and increase its damping rate by an amount γ (c.f. Eq. (2.12)). To make things quantitative, we let neq b. Filter plus photomultiplier denote the average number of quanta in the filter circuit prior to coupling to Fˆ(t), i.e. We now turn to quantum spectrum analyzers involving a square law detector we can accurately model– a photo- 1 multiplier. As a first example of such a system, consider a n = , (B17) eq photomultiplier with a narrow band filter placed in front exp ~Ω − 1 kB Teq of it. The mean photocurrent is then given by

+∞ and let neff represent the Bose-Einstein factor associated Z hIi = dω |f[ω]|2r[ω]S [ω], (B22) with the effective temperature Teff [Ω] of the noise source VV −∞ Fˆ(t), where f is the filter (amplitude) transmission function 1 defined previously and r[ω] is the response of the pho- neff = . (B18) exp ~Ω − 1 todetector at frequency ω, and SVV represents the elec- kB Teff [Ω] tric field spectral density incident upon the photodetector. Naively one thinks of the photomultiplier as a square One then finds (Gavish et al., 2000; Lesovik and Loosen, law detector with the square of the electric field repre- 1997): senting the optical power. However, according to the γ Glauber theory of (ideal) photo-detection (Gardiner and ∆hEi = Ω · (n − n ) (B19) ~ γ + γ eff eq Zoller, 2000; Glauber, 2006; Walls and Milburn, 1994), 0 photocurrent is produced if, and only if, a photon is This equation has an extremely simple interpretation: absorbed by the detector, liberating the initial photo- the first term results from the expected heating effect electron. Glauber describes this in terms of normal or- of the noise, while the second term results from the dering of the photon operators in the electric field auto- noise source having increased the circuit’s damping by correlation function. In our language of noise power at an amount γ. Re-expressing this result in terms of the positive and negative frequencies, this requirement be- symmetric and anti-symmetric in frequency parts of the comes simply that r[ω] vanishes for ω > 0. Approximat- quantum noise spectral density SFF [Ω], we have: ing the narrow band filter centered on frequency ±ω0 as in Eq. (A26), we obtain ¯ 1 SFF (Ω) − neq + 2 (SFF [Ω] − SFF [−Ω]) ∆hEi = hIi = r[−ω0]SVV[−ω0] (B23) 2m (γ0 + γ) (B20) which shows that this particular realization of a quantum spectrometer only measures electric field spectral density We see that ∆hEi is in general not simply proportional at negative frequencies since the photomultiplier never to the symmetrized noise S¯FF [Ω]. Thus, the ‘filter plus emits energy into the noise source. Also one does not diode’ spectrum analyzer does not simply measure the see in the output any ‘vacuum noise’ and so the output symmetrized quantum noise spectral density. We stress (ideally) vanishes as it should at zero temperature. Of that there is nothing particularly quantum about this re- course real photomultipliers suffer from imperfect quan- sult. The extra term on the RHS of Eq. (B20) simply tum efficiencies and have non-zero dark current. Note reflects the fact that coupling the noise source to the fil- that we have assumed here that there are no additional ter circuit could change the damping of this circuit; this fluctuations associated with the filter circuit. Our re- could easily happen in a completely classical setting. As sult thus coincides with what we found in the previous long as this additional damping effect is minimal, the subsection for the ‘filter plus diode’ spectrum analyzer second term in Eq. (B20) will be minimal, and our spec- (c.f. Eq. (B20), in the limit where the filter circuit is ini- trum analyzer will (to a good approximation) measure tially at zero temperature (i.e. neq = 0). the symmetrized noise. Quantitatively, this requires:

neﬀ neq. (B21) c. Double sideband heterodyne power spectrum

We now see where quantum mechanics enters: if the noise At RF and microwave frequencies, practical spectrome- to be measured is close to being zero point noise (i.e. ters often contain heterodyne stages which mix the initial 8 frequency down to a lower frequency ωIF (possibly in the We will see in the following that the physical oscillations classical regime). Consider a system with a mixer and lo- of this signal in a transmission line are precisely the sinu- cal oscillator at frequency ωLO that mixes both the upper soidal oscillations of a simple harmonic oscillator. Com- sideband input at ωu = ωLO +ωIF and the lower sideband parison of Eq. (C1) with x(t) = x0 cos ωt+(p0/Mω) sin ωt input at ωl = ωLO − ωIF down to frequency ωIF. This shows that we can identify the quadrature amplitude X can be achieved by having a Hamiltonian with a 3-wave with the coordinate of this oscillator and thus the quadra- mixing term which (in the rotating wave approximation) ture amplitude Y is proportional to the momentum con- is given by jugate to X. Quantum mechanically, X and Y become operators Xˆ and Yˆ which do not commute. Thus their † † † † † † V = λ[âIFaˆlaˆLO +a ÎFaˆl aˆLO] + λ[âIFaûaˆLO +a ÎFaûaˆLO] quantum fluctuations obey the Heisenberg uncertainty (B24) relation. The interpretation of this term is that of a Raman pro- Ordinarily (e.g., in the absence of squeezing), the phase cess. Notice that there are two energy conserving pro- choice defining the two quadratures is arbitrary and so cesses that can create an IF photon which could then their vacuum (i.e. zero-point) fluctuations are equal activate the photodetector. First, one can absorb an LO photon and emit two photons, one at the IF and one at XZPF = YZPF. (C2) the lower sideband. The second possibility is to absorb Thus the canonical commutation relation becomes an upper sideband photon and create IF and LO photons. ˆ ˆ 2 Thus we expect from this that the power in the IF chan- [X, Y ] = iXZPF. (C3) nel detected by a photomultiplier would be proportional We will see that the fact that X and Y are canoni- to the noise power in the following way cally conjugate has profound implications both classically I ∝ S[+ω ] + S[−ω ] (B25) and quantum mechanically. In particular, the action of l u any circuit element (beam splitter, attenuator, amplifier, since creation of an IF photon involves the signal source etc.) must preserve the Poisson bracket (or in the quan- either absorbing a lower sideband photon from the mixer tum case, the commutator) between the signal quadra- or the signal source emitting an upper sideband photon tures. This places strong constraints on the properties into the mixer. In the limit of small IF frequency this of these circuit elements and in particular, forces every expression would reduce to the symmetrized noise power amplifier to add noise to the signal.

I ∝ S[+ωLO] + S[−ωLO] = 2S¯[ωLO] (B26) 1. Transmission lines and classical input-output theory which is the same as for a ‘classical’ spectrum analyzer with a square law detector (c.f. Appendix A.3). For equi- We begin by considering a coaxial transmission line modeled as a perfectly conducting wire with inductance librium noise spectral density from a resistance R0 derived in Appendix D we would then have per unit length of ` and capacitance to ground per unit length c as shown in Fig. 1. If the voltage at position x at time t is V (x, t), then the charge density is q(x, t) = SVV[ω] + SVV[−ω] = 2R0~|ω|[2nB(~|ω|) + 1], (B27) cV (x, t). By charge conservation the current I and the Assuming our spectrum analyzer has high input charge density are related by the continuity equation impedance so that it does not load the noise source, this ∂ q + ∂ I = 0. (C4) voltage spectrum will determine the output signal of the t x analyzer. This symmetrized quantity does not vanish at The constitutive relation (essentially Newton’s law) gives zero temperature and the output contains the vacuum the acceleration of the charges noise from the input. This vacuum noise has been seen in experiment. (Schoelkopf et al., 1997) `∂tI = −∂xV. (C5) We can decouple Eqs. (C4) and (C5) by introducing left and right propagating modes Appendix C: Modes, Transmission Lines and Classical Input/Output Theory V (x, t) = [V → + V ←] (C6)

1 → ← In this appendix we introduce a number of important I(x, t) = [V − V ] (C7) Zc classical concepts about electromagnetic signals which p are essential to understand before moving on to the study where Zc ≡ `/c is called the characteristic impedance of their quantum analogs. A signal at carrier frequency of the line. In terms of the left and right propagating ω can be described in terms of its amplitude and phase modes, Eqs. (C4) and C5 become or equivalently in terms of its two quadrature amplitudes → → vp∂xV + ∂tV = 0 (C8) ← ← s(t) = X cos(ωt) + Y sin(ωt). (C1) vp∂xV − ∂tV = 0 (C9) 9 √ where vp ≡ 1/ `c is the wave phase velocity. These indicating that the transmission line acts as a simple re- equations have solutions which propagate by uniform sistor which, instead of dissipating energy by Joule heat- translation without changing shape since the line is dis- ing, carries the energy away from the system as propa- persionless gating waves. The fact that the line can dissipate energy despite containing only purely reactive elements is a con- → x V (x, t) = Vout(t − ) (C10) sequence of its infinite extent. One must be careful with v p the order of limits, taking the length to infinity before ← x V (x, t) = Vin(t + ), (C11) allowing time to go to infinity. In this way the outgoing vp waves never reach the far end of the transmission line and where Vin and Vout are arbitrary functions of their argu- reflect back. Since this is a conservative Hamiltonian sys- ments. For an infinite transmission line, Vout and Vin are tem, we will be able to quantize these waves and make a completely independent. However for the case of a semi- quantum theory of resistors (Caldeira and Leggett, 1983) infinite line terminated at x = 0 (say) by some system S, in Appendix D. The net power flow carried to the right these two solutions are not independent, but rather re- by the line is lated by the boundary condition imposed by the system. We have 1 2 2 P = [Vout(t) − Vin(t)]. (C16) Zc V (x = 0, t) = [Vout(t) + Vin(t)] (C12) 1 The fact that the transmission line presents a dissipa- I(x = 0, t) = [V (t) − V (t)], (C13) Z out in tive impedance to the system means that it causes damp- c ing of the system. It also however opens up the possibility from which we may derive of controlling the system via the input field which partially determines the voltage driving the system. From V (t) = V (t) + Z I(x = 0, t). (C14) out in c this point of view it is convenient to eliminate the output field by writing the voltage as 0 d x V (x = 0, t) = 2Vin(t) + ZcI(x = 0, t). (C17) V (x,t) a) As we will discuss in more detail below, the first term I(x,t) Zc , vp drives the system and the second damps it. From V(x,t) Eq. (C14) we see that measurement of the outgoing field V (x,t) can be used to determine the current I(x = 0, t) injected L LL by the system into the line and hence to infer the system b) dynamics that results from the input drive field. CCC As a simple example, consider the system consisting of an LC resonator shown in Fig. (1 c). This can be viewed a as a simple harmonic oscillator whose coordinate Q is the charge on the capacitor plate (on the side connected to I ˙ Vout L0). The current I(x = 0, t) = Q plays the role of the c) velocity of the oscillator. The equation of motion for the V oscillator is readily obtained from

Vin + + Q = C0[−V (x = 0 , t) − L0I˙(x = 0 , t)]. (C18) FIG. 1 a) Coaxial transmission line, indicating voltages and Using Eq. (C17) we obtain a harmonic oscillator damped currents as defined in the main text. b) Lumped element representation of a transmission line with capacitance per unit by the transmission line and driven by the incoming length c = C/a and inductance per unit length ` = L/a. c) waves Discrete LC resonator terminating a transmission line. ¨ 2 ˙ 2 Q = −Ω0Q − γQ − Vin(t), (C19) L0 √ If the system under study is just an open circuit so 2 where the resonant frequency is Ω0 ≡ 1/ L0C0. Note that I(x = 0, t) = 0, then Vout = Vin, meaning that the outgoing wave is simply the result of the incoming wave that the term ZcI(x = 0, t) in Eq. (C17) results in the reflecting from the open circuit termination. In general linear viscous damping rate γ ≡ Zc/L0. however, there is an additional outgoing wave radiated If we solve the equation of motion of the oscillator, we by the current I that is injected by the system dynamics can predict the outgoing field. In the present instance of into the line. In the absence of an incoming wave we have a simple oscillator we have a particular example of the general case where the system responds linearly to the V (x = 0, t) = ZcI(x = 0, t), (C15) input field. We can characterize any such system by a 10 complex, frequency dependent impedance Z[ω] defined system and none is reflected. The other two limits of in- by terest are open circuit termination with Z = ∞ for which r = +1 and short circuit termination Z = 0 for which V (x = 0, ω) Z[ω] = − . (C20) r = −1. I(x = 0, ω) Finally, if the system also contains an active device which has energy being pumped into it from a separate Note the peculiar minus sign which results from our def- external source, it may under the right conditions be de- inition of positive current flowing to the right (out of the scribed by an effective negative resistance Re Z[ω] < 0 system and into the transmission line). Using Eqs. (C12, over a certain frequency range. Eq. (C22) then gives C13) and Eq. (C20) we have |r| ≥ 1, implying |Vout| > |Vin|. Our system will thus act like the one-port amplifier discussed in Sec. V.D: it am- V [ω] = r[ω]V [ω], (C21) out in plifies signals incident upon it. We will discuss this idea where the reflection coefficient r is determined by the of negative resistance further in Sec. C.4; a physical real- impedance mismatch between the system and the line ization is provided by the two-port reflection parametric and is given by the well known result amplifier discussed in Appendix V.C.

Z[ω] − Zc r[ω] = . (C22) 2. Lagrangian, Hamiltonian, and wave modes for a Z[ω] + Zc transmission line If the system is constructed from purely reactive (i.e. lossless) components, then Z[ω] is purely imaginary and Prior to moving on to the case of quantum noise it the reflection coefficient obeys |r| = 1 which is consistent is useful to review the classical statistical mechanics of with Eq. (C16) and the energy conservation requirement transmission lines. To do this we need to write down of no net power flow into the lossless system. For exam- the Lagrangian and then determine the canonical mo- ple, for the series LC oscillator we have been considering, menta and the Hamiltonian. Very conveniently, the sys- we have tem is simply a large collection of harmonic oscillators (the normal modes) and hence can be readily quantized. 1 Z[ω] = + jωL0, (C23) This representation of a physical resistor is essentially the jωC0 one used by Caldeira and Leggett (Caldeira and Leggett, 1983) in their seminal studies of the effects of dissipation where, to make contact with the usual electrical engi- on tunneling. The only difference between this model neering sign conventions, we have used j = −i. If the and the vacuum fluctuations in free space is that the rel- damping γ of the oscillator induced by coupling it to the ativistic bosons travel in one dimension and do not carry transmission line is small, the quality factor of the reso- a polarization label. This changes the density of states as nance will be high and we need only consider frequencies √ a function of frequency, but has no other essential effect. near the resonance frequency Ω ≡ 1/ L C where the 0 0 0 It is convenient to define a flux variable (Devoret, 1997) impedance has a zero. In this case we may approximate 2 Z t ϕ(x, t) ≡ dτ V (x, τ), (C26) Z[ω] ≈ 2 [Ω0 − ω] = 2jL0(ω − Ω0) (C24) jC0Ω0 −∞ which yields for the reflection coefficient where V (x, t) = ∂tϕ(x, t) is the local voltage on the transmission line at position x and time t. Each segment of ω − Ω + jγ/2 r[ω] = 0 (C25) the line of length dx has inductance ` dx and the voltage ω − Ω0 − jγ/2 drop along it is −dx ∂x∂tϕ(x, t). The flux through this inductance is thus −dx ∂xϕ(x, t) and the local value of showing that indeed |r| = 1 and that the phase of the the current is given by the constitutive equation reflected signal winds by 2π upon passing through the resonance. 4 1 I(x, t) = − ∂ ϕ(x, t). (C27) Turning to the more general case where the system ` x also contains lossy elements, one finds that Z[ω] is no longer purely imaginary, but has a real part satisfying The Lagrangian for the system is Re Z[ω] > 0. This in turn implies via Eq. (C22) that Z ∞ Z ∞ |r| < 1. In the special case of impedance matching c 2 1 2 Lg ≡ dx L(x, t) = dx (∂tϕ) − (∂xϕ) , Z[ω] = Zc, all the incident power is dissipated in the 0 0 2 2` (C28) The Euler-Lagrange equation for this Lagrangian is simply the wave equation 4 For the case of resonant transmission through a symmetric cav- 2 2 2 ity, the phase shift only winds by π. vp∂xϕ − ∂t ϕ = 0. (C29) 11

The momentum conjugate to ϕ(x) is simply the charge where for simplicity we have taken the ﬁelds to obey pe- density riodic boundary conditions on a length L. Thus we have (in a form which anticipates the full quantum theory) δL q(x, t) ≡ = c∂ ϕ = cV (x, t) (C30) δ∂ ϕ t 1 X t H = (A∗A + A A∗) . (C34) 2 k k k k and so the Hamiltonian is given by k Z The classical equation of motion (C29) yields the simple 1 2 1 2 H = dx q + (∂xϕ) . (C31) result 2c 2`

We know from our previous results that the charge ∂tAk = −iωkAk. (C35) density consists of left and right moving solutions of ar- Thus bitrary ﬁxed shape. For example we might have for the right moving case q(x, t) r q(t−x/v ) = α cos[k(x−v t)]+β sin[k(x−v t)]. (C32) c X p k p k p = eikx A (0)e−iωkt + A∗ (0)e+iωkt (C36) 2L k −k A confusing point is that since q is real valued, we see k r that it necessarily contains both eikx and e−ikx terms c X h i = A (0)e+i(kx−ωkt) + A∗(0)e−i(kx−ωkt) . even if it is only right moving. Note however that for 2L k k k k > 0 and a right mover, the eikx is associated with the (C37) positive frequency term e−iωkt while the e−ikx term is associated with the negative frequency term e+iωkt where We see that for k > 0 (k < 0) the wave is right (left) ω ≡ v |k|. For left movers the opposite holds. We can k p moving, and that for right movers the eikx term is asso- appreciate this better if we deﬁne ciated with positive frequency and the e−ikx term is associated with negative frequency. We will return to this Z ( r 2 ) 1 −ikx 1 k Ak ≡ √ dx e √ q(x, t) − i ϕ(x, t) in the quantum case where positive (negative) frequency L 2c 2` will refer to the destruction (creation) of a photon. Note (C33) that the right and left moving voltages are given by

r 1 X h i V → = A (0)e+i(kx−ωkt) + A∗(0)e−i(kx−ωkt) (C38) 2Lc k k k>0 r 1 X h i V ← = A (0)e+i(kx−ωkt) + A∗(0)e−i(kx−ωkt) (C39) 2Lc k k k<0 The voltage spectral density for the right moving waves is thus

2π X S→ [ω] = {hA A∗iδ(ω − ω ) + hA∗A iδ(ω + ω )} (C40) VV 2Lc k k k k k k k>0 The left moving spectral density has the same expression but k < 0. Using Eq. (C16), the above results lead to a net power flow (averaged over one cycle) within a frequency band defined by a pass filter G[ω] of

vp X P = P → − P ← = sgn(k)[G[ω ]hA A∗i + G[−ω ]hA∗A i] . (C41) 2L k k k k k k k

3. Classical statistical mechanics of a transmission line thermal equilibrium at temperature T . Since each mode k is a simple harmonic oscillator we have from Eq. (C34) and the equipartition theorem Now that we have the Hamiltonian, we can consider ∗ the classical statistical mechanics of a transmission line in hAkAki = kBT. (C42) 12

Using this, we see from Eq. (C40) that the right moving a) in which a real resistor has in parallel a random cur- voltage signal has a simple white noise power spectrum. rent generator representing thermal noise fluctuations of Using Eq. (C41) we have for the right moving power in the electrons in the resistor. This is the essence of the a bandwidth B (in Hz rather than radians/sec) the very fluctuation dissipation theorem. simple result In order to make a quantitative analysis in terms of the power flowing in the two lines, voltage is not the best vp X P → = hG[ω ]A∗A + G[−ω ]A A∗i variable to use since we are dealing with more than one 2L k k k k k k k>0 value of line impedance. Rather we define incoming and k T Z +∞ dω outgoing fields via = B G[ω] 2 2π −∞ 1 ← Ain = √ Vc (C44) = kBTB. (C43) Zc 1 → where we have used the fact mentioned in connection Aout = √ Vc (C45) with Eq. (A26) and the discussion of square law detec- Zc tors that all passive filter functions are symmetric in fre- 1 → Bin = √ V (C46) quency. R R One of the basic laws of statistical mechanics is Kirch- 1 ← Bout = √ V (C47) hoff’s law stating that the ability of a hot object to emit R R radiation is proportional to its ability to absorb. This follows from very general thermodynamic arguments con- Normalizing by the square root of the impedance allows cerning the thermal equilibrium of an object with its ra- us to write the power flowing to the right in each line in diation environment and it means that the best possible the simple form emitter is the black body. In electrical circuits this princi- 2 2 ple is simply a form of the fluctuation dissipation theorem Pc = (Aout) − (Ain) (C48) 2 2 which states that the electrical thermal noise produced PR = (Bin) − (Bout) (C49) by a circuit element is proportional to the dissipation it introduces into the circuit. Consider the example of a ter- The out fields are related to the in fields by the s matrix minating resistor at the end of a transmission line. If the A A resistance R is matched to the characteristic impedance out = s in (C50) Zc of a transmission line, the terminating resistor acts Bout Bin as a black body because it absorbs 100% of the power incident upon it. If the resistor is held at temperature T Requiring continuity of the voltage and current at the it will bring the transmission line modes into equilibrium interface between the two transmission lines, we can solve at the same temperature (at least for the case where the for the scattering matrix s: transmission line has finite length). The rate at which the +r t equilibrium is established will depend on the impedance s = (C51) t −r mismatch between the resistor and the line, but the final temperature will not. where A good way to understand the fluctuation-dissipation theorem is to represent the resistor R which is terminat- R − Z r = c (C52) ing the Zc line in terms of a second semi-infinite trans- R + Zc mission line of impedance R as shown in Fig. (2). First √ 2 RZc consider the case when the R line is not yet connected to t = . (C53) R + Zc the Zc line. Then according to Eq. (C22), the open termi- 2 nation at the end of the Zc line has reflectivity |r| = 1 Note that |r|2 + |t|2 = 1 as required by energy conserva- so that it does not dissipate any energy. Additionally tion and that s is unitary with det (s) = −1. By moving of course, this termination does not transmit any sig- the point at which the phase of the Bin and Bout fields nals from the R line into the Zc. However when the are determined one-quarter wavelength to the left, we two lines are connected the reflectivity becomes less than can put s into different standard form unity meaning that incoming signals on the Zc line see a source of dissipation R which partially absorbs them. +r it s0 = (C54) The absorbed signals are not turned into heat as in a it +r true resistor but are partially transmitted into the R line which is entirely equivalent. Having opened up this port which has det (s0) = +1. for energy to escape from the Zc system, we have also As mentioned above, the energy absorbed from the Zc allowed noise energy (thermal or quantum) from the R line by the resistor R is not turned into heat as in a line to be transmitted into the Zc line. This is com- true resistor but is is simply transmitted into the R line, pletely equivalent to the effective circuit shown in Fig. (3 which is entirely equivalent. Kirchhoff’s law is now easy 13

R V Z voltage noise at the open termination of a semi-infinite R Vc c transmission line with Zc = R. For an open termination V V → = V ← so that the voltage at the end is given by R Vc V = 2V ← = 2V → (C59) FIG. 2 (Color online) Semi-infinite transmission line of impedance Zc terminated by a resistor R which is represented and thus using Eqs. (C40) and (C42) we find as a second semi-infinite transmission line. → SVV = 4SVV = 2RkBT (C60) which is equivalent to Eq. (C57). to understand. The energy absorbed from the Zc line by R, and the energy transmitted into it by thermal fluctua- 4. Amplification with a transmission line and a negative tions in the R line are both proportional to the absorption resistance coefficient 4RZ We close our discussion of transmission lines by fur- A = 1 − |r|2 = |t|2 = c . (C55) (R + Z )2 ther expanding upon the idea mentioned at the end of c App. C.1 that one can view a one-port amplifier as a transmission line terminated by an effective negative resistance. The discussion here will be very general: we will IN R explore what can be learned about amplification by simply extending the results we have obtained on transmission lines to the case of an effective negative resistance. Our general discussion will not address the important is- S sues of how one achieves an effective negative resistance II R VN SVV over some appreciable frequency range: for such ques- tions, one must focus on a specific physical realization, such as the parametric amplifier discussed in Sec. V.C. We start by noting that for the case −Zc < R < 0 the a) b) power gain G is given by G = |r|2 > 1, (C61) 0 FIG. 3 Equivalent circuits for noisy resistors. and the s matrix introduced in Eq. (C54) becomes √ √ G ± G − 1 s0 = − √ √ (C62) ± G − 1 G The requirement that the transmission line Zc come to equilibrium with the resistor allows us to readily compute where the sign choice depends on the branch cut chosen the spectral density of current fluctuations of the random in the analytic continuation of the off-diagonal elements. current source shown in Fig. (3 a). The power dissipated This transformation is clearly no longer unitary (because in Zc by the current source attached to R is there is no energy conservation since we are ignoring the Z +∞ 2 work done by the amplifier power supply). Note however dω R Zc 0 P = SII [ω] 2 that we still have det (s ) = +1. It turns out that this −∞ 2π (R + Zc) naive analytic continuation of the results from positive to (C56) negative resistance is not strictly correct. As we will show in the following, we must be more careful than we have For the special case R = Z we can equate this to the c been so far in order to insure that the transformation right moving power P → in Eq. (C43) because left moving from the in fields to the out fields must be canonical. waves in the Z line are not reflected and hence cannot c In order to understand the canonical nature of the contribute to the right moving power. Requiring P = transformation between input and output modes, it is P → yields the classical Nyquist result for the current necessary to delve more deeply into the fact that the noise of a resistor two quadrature amplitudes of a mode are canonically 2 conjugate. Following the complex amplitudes defined S [ω] = k T (C57) II R B in Eqs. (C44-C47), let us define a vector of real-valued or in the electrical engineering convention quadrature amplitudes for the incoming and outgoing fields 4  in   out  SII [ω] + SII [−ω] = kBT. (C58) XA XA R in out in  XB  out XB ~q =  in  , ~q =  out . (C63) We can derive the equivalent expression for the volt-  YB   YB  in out age noise of a resistor (see Fig. 3 b) by considering the YA YA 14

The Poisson brackets amongst the diﬀerent quadrature the phases of the various input and output waves are amplitudes is given by measured.

in in {qi , qj } ∝ Jij, (C64) Another allowed form of the scattering matrix is: or equivalently the quantum commutators are

[qin, qin] = iX2 J , (C65) i j ZPF ij  + cosh θ + sinh θ 0 0  + sinh θ + cosh θ 0 0 where s˜0 = −   .  0 0 + cosh θ − sinh θ   0 0 0 +1  0 0 − sinh θ + cosh θ 0 0 +1 0 J ≡   . (C66) √ (C72)  0 −1 0 0  If one takes cosh θ = G, this scattering matrix is −1 0 0 0 essentially the canonically correct formulation of the negative-resistance scattering matrix we tried to write in In order for the transformation to be canonical, the same Eq. (C62). Note that the off-diagonal terms have changed Poisson bracket or commutator relations must hold for sign for the Y quadrature relative to the naive expression the outgoing field amplitudes in Eq. (C62) (corresponding to the other possible analytic continuation choice). This is necessary to satisfy [qout, qout] = iX2 J . (C67) i j ZPF ij the symplecticity condition and hence make the transfor- 0 In the case of a non-linear device these relations would mation canonical. The scattering matrixs ˜ can describe amplification. Unlike the beam splitter scattering ma- apply to the small fluctuations in the input and output 0 0 fields around the steady state solution. Assuming a linear trixs ˜ above,s ˜ is not unitary (even though dets ˜ = 1). device (or linearization around the steady state solution) Unitarity would correspond to power conservation. Here, we can define a 4 × 4 real-valued scattering matrixs ˜ in power is not conserved, as we are not explicitly tracking analogy to the 2 × 2 complex-valued scattering matrix s the power source supplying our active system. in Eq. (C51) which relates the output fields to the input fields The form of the negative-resistance amplifier scattering matrixs ˜0 confirms many of the general statements we out in qi =s ˜ijqj . (C68) made about phase-preserving amplification in Sec. V.B. First, note that the requirement of finite gain G > 1 and Eq. (C67) puts a powerful constraint on on thes ˜ matrix, phase preservation makes all the diagonal elements ofs ˜0 namely that it must be symplectic. That is,s ˜ and its (i.e. cosh θ ) equal. We see that to amplify the A mode, transpose must obey it is impossible to avoid coupling to the B mode (via the sinh θ term) because of the requirement of symplecticity. sJ˜ s˜T = J. (C69) We thus see that it is impossible classically or quantum mechanically to build a linear phase-preserving amplifier From this it follows that whose only effect is to amplify the desired signal. The presence of the sinh θ term above means that the output dets ˜ = ±1. (C70) signal is always contaminated by amplified noise from at least one other degree of freedom (in this case the B This in turn immediately implies Liouville’s theorem mode). If the thermal or quantum noise in A and B that Hamiltonian evolution preserves phase space volume are equal in magnitude (and uncorrelated), then in the (since dets ˜ is the Jacobian of the transformation which limit of large gain where cosh θ ≈ sinh θ, the output noise propagates the amplitudes forward in time). (referred to the input) will be doubled. This is true for Let us further assume that the device is phase preserv- both classical thermal noise and quantum vacuum noise. ing, that is that the gain or attenuation is the same for both quadratures. One form for thes ˜ matrix consistent with all of the above requirements is The negative resistance model of an amplifier here gives us another way to think about the noise added by  + cos θ sin θ 0 0  an amplifier: crudely speaking, we can view it as being sin θ − cos θ 0 0 directly analogous to the fluctuation-dissipation theorem s˜ =   . (C71)  0 0 − cos θ sin θ  simply continued to the case of negative dissipation. Just 0 0 sin θ + cos θ as dissipation can occur only when we open up a new channel and thus we bring in new fluctuations, so ampli- This simply corresponds to a beam splitter and is the fication can occur only when there is coupling to an ad- equivalent of Eq. (C51) with r = cos θ. As mentioned in ditional channel. Without this it is impossible to satisfy connection with Eq. (C51), the precise form of the scat- the requirement that the amplifier perform a canonical tering matrix depends on the choice of planes at which transformation. 15

Appendix D: Quantum Modes and Noise of a Transmission In a similar fashion, we have: Line r Z ∞ dω ωZ h i Vˆ ←(t) = ~ c ˆb←[ω]e−iωt + h.c. (D9) 1. Quantization of a transmission line 0 2π 2 r vp X Recall from Eq. (C30) and the discussion in Appendix ˆb←[ω] ≡ 2π ˆb δ(ω − ω ) (D10) L k k C that the momentum conjugate to the transmission line k<0 ﬂux variable ϕ(x, t) is the local charge density q(x, t). ˆ→ ˆ← Hence in order to quantize the transmission line modes One can easily verify that among the b [ω], b [ω] opera- we simply promote these two physical quantities to quan- tors and their conjugates, the only non-zero commutators tum operators obeying the commutation relation are given by:

0 0 † † [ˆq(x), ϕˆ(x )] = −i~δ(x − x ) (D1) ˆb→[ω], ˆb→[ω0] = ˆb←[ω], ˆb←[ω0] = 2πδ(ω − ω0) from which it follows that the mode amplitudes defined (D11) in Eq. (C33) become quantum operators obeying We have taken the continuum limit L → ∞ here, allowing us to change sums on k to integrals. We have thus ob- ˆ ˆ† [Ak0 , Ak] = ~ωkδkk0 (D2) tained the description of a quantum transmission line in and we may identify the usual raising and lowering oper- terms of left and right-moving frequency resolved modes, ators by as used in our discussion of amplifiers in Sec. VI (see Eqs. 6.2). Note that if the right-moving modes are fur- ˆ p ˆ Ak = ~ωk bk (D3) ther taken to be in thermal equilibrium, one finds (again, in the continuum limit): ˆ where bk destroys a photon in mode k. The quantum † form of the Hamiltonian in Eq. (C34) is thus ˆ→ ˆ→ 0 0 b [ω] b [ω ] = 2πδ(ω − ω )nB(~ω) (D12a) X 1 ˆ†ˆ † H = ~ωk bkbk + . (D4) ˆ→ ˆ→ 0 0 2 b [ω] b [ω ] = 2πδ(ω − ω ) [1 + nB( ω)] k ~ For the quantum case the thermal equilibrium expression . (D12b) then becomes We are typically interested in a relatively narrow band ˆ† ˆ hAkAki = ~ωknB(~ωk), (D5) of frequencies centered on some characteristic drive or resonance frequency Ω0. In this case, it is useful to work which reduces to Eq. (C42) in the classical limit ω ~ k in the time-domain, in a frame rotating at Ω0. Fourier kBT . transforming 5 Eqs. (D8) and (D10), one finds: We have seen previously in Eqs. (C6) that the volt- r age fluctuations on a transmission line can be resolved vp X ˆb→(t) = e−i(ωk−Ω0)tˆb (0), (D13a) into right and left moving waves which are functions of a L k combined space-time argument k>0 r vp X x x ˆb←(t) = e−i(ωk−Ω0)tˆb (0). (D13b) V (x, t) = V →(t − ) + V ←(t + ). (D6) L k vp vp k<0 Thus in an infinite transmission line, specifying V → ev- These represent temporal right and left moving modes. erywhere in space at t = 0 determines its value for all Note that the normalization factor in Eqs. (D13) has been times. Conversely specifying V → at x = 0 for all times chosen so that the right moving photon flux at x = 0 and fully specifies the field at all spatial points. In prepa- time t is given by ration for our study of the quantum version of input- hN˙ i = hˆb†→(t)ˆb→(t)i (D14) output theory in Appendix E, it is convenient to extend Eqs. (C38-C39) to the quantum case (x = 0): In the same rotating frame, and within the approxima- r tion that all relevant frequencies are near Ω0, Eq. (D7) 1 X p h i Vˆ →(t) = ω ˆb e−iωkt + h.c. becomes simply: 2Lc ~ k k k>0 r h i r ˆ → ~Ω0Zc ˆ→ ˆ†→ Z ∞ dω ωZ h i V (t) ≈ b (t) + b (t) (D15) = ~ c ˆb→[ω]e−iωt + h.c. (D7) 2 0 2π 2 In the second line, we have defined: r 5 As in the main text, we use in this appendix a convention which ˆ→ vp X ˆ b [ω] ≡ 2π bkδ(ω − ωk) (D8) differs from the one commonly used in quantum optics:a ˆ[ω] = L R +∞ +iωt † † R +∞ +iωt † k>0 −∞ dt e aˆ(t) anda ˆ [ω] = [â[−ω]] = −∞ dt e aˆ (t). 16

We have already seen that using classical statistical For our present purposes where we are interested in mechanics, the voltage noise in equilibrium is white. just a single narrow frequency range centered on Ω0, a The corresponding analysis of the temporal modes using convenient windowed transform kernel for smoothing the Eqs. (D13) shows that the quantum commutator obeys quantum noise is simply a box of width ∆t representing the ﬁnite integration time of our detector. In the frame ˆ→ ˆ†→ 0 0 [b (t), b (t )] = δ(t − t ). (D16) rotating at Ω0 we can deﬁne

In deriving this result, we have converted summations Z tj+1 ˆ→ 1 ˆ→ over mode index to integrals over frequency. Further, Bj = √ dτ b (τ) (D18) ∆t because (for finite time resolution at least) the integral is tj dominated by frequencies near +Ω0 we can, within the where tj = j(∆t) denotes the time of arrival of the jth Markov (Wigner Weisskopf) approximation, extend the temporal mode at the point x = 0. Recall that ˆb→ has lower limit of frequency integration to minus infinity and a photon flux normalization and so Bˆ→ is dimensionless. thus arrive at a delta function in time. If we further take j From Eq. (D16) we see that these smoothed operators the right moving modes to be in thermal equilibrium, obey the usual bosonic commutation relations then we may similarly approximate: ˆ→ ˆ†→ ˆ†→ 0 ˆ→ 0 [Bj , B ] = δjk. (D19) hb (t )b (t)i = nB(~Ω0)δ(t − t ) (D17a) k ˆ→ ˆ†→ 0 0 hb (t)b (t )i = [1 + nB( Ω0)] δ(t − t ). (D17b) † ~ The state Bj |0i has a single photon occupying basis ˆ → mode j, which is centered in frequency space at Ω0 and in Equations (D15) to (D17b) indicate that V (t) can be time space on the interval j∆t < t < (j + 1)∆t (i.e. this treated as the quantum operator equivalent of white temporal mode passes the point x = 0 during the jth noise; a similar line of reasoning applies mutatis mutan- time interval.) This basis mode is much like a note in a dis to the left moving modes. We stress that these re- musical score: it has a certain specified pitch and occurs sults rely crucially on our assumption that we are dealing at a specified time for a specified duration. Just as we with a relatively narrow band of frequencies in the vicin- can play notes of different frequencies simultaneously, we ity of Ω0; the resulting approximations we have made can define other temporal modes on the same time in- are known as the Markov approximation. As one can al- terval and they will be mutually orthogonal provided the ready see from the form of Eqs. (D7,D9), and as will be angular frequency spacing is a multiple of 2π/∆t. The discussed further, the actual spectral density of vacuum result is a set of modes Bm,p labeled by both a frequency noise on a transmission line is not white, but is linear in index m and a time index p. p labels the time interval as frequency. The approximation made in Eq. (D16) treats before, while m labels the angular frequency: it as a constant within the narrow band of frequencies of interest. If the range of frequencies of importance is 2π ω = Ω + m (D20) large then the Markov approximation is not applicable. m 0 ∆t The result is, as illustrated in Fig. (4), a complete lat- 2. Modes and the windowed Fourier transform tice of possible modes tiling the frequency-time phase space, each occupying area 2π corresponding to the time- While delta function correlations can make the quan- frequency uncertainty principle. tum noise relatively easy to deal with in both the time We can form other modes of arbitrary shapes centered and frequency domain, it is sometimes the case that it on frequency Ω0 by means of linear superposition of our is easier to deal with a ‘smoothed’ noise variable. The basis modes (as long as they are smooth on the time scale introduction of an ultraviolet cutoff regulates the math- ∆t). Let us define ematical singularities in the noise operators evaluated at X ˆ→ equal times and is physically sensible because every real Ψ = ψjBj . (D21) measurement apparatus has finite time resolution. A sec- j ond motivation is that real spectrum analyzers output a time varying signal which represents the noise power in This is also a canonical bosonic mode operator obeying a certain frequency interval (the ‘resolution bandwidth’) † averaged over a certain time interval (the inverse ‘video [Ψ, Ψ ] = 1 (D22) bandwidth’). The mathematical tool of choice for dealing provided that the coefficients obey the normalization con- with such situations in which time and frequency both dition appear is the ‘windowed Fourier transform’. The windowed transform uses a kernel which is centered on some X 2 |ψj| = 1. (D23) frequency window and some time interval. By summa- j tion over all frequency and time windows it is possible to invert the transformation. The reader is directed to We might for example want to describe a mode which is (Mallat, 1999) for the mathematical details. centered at a slightly higher frequency Ω0 + δΩ (obeying 17

Δt ω area = 2π amplifiers that ‘the noise temperature corresponds to a mode occupancy of X photons’. This simply means that the photon flux per unit bandwidth is X. Equivalently m the flux in bandwidth B is Bmp 2π/Δt X N˙ = (B∆t) = XB. (D27) ∆t

ω=Ω0 t The interpretation of this is that X photons in a temporal mode of duration ∆t pass the origin in time ∆t. Each 1 mode has bandwidth ∼ ∆t and so there are B∆t inde- -m B-mp pendent temporal modes in bandwidth B all occupying the same time interval ∆t. The longer is ∆t the longer it takes a given mode to pass the origin, but the more such modes fit into the frequency window. p As an illustration of these ideas, consider the following FIG. 4 (Color online) Schematic figure indicating how the elementary question: What is the mode occupancy of a various modes defined by the windowed Fourier transform tile laser beam of power P and hence photon flux N˙ = P ? Ω0 the time-frequency plane. Each individual cell corresponds to We cannot answer this without knowing the coherence~ a different mode, and has an area 2π. time or equivalently the bandwidth. The output of a good laser is like that of a radio frequency oscillator–it has essentially no amplitude fluctuations. The frequency (δΩ)(∆t) << 1) and spread out over a large time interval is nominally set by the physical properties of the oscillator, but there is nothing to pin the phase which conse- T centered at time T0. This could be given for example by quently undergoes slow diffusion due to unavoidable noise perturbations. This leads to a finite phase coherence time 2 (j∆t−T0) τ and corresponding frequency spread 1/τ of the laser − 2 −i(δΩ)(j∆t) ψj = N e 4T e (D24) spectrum. (A laser beam differs from a thermal source where N is the appropriate normalization constant. that has been filtered to have the same spectrum in that The state having n photons in the mode is simply it has smaller amplitude fluctuations.) Thus we expect that the mode occupancy is X = Nτ˙ . A convenient ap- 1 n √ Ψ† |0i. (D25) proximate description in terms of temporal modes is to n! take the window interval to be ∆t = τ. Within the jth interval we take the phase to be a (random) constant ϕj The concept of ‘wave function of the photon’ is fraught so that (up to an unimportant normalization constant) with dangers. In the very special case where we re- we have the coherent state strict attention solely to the subspace of single photon √ iϕj ˆ†→ Y Xe Bj Fock states, we can usefully think of the amplitudes {ψj} e |0i (D28) as the ‘wave function of the photon’ (Cohen-Tannoudji j et al., 1989) since it tells us about the spatial mode which which obeys is excited. In the general case however it is essential to √ ˆ→ iϕk keep in mind that the transmission line is a collection of hBk i = Xe (D29) coupled LC oscillators with an infinite number of degrees and of freedom. Let us simplify the argument by considering ˆ†→ ˆ→ a single LC oscillator. We can perfectly well write a wave hBk Bk i = X. (D30) function for the system as a function of the coordinate (say the charge q on the capacitor). The ground state wave function χ0(q) is a gaussian function of the coor- 3. Quantum noise from a resistor dinate. The one photon state created by Ψ† has a wave function χ1(q) ∼ qχ0(q) proportional to the coordinate Let us consider the quantum equivalent to Eq. (C60), times the same gaussian. In the general case χ is a wave SVV = 2RkBT , for the case of a semi-infinite transmis- functional of the charge distribution q(x) over the entire sion line with open termination, representing a resistor. transmission line. From Eq. (C27) we see that the proper boundary con- Using Eq. (D17a) we have dition for the ϕ field is ∂xϕ(0, t) = ∂xϕ(L, t) = 0. (We have temporarily made the transmission line have a large ˆ†→ ˆ→ but finite length L.) The normal mode expansion that hBj Bk i = nB(~Ω0)δjk (D26) satisfies these boundary conditions is independent of our choice of the coarse-graining time win- r ∞ 2 X dow ∆t. This result allows us to give meaning to the ϕ(x, t) = ϕ (t) cos(k x), (D31) L n n phrase one often hears bandied about in descriptions of n=1 18

πn where ϕn is the normal coordinate and kn ≡ L . Sub- Again, the step function tells us that the resistor can only stitution of this form into the Lagrangian and carrying absorb energy, not emit it, at zero temperature. out the spatial integration yields a set of independent If we use the engineering convention and add the noise harmonic oscillators representing the normal modes. at positive and negative frequencies we obtain ∞ X c 2 1 2 2 ~ω Lg = ϕ˙ n − knϕn . (D32) SVV[ω] + SVV[−ω] = 2Zc~ω coth (D39) 2 2` 2kBT n=1 From this we can ﬁnd the momentum operatorp ˆ canon- for the symmetric part of the noise, which appears in the n quantum ﬂuctuation-dissipation theorem (cf. Eq. (2.16)). ically conjugate to the coordinate operatorϕ ˆn and quantize the system to obtain an expression for the operator The antisymmetric part of the noise is simply representing the voltage at the end of the transmission S [ω] − S [−ω] = 2Z ω, (D40) line in terms of the mode creation and destruction oper- VV VV c~ ators yielding ∞ r ˆ X ~Ωn ˆ† ˆ V = i(b − bn). (D33) SVV[ω] − SVV[−ω] ~ω Lc n = tanh . (D41) n=1 SVV[ω] + SVV[−ω] 2kBT

The spectral density of voltage ﬂuctuations is then found This quantum treatment can also be applied to any to be arbitrary dissipative network (Burkhard et al., 2004; De- ∞ voret, 1997). If we have a more complex circuit con- 2π X ~Ωn SVV[ω] = nB( Ωn)δ(ω + Ωn) taining capacitors and inductors, then in all of the above L c ~ n=1 expressions, Zc should be replaced by Re Z[ω] where Z[ω]

+[nB(~Ωn) + 1]δ(ω − Ωn) , (D34) is the complex impedance presented by the circuit. In the above we have explicitly quantized the stand- where nB(~ω) is the Bose occupancy factor for a photon ing wave modes of a ﬁnite length transmission line. We with energy ~ω. Taking the limit L → ∞ and converting could instead have used the running waves of an inﬁnite the summation to an integral yields line and recognized that, as the in classical treatment in Eq. (C59), the left and right movers are not independent. SVV(ω) = 2Zc~|ω| nB(~|ω|)Θ(−ω)+[nB(~|ω|)+1]Θ(ω) , The open boundary condition at the termination requires (D35) V ← = V → and hence b→ = b←. We then obtain where Θ is the step function. We see immediately that at zero temperature there is no noise at negative frequen- → SVV [ω] = 4SVV [ω] (D42) cies because energy can not be extracted from zero-point motion. However there remains noise at positive frequen- and from the quantum analog of Eq. (C40) we have cies indicating that the vacuum is capable of absorbing energy from another quantum system. The voltage spec- 4~|ω| SVV [ω] = {Θ(ω)(nB + 1) + Θ(−ω)nB} tral density at both zero and non-zero temperature is 2cvp plotted in Fig. (1). = 2Z |ω| {Θ(ω)(n + 1) + Θ(−ω)n } Eq. (D35) for this ‘two-sided’ spectral density of a re- c~ B B sistor can be rewritten in a more compact form (D43)

2Zc~ω in agreement with Eq. (D35). SVV[ω] = , (D36) 1 − e−~ω/kBT which reduces to the more familiar expressions in various Appendix E: Back Action and Input-Output Theory for limits. For example, in the classical limit kBT ~ω the Driven Damped Cavities spectral density is equal to the Johnson noise result6 A high Q cavity whose resonance frequency can be SVV[ω] = 2ZckBT, (D37) parametrically controlled by an external source can act in agreement with Eq. (C60). In the quantum limit it as a very simple quantum amplifier, encoding informa- reduces to tion about the external source in the phase and amplitude of the output of the driven cavity. For example, in an optical cavity, one of the mirrors could be move- SVV[ω] = 2Zc~ωΘ(ω). (D38) able and the external source could be a force acting on that mirror. This defines the very active field of optome- chanics, which also deals with microwave cavities cou- 6 Note again that in the engineering convention this would be pled to nanomechanical systems and other related setups SVV[ω] = 4ZckBT . (Arcizet et al., 2006; Brown et al., 2007; Gigan et al., 19

2006; Harris et al., 2007; Höhberger-Metzger and Kar- where the first term is the ‘classical part’ of the mode am- rai, 2004; Marquardt et al., 2007, 2006; Meystre et al., plitude ψ(t) =ae ¯ −iωLt determined by the strength of the 1985; Schliesser et al., 2006; Teufel et al., 2008; Thomp- drive field, the damping of the cavity and the detuning son et al., 2008; Wilson-Rae et al., 2007). In the case of a ∆, and d is the quantum part. By definition, microwave cavity containing a qubit, the state-dependent polarizability of the qubit acts as a source which shifts aˆ|ψi = ψ|ψi (E5) the frequency of the cavity (Blais et al., 2004; Schuster so the coherent state is annihilated by dˆ: et al., 2005; Wallraff et al., 2004). The dephasing of a qubit in a microwave cavity and dˆ|ψi = 0. (E6) the fluctuations in the radiation pressure in an optical cavity both depend on the quantum noise in the number That is, in terms of the operator dˆ, the coherent state of photons inside the cavity. We here use a simple equa- looks like the undriven quantum ground state . The dis- tion of motion method to exactly solve for this quantum placement transformation in Eq. (E4) is canonical since noise in the perturbative limit where the dynamics of the [â, aˆ†] = 1 ⇒ [d,ˆ dˆ†] = 1. (E7) qubit or mirror degree of freedom has only a weak back action effect on the cavity. Substituting the displacement transformation into In the following, we first give a basic discussion of Eq. (E3) and using Eq. (E6) yields the cavity field noise spectrum, deferring the detailed ˆ ˆ† microscopic derivation to subsequent subsections. We Snn(t) =n ¯hd(t)d (0)i, (E8) then provide a review of the input-output theory for wheren ¯ = |a¯|2 is the mean cavity photon number. If we driven cavities, and employ this theory to analyze the set the cavity energy damping rate to be κ, such that the important example of a dispersive position measurement, amplitude damping rate is κ/2, then the undriven state where we demonstrate how the standard quantum limit obeys can be reached. Finally, we analyze an example where † +i∆t − κ |t| a modified dispersive scheme is used to detect only one hdˆ(t)dˆ (0)i = e e 2 . (E9) quadrature of a harmonic oscillator’s motion, such that this quadrature does not feel any back-action. This expression will be justified formally in the subsequent subsection, after introducing input-output theory. We thus arrive at the very simple result 1. Photon shot noise inside a cavity and back action i∆t− κ |t| Snn(t) =ne ¯ 2 . (E10) Consider a degree of freedomz ˆ coupled parametrically The power spectrum of the noise is, via the Wiener- with strength A to the cavity oscillator Khinchin theorem (Appendix A.2), simply the Fourier transform of the autocorrelation function given in Hˆ = ω (1 + Azˆ) [â†aˆ − haˆ†aî] (E1) int ~ c Eq. (E10) where following Eq. (3.12), we have taken A to be dimen- Z +∞ κ sionless, and usez ˆ to denote the dimensionless system iωt Snn[ω] = dt e Snn(t) =n ¯ 2 2 . variable that we wish to probe. For example,z ˆ could −∞ (ω + ∆) + (κ/2) represent the dimensionless position of a mechanical os- (E11) cillator As can be seen in Fig. 5a, for positive detuning ∆ = ω − ω > 0, i.e. for a drive that is blue-detuned with xˆ L c zˆ ≡ . (E2) respect to the cavity, the noise peaks at negative ω. This xZPF means that the noise tends to pump energy into the degree of freedomz ˆ (i.e. it contributes negative damping). We have subtracted the haˆ†aî term so that the mean For negative detuning the noise peaks at positive ω cor- force on the degree of freedom is zero. To obtain the full responding to the cavity absorbing energy fromz ˆ. Basi- Hamiltonian, we would have to add the cavity damping cally, the interaction withz ˆ (three wave mixing) tries to and driving terms, as well as the Hamiltonian governing Raman scatter the drive photons into the high density of the intrinsic dynamics of the systemz ˆ. From Eq. (3.18) states at the cavity frequency. If this is uphill in energy, we know that the back action noise force acting onz ˆ is thenz ˆ is cooled. proportional to the quantum fluctuations in the number As discussed in Sec. B.2 (c.f. Eq. (2.8)), at each fre- of photonsn ˆ =a ˆ†aˆ in the cavity, quency ω, we can use detailed balance to assign the noise † † † 2 Snn(t) = haˆ (t)â(t)â (0)â(0)i − haˆ (t)â(t)i . (E3) an effective temperature Teff [ω]:

Snn[ω] For the case of continuous wave driving at frequency = e~ω/kB Teff [ω] ⇔ ωL = ωc + ∆ detuned by ∆ from the resonance, the Snn[−ω] cavity is in a coherent state |ψi obeying ~ω kBTeﬀ [ω] ≡ (E12) h Snn[ω] i −iωLt ˆ log aˆ(t) = e [¯a + d(t)] (E4) Snn[−ω] 20

(a) As can be seen in Fig. 5, the asymmetry in the noise (b) changes sign with detuning, which causes the effective 4 4 temperature to change sign. First we discuss the case of a positive Teff , where this mechanism can be used to laser cool an oscillating me- 0 chanical cantilever, provided Teff is lower than the in- 2 trinsic equilibrium temperature of the cantilever. (Ar- cizet et al., 2006; Brown et al., 2007; Gigan et al., 2006; -4 Harris et al., 2007; Höhberger-Metzger and Karrai, 2004; 0 Marquardt et al., 2007; Schliesser et al., 2006; Thompson -6 06-6 0 6 et al., 2008; Wilson-Rae et al., 2007). A simple classical Noise power Frequency argument helps us understand this cooling effect. Sup- (c) pose that the moveable mirror is at the right hand end 3 of a cavity being driven below the resonance frequency. If the mirror moves to the right, the resonance frequency will fall and the number of photons in the cavity will rise. There will be a time delay however to fill the cavity and 2 so the extra radiation pressure will not be fully effective - in doing work on the mirror. During the return part of the oscillation as the mirror moves back to the left, the time delay in emptying the cavity will cause the mirror to have to do extra work against the radiation pressure. At 1 the end of the cycle it ends up having done net positive work on the light field and hence is cooled. The effect can therefore be understood as being due to the introduction of some extra optomechanical damping. The signs reverse (and Teff becomes negative) if the

Noise temperature 0 0 1 2 3 4 5 6 cavity is driven above resonance, and consequently the Frequency cantilever motion is heated up. In the absence of intrinsic mechanical losses, negative values of the effective FIG. 5 (Color online) (a) Noise spectrum of the photon num- temperature indicate a dynamical instability of the can- ber in a driven cavity as a function of frequency when the tilever (or population inversion in the case of a qubit), cavity drive frequency is detuned from the cavity resonance where the amplitude of motion grows until it is finally by ∆ = +3κ (left peak) and ∆ = −3κ (right peak). (b) Ef- stabilized by nonlinear effects. This can be interpreted fective temperature Teff of the low frequency noise, ω → 0, as a function of the detuning ∆ of the drive from the cavity as negative damping introduced by the optomechanical resonance. (c) Frequency-dependence of the effective noise coupling and can be used to create parametric amplifica- temperature, for different values of the detuning. tion of mechanical forces acting on the oscillator. Finally, we mention that cooling towards the quantum ground state of a mechanical oscillator (where phonon numbers become much less than one), is only possible or equivalently (Marquardt et al., 2007; Wilson-Rae et al., 2007) in the “far-detuned regime”, where −∆ = ω κ (in contrast Snn[ω] − Snn[−ω] = tanh(β~ω/2). (E13) to the ω κ regime discussed above). Snn[ω] + Snn[−ω] Ifz ˆ is the coordinate of a harmonic oscillator of frequency ω (or some non-conserved observable of a qubit with level 2. Input-output theory for a driven cavity splitting ω), then that system will acquire a temperature Teff [ω] in the absence of coupling to any other environ- The results from the previous section can be more for- ment. In particular, if the characteristic oscillation fre- mally and rigorously derived in a full quantum theory quency of the systemz ˆ is much smaller than κ, then we of a cavity driven by an external coherent source. The have the simple result theory relating the drive, the cavity and the outgoing 1 2 S [ω] − S [−ω] waves radiated by the cavity is known as input-output = lim nn nn theory and the classical description was presented in Ap- kBTeff ω→0+ ~ω Snn[ω] + Snn[−ω] pendix C. The present quantum discussion closely fol- d ln S [ω] = 2 nn lows standard references on the subject (Walls and Mil- d~ω burn, 1994; Yurke, 1984; Yurke and Denker, 1984). The 1 −4∆ crucial feature that distinguishes such an approach from = . (E14) ~ ∆2 + (κ/2)2 many other treatments of quantum-dissipative systems 21 is the goal of keeping the bath modes instead of trac- Note that the most general linear coupling to the bath ˆ† † ˆ ing them out. This is obviously necessary for the situa- modes would include terms of the form bqaˆ and bqa but tions we have in mind, where the output field emanating these are neglected within the rotating wave approxima- from the cavity contains the information acquired during tion because in the interaction representation they os- a measurement of the system coupled to the cavity. As cillate at high frequencies and have little effect on the we learned from the classical treatment, we can elimi- dynamics. nate the outgoing waves in favor of a damping term for The Heisenberg equation of motion (EOM) for the the system. However we can recover the solution for the bath variables is outgoing modes completely from the solution of the equa- i tion of motion of the damped system being driven by the ˆ˙ ˆ ˆ ˆ ∗ bq = [H, bq] = −iωqbq + fq aˆ (E20) incoming waves. ~ In order to drive the cavity we must partially open one We see that this is simply the EOM of a harmonic oscil- of its ports which exposes the cavity both to the external lator driven by a forcing term due to the motion of the drive and to the vacuum noise outside which permits en- cavity degree of freedom. Since this is a linear system, ergy in the cavity to leak out into the surrounding bath. the EOM can be solved exactly. Let t0 < t be a time in We will formally separate the degrees of freedom into in- the distant past before any wave packet launched at the ternal cavity modes and external bath modes. Strictly cavity has reached it. The solution of Eq. (E20) is speaking, once the port is open, these modes are not distinct and we only have ‘the modes of the universe’ (Gea- Z t ˆ −iωq(t−t0)ˆ −iωq(t−τ) ∗ Banacloche et al., 1990a,b; Lang et al., 1973). However bq(t) = e bq(t0) + dτ e fq aˆ(τ). t0 for high Q cavities, the distinction is well-defined and we (E21) can model the decay of the cavity in terms of a spon- The first term is simply the free evolution of the bath taneous emission process in which an internal boson is while the second represents the waves radiated by the destroyed and an external bath boson is created. We cavity into the bath. assume a single-sided cavity. For a high Q cavity, this The EOM for the cavity mode is physics is accurately captured in the following Hamilto- nian ˙ i X ˆ aˆ = [Hˆsys, aˆ] − fqbq. (E22) ~ q Hˆ = Hˆsys + Hˆbath + Hînt. (E15)

The bath Hamiltonian is Substituting Eq. (E21) into the last term above yields X X ˆ X ˆ†ˆ ˆ −iωq(t−t0)ˆ Hbath = ~ωqbqbq (E16) fqbq = fqe bq(t0) q q q Z t X 2 −i(ωq−ωc)(t−τ) +iωc(τ−t) where q labels the quantum numbers of the independent + |fq| dτ e [e aˆ(τ)], (E23) harmonic oscillator bath modes obeying q t0

ˆ ˆ† where the last term in square brackets is a slowly varying [bq, bq0 ] = δq,q0 . (E17) function of τ. To simplify our result, we note that if Note that since the bath terminates at the system, there the cavity system were a simple harmonic oscillator of is no translational invariance, the normal modes are frequency ωc then the decay rate from the n = 1 single standing not running waves, and the quantum numbers photon excited state to the n = 0 ground state would be q are not necessarily wave vectors. given by the following Fermi Golden Rule expression The coupling Hamiltonian is (within the rotating wave X 2 approximation) κ(ωc) = 2π |fq| δ(ωc − ωq). (E24) q X h i Hˆ = −i f aˆ†ˆb − f ∗ˆb†aˆ . (E18) int ~ q q q q From this it follows that q Z +∞ dν X For the moment we will leave the system (cavity) Hamil- κ(ω + ν)e−iν(t−τ) = |f |2e−i(ωq−ωc)(t−τ). 2π c q tonian to be completely general, specifying only that it −∞ q consists of a single degree of freedom (i.e. we concentrate (E25) on only a single resonance of the cavity with frequency We now make the Markov approximation which assumes ωc) obeying the usual bosonic commutation relation that κ(ν) = κ is a constant over the range of frequencies relevant to the cavity so that Eq. (E25) may be repre- † [ˆa, aˆ ] = 1. (E19) sented as

X 2 −i(ωq−ωc)(t−τ) (N.B. this does not imply that it is a harmonic oscilla- |fq| e = κδ(t − τ). (E26) tor. We will consider both linear and non-linear cavities.) q 22

Using the bath modes as non-propagating while the cavity flies along the bath (taken to be 1D) at a velocity v. The Z x0 1 system then only interacts briefly with the local mode dx δ(x − x0) = (E27) −∞ 2 positioned at x = vt before moving on and interacting with the next local bath mode. We will elaborate on this we obtain for the cavity EOM view further at the end of this subsection. i κ The expression for the power Pin (energy per time) ˙ X −iωq(t−t0)ˆ aˆ = [Hˆsys, aˆ] − aˆ − fqe bq(t0). (E28) impinging on the cavity depends on the normalization ~ 2 ˆ q chosen in our definition of bin. It can be obtained, for example, by imagining the bath modes ˆb to live on a one- The second term came from the part of the bath motion q dimensional waveguide with propagation velocity v and representing the wave radiated by the cavity and, within length L (using periodic boundary conditions). In that the Markov approximation, has become a simple linear case we have to sum over all photons to get the average damping term for the cavity mode. Note the important power flowing through a cross-section of the waveguide, factor of 2. The amplitude decays at half the rate of the D E P ˆ†ˆ intensity (the energy decay rate κ). Pin = q ~ωq(vp/L) bqbq . Inserting the definition for ˆ Within the spirit of the Markov approximation it is bin, Eq. (E31), the expression for the input power carried p 2 further convenient to treat f ≡ |fq| as a constant and by a monochromatic beam at frequency ω is define the density of states (also taken to be a constant) by Dˆ† ˆ E Pin(t) = ~ω bin(t)bin(t) (E33) X ρ = δ(ω − ω ) (E29) c q Note that this has the correct dimensions due to our q ˆ √ choice of normalization for bin (with dimensions ω). In so that the Golden Rule rate becomes the general case, an integration over frequencies is needed (as will be discussed further below). An analogous for- κ = 2πf 2ρ. (E30) mula holds for the power radiated by the cavity, to be discussed now. ˆ We can now define the so-called ‘input mode’ The output mode bout(t) is radiated into the bath and evolves freely after the system interacts with ˆb (t). If 1 X in ˆb (t) ≡ √ e−iωq(t−t0)ˆb (t ) . (E31) the cavity did not respond at all, then the output mode in 2πρ q 0 q would simply be the input mode reflected off the cavity mirror. If the mirror is partially transparent then For the case of a transmission line treated in Appendix the output mode will also contain waves radiated by the ˆ→ D, this coincides with the field b moving towards the cavity (which is itself being driven by the input mode cavity [see Eq. (D13a)]. We finally have for the cavity partially transmitted into the cavity through the mirror) EOM and hence contains information about the internal dy- √ namics of the cavity. To analyze this output field, let ˙ i ˆ κ ˆ aˆ = [Hsys, aˆ] − aˆ − κ bin(t). (E32) t > t be a time in the distant future after the input 2 1 ~ field has interacted with the cavity. Then we can write Note that when a wave packet is launched from the bath an alternative solution to Eq. (E20) in terms of the final towards the cavity, causality prevents it from knowing rather than the initial condition of the bath about the cavity’s presence until it reaches the cavity. Z t1 Hence the input mode evolves freely as if the cavity were ˆ −iωq (t−t1)ˆ −iωq(t−τ) ∗ bq(t) = e bq(t1) − dτ e fq aˆ(τ). not present until the time of the collision at which point t ˆ it begins to drive the cavity. Since bin(t) evolves under (E34) the free bath Hamiltonian and acts as the driving term in Note the important minus sign in the second term as- the cavity EOM, we interpret it physically as the input sociated with the fact that the time t is now the lower mode. Eq. (E32) is the quantum analog of the classi- limit of integration rather than the upper as it was in cal equation (C19), for our previous example of an LC- Eq. (E21). oscillator driven by a transmission line. The latter would Defining also have been first order in time if as in Eq. (C35) we 1 X had worked with the complex amplitude A instead of the ˆb (t) ≡ √ e−iωq(t−t1)ˆb (t ), (E35) out 2πρ q 1 coordinate Q. q Eq. (E31) for the input mode contains a time label just as in the interaction representation. However it is we see that this is simply the free evolution of the bath best interpreted as simply labeling the particular linear modes from the distant future (after they have interacted combination of the bath modes which is coupled to the with the cavity) back to the present, indicating that it is system at time t. Some authors even like to think of indeed appropriate to interpret this as the outgoing field. 23

Proceeding as before we obtain incoming wave which is promptly reflected from the cavity and the field radiated by the cavity. The naive expres- √ ˙ i κ ˆ sion becomes correct after the drive has been switched off aˆ = [Hˆsys, aˆ] + aˆ − κ bout(t). (E36) ~ 2 (where ignoring the effect of the incoming vacuum noise, ˆ √ Subtracting Eq. (E36) from Eq. (E32) yields we would have bout = κaˆ). We note in passing that for a driven two-sided cavity with coupling constants κL and √ ˆ ˆ κR (where κ = κL + κR), the incoming power sent into bout(t) = bin(t) + κ aˆ(t) (E37) the left port is related to the photon number by which is consistent with our interpretation of the out- P = ωκ2/(4κ ) aˆ†aˆ . (E46) going field as the reflected incoming field plus the field ~ L radiated by the cavity out through the partially reflecting Here for κL = κR the interference effect completely elim- mirror. inates the reflected beam and we have in contrast to The above results are valid for any general cavity Eq. (E45) Hamiltonian. The general procedure is to solve Eq. (E32) κ P = ω aˆ†aˆ . (E47) fora ˆ(t) for a given input field, and then solve Eq. (E37) ~ 2 to obtain the output field. For the case of an empty cavity we can make further progress because the cavity mode Eq. (E39) can also be solved in the time domain to is a harmonic oscillator obtain

−(iωc+κ/2)(t−t0) ˆ † aˆ(t) = e aˆ(t0) Hsys = ~ωcaˆ a.ˆ (E38) √ Z t − κ dτ e−(iωc+κ/2)(t−τ)ˆb (τ). (E48) In this simple case, the cavity EOM becomes in t0 √ ˙ κ ˆ If we take the input field to be a coherent drive at fre- aˆ = −iωcaˆ − aˆ − κ bin(t). (E39) 2 quency ωL = ωc + ∆ so that its amplitude has a classical and a quantum part Eq. (E39) can be solved by Fourier transformation, yield- ˆ −iωLt ¯ ˆ ing bin(t) = e [bin + ξ(t)] (E49) √ and if we take the limit t → ∞ so that the initial tran- κ ˆ 0 aˆ[ω] = − bin[ω] (E40) sient in the cavity amplitude has damped out, then the i(ωc − ω) + κ/2 √ solution of Eq. (E48) has the form postulated in Eq. (E4) ˆ = − κχc[ω − ωc]bin[ω] (E41) with √ κ and a¯ = − ¯b (E50) −i∆ + κ/2 in ˆ ω − ωc − iκ/2 ˆ bout[ω] = bin[ω] (E42) and (in the frame rotating at the drive frequency) ω − ωc + iκ/2 √ Z t which is the result for the reflection coefficient quoted in dˆ(t) = − κ dτ e+(i∆−κ/2)(t−τ)ξˆ(τ). (E51) Eq. (3.13). For brevity, here and in the following, we will −∞ sometimes use the susceptibility of the cavity, defined as Even in the absence of any classical drive, the input field delivers vacuum fluctuation noise to the cavity. No- 1 χc[ω − ωc] ≡ (E43) tice that from Eqs. (E31, E49) −i(ω − ωc) + κ/2 ˆ ˆ† 0 ˆ ˆ† 0 [bin(t), bin(t )] = [ξ(t), ξ (t )] For the case of steady driving on resonance where ω = ωc, 1 X 0 the above equations yield = e−i(ωq−ωL)(t−t ) 2πρ √ q κ ˆb [ω] = aˆ[ω]. (E44) = δ(t − t0), (E52) out 2 which is similar to Eq. (D16) for a quantum transmission In steady state, the incoming power equals the outgoing line. This is the operator equivalent of white noise. Using power, and both are related to the photon number inside Eq. (E48) in the limit t0 → −∞ in Eqs. (E4,E51) yields the single-sided cavity by [â(t), aˆ†(t)] = [dˆ(t), dˆ†(t)] D E κ ˆ† ˆ † t t P = ~ω bout(t)bout(t) = ~ω aˆ (t)â(t) (E45) Z Z 4 = κ dτ dτ 0 e−(−i∆+κ/2)(t−τ) −∞ −∞ Note that this does not coincide with the naive expecta- −(+i∆+κ/2)(t−τ 0) 0 † e δ(τ − τ ) tion, which would be P = ~ωκ aˆ aˆ . The reason for this discrepancy is the the interference between the part of the = 1 (E53) 24 as is required for the cavity bosonic quantum degree of this circumstance it is useful to introduce the quantum freedom. We can interpret this as saying that the cavity Wiener increment related to Eq. (D18) zero-point fluctuations arise from the vacuum noise that enters through the open port. We also now have a simple Z t+∆t dWc ≡ dτ ξˆ(τ) (E58) physical interpretation of the quantum noise in the num- t ber of photons in the driven cavity in Eqs. (E3,E8,E11). It is due to the vacuum noise which enters the cavity which obeys through the same ports that bring in the classical drive. † The interference between the vacuum noise and the clas- [dWc , dWc ] = ∆t. (E59) sical drive leads to the photon number fluctuations in the In the interaction picture (in a displaced frame in cavity. ˆ which the classical drive has been removed) the Hamilto- In thermal equilibrium, ξ also contains thermal radi- nian term that couples the cavity to the quantum noise ation. If the bath is being probed only over a narrow of the environment is from Eq. (E18) range of frequencies centered on ωc (which we have assumed in making the Markov approximation) then we √ † † Vˆ = −i~ κ(â ξˆ− aˆξˆ ). (E60) have to a good approximation (consistent with the above commutation relation) Thus the time evolution operator (in the interaction picture) on the jth short time interval [tj, tj + ∆t] is hξˆ†(t)ξˆ(t0)i = Nδ(t − t0) (E54) √ ˆ κ(â dWc†−aˆ† dWc) hξˆ(t)ξˆ†(t0)i = (N + 1)δ(t − t0) (E55) Uj = e (E61) Using this we can readily evolve the incoming temporal where N = nB(~ωc) is the thermal equilibrium occupation number of the mode at the frequency of interest. We mode forward in time by a small step ∆t √ can gain a better understanding of Eq. (E54) by Fourier 0 ˆ † ˆ transforming it to obtain the spectral density dWc = U dWcU ≈ dWc + κ∆t a.ˆ (E62)

+∞ Recall that in input-output theory we formally defined Z 0 S[ω] = dt hξˆ†(t)ξˆ(t0)ieiω(t−t ) = N. (E56) the outgoing field as the bath field far in the future prop- −∞ agated back (using the free field time evolution) to the present, which yielded As mentioned previously, this dimensionless quantity is √ the spectral density that would be measured by a photo- ˆ ˆ bout = bin + κa.ˆ (E63) multiplier: it represents the number of thermal photons passing a given point per unit time per unit bandwidth. Eq. (E62) is completely equivalent to this. Thus we con- Equivalently the thermally radiated power in a narrow firm our understanding that the incoming field is the bath bandwidth B is temporal mode just before it interacts with the cavity and the outgoing field is the bath temporal mode just after it P = ~ωNB. (E57) interacts with the cavity. This leads to the following picture which is especially One often hears the confusing statement that the noise useful in the quantum trajectory approach to conditional added by an amplifier is a certain number N of photons quantum evolution of a system subject to weak continu- (N = 20, say for a good cryogenic HEMT amplifier op- ous measurement (Gardiner et al., 1992; Walls and Mil- ¯ erating at 5 GHz). This means that the excess output burn, 1994). On top of the classical drive bin(t), the bath noise (referred back to the input by dividing by the power supplies to the system a continuous stream of “fresh” har- gain) produces a flux of N photons per second in a 1 Hz monic oscillators, each in their ground state (if T = 0). bandwidth, or 106N photons per second in 1 MHz of Each oscillator with its quantum fluctuation dWc inter- bandwidth (see also Eq. (D27)). acts briefly for a period ∆t with the system and then We can gain further insight into input-output theory is disconnected to propagate freely thereafter, never in- ˆ by using the following picture. The operator bin(t) repre- teracting with the system again. Within this picture it sents the classical drive plus vacuum fluctuations which is useful to think of the oscillators arrayed in an infinite are just about to arrive at the cavity. We will be able stationary line and the cavity flying over them at speed to show that the output field is simply the input field vp and touching each one for a time ∆t. a short while later after it has interacted with the cavity. Let us consider the time evolution over a short time period ∆t which is very long compared to the inverse 3. Quantum limited position measurement using a cavity bandwidth of the vacuum noise (i.e., the frequency scale detector beyond which the vacuum noise cannot be treated as constant due to some property of the environment) but very We will now apply the input-output formalism intro- short compared to the cavity system’s slow dynamics. In duced in the previous section to the important example 25 of a dispersive position measurement, which employs a and similarly for the cavity field following Eq. (E4) cavity whose resonance frequency shifts in response to the motion of a harmonic oscillator. This physical sys- aˆ =a ¯ + d.ˆ (E70) tem was considered heuristically in Sec. III.B.3. Here we will present a rigorous derivation using the (linearized) Substituting these expressions into the equation of mo- equations of motion for the coupled cavity and oscillator tion, we find that the constant classical fields obey system. √ κ Let the dimensionless position operator a¯ = − ¯b (E71) κ/2 − i∆ in 1 zˆ = xˆ = [ˆc† +c ˆ] (E64) and the new quantum equation of motion is, after ne- x ZPF glecting a small term dˆzˆ: be the coordinate of a harmonic oscillator whose energy √ ˆ˙ ˆ κ ˆ ˆ is d = +i∆d − iAωca¯zˆ − d − κξ. (E72) 2 † HM = ~ωMcˆ cˆ (E65) The quantum limit for position measurement will be reached only at zero detuning, so we specialize to the and whose position uncertainty in the quantum ground case ∆ = 0. We also choose the incoming field amplitude p 2 state is xZPF = h0|xˆ |0i. and phase to obey This Hamiltonian could be realized for example by mounting one of the cavity mirrors on a flexible cantilever q ¯b = −i N,˙ (E73) (see the discussion above). in When the mirror moves, the cavity resonance fre- so that quency shifts, s N˙ ω˜c = ωc[1 + Azˆ(t)] (E66) a¯ = +2i , (E74) κ where for a cavity of length L, A = −xZPF/L. Assuming that the mirror moves slowly enough for the where N˙ is the incoming photon number flux. The quan- cavity to adiabatically follow its motion (i.e. Ω κ), the tum equation of motion for the cavity then becomes outgoing light field suffers a phase shift which follows the ˙ κ √ changes in the mirror position. This phase shift can be dˆ= +gzˆ − dˆ− κξ,ˆ (E75) 2 detected in the appropriate homodyne set up as discussed in Sec. III.B, and from this phase shift we can determine where the opto-mechanical coupling constant is propor- the position of the mechanical oscillator. In addition to tional to the laser drive amplitude the actual zero-point fluctuations of the oscillator, our s measurement will suffer from shot noise in the homodyne N˙ √ signal and from additional uncertainty due to the back g ≡ 2Aωc = Aωc n.¯ (E76) action noise of the measurement acting on the oscillator. κ All of these effects will appear naturally in the derivation and below. We begin by considering the optical cavity equation N˙ n¯ = |a¯|2 = 4 (E77) of motion based on Eq. (E32) and the optomechanical κ coupling Hamiltonian in Eq. (E1). These yield is the mean cavity photon number. Eq. (E75) is easily κ √ aˆ˙ = −iω (1 + Azˆ)â − aˆ − κˆb . (E67) solved by Fourier transformation c 2 in 1 n √ o dˆ[ω] = gzˆ[ω] − κξˆ[ω] . (E78) Let the cavity be driven by a laser at a frequency ωL = [κ/2 − iω] ωc +∆ detuned from the cavity by ∆. Moving to a frame rotating at ωL we have Let us assume that we are in the limit of low mechanical frequency relative to the cavity damping, Ω κ, √ ˙ κ ˆ so that the cavity state adiabatically follows the motion aˆ = +i(∆ − Aωczˆ)â − aˆ − κbin. (E68) 2 of the mechanical oscillator. Then we obtain to a good approximation and we can write the incoming field as a constant plus white noise vacuum fluctuations (again, in the rotating 2 n √ o dˆ[ω] = gzˆ[ω] − κξˆ[ω] (E79) frame) κ 2 n √ o ˆ ¯ ˆ dˆ†[ω] = gzˆ[ω] − κξˆ†[ω] (E80) bin = bin + ξ (E69) κ 26

The mechanical oscillator equation of motion which is identical in form to that of the optical cavity

γ0 √ i ∂tcˆ = −[ + iΩ]ˆc − γ0ηˆ(t) + [Hînt, cˆ(t)], (E81) 2 ~ where Hînt is the Hamiltonian in Eq. (E1) andη ˆ is the mechanical vacuum noise from the (zero temperature) bath which is causing the mechanical damping at rate γ0. Using Eq. (E70) and expanding to first order in small fluctuations yields the equation of motion linearized about the steady state solution γ √ g ∂ cˆ = −[ 0 + iΩ]ˆc − γ ηˆ(t) + 2√ [ξˆ(t) − ξˆ†(t)]. (E82) t 2 0 κ It is useful to consider an equivalent formulation in which we expand the Hamiltonian in Eq. (E1) to second order in the quantum fluctuations about the classical solution ˆ ˆ† ˆ ˆ Hint ≈ ~ωcd d +x ˆF, (E83) FIG. 6 (Color online) Phasor diagram for the cavity ampli- where the force (including the coupling A) is (up to a tude showing that (for our choice of parameters) the imag- sign) inary quadrature of the vacuum noise ξˆ interferes with the classical drive to produce photon number fluctuations while g the real quadrature produces phase fluctuations which lead to Fˆ = −i ~ [dˆ− dˆ†]. (E84) xZPF measurement imprecision. The quantum fluctuations are illustrated in the usual fashion, depicting the Gaussian Wigner Note that the radiation pressure fluctuations (photon density of the coherent state in terms of color intensity. shot noise) inside the cavity provide a forcing term. The state of the field inside the cavity in general depends on the past history of the cantilever position. However for this special case of driving the cavity on resonance, the phase noise which determines the measurement impreci- dependence of the cavity field on the cantilever history is sion (shot noise in the homodyne signal). This will be such that the latter drops out of the radiation pressure. discussed further below. To see this explicitly, consider the equation of motion for The solution for the cantilever position can again be the force obtained from Eq. (E75) obtained by Fourier transformation. For frequencies small on the scale of κ the solution of Eq. (E85) is ˙ κ g √ Fˆ = − Fˆ + i ~ κ[ξˆ− ξˆ†]. (E85) 2i~g n † o 2 xZPF Fˆ[ω] = √ ξˆ[ω] − ξˆ [ω] (E86) xZPF κ Within our linearization approximation, the position of the mechanical oscillator has no effect on the radiation and hence the back action force noise spectral density is pressure (photon number in the cavity), but of course it at low frequencies does affect the phase of the cavity field (and hence the 2 2 outgoing field) which is what we measure in the homo- 4~ g SFF [ω] = 2 (E87) dyne detection. xZPFκ Thus for this special casez ˆ does not appear on the RHS of either Eq. (E85) or Eq. (E82), which means that in agreement with Eq. (3.18). there is no optical renormalization of the cantilever fre- Introducing a quantity proportional to the cantilever quency (‘optical spring’) or optical damping of the can- (mechanical) susceptibility (within the rotating wave ap- tilever. The lack of back-action damping in turn implies proximation we are using) that the effective temperature T of the cavity detector eff 1 is infinite (cf. Eq. (2.8)). For this special case of zero χM[ω − Ω] ≡ , (E88) −i(ω − Ω) + γ0 detuning the back action force noise is controlled by a 2 single quadrature of the incoming vacuum noise (which we find from Eq. (E82) interferes with the classical drive to produce photon number fluctuations). This is illustrated in the cavity ampli- i ˆ tude phasor diagram of Fig. (6). We see that the vac- zˆ[ω] =z ˆ0[ω] − xZPF {χM[ω − Ω] − χM[ω + Ω]} F [ω], ~ uum noise quadrature ξˆ+ ξˆ† conjugate to Fˆ controls the (E89) 27 where the equilibrium fluctuations in position are given in agreement with Eq. (3.10). by Notice also from Eq. (E94) that the quadrature of the √ † vacuum noise which leads to the measurement impreci- zˆ0[ω] ≡ − γ0 χM[ω − Ω]ˆη[ω] + χM[ω + Ω]ˆη [ω] . sion is conjugate to the one which produces the back (E90) action force noise as illustrated previously in Fig. (6). We can now obtain the power spectrum Szz describing Recall that the two quadratures of motion of a harmonic the total position fluctuations of the cantilever driven by oscillator in its ground state have no classical (i.e., sym- the mechanical vacuum noise plus the radiation pressure metrized) correlation. Hence the symmetrized cross cor- shot noise. From Eqs. (E89, E90) we find relator

Sxx[ω] S¯IF [ω] = 0 (E98) 2 = Szz[ω] xZPF 2 vanishes. Because there is no correlation between the = γ0|χ[ω − Ω]| (E91) output imprecision noise and the forces controlling the 2 xZPF 2 position fluctuations, the total output noise referred back + |χM[ω − Ω] − χM[ω + Ω]| SFF . ~2 to the position of the oscillator is simply ¯ ¯ ¯I Note that (assuming high mechanical Q, i.e. γ0 Ω) the Sxx,tot[ω] = Sxx[ω] + Sxx (E99) equilibrium part has support only at positive frequencies S¯0 [Ω] 2 while the back action induced position noise is symmetric = S¯0 [ω] 1 + xx S¯ + ~ . xx 2 FF ¯ in frequency reflecting the effective infinite temperature ~ 4SFF of the back action noise. Symmetrizing this result with This expression again clearly illustrates the competition respect to frequency (and using γ0 Ω) we have between the back action noise proportional to the drive laser intensity and the measurement imprecision noise S¯0 [Ω] ¯ ¯0 xx ¯ which is inversely proportional. We again emphasize that Sxx[ω] ≈ Sxx[ω] 1 + 2 SFF , (E92) ~ all of the above relations are particular to the case of zero ¯0 detuning of the cavity drive field from the cavity. where Sxx[ω] is the symmetrized spectral density for position fluctuations in the ground state given by Eq. (3.54). The total output noise at some particular frequency Now that we have obtained the effect of the back action will be a minimum at some optimal drive intensity. The noise on the position fluctuations, we must turn our at- precise optimal value depends on the frequency chosen. tention to the imprecision of the measurement due to shot Typically this is taken to be the mechanical resonance noise in the output. The appropriate homodyne quadra- frequency where we find that the optimal coupling leads ture variable to monitor to be sensitive to the output to an optimal back action noise phase shift caused by position fluctuations is 2 2γ S¯ = ~ = ~ 0 . (E100) FF,opt ¯0 2 ˆ ˆ ˆ† 2Sxx[Ω] 4xZPF I = bout + bout, (E93) This makes sense because the higher the damping the less which, using the input-output results above, can be writ- susceptible the oscillator is to back action forces. At this ten optimal coupling the total output noise spectral density Iˆ = −(ξˆ+ ξˆ†) + λx.ˆ (E94) at frequency Ω referred to the position is simply twice the vacuum value We see that the cavity homodyne detector system acts ¯ ¯0 as a position transducer with gain Sxx,tot[Ω] = 2Sxx[Ω], (E101) 4g in agreement with Eq. (3.62). Evaluation of Eq. (E100) λ = √ . (E95) at the optimal coupling yields the graph shown in xZPF κ Fig. (6). The background noise floor is due to the The first term in Eq. (E94) represents the vacuum noise frequency independent imprecision noise with value 1 ¯0 that mixes with the homodyne local oscillator to produce 2 Sxx[Ω]. The peak value at ω = Ω rises a factor of three the shot noise in the output. The resulting measurement above this background. imprecision (symmetrized) spectral density referred back We derived the gain λ in Eq. (E95) by direct solution to the position of the oscillator is of the equations of motion. With the results we have derived above, it is straightforward to show that the Kubo 1 ¯I formula in Eq. (4.3) yields equivalent results. We have Sxx = 2 . (E96) λ already seen that the classical (i.e. symmetrized) correla- Comparing this to Eq. (E87) we see that we reach the tions between the output signal Iˆ and the force Fˆ which quantum limit relating the imprecision noise to the back couples to the position vanishes. However the Kubo for- action noise mula evaluates the quantum (i.e. antisymmetric) corre- 2 lations for the uncoupled system (A = g = 0). Hence we S¯I S¯ = ~ (E97) have xx FF 4 28

i ˆ ˆ† 2i~g ˆ ˆ† χIF (t) = − θ(t) − ξ(t − δt) + ξ (t − δt) , √ ξ(0) − ξ (0) , (E102) ~ xZPF κ 0

where δt is a small (positive) time representing the delay ture takes the simple form corresponding to Eq. (5.102b): between the time when the vacuum noise impinges on the cavity and when the resulting outgoing wave reaches ˜ ˆ ˆ† iδ −iδ † Hint = ~A d + d e cˆ + e cˆ the homodyne detector. (More precisely it also compen- ˆ sates for certain small retardation effects neglected in the † Xδ = ~A˜ dˆ+ dˆ , (E105) limit ω κ used in several places in the above deriva- xZPF tions.) Using the fact that the commutator between the two quadratures of the vacuum noise is a delta function, where Fourier transformation of the above yields (in the limit p Nκ˙ ω δt 1 the desired result A˜ = A · ω , (E106) c 4Ω χ [ω] = λ. (E103) IF and in the second line, we have made use of the definition of the quadrature operators Xˆδ, Yˆδ given in Eqs. (5.92). Similarly we readily find that the small retardation The form of Hint was discussed heuristically in the main causes the reverse gain to vanish. Hence all our results text in terms of Raman processes where photons are re- are consistent with the requirements needed to reach the ¯ moved from the classical drive bin and either up or down standard quantum limit. converted to the cavity frequency via absorption or emis- Thus with this study of the specific case of an oscillator sion of a mechanical phonon. Alternatively, we can think parametrically coupled to a cavity, we have reproduced of the drive yielding a time-dependent cavity-oscillator all of the key results in Sec. V.E derived from completely coupling which “follows” the Xδ quadrature. Note that general considerations of linear response theory. we made crucial of use of the good cavity limit (κ Ω) to drop terms in Hînt which oscillate at frequencies ±2Ω. These terms represent Raman sidebands which are away 4. Back-action free single-quadrature detection from the cavity resonance by a distance ±2Ω. In the good cavity limit, the density of photon states is negligible so far off resonance and these processes are suppressed. We now provide details on the cavity single-quadrature Similar to Eqs. (E39) and (E81), the Heisenberg equa- detection scheme discussed in Sec. V.H.2. We again con- tions of motion (in the rotating frame) follow directly sider a high-Q cavity whose resonance frequency is mod- from H and the dissipative terms in the total Hamil- ulated by a high-Q mechanical oscillator with co-ordinate int tonian: xˆ (cf. Eqs. (E1) and (E64)). To use this system for ampli- κ √ fication of a single quadrature, we will consider the typi- ˆ ˆ ˆ iωct iδ −iδ † ∂td = − d − κξ(t)e − iA˜ e cˆ + e cˆ (E107a) cal case of a fast cavity (ωc Ω), and take the “good cav- 2 γ √ h i ity” limit, where Ω κ. As explained in the main text, ∂ cˆ = − 0 cˆ − γ ηˆ(t)eiΩt − ie−iδ A˜ dˆ+ dˆ† − f(t) the crucial ingredient for single-quadrature detection is t 2 0 to take an amplitude-modulated cavity drive described (E107b) ¯ by the classical input field bin given in Eq. (5.101). As before (cf. Eq. (E4)), we may write the cavity annihila- As before, ξˆ(t) represents the unavoidable noise in the tion operatora ˆ as the sum of a classical piecea ¯(t) and a cavity drive, andη ˆ(t), γ0 are the noisy force and damp- quantum piece dˆ; only dˆ is influenced by the mechanical ing resulting from an equilibrium bath coupled to the oscillator.a ¯(t) is easily found from the classical (noise- mechanical oscillator. Note from Eq. (E107a) that as an- free) equations of motion for the isolated cavity; making ticipated, the cavity is only driven by one quadrature of use of the conditions ωc Ω κ, we have the oscillator’s motion. We have also included a driving force F (t) on the mechanical oscillator which has some p narrow bandwidth centered on the oscillator frequency; Nκ˙ a¯(t) ' cos (Ωt + δ) e−iωct (E104) this force is parameterized as: 2Ω 2~ −iΩt −iδ To proceed with our analysis, we work in an interac- F (t) = Re f(t)e e (E108) xZPF tion picture with respect to the uncoupled cavity and oscillator Hamiltonians. Making standard rotating-wave where f(t) is a complex function which is slowly varying approximations, the Hamiltonian in the interaction pic- on the scale of an oscillator period. 29

The equations of motion are easily solved upon Fourier be the homodyne signal integrated up to time t as in transformation, resulting in: Sec. III.B. We would like to understand the relationship between the signal-to-noise ratio defined in Eq. (3.23), " and the rate at which information about the state of the ˆ ∗ Xδ[ω] = −xZPF · χM [ω] i (f [−ω] − f[ω]) (E109a) qubit is being gained. The probability distribution for I conditioned on the state of the qubit σ = ±1 is # √ + γ eiδηˆ(ω + Ω) + e−iδηˆ†(ω − Ω) 1 −(I − σθ t)2 0 p(I|σ) = √ exp 0 . (F1) 2πS t 2S t " θθ θθ ∗ Yˆδ[ω] = ixZPF · χM [ω] (−i)(f[ω] + f [−ω]) (E109b) Based on knowledge of this conditional distribution, we now present two distinct but equivalent approaches to √ iδ −iδ † giving an information theoretic basis for the definition of + γ0 e ηˆ(ω + Ω) − e ηˆ (ω − Ω) # the measurement rate. √ ˆ ˆ† −2iAχ˜ c[ω] κ ξ(ω + ωc) + ξ (ω − ωc) 1. Method I where the cavity and mechanical susceptibilities χc, χM are defined in Eqs. (E43) and (E88). Suppose we start with an initial qubit density matrix ˆ As anticipated, the detected quadrature Xδ is com- 1 0 ρ = 2 . (F2) pletely unaffected by the measurement: Eq. (E109a) is 0 0 1 identical to what we would have if there were no coupling 2 between the oscillator and the cavity. In contrast, the After measuring for a time t, the new density matrix conjugate quadrature Yˆδ experiences an extra stochastic conditioned on the results of the measurement is force due to the cavity: this is the measurement back- p+ 0 action. ρ1 = (F3) ˆ 0 p− Turning now to the output field from the cavity bout, we use the input-output relation Eq. (E37) to find in the where it will be convenient to parameterize the two prob- lab (i.e. non-rotating) frame: abilities by the polarization m ≡ Tr(σzρ1) by ˆ ¯ −i(ω − ωc) − κ/2 ˆ 1 ± m bout[ω] = bout[ω] + ξ[ω] p± = . (F4) −i(ω − ωc) + κ/2 2 ˜√ A κ ˆ The information gained by the measurement is the en- −i χc[ω − ωc] · Xδ(ω − ωc) 7 xZPF tropy loss of the qubit (E110) I = Tr(ρ1 ln ρ1 − ρ0 ln ρ0). (F5) The first term on the RHS simply represents the output We are interested in the initial rate of gain of information field from the cavity in the absence of the mechanical 2 at short times θ0t Sθθ where m will be small. In this oscillator and any fluctuations. It will yield sharp peaks limit we have at the two sidebands associated with the drive, ω = ωc ± Ω. The second term on the RHS of Eq. (E110) represents m2 I ≈ . (F6) the reflected noise of the incident cavity drive. This noise 2 will play the role of the “intrinsic ouput noise” of this We must now calculate m conditioned on the measure- amplifier. ment result I Finally, the last term on the RHS of Eq. (E110) is the X amplified signal: it is simply the amplified quadrature m ≡ σp(σ|I). (F7) ˆ I Xδ of the oscillator. This term will result in a peak in σ the output spectrum at the resonance frequency of the From Bayes theorem we can express this in terms of cavity, ωc. As there is no back-action on the measured p(I|σ), which is the quantity we know, Xˆδ quadrature, the added noise can be made arbitrarily ˙ small by simply increasing the drive strength N (and p(I|σ)p(σ) ˜ hence A). p(σ|I) = P 0 0 . (F8) σ0 p(I|σ )p(σ )

Appendix F: Information Theory and Measurement Rate 7 It is important to note that we use throughout here the physi- Suppose that we are measuring the state of a qubit cist’s entropy with the natural logarithm rather than the log base via the phase shift ±θ0 from a one-sided cavity. Let I(t) 2 which gives the information in units of bits. 30

Using Eq. (F1) the polarization is easily evaluated exactly the same result as Eq. (F11). (Here hI(t)iσ is the mean value of I given that the qubit is in state σ.) Iθ0 mI = tanh . (F9) Sθθ Appendix G: Number Phase Uncertainty The information gain is thus

1 Iθ I2 θ 2 In this appendix, we briefly review the number-phase I = tanh2 0 ≈ 0 (F10) I 2 S 2 S uncertainty relation, and from it we derive the relation- θθ θθ ship between the spectral densities describing the photon where the second equality is only valid for small |m|. number fluctuations and the phase fluctuations. Con- Ensemble averaging this over all possible measurement sider a coherent state labeled by its classical amplitude results yields the mean information gain at short times α 1 θ2 |α|2 I ≈ 0 t (F11) |αi = exp − exp{αaˆ†}|0i. (G1) 2 Sθθ 2 which justifies the definition of the measurement rate This is an eigenstate of the destruction operator given in Eq. (3.24). aˆ|αi = α|αi. (G2)

2. Method II It is convenient to make the unitary displacement transformation which maps the coherent state onto a new vac- An alternative information theoretic derivation is to uum state and the destruction operator onto consider the qubit plus measurement device to be a sig- aˆ = α + dˆ (G3) naling channel. The two possible inputs to the channel are the two states of the qubit. The output of the chan- where d annihilates the new vacuum. Then we have nel is the result of the measurement of I. By toggling the qubit state back and forth, one can send information N¯ = hNî = h0|(α∗ + dˆ†)(α + dˆ)|0i = |α|2, (G4) through the signal channel to another party. The channel is noisy because even for a fixed state of the qubit, and the measured values of the signal I have intrinsic fluctu- (∆N)2 = h(Nˆ − N¯)2i = |α|2h0|dˆdˆ†|0i = N.¯ (G5) ations. Shannon’s noisy channel coding theorem (Cover and Thomas, 1991) tells us the maximum rate at which Now define the two quadrature amplitudes information can be reliably sent down the channel by tog- 1 gling the state of the qubit and making measurements of Xˆ = √ (â +a ˆ†) (G6) I. It is natural to take this rate as defining the measure- 2 ment rate for our detector. i Yˆ = √ (â† − aˆ). (G7) The reliable information gain by the receiver on a noisy 2 channel is a quantity known as the ‘mutual information’ of the communication channel (Clerk et al., 2003; Cover Each of these amplitudes can be measured in a homodyne and Thomas, 1991) experiment. For convenience, let us take α to be real and positive. Then ( ) Z +∞ X √ R = − dI p(I) ln p(I) − p(σ)[p(I|σ) ln p(I|σ)] hXî = 2α (G8) −∞ σ (F12) and The first term is the Shannon entropy in the signal I ˆ when we do not know the input signal (the value of the hY i = 0. (G9) qubit). The second term represents the entropy given If the phase of this wave undergoes a small modulation that we do know the value of the qubit (averaged over due for example to weak parametric coupling to a qubit the two possible input values). Thus the first term is then one can estimate the phase by signal plus noise, the second is just the noise. Subtracting the two gives the net information gain. Expanding this hYˆ i expression for short times yields hθi = . (G10) hXî 1 (hI(t)i − hI(t)i )2 R = + − This result is of course only valid for small angles, θ 1. 8 S t θθ For N¯ 1, the uncertainty will be 2 θ0 = t ˆ 2 1 ˆˆ† 2Sθθ 2 hY i 2 h0|dd |0i 1 (∆θ) = = ¯ = ¯ . (G11) = Γmeast (F13) (hXî)2 2N 4N 31

Thus using Eq. (G5) we arrive at the fundamental quan- where ξˆ(t) is the vacuum noise obeying tum uncertainty relation ˆ ˆ† 0 0 1 [ξ(t), ξ (t )] = δ(t − t ). (G14) ∆θ∆N = . (G12) 2 We are using a flux normalization for the field operators Using the input-output theory described in Ap- so pendix E we can restate the results above in terms of noise spectral densities. Let the amplitude of the field ˙ ˆ† ˆ ¯ 2 coming in to the homodyne detector be N = hbinbini = |bin| (G15) ˆ ¯ ˆ bin = bin + ξ(t) (G13) and

2 ˙ ˙ ˙ ¯∗ ˆ† ¯ ˆ ¯∗ ˆ† ¯ ˆ ¯ 4 ˙ hN(t)N(0)i − N = h0|(bin + ξ (t))(bin + ξ(t))(bin + ξ (0))(bin + ξ(0))|0i − bin = Nδ(t). (G16)

From this it follows that the shot noise spectral density able to reach the weak-coupling (i.e. op-amp) quantum is limit. Note that quantum amplifiers with feedback are also treated in Courty et al. (1999); Grassia (1998). ˙ SN˙ N˙ = N. (G17) On a heuristic level, we can understand the need for either reflections or reverse gain to reach the quantum Similarly the phase can be estimated from the quadra- limit. A problem with the “minimal” amplifier of the ture operator last subsection was that its input impedance was too low in comparison to its noise impedance Z ∼ Z . From i(ˆb† − ˆb ) (ξˆ† − ξˆ) N a θˆ = in in = hθî + i (G18) general expression for the input impedance, Eq. (6.7d), ˆ† ˆ 2¯b hbin + bini in we see that having non-zero reverse gain (i.e. s12 6= 0) and/or non-zero reflections (i.e. s 6= 0 and/or s 6= 0) which has noise correlator 11 22 could lead to Zin Za. This is exactly what occurs 1 when feedback is used to reach the quantum limit. Keep hδθˆ(t)δθˆ(0)i = δ(t) (G19) ˙ in mind that having non-vanishing reverse gain is danger- 4N 0 ous: as we discussed earlier, an appreciable non-zero λI corresponding to the phase imprecision spectral density can lead to the highly undesirable consequence that the amplifier’s input impedance depends on the impedance 1 Sθθ = . (G20) of the load connected to its output (cf. Eq. (6.6)). 4N˙

We thus arrive at the fundamental quantum limit relation 1. Feedback using mirrors p 1 SθθS ˙ ˙ = . (G21) To introduce reverse gain and reflections into the “min- NN 2 imal” two-port bosonic amplifier of the previous subsection, we will insert mirrors in three of the four arms lead- Appendix H: Using feedback to reach the quantum limit ing from the circulator: the arm going to the input line, the arm going to the output line, and the arm going In Sec. (VI.B), we demonstrated that any two port am- to the auxiliary “cold load” (Fig. 7). Equivalently, one plifier whose scattering matrix has s11 = s22 = s12 = 0 could imagine that each of these lines is not perfectly will fail to reach quantum limit when used as a weakly impedance matched to the circulator. Each mirror will coupled op-amp; at best, it will miss optimizing the quan- be described by a 2 × 2 unitary scattering matrix: tum noise constraint of Eq. (5.88) by a factor of two. ˆ Reaching the quantum limit thus requires at least one aˆj,out bj,in ˆ = Uj · (H1) of s11, s22 and s12 to be non-zero. In this subsection, bj,out aˆj,in we demonstrate how this may be done. We show that cos θj − sin θj Uj = (H2) by introducing a form of negative feedback to the “min- sin θj cos θj imal” amplifier of the previous subsection, one can take advantage of noise correlations to reduce the back-action Here, the index j can take on three values: j = z for the current noise SII by a factor of two. As a result, one is mirror in the input line, j = y for the mirror in the arm 32

cold amount of reflection in the cold load line with the correct load phase, then we can reach the quantum limit, irrespective of the mirrors in√ the input and output lines. In particular, if sin θy = −1/ G, our amplifier optimizes the quantum by,in by,out noise constraint of Eq. (5.88) in the large gain (i.e. large G) limit, independently of the values of θx and θy. Note mirror Y that tuning θy to reach the quantum limit does not have a catastrophic impact on other features of our amplifier. ay,out ay,in One can verify that this tuning only causes the voltage gain λ and power gain G to decrease by a factor of input line output line V P two compared to their θ = 0 values (cf. Eqs. (I60) and bz,out az,in ax,out bx,in y (I64)). This choice for θy also leads to Zin Za ∼ ZN (cf. (I62)), in keeping with our general expectations. Physically, what does this precise tuning of θ corre- bz,in Z az,out ax,in X bx,out y spond to? A strong hint is given by the behaviour of the 1/2 1/2 † ¯ cin cout=G cin+(G-1) v in amplifier’s cross-correlation noise SVI [ω] (cf. Eq. (I65c)). In general, we find that S¯VI [ω] is real and non-zero. ideal √ However, the tuning sin θy = −1/ G is exactly what is 1-port ¯ amp. needed to have SVI vanish. Also note from Eq. (I65a) that this special tuning of θy decreases the back-action current noise precisely by a factor of two compared to v v in out its value at θy = 0. A clear physical explanation now emerges. Our original, reflection-free amplifier had cor- FIG. 7 Schematic of a modified minimal two-port amplifier, relations between its back-action current noise and out- where partially reflecting mirrors have been inserted in the put voltage noise (cf. Eq. (6.18c)). By introducing nega- input and output transmission lines, as well as in the line tive feedback of the output voltage to the input current leading to the cold load. By tuning the reflection coefficient (i.e. via a mirror in the cold-load arm), we are able to of the mirror in the cold load arm (mirror Y ), we can in- use these correlations to decrease the overall magnitude duce negative feedback which takes advantage of correlations ˜ between current and voltage noise. This then allows this sys- of the current noise (i.e. the voltage fluctuations V par- tem to reach the quantum limit as a weakly coupled voltage tially cancel the original current fluctuations I˜). For an op amp. See text for further description optimal feedback (i.e. optimal choice of θy), the current noise is reduced by a half, and the new current noise is not correlated with the output voltage noise. Note that this is indeed negative (as opposed to positive) feedback– going to the cold load, and j = x for the mirror in the it results in a reduction of both the gain and the power output line. The mode aj describes the “internal” mode gain. To make this explicit, in the next section we will which exists between the mirror and circulator, while the map the amplifier described here onto a standard op-amp mode bj describes the “external” mode on the other side with negative voltage feedback. of the mirror. We have taken the Uj to be real for convenience. Note that θj = 0 corresponds to the case of no mirror (i.e. perfect transmission). 2. Explicit examples It is now a straightforward though tedious exercise to construct the scattering matrix for the entire system. To obtain a more complete insight, it is useful to go From this, one can identify the reduced scattering matrix back and consider what the reduced scattering matrix of s appearing in Eq. (6.3), as well as the noise operators our system looks like when θy has been tuned to reach the quantum limit. From Eq. (I58), it is easy to see that Fj. These may then in turn be used to obtain the op-amp description of the amplifier, as well as the commutators at the quantum limit, the matrix s satisfies: of the added noise operators. These latter commutators s = −s (H3a) determine the usual noise spectral densities of the ampli- 11 22 1 fier. Details and intermediate steps of these calculations s12 = s21 (H3b) may be found in Appendix I.6. G As usual, to see if our amplifier can reach the quantum The second equation also carries over to the op-amp pic- limit when used as a (weakly-coupled) op-amp , we need ture; at the quantum limit, one has: to see if it optimizes the quantum noise constraint of 1 Eq. (5.88). We consider the optimal situation where both 0 λI = λV (H4) † G the auxiliary modes of the amplifier (ûin andv în) are in the vacuum state. The surprising upshot of our analysis One particularly simple limit is the case where there (see Appendix I.6) is the following: if we include a small are no mirrors in the input and output line (θx = θz = 0), 33 only a mirror in the cold-load arm. When this mirror√ is tuned to reach the quantum limit (i.e. sin θy = −1/ G), Va + the scattering matrix takes the simple form: Gf Vb √ - 0 1/ G R1 R2 s = √ (H5) G 0 In this case, the principal effect of the weak mirror in the cold-load line is to introduce a small amount of reverse gain. The amount of this reverse gain is exactly FIG. 8 Schematic of a voltage op-amp with negative feedback. what is needed to have the input impedance diverge (cf. Eq. (6.7d)). It is also what is needed to achieve an optimal, noise-canceling feedback in the amplifier. To 3. Op-amp with negative voltage feedback see this last point explicitly, we can re-write the amplifier’s back-action current noise (I˜) in terms of its original ˜ ˜ We now show that a conventional op-amp with feed- noises I0 and V0 (i.e. what the noise operators would have back can be mapped onto the amplifier described in been in the absence of the mirror). Taking the relevant the previous subsection. We will show that tuning the limit of small reflection (i.e.r ˜ ≡ sin θy goes to zero as strength of the feedback in the op-amp corresponds to |G| → ∞), we find that the modification of the current tuning the strength of the mirrors, and that an optimally noise operator is given by: tuned feedback circuit lets one reach the quantum limit. √ ˜ This is in complete correspondence to the previous sub- ˜ ˜ 2 Gr˜ V0 I ' I0 + √ (H6) section, where an optimal tuning of the mirrors also lets 1 − Gr˜ Za one reach the quantum limit. As claimed, the presence of a small amount of reflection More precisely, we consider a scattering description of r˜ ≡ sin θy in the cold load arm “feeds-back” the original a non-inverting op-amp amplifier having negative voltage ˜ voltage noise of√ the amplifier V0 into the current. The feedback. The circuit for this system is shown in Fig. 8. choicer ˜ = −1/ G corresponds to a negative feedback, A fraction B of the output voltage of the amplifier is fed and optimally makes use of the fact that I˜0 and V˜0 are back to the negative input terminal of the op-amp. In correlated to reduce the overall fluctuations in I˜. practice, B is determined by the two resistors R1 and R2 While it is interesting to note that one can reach the used to form a voltage divider at the op-amp output. The quantum limit with no reflections in the input and output op-amp with zero feedback is described by the “ideal” arms, this case is not really of practical interest. The amplifier of Sec. VI.B: at zero feedback, it is described 0 by Eqs. (6.11a)- (6.11d). For simplicity, we consider the reverse√ current gain in this case may be small (i.e. λI ∝ 1/ G), but it is not small enough: one finds that because relevant case where: of the non-zero λ0 , the amplifier’s input impedance is I Z R ,R Z (H9) strongly reduced in the presence of a load (cf. Eq. (6.6)). b 1 2 a There is a second simple limit we can consider which is In this limit, R1 and R2 only play a role through the more practical. This is the limit where reflections in the feedback fraction B, which is given by: input-line mirror and output-line mirror are both strong. 1/8 Imagine we take θ = −θ = π/2 − δ/G . If again R2 z√ x B = (H10) we set sin θy = −1/ G to reach the quantum limit, the R1 + R2 scattering matrix now takes the form (neglecting terms √ Letting Gf denote the voltage gain at zero feedback which are order 1/ G): (B = 0), an analysis of the circuit equations for our op- +1 0 amp system yields: s = 2 1/4 (H7) δ G −1 2 Gf λV = (H11a) In this case, we see that at the quantum limit, the reflec- 1 + B · Gf tion coefficients s and s are exactly what is needed 11 22 0 B to have the input impedance diverge, while the reverse λI = (H11b) 1 + B · Gf gain coefficient s12 plays no role. For this case of strong Zb reflections in input and output arms, the voltage gain is Zout = (H11c) reduced compared to its zero-reflection value: 1 + B · Gf r 2 Zin = (1 + B · Gf )Za (H11d) Zb δ 1/4 2 λV → G (H8) Gf /2 Za 2 GP = (H11e) B · Gf + 2Zb/Za The power gain however is independent of θx, θz, and is still given by G/2 when θy is tuned to be at the quantum Again, Gf represents the gain of the amplifier in the ab- limit. sence of any feedback, Za is the input impedance at zero 34 feedback, and Zb is the output impedance at zero feed- keeping B non-zero but finite, one obtains: back. Transforming this into the scattering picture yields a 2! ¯ ¯ 2 2B 1 scattering matrix s satisfying: SVV SII → (~ω) 1 − + O (H16) Gf Gf BGf (Za − (2 + BGf )Zb) s11 = −s22 = − 2 (H12a) BGf Za + (2 + BGf ) Zb Thus, for a fixed, non-zero feedback ratio B, it is possible √ 2 Z Z G (1 + BG ) to reach the quantum limit. Note that if B does not tend s = − a b f f (H12b) to zero as G tends to infinity, the voltage gain of this 21 BG Z + (2 + BG )2Z f f a f b amplifier will be finite. The power gain however will be B proportional to Gf and will be large. If one wants a s12 = s21 (H12c) Gf large voltage gain, one could set B to go to zero with 1 Gf i.e. B ∝ √ . In this case, one will still reach Note the connection between these equations and the nec- Gf essary form of a quantum limited s-matrix found in the the quantum limit in the large Gf limit, and the voltage p previous subsection. gain will also be large (i.e. ∝ Gf ). Note that in all Now, given a scattering matrix, one can always find a these limits, the reflection coefficients s11 and s22 tend minimal representation of the noise operators Fa and Fb to −1 and 1 respectively, while the reverse gain tends which have the necessary commutation relations. These to 0. This is in complete analogy to the amplifier with are given in general by: mirrors considered in the previous subsection, in the case where we took the reflections to be strong at the input ˆ p 2 2 2 † and at the output (cf. Eq. (H7)). We thus see yet again Fa = 1 − |s11| − |s12| + |l| · uîn + l · vîn(H13) how the use of feedback allows the system to reach the ˆ p 2 2 † Fb = |s21| + |s22| − 1 · vîn (H14) quantum limit. s s∗ + s s∗ l = 11 21 12 22 (H15) Appendix I: Additional Technical Details p 2 2 |s21| + |s22| − 1 This appendix provides further details of calculations Applying this to the s matrix for our op-amp, and then presented in the main text. † taking the auxiliary modesu în andv în to be in the vacuum state, we can calculate the minimum allowed S¯VV and S¯II for our non-inverting op-amp amplifier. One can then calculate the product S¯VV S¯II and compare against 1. Proof of quantum noise constraint the quantum-limited value (S¯IV is again real). In the case of zero feedback (i.e. B = 0), one of course finds Note first that we may write the symmetrized Iˆ and Fˆ that this product is twice as big as the quantum limited noise correlators defined in Eqs. (4.4a) and (4.4b) as sums value. However, if one takes the large Gf limit while over transitions between detector energy eigenstates:

¯ X ˆ 2 SFF [ω] = π~ hi|ρˆ0|ii · |hf|F |ii| [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)] (I1) i,f ¯ X ˆ 2 SII [ω] = π~ hi|ρˆ0|ii · |hf|I|ii| [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)] (I2) i,f

Here,ρ ˆ0 is the stationary density matrix describing the To proceed, we fix the frequency ω > 0, and let the state of the detector, and |ii (|fi) is a detector energy index ν label each transition |ii → |fi contributing to the eigenstate with energy Ei (Ef ). Eq. (I1) expresses the noise. More specifically, ν indexes each ordered pair of noise at frequency ω as a sum over transitions. Each detector energy eigenstates states {|ii, |fi} which satisfy transition starts with an an initial detector eigenstate Ef − Ei ∈ ±~[ω, ω + dω] and hi|ρ0|ii= 6 0. We can now |ii, occupied with a probability hi|ρ0|ii, and ends with a consider the matrix elements of Iˆ and Fˆ which contribute final detector eigenstate |fi, where the energy difference to S¯II [ω] and S¯FF [ω] to be complex vectors ~v and ~w. between the two states is either +~ω or −~ω . Further, each transition is weighted by an appropriate matrix element. 35

Letting δ be any real number, let us deﬁne: for each pair of initial and ﬁnal states |ii, |fi contributing to S¯ [ω] and S¯ [ω] (cf. Eq. (I1)). Note that this not ˆ FF II [~w]ν = hf(ν)|F |i(ν)i (I3) the same as requiring Eq. (I13) to hold for all possible ( −iδ ˆ states |ii and |fi. This proportionality condition in turn e hf(ν)|I|i(ν)i if Ef(ν) − Ei(ν) = +~ω, [~v] = implies a proportionality between the input and output ν eiδhf(ν)|Iˆ|i(ν)i if E − E = − ω. f(ν) i(ν) ~ (unsymmetrized) quantum noise spectral densities: (I4) 2 SII [ω] = |α| SFF [ω] (I14) Introducing an inner product h·, ·iω via: It thus also follows that the imaginary parts of the input ~ X ∗ h~a, biω = π hi(ν)|ρˆ0|i(ν)i · (aν ) bν , (I5) and output susceptibilities are proportional: ν 2 Im χII [ω] = |α| Im χFF [ω], (I15) we see that the noise correlators S¯II and S¯FF may be written as: as well as the symmetrized input and output noise (i.e. Eq. (4.18)). Finally, one can also use Eq. (I13) to relate S¯ [ω]dω = h~v,~vi (I6) II ω the unsymmetrized I-F quantum noise correlator SIF [ω] S¯FF [ω]dω = h~w, ~wiω (I7) to SFF [ω]: (cf. Eq. (4.7)): ( We may now employ the Cauchy-Schwartz inequality: e−iδα∗S [ω] if ω > 0, S [ω] = FF (I16) 2 IF eiδα∗S [ω] if ω < 0 h~v,~viωh~w, ~wiω ≥ |h~v, ~wiω| (I8) FF

A straightforward manipulation shows that the real Note that SFF [ω] is necessarily real and positive. part of h~v, ~wiω is determined by the symmetrized cross- Finally, for a detector with quantum-ideal noise prop- correlator S¯IF [ω] defined in Eq. (4.4c): erties, the magnitude of the constant α can be found from Eq. (4.18). The phase of α can also be determined from: iδ Re h~v, ~wiω = Re e S¯IF [ω] dω (I9) −Im α ~|χ˜IF |/2 ˜ In contrast, the imaginary part of h~v, ~wiω is independent = p cos δ0 (I17) |α| S¯ S¯ of S¯IF ; instead, it is directly related to the gain χIF and II FF reverse gain χFI of the detector: For zero frequency or for a large detector effective temperature, this simplifies to: ~ iδ ∗ Im h~v, ~wiω = Re e χIF [ω] − [χFI [ω]] dω (I10) 2 −Im α χ˜ /2 = ~ IF (I18) Substituting Eqs. (I10) and (I9) into Eq. (I8), one im- p |α| S¯II S¯FF mediately finds the quantum noise constraint given in ∗ Eq. (4.16). As in the main text, we letχ ˜IF = χIF −χFI . Note importantly that to have a non-vanishing gain Maximizing the RHS of this inequality with respect to and power gain, one needs Im α 6= 0. This in turn places the phase δ, one finds that the maximum is achieved for a very powerful constraint on a quantum-ideal detectors: ˜ δ = δ0 = − arg(˜χIF ) + δ0 with all transitions contributing to the noise must be to final states |fi which are completely unoccupied. To see this, ¯ ˜ |SIF | sin 2φ imagine a transition taking an initial state |ii = |ai to tan 2δ0 = − ¯ (I11) (~/2)|χ˜IF | + |SIF | cos 2φ a final state |fi = |bi makes a contribution to the noise. For a quantum-ideal detector, Eq. (I13) will be satisfied: where φ = arg(S¯IF χ˜IF ∗). At δ = δ0, Eq. (4.16) becomes the final noise constraint of Eq. (4.11). hb|Iˆ|ai = e±iδαhb|Fˆ|ai (I19) The proof given here also allows one to see what must be done in order to achieve the “ideal” noise condition where the plus sign corresponds to Eb > Ea, the mi- of Eq. (4.17): one must achieve equality in the Cauchy- nus to Ea > Eb. If now the final state |bi was also oc- Schwartz inequality of Eq. (I8). This requires that the cupied (i.e. hb|ρˆ0|bi= 6 0), then the reverse transition vectors ~v and ~w be proportional to one another; there |i = bi → |f = ai) would also contribute to the noise. must exist a complex factor α (having dimensions [I]/[F ]) The proportionality condition of Eq. (I13) would now re- such that: quire:

~v = α · ~w (I12) ha|Iˆ|bi = e∓iδαha|Fˆ|bi (I20)

Equivalently, we have that As Iˆ and Fˆ are both Hermitian operators, and as α must have an imaginary part in order for there to be gain, we ( iδ e αhf|F |ii if Ef − Ei = +~ω have a contradiction: Eq. (I19) and (I20) cannot both be hf|I|ii = −iδ (I13) e αhf|F |ii if Ef − Ei = −~ω. true. It thus follows that the final state of a transition 36 contributing to the noise must be unoccupied in order proportionality relation in Eq. (4.8b) forχ ˜IF yields: for Eq. (I13) to be satisfied and for the detector to have ideal noise properties. Note that this necessary asym- α χ˜IF = (SFF [ω] − SFF [−ω]) metry in the occupation of detector energy eigenstates ~ immediately tells us that a detector or amplifier cannot = 2α [−Im χFF [ω]] (I22) reach the quantum limit if it is in equilibrium. Using this result and the relation between χFF and χII in Eq. (I15), we can write the power gain in the absence of reverse gain, G (c.f. Eq. (5.52)), as 2. Proof that a noiseless detector does not amplify P 2 With the above results in hand, we can now prove as- GP = 1/ |1 − χFI /χIF | (I23) sertions made in Sec. IV.A.4 that detectors which evade the quantum noise constraint of Eq. (4.11) and simply If the reverse gain vanishes (i.e. χFI = 0), we immedi- satisfy ately find that GP = 1: the detector has a power gain of one, and is thus simply a transducer. If the reverse gain S¯ S¯ = |S¯ |2 (I21) is non-zero, we must take the expression for GP above FF II IF and plug it into Eq. (5.53) for the power gain with re- are at best transducers, as their power gain is limited to verse gain, GP,rev. Some algebra again yields that the being at most one. full power gain is at most unity. We again have the conclusion that the detector does not amplify. The first way to make the RHS of Eq. (4.11) vanish is ∗ to have χIF = χFI . We have already seen that whenever this relation holds, the detector power gain cannot be any larger than one (c.f. Eq. (5.53)). Now, imagine 3. Simplifications for a quantum-limited detector that the detector also has a minimal amount of noise, i.e. Eq. (I21) also holds. This latter fact implies that the In this appendix, we derive the additional constraints proportionality condition of Eq. (I13) also must hold. In on the property of a detector that arise when it satisfies this situation, the detector must have a power gain of the quantum noise constraint of Eq. (4.17). We focus on unity, and is thus a transducer. There are two possibil- the ideal case where the reverse gain χFI vanishes. ities to consider here. First, S¯FF and S¯II could both To start, we substitute Eq. (I16) into Eqs. (4.8b)- 2 be non-zero, but perfectly correlated: |S¯IF | = S¯FF S¯II . (4.8a); writing SFF [ω] in terms of the detector effective In this case, the proportionality constant α must be real temperature Teff (cf. Eq. (2.8)) yields: (c.f. Eq. (I17)). Using this fact along with Eqs. (I16) and λ[ω] (4.8b), one immediately finds that SFF [ω] = SFF [−ω]. ~ = −e−iδ [−Im χ [ω]] (I24) This implies the back-action damping γ associated with 2 ~ FF the detector input vanishes (c.f. Eq. (2.12)). It thus fol- ω (Im α) coth ~ + i (Re α) lows immediately from Eq. (5.52) and Eq. (5.53) that the 2kBTeff power gain G (defined in a way that accounts for the P,rev S¯ [ω] = e−iδ [−Im χ [ω]] (I25) reverse gain) is exactly one. The detector is thus simply a IF ~ FF ∗ ω transducer. The other possibility here is that χIF = χFI (Re α) coth ~ − i (Im α) and one or both of S¯II , S¯FF are equal zero. Note that if 2kBTeff the symmetrized noise vanishes, then so must the asymmetric part of the noise. Thus, it follows that either the To proceed, let us write: damping induced by the detector input, γ, or that in- λ ˜ duced by the output, γout (c.f. Eq. (5.48)) (or both) must e−iδ = e−iδ (I26) be zero. Eqs. (5.52) and (5.53) then again yield a power |λ| gain GP,rev = 1. We thus have shown that any detector ∗ ¯ ¯ ¯ 2 which has χFI = χIF and satisfies SFF SII = |SIF | must The condition that |λ| is real yields the condition: necessarily be a transducer, with a power gain precisely equal to one. Re α ω tan δ˜ = tanh ~ (I27) A second way to make the RHS of Eq. (4.11) vanish is Imα 2kBTeff to have S¯IF /χ˜IF be purely imaginary and larger in magnitude than ~/2. Suppose this is the case, and that the We now consider the relevant limit of a large detector detector also satisfies the minimal noise requirement of power gain GP . GP is determined by Eq. (5.56); the only Eq. (I21). Without loss of generality, we takeχ ˜IF to be way this can become large is if kBTeff /(~ω) → ∞ while real, implying that S¯IF is purely imaginary. Eqs. (I10) Im α does not tend to zero. We will thus take the large iδ and (I11) then imply that the phase factor e appear- Teff limit in the above equations while keeping both α and ing in the proportionality relation of Eq. (I16) is purely the phase of λ fixed. Note that this means the parameter ˜ imaginary, while the constant α is purely real. Using this δ must evolve; it tends to zero in the large Teff limit. In 37 this limit, we thus find for λ and S¯IF : Using this lowest-order self energy, Eq. (I33) yields:

" " 2## R ~ ~λ[ω] −iδ ~ω G [ω] = = −2e k T γ[ω] (Im α) 1 + O 2 2 2 R B eff m(ω − Ω ) − A Re D [ω] + iω(γ0 + γ[ω]) 2 kBTeff (I36) (I28) A R ∗ G [ω] = G [ω] (I37) −iδ ~ω S¯IF [ω] = 2e kBTeff γ[ω] (Re α) 1 + O K R kBTeff G [ω] = −2iIm G [ω] × ω ω (I29) γ0 coth ~ + γ[ω] coth ~ 2kBTbath 2kBTeff

Thus, in the large power-gain limit (i.e. large Teff γ0 + γ[ω] ¯ limit), the gain λ and the noise cross-correlator SIF have (I38) the same phase: S¯IF /λ is purely real. where γ[ω] is given by Eq. (2.12), and Teff [ω] is defined by Eq. (2.8). The main effect of the real part of the retarded 4. Derivation of non-equilibrium Langevin equation Fˆ Green function DR[ω] in Eq. (I36) is to renormalize the oscillator frequency Ω and mass m; we simply incorporate In this appendix, we prove that an oscillator weakly these shifts into the definition of Ω and m in what follows. coupled to an arbitrary out-of-equilibrium detector is de- If Teff [ω] is frequency independent, then Eqs. (I36) scribed by the Langevin equation given in Eq. (5.41), an - (I38) for Gˇ corresponds exactly to an oscillator cou- equation which associates an effective temperature and pled to two equilibrium baths with damping kernels γ0 damping kernel to the detector. The approach taken here and γ[ω]. The correspondence to the Langevin equa- is directly related to the pioneering work of Schwinger tion Eq. (5.41) is then immediate. In the more general (Schwinger, 1961). case where Teff [ω] has a frequency dependence, the cor- We start by defining the oscillator matrix Keldysh relators GR[ω] and GK [ω] are in exact correspondence green function: to what is found from the Langevin equation Eq. (5.41): K K R G [ω] corresponds to symmetrized noise calculated from ˇ G (t) G (t) R G(t) = A (I30) Eq. (5.41), while G [ω] corresponds to the response co- G (t) 0 efficient of the oscillator calculated from Eq. (5.41). This again proves the validity of using the Langevin equation where GR(t − t0) = −iθ(t − t0)h[ˆx(t), xˆ(t0)]i, GA(t − t0) = Eq. (5.41) to calculate the oscillator noise in the presence iθ(t0 −t)h[ˆx(t), xˆ(t0)]i, and GK (t−t0) = −ih{xˆ(t), xˆ(t0)}i. of the detector to lowest order in A. At zero coupling to the detector (A = 0), the oscillator is only coupled to the equilibrium bath, and thus Gˇ0 has the standard equilibrium form: 5. Linear-response formulas for a two-port bosonic amplifier ! ~ω ~ −2Im g0[ω] coth 2k T g0[ω] Gˇ0[ω] = B bath In this appendix, we use the standard linear-response ∗ m g0[ω] 0 Kubo formulas of Sec. V.F to derive expressions for (I31) 0 the voltage gain λV , reverse current gain λI , input where: impedance Zin and output impedance Zout of a two-port 1 bosonic voltage amplifier (cf. Sec. VI). We recover the g [ω] = (I32) 0 ω2 − Ω2 + iωγ /m same expressions for these quantities obtained in Sec. VI 0 from the scattering approach. We stress throughout this and where γ0 is the intrinsic damping coefficient, and appendix the important role played by the causal struc- Tbath is the bath temperature. ture of the scattering matrix describing the amplifier. We next treat the effects of the coupling to the detector In applying the general linear response formulas, we in perturbation theory. Letting Σˇ denote the correspond- must bear in mind that these expressions should be ap- ing self-energy, the Dyson equation for Gˇ has the form: plied to the uncoupled detector, i.e. nothing attached to

A the detector input or output. In our two-port bosonic ˇ −1 ˇ −1 0 Σ [ω] G[ω] = G0[ω] − R K (I33) voltage amplifier, this means that we should have a Σ [ω]Σ [ω] short circuit at the amplifier input (i.e. no input voltage, V = 0), and we should have open circuit at the out- To lowest order in A, Σ[ˇ ω] is given by: a put (i.e. Ib = 0, no load at the output drawing current). Σ[ˇ ω] = A2Dˇ[ω] (I34) These two conditions define the uncoupled amplifier. Us- A2 Z ing the definitions of the voltage and current operators ≡ dt eiωt (I35) (cf. Eqs. (6.2a) and (6.2b)), they take the form: ~ 0 iθ(−t)h[Fˆ(t), Fˆ(0)]i aîn[ω] = −aôut[ω] (I39a) ˆ ˆ −iθ(t)h[Fˆ(t), Fˆ(0)]i −ih{Fˆ(t), Fˆ(0)}i bin[ω] = bout[ω] (I39b) 38

The scattering matrix equation Eq. (6.3) then allows us where we have defined the real function Λaa[ω] for ω > 0 to solve fora în anda ôut in terms of the added noise via: operators Fâ and Fˆb. h † 0 i 0 1 − s s aîn[ω], aîn(ω ) = 2πδ(ω − ω )Λaa[ω] (I44) aˆ [ω] = − 22 Fˆ [ω] − 12 Fˆ [ω] (I40a) in D a D b s 1 + s It will be convenient to also define Λaa[ω] for ω < 0 ˆb [ω] = − 21 Fˆ [ω] + 11 Fˆ [ω] (I40b) in D a D b via Λaa[ω] = Λaa[−ω]. Eq. (I43) may then be written as: ∞ ∞ 0 0 where D is given in Eq. (6.8), and we have omitted writ- 2 Z Z dω ω 0 Y [ω] = dt Λ (ω0)ei(ω−ω )t ing the frequency dependence of the scattering matrix. in,Kubo Z 2π ω aa Further, as we have already remarked, the commutators a 0 −∞ Z ∞ 0 0 of the added noise operators is completely determined by Λaa[ω] i 0 ω Λaa(ω )/Za = + P dω 0 the scattering matrix and the constraint that output op- Za πω −∞ ω − ω erators have canonical commutation relations. The non- (I45) vanishing commutators are thus given by: Next, by making use of Eq. (I40a) and Eqs. (I41) for h ˆ ˆ† 0 i 0 2 2 Fa[ω], Fa(ω ) = 2πδ(ω − ω ) 1 − |s11| − |s12| the commutators of the added noise operators, we can (I41a) explicitly evaluate the commutator in Eq. (I44) to calculate Λaa[ω]. Comparing the result against the result h ˆ ˆ† 0 i 0 2 2 Fb [ω], Fb (ω ) = 2πδ(ω − ω ) 1 − |s21| − |s22| Eq. (6.7d) of the scattering calculation, we find: (I41b) Λaa[ω] h ˆ ˆ† 0 i 0 ∗ ∗ = Re Yin,scatt[ω] (I46) Fa[ω], Fb (ω ) = −2πδ(ω − ω )(s11s21 + s12s22) Za (I41c) where Yin,scatt[ω] is the input admittance of the ampli- The above equations, used in conjunction with fier obtained from the scattering approach. Returning to Eqs. (6.2a) and (6.2b), provide us with all all the infor- Eq. (I45), we may now use the fact that Yin,scatt[ω] is mation needed to calculate commutators between current an analytic function in the upper half plane to simplify and voltage operators. It is these commutators which en- the second term on the RHS, as this term is simply a ter into the linear-response Kubo formulas. As we will Kramers-Kronig integral: see, our calculation will crucially rely on the fact that 1 Z ∞ ω0Λ (ω0)/Z the scattering description obeys causality: disturbances 0 aa a P dω 0 at the input of our system must take some time before πω −∞ ω − ω they propagate to the output. Causality manifests itself 1 Z ∞ ω0Re Y (ω0) = P dω0 in,scatt in the energy dependence of the scattering matrix: as a 0 πω −∞ ω − ω function of energy, it is an analytic function in the upper = Im Y [ω] (I47) half complex plane. in,scatt It thus follows from Eq. (I45) that input impedance calculated from the Kubo formula is equal to what we found a. Input and output impedances previously using the scattering approach. The calculation for the output impedance proceeds in Eq. (5.84) is the linear response Kubo formula for the same fashion, starting from the Kubo formula given the input impedance of a voltage amplifier. Recall in Eq. (5.85). As the steps are completely analogous to that the input operator Qˆ for a voltage amplifier is re- the above calculation, we do not present it here. One lated to the input current operator Iâ via −dQ/dtˆ = ˆ again recovers Eq. (6.7c), as found previously within the Ia (cf. Eq. (5.83)). The Kubo formula for the input scattering approach. impedance may thus be re-written in the more familiar form: Z ∞ i i iωt b. Voltage gain and reverse current gain Yin,Kubo[ω] ≡ − dth[Iâ(t), Iâ(0)]ie (I42) ω ~ 0 Within linear response theory, the voltage gain of the where Y [ω] = 1/Z [ω], in in amplifier (λ ) is determined by the commutator between Using the defining equation for Iˆ (Eq. (6.1b)) and V a the “input operator” Qˆ and Vˆ (cf. Eq. (4.3); recall that Eq. (I39a) (which describes an uncoupled amplifier), we b Qˆ is defined by dQ/dtˆ = −Iâ. Similarly, the reverse obtain: 0 current gain (λI ) is determined by the commutator be- Y [ω] = (I43) in,Kubo tween Iâ and Φ,ˆ where Φˆ is defined via dΦˆ/dt = −Vˆb Z ∞ Z ∞ 0 0 2 iωt dω ω 0 −iω0t iω0t (cf. Eq. (4.6)). Similar to the calculation of the input dte Λaa(ω ) e − e Za 0 0 2π ω impedance, to properly evaluate the Kubo formulas for 39 the gains, we must make use of the causal structure of dependence in the scattering in our amplifier arises from the scattering matrix describing our amplifier. the fact that there are small transmission line “stubs” of Using the defining equations of the current and volt- length a attached to both the input and output of the age operators (cf. Eqs. (6.1a) and (6.1b)), as well as amplifier (these stubs are matched to the input and out- Eqs. (I39a) and (I39b) which describe the uncoupled am- put lines). Because of these stubs, a wavepacket incident plifier, the Kubo formulas for the voltage gain and reverse on the amplifier will take a time τ = 2a/v to be either current gain become: reflected or transmitted, where v is the characteristic velocity of the transmission line. This situation is described r Zb by a scattering matrix which has the form: λV,Kubo[ω] = 4 (I48) Za 2iωa/v ∞ ∞ 0 s[ω] = e · s¯ (I53) Z Z dω 0 × dt eiωtRe Λ (ω0)e−iω t 2π ba 0 0 wheres ¯ is frequency-independent and real. To further r 0 Zb simplify things, let us assume thats ¯11 =s ¯22 =s ¯12 = 0. λI,Kubo[ω] = −4 (I49) Za Eqs. (I51) then simplifies to ∞ ∞ 0 Z Z 0 iωt dω 0 iω t iωτ × dt e Re Λba(ω )e Λba[ω] = = s21[ω] =s ¯21e (I54) 0 0 2π where the propagation time τ = 2a/v We then have: where we define the complex function Λba[ω] for ω > 0 via: r Z ∞ Zb λ [ω ] = 2 s¯ dt eiω0tδ(t − τ) (I55) hˆ † 0 i 0 V 0 21 bin[ω], aîn(ω ) ≡ (2πδ(ω − ω )) Λba[ω] (I50) Za 0 r Z Z ∞ b iω0t λI [ω0] = −2 s¯21 dt e δ(t + τ) (I56) We can explicitly evaluate Λba[ω] by using Eqs. (I40a)- Za 0 (I41) to evaluate the commutator above. Comparing the result against the scattering approach expressions for the If we now do the time integrals and then take the limit gain and reverse gain (cf. Eqs. (6.7a) and (6.7b)), one τ → 0+, we recover the results of the scattering approach finds: (cf. Eqs. (6.7a) and (6.7b)); in particular, λI = 0. Note ∗ that if we had set τ = 0 from the outset of the calculation, Λ [ω] = λ [ω] − λ0 [ω] (I51) ba V,scatt I,scatt we would have found that both λV and λI are non-zero! Note crucially that the two terms above have different analytic properties: the first is analytic in the upper half 6. Details for the two-port bosonic voltage amplifier with plane, while the second is analytic in the lower half plane. feedback This follows directly from the fact that the scattering matrix is causal. In this appendix, we provide more details on the calcu- At this stage, we can proceed much as we did in the lations for the bosonic-amplifier-plus-mirrors system dis- calculation of the input impedance. Defining Λ [ω] for ba cussed in Sec. H. Given that the scattering matrix for ω < 0 via Λ [−ω] = Λ∗ [ω], we can re-write Eqs. (I48) ba ba each of the three mirrors is given by Eq. (H2), and that and (I49) in terms of principle part integrals. we know the reduced scattering matrix for the mirror-free Z ∞ 0 system (cf. Eq. (6.10)), we can find the reduced scattering Zb i 0 Λba(ω ) λV,Kubo[ω] = Λba[ω] + P dω 0 matrix and noise operators for the system with mirrors. Za π −∞ ω − ω One finds that the reduced scattering matrix s is now Z ∞ 0 0 Zb i 0 Λba(ω ) given by: λI,Kubo[ω] = − Λba[ω] + P dω 0 Za π −∞ ω + ω 1 (I52) s = × (I57) M √ Using the analytic properties of the two terms in Eq. (I51) sin θ√z + G sin θx sin θy − cos θ√x cos θz sin θy for Λba[ω], we can evaluate the principal part integrals G cos θx cos θz sin θx + G sin θy sin θz above as Kramers-Kronig relations. One then finds that the Kubo formula expressions for the voltage and cur- where the denominator M describes multiple reflection rent gain coincide precisely with those obtained from the processes: scattering approach. √ While the above is completely general, it is useful to M = 1 + G sin θx sin θz sin θy (I58) go through a simpler, more specific case where the role of causality is more transparent. Imagine that all the energy Further, the noise operators are given by: 40

√ Fa 1 cos θy cos θz G − 1 cos θz sin θx sin θy uin = √ √ † (I59) Fb M − G cos θx cos θy sin θz G − 1 cos θx vin

The next step is to convert the above into the op-amp output line (e.g. θx): representation, and find the gains and impedances of the amplifier, along with the voltage and current noises. The 2~ω 1 − sin θz S¯II = × voltage gain is given by: Za 1 + sin θz   √ r G sin2 θ + cos(2θ ) ZB 2 G 1 + sin θx 1 − sin θz  y y  (I65a) λV = √ · (I60) √ 2 Z cos θ cos θ   A 1 − G sin θy x z G sin θy − 1 ¯ 1 + sin θz while the reverse gain is related to the voltage gain by SVV = ~ωZa × the simple relation: 1 − sin θz 3 + cos(2θ ) 1 y − (I65b) 0 sin θy 4 2G λ = − √ λV (I61) √ I 2 G G(1 − 1/G) sin θy + cos θy S¯VI = √ (I65c) 1 − G sin θy The input impedance is determined by the amount of reflection in the input line and in the line going to the As could be expected, introducing reflections in the input cold load: line (i.e. θz 6= 0) has the opposite effect on S¯II versus ¯ √ SVV : if one is enhanced, the other is suppressed. 1 − G sin θy 1 + sin θz It thus follows that the product of noise spectral Zin = Za √ · (I62) densities appearing in the quantum noise constraint of 1 + G sin θy 1 − sin θz Eq. (5.88) is given by (taking the large-G limit):

Similarly, the output impedance only depends on the ¯ ¯ 2 SII SVV 2 1 + G sin θy amount of reflection in the output line and the in the = 2 − sin θy · (I66) ( ω)2 √ 2 cold-load line: ~ 1 − G sin θy √ 1 + G sin θy 1 + sin θx Note that somewhat amazingly, this product (and hence Zout = Zb √ · (I63) 1 − G sin θy 1 − sin θx the amplifier noise temperature) is completely independent of the mirrors in the input and output arms (i.e. θz √ and θ ). This is a result of both S¯ and S¯ having Note that as sin θ tends to −1/ G , both the input x VV II y no dependence on the output mirror (θ ), and their hav- admittance and output impedance tend to zero. x ing opposite dependencies on the input mirror (θz). Also Given that we now know the op-amp parameters of our note that Eq. (I66) does indeed reduce to the result of amplifier, we can use Eq. (6.5) to calculate the amplifier’s the last subsection: if θy = 0 (i.e. no reflections in the power gain G . Amazingly, we find that the power gain P line going to the cold load), the product S¯II S¯VV is equal is completely independent of the mirrors in the input and to precisely twice the quantum limit value of (~ω)2. For output lines: √ sin(θy) = −1/ G, the RHS above reduces to one, implying that we reach the quantum limit for this tuning of G the mirror in the cold-load arm. GP = 2 (I64) 1 + G sin θy

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