Cavity Quantum Optomechanics with Ultracold Atoms

Home , Optomechanics, Ultracold atom

Kater Whitney Murch

B.A. (Reed College) 2002 M.A. (University of California, Berkeley) 2007

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy

Physics

in the

GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY

Committee in charge: Professor Dan M. Stamper-Kurn, Chair Professor Irfan Siddiqi Professor Birgitta Whaley

Spring 2008 The dissertation of Kater Whitney Murch is approved:

Chair Date

Date

University of California, Berkeley

Spring 2008 Cavity Quantum Optomechanics with Ultracold Atoms

Abstract

Cavity Quantum Optomechanics with Ultracold Atoms

Kater Whitney Murch Doctor of Philosophy in Physics

University of California, Berkeley

Professor Dan M. Stamper-Kurn, Chair

A common goal of recent research is the elucidation and control over quantum mechanical behavior in ever-larger physical systems. In this thesis I present an alternative target for investigating the quantum motion of macroscopic bodies: the collective motion of an ultracold atomic gas trapped within a high-finesse Fabry-Perot optical cavity in the single- atom strong-coupling regime of cavity quantum electrodynamics (CQED). When ultracold atoms are trapped in the Lamb-Dicke regime, the cavity-mode structure selects a single collective degree of freedom that is at once actuated by the optical forces from cavity probe light and measured by the cavity’s optical properties. Dispersive optical bistability arising from collective motion of the atomic medium was observed. Measurement of the collective motion was subject to quantum measurement backaction by the quantum force fluctuations of the cavity optical field. The strength and spectrum of these backaction force fluctuations was measured by quantifying the cavity-light-induced heating rate of the intracavity atomic ensemble, finding quantitative agreement with the expected heating rate from quantum optical fluctuations. Dynamical phenomena in the optomechanical system were explored experimentally and theoretically. Quantum limited measurements are discussed and were explored experimentally. The application of these quantum limited measurements to precise measurements of the gravitational inverse-square law using betatron resonances is discussed.

Professor Dan M. Stamper-Kurn Dissertation Committee Chair 2 i

To all my teachers who broke the mold:

Don Jolley, David Lapp, Nicholas Wheeler ii

Contents

List of Figures iv

List of Tables vi

1 Introduction 3 1.1 Measurement and the quantum limit ...... 3 1.2 Cavity QED ...... 5 1.3 Optomechanics ...... 7 1.4 The next 100 pages ...... 8

2 Cavity-optomechanics with cold atoms 10 2.1 A model optomechanical system ...... 10 2.2 Many atom CQED ...... 12 2.3 Toward collective variables ...... 14 2.4 Normal modes of the system ...... 17 2.5 Comparison to traditional CQED: Cavity quantum optomechanics . . . . . 19

3 The experiment 21 3.1 The optical ﬁelds ...... 21 3.1.1 An imperfect cavity ...... 23 3.1.2 Cavity-QED parameters ...... 24 3.1.3 Photon detection eﬃciency ...... 24 3.2 The cavity lock chain ...... 25 3.2.1 The transfer cavity ...... 25 3.2.2 The science cavity ...... 28 3.2.3 Noise ...... 29 3.2.4 The temperature of photo-detection ...... 32 3.3 The FORT ...... 33 3.4 The atoms ...... 34 3.4.1 Condensation, collisions and the loitering at each lattice site . . . . . 34 3.5 Distribution of atoms in the lattice ...... 35 3.6 Experimental tricks ...... 37 3.6.1 Frequency sweeps ...... 37 3.6.2 Wait and see ...... 40 iii

3.6.3 Absorption imaging and time of ﬂight ...... 40

4 Nonlinear optics from collective motion 42 4.1 Adiabatic collective motion ...... 44 4.2 The bistable potential ...... 47 4.3 Connection to experimental measures ...... 49 4.4 Granularity ...... 51 4.5 Nonlinearities at very low photon number ...... 52

5 Quantum measurement backaction 55 5.1 The two faced nature of light ...... 55 5.2 A model quantum measurement ...... 56 5.3 Backaction heating ...... 60 5.3.1 A quantum limited amplifier ...... 63 5.4 Heating from incoherent scattering ...... 66 5.5 Measuring backaction heating by the evaporative loss of trapped atoms . . 67 5.5.1 Line rates ...... 68 5.5.2 Technical sources of heating ...... 70 5.5.3 Quantitative interpretation of the cavity line shape ...... 73 5.6 Off cavity-resonance heating ...... 76 5.7 Connection to quantum optics: an intracavity fluctuation bolometer . . . . 79

6 Collective motion 83 6.1 “Kick and watch” ...... 84 6.2 The optomechanical frequency shift ...... 86 6.3 Ampliﬁcation, damping, and saturation ...... 88 6.4 Toward quantum limited measurement ...... 89 6.4.1 Parameter estimation ...... 90 6.4.2 The “kick” ...... 94 6.4.3 Evolution ...... 95 6.5 Backaction induced phase diﬀusion ...... 101 6.6 The right left and center of cavity resonance ...... 103 6.7 The quantum–classical boundary: granularity revisited ...... 103 6.8 What? Who? Which way photons leave the cavity ...... 106

7 Betatron motion in the ultracold atom storage ring 109 7.1 Forming a storage ring for cold atoms ...... 111 7.2 Betatron resonances ...... 113 7.3 Modeling betatron resonances ...... 118 7.4 Dispersion Management ...... 118

8 A proposal to test the gravitational inverse-square law 129

Bibliography 134 iv

List of Figures

1.1 Nature’s rulers ...... 4 1.2 Cavity QED ...... 5

2.1 The basic optomechanical system ...... 11 2.2 The spectrum of the many atom-cavity system...... 13 2.3 The atoms-cavity system ...... 15

3.1 Schematic of the cavity lock chain ...... 26 3.2 Laser stabilization ...... 27 3.3 The Voigt proﬁle ...... 29 3.4 Characterization of the science cavity lock ...... 32 3.5 Spatial dependence of the atom cavity coupling ...... 36 3.6 Absorption imaging versus cavity based atom counting ...... 36 3.7 Schematic of a typical experiment ...... 38 3.8 Two types of experiments ...... 39 3.9 Time-of-ﬂight temperature measurement ...... 40

4.1 Asymmetric cavity lineshapes due to collective motion ...... 46 4.2 Dispersive bistability ...... 48 4.3 Bistable ringing ...... 49 4.4 Nonlinear and bistable cavity lineshapes ...... 50 4.5 Observation of collective adiabatic motion ...... 51 4.6 Optical nonlinearity at low photon number ...... 52 4.7 Nonlinearity withn ¯ ∼ 0.1 ...... 53

5.1 Basics of a quantum measurement ...... 59 5.2 Quantum limited ampliﬁers ...... 65 5.3 Study of thermal and evaporative equilibration ...... 69 5.4 Temporal cavity lineshapes ...... 71 5.5 Line-time measurements ...... 72 5.6 Cavity-based observation of evaporative atomic losses ...... 75 5.7 Cavity-heating of the collective atomic mode ...... 76 5.8 Controlled dose experiment ...... 78 5.9 Branching ratios ...... 79 v

5.10 The ﬂuctuation bolometer ...... 81

6.1 Kick and watch experiment ...... 84 6.2 The optomechanical frequency shift ...... 87 6.3 The nonlinear optomechanical frequency shift ...... 88 6.4 Study of data analysis techniques ...... 92 6.5 Amplitude estimates ...... 93 6.6 A classical kick for a quantum displacement ...... 96 6.7 Relevant timescales for measurement ...... 99 6.8 The kick and watch experiment ...... 101 6.9 Time resolved quantum measurement ...... 102 6.10 Spectrum of the cavity resonance ...... 104 6.11 Nonlinearity in the granular regime ...... 105

7.1 Forming a circular magnetic storage ring for ultracold atoms ...... 112 7.2 An ultracold-atom storage ring ...... 114 7.3 Computer simulation of a νz = 5 betatron resonance ...... 119 7.4 Basis of dispersion management with betatron resonances ...... 120 7.5 Stopbands for νr = 5 radial and νz = 5 axial betatron resonances...... 121 7.6 Dispersion management of matter waves in a storage ring ...... 123 7.7 Matter wave dispersion at an axial betatron resonance ...... 124 7.8 Axial betatron motion ...... 125 7.9 Tuning the resonant velocities of betatron resonances ...... 126 7.10 Deliberate variation of the ν = 4 radial resonance ...... 127 7.11 Manipulating the νr = 4 betatron resonance ...... 128

8.1 Excluded regions for Yukawa additions to Newtonian gravity ...... 132 vi

List of Tables

3.1 Parameters and deﬁnintions ...... 22

4.1 Common terms ...... 43

5.1 Account of errors ...... 77 vii

Acknowledgments

Sometime about five years ago I rode my bicycle the three miles up hill to the UC Berkeley campus for the first time. By some fortune, my soon to be good friend and mentor, Kevin Moore, had arrived to work spuriously early that day, saving me hours of wandering in the contorted hallways of the physics department. During that first summer I was overwhelmed with the eloquence, tenacity, brilliance, and patience of the Stamper-Kurn group. Today, riding into to work, eight thousand miles later, I have the opportunity to reflect on all those years and thank those who held my hand along the way. At the top of the list is my advisor, Dan M. Stamper-Kurn. Dan has always given me the right blend of encouragement, from the overt demands of why don’t you get this done by tomorrow?, to flattering (and probably undeserved) compliments, to setting an example of dauntingly unachievable genius. I have always told people that there is something oppressive about having an advisor who is always right, but then again, he is surely the person you want to be leading your team. The majority of my time (or at least the productive time) at Berkeley was spent with three outstanding colleagues, Kevin Moore, Tom Purdy, and Subhadeep Gupta. I am forever indebted to Kevin, not just for the inclusion of indie music in to my quirky tastes, but for the two whole years he spent toiling away in lab with prototype experiments, soldering electronics, machining parts and generally building our whole lab. Kevin, Deep and I worked well as a group, with Deep and Kevin bantering back and forth in Lebowski reference, and me trying desperately to keep our experiments from dragging too late into the evening. Deep and Kevin have been amazing mentors, advisors and friends. Tom Purdy, who once mentioned that he didn’t like to use a particular type of function generator because he didn’t know exactly what was inside, is responsible for most every thing in our lab which works well, from the millimeter scale electromagnets which hold our atoms, to the design of the electronics which lock our cavity, to the computer system which makes everything happen. The Stamper-Kurn group has been an amazing environment to work and learn in. Cooperative and constructive, our unity ranges from the daily trips to eat curry for lunch to the post-group-meeting game of soccer. Everybody has worked relentlessly to shape this group: Lorraine Sadler tirelessly tried to make our group meetings and journal club useful, Ananth Chikkatur instituted the group-lunch-every-single-day, Kevin Moore viii realized the importance of indoor bicycle parking (or at least I give him credit for it), Mukund Vengalattore lead the S-K group soccer team, and Anton Ottl¨ brought the brass.

Through all the great mentors and teachers I have had, three stand out in particular. My sixth grade teacher, Don Jolley boiled road-kill opossums, rabbits, and raccoons in class so that we students could assemble the bones of these unfortunate animals. It was art, anatomy, math, ecology, and history in the same sentence. There were no boundaries to where our class went, over the years Don’s class has moved from scrubbing stinky cow bones to assembling museum quality marine mammal skeletons, then on to paleontology and geology in Utah and large format pinhole camera photography in Joshua Tree. My first physics class in high school was taught by a projectile crazed, gun toting libertarian, David Lapp. Predictably, real guns were fired in the classroom on more than one occasion –to learn about conservation of momentum. David sparked my interest in physics, music, and food. After grilling hamburgers during the study hall period of class, and the subsequent fire-alarm-induced evacuation of the building we would spend the lunch hour jamming on the musical instruments students had made for the class that year. Nicholas Wheeler, my undergraduate thesis advisor, was by all accounts a very old school professor of physics. When teaching a class he would not assign a single text book, but would assume that you would naturally check out every book on the subject from the library and read them all in supplement to his beautifully crafted notes. As my thesis advisor he taught me to think in whole sentences; to approach problems from their beginning and work them logically to their end. Beyond academia, I owe special thanks to my friends and family. My parents Sarah and Don, grandparents Carol, Dexter, Bob, and Lou, my brother Mickey. Living in Berkeley, I have found an amazing community of friends, which range from academic colleagues, Andrew Essin, and Hal Haggard, to my dear housemates without whom life would be rather miserable, Marty, Amber, and Hal. Finally, and most importantly, the love of my life, my best friend, and my wife, Becky Bart. From moving into a run-down house in a sketchy neighborhood in a foreign city, to commuting one hundred and twenty miles each day, Becky has sacrificed more than I can possibly list to make our life as blissful as it is. She rightly deserves credit for more than half of everything I have done. 1

Foreward

I distinctly remember, somewhere in the back of my Father’s dusty work shop, using an old nail to poke holes into the lid of a mason jar. For the third or fourth time that week I was making a quart sized terrarium so that I would have a good place to store all the bugs which I was about to collect. We found lots of bugs back in those days. Lifting up boards which were left on the ground for a long time we would find centipedes, millipedes, potato bugs and sometimes snakes. The gray-blue “rolley poley” bugs were hardly considered to be part of the bug kingdom, far too common and antisocial for my tastes. I was much less afraid of bugs back then than I am now. I was on many occasions bitten by snakes, stung by bees, by centipedes, by hornets. These attacks were not unpro- voked. I somehow learned that honey bees won’t sting you as long as you cup them tightly in your hand in total darkness, (a fact I still have difficulty trusting) they still won’t sting you as you shake them vigorously (preparing them to meet my little brother of course). With a new jar and slightly permeable lid in hand I would collect whatever bugs I could find, getting them ready for the culmination of my eight year old scientific interests; the fight. It was unsurprising how difficult it was to get two different bugs from different parts of the bug kingdom to interact with one another considering how they so peacefully coexisted under the same board before being placed in my jar. It soon became obvious that the size of the jar was the only lever with which I could press on the boring state of affairs. Quart jars begat pint jars and pint jars were quickly succeeded by cup jars, and then eventually test tubes and tiny glass cuevettes were used. If the bugs were confined closely together, then something had to happen. Twenty years later I’m still performing the same cruel experiments, only on a conceptually simpler scale. These experiments have to do with the ability to trap single 2 photons and small collections of atoms in a small jar, forcing something interesting to happen. And in true karmic revenge for all my sadistic experiments as a youth, the cruelty is now turned on me; keeping me up at all hours of the day and night trying to get my fancy jar to work properly. I had the pleasure of sharing the drudgery of trying to get our fancy “jar” for single photons and atoms to work with three fine colleagues. In the early summer of 2003, Tom Purdy, myself, and then Subhadeep Gupta joined Kevin Moore on the project. Kevin had already spent a productive year drawing up plans, testing prototypes, and basically laying the foundation for what had already earned the name “E2”. It took about three and a half years of stops, starts and u-turns to finally get this fancy atom-photon jar to do what it intended, and then another year to understand what we were seeing. Exactly what we were seeing is the subject of the following 120 pages, so I’ll save the surprise at this moment and continue speaking abstractly. Flash forward a year or so, and Kevin Moore was busily studying medical imaging and radiation therapy at Washington University, Deep had moved to Seattle to start his own experiments at the University of Washington, and I was left somewhat to my own devices. On somewhat of a whim, I let Tom Purdy drag me to a section of the Optical Society of America’s Frontiers in Optics meeting on the subject of radiation pressure, cooling and quantum cantilevers; the budding field of optomechanics. As the session unfolded, I sat in the back row, my jaw on the floor and wrote furiously on scraps of paper. To my amazement, I had the whole field of optomechanics somehow trapped in my little jar. 3

Chapter 1

Introduction

1.1 Measurement and the quantum limit

When a clever physicist wants to measure a distance carefully, she uses the beautifully simple ruler which nature has afforded her. Waves of light are conveniently endowed with regular markings at every wavelength and a phase which increases linearly with the distance traveled. A readout of this phase is achieved by interference with some reference beam of light, a property which hinges on the wave nature of light. While this basic interferometric measurement is simple and has been used in studies ranging from Michelson’s searches for an ether, to current searches for gravity waves, and applications ranging from clocks to compact disc players, quantum mechanics places firm limits on what, and how well things may be measured. Quantum mechanics extends classical mechanics stating that for variables which are cannonically conjugate, an uncertainty relation is implied between the two. Hence, the mantra: you can’t know the position and momentum of a particle at the same time. For two operators, A and B, the rms fluctuations of these quantities, δA = phA2i − hAi2, obey the uncertainty relation,

1 δAδB ≥ [A, B] . (1.1) 2 At a fundamental level, measurements are limited by this uncertainty principle; ﬂuctuations and uncertainty play a critical role in its description. Beginning with Plank’s work on the blackbody spectrum and with Einstein’s study of the photoelectric eﬀect it was known that light was in some way quantized. While these 4

0π 2π 4π 6π 2ft 1ft 0ft

Figure 1.1: Nature’s rulers. Feet are compared to optical wavelengths (or phase) which might be read out in an interferometer. connections were established, the study of optics evolved separately from the development of quantum mechanics until the famous experiments of Hanbury Brown and Twiss. Their intensity interferometry experiment studied the correlation of fluctuations between two detectors, establishing that optics was fundamentally quantum mechanical and necessitated discussion of fluctuations. In response to Hanbury Brown and Twiss’ seminal experiment R. Glauber developed a true quantum theory of optics, which described photon anti-bunching (first observed in 1977 [1]), coherent states, and squeezed light (first observed in 1985 [2]). Just as a harmonic oscillator can be decomposed into two conjugate quadratures of motion which evolve as sin ωt and cos ωt, the real and imaginary parts of a monochro- matic plane wave may also be decomposed into oscillating quadratures of the electric field. The coherent state introduced by Glauber split the necessary uncertainty relation between these two quadratures evenly between the two states. But, in general uncertainty can be “squeezed” from one quadrature to another, effectively beating the standard quantum limit, which assumes that uncertainty is shared equally between the two quadratures. Experiments in quantum optics naturally require some sort interaction to measure or control properties of optical quanta. This can be achieved, for example with the resonant interactions with atoms, or at high intensities with bulk materials. Quantum optics has provided many precise tests of quantum theory largely due to the ability to create experimental systems which were simple enough to be theoretically tractable, but at the same time realistic enough to be studied experimentally [3]. 5

Figure 1.2: Basic elements of Cavity QED. Light is conﬁned within an optical cavity that is here formed by two mirrors in a Fabry-Perot conﬁguration. Atoms are trapped within the cavity and interact strongly with the near resonant cavity light. The cavity decay rate is denoted κ, the atomic decay rate Γ, and atom cavity coupling rate, g.

1.2 Cavity QED

Cavity Quantum Electrodynamics (CQED) got its start when Purcell noted in his calculations an enhancement in the spontaneous emission rate of an atom when placed near a conducting surface [4]. The effect was understood as a modification of the density of states of the electromagnetic vacuum, hinging on the quantization of the electromagnetic field. When atoms are placed within a resonator, the modification of the density of states of the electromagnetic vacuum can enhance or suppress [5] spontaneous emission by the 3 2 Purcell factor, 3Qλ /(4π Vc), where Q is the quality factor of the resonator, λ is the optical wavelength and Vc is the volume of the resonator. The modification of spontaneous emission was first observed in dye molecules which were embedded in a polymer on a mirror [6], and later with atoms [7, 8]. Excellent histories of the beautiful field of cavity QED have been written [9, 10, 11]. This discussion aims to give the unfamiliar reader only a faint whiff of the subject. Important concepts will be developed more completely in Chapter 2. Studies in the field of cavity QED have been pursued along many parallel trajectories. These include nuclear magnetic resonance experiments enhanced by coupling to a resonant circuit [12], microwave cavity experiments coupled to atoms in high Rydberg states [7], superconducting microwave circuits [13], and optical photons coupled to atoms [14]. All these cavity QED approaches contain the same essential elements sketched in Figure 1.1, which include an atom cavity coupling rate g, a resonator field decay rate κ, 6 and an atomic decay rate Γ. When only single atoms are present a situation known as single-atom-strong-coupling is attained if the coupling greatly exceeds the decay, g κ, Γ. This regime of strong coupling will be treated more carefully in chapter 2. For now, I’ll mention that the lowest lying excitations of this system do not simply consist of “an atom in its excited state” or a “single photon in the resonator,” but instead these two systems are necessarily intertwined. The lowest excitations are instead given by dressed states which are the symmetric and antisymmetric combinations of these possibilities. When a resonance of the cavity is degenerate with an allowed atomic transition, the energy of these two states is split by an amount 2~g, and can be spectroscopically distinguished [15, 16, 17]. The total atom-cavity coupling can be increased by including more atoms in the system. In fact, the first strong coupling experiments were performed with many atom systems [18, 19]. The achievement of single atom strong coupling in the optical domain has made possible single atom detectors [14, 20, 21], lasers [22], and the atom cavity microscope [23]. Microwave and optical CQED also differ in essential ways. In microwave CQED, because microwave photons are difficult to detect, information about the system is gathered from the atomic state of atoms transiting the cavity [24] which may be detected with very high efficiency by ionization. In optical cavity QED the photons transmitted through the cavity are themselves detected, conveying simply, information about the cavity field. A second important difference pertains to the forces on atoms within a cavity. The Rydberg atoms used in microwave CQED experiments have similar masses to the atoms used for optical CQED, yet microwave photons have much less momentum than optical photons. In microwave CQED, forces on atoms result in a momentum impulse which is always much less than the atom’s thermal momentum. In contrast, these impulses can be quite significant in the optical domain, especially compared to the rms momentum of laser cooled atoms, allowing the trapping of a single atom by a single photon [23] and cooling [25, 26, 27, 28, 29]. Recent experiments which utilize superconducting circuits [30] or very large numbers of atoms [31, 32, 33, 34] attain a regime of “collective-dispersive-strong-coupling1”, √ 2 where Ng /∆ca Γ, κ, and ∆ca = ωc − ωa Ng is the detuning between the atomic cavity resonance. This regime of dispersive strong coupling is the subject of this thesis. Here the mechanical action of a large number of atoms couples to the resonance of the

1If this sounds unfamiliar, it is because I have just coined the term. Ref. [30] introduced the term dispersive strong coupling in the context of a single superconducting qubit, so the many atom case is a natural extension. 7 cavity, and equivalently atoms in the cavity experience a dipole force proportional to the intracavity intensity.

1.3 Optomechanics

An optomechanical system is any system which includes coupling between an optical and mechanical element. A simple example, sometimes referred to as cavity-optomechanics consists of a Fabry-Perot cavity where one mirror is fixed in space and the other is confined harmonically. In this system a displacement of the mirror naturally changes the resonance frequency of the cavity, and in turn, a radiation pressure force is exerted on the mirror by the light in the cavity. These effects are two sides of the same coin, a higher “coupling” between the mechanical and optical components implies a larger radiation pressure force, and a higher sensitivity to the mechanical element’s position. These optomehcani- cal systems span a vast array of sizes and implementations, ranging from the kilometer and kilogram scale of the Laser Interferometer Gravitational wave Observatory, to micro- and nano-fabricated cantilevers, yet are described by the same Hamiltonian. Other types of physical systems such as electro-mechanical systems are very similar to optomechanical systems, since they simply contain a mechanical degree of freedom coupled to an electrical resonator whose frequency is tuned by the position of the mechanical object. Optomechanical systems exhibit a number of interesting features. The position of the mechanical object typically couples linearly to the resonator’s frequency, ωc and the force on the mechanical object is linearly dependent on the intracavity intensity. For a constant drive at frequency ωp the intracavity intensity is a Lorentzian function of the detuning, ∆ = ωp − ωc. The spectrum of this driven system is inherently nonlinear and for some parameters is bistable. Early experiments observing bistability in such a system were performed in the group of H. Walther at the Max Planck Institute for Quantum Optics [35]. Another natural consequence of a system which contains an element which responds to the intensity of the resonator is ponderomotive squeezing [36, 37]. When an intensity fluctuation occurs, the movable mirror is displaced, changing the phase of the cavity. For some parameters, this coupling generates squeezing at a certain angle in phase space [38]. Mechanical resonators which can be suitably isolated from the environment are predicted to exhibit entanglement between the mechanical degree of freedom and the cavity field [39, 40] . The observation of entanglement and superposition states of a macroscopic 8 object has been proposed to test theories which may connect gravity to quantum mechanics [41]. In the model optomechanical system consisting of a Fabry-Perot cavity (see Fig. 2.1), the size of the coupling relative to dissipation mechanisms is set by the quality factors of the optical and the mechanical elements. For high quality resonators, losses to the environment are reduced, producing a natural setting for quantum limited and quantum non-demolition measurements [42, 43, 44, 45, 46, 47, 48]. These quantum limited and nondemolition measurements are of interest in all areas of physics, ranging from the detection of gravity waves [49] to navigation and precise tests of fundamental physics [50]. For many experiments it is desireable to be able to place the optical resonator in the ground state. Such a situation has been achieved, either trivially in the optical domain or with the aid of cryogenics in the microwave domain. However, at present many proposals and experiments are limited by thermal motion of the mechanical degree of freedom. Ground state mechanical resonators are currently being pursued on two fronts. The first front is to push the resonator frequency to ever higher frequencies [51], for example a GHz resonator [51] has an occupation factor of ∼ 1 at a temperature of 50 mK, a temperature achievable in dilution refrigerators. A second front is the use of radiation pressure cooling. First proposed by Braginsky in 2002 [52], powerful schemes to cool a single mode of a mechanical resonator below its ambient temperature have been demonstrated [53, 54, 55, 56, 57, 44, 58]. It has been recently shown that an approach on both fronts, high frequency resonators and cooling, are required to reach the ground state of motion [59, 60]. Some of the most impressive advances in the field of optomechanics have come from its agile stunt double, the electromechanical system. Often refered to as micro- or nano-electro-mechanical systems (MEMS and NEMS respectively), these systems couple microwave resonators to mechanical elements [61, 56, 62].

1.4 The next 100 pages

This thesis aims to connect the ﬁelds of cavity QED and optomechanics. In the next chapter a connection will be drawn between a model optomechanical system and the many-atoms CQED system. To make this discussion as clear as possible, the necessary experimental details for the statements in Chapter 2 are relegated to Chapter 3. Approx- imations will be made as explicitly as possible. The following 4th, 5th and 6th chapters 9 form the body of this work: nonlinear optics arising from collective adiabatic motion, the observation of quantum measurement backaction, and collective motion. The 7th chapter, on betatron motion in an ultracold atom storage ring, may appear a bit of an outlier in the atmosphere of this thesis, so a ﬁnal concluding chapter ties it all together with a proposal to measure gravity at short distances by measuring betatron motion at the quantum limit. 10

Chapter 2

Cavity-optomechanics with cold atoms

This chapter discusses and develops the basis of the cold atoms-CQED system. In May of 2006, I, somewhat on a whim, tuned the resonance of the cavity to occur some 30 GHz from the atomic resonance, rather than the few GHz range we had been studying. As it turned out, at this detuning, and as our cavity resonance was eventually tuned even further from the atomic resonance, our experiment entered into the world of cavity optomechanics, a whole subﬁeld of physics. We were completely unaware of the connection between our atoms- cavity soup and the mirrors glued to cantilevers which exemplify cavity optomechanics, and had the pleasure to explore and develop many of these ideas ourselves.

2.1 A model optomechanical system

As discussed in the introduction, a huge variety of optomechanical systems span a vast range of scales, yet can all be described by the same Hamiltonian. Figure 2.1 indicates such a system. A stable cavity is formed by one fixed mirror and one harmonically confined movable mirror. The resonance frequency of the cavity, ωc is tuned by the position of the movable mirror. The classical dynamics of such a system are surely very interesting and will be discussed throughout this thesis, however we are interested in a regime of quantum optomechanics, where we consider the movable mirror to be necessarily described quantum- mechanically, and coupled correspondingly to a quantized electromagnetic field. Defining † † creation and annihilation operators (a and a) for the mirror’s position as Z = Zho(a + a ), 11

z0 z

Figure 2.1: The basic optomechanical system. A stable optical mode is formed between one ﬁxed mirror (left) and a movable mirror (at position Z) contained in a harmonic potential (equilibrium position Z0). The radiation pressure force on the movable mirror is proportional to the intensity of light in the cavity and the resonance frequency of the cavity is proportional to the position of the movable mirror.

p with Zho = ~/2Mωz. The mass of the mirror is M and ωz is the frequency of harmonic conﬁnement. The system is described by the following Hamiltonian,

† H = ~ωc(1 − Z/l)n + ~ωz(a a) + HΓM + HIN , (2.1) where the cavity resonance frequency ωc, photon number operator n, and cavity length l, have been introduced. The term HIN contains details of the cavity drive and decay which will be discussed later. HΓM describes the decay rate, or ﬁnite Q of the mechanical resonator. Examining (2.1) we recognize a collection of terms which may be interpreted as a

force; the radiation pressure force per photon, Frp = ~ωc/l. With this new notation, the Hamiltonian reads,

† H = ~ωcn − FrpZn + ~ωM (a a) + HΓM + HIN , (2.2)

In this chapter the connection between the cold atoms-CQED system and (2.2) will be addressed. In some limits the seemingly very complicated system of many atom CQED is clearly described by (2.2). 12

2.2 Many atom CQED

The interaction of N atoms with a single mode of a cavity is described by the Tavis-Cummings Hamiltonian [63]:

N N ωa X X H = ~ σz + ω c†c + g(r ) c†σ− + cσ+ + H + H . (2.3) 2 i ~ c ~ i i i Γ,κ IN i=1 i z ± th Here, {σi , σi } are the conventionally deﬁned Pauli spin operators of the i atom, and c†(c) are the creation(annihilation) operator for a photon in the cavity mode1. The atomic

resonance frequency is denoted ωa. The atom cavity coupling rate, g(r) depends on the position of the ith atom. I’ll leave all the interesting details of the cavity-QED system to the countless theses ([64, 65, 66, 67] to name a few), review articles [3, 68, 10, 11], and excellent books [9, 69] that have come before, and jump to the conclusions that I need. Ignoring (for now) the ﬁnal term in the Hamiltonian, the system admits two bright eigenstates excited by cavity transmission [70]. These eigenfrequencies are, v u N ωa + ωc u∆ca 2 X ω = + t + g2(r ), (2.4) + 2 2 i i=1 v u N ωa + ωc u∆ca 2 X ω = − t + g2(r ). (2.5) − 2 2 i i=1

The atom-cavity coupling rate g(r), has a maximum value g0 and spatial distribution dictated by the cavity mode. The atom-cavity detuning is ∆ca = ωc − ωa. s 2 d ωc g0 = , (2.6) 2~oVc is a product of the dipole matrix element d, and the electric ﬁeld per photon. Here, o is

the permittivity of free space and Vc is the mode volume of the cavity. Making g0 as big as possible (compared to cavity, κ, and atomic, Γ decay rates) is a prerequisite for single atom strong coupling, and allows us to largely neglect decay (HΓ,κ) in (2.3). The two knobs which traditionally tune the coupling rate are the mode volume [14] and the dipole matrix element [71]. Figure 2.2 sketches the spectrum of the many atom cavity. There are clearly two distinct limits of the system. Familiar to most cavity-QED work with small samples of atoms

1Get this straight everybody: c† is for cavity, and a† is for atom—I know that we’re swimming up stream here, but sometimes you have to be stubborn. 13

Δ ca (GHz)

(GHz) Δ ca

Δ (GHz) Cavity detuning from atomic resonance ca

Figure 2.2: The spectrum of the many atom-cavity system. In the dispersive limit, the two resonances are distinctly “cavity-like” and “atom-like”. The approximation (2.7) is clearly g2 very good for ∆ > 30 GHz. Here, the sum, PN g2(r ) = N 0 , and N = 5 × 104, and ca i=1 i 2∆ca 2 g0 = 2π × 14.4MHz. 14

is the resonant regime. When ∆ca = 0 the spectrum exhibits the familiar “vacuum-Rabi splitting”, where the cavity resonance is split into two resonances for each bright eigenstate, q PN 2 separated by 2 i=1 g (ri). The other clear limit is the dispersive limit, when the resonance of the cavity is √ tuned far from the atomic resonance, ∆ca Ng. In this limit, the two resonances occur at: N 2 X g (ri) ω = ω + (2.7) 1 c ∆ i=1 ca N 2 X g (ri) ω = ω − (2.8) 2 a ∆ i=1 ca

These two resonances are distinctly “cavity like” (ω1) and “atom like” (ω2). Except for a few sections in latter chapters, we’ll work almost exclusively in this dispersive limit.

2.3 Toward collective variables

To make this section as clear as possible many of the experimental details which are necessary to make the following approximations are presented in the following chapter. Figure 2.3 sketches the geometry of the atom cavity system. A large number of atoms are distributed over many sites of an one dimensional intracavity optical lattice which is formed −1 at wavevector kt = 2π/850 nm , which is diﬀerent than the probe wavevector kp = 2π/780 nm−1. In the dispersive limit, the eﬀect of N atoms is to shift the cavity resonance by an amount,

N 2 X g (ri) . (2.9) ∆ i=1 ca 2 th The spatially dependent coupling g (ri) of the i atom, for the TEM00 mode of the cavity is, 2 2 2 2 2 xi + yi g (ri) = g0 sin (kpzi) exp −2 2 , (2.10) w(zi) p 2 2 where, w(z) = w0 1 + (2z/(kpw0) is the probe beam width, and w0 is the beam waist. The radial distribution function of the ith atom is, 2 2 xi yi f(ri) = exp − 2 + 2 (2.11) σx σy p 2 σx = σy = kBT/(2mωx) is the gaussian full width of the atomic distribution. 15

z z y Top cavity y 100 μm x mirroorr x

z 10 μm x

1mm Bottom cavity Coils for magnetic con!nement mirror

zi δzi m) μ x (

z (μm)

Figure 2.3: Detailed schematic of the atoms-cavity system. Top: successive scale views of the cavity system. Bottom: enhanced view of the cavity lattice. The 1/e width of the atomic distribution (back) is small compared to variations in the probe intensity (gray linear density plot). At each location, harmonic confinement is provided by the lattice at 850 nm, and the atomic distribution occupies the ground state of the ωz confinement. Each atom’s position is given by the location of the minimum of the harmonic confinement,z ¯i plus a deviation from that minimum, δzi. 16

2 Approximation #1: the coupling g (ri) depends only on the zi position. Justiﬁcation: the radial extent of the atomic distribution function is very small when compared to the beam waist of the probe light.

2 σx 2 = 0.04 1. (2.12) w0

Returning to (2.10), under approximation # 1 our system is eﬀectively one dimensional. I’ll write the position of each atom as given by its equilibrium position,z ¯i and δzi, the position operator for the deviation away from that minimum position, zi =z ¯i − δzi.

N 2 N 2 X g (zi) X g = 0 sin2(k (¯z − δz )). (2.13) ∆ ∆ p i i i=1 ca i=1 ca

Approximation # 2: the position deviation of each atom away from its equilibrium is very small compared to the probe wavevector, kpδzi 1.

Justiﬁcation: atoms are in the ground state of the ωz oscillator since kBT ~ωz. The harmonic oscillator length zho is much less then the inverse wavevector of the probe,

zhokp = 0.3 1. (2.14)

The careful reader will note that this is not the strongest small parameter assumption. Higher order corrections to this approximation, responsible for “lattice dephasing” are discussed further in Chapter 6.

Expanding equation 2.13,

N X g2 2 0 cos(k δz ) sin(k z¯ ) − cos(k z¯ ) sin(k δz ) , (2.15) ∆ p i p i p i p i i=1 ca N X g2 2 ' 0 sin(k z¯ ) − k δz cos(k z¯ ) , (2.16) ∆ p i p i p i i=1 ca N X g2 ' 0 sin2(k z¯ ) − 2k δz cos(k z¯ ) sin(k z¯ ), (2.17) ∆ p i p i p i p i i=1 ca N X g2 = 0 sin2(k z¯ ) − k δz sin(2k z¯ ). (2.18) ∆ p i p i p i i=1 ca 17

Guided by (2.18), I’ll make the following identiﬁcations,

N X g2 ∆ ≡ 0 sin2(k z¯ ), (2.19) N ∆ p i i=1 ca N 1 X Z ≡ δz sin(2k z¯ ), (2.20) N i p i eﬀ i=1 N X 2 Neﬀ ≡ sin (2kpz¯i). (2.21) i=1

Here, ∆N is the shift in the cavity resonance with all the atoms localized at their potential minima. The collective position Z is normalized sensibly to the eﬀective number of atoms, which counts the number of atoms who’s motion couples to the cavity. With 2 Fdp = ~kpNeﬀg0/∆ca, our sum can now be written simply,

N 2 X g (ri) = ∆ − F Z/ . (2.22) ∆ N dp ~ i=1 ca

The collective position Z is a weighted sum of the position operator for each atom, δzi, † and is therefore necessarily an operator as well. We can write, Z = Zho(a + a ), where p Zho = ~/2Mωz is the harmonic oscillator length for a mass, M = mNeff particle. The conjugate variable of the collective position is the collective momentum, which is defined to th satisfy proper commutation relations, with pi the momentum operator for the i atom, N 1 X P ≡ sin(2k z¯ )p . (2.23) N p i i eff i=1 The Hamiltonian for our system of atoms distributed within the lattice is now,

† H = ~ωcn + ~∆N n − FdpZn + ~ωM (a a) + HΓM + HIN . (2.24)

Noting that the term ∆N only shifts the cavity resonance, but does not couple any dynamics, 0 we write the shifted cavity resonance, ωc = ωc + ∆N :

0 † H = ~ωcn − FdpZn + ~ωM (a a) + HΓM + HIN . (2.25)

which is identical to (2.2).

2.4 Normal modes of the system

For the N atom soup which couples to the cavity there are 3N motional modes. Of these modes, the N axial modes are nominally degenerate with the collective mode 18 introduced in this chapter. As this section shows, when the cavity is probed, an eﬀect known as the optical spring promotes the collective mode to be the only eigenmode of the system which is non-degenerate. The atoms-cavity system exhibits two relevant features for the following discussion:

(1) harmonic conﬁnement ωz assumed to be identical for all atoms, and (2) a dipole force proportional to the intracavity photon number which depends on the collective position. This latter eﬀect will be discussed in detail in Chapters 4 and 6. In this sense, this discussion occurs far too early in this thesis, but is presented here to convince the wary reader that the atoms-cavity soup really is a typical optomechanical system. The equation of motion for the ith atom is then,

¨ 2 ∂ mδzi = −mωz δzi + FdpnZ,¯ (2.26) ∂δzi 2 ∂n¯ n¯ sin(2kpz¯i) = −mωz δzi + Fdp Z + . (2.27) ∂δzi Neﬀ

A constant force fi = (Fdp/Neﬀ)¯n sin(2kpz¯i) will simply displace each atom’s position by an 2 2 amount fi/(mωz ). Redeﬁning δzi → z¯i−zi−fi, we can ignore this constant force . Introduc- ing the optical spring constant, kopt = Fdp∂n/∂Z¯ (see Chapter 6), and the optomechanical 2 frequency shift ωopt = kopt/(M) we have,

2 N ωopt X δz¨ = −ω2δz + sin(2k z¯ ) δz sin(2k z¯ ). (2.28) i z i N p i i p i eﬀ i This is somewhat clearer when written in matrix form,

 ¨          δz1 δz1 a1 0 ··· 0 a1 a2 ··· aN δz1  ¨    2        δz2   δz2  ω  0 a2 ··· 0   a1 a2 ··· aN   δz2    2   opt        .  = −ωz I  .  +  . . . .   . . . .   .   .   .  Neﬀ  . . .. .   . . .. .   .            ¨ δzN δzN 0 0 ··· aN a1 a2 ··· aN δzN (2.29)

Where I is the N × N identity matrix and ai = sin(2kpz¯i). The ﬁrst thing to notice is that

when the probe light is oﬀ, ωopt = 0, and there are N degenerate modes with eigenfrequency

ωz. The collective mode introduced in the chapter is trivially a normal mode of this system.

2For now, we’ll ignore this constant force, but return to the interesting nonlinear eﬀects induced by it in Chapter 4 19

It’s easy to check if the collective mode corresponds to a normal mode when the atoms-cavity system is probed. This collective mode corresponds to the vector,   a1    a2    Z =  .  . (2.30)  .    aN

Trying this collective mode out

 2 2 2 2 2  a1(a1 − Neffωz /ωopt) + a1a2 + ··· + a1aN 2  2 2 2 2 2  ω  a a2 + a2(a − Neffω /ω ) + ··· + a2a  opt  1 2 z opt N  M × Z =  .  , (2.31) Neff  .    2 2 2 2 2 a1aN + a2aN + ··· + aN (aN − Neffωz /ωopt)

2 2 2 noting that Neﬀ = a1 +a2 +...+aN , we ﬁnd that the collective mode is in fact an eigenmode 2 2 2 of the system which oscillates at an eigenfrequencyω ¯ = ωz − ωopt.

2 2 M × Z = (ωopt − ωz )Z. (2.32)

The collective mode is also the only non-degenerate mode of the system. This point is clearly illustrated by rewriting the equation of motion as,       δz¨1 δz1 a1  ¨    2    δz2   δz2  ω  a2    2   opt    .  = −ωz I  .  + a1 a2 ··· aN  .  . (2.33)  .   .  Neﬀ  .        ¨ δzN δzN aN

Any mode which is orthogonal to Z will not be an eigenmode of the second term in 2.33 2 and will have and eigenvalue −ωz .

2.5 Comparison to traditional CQED: Cavity quantum optomechanics

Working in the dispersive regime of many-atom CQED affords a number of simplifying benefits. A common experimental difficulty in few-atom CQED experiments is the contribution of the AC stark shift of the trapping potential. For 100 µK Rubidium atoms, 20 the AC stark shift induces variations in the atomic resonance frequency by roughly 2 MHz, a significant shift compared to the natural linewidth. For atomic species which afford the use of a ”magic wavelength”, this difficult may be avoided [72], but for 87Rb, experiments have relied on numerical simulation to account for these effects [73]. In the dispersive regime, √ when ∆ca Ng Γ, changes in the detuning, ∆ca due to the stark shift of the trapping potential are completely negligible. The full glory of the Jaynes-(or Tavis)-Cummings Hamiltonian is a beautiful quantum system and has allowed the study of many properties which are exclusively described by quantum mechanics. A common goal of recent research is the elucidation of quantum properties in systems which are typically described adequately with classical mechanics. To date, no experiments have been performed which demonstrate exclusively quantum phenomena with macroscopic mechanical systems. To this end, using the innately quantum mechanical system of strong coupling CQED in its dispersive (and ever more classical) limit appears to be an excellent laboratory of the quantum classical boundary. The choice of the title “Cavity Quantum Optomechanics with Ultracold Atoms” for this thesis may appear brash or boastful to the cautious reader. The term (CQOM), introduced in a review article [74] authored by K. Vahala and T. Kippenberg, refers to a regime of optomechanics where mechanical phonon occupancies are reduced below unity and quantum phenomena of these mechanical systems are observed. The cold atoms-CQED system presented in this thesis makes significant steps into this new regime; achieving photon occupancies as low as 0.06, the lowest reported for an optomechanical system, and detailing experiments performed in the “granular” or quantum regime, where the impulse of a single photon displaces the mechanical element by more than its ground state rms momentum fluctuations. 21

Chapter 3

The experiment

This chapter discusses some of the experimental and procedural details for the cold-atom cavity-QED machine. The lion’s share of the experimental details and design considerations are contained in the thesis of K. L. Moore [67]. I have attempted to avoid significant overlap with the material presented in that work, but for clarity some details are reproduced here. Our experiment is composed of two clear sub-systems; the atoms, and the cavity. More than half of the laboratory, and equally as much of the headache belongs to the half of the atom cavity system which is responsible for producing cold samples of Rubidium atoms. Briefly, we formed a magneto-optical trap from a Zeeman slowed source of 87Rb atoms. This magneto-optical trap of approximately 1010 atoms was magnetically trapped without any fancy laser cooling or optical pumping, translated ∼ 6 cm and subjected to a first stage of forced RF evaporation. At a temperature of ∼ 10 µK, the remaining ∼ 2 × 106 atoms were again translated and loaded into a time orbiting trap. Following evaporation in this TOP trap, cold atoms were again moved into the mode of the cavity and trapped within an intracavity optical dipole trap. This chapter picks up the story in more detail at this point, where the two subsystems of cold atoms and cavities are necessarily intertwined.

3.1 The optical ﬁelds

Two curved mirrors with appropriate radii and separation form the simplest type of optical cavity, which may support many of the Hermite-Gaussian modes of the electromagnetic ﬁeld. The mirrors which are used for the science in this thesis are considered to 22

Table 3.1: Parameters and deﬁnintions

m : Atomic mass of 87Rb ...... 1.44 × 10−25kg N : The number of trapped atoms Neff : Effective number of atoms ...... ∼ N/2 M : Collective mass ...... mNeff ω{x,y,z} : Trap frequency in FORT along {x, y, z} axis . . . . . 2π × {0.39, 0.39, 50} kHz a (a†) : Collective position annihilation(creation) operator † Z : Collective position operator ...... Zho(a + a ) p Zho : Collective oscillator length ...... ~/2Mωz n : Cavity photon number operator ...... c†c n¯ : Average number of cavity photons κ : Cavity half linewidth ...... 2π × 0.66 MHz Γ : Spontaneous emission rate ...... 2π × 2.99 MHz g0 : Maximum atom cavity coupling rate ...... 2π × 15.6 MHz η : Photon detection efficiency...... 0.05(1) l : Mirror separation ...... 194 µm r : Mirror radius of curvature ...... 5 cm T(t,b) : Cavity mirror transmission (top and bottom) . . . . (1.45, 1.60) ppm L : Cavity losses per mirror ...... 3.8 ppm F : Cavity finesse ...... 556000 FSR : Cavity free spectral range ...... 2π × 780 GHz T : Equilibrium temperature of the trapped atoms . . 0.8(1) µK w0 : Cavity mode waist ...... 24 µm 4 3 Vc : Cavity mode volume ...... 8.27 × 10 µm ωc : Bare cavity resonance frequency ∆ca : Cavity atom detuning...... ωc − ωa ωa : Rubidium D2 resonance frequency ...... 2π × 384 THz ωp : Probe frequency 0 ωc : Shifted cavity resonance frequency ωc + ∆N (1 + kZ) 0 ∆ : Detuning from the shifted cavity resonance ...... ωp − ωc Ut : FORT trap depth...... kB × 6 µK −1 kp : Probe wavevector ...... 2π/780 nm −1 kt : FORT wavevector ...... 2π/850 nm ¯ N{x,y,z} : Average phonon number ...... {42, 42, 0.07} σ{x,y,z} : Gaussian width of the atoms at each lattice site . {3.6, 3.6, 0.048} µm 23 form a near-planar resonator since their radius of curvature r = 5 cm l = 200 µm, the cavity lengh. In this case the Gaussian beam waist is given by, [75] r 2 2l r w0 = (3.1) kp 2l

2 The Rayleigh range for light at 780 and 850 nm, πwo/λ ' 2 mm is long compared to the length of the cavity.

3.1.1 An imperfect cavity

No optical cavity can store light indefinitely1, losses and transmission lead to a decay of the field within the cavity. The finesse of a cavity is dictated by the total losses (2L) and transmission (2T ) of both mirrors.

2π F = (3.2) 2L + 2T

The ﬁnesse of the cavity relates directly to the cavity half-linewidth, κ and its length,

FSR c/2l F = = , (3.3) 2κ 2κ where FSR is the free spectral range. The length of the cavity was measured initially by measuring the free spectral range, and later by probing the cavity with two wavelengths of light simultaneously. The cavity half-linewidth (from here on out just linewidth) was measured by cavity ringdown [76]. For light near 780 nm, repeated measurements have produced cavity ringdown times of τ = 1/(2κ) = 120 ns, corresponding to a linewidth κ = 2π × 0.66 MHz. Because the cavity is formed from dielectric coatings optimized to produce the smallest transmissions near 780 nm, the ﬁnesse of the cavity for light at other

wavelengths is lower. At λt ' 850 nm, the wavelength of light used for trapping the atoms

and stabilizing the length of the cavity, the ﬁnesse is reduced to F850 = 36000, small

compared to F780 = 556000. A full characterization of the cavity mirrors would measure the losses and transmission of each mirror individually. We measured (L) and (T ) as suggested by C. Hood in Ref. [77], by making transmission and reflection efficiency measurements from both sides of the cavity. Together with the measurement of the finesse, these measurements determined

1Though, some have come pretty close, the latest achievements from the microwave cavity-QED group at ENS boasts a cavity lifetime of 400 ms.[24] 24 the losses and transmissions of each mirror. For simplicity we typically quote the average value of each: L = 3.8 ppm, and T = 1.5 ppm. For light at 850 nm, the losses are similar, and the transmissions are much larger T850 ∼ 83 ppm.

3.1.2 Cavity-QED parameters

In this thesis, a 194 µm cavity produces a mode volume of 8.27 × 104 µm3. For the cycling transition in 87Rb, the atom cavity coupling rate is then 2π × 15.6 MHz. Every experiment, however, is performed with atoms in the |F = 1, mF = −1i ground state with the cavity significantly detuned from the atomic transition, rendering the excited state hyperfine structure indistinguishable. Summing over the Clebsh-Gordon coefficients for transitions to each excited hyperfine state, and with σ+ probe on the D2 line, the p appropriate maximum atom cavity coupling rate, g0 = 2π × 5/6 15.6 MHz. The atom cavity coupling rate is larger than the cavity field decay rate, κ = 2π × 0.66 MHz and the atomic linewidth, Γ = 2π × 2.99 MHz, meeting the requirements for single atom strong 2 coupling. The single atom cooperativity, C = g0/(2κΓ) = 51.

3.1.3 Photon detection eﬃciency

The efficiency of detecting an intracavity photon is denoted by the parameter η. The accuracy of every quantitative measurement involving the cavity hinges on this detection efficiency, and as we will see, almost unavoidably the detection efficiency is very low. The detected photon flux (CPS) is related to the average intracavity photon number (¯n):n ¯ = CPS/(2κη). We express η as a product of its various sources:

η = ηcav × ηﬁlters × ηcoupling × ηdet. (3.4)

Imagining that we stuck a photon inside of our (assumed symmetric) cavity, and asked the probability that the photon exits the left side of the cavity. Four possible fates await this impatient photon; it could be transmitted through either the right or left mirror, or lost on the surface of the right or left mirror2. The probability that the photon leaves

2This statement is actually a bit of misleading: imagine a perfect cavity with no losses, when probed on resonance, all the light coupled into the cavity from one side exits out the other side. In this case, light in the cavity only goes out one mirror, not both. Yet, a photon placed inside the cavity will go out either mirror. This subtlety is addressed carefully in section 6.8 25 the left side of the cavity is then,

T η = = 0.135. (3.5) cav 2T + 2L

Downstream of the cavity, a variety of filters are used to separate the probe light from the trapping/stabilization beams and background room light. Two dichroic mirrors which reflect 780 nm light with 99% efficiency separate the probe from the trapping laser. A narrowband interference filter is used to filter out background room light and remaining trapping light, while transmitting probe light with 50 − 80% efficiency, ηfilters ∼ 0.6. The Single Photon Counting Modules (SPCMs) which we use to detect cavity light have a measured quantum efficiency of ηdet ∼ 0.5. There is also a possibility that not all the probe light is coupled into the SPCMs, or that some of it clips on various mirrors and apertures along its path. Experimentally we can measure the product ηfilters · ηcoupling · ηdet. These measurements were made by locking the cavity in reflection and probing the cavity at a level which resulted in a cavity transmission of approximately 1 µW, measurable with a calibrated photodiode directly at the output of the cavity. A OD = 5.6 filter was then placed before the probe fiber and the photon count rate was measured. A photon detection rate of 1 s−1 corresponds to a power of 2.54×10−19 W. Given the filter OD and the detected count rate, the efficiency of photo detection was determined to an accuracy of approximately 20%.

3.2 The cavity lock chain

The stabilization of the high ﬁnesse cavity requires a number of feedback loops and frequency handshakes. Aside from the day to day challenge of keeping an aging atom- cooling apparatus humming smoothly, locking the cavity and stabilizing the probe laser(s) posed a signiﬁcant hurtle and continues to occupy much of our attention. A schematic of the lock chain is provided in Figure 3.1.

3.2.1 The transfer cavity

The heart of our cavity lock chain is a stable length reference between the 780 nm probe and the 850 nm stabilization laser. This length reference was a 30 cm Fabry-Perot “transfer cavity”, to which both the probe and stabilization lasers were locked. 26

780 IF SPCMSPCM Science cavity ll=194=194 μm F = 556,000556,000 (780)& 36,00036,000 (850(850)) Dichroic mirrors λ/4 50/50 APDAPD 850 IFIF

APDAPD 780780 re"ection SC lock BW = 10 kHz Intensity stabilization laser

AOMOM aser l

80 MHz e AOOMM ode d AOM 22x80x80 MHz i

AOOMM d 2x80 MHz TWM io 850 lock + 242400 kkHHz d 22x200x200 MHz + 240 kHz 0.0.11 - 1 GGHHz box +12 MHz BW = 10 kHz nm 780 nm diode laser 50 nm 850 nm diode laser 8 OOII OOII ODOD 4 ND #lter to wavelenl gth meter 780 lock box BW = 10 kHz PD PD

780 IF 885050 IF

Transfer cavity l = 30 cm F = 10,00010,000 DichrDichroic mirror pick o! to modemode imagimaginging cameracamera

Figure 3.1: A simpliﬁed schematic of the cavity lock chain. The actual number of optical elements is 5 times larger. Even then, this schematic probably gives the reader vertigo. The chain of locks is separated into 4 distinct regions of the optical table. The stable length reference of the transfer cavity is used to control the frequency diﬀerence between the 780 nm laser and the 850 nm laser. The cavity length is stabilized to the 850 nm laser, and then by virtue of the three locks, stable with reference to the the probe laser. Acousto- Optic Modulators (AOMs) are used to shift frequencies and add sidebands to probe and stabilization beams, the Traveling Wave Modulator (TWM) places a variable sideband on the stabilization light. A Single Photon Counting Module (SPCM) is used to detect the probe photons transmitted through the cavity. An Avalanche Photo-Doide is (APD) is used to detect the stabilization light. For the transfer cavity locks, NewFocus Photo-Diodes (PDs) are used. Optical Isolators (OI) and Interference Filters (IF) are also shown. 27

0.1

0.01

0.001

1 2 3 4 5 6 5 10 15 20 25 30 time (μs) frequency (MHz)

Figure 3.2: Ring down measurements for the transfer cavity (left) show the decay of the transmitted intensity. The cavity ﬁeld amplitude decay rate for 780 nm (upper curve) is 50 kHz. For light at 850 nm, the decay rate is 200 kHz. The passive linewidth of the probe laser (right) is assessed by beating two diﬀerent probe lasers together. For 20 µs of integration, high frequency noise broadens the laser beat note several MHz.

While performing early experiments which required a good absolute length reference, the transfer cavity was itself locked to a Rb vapor cell. Working far from any atomic transitions required excellent passive stability of the transfer cavity. The transfer cavity was therefore constructed from materials with a low coefficient of thermal expansion. The combined thermal expansion of the 30 cm fused quartz tube and Invar mirror mounts resulted in linear thermal expansion of 2 × 10−7 m/◦C, corresponding to a frequency drift of 260 MHz/◦C. This frequency stability, owing to the ∼ 2◦C temperature fluctuations in the lab was sufficient.

The measured half linewidth of the transfer cavity was κtc(780) = 2π × 50 kHz and

κtc(850) = 2π × 200 kHz. The passive frequency stability of our diode lasers, on the order of 5 − 10 MHz (see Fig. 3.2 ), made it difficult to lock them robustly to the relatively narrow transfer cavity. We succeeded in locking the lasers in transmission by applying 200 kHz FM sidebands, but with a modulation depth corresponding to several MHz, effectively broadening the narrow error signal. This method was insufficient to narrow the lasers substantially through feedback since, in transmission, the bandwidth of feedback is limited by cavity linewidth. 28

3.2.2 The science cavity

We locked the cavity system with the following procedure. First the wavelength of the probe was tuned to the desired atom cavity detuning, and locked to the nearest TEM0,0 mode of the transfer cavity. The passive stability of this cavity was suﬃcient to ﬁx the probe frequency to the desired atom-cavity detuning. A camera was used to establish that the proper mode was selected, thought it proved unnecessary to work exclusively with the

TEM0,0 mode. Variable sidebands (between 100 and 1000 MHz) were placed on the cavity stabilization light with a Traveling Wave Modulator (TWM). Next, the wavelength of one

of the stabilization laser’s sidebands was tuned to occupy a diﬀerent TEM0,0 mode of the science cavity. The stabilization laser was locked to a mode of the transfer cavity (usually

the TEM0,0 mode). Once the science cavity was locked to one of the ∼ 100 MHz sidebands, the frequency of this sideband was slowly adjusted to bring the probe into exact resonance with the cavity. After this exhaustive procedure resulted in a “locked cavity” the overall frequency stability of the lock chain could be evaluated. If we consider that each lock only stabilizes the relative frequency between two things to a level of ∆ν, then the three locks should lead to broadening of the frequency difference between the probe and cav- p 2 2 2 ity ∆ν = ∆ν780 + ∆ν850 + ∆νsc, where these are respectively the frequency stability of the probe–transfer cavity lock, the stabilization laser–transfer cavity lock, and the science cavity–stabilization laser lock. The resulting frequency broadening contributes a Gaussian width to the native, Lorentzian cavity lineshape. Such a convolved lineshape is known as a Voigt profile, as shown in Figure 3.3. Voigt profiles are well known from spectroscopy, where a spectroscopic line is broadened by the doppler effect. A Voigt profile is given by a convolution of a Gaussian and a Lorentzian.

2 2 G(δ, ∆ν) = e−δ /2∆ν , (3.6) 1 L(δ, κ) = , (3.7) 1 + (δ/κ)2 Z ∞ V (∆, ∆ν, κ) = dδ G(δ, ∆ν)L((∆ − δ), κ). (3.8) −∞ Unfortunately (3.8) is not an analytic function, but for most purposes, making a numeric function will suffice. A good agreement with the measured transmission spectrum is achieved with the profile, V (∆/MHz, ∆ν/2π = 1.1, κ/2π = 0.66). While this profile characterizes 29

Lo 0.1 re nt zi an ansmission r T Gauss !t 0.01 Voigt

-20 -10 0 10 20 Δ/2π (MHz)

Figure 3.3: The observed cavity transmission (dots, average of 100 traces) is well approx- imated by a Voigt profile (Solid line). Both Lorentzian and Gaussian fits are poor. For this measurement, the probe frequency was swept at a rate of 200 MHz/s, chosen to take ∼ 20 ms to cross the width of the resonance, a relevant time scale for the measurements presented in Chapter 5. our system well, and gives excellent agreement between experiment and theory, it indicates that at some level, the atom cavity dynamics are influenced by technical fluctuations.

3.2.3 Noise

We divide noise into two categories, there is the fundamental noise, such as shot noise, who’s origin is necessarily quantum mechanical, is interesting to study (Chapter 5), and limits measurements at a fundamental level. Then there is technical noise, which can in principle be avoided if we just work a little harder. Technical noise might include Johnson noise in our electronics, magnetic field noise from the nearby subway or radio-frequency noise from the low quality AC power provided to the lab. Technical noise can be avoided by reducing it (through filtering, shielding and feedback) to a level which is less than shot noise. Referring now specifically to technical noise, I make a distinction between “measurement” and “dynamical noise”. Measurement noise refers to noise which is between the cavity bandwidth (set by κ) and the bandwidth of mechanical motion. Measurement noise doesn’t affect the dynamics of our ensemble of atoms, but adds noise to the measurement of the mechanical system. Dynamical noise refers to noise within the bandwidth of motion, and thereby contaminates the dynamics. 30

• Measurement noise. A study of the collective motion requires a measurement which

integrated for a time which is less than the timescale of mechanical motion, ∼ 1/ωz. For example, imagine that the measurement signal is contaminated by some technical noise. Technical noise which is above the bandwidth of the cavity will be effectively filtered by the cavity and is irrelevant. The noise on this remaining signal will add to the shot noise on the detected photons. It is desireable to have technical noise at a level which is less than shot noise. Measurement noise can be either due to intensity noise, or in our case, because of the frequency selectivity of the cavity, to frequency noise. In order to be “shot noise limited”, technical frequency fluctuations must then result in intensity fluctuations below the level of shot noise intensity fluctuations. For

higher photon ﬂuxes, the frequency stability required above ωz, but below κ is, κ ∆νmeas ≤ √ p . (3.9) (3 3/8) ηn/ω¯ z

• Dynamical noise. Rather than spoil the detection of collective motion, technical noise can contaminate the collective motion itself. To see how this might happen, consider the opto-mechanical frequency shift (see Chapter 6), which results in a dependence of the mean oscillation frequencyω ¯ on the intracavity intensity. After a time τ, the phase of the collective motion will accumulate an error, Z τ δφ(τ) =ωτ ¯ − dt ω¯(n(t)), (3.10) 0 While this phase error is interesting in and of itself, here we only desire that the phase error accumulated from technical fluctuations of the intracavity intensity be smaller than the error from shot noise technical flucutaions. This requires that the intensity fluctuations be primarily due to shot noise. The relevant time scale τ for which the phase is integrated is on the order of 10 ms, since 10 ms is roughly the collisional time scale, and the damping time for atomic motion. At an intracavity intensity ofn ¯ = 1, about 4 × 103 photons transit the cavity over the 10 ms. Shot noise then corresponds to an intensity fluctuation of about 1.6%. In order to be shot noise limited, technical frequency fluctuations must then result in intensity fluctuations below that level. The frequency fluctuations in the range of bandwidth between the integration time τ and

the motional timescale ωz,

Z ωz 2 (∆ν) = Sνν(ω)dω, (3.11) 1/τ 31

where we deﬁne the spectral density of frequency ﬂuctuations, Sνν(ω). Frequency

ﬂuctuations Sνν(ω) result in changes of intensity at the most sensitive part of the √ cavity line of (3 3/8)Sνν(ω)/κ, integrating over the relevant bandwidth the required frequency stability to have “shot noise limited” dynamical noise is, κ ∆νdynam ≤ √ √ . (3.12) (3 3/8) κnτ¯

Forn ¯ = 1 and τ = 10 ms, the requirement is ∆νdynam < 16 kHz. The experimental requirements for shot noise limited cavity dynamics are threefold. (1) The linewidth and stability of the probe laser must be less than ∆ν. (2) Cavity length ﬂuctuations must be reduced to keep the frequency stability of the cavity below ∆ν. (3) The center frequency of the 850 laser must be stabilized to better than ∆ν, and quiet enough to permit locking the length of the cavity to the requirements of (2).

We have found that for sweep rates which are roughly 2π × (κ/10) MHz/ms, technical broadening contributes a Gaussian width of 1.1 MHz to the science cavity resonance. For these sweeps, the bandwidth τ ∼ 10 ms.

Z ωz ∆νsc = Sνν(ω)dω = 1.1 MHz (3.13) 1/τ This 1.1 MHz frequency stability is much larger than the goal of ∆ν = 16 kHz, and cavity dynamics are not shot noise limited, but in fact dominated by technical noise. The weakest link (in terms of noise susceptibility) in the cavity lock chain is the science cavity lock. Since the cavity length is only actuated by piezoelectric control, the bandwidth of this servo is limited. Figure 3.4 shows the science cavity servo bandwidth and piezo response. The signal to noise of this lock is limited by the detector noise of the avalanche photodiodes used for detecting the transmitted light. At typical light levels of √ 190 nW, the corresponding 14 pW/ Hz of noise, limits the signal to noise ratio of the lock, (with a bandwidth of 30 kHz) to 75. Given this signal to noise ratio, and turning a deaf ear to the limited servo bandwidth, we would expect to lock the cavity to κ850/75 = 180 kHz, a factor of 10 from our desired level. To see how good we might do in the future, consider the shot noise limited minimum frequency resolution of a heterodyne detection setup. On resonance, we have a 2 frequency resolution of ∆ν = κ850/(4η850n¯850τ), where 1/τ is the bandwidth of the measurement [78]. We normally ﬁlter our error signal at 30 kHz, so τ ∼ 33 µs. Our 190 nW 32

1 1 esponse (arb) o r ez i

P 0.1

102 103 104 0.1 1 10 3 3 Frequency (Hz) x10 Frequency (Hz) x10 -20 0

-40 -20

-40

dBm (arb) -60 dBm (arb) -60 -80 -80 -100 0 5 10 15 20 0 5 10 15 20 3 3 Frequency (Hz) x10 Frequency (Hz) x10

Figure 3.4: Characterization of the science cavity lock. The piezo response (upper right) has a prominent resonance around 25 kHz. The science cavity servo gain (upper left). The residual noise of the science cavity lock with and without liquid nitrogen ﬂow (gray line bottom left and right) was obtained after using a notch ﬁlter to knock out the prominent piezo resonance. The spectrum of intensity noise (black curves bottom) on light transmitted through the cavity was obtained by plugging the output of the single photon counters directly into the spectrum analyzer.

7 detection level corresponds ton ¯850τ = 2.7 × 10 photons, giving the shot noise limited ∆ν = 1.7 kHz, within our requirements.

3.2.4 The temperature of photo-detection

The practice of assigning temperatures to amplifiers and measurement devices is commonplace in the field of microwave electronics, but curiously absent when talking about optical photons. The single photon counting modules which we use constitute a “phase sensitive”, (also called, “phase non-preserving”) amplifier which adequately amplify a single photon click to a level which is well above the electron shot noise of the signal. One 33 can amplify a signal with a phase non-preserving amplifier without necessarily adding noise. The imperfection of our photo detectors arises in that (1) they are not useful in making measurements of the phase of light transmitted through the cavity, and (2) they exhibit a noise floor of dark counts. The single photon counting modules have a dark count rate of 250 s−1. These extra counts add noise to our signal and can be recast as an effective detector temperature.

The dark count rate Rdc, corresponds to an intracavity photon number ofn ¯ = Rdc/(2ηκ) ' 6 × 10−4. To assign a temperature to this, we consider how hot the cavity would have to be in order to have a thermal occupation of photons at this level, 1 Rdc/(2ηκ) = (3.14) e~ωc/(kT ) − 1 With T = 2500 ◦K this seems mighty hot, but should be considered nearly zero-temperature ◦ when compared to the effective temperature of a 780 nm photon, ~ωc/k ∼ 18, 000 K. There is a statement that a system comes to equilibrium with the temperature of its amplifier which sounds like a pretty absurd idea. But imagine that an experimenter somehow closed the loop, using measurements to feedback to and cool the system. The dark counts would then add noise to that feedback and heat the collective motion to a temperature T/G, where G is the gain. The gain is certainly very large since tiny mechanical motions are “amplified” into real optical photons.

3.3 The FORT

The use of a Far Off-Resonant optical dipole Trap (FORT) in CQED experiments[72] is a natural specialization of its application to cold atoms [79, 80, 81]. Two critical features of an optical trap contribute to the lifetime of atoms confined within, and guided our choice of wavelength for a FORT. Spontaneous scattering can be reduced relative to the depth of the optical trap by increasing the detuning. Intensity fluctuations of the FORT also parametrically heat the trapped atoms, and can be driven by laser frequency fluctuations when resonating with a narrow cavity. For the mirrors which form our cavity, the dielectric coatings create a lower finesse cavity, which is less sensitive to this frequency noise. The AC stark shift of the ground state due to a far-detuned laser field,

2 2 Γ I X |ceg| U = ~ . (3.15) t 2 I ∆ sat e ta 34

13 Since the detuning ∆ta of the trapping laser is very large (3.2×10 ) Hz compared to the 6.8 GHz hyperfine splitting, we only consider fine structure, summing over the Clebsch-Gordon coefficients for the D1 and D2 transitions [82]. This produces a ground state energy shift, 2 ~Γ I 2 1 Ut = + , (3.16) 2 Isat 3∆D2 3∆D1 I = ~ (2π × 0.31 Hz) (3.17) Isat The FORT was formed with the same 850 nm light which was used to stabilize the length of the science cavity and consists of standing wave mode of the cavity, described by the TEM0,0 mode,

2 2 −2( x +y ) w2 2 Ut(x, y, z) = Ut e 0 cos (ktz). (3.18)

At one of the trap minima, for small deviations, 2 2 2x 2y 2 2 Ut(x, y, z) = −Ut 2 + 2 + kt z , (3.19) w0 w0 m = ω2x2 + ω2y2 + ω2z2. (3.20) 2 x y z

q −4Ut p 2 We identify ωx,y = mw2 , and ωz = −2kt Ut/m. Notably, the trap frequency along the √ 0 z axis is ktw0/ 2 ' 125 larger than along the x or y axes. The FORT was loaded most reliably by translating a cold sample of atoms T ∼ 1 µK to overlap with the standing wave mode of the FORT, with the trap depth of the FORT as low as possible. With the atoms overlapping the mode of the FORT, the optical intensity was ramped up over 25 ms to form a trap of ∼ 6µK depth. After 50 ms of equilibration, the TOP trap was suddenly turned off. Evaporative cooling quickly reduced the temperature of the trapped atoms. This final temperature appeared to be independent of the initial temperature of the atoms loaded into the lattice. The main effect of tweaking the procedure for loading atoms into the FORT was to vary the atom number, rather than the temperature of the gas.

3.4 The atoms

3.4.1 Condensation, collisions and the loitering at each lattice site

Distributing 50000 atoms equally among 200 lattice sites produces 250 atoms per lattice site. For such a system, with ωz = 2π × 50 kHz, the BEC transition temperature 35 is 0.6 µK, this is quite close to the measured temperature for atoms in the lattice, 0.8 µK. While it would sound impressive to throw our hands up in victory and declare BEC-CQED!! [33, 34], I’m not sure what exactly this would mean. Nearly two full years after our group first announced our achievement of placing a BEC within a high finesse cavity at the 2006 DAMOP, followed shortly by two other groups’ announcements in the litterature [33, 34], no study which links definitive features of BECs (i.e. superfudity, phase coherence, single quantum wavefunction) to cavity-QED observations has been performed. I’m not even sure where one would look for such phenomena. The peak of the atomic distribution at each lattice site is 7.4 × 1013 cm−3. The thermal collision rate is 550 Hz. The tunneling rate between adjacent sites of the optical lattice is worth consideration: atoms tunnel at a rate: r h Z z2 2m i γ = ωz exp −2 dz 2 (V (z) − kT ) , (3.21) z1 ~ 1 p π z1 = arcsin kT/Ut, z2 = − z1. (3.22) kt kt For parameters in our system, γ = 2π × 10−6 Hz, so tunneling between sites is not a significant effect. The size of atomic cloud at each site is dictated by the size of the ground state along the z-axis, and by the cloud’s temperature along the {x, y}-axes. The extent of the the 2 2 cloud at each site is given by a gaussian distribution with full widths; G(x) = exp(−2x /σx),

2 kT 2 σx,y = 2 = (2.5 µm) (3.23) 2mωx,y 2 2 ~ 2 σz = zho = = (34 nm) (3.24) 2mωz Clearly the atomic cloud at each lattice site is very well localized. The localization of the √ cloud along the z-axis is quantiﬁed by the Lamb-Dicke parameter, ηLD = kpσz/ 2 = 0.27

3.5 Distribution of atoms in the lattice

The atom cavity coupling rate is diﬀerent at the location of each potential minima, but as the atoms at that lattice site are very well localized, we can assign a single coupling rate to all of the atoms at that site. If the atoms are evenly distributed among a large number 2 of lattice sites, the atom induced cavity shift will be ∆N = (N/2)g0/∆ca. and Neﬀ = N/2. The distribution of the atoms across many lattice sites was assessed by measuring the 36 2 0 2 g (z)/g l a i t n e t o p p a r t 2 0 /g )

i oupling (g c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 site number

Figure 3.5: The atom-cavity coupling rate is different at the potential minimum of every lattice site. The effective number of atoms counts with higher weight the atoms which are 2 2 located at the gradient of the coupling, where g /g0 = 1/2. If many sites are considered, 2 ∆N = Ng0/(2∆ca) and Neff = N/2.

3 80x10

Atom number 20

20 40 60 80 100 !N/2" (MHz)

Figure 3.6: Measured relationship between the total number of atoms trapped in the lattice as assessed with absorption imaging (left axis), and the atom induced shift of the cavity resonance. The plotted line is the predicted relationship based on an equal distribution of 2 atoms in the lattice, ∆N = Ng0/(2∆ca). 37

2 relationship between the atom induced shift of the cavity resonance, ∆ = PN g (ri) , N i=1 ∆ca and the total number of atoms, as counted by absorption imaging (Fig. 3.9). We take the agreement of the measured relationship with the prediction as conﬁrmation of an equal distribution of atoms within the lattice. Other “parasitic” arrangements of atoms could also produce such a measured relationship. For example if Neﬀ = N, then all the atoms are localized at regions which are maximally sensitive to their position, but only couple halfway 2 and ∆N = (N/2)g0/∆ca. To rule out such arrangements I’ll appeal to Chapters 4 and 5, where quantitative agreement with predictions requiring equal distribution can be found.

3.6 Experimental tricks

A typical experiment is detailed in Figure 3.7. A Count-Rate-to-Voltage-Converter (CRVC) built by K. L. Moore was used to quickly convert the digital signal to an analog signal with a bandwidth of about 1 kHz. This signal was monitored on a digital oscilloscope. In a typical experiment a few hundred thousand atoms were loaded into the FORT, inducing a large shift in the cavity resonance. We would simply detune the probe frequency from the bare cavity resonance by a few tens of MHz and wait for some sort of signal. After many seconds, owing to the lifetime of atoms in the optical trap, a discernible transmission through the cavity would increase the count rate of the CRVC and ultimately trigger the scope. This scope trigger could then be used for many other purposes, e.g to trigger a computer based oscilloscope to capture and store data. We found that triggering on the signal from the CRVC resulted in number ﬂuctu- ations that were dictated by atom shot noise, essentially preparing the system with a very well deﬁned number of atoms. To utilize this prepared system, a whole tangle of conditional operations were sewn together with simple logic operations and 555 timers rivaling in complexity the computer control system itself.

3.6.1 Frequency sweeps

A natural way to probe the properties of the atom-cavity system is in the transmission or reﬂection of the probe laser from the cavity. For frequency sweeps, a linear chirp is imposed on the frequency of the probe laser with an acousto-optic modulator. Usually we impose frequency chirps which probe the entire resonance of the cavity over a timescale which is fast compared to the lifetime of atoms in the optical trap, yet slow enough that 38

CRVC SPCM Science cavity digital oscilloscope scope trigger out/ GageScope trigger in

SRS pulse gen.

to A AB B C CD D frequency multiplexer

SRS func. gen. GageScope trigger

probe VCO

dose frequency

frequency

intensity multiplexer intensity

intensity 1 A C D B time intensity 2

Figure 3.7: Schematic of a typical experiment. Transmitted intensity from the science cavity (top) detected with a Single Photon Counting Module (SPCM), is both digitally captured by a computer based digital storage oscilloscope, and converted to a voltage signal with a real–time Count Rate to Voltage Converter (CRVC). The raw data from the SPCM is digitized before analysis. Further experiments were triggered oﬀ the cavity resonance as resolved by the CRVC (bottom). Shown here is the process used for measuring the oﬀ- cavity-resonance heating rate (see section 5.6), which involves changing the probe intensity, and frequency for a variable duration, and then sweeping the probe frequency. This process was coordinated primarily with a single SRS DG535 pulse generator. 39

Δ N

ωc ωp ωp ωp ωp ωp

Δ N(t)

ωc ωp

Figure 3.8: Two types of experiments. In a probe sweep experiment(top), the frequency of the probe laser is swept rapidly across the shifted resonance of the cavity. In a wait and see type experiment(bottom), a decrease in the total number of atoms causes ∆N to decrease, eventually resulting in a discernible transmission. 40

m) 50 μ

radial width ( 20

0 1 2 3 4 time (ms)

Figure 3.9: Time-of-flight temperature measurement. Absorption images (left, shown are 0, 2 and 4 ms TOF) were axially integrated and fit to a Gaussian distribution to find the width along the x axis. the intracavity intensity changes adiabatically with respect to the motional timescales of the trapped atoms. Frequency sweeps are sensitive to dynamics which occur above a cutoff frequency dictated roughly by the temporal cavity linewidth. For a frequency sweep rate, r, this cutoff frequency is r/κ. For example, to freeze out atomic motion, we would need r ∼ κωz/2π.

3.6.2 Wait and see

For a ﬁxed probe frequency, the observed lineshape is dictated primarily by the rate of change in atom number, but also by changes in the collective position. A solid understanding of these slow dynamics of the collective position allows for robust interpretation of the transmission trace as a measurement of number. These atom–loss driven lineshapes are analyzed in detail in Chapter 5.

3.6.3 Absorption imaging and time of ﬂight

Orienting the optical access along they ˆ-axis (see Fig. 2.3) allowed us to probe the position and momentum distribution of the atomic ensemble. Limited time-of-flight prevented very accurate measures of the temperature of the ensemble. Poor imaging resolution (> 10 µm) and the size broadening due to absorption imaging, made temperature analysis through ballistic expansion the most sensitive measure of the number of atoms. To image atoms in the |F = 1i state we first pumped the atoms to the |F = 2i state 41 to establish a cycling transition using a 50 µs pulse of light, counter-propagating to the imaging axis. Absorption images were exposed for 130 µs, with an imaging magnification of 4.3. The width of the atom cloud expanded ballistically from a finite size, leading to the p 2 2 observed width, σx(t) = σ0 + (vt) , with the observed velocity relating to the tempera- 2 ture as T = (v/vrec) Trec, with vrec = 0.59 cm/s, and Trec = 180 nK, the recoil velocity and temperature for 87Rb. 42

Chapter 4

Nonlinear optics from collective motion

Calling something nonlinear is like calling every animal at the zoo a nonzebra. — modern physics proverb

This chapter discusses nonlinear optical phenomena arising from the collective motion of atoms within the cavity. Owing to the long lived motional coherence of ultracold atoms, strongly nonlinear optical phenomena were observed even when the cavity contained far less than one photon on average. Portions of this work were discussed in the publication:

• S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, Cavity Nonlinear Optics at Low Photon Numbers from Collective Atomic Motion, Phys. Rev. Lett. 99, 213601 (2007)

One normally thinks of nonlinear optics as a ﬁeld of physics which pertains to and requires very powerful lasers operating at high optical intensities. Whole sections of the library are devoted to these nonzebras. In contrast to this high intensity regime, optical nonlinearities which occur at low intensities, or equivalently low average photon numbers n¯ in an optical resonator, may have applications ranging from optical communication to quantum computation [83, 84]. For optical nonlinearities in the regimen ¯ ≤ 1, one requires materials with optical properties which are altered by interaction with even single photons and in which, this alteration lingers long enough to inﬂuence the interaction of subsequent photons with the material. 43

Cavity QED systems which attain the condition of collective strong coupling, NC = N(g2/2κΓ) 1 satisfy such requirements. In these systems, atomic saturation on an optical transition induces nonlinear effects such as absorptive optical bistability [85, 86, 87], cross phase modulation [88] and photon blockade[89]. Nonlinear effects withn ¯ ≤ 1 require that the coherence of the atomic excited state persist longer than the residence time of photons in the cavity. Cold atomic gasses present a new source of long-lived coherence in their motional degrees of freedom. This coherence has led, for example, to the observation of superradiant light scattering [90, 91, 92]. For atoms which are unconfined, thermal motion limits the coherence time of the atom-light interaction [91]. This thermal decoherence p rate, Γc = 2kp kT/m, can be further evaded if atoms are confined within the Lamb-Dicke regime, ηLD 1, which impedes thermal motion from allowing atoms to stray too far compared to an optical wavelength. With atoms confined tightly within the Lamb-Dicke regime, other sources of damping, e.g. collisions, aharmonicities, and atom loss, (to name a few) lead to decoherence, but at a rate, ΓM , which in principle may be much smaller than the decay rate of the atomic excited state.

Table 4.1: Common terms

g0 : Maximum atom cavity coupling rate

∆ca : Atom cavity detuning ...... ∆ca = ωc − ωa PN 2 Neff : Effective atom number ...... Neff = i=1 sin (2kpz¯i) 2 Fdp : Single photon dipole force ...... Fdp = ~kpNeffg0/∆ca g2 ∆ : Atom induced shift of the cavity resonance ∆ = PN 0 sin2(k z¯ ) N N i ∆ca p i p Zho : Collective oscillator length ...... Zho = ~/(2Mωz)

M : Collective mass ...... M = Neﬀm

: Granularity parameter ...... = FdpZho/(~κ) 2 2 2 α : Kerr parameter ...... α = ~kpg0/(∆camωz )

β : Line bending kerr parameter ...... β = αn¯max∆N /κ 2 ...... β = n¯max2κ/ωz 2 ωrec : Recoil frequency ...... ωrec = ~kp/(2m) 44

4.1 Adiabatic collective motion

As discussed in Chapter 2, the interaction of the collective mode with the cavity mode is captured by the Hamiltonian,

† H = ~ωcn + ~∆N n − FdpZn + ~ωM (a a) + HΓM + HIN . (4.1)

The interaction of the collective position and the intracavity intensity leads to nonlinear response of the intracavity intensity and for sufficient drive, optical bistability and hystere- sis. In the context of a radiation pressure force, this nonlinearity was first observed in H. Walther’s group in 1983 [35]. To get a handle on this system, we assume for now (but will relax in later chapters) that the collective position adiabatically follows the intracavity intensity. Such an assumption entails that: (1) the intracavity intensity change slowly compared to the axial trap frequency, and (2) that the system is in the non-granular regime1, in which, the momentum impulse imparted by a single photon is small compared to the ground state rms momentum fluctuations, and therefore negligible. 2 The dipole force displaces the collective position to Z = Fdpn/¯ (Mωz ). In turn,

this displacement shifts the resonance of the cavity by FdpZ/~. If the cavity is probed with

light at frequency ωp, the intracavity intensity is then,

n¯ n¯ = max (4.2) ∆pc−∆N +FdpZ/~) 2 1 + 2 κ Z=Fdpn/¯ (Mωz )

n¯ = max . (4.3) ∆ −∆ +F 2 n/¯ ( Mω2) pc N dp ~ z 2 1 + κ

Here, nmax is the intracavity intensity on resonance. As the intracavity intensity depends on itself, we clearly have a recipe for nonlinearity. Things get a bit nicer if we assume right now that atoms are equally distributed in the optical lattice, so that Neﬀ = N/2 = 2 2 2 2 ∆N /(g0/∆ca). We’ll substitute the dimensionless kerr parameter , α = Fdp/(~Mωz ∆N ) = 2 2 2 ~kpg0/(∆camωz ), n¯ n¯ = max . (4.4) ∆pc−∆N (1−αn¯) 2 1 + κ 1We’ll address this more carefully in section 4.4. 2The kerr parameter α is the same as the dimensionless parameter in [31], but, since will be used to denote the granularity parameter, we have renamed the kerr parameter α. 45

We can identify the intensity dependence of the cavity resonance as Kerr nonlinearity, with the intensity dependent refractive index of the intracavity medium, 1 + ∆N (1 −

αn¯) = n0 + n2Ip. The value of n2 = αI0∆N , where I0 is the intensity of a single probe photon, is indeed very large compared to other dispersive material systems, but does not compare to values obtained in BEC slow light systems [93]. Figure 4.1 illustrates how asymmetric spectra are observed in the transmission of the nonlinear cavity. In the situation depicted, the cavity is “red-detuned” (i.e. ∆ca < 0) from the atomic resonance, inducing a force which displaces the collective position in the negative Z direction, and leading to an increase in the absolute shift of the cavity resonance. If the probe is swept from the bare resonance with a negative frequency chirp, (away from the bare resonance), then the increasing intensity of light causes the shifted cavity resonance to move further from the bare resonance, elongating that side of the line. When the intensity in the cavity reachesn ¯max, the atoms are at their maximal displacement. As the intensity in the cavity decreases, the cavity shift reduces resulting in a steeper side of the cavity line. Less precisely: red-detuned probe light forms an attractive potential, so as the probe light increases in the cavity, atoms move to locations of higher probe intensity and interact more strongly, causing the cavity resonance to shift further from the bare resonance. When a sweep begins at the bare resonance, the cavity resonance “runs away” from the probe sweep, is eventually overcome by it and shirks back to its original position. Equation 4.4 is a cubic equation in the intracavity intensity, indicating for some parameter regimes there may exist three real solutions for the intracavity intensity for a constant drive and detuning. Writing (4.4) clearly as a cubic,

2 3 a0 + a1n¯ + a2n¯ + a3n¯ = 0, (4.5)

a0 = −n¯max, ∆ 2∆ ∆ ∆2 a = 1 + pc 2 − pc N + N 1 κ κ2 κ2 2α∆ ∆ ∆2 a = N pc − 2α N , 2 κ2 κ2 ∆2 a = α2 N . 3 κ2

Unhampered by these burgeoning expressions, we can plow ahead algebraically to ﬁnd under what conditions there exist three real roots to (4.5). While these details are a delightful way to expend many pads of paper, it is more intuitive to consider the dimensionless “line 46 ansmission y tr vit a C

Probe detuning, Δ pc (MHz)

Δ , (MH

obe detuning r

P ansmission y tr vit a C

-54 -53 -52 -51 -50 -49

Detuning from bare resonance (MHz)

Figure 4.1: The origin of asymmetric cavity lineshapes due to atomic motion. As the probe detuning is swept across the cavity resonance, the atoms move to the new potential minima. This movement induces a small change in the position of the shifted cavity resonance, resulting in the intracavity intensity, (circles). Here, N = 50000, nmax = 2, ∆ca = −2π × 100 GHz, and ωz = 2π × 50 kHz. 47

bending kerr-parameter”, β = (α∆N /κ)¯nmax. The line bending parameter β gives the maximum nonlinear shift of the cavity resonance in units of the cavity half-linewidth. Once the maximum shift of the cavity resonance is somewhat larger than the cavity half-linewidth, the increasing asymmetry of the cavity transmission will result in bistability. Algebraic √ analysis of (4.5) indicates that for β > 8 3/3 ∼ 1.54, there are three real roots to (4.5) resulting in optical bistabilty.

4.2 The bistable potential

There is another productive way of understanding optical bistability in terms of an optomechanical potential. To obtain the optomechanical potential we integrate the dipole force over a displacement of the collective position, Z Z Fdpn¯max Udp = dZ Fdpn¯ = dZ , (4.6) ∆pc−∆N (1−kpZ) 2 1 + κ ∆ − ∆ (1 − k Z) =n ¯ F κ arctan pc N p , (4.7) max dp κ gives the dipole force contribution to the atoms’ harmonic potential. The total potential which the atoms encounter is then [59],

1 U = U + Mω2x2. (4.8) tot dp 2 z

This modiﬁed potential simply reiterates our previous discussion of the kerr nonlinearity, but has a particularly nice representation of the bistable regime. For parameters which exhibit bistability, the total potential is a double well containing two potential minima. As shown in Figure 4.2, under some conditions of our experiments, this double well potential contains two minima separated by a few nm, and a barrier of a few hundred µK. It is remarkable to note that for just a single ground state trapped atom, the p harmonic oscillator length zho = ~/2mωz = 34 nm; yet, the collective coupling of the ensemble results in features of the collective position which are much smaller than the wave-function of a single atom. The mention of a double well potential elicits images of ethereal cats lounging unwatched in two places at once, or, of quantum tunneling and Josephson junctions from the depths of most readers’ minds. This bistable potential, however, exists only in the presence of the cavity probe which continuously records the position of the atoms, preventing such 48

(a) (b) (c) ) K μ ential ( pot

(nm) (d) (e) (f)

Probe detuning, Δ pc (MHz)

Figure 4.2: Dispersive bistability. The potential landscape for ∆pc = −2π×(52–60) is shown in panes (a-e). In pane (f) the average number of photonsn ¯ is plotted as a function of the probe detuning. If the probe is swept with negative chirp (i.e. (a) to (e), the upper (gray) branch of bistability is observed. With an opposite sweep ((e) to (a)), the lower branch (black) is observed. The unstable branch (dashed) is not observed. The location of the atomic ensemble is shown by arrows in (c) and (d). The parameters used these plots are N = 50000, n¯max = 20, ∆ca = 2π × −100 GHz, and ωz = 2π × 50 kHz. 49

250

200

150 n

100

0 2.4 2.8 3.2 3.6 Time (ms)

Figure 4.3: Ringing in the bistable potential. A wait and see experiment was conducted at ∆ca = −2π × 93 GHz, with nmax ' 200. The termination of the upper branch of bistability exhibited clear oscillations of the collective motion. unwatched behavior. If you will, the probe ﬁeld is entangled with the position and dynamics of the atoms. Shortly thereafter, however, the probe ﬁeld is very destructively measured on our photodetectors, razing such a diaphanous entanglement.

4.3 Connection to experimental measures

Chapter 3 discusses in some detail the excess technical noise which contaminates the cavity signal. In the adiabatic limit which we are considering the collective position is simply “along for the ride” with the technical variations of the probe cavity detuning. These ﬂuctuations were accounted for by replacing the Lorentzian cavity lineshape with a

Voigt profile,n ¯ = V (σ, κ, n¯max, ∆), where ∆ = ∆pc − ∆N (1 − αn¯), is the detuning from the atom (and position) shifted cavity resonance. The expected cavity transmission can then be calculated numerically accounting for these fluctuations. The nonlinear response of the collective position was probed by sweeping the frequency of the probe laser linearly across the cavity resonance at a rate of a few MHz/ms. This sweep rate was sufficiently slow so that the trapped atoms adiabatically followed the varying intracavity optical potential, yet sufficiently fast so that atom loss from probe- 50

0.4

n n 5 0.2

0.0 0 -155 -150 -145 -140 -75 -70 -65

Δ pc /2π (MHz) Δ pc /2π (MHz)

Figure 4.4: Observed nonlinear and bistable cavity lineshapes observed in sweep experiments. The cavity line becomes increasingly asymmetric as the input intensity is increased. All traces, except the one with highest nmax are an average of three traces. The triggering technique (see Chapter 3) used to obtain these spectra resulted in variations of up to 1 MHz in ∆N /2π between measurements. Model lineshapes (black lines) were obtained using the discussed Voigt profile and β = {0.37, 1.3, 3.7} determined from the observedn ¯max and the common ∆N = 2π × 148 MHz, ∆ca = −2π × 30 GHz and ωz = 2π × 42 kHz. The lower (gray) and upper (black) branches of optical bistability were observed in a single experiment with probe light swept with opposite chirps (±6 MHz/ms) across the cavity resonance, and match the expected behavior with β = 9.5, ∆ca = −2π × 101 GHz, and ωz = 2π × 42 kHz. The upper branch of bistabilty (black) has been shifted by 1.7 MHz to account for atom loss which occurred during the 25 ms duration of the sweep. induced heating was negligible. These experiments were performed on nearly identically prepared samples using the triggered sweep technique introduced in section 3.6.1. Figure 4.4 displays such experimental sweeps both in the deeply bistable regime (β = 9.5) and with less significant nonlinearity. Notably, nonlinearity was observed with significantly less than one photon in the cavity. The collective adiabatic motion of the atoms was also observed. To do this we looked for a systematic change in the resonance of the cavity with the probe intensity. To verify that the collective position was displaced by the intracavity intensity, we made simultaneous measurements of the cavity resonance (by probe transmission) and the total atom number with absorption imaging. With ∆ca = 100 GHz, images were triggered to be taken at different portions of the line in a wait and see experiment. As shown in Figure 4.5 the collective position of the atoms varied roughly as expected. 51

n N Δ

Z (nm) (1-kZ)/ N Δ

image

Figure 4.5: Observation of collective adiabatic motion. The intensity dependent line shift was observed by comparing the shift of the cavity resonance, ∆N (1 + kpZ) to the expected resonance shift with no motion, ∆N , as calculated by collocated absorption images and the known distribution of atoms in the lattice. A schematic of the experiment (left); absorption images were performed at various ∆ by triggering absorption images at diﬀerentn ¯. Given that ∆ = ∆pc − ∆N + FdpZ/~, Z was obtained from the known ∆pc and expected ∆N = 2 Ng0/(2∆ca) from atom counting. Here, ∆ca = 100 GHz. Probe light induced atom loss between the trigger of the line and the imaging of the atoms (a 2 ms delay) reduced the observed trend (black line) from the expected trend (dashed line).

4.4 Granularity

Already the concept of granularity has been mentioned a few times in this chapter. Here we deﬁne the dimensionless “granularity” parameter,

F Z k N g2Z ≡ dp ho = p eff 0 ho , (4.9) ~κ κ∆ca that quantifies the coupling between quantum fluctuations of the collective atomic and optical fields. To characterize this parameter, consider the impulse Fdp/2κ imparted to the collective momentum from the force of a single photon, which acts for the average lifetime of a photon in the cavity (2κ)−1. For < 1, this impulse is smaller than the zero-point momentum fluctuations of rms magnitude Pho = ~/(2Zho), thus, the effects of optical force fluctuations on the atomic ensemble are adequately described by coarse graining. Likewise, the transient displacements induced by this impulse will shift the cavity resonance by an amount that is much smaller than κ. So far in this chapter we have only discussed the non-granular regime. 52 > 1

ε > 1 β y me i g e r r ula n

a ong nonlinearit

Gr Str n

less than one photon per trap period

Δca /2π (GHz)

Figure 4.6: The kerr-nonliearity “phase diagram”, shows the transition from weak β < 1 to strong kerr-nonlinearity as a function of the atom cavity detuning for typical parameters of the experiment, N = 50000, ωz = 2π × 50 kHz. Below about 13.6 GHz, the cavity system is in the granular regime. For average intracavity photon numbern ¯ < 0.04, less than one photon “visits” the cavity per trap period.

4.5 Nonlinearities at very low photon number

As the atom-cavity detuning is decreased, the dipole force per photon Fdp ∝ 1/∆ca increases correspondingly, allowing for signiﬁcant nonlinearity at lower and lower average photon number. At some point the dispersive limit of cavity QED becomes invalid and we are forced to work with the dressed state picture introduced in Chapter 2. Here, the two √ resonances are no longer distinctly cavity- and atom-like but are split by Ng0, a large amount. We imagine probing the system at one of these resonances, where the other is unexcited. Starting with Eq. 2.4 we can calculate the force given the corresponding term in the Hamiltonian: ! ω + ω q H = n a c + (∆ /2)2 + Ng2/2(1 − kZ) + other terms, (4.10) ~ 2 ca 0 2 1 ∂H −~kNg0/2 Fdp = = . (4.11) n ∂Z Z=0 p 2 2 2 Ng0/2 + (∆ca/2) 53

0.20

0.15

n 0.10

0.05

0.00 0.20

0.15

n 0.10

0.05

0.00 -20 -10 0 10 20 -20 -10 0 10 20 time (ms) time (ms)

Figure 4.7: Frequency sweeps applied at low photon number. Here, ∆ca = −2π × 10 GHz, Neﬀ = 10000. Over the 50 ms of data, the probe frequency was swept −∆pc/2π = 180 → 240 → 180 MHz. Probe light induced loss causes the tail of the second sweep to be extended. Bistability is indicated from the higher intracavity photon number observed for the upper branch (ﬁrst) sweep.

Expanding the resonance frequency, ω+ for small displacements, ! ω + ω q kZ Ng2/2 ω ' a c + (∆ /2)2 + Ng2/2 1 − 0 . (4.12) + ca 0 2 2 2 2 (∆ca/2) + Ng0/2

We’ll set β ≥ 1 as the condition for strong nonlinearity. In order for the dipole force induced displacement to be larger that the cavity linewidth,

−kZ Ng2/2 κ ≤ 0 . (4.13) p 2 2 2 (∆ca/2) + Ng0/2 Fdpn¯nl Z= 2 (Mωz )

2 Solving forn ¯nl and substituting the recoil frequency ωrec = ~k /2m, the minimum photon number for nonlinearity,

2 2 2 ωz κ Ng0/2 + (∆ca/2) n¯nl ≥ 4 2 2 (4.14) ωrecg0 Ng0

−4 For ∆ca → 0 this number reaches a limiting value ofn ¯nl ' 10 for the minimum ωz = 2ωrec.

Much before that limit, already atn ¯nl < ωz/2κ ∼ 0.03, reached at |∆ca| ≤ 2π × 15 GHz for our experimental conditions, the optical nonlinearity stems from the passage through the cavity of less than one photon per trap oscillation period. In this regime 54 our foregoing adiabatic approximation is no longer valid. Rather, the momentum impulse imparted by an itinerant photon induces a transient oscillation of the atomic medium, shifting the atomic resonance in chorus. Succeeding photons will be either on or off resonance with the cavity contingent on the period of elapsed time before the previous photon’s rever- berations. In this regime we would say that the atomic and cavity fields become entangled, the state of the cavity field and the atomic motion are necessarily dependent on each other and should be described by a density matrix for the total quantum state. Such nonlinearities were observed in the granular regime, however, a proper treatment requires substantial material from later chapters and is therefore relegated to Chapter 6. The nonlinear response discussed in this chapter occurs at average intracavity photon number much less than one owing to the long lived motional coherence of an ultracold gas. If we consider an unconfined atomic gas, the thermal coherence time is limited to the time it takes thermal motion to displace atoms significantly compared to the optical wavelength. For the temperature of our intracavity atomic gas this thermal coherence time, p tc = 2kp kBT/m = 7 µs would limit nonlinearities ton ¯ > 0.02. Rather, we have the fortune of pinning down atoms tightly, and effectively circumventing the thermal decoherence limit. A finite temperature of the gas still limits the coherence time of collective atomic motion. Most clearly, this appears as a broadening of the frequency of harmonic confinement in the radial direction of the cloud. Taking the HWHM (half-width at half-max) width of the atomic distribution, σHWHM = 1.5 µm, at this location, the intensity of the TEM00 mode is reduced by 1%. For ωz = 50 kHz, this amounts to a trap frequency reduction of 500 Hz over the size of the cloud. From this finite temperature effect, the damping rate of mechanical motion is expected to be approximately 500 Hz. Reducing the temperature of the gas, or using condensed samples of atoms could reduce this damping rate further. 55

Chapter 5

Quantum measurement backaction

This chapter discusses the heating of atoms due to force ﬂuctuations induced by the photon shot noise in a cavity. This heating constitutes the backaction of quantum position measurement. This observation was the ﬁrst quantitative study of backaction on a macroscopic mechanical resonator at the standard quantum limit. Portions of the work in this chapter was discussed in the publication:

• K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Observation of quantum-measurement backaction with an ultracold atomic gas, Nature Physics, published online May 18, 2008

5.1 The two faced nature of light

We are all familiar with the wave–particle duality of photons. Electromagnetic radiation is both described by a wave with measurable phase yet undetermined extent, and as a particle with definite location, but no semblance of phase. The wave nature of light is often used as a ruler to measure objects, utilizing the phase of the beam to measure distance. The particle nature of light, however is always present to impose some sort of disturbance on that distance. The Hiesenberg uncertainty principle poses a severe and hard limit on precision measurements, interferometry, and information technology to name a few. And while the mantra, you can’t know the position and momentum of a particle at the same time reminds us of this principle, its essence it more subtle. After a measurement of the position of some object to a precision δZ, the momentum of that particle would be 56 inevitably disturbed in an unpredictable way by an amount δP = ~/(2δZ). Just how this disturbance is applied gets to the heart of quantum measurement backaction. In our experiment we use the wave nature of coherent laser light to detect the phase shifts induced by collective motion of an intracavity atomic medium. These phase measurements have an associated “phase-imprecision spectral density” – complicated prose to refer to the well known fact that your measurement of phase is subject to shot noise. Associated with the sensitivity of the phase of light to the collective position is a force on the collective motion due to the intensity of light. Fluctuations of the intensity of this light, e.g. shot noise, obey a similar “photon number fluctuation spectral density”. One effect of this fluctuating force is to induce momentum diffusion. The number of photons and the phase of light are canonically conjugate implying an uncertainty relationship between them,

1 (∆n)(∆φ) ≥ , (5.1) 2 and correspondingly the phase-imprecision spectral density Sφφ is related to the spectral density of photon number ﬂuctuations, Snn by the Heisenberg uncertainty principle, 1 pS S ≥ . (5.2) φφ nn 2

Momentum diﬀusion due to photon number ﬂuctuations increases the energy of the collective motion, which ultimately couples to the many other degenerate modes of the atomic gas, increasing the total thermal energy. We measured this backaction induced heating through evaporative atom loss. This chapter begins with a model discussion of quantum measurement by way of an example, then moves to section devoted to a quantum derivation of the backaction heating of an optomechanical system. The second half of the chapter details the measurement and analysis of systematics. A third half of the chapter connects these measurements to quantum optics.

5.2 A model quantum measurement

Theoretical physicists often talk about measurement of some quantum property by sketching on the blackboard a galvanometer and declaring that the meter is used to measure something about a quantum system. While this picture of a measurement is a splendid kickoﬀ for a discussion of its implications, the details of how a galvanometer might 57 measure some property of a quantum system are quite interesting and subtle on their own. Here, through example we seek to illuminate some of these details by way of an example. Information about the atoms’ collective motion is probed, and conveyed to the detector by photons. In practice we detect these photons with single photon detectors and analyze the detected stream appropriately, but this measurement is not necessarily the most optimal measure of the collective motion. Imagine, rather, that we measure the phase of light transmitted through the cavity in a homodyne measurement. Photons are registered at two detectors at rates R1 and R2,

Rlo + Rc p R = + R R sin φ , (5.3) 1 2 lo c Rlo + Rc p R = − R R sin φ . (5.4) 2 2 lo c

Adjusting the phase diﬀerence, φ between the local oscillator, Rlo and the cavity transmission Rc, to be near zero, the phase is determined by counting photons for a time τ, (R − R )τ φ ' √1 2 (5.5) 2 RcRloτ The error of the phase estimate, δφ is due to shot noise on the total number of detected photons, N = (R1 + R2)τ, and can be determined by √ δφ N δφ = δN = √ , (5.6) δN 2 RcRloτ hence,

2 (R1 + R2)τ (δφ) = 2 . (5.7) 4RcRloτ

If the local oscillator is very bright, then R1 + R2 'Rlo, and, Rc = 2ηnκ¯ (recall that η is the photon detection eﬃciency), 1 (δφ)2 = . (5.8) 8ηnκτ¯ The phase of light transmitted through the cavity relates to the collective position as, ∆ ∆ − ∆ + F Z/ tan φ = = pc N dp ~. (5.9) κ κ

From which, assuming that we probe on resonance with Z near zero, i.e. ∆pc = ∆N , dZ κ κ = ~ sec2 φ ' ~ . (5.10) dφ Fdp Fdp 58

The uncertainty of a measurement of the collective position then decreases in time as the fractional shot noise on the detectors decreases. 2 2 dZ 2 ~ κ/F (δZ)2 = (δφ)2 = dp . (5.11) meas dφ 8ηnτ¯

2 2 2 If this were the end of the story, after a time τ = ~ κ/(8FdpηnZ¯ ho) the measurement would measure the collective position better than the standard quantum limit. In principle, this is okay as long as the other quadrature of motion is allowed to become correspondingly more uncertain. In practice, unless special care is made to construct a so called “backaction evading” measurement [43, 94, 95, 96], backaction adds noise to the object of inquiry. After the attentive reader has proceeded four pages into the next section, they will be thoroughly convinced that force ﬂuctuations of the intracavity shot noise lead to heating of the collective

motion at a maximal rate (on resonance assuming ωz κ),

F 2 n¯ R = dp . (5.12) c 2Mκ Utilizing the virial theorem for a harmonic potential, heating of the collective mode leads to an increase in the uncertainty of the collective position,

F 2 nτ¯ 2 dp (δZ)heat = 2 2 . (5.13) 2M κωz The total uncertainty is then,

2 2 2 (δZ) = (δZ)heat + (δZ)meas. (5.14)

The optimal measurement occurs when the total uncertainty is minimized,

2 2 2 d h Fdp ~ κ i 2 2 2 N + 2 = 0, (5.15) dN 4M ωz κ η 4FdpN N = 2κηnτ¯ is the number of photons which have been detected. Minimum uncertainty occurs when,

2 2 2 √ ~ κ . Fdp 1/2 η N = 2 2 2 2 = 2 (5.16) 4Fdp 4M ωz κ η 2 Plugging this in, the minimum uncertainty which can be attained,

δZ 2 1 = √ (5.17) Zho η 59 2 ho 2 Z)/(Z ) δ (

detected photons

Figure 5.1: Basics of a standard quantum measurement. As the photon fluence increases, the shot-noise imprecision of the measurement is decreased, and eventually offset by backaction. The line with negative slope shows the decreasing measurement uncertainty, the rising line shows the increase in the uncertainty due to backaction, and he dashed line shows the total 4 uncertainty. Here, with an optimal measurement η = 1, ∆ca = 2π×100 GHz, Neff = 2×10 , and ωz = 2π × 50 KHz, the collective position is measured at the quantum limit after a fluence of 7 photons. 60

Figure 5.1 shows this beautiful result. In the instance of an ideal measurement, where all the photons are detected (η = 1), the backaction heating perfectly cancels and eventually oﬀsets the reduced shot noise measurement imprecision. This illustrates the two faced nature of light; not only can light be used to make measurements of position, but the lawful half of its dual personality dutifully enforces the uncertainty principle.

5.3 Backaction heating

To get the details of heating and cooling dynamics for a quantum mechanical degree of freedom coupled to a cavity, one must necessarily look for a derivation which treats the system as such. Other approaches have worked out the rates of cavity heating and cooling though powerful, but opaque methods such as the fluctuation dissipation [59] and the quantum regression theorem [97]. Here, I present an “experimentalist’s” approach to the problem, which looks for small parameters, and simplifications which may be obtained. Rather than investigate the adiabatic and coherent effects of Chapter 4, we are interested in fluctuations of the intracavity field and of the collective position. As such, we neglect the constant average optical force ofn ¯ cavity photons by redefining Z → Z − ∆Z = † 0 2 Zho(a + a), and ωc = ωc + ∆N − Fdp∆Z/~, where ∆Z = Fdpn/Mω¯ z is the probe light induced displacement. With these substitutions, we have the Hamiltonian describing the collective mode–cavity system:

0 † † H = ~ωcn − FdpZho(a + a)(n − n¯) + ~ωza a + HIN . (5.18)

From Eq. 5.18 we obtain equations of motion for a and for the cavity ﬁeld annihilation operator c, where as usual n = c†c, da = −iω a + iκ(n − n¯), (5.19) dt z dc √ = −iω0 c + iκ(a† + a)c − κc + 2κc ., (5.20) dt c in

Here cin represents the coherent-state input field that drives the cavity. The granularity parameter is clearly responsible for the coupling between the atomic and cavity fields. The basic strategy for the next few pages will be to assume that this coupling is small, and solve for the atom cavity dynamics iteratively. We can now express the atomic field operator as, t Z 0 a(t) = e−iωzta(0) + iκ dt0e−iωz(t−t )n(t0) − n¯. (5.21) 0 61

From here, we evaluate the rate of change of the atomic energy:

d d d (a†a) = a† a(t) + a†(t) a (5.22) dt dt t dt t h † i † h † i = iωza (t) − iκ(n(t) − n¯) a(t) + a (t) −ωza (t) + iκ(n(t) − n¯) (5.23)

" t # Z 0 = 2κ22 Re dt0(n(t) − n¯)(n(t0) − n¯)e−iωz(t−t ) 0 + iκ a†(0)(n(t) − n¯)eiωzt − (n(t) − n¯)a(0)e−iωzt . (5.24)

For the sake of evaluating the cavity ﬁeld evolution we restrict our treatment to times which are short compared to the timescale over which the atomic motion is significantly varied by interaction with the light. Under this ansatz we approximate Eq. 5.21 as

a(t) ' e−iωzta(0). (5.25)

Inserting this solution for the atomic ﬁeld operator into the equation of motion for the cavity ﬁeld, (5.20) we have the following for the frequency components of c: √ 0 † −iωc(ω) = −iωcc(ω) − κc(ω) + 2κcin(ω) + iκ a(0)c(ω − ωz) + a (0)c(ω + ωz) . (5.26)

0 −1 Deﬁning L(ω) = (1 − i(ω − ωc)/κ) , we obtain L(ω)h√ i c(ω) = 2κc (ω) + i a(0)c(ω − ω ) + a†(0)c(ω + ω ) . (5.27) κ in z z

We can solve this equation iteratively, and at ﬁrst order in the granularity parameter we obtain

L(ω)h√ √ c(ω) = 2κc (ω) + i 2κ a(0)L(ω − ω )c (ω − ω ) κ in z in z † 2i +a (0)L(ω + ωz)cin(ω + ωz) + O |a(0)| . (5.28)

In the non-granular regime 1, and assuming small values of a(0), i.e. that the atoms are suﬃciently cold, we neglect terms of order 2 or higher. Utilizing the truncated expressions for the ﬁeld operator, (5.28), we can evaluate 62 the intracavity photon number operator,

1 Z n(t) = dω dω ei(ω1−ω2)t c†(ω )c(ω ) (5.29) 2π 1 2 1 2 " 1 Z L∗(ω )L(ω ) = dω dω ei(ω1−ω2)t 1 2 2κ c† (ω )c (ω )+ 2π 1 2 κ2 in 1 in 2

† † icin(ω1) a(0)L(ω2 − ωz)cin(ω2 − ωz) + a (0)L(ω2 + ωz)cin(ω2 + ωz) − # † ∗ † ∗ † i a (0)L (ω1 − ωz)cin(ω1 − ωz) + a(0)L (ω1 + ωz)cin(ω1 + ωz) cin(ω2) . (5.30)

With the above normally ordered product of operators cin we are justiﬁed in replacing: √ cin(ω) → πκnmax δ(ω − ωp), (5.31) † √ cin(ω) → πκnmax δ(ω − ωp), (5.32)

The intracavity photon number operator is then, h −iωzt † +iωzt n(t) =¯n 1 + i a(0)L(ωp + ωz)e + a (0)L(ωp − ωz)e − i † ∗ +iωzt ∗ −iωzt i a (0)L (ωp + ωz)e + a(0)L (ωp − ωz)e . (5.33)

2 Here we have substitutedn ¯ = nmax|L(ωp)| . We are now in a position to evaluate the heating rate:

d D d E E = ω a†a (5.34) dt ~ z dt " t # Z 0 2 2 0 0 −iωz(t−t ) = 2~ωzκ Re dt h n(t) − n¯ n(t ) − n¯ ie + 0 D E † iωzt −iωzt i~ωzκ a (0) n(t) − n¯ e − n(t) − n¯ a(0)e . (5.35)

Addressing the ﬁrst term ﬁrst; for a linear cavity driven by a constant coherent state input, we substitute the relation,

0 hn(τ)n(0)i − hn(τ)i2 =ne ¯ i(ωp−ωc)τ−κτ . (5.36)

Assuming the system is in a steady state, in that hn(t)n(t0)i = hn(t − t0)n(0)i, and substi- tutingn ¯2 = hn(τ)i2 we obtain for the ﬁrst half of the heating rate,

2 1 2 2 (−) 2~ωzκ n¯ 0 2 2 = ~ωzκ [Snn (ωz)]. (5.37) 1 + (ωp − ωc − ωz) /κ 63

Here we have introduced the spectral density of photon number ﬂuctuations [59],

2¯nκ S(±)(ω) = (5.38) nn κ2 + (∆ ± ω)2)

0 where, ∆ = ωp − ωc is the detuning of the probe from from the atoms shifted cavity resonance. The second term in Eq. 5.35 accounts for the eﬀect of transient atomic motion on the cavity ﬁeld. To evaluate this term we take the time average over an atomic oscillation period. D E iκ a†(0)n(t) − n¯eiωzt − n(t) − n¯a(0)e−iωzt

2 ∗ ∗ † =n ¯ κ L(ωp + ωz) − L (ωp − ωz) + L(ωp − ωz) − L (ωp + ωz) ha (0)a(0)i

2 2 (−) (+) † = κ Snn (ωz) − Snn (ωz) ha ai. (5.39)

These terms represent cavity cooling/anti-cooling. In total, the change in energy is, " # d E = ω κ22 S(−)(ω ) + S(−)(ω ) − S(+)(ω ) ha†ai . (5.40) dt ~ z nn z nn z nn z

When the condition T ~ωz/kB is satisﬁed, the mean vibrational quantum number ha†ai 1 and the latter two terms of (5.40) which represent coherent damping or am- pliﬁcation [98, 26, 99, 59, 60] may be neglected. In this case the collective atomic motion is 2 2 (−) heated at a rate Rc = ~ωzκ Snn .

5.3.1 A quantum limited ampliﬁer

Equation 5.40 is a very interesting expression. It contains all the aspects of back- (−) † action that are of current experimental interest. Referring to the term Snn (ωz)ha ai as anti-cooling is actually a bit of a misnomer, rather it represents amplification of the atomic energy, i.e. a phase preserving linear amplifier. In order for a linear amplifier to be phase preserving, it must amplify both quadra-

tures of motion (Z and P/(Mωz)) equally.

Zout = gZin,Pout = gPin. (5.41)

2 Unfortunately this type of ampliﬁer does not exist, since [Zout,Pout/(Mωz)] = g i~/(Mωz), unless g2 ≤ 1 the uncertainty principle is violated. Incidentally, we can see that a squeezing 64

ampliﬁer which ampliﬁes one quadrature, Zout = gZin, while attenuating the other Pout =

(1/g)Pin is allowed. The only way to construct a phase-preserving linear ampliﬁer is to consider other modes of the system [100]. For the annihilation operator, a = (Z/Zho + † iP/Pho), which entails, [a, a ] = 1, we can build an ampliﬁer which gives,

p 2 † aout = gain + g − 1bin. (5.42)

† The term bin refers to another mode of the system, and is cleverly combined to cancel out the terms which foiled (5.41).

† 2 † 2 † [aout, aout] = g [ain, ain] + (g − 1)[bin, bin] (5.43) = 1 (5.44)

Equation (5.42) describes a phase preserving linear ampliﬁer which obeys the uncertainty principle. This ampliﬁer necessarily adds more uncertainty than (5.41) which would have 2 2 2 give the relation (δZout) = g (δZin) . To see this, consider the position uncertainty.

2 2 2 2 † 2 † 2 (δZin) = hZini − hZini = Zho h(ain + ain) i − hain + aini . (5.45)

† After a few minutes of pushing a’s and a ’s around and using the fact that [ain, bin] = † † [ain, bin] = 0, applying the ampliﬁer (5.42) to Zin = Zho(ain + ain) gives,

2 2 2 2 2 (δZout) = g (δZin) + (g − 1)Zho, (5.46)

† 2 where I have assumed that the mode b is in a minimum uncertainty state, h(bin + bin) i − † 2 hbin + bini = 1. Equation 5.46 represents the sad state of aﬀairs in the business of amplifying quantum signals. Assuming that we start with a minimum uncertainty state,

δZin = Zho, our ampliﬁer “adds” uncertainty to our initial signal,

(δZ )2 − g2(δZ )2 2 out in = g2 − 1, (5.47) Zho which is the quantum limit to phase preserving linear amplifiers, and was worked out in general by C. Caves in 1982 [101]. It turns out that equation 5.40 in fact represents a near-ideal quantum amplifier. The dynamical backaction terms represent coherent amplification (i.e. a phase preserving linear amplifier) or damping of the collective motion . The measurement backaction simply 65 ) ) ω ω P/(M P/(M

ω ω Z Z

Figure 5.2: The momentum and position quadratures of motion are shown before and after amplification. Two types of amplifiers are considered. A phase preserving linear amplifier (left) adds more uncertainty than would be expected (dashed) by simply amplifying the quantum state and initial (minimum uncertainty) phase space area. A phase non- preserving amplifier (right) amplifies one quadrature while attenuating the other, squeezing the uncertainty. adds noise to the collective motion at a random phase1. If at time t = 0 the probe light is turned on, then after a time τ the collective energy would be amplified by a gain g2 = 2 2 (−) (+) 2 2 (−) κ (Snn (ωz) − Snn (ωz) τ + 1, and measurement backaction would add κ Snn (ωz)τ of

noise. In the resolved sideband regime ωz κ, and at ∆ = ωz, where the gain approaches 2 2 2 (−) g ' κ Snn (ωz)τ + 1, the uncertainty of the collective mode is increased by an amount 2 2 (−) 2 κ Snn (ωz) = g − 1, the minimum amount allowed by the uncertainty principle. The ﬁrst term in (5.40) represents the quantum backaction of measurements of the position of the collective motion. The backaction heating is largest at the cavity resonance

(for slow motion, i.e. ωz κ) and in accord, this is where a measurement of the collective position is strongest. Under constant drive the electric ﬁeld transmitted through a cavity

is κEin/(κ − i∆). Small deviation of the detuning, δ∆, will result in transmitted electric ﬁeld,

κE i E = in 1 + δ∆ . (5.48) out κ − i∆ κ − i∆

For a ﬁxed transmission through the cavity the magnitude of the change of the electric ﬁeld, 2 2 −1 2 (δEout) = (1 + (∆/κ) ) (δ∆/κ) is given by a Lorentzian, and is biggest on resonance.

1This point is not clear from simply staring at (5.40) since this expression only illustrates how the expectation value grows. 66

5.4 Heating from incoherent scattering

So far we have neglected the force fluctuations on the atoms associated with incoherent scattering. As in free space, spontaneous emission by atoms driven by laser light leads to momentum diffusion due to both recoil kicks and fluctuations of the optical dipole force. The less subtle of the two heating rates is the random recoil kicks due to spontaneous emission. These recoil kicks give a momentum impulse ~kp, [102, 103] at a rate proportional to the excited state population. The saturation parameter for the ith atom 2 2 2 is 2g0n/¯ ∆ca sin (kpz¯i). Spontaneous emission then leads to momentum diffusion (in three dimensions) at a rate,

2 d 2 2 2 2g0n¯ 2 Dse = hpi = ~ kp 2 Γ sin kpz. (5.49) dt ∆ca Atoms also undergo transitions between dressed states in the intracavity standing wave. As an atom absorbs and then reemits photons into the standing wave, the intracavity dipole potential from the probe light makes sudden jumps in strength. Where there is substantial gradient of the probe standing wave, these transitions result in a fluctuating force which also leads to momentum diffusion: 2 2 2 2g0n¯ 2 Dff = ~ kp 2 Γ cos kpz. (5.50) ∆ca The sum of these two “free-space” sources of momentum diffusion leads to a heating rate 2 2 2 2 of, Rfs = 1/(m)~ kpΓg0n/¯ ∆ca. These terms are labeled “free-space” because they would occur identically in a free space standing wave of light with no cavity boundary conditions present. It is interesting to compare this heating rate to the backaction heating rate, (5.40). 2 At its maximum, ∆ = ωz, the backaction heats the collective mode rate, Rc = ~ωzκ = 2 2 Neffg0/(2Γκ)Rfs. Recognizing the single atom cooperitivity, C = g0/(2Γκ), we see that at its maximum, backaction induces heating at a per atom rate which is C ∼ 50 times larger than the effect from spontaneous emission. Returning our attention briefly to the discussion in section 5.2, the free space heating also adds uncertainty to any measurement. After a time τ, free space heating adds 2 2 uncertainty to the position of each atom, (δzi) = Rfsτ/(mωz ). These uncertainties are uncorrelated between lattice sites and are added in quadrature.

N 2 2 2 X Rfsτ sin (2kpz¯i) Rfsτ Fdpτ (δZ) (δZ)2 = = = = heat (5.51) fs mω2 N 2 N mω2 2N 3 κCm2ω2 N C i z eff eff z eff z eff 67

This is an important statement; atom cavity systems which attain the collective strong coupling regime, deﬁned by NC 1, are capable of being backaction limited, it is not necessary to work in the regime of single atom strong coupling. Including this source of uncertainty in our analysis, the new minimum uncertainty is, p δZ 2 1 + 1/(NeﬀC) = √ (5.52) Zho η

Which for C ' 50, and N ∼ 104, is not a signiﬁcant change. There is a fundamental diﬀerence between the “free-space” heating and backaction. Uncertainty gained from the “free-space” contribution is unavoidable, whereas backaction can in principle be evaded [43, 95], canceled [104], or reduced by feedback.

5.5 Measuring backaction heating by the evaporative loss of trapped atoms

Measurement backaction heats the collective mode of the ensemble. Energy of this mode is eventually coupled to the many other degrees of freedom of the ensemble, resulting in an increase in the total thermal energy. As discussed in Chapter 4, the finite size of the atomic distribution in thex ˆ andy ˆ directions broadens the frequency of harmonic confinement and collisions cause energy of the collective mode to be quickly distributed among the 3N − 1 other modes of the system. In turn, the total thermal energy of the ensemble is kept constant by evaporation. When atoms are ejected from the trap, either by an RF spin flip as with forced RF evaporation, or by the chance that they end up at the lucky end of the Boltzmann distribution with enough kinetic energy to escape, they carry away on average an amount of energy equal to the depth of the trap, Ut. The factor ηe, not to be confused with our detector efficiency, is used to denote the ratio by which the depth of the trap exceeds the thermal energy of the ensemble, ηe = Ut/(kT ). Given ηe, one can calculate the probability of an atom escaping by positing a cutoff on the Boltzmann distribution, where atoms which have an energy higher than Ut are removed. A general rule of thumb for traps with large ηe is that the rate of evaporated atoms scales as [105],

−ηe N˙ ∼ Nηee , (5.53) 68

The rate of evaporation increases, increasingly with decreasing ηe. In principle, the temperature of the ensemble would sink to zero as all the energy of the ensembe was removed through evaporation, but in practice, the evaporation rate UtN˙ balances some ﬁnite heating rate. If the heating rate is increased substantially, the thermal energy of the sample only need increase slightly to increase the evaporation rate to balance the new heating rate. For this reason we relate an observed loss of atoms to the induced, per atom heating rate as R = −(U/N)N˙ . The number of atoms is conveniently related to the shift of the cavity resonance,

∆N . The absolute number of atoms is easily assessed by sweeping the probe across the resonance. The rate N˙ is assessed by monitoring the transmission of the probe change in time with the number of atoms. The intracavity intensity is related to the number of atoms as, (setting the collective position to zero, and assuming atoms to be equally distributed in the lattice)

nmax n¯(t) = 2 . (5.54) ∆pc−N(t)g0 /(2∆ca) 2 1 + κ Depending on the transmission, the number of atoms is,

2∆pc∆ca 2κ∆ca p N(t) = 2 ± 2 nmax/n¯(t) − 1. (5.55) g0 g0 The number of atoms almost always decreases monotonically in time, and we take the appropriate sign of the square root to reﬂect this. This section contains studies which utilize the relationship between the observed intracavity intensity and the number of atoms. These studies were conducted on two levels of detail, ﬁrst, “line rates”, a rough measure of atom loss rates are presented. These line rates were used to study the scaling of the atom loss rate with the atom-cavity detuning and intracavity photon number. Second, under some circumstances, the cavity lineshape was interpreted in detail to study the loss rate as a function of the cavity detuning.

5.5.1 Line rates

If we assume that atoms are lost at a constant rate N˙ , then the temporal linewidth of the line, i.e. how long it takes for the resonance to move across the ﬁxed probe frequency due to atom loss, has a useful meaning. The temporal half-width at half-max, τhw = ˙ 2 ˙ 2κ∆ca/(Ng0), relates to the loss rate of atoms. In practice, N depends on the detuning 69

nmax

2.5 n(τ) 2.0 ) τ 1.5 n(

1.0

0.5

0 5 10 15 20 25 τ (ms) 0 τ

40.5

40.0

39.5 N Δ ( τ ) 39.0

38.5

38.0

0 10 20 τ (ms) 30 40 50

Figure 5.3: Study of thermal and evaporative equilibration. Inset shows a basic schematic of the experiment. The cavity line is interrupted by extinguishing the probe for a variable time τ. It is assumed that some probe induced heating will cause atoms to eventually be evaporated from the trap leading to a loss of atoms, and shift of the cavity resonance during the variable time τ. To measure the loss of atoms during this time period the level 2 n¯(τ) was recorded, and noting thatn ¯(τ) = nmax/(1 + (∆/κ) ), we extract ∆N (τ) = ∆pc ± p κ nmax/n¯(τ) − 1, with the appropriate roots taken assuming ∆ decreases monotonically. The number of atoms is proportional to ∆N (τ). Noting that ∆N (τ) is plotted on a log scale, and will eventually decay to zero, two distinct rates are clearly evident. The initial fast loss of atoms is attributed to the time scale for thermal equilibration and evaporation resulting from the induced heating at times prior to τ = 0. The latter, slow decrease in ∆N relates to the background loss rate. 70 from the cavity resonance and the intracavity photon number. To examine this eﬀect we posit a rate which scales as,

N˙ Acnmax Afsnmax = + + Abg. (5.56) N 1 + (∆/κ)22 1 + (∆/κ)2

This is the expected heating rate for the atoms-cavity system and combines the rate which is expected for backaction with the rate from incoherent scattering and a background loss 2 2 rate. Note that the cavity detuning ∆ = ∆pc − Ng0/(2∆ca), varies as , d∆ g2 = − 0 N.˙ (5.57) dt 2∆ca This is a diﬀerential equation which can be solved easily.

Z κ d∆ Z τhw g2 = 0 dt (5.58) 0 N˙ (∆) 0 2∆ca

We already posited a functional form for N˙ (∆), so

Z κ A n A n −1 τ Ng2/(2∆ ) = d∆ c max + fs max + A , (5.59) hw 0 ca 2 2 bg 0 1 + (∆/κ)2 1 + (∆/κ)

Hence, the “line time” is inversely proportional to the loss rates. In particular, we expect that the backaction induced loss rate dominates signiﬁcantly over the other loss rates, and we equate directly, with Ac = RfsC/Ut,

2∆ca 28κUt τhw = 2 . (5.60) Ng0 15RfsCnmax This measure works well for situations where non-linearities are very slight. In the bistable regime, the upper branch may extend the line width much further than dictated by the loss rate, yet when following the lower branch of bistability this eﬀect is less signiﬁcant. We found in practice that the line time measured loss rates bigger than expected, but with scaling that agreed fairly well with the expected trends. The disagreement by a slight numerical factor of 2 remains unsolved.

5.5.2 Technical sources of heating

To interpret our measurement of heating as quantum backaction it was necessary to establish that technical sources of heating did not contribute to our observed heating

2Note that we’re ignoring collective motion here. The kerr eﬀect will be included in section 5.5.3. 71

14 12 10 n 8 6 4 max 2 easing n 0 incr 0 50 100 150 200 250 time (ms)

Figure 5.4: Temporal cavity lineshapes taken at increasing nmax. The temporal width of the observed spike in transmission is inversely proportional to the loss rate of atoms. Here ∆ca = 2π × 294 GHz, Neff = 8700, representative data traces used in Figure 5.5 to measure the dependence of the heating rate on the intracavity intensity. rate. Classical intensity fluctuations either due to frequency noise, or fluctuations in the intensity of the probe would also heat the atoms. Force fluctuations induce fluctuations in the trap center, which induces heating at a rate [106]

dE π 1 = S (ω ), (5.61) dt 2 M ﬀ z

where Sff(ωz) is the one-sided power spectrum of force fluctuations. This spectrum is given by, Z ∞ 2 2 Sff(ωz) = Fdp dt cos ωzthn(t)n(0)i (5.62) π 0 which should start to look reminiscent of terms encountered in section 5.3. Here we consider the possibility that the correlation hn(t)n(0)i is classical in nature. We can clearly expect that the correlation, and in consequence the heating rate scales asn ¯2 and not asn ¯ as seen in (5.36). This is in its very nature the difference between quantum and classical fluctuations. Shot noise can be seen as a beat-note between a coherent state in one mode of the electromagnetic field with the vacuum in the modes at all other frequencies. Half of the two time correlation comes from the vacuum, (which gives the Lorentzian line shape for the backaction heating), and half from the coherent state. 72

3 40

z) 10 x10 z)

30 1 (dN/dt)/(nN) (H

(dN/dt)/(n) (H 20

0.1 2 3 4 5 6 7 8 9 5 10 15 100 N x10 3 Δ(GHz)

200

150 z)

100

(dN/dt)/N (H 50

10 20 30 40 n

Figure 5.5: Line-time measurements: the per photon loss rate N/n˙ max was deduced from the measurements of the line time (see Fig. 5.4). The τhw was determined by fitting transmission traces to a Gaussian distribution (which fits more reliably). The loss rate was determined ˙ 2 from the line time as, N/(Nnmax) = 59∆caκ/(15Ng0τhw). Measurements of atom loss rate vs atom number, (shown in the upper left for ∆ca = 46 GHz) were compiled for many different detunings (upper left). The expected loss rate from backaction heating (black solid line) disagrees with the prediction by a factor of four. The loss rate versusn ¯ shows slight quadratic dependence, indicating a ∼ 4% contribution of technical heating atn ¯ = 1. At high photon levels, τhw approaches the re-equilibration time scale of ∼ 1 ms, and the observed loss rate is reduced. These points are ignored in the fit. 73

The diﬀerence between classical, technical heating is then measured in the scaling of the heating rate with photon number. As shown in ﬁgures 5.4 and 5.5, to make this measurement we operated at ∆ca = 294 GHz, with N = 17, 000, and measured the line- time forn ¯ ranging from much less than one, up ton ¯ = 40. Ignoring line-times which were comparable to the 1 ms equilibration time, we found that the heating rate scaled as 7(1)¯n + 0.26(6)¯n2, which indicates that forn ¯ = 1, backaction heating dominates over technical heating by a factor of 27.

5.5.3 Quantitative interpretation of the cavity line shape

So far, by analyzing the temporal linewidth of cavity lineshapes we have demonstrated that cavity induced heating has the appropriate scaling with: (1) atom number, (2) atom-cavity detuning, and (3) intracavity photon number, establishing that quantum ﬂuc- tuations are responsible for the large majority of the observed heating. In this subsection, I discuss how, by analyzing the cavity transmission more carefully, we were able to measure the cavity induced heating rate as a function of the detuning, ∆. The cavity transmission line shape contains, in principle, much more information than just the line-time, which is related to the integral of the heating which occurs across the resonance. For line-times which are signiﬁcantly longer than the 1 ms re-equilibration time scale, at every point the line shape conveys information about the number of atoms, its rate of change, and the intracavity photon number.

The Voigt proﬁle

As discussed in Chapter 3, technical noise broadens the cavity transmission. This was well described by a Voigt profile with Gaussian and Lorentzian widths, σ = 2π × 1.1 MHz, and κ = 2π × 0.66 MHz respectively. Based on the assumption that the number of atoms did not vary significantly over the duration of the line (it varies by 4 × 103 out of 4 × 104 over the course of the line) the magnitude of the Kerr effect was constant across the duration of the line. We accounted for the Kerr nonlinearity by using a skewed Voigt profile and used this shifted profile to convert our transmission into ∆(t). We scaled the transmission curve by nmax, and subtracted the background light level as determined from the wings of the transmission trace. The shifted Voigt profile is a numeric function and can not be inverted, rather we numerically compared the observed intensity to the Kerr-shifted 74

Voigt proﬁle to determine ∆(t) from the matching intensity. The trace ∆(t) is relates directly to the total number of atoms, from its rate of change the per-atom loss rate is then,

1 dN d∆(t)/dt = . (5.63) N dt ∆pc − ∆(t) After subtracting the background loss rate, the per-atom loss rate is normalized to the intracavity photon number, which is small at the wings of the measurement coincidentally where the signal to noise is bad. This limits the measurement of the per-photon heating rate to regions which are near the cavity resonance.

Measurement of backaction heating

We measured the spectral dependence of backaction induced heating by quantitatively interpreting the temporal cavity lineshape. With ∆ca = 100 GHz, and ∆pc = 2π × 40

MHz, the large initial number of atoms shifted the cavity resonance by ∆N = 2π × 100 MHz, producing an initially negligible transmission owing to the large detuning between the probe and cavity resonance frequencies. A background loss rate of approximately 1 Hz eventually brought the atoms-cavity resonance near the probe frequency, leading to discernible transmission (Fig. 5.6). The atom loss rate was determined by ﬁtting 12 ms sections of the extracted number to a line. From the observed loss rate we determined the per-atom heating rate of the trapped atomic sample as R = −(U/N)dN/dt (Fig. 5.7).

Atoms exposed to cavity-resonant light were heated at a per atom rate that is R/Rfs ' 40 times larger than that of atoms exposed to a standing wave of light of equal intensity in free space. The cavity-induced heating was abated for light detuned from the cavity resonance. While this cavity-enhanced diffusion has been inferred from the lifetime [17] and spectrum [107] of single atoms in optical cavities, our measurements are performed under experimental conditions that allow its direct quantification, and a proper interpretation of its physical origin as an enforcement of the Heisenberg uncertainty principle. To compare our measurement of backaction induced heating to our theoretical predictions we convolved the expected spectral dependence with with the same Gaussian kernel used in the Voigt profile. Flucutations of the probe-cavity detuning reduce the maximum expected heating. This comparison is shown in Figure 5.7. The measured heating rate agrees well with the prediction for measurement backaction. 75

2 n

6 5 4 3 Avg. photon num.

45 -5 Δ /2 ) 3 π (MHz) 40 0 N (10

35 5

-100 -50 0 50 100 time (ms)

Figure 5.6: Cavity-based observation of evaporative atomic losses due to cavity-light- induced diffusive heating. The intracavity photon number (top),n ¯ (points, average of 30 measurements) is monitored as the atom number is reduced by evaporation, and the cavity resonance is brought across the fixed probe frequency. The expectedn ¯(t) exclud- ing (dashed) or including (solid) cavity-enhanced diffusive heating are shown. The atom number N(t) is inferred from the measured photon number based on the cavity lineshape (bottom). Atoms are lost at a background rate of 0.9(1) s−1 per atom away from the cavity resonance, and thrice faster near resonance. 76

40 s f

R/R 20

0 -4 0 4 Δ/2π (MHz)

Figure 5.7: Cavity-heating of the collective atomic mode in a strongly coupled Fabry-Perot cavity over the spontaneous emission dominated free space heating. The measured ratio R/Rfs is shown with 1σ statistical error bars. Systematic errors, at a level of 23% at the cavity resonance, arise from uncertainty in the background loss rate, the background light level, and overall photo-detection eﬃciency. Grey line shows theoretical prediction with no adjustable parameters. Dashed line shows an upper bound on the oﬀ-resonance heating rate based on measurements at ∆ca =2π×29.6 GHz and ∆=2π×40 MHz.

Budget of errors

Nearly any quantitative cavity based measurement is subject to systematic uncertainty in the detection efficiency. For our experiment, η proved to be a difficult quantity to measure because the power at which the SPCM saturates is so low that it cannot be measured by any other means. For each measurement, uncertainties arise from the following: (1) the exact OD of the filter, and its effect on the fiber coupling, (2) the alignment of the light into the Newport power meter head, (3) thermo– or opto–mechanical effects of the high probe level on the cavity lock, and (4) how effectively the Gage and Igor software count each photon. For each measurement we performed a single measurement of η, but assign systematic uncertainty to our value of η as the standard deviation of repeated η measurements over a time-scale of a single data run.

5.6 Oﬀ cavity-resonance heating

We also measured the atom heating rate due to intracavity light that is far from the cavity resonance. In this case, the backaction heating is signiﬁcantly reduced and one should observe the spontaneous-emission-dominated heating of atoms in free space. To make 77

Table 5.1: Account of errors

Source Value Statistical Systematic background light level 0.175 0.04 η systematics averagen ¯ 1.94 0.03(ﬁt) 0.02(bgrnd), η trapdepth 6.6 µK 0.7 0.6 averaging overlap shift by ∼ 20 ms 0.4 ms 0 background loss rate 0.88 12% triggerpoint 40 MHz 1 MHz

∆ca 100.3 GHz 1 GHz 100 MHz dN/dt ﬁt 10% at peak

ηcoupling 0.27(for Fig. 5.6) 22% clipping (< 10%)

ηcav 0.135 0.007 power meter calibration a measurement of this fairly insubstantial heating rate, a method of controlled irradiation was developed. Experiments were conditioned on the observation a cavity transmission which surpassed a threshold level. For constant probe intensity this established a certain number of atoms in the cavity close to the atom shot noise level [67]. With this well determined number of atoms, the probe laser frequency was rapidly switched to a detuning of ∆ = 2π × 40 MHz from the cavity resonance, and increased in intensity manyfold (a factor of 5 × 104) to populate the cavity withn ¯ = 2 off-resonant photons. After a variable “dose” of light, the intensity of the probe was reduced, and its frequency swept to determine the shifted position of the resonance. From the decay rate of N, we observed a probe-light-induced per-atom loss rate that, if ascribed completely to diffusive heating of the atomic sample, yields a heating rate of R/Rfs =2.9(7), far smaller than that observed at the cavity resonance. Yet, these losses exceeded those expected based on diffusion from Rayleigh scattering. This discrepancy may be explained by additional effects of Raman scattering. Atoms may be scattered out of the F = 1 ground state with a probability indicated in Figure 5.9. These atoms which end up |F = 2, m = −1i state may result in the loss of two atoms later on due to an inelastic collision, or at least couple to the cavity with a different strength. These Raman scattering events occur less frequently (with a branching ratio of 0.13), but may result in the 78

(a) (c)

power 8 7 6 Probe

r 5

frequency e b ω m 4

c u

trigger dose detect n m 3 o

t y t A i 1 (b) v a

C 4 3

n 2x10 i

s 2 n o t 0.1 o h

P 50 100 150 200 250 300 4 dose time (ms)

-50 0 50 100 Time (ms)

Figure 5.8: Controlled dose experiment for measuring the heating rate for light which is detuned from the cavity resonance. (a) Schematic of the switching of the probe intensity and frequency. (b) At ∆ca = 2π ×29.6 GHz and with N ' 9000 atoms we performed a measurement of the probe induced heating. When the cavity transmission surpassed a threshold level ofn ¯ = 0.1 the frequency was switched to detune the probe light from the cavity resonance. The intensity was increased by a factor of 5 × 104 to populate the cavity withn ¯ = 2 oﬀ resonant photons. After a variable dose, the frequency of the probe was swept at a low intensity to probe the resonance of the cavity thereby measuring the remaining number. We recorded the remaining number for experiments with and without a dose of heating. The background loss rate of approximately 1 Hz was comparable to the additional heating. 79

-2 -1 0 1 -2 F=2

F=1 -1 0 1 0 F=0

25 5

F=2 -2 -1 0 1 -2 F=1 -1 0 1

Figure 5.9: Relevant transition strengths for atoms probed with σ+ polarization on the D2: 2 2 S1/2− P3/2 transition [108]. Atoms may be scattered out of the F = 1 ground state.

loss of an atom after only one scattering event, rather than the many Rayleigh scattering events required to eject an atom from the trap. The details of how atoms are eventually scattered into the many available states, and how the lifetime of these states and inelastic collisions between them lead to the observed heating rate for oﬀ-cavity resonance light are not particularly relevant to the message of this thesis. Our measurements indicate that these other eﬀects are constrained to contribute negligibly to the backaction dominated heating on resonance.

5.7 Connection to quantum optics: an intracavity ﬂuctua- tion bolometer

Our backaction dominated cavity system may also be used to sense properties of the intracavity field. As shown in (5.40), when working in the non-granular regime, and with the collective mode near its ground state so that the effects of dynamical backaction may be neglected, we may interpret the measured backaction heating as a direct measurement of the spectrum of photon number fluctuations in a driven cavity, a quantity of fundamental

interest in quantum optics. From the measured maximum heating rate of R/Rfs = 43(10) we find the spectral noise power of photon fluctuations in a resonantly driven cavity to be −7 −7 Snn/n¯ = 4.0(9) × 10 s, in agreement with the predicted Snn/n¯ = 2/κ = 4.8 × 10 s. Specifically, working in the non-granular regime, and with atoms in their mechanical ground state, the atoms-cavity system serves as an intracavity fluctuation bolometer. I’ll 80 note that this bolometer has already been used in determining that shot noise dominated over technical noise for our measurement of the backaction heating. One could, demonstrate this fact down stream of the cavity with traditional photodetectors. But, because the efficiency of photo detection is fairly low, one would have to perform these measurements, and demonstrate that shot noise dominates over techincal noise at photon levels η−1 higher than the nmax = 2 used in our measurement. Likewise, quadrature squeezed light is only degraded by the finite detection efficiency, but could be studied in-situ with this bolometer. While the resolution obtained in Figure 5.7 may not look particularly impressive, and the contorted difficulty of using an ultracold atomic gas to measure something easily obtained by traditional means appears sadistic; we note that these fluctuations are not visible in the coherent light transmitted through the cavity, for which the shot-noise spectrum remains white. The modified spectrum of photon number fluctuations for coherent light exciting a cavity is often explained as a filtering of the white shot noise of free space by a resonant filter. This leads to the misconception that light transmitted through a cavity should also have a modified spectrum of fluctuations. Rather, we interpret shot noise as a beat note between a coherent state occupying a single mode of the electromagnetic vacuum and the vacuum fluctuations in other modes. In free space, where the density of states of the electromagnetic vacuum is flat, we expect a beat note of the same strength at all frequencies, hence the term white noise. In a cavity, the density of states of the electromagnetic vacuum is modified by the boundary conditions, and gives rise to a colored spectrum of shot noise. Justification of this claim exists elusively in the literature. It is pointed out in passing [110] that the two time correlation for thermal light is the same up and down stream of the cavity. But, the point is concealed by a statement in my favorite quantum optics text that “the photon statistics of the output field will also be the same as of the intracavity field” [111]. This statement is correct only for normally ordered photon operators, and does not apply to the two time correlations, e.g. (5.36), which is not a normally ordered product of operators. The essential difference between the fields inside and outside the cavity are contained in the commutation relations for the photon creation and annihilation operators. For 81

laser beameam vacuumm inin other modes spectrum analyzer

Figure 5.10: The fluctuation bolometer. Atoms serve as an in-situ heterodyne detector of cavity-enhanced fluctuations of the electromagnetic field. The atoms are a non-destructive probe of the intracavity photon number and are sensitive to abnormally ordered products of operators which are not visible in the light transmitted through the cavity.

a two sided cavity [111], √ bout(t) + bin(t) = κc(t), (5.64) √ dout(t) + din(t) = κc(t), (5.65) √ √ κbin(ω) + κdin(ω) c(ω) = 0 . (5.66) κ − i(ω − ωc)

The operators bout, dout, bin, din are photon annihilation operators for the outgoing and in- going ﬁelds on either side of the cavity, and c is again the cavity ﬁeld annihilation operator. The known commutation relations are,

h † i h † i bin(ω1), bin(ω2) = δ(ω1 − ω2), bin(t1), bin(t2) = δ(t1 − t2), (5.67)

and similarly for din. To examine the spectrum of photon ﬂuctuations inside the cavity we 82 calculate the commutation relation for the cavity ﬁeld operator:

Z −iω1t1 iω2t2 h † i dω1 dω2 e e c(t1), c (t2) = 0 0 × 2π κ − i(ω1 − ωc) κ + i(ω2 − ωc) h√ √ √ † √ † i κbin(ω1) + κdin(ω1), κbin(ω2) + κdin(ω2) (5.68)

Z −iω1t1 iω2t2 dω1 dω2 e e = 0 0 2κδ(ω1 − ω2) (5.69) 2π κ − i(ω1 − ωc) κ + i(ω2 − ωc) Z dω 2κ iω(t2−t1) = 2 0 2 e (5.70) 2π κ + (ω − ωc) 0 = eiωc(t2−t1)−κ|t2−t1|. (5.71)

From this we obtain the two-time correlation in Eq. 5.36. Now, for the cavity output, (say,

dout), " h i Z dω dω κb (ω ) + κd (ω ) † 1 2 −iω1t1+iω2t2 in 1 in 1 dout(t1), dout(t2) = e 0 − din(ω1), 2π κ − i(ω1 − ωc) † † # κbin(ω2) + κdin(ω2) † 0 − din(ω2) κ + i(ω2 − ωc) (5.72)

Z dω dω 2κ2δ(ω − ω ) 1 2 −iω1t1+iω2t2 1 2 = e 0 0 − 2π (κ − i(ω1 − ωc))(κ + i(ω2 − ωc)) ! κδ(ω1 − ω2) κδ(ω1 − ω2) 0 − 0 (5.73) κ − i(ω1 − ωc) κ + i(ω2 − ωc) Z dω = eiω(t2−t1) = δ(t − t ). (5.74) 2π 1 2

The commutation relations for ﬁelds outside the cavity are the same as for light entering the cavity, and do not carry any evidence of the photon number dynamics (5.71) inside the cavity. 83

Chapter 6

Collective motion

Chapter 4 discussed much of the material covered in the following publication,

• S. Gupta, K. L. Moore, K. W. Murch, and D. M. Stamper-Kurn, Cavity Nonlinear Optics at Low Photon Numbers from Collective Atomic Motion, Phys. Rev. Lett. 99, 213601 (2007)

This chapter addresses aspects of collective motion observed in the aforementioned publication. Because the atom-cavity system is “quantum backaction limited” (Chapter 5), we might expect position measurements to be nearly quantum limited. Figure 4 of [31] confirmed this expectation. As this approach to the quantum limit of a position measurement is one of the closest ever reported, details of that observation are further developed in this chapter. So far, discussion of collective motion has been limited to the adiabatic displacements of Chapter 4 and the heating of that collective motion treated in Chapter 5. Beyond these two studies, the optomechanical system contains a rich set of dynamical phenomena which are introduced and discussed in this chapter. An important aspect of studying motion in the optomechanical system is simply observing such motion, which as we know should be constrained by the uncertainty principle, and subject to measurement backaction. As a preamble to this chapter about collective motion I’ll first introduce the “kick and watch” experiment. These experiments were first conducted to test Deep Gupta’s hypothesis (which we now know to be correct) that some sort of light induced displacement was responsible for the observed asymmetric cavity spectra. If the collective position could be displaced adiabatically, then it could surely be displaced diabatically and observed thereafter. In hindsight, the fact that those first experiments worked at all is surprising. Had we 84

n FT amp (arb.)

frequency (kHz) time (μs)

Figure 6.1: Kick and watch experiment. Shortly after t = 0 the intensity of the probe laser is increased significantly to displace the collective position. Resulting oscillations can be observed in the Fourier spectral amplitude (left), or in real time (right). For this experiment ∆ca = −2π × 462 GHz, ωz = 2π × 46 kHz, and Neff = 46000. thought carefully about the experiment which was published as the last figure in [31], we would have likely concluded that the experiment would not work; for reasonable settings of the experiment, as discussed in Chapter 5 a maximum signal to noise ratio would be expected after just a handful of detected photons, with the signal degrading quickly thereafter. Instead, as usual, the three of us tried the experiment to see if it would work, only months later finding the results to be quite confusing.

6.1 “Kick and watch”

To perform the kick and watch experiment shown in Figure 6.1, the transmission of the cavity was monitored at a ﬁxed probe detuning. When the transmission surpassed a threshold level, the probe intensity is extinguished for a 10 ms interval. During this 10 ms interval any excitation of collective motion was expected to damp out returning the collective mode to its ground state. After this delay, the probe intensity was turned on very rapidly to a high level to diabatically shift the potential minima. After this kick, the probe intensity remained at a high level, and collective motion modulated the transmission of this light. 85

Such an experiment is shown in Figure 6.1. This representative experiment was performed at a setting where oscillations of the collective motion were clearly visible in real time and also in the Fourier Spectral amplitude of a longer section of data. Recall that the cavity transmission depends on the collective position as,

n¯ n¯ = max . (6.1) ∆pc−∆N +FdpZ/~) 2 1 + κ After the probe light has been turned on to a high level, the initial displacement, and 2 subsequent amplitude of collective motion is Z0 = Fdpn/¯ (Mωz ). After an initial kick the collective position oscillates in the self consistent potential described in Chapter 4. A few things are noteworthy about this oscillation: (1) the collective mode oscillates about the displaced trap minima with amplitude equal to that displacement, (2) the frequency of oscillation is shifted by the “optomechanical frequency shift” [112, 113], (3) coherent amplification or damping known as dynamical backaction either amplify or damp the collective motion [114, 36, 115, 99], (4) random kicks from photon number fluctuations (measurement backaction) affect the amplitude and phase of oscillation. Focusing our attention first on the observation of minimum motion, small displacements of the collective position about its shifted minima modulate the amplitude of light transmitted through the cavity. As in section 5.3, I’ll redefine the collective position to

refer to motions about its displacement, Z → Z − Z0 and use the detuning from the shifted 0 cavity resonance, ∆ = ωp − ωc,

2 2 nmaxκ nmaxκ n¯(t) = 2 = 2 κ2 + ∆ + F Z/ 2 2 (FdpZ/~) +2FdpZ∆/~ dp ~ (κ + ∆ ) 1 + κ2+∆2 2 2 2FdpZ(t)∆/( κ ) FdpZ(t)/(~κ) ' n¯ 1 − ~ − . (6.2) 1 + (∆/κ)2 1 + (∆/κ)2

By monitoring the amplitude of the transmitted light we are most sensitive to the atomic √ motion when the detuning, ∆ = ±κ/ 3. Rather than carry around a burgeoning number of prefactors, in this chapter I’ll generally evaluate expressions which depend on the detuning at ∆ = ±κ. At this detuning, the intracavity photon number varies in time, for small motions, as,

n F Z(t) n¯(t) = max 1 ∓ dp . (6.3) 2 ~κ . 86

The classical dynamics of the collective position–cavity system are captured by the coupled rate equations for the collective position and cavity ﬁeld, α [99, 116],

FdpZ α˙ = i ∆ + − κ α − iαL (6.4) ~ F Z¨ = −ω2Z + dp |α|2 − γZ˙ (6.5) z M 2 2 Where αL characterizes the drive of the cavity, so that nmax = 16αL/κ . These coupled equations exhibit all the classical characteristics which will be discussed in this chapter. For example, if the average position does not move, hZï = 0, then the force due to the 2 2 intracavity photons displaces the collective positon, Fdph|α| |i = Mωz hZi. This collective displacement was discussed in detail in Chapter 4. Second, this system of equations exhibit self induced oscillations. Stable oscillations are obtained when the power input due to the cavity drive balances the dissipation. An analysis of these stable oscillations [99] shows that for some parameters there exists a multiplicity of stable oscillation amplitudes. In principle we can plug (6.4 - 6.5) into a computer and simulate the optomechanical dynamics. Much intuition can be gained, however, by studying certain limits, breaking the dynamics down into a number of different effects.

6.2 The optomechanical frequency shift

At the side of the cavity resonance, the intracavity intensity and consequently the force depend linearly on displacements of the collective position. In the quasistatic regime, i.e. for motions which are slow compared to cavity response, the spring constant

kopt = d(Fdpn¯(Z))/dZ [112, 113, 38],

2 d nmax −2nmaxFdp ∆ + FdpZ/~ kopt = Fdp = (6.6) ∆+FdpZ/ 2 2 2 dZ 1 + ~ ~κ ∆+FdpZ/~ 2 κ 1+ κ −2¯nF 2 ∆/κ ' dp , ~κ 1 + (∆/κ)2 where we have assumed small motions of the collective position. The optomechanical fre- 2 quency shift, ωopt = kopt/(M) is then, −2¯nF 2 2 dp ∆/κ ωopt = (6.7) ~κM 1 + (∆/κ)2 2 = ∓2κ ωzn.¯ 87

56 n

x10 52 -15 -10 -5 0 5 10 z Time (ms) ω 10 50 8

48 n 6 4 46 2 0 2 4 6 8 10 12 14 16 18 -15 -10 -5 0 5 10 n Time (ms)

Figure 6.2: The optomechanical frequency shift: A grossly exaggerated schematic of the experiment (upper right) shows that after a delay of 10 ms, the probe light is turned on diabatically and the resulting oscillation is observed. The observed oscillation varies with detuning from the cavity resonance and the intracavity photon number. Analysis of the actual data trace (lower right) was performed by taking the peak of the Fourier spectral amplitude from 750 µs sections of the digitized photon record. The peak in the Fourier spectrum and average intracavity photon number for each section of data are plotted parametrically (left). The observed oscillation frequency was shifted from ωz ∼ 2π ×45 kHz to ωz ∼ 2π × 58 kHz. Data was taken with Neﬀ = 20, 000 atoms and ∆ca = −2π × 92 GHz, and 60 independent runs of the experiment have been combined.

The second equality is evaluated for ∆ = ±κ, and I have plugged in the granularity pa-

rameter = FdpZho/(~κ). This is valid only in the quasistatic regime, where motions are slow compared to the response of the cavity, a fair approximation for this thesis. When the atomic motion is comparable to, or faster than the response of the cavity, the response is necessarily weaker. The optical spring constant is in general [112, 38],

1 + (∆/κ)21 + (∆/κ)2 − (ω/κ)2 Kopt(ω) = kopt × . (6.8) 1 + (∆/κ)2 − (ω/κ)22 + 4(ω/κ)2

The optomechanical frequency shift was observed in a kick and watch experiment (Fig. 6.2). After an initial kick, the oscillation frequency was determined from the peak of the Fourier spectral amplitude. The data was analyzed in 750 µs segments. The oscillation 88

3 40x10 30

-27 35

x10 20

) 30 J

10 1/T (Hz) 25 ential (

pot 20 0

-9 -20 0 20 10 20 30 40 50x10 Z (nm) Amplitude (nm)

Figure 6.3: The nonlinear optomechanical frequency shift. For certain potentials (as shown on left), the period of motion depends on the amplitude. For the simulation shown here, n¯ = 6, ∆ca = 2π × −100 GHz, Neﬀ = 17000, and ∆pc = 2π × −40 MHz. The modeled fundamental frequency (right) of oscillation. The sensitivity to the amplitude of motion is as high as 5.8×1012 Hz/m. With a frequency resolution of ∼ 1 kHz the amplitude resolution is less than Zho.

frequency is nominally a function of the detuning, the number of atoms, and nmax. Instead of extracting these quantities, the intracavity photon number is used as a proxy, and the observed oscillation frequency is plotted versus the average intracavity photon number. When collective motion is not necessarily small, the frequency is a function of the collective amplitude (see Eq. 6.6). For certain parameters the frequency of oscillation changes very sensitively with the amplitude of motion. Given the potential shown in Figure 6.3, at a speciﬁc amplitude of motion, the period of oscillation drops signiﬁcantly.

6.3 Ampliﬁcation, damping, and saturation

Optomechanical cooling has been recently demostrated by a large number of groups [117, 118, 57, 54, 53, 55, 112, 44] and is the workhorse of many experiments striving to attain the mechanical ground state. Cavity-cooling’s foil, causes ampliﬁcation, and instability of motion [114, 36, 115, 99] and has been recently observed [119, 120, 121, 122]. In this regime of instability, the damping coeﬃcient becomes negative, driving oscillations to ever higher amplitudes of oscillation. Eventually motion is large enough that other sources of mechanical damping balance the driving, and the amplitude of oscillation saturates. 89

Some intuition can be gained by way of a simple model. Because of the ﬁnite residence time of photons in the cavity, the intracavity photon number does not depend on the instantaneous collective postion. The cavity intensity lags behind the collective motion by a time 1/(2κ). The force on the collective mode then depends on location of atoms in ˙ the past, Z|t=−1/(2κ) ' Z − Z/(2κ),

Z˙ F 0 = F n¯Z − . (6.9) dp dp 2κ Rewriting, and expanding, Z˙ ∂n¯ F 0 'F n¯(Z) − . (6.10) dp dp 2κ ∂Z One can see that a velocity dependent term, (1/2κ)∂n/∂Z¯ would lead to damping or amplification depending on the sign of ∂n/∂Z¯ . While this simple model gives the correct dependence on the important prefactors, it fails to get the scaling with the cavity detuning correct. For light which is nearly resonant with the cavity, it is correct that the cavity responds with a time scale given by 1/(2κ). Light which is significantly detuned from the cavity resonates less strongly and the time scale for cavity response is shorter. One might say that for off-resonant light, it takes fewer intracavity round trips to accumulate the phase shift of π necessary for destructive interference with the input light. Returning to equation 5.40, dynamical backaction either amplifies or damps collective motion at a rate [59, 112, 52],

1 d 2 2 (−) (+) γd = † E = κ Snn (ωz) − Snn (ωz) . (6.11) ~ωzha ai dt

In the unresolved sideband, or quasi-static limit, i.e. ωz κ the damping rate reduces to,

2 −∆/κ ωz γd ' 8¯n . (6.12) 1 + (∆/κ)22 The damping rate caries an extra Lorentzian scaling factor compared to the optomechanical frequency shift. Using multiple probe beams, a stable optomechanical trap which is both damped and has a positive spring constant may be created [112].

6.4 Toward quantum limited measurement

Figure 5.1 presented the archetype of a quantum measurement. In that example, a shot noise limited measurement was held to the quantum limit by backaction. The 90 exchange between “measurement” and “backaction” is beautifully simple, but has not yet been observed. We tried for several months to perform such a measurement, and this section provides some homage to all that hard work. We made measurements of amplitude quadrature of light transmitted through the cavity to determine the amplitude and phase of collective motion. For small collective motions, this amounts to measuring the frequency, amplitude, and phase of a small modulation on the detected stream of photons. The shot noise limit had a simple interpretation when we considered the homodyne measurement in Chapter 5. The interpretation of measurements which rely on the single photon counters we use is somewhat more subtle.

6.4.1 Parameter estimation

Given a set of data taken by the competent hands of one of my fellow lab mates, my job is to analyze it properly and efficiently. If I’m looking to measure, for example, the amplitude and phase of a sinusoidal signal, I may choose from any number of analysis routines, properly referred to as estimators. One option is to simply fit a sine wave to the data set in question, another option is to take the Fourier transform of the data, and more options can be found in the numerous papers which have developed estimators. The good news (for me) is that there exists a limit, given the signal to noise of a set of data which dictates how good a deterministic parameter can be estimated. This limit is known as the Cramér-Rao bound [123, 124], and means that if my estimator achieves this bound, I need not look any further. For an undamped sinusoid, with (A, ω, φ) the amplitude, frequency and phase respectively, the Cramér-Raobound states [125],

2N + 1 1 (δφ)2 ≥ , (6.13) N(N − 1) SNR 2N + 1 A2 (δA)2 ≥ , (6.14) N(N − 1) SNR 6 1 (δω)2 ≥ , (6.15) N(N 2 − 1) SNR

where, N is the number of data points and SNR = A2/σ2 is the signal to noise ratio. In our experiments the detected photons are recorded as a data trace of 0’s and 1’s at a sample rate r. For each data point, an average number of 2κηn/r¯ photons are registered, for which the shot noise variance is σ2 = 2κηn/r¯ . On top of this average ﬂux is a sinusoidal variation 91 due to collective motion. For ∆ = κ, collective motion induces a variation of amplitude

FdpZ/(~κ) × 2κηn/r¯ . The signal to noise ratio is then, F Z 2 SNR = dp 2κηn/r.¯ (6.16) ~κ This is a confusing result. It states that the signal to noise ratio decreases as the data is sampled more frequently. This is true, but the Cram´er-Rao bound also decreases with the number data points, since N = rτ, where τ is the measurement time. Putting this all together, the minimum phase and amplitude uncertainties are only related to the number of detected photons, call this N = 2κηnτ¯ .

δA2 2 κ 2 = (δφ)2 ' ~ . (6.17) A N FdpZ As an aside, recall our previous discussion of quantum limited measurement in Chapter 5. There, we found that with an ideal measurement on resonance, the measurement un- 2 2 2 2 certainty, (δZ)meas = ~ κ /(4NFdp). Using detected single photons and the Cram´er-Rao bound,

2 2 2 2~ κ (δZ)meas = 2 . (6.18) NFdp Here, we basically recover the “shot noise limit” applicable to to a measurement at the side of the cavity resonance with single photon counters. This limit is a factor of eight worse than the homodyne measurement on resonance of the cavity. We would, of course expect that the measurement uncertainty would be higher than a measurement on resonance, since the backaction at ∆ = κ is a factor of two smaller than at the cavity resonance. These remaining factors of two are somewhat elusive; a derivation of the Cramér-Raobound which does not [126] include the possibility of damping as was used in Ref. [125] finds the bound to be a factor of four lower than stated in (6.14). The punch line: by measuring at the side of the cavity resonance you can get within a factor of p1/η or p4/η of the standard quantum limit. Methods of analysis for our detected stream of photons were tested against the Cramér-Rao bound by simulating fluxes of data. These simulated data runs were generated using a Mersenne Twister Poisson random number generator to simulate the coherent state injected into the cavity, and a linear congruential random number generator to simulate cavity decay. The cavity detuning was modulated with a controlled amplitude and phase to 92

(a) (b) ) d a r ( z) Δφ CR bound (H δω /2 π

ω/2π (kHz) modulation amplitude (MHz)

Figure 6.4: Study of data analysis techniques. 10 Computer simulated data sets were generated with nmax = 2, and analyzed to study the standard deviation of estimator errors. (a) The phase uncertainty from the LLS√ estimator is compared to the Cram´er-Rao bound for various modulation amplitudes (∝ SNR) of the detuning. (b) The frequency error was determined by ﬁnding the peak in the Fourier spectral amplitude for simulated data sets with random frequency in the range 45 − 50 kHz. “Resolution enhancement” (Bottom curve) provided a better frequency estimate. (c) The LLS estimates of phase (left axis) and amplitude (right) depend on errors in the frequency estimate. (d) The frequency estimate from the FT peak is shown versus modulation amplitude and compared to the CR bound. The frequency estimates are a factor of > 107 from the CR bound. 93

CR bound

A (MH

number of detected photons

Figure 6.5: Simulated amplitude error based from the LLS estimator. A modulation of the cavity resonance of 0.2 MHz was simulated with random phase. The simulated measurement subtends only a few oscillation periods of the collective motion. simulate collective motion. Simulated streams of detected photons had the expected scaling and magnitude of shot noise fluctuations. The Fast Fourier Transform (FFT) included in Igor’s software package as well as a Linear Least Squares (LLS) estimator were studied. For these studies 10 simulated experiments with random phase and frequency were analyzed. Estimators are subject to bias (for example always guessing low) and uncertainty. For all estimates, the LLS and FFT exhibited no bias. The estimator uncertainties, (δφ, δA, δω) were determined from the standard deviation of estimate errors for the 10 simulated experiments. Studies of both estimators are included in Figure 6.4. The FFT defines the phase in some nontrivial way, or does a terrible job as an estimator. If the frequency of oscillation is known, the LLS estimator can be used to efficiently determine the amplitude and phase of oscillation and performs at the CR bound. When 1, the frequency of collective motion is primarily determined by the intensity of the optical dipole trap which was easily monitored and stabilized, but also changed by fluctuations in the probe intensity and detuning owing to the optomechanical frequency shift. Frequency estimators generally fail to meet the CR bound when the signal to noise is near or below unity, as is always the case for our studies. The LLS methods of phase and amplitude estimation are sensitive to errors in the frequency of analysis. So far, we have established that, provided the frequency of oscillation can be determined accurately enough, the amplitude of motion can be measured within a small 94 factor of the quantum limit, since the stream of detected photons can be used to determine the amplitude of motion near the CR bound. The weak link is of course, determining the frequency accurately, since our method of estimating the frequency is a factor of 107 from the CR bound. It is not actually necessary to determine the frequency to the accuracy of the CR bound. From Figure 6.4 (d), we can see that if a long stretch of data is used to estimate the frequency, it can be determined to the 100 Hz level. From Figure 6.4 (c) we can see a 100 Hz frequency inaccuracy would produce a systematic error of the LLS amplitude estimate which is limited to the level of 4%. Our measurement can tolerate this systematic error as long as it is less than the amplitude fluctuations corresponding to the zero point rms position fluctuations of the collective mode, Zho/Z(0). This means that given the unimpressive frequency estimation, experiments should be limited to observing collective oscillations with Z(0) < 25Zho.

6.4.2 The “kick”

We conducted actual studies of quantum limited measurement in a manner similar to that which was used to study our estimation routines. That is, to repeatedly prepare collective displacements (the kick) and measure the evolution. As longer sections of the recorded data are included in the analysis, uncertainty in the quadratures of collective motion would be expected to decrease with the shot noise limit and then increase due to measurement backaction. Uncertainty estimates are then derived from the standard deviation of the amplitude and phase estimates derived from many iterations of the experiment. This measurement technique hinges on the ability to prepare initial displacements which vary between experiments by an amount which is close to or below the quantum limit. The ability to prepare states which are repeatedly displaced by an amount which is known better than the quantum limit hinges on a “many–to–one” correspondence between the number of light quanta used for the kick and the size of the displacement. That said, the pulses of light which we used to induce a collective displacement were very weak pluses in a classical sense, and are therefore have large shot noise fluctuations. If we consider the size of the initial displacement to be determined by the intracavity intensity during the first 2 2π/(4ωz) seconds of the kick, the initial displacement Z(0) = Fdpn/¯ (Mωz ) has fluctuations of r Fdpn¯ ωz δZ(0) = 2 . (6.19) Mωz πκn¯ 95

In order to compare successive experiments and build statistics on the accuracy of our measurement, the ﬂuctuations of this initial displacement must be less than the zero point rms position ﬂuctuations,

2 2 Zho > δZ(0) , (6.20) nκ¯ 2 < 1. (6.21) πωz It turns out that this is a pertinent requirement for experiments. Most experiments we perform satisfy this inequality forn ¯ = 1, but not for displacements which are larger than

Zho (see Fig. 6.6). This limit hinges on an assumption that the intensity fluctuations of the kick are due to shot noise, and is in this sense invalid. Especially at the side of the cavity resonance, technical frequency fluctuations induce classical intensity fluctuations of the probe light which hugely dominate over shot noise. One option is to stabilize the size of the kick through post selection. After many iterations of the experiment, the ones with kicks of adequately similar magnitude are chosen for further analysis. Here, because the efficiency of detecting photons is small, we reach a more stringent limit, that the shot noise on the detected photons have fluctuations corresponding to displacements which deviated

by less than Zho, nκ¯ 2 < 1. (6.22) πωzη Which is a much more stringent limit and indicates that the atom cavity detuning must be very large if we are to be certain about the size of the kick to the quantum limit.

6.4.3 Evolution

After a kick has been imparted on the collective position the resulting evolution is not simply given by the evolution of a simple harmonic oscillator, but contains many rich and convoluting dynamics. In this section, these dynamics are addressed and pertinent timescales are given with their dependence on cavity detuning and evaluated at ∆ = −κ.

• Dynamical backaction. Damping or ampliﬁcation cause the amplitude of the collective displacement to be attenuated or ampliﬁed with an exponential time scale, 2 1 1 1 + (∆/κ)2 τd = ' 2 , (6.23) γd 8¯n ωz −∆/κ 1 τd ∆=−κ = 2 . (6.24) 2¯n ωz 96

ho ick < Z

ick (n) shot noise of k k ho ick < Z ho d k te Z(0) =25 Z ec

ho Z(0) = Z shot noise of det

-Δca /2π (GHz)

Figure 6.6: An ideal kick produces a displacement, Zho < Z(0) < 25Zho. It is also desirable to make repeated displacements which are reproducible to a level better that Zho. This may be achieved by using a shot noise limited pulse, or by post selection. 97

• Atom loss. Cavity heating leads to atom number loss and shifts the resonance of cavity. The time it takes for heating to shift the cavity resonance by κ,

2 2 τheat ' 4Ut∆ca/(¯n~ωz g0). (6.25) While the shift of the cavity resonance due to atom loss can be easily ameliorated by shifting the probe frequency in chorus with the resonance, it is a bit inconvenient, or at least embarrassing to have an optomechanical element which evaporates away during the course of an experiment. We therefore desire to perform experiments which

take much less less time than τheat.

• Measurement time. The optimal measurement time for a nearly quantum limited 2 2 2 measurement occurs when (δZ) = (δZ)heat + (δZ)meas is minimized. When counting photons at the side of the cavity resonance, 2 2 2 2 2 1 ~ κ 1 + (∆/κ) (δZ)meas = 2 2 , (6.26) 4ηv κηnτ¯ Fdp ∆/κ

ηv ∼ 0.4 is another possible ineﬃciency in a realistic measurement and accounts for the broadening of the cavity linewidth due to technical ﬂuctuations. Minimum uncertainty 2 2 2 2 occurs when the total uncertainty is minimized. Using (δZ)heat = Fdpnτ/¯ (2M κωz )× 1/(1 + (∆/κ)2), the total uncertainty is minimized after,

23/2 Mωz~κ 1 + (∆/κ) τmeas = −√ √ . (6.27) 2 ∆/κ 2Fdpnη¯ v η 1 2 τmeas ∆=−κ = √ 2 . (6.28) η ηv nκ¯ This is the measurement time which provides the best signal to noise. The minimum uncertainty attained at this ideal measurement time is, √ 1/2 δZ 2 2 1 + (∆/κ)2 = √ . (6.29) Zho ηv η |∆/κ| δZ 2 2 = √ . (6.30) Zho ∆=−κ ηv η • Mechanical damping. The radial width of the atomic cloud contributes to a broadening of the mechanical resonance frequency. At a the radial half-width, the trap 0 2 frequency is reduced to ω ' ωz(1 − kT/(2mωxwo). Because of this, the inverse mechanical damping rate is then, 2 2mωxwo τd ' . (6.31) ωzkT 98

• Lattice dephasing. There are two types of frequency shifts which aﬀect the collective oscillation. The optomechanical frequency shift which is a general property of optomechanical systems has already been discussed. Another frequency shift occurs as a second order expansion in the deﬁnition of the collective mode, and addresses the weakness of approximation #2 made in Section 2.3. Returning to Section 2.3, and keeping terms to second order in the displacement operator; the potential energy of the ith atom is,

2 2 mδzi ~g0n¯ h 2 2 2 2 i U ' + sin (kpz¯i) − kpδzi sin(2kpz¯i) + kpδzi cos (kpz¯i) , (6.32) 2 ∆ca so, 2F nk¯ ω2 = ω2 + dp p cos(2k z¯ ). (6.33) i z M p i The trap frequency at individual sites is shifted from location to location causing dephasing, and ultimately re-phasing of the collective motion. If two oscillators evolve p 2 2 2 at frequencies ωz and ωz + (δω) , with (δω) = Fdpnk¯ p/M, they dephase after a time,

2ωzM τδφ ' . (6.34) n¯Fdpkp

• Frequency chirps. Fluctuations in the intracavity intensity induce changes in the magnitude of the optomechanical frequency shift. Variations in the frequency of oscillation makes analysis of the transmitted signal substantially more complicated. One tactic is to require that these ﬂuctuations produce a small contamination and can thereby be ignored. Fortuitously, estimates of the amplitude are somewhat insensitive to frequency errors. To get some sense of the magnitude of this eﬀect, consider the phase error accumulation due to a intracavity intensity variationn ¯(t) =n ¯ + δn¯(t) s Z τ F 2 n¯ − δn¯(t) 2 dp δφ(τ) = −ωτ¯ + dt ωz ∓ (6.35) 0 M~κ Z τ F δn¯(t) ' − dt dp (6.36) 0 2M~κω¯

2 2 2 where,ω ¯ = ωz ∓ Fdpn/¯ (M~κ) is the average mechanical oscillation frequency. I have assumed that the variations in the optomechanical frequency are small compared to the total trap frequency. This is no small assumption, the optomechanical frequency

r a

l damping

nu n

a a

r r

g g

s s tion o l ac m o t a

timescale (s) dynamical back trap period e dephasing ement time tic ur latticat s

optimal measurmeasu exp. range

−Δ ca Δ /2ca GHπ (GzHz)

Figure 6.7: Relevant timescales for measurement. Assuming a measurement with ∆ = −κ, n¯ = 50, ωz = 2π × 46 kHz, and Neﬀ = 46000 the relevant timescales are displayed as a function of detuning. Ideally the timescale for an optimal measurement is longer than the trap period (which includes the optomechanical frequency shift), but shorter than the timescales for dephasing, and atom loss, 300 < ∆ca/(2π) < 2000 GHz. 100

shift can easily be of the same order as ωz, so phase diﬀusion due to variations in the intracavity intensity can be signiﬁcant.

Examining all of these different dynamical effects we can explore parameter regions which are relevant to study of each phenomena. Figure 6.7 displays these timescales as a function of the atom-cavity detuning for particular values of the experiment. Notably, the optimal measurement time is always close to the timescale for amplification or damping of 1 √ motion because 4ωz ∼ ηκ. Above detunings in the 1000 GHz range lattice dephasing dominates over the timescale for optimal measurement. Referring to Figure 6.6, detunings above 100 GHz are classical enough to provide repeatable displacements which are greater than the harmonic oscillator length. Working in a parameter regime where an optimal quantum measurement takes a time which is longer than the period of motion, we can attempt to make a “real time” quantum measurement. Working with ∆ca = −2π × 462 GHz and Neff = 46000 we performed kick and watch experiments. In these experiments, after triggering of a threshold transmission (to establish a well known number of atoms) the probe light was extinguished for 10 ms, and then turned at a very large value. As shown in Figure 6.8 the details of the kick in each experiment varied wildly from shot to shot, and even further, the resulting steady state amplitude of motion, as determined from the Fourier spectral amplitude did not correlate at all with this initial displacement. It was important to establish a very large “nearly classical” amplitude of oscillation to estimate the frequency of oscillation reliably. Of 49 iterations of the experiment, only 27 iterations resulted in a steady state amplitude of oscillation which was large enough to result in a peak in the Fourier spectral amplitude which was resolved with sufficient signal to noise. That is, on many occasions the initial displacement failed to result in an oscillation which persisted over the first 750 µs of data. Without a reliable way to determine the frequency of oscillation, these 22 data runs were eliminated from further analysis. Using the frequency estimate from the peak in the Fourier spectral amplitude, LLS phase and amplitude estimates were taken from successively longer segments of cavity transmission. For each experiment a time resolved estimate of the amplitude and phase of oscillation was generated. These traces were then averaged across the 27 “good” experimental runs to generate statistics about the measurement.

1 You will notice that I have ignored the contribution of the Voigt proﬁle, ηv here. My suspicion is that this broadening of the line also decreases the eﬀect of dynamical backaction. 101

24 ) ho 20 15-20 μs

10-15 μs 16 n

5-10 μs amplitude (Z/Z 12 100 200 300 experiment number initial displacement (Zho)

Figure 6.8: The kick and watch experiment. The initial displacement varied significantly between experiment iterations (left). The intracavity photon number is shown for the first 20 µs of the experiment. On most iterations the displacement was initiated at t = 15 µs but this turn-on varied from run to run. The successive phase of oscillation then varied from experiment to experiment. The steady state amplitude of oscillation (as determined from the peak of the Fourier spectral amplitude over the first 750 µs of data) did not correlate with the magnitude of the initial displacement.

Figure 6.9 displays this measurement. Due to errors in the frequency estimate (resulting from the very poor performance of the FT based estimation) the phase estimates varied wildly, with variance δφ ∼ π. Amplitude estimates are less sensitive to errors in the frequency estimate and resulted in smaller shot to shot variances. The variance of the ensemble of measurements exhibited a pronounced dip after approximately 10 µs of

measurement, corresponding to a variance of δZ ' 4Zho. This variance is impressively small, in fact, it is just as small as we could ever expect it to be. The distribution of initial displacements, based on analysis of the ﬁrst 5 µs of cavity transmission (Fig. 6.8), varied between 50 and 300 × Zho, much more than the quantum limited variance of 4Zho. The reason for this apparent contradiction is at present not clear.

6.5 Backaction induced phase diﬀusion

Intracavity photon number fluctuations disturb the collective momentum, inducing momentum diffusion, which ultimately causes the position certainty to diffuse in a harmonic trap. The frequency of collective motion is also affected by the intracavity intensity and its fluctuations. Considering many trajectories of the phase of collective motion (6.36), the 102 ho ho Z)/(Z ) δ ( (Z)/(Z )

time (μs) time (s)

Figure 6.9: Attempt at a time resolved quantum measurement. With ∆ca = −2π × 462.6 GHz, Neff = 46000, ωz = 2π × 46 kHz andn ¯ ∼ 40, a total of 49 experiment traces were compiled. Of these 49 experiments, 27 which exhibited a clear peak in the Fourier spectral amplitude were analyzed separately to determine the amplitude and phase of an oscillation at a frequency determined by the peak location of the Fourier spectral amplitude. The estimates for the amplitude and phase were averaged to determine the distribution of measurements obtained, and therefore the uncertainty of an individual measurement. The measurement uncertainty exhibits a pronounced dip after approximately 10 µs of measurement. The gray line depicts the expected measurement uncertainty. Its thickness encompasses uncertainty in the atom cavity detuning and intracavity photon number. 103 phase of collective oscillation diffuses as, d Z t hδφi2 = κ24 dt0 hn(t)n(t0)i. (6.37) dt 0 This harkens back to derivation of backaction heating from Chapter 5. We can jump right to evaluating the autocorrelation to find that, d hδφi2 = κ24S(−)(¯ω). (6.38) dt nn Which is clearly an effect which is second order to the effect of measurement backaction.

6.6 The right left and center of cavity resonance

This chapter has focused on effects which are present and relevant to measurements which are performed at the side of the cavity resonance. The interest in the side of the cavity resonance are largely due to the happenstance that we installed single photon counters, rather than a homodyne detection system in the experiment. As illustrated in Figure 6.10, effects such as amplification, and optical springs do not occur on cavity resonance. Homodyne measurements have the advantage of performing measurements of the cavity phase on cavity resonance and are insensitive to these effects. As discussed earlier, these measurements can in principle reach the quantum limit, though technical details which might hinder the performance of measurements on-resonace have not yet been explored. There may, however be significant benefits to measurements which are conducted at the side of the cavity resonance. Dynamical backaction may be used to amplify small quantum signals near (in the unresolved sideband regime) the quantum limit, which would help overcome technical hurtles such as the low quantum efficiency and measurement processing. Likewise, as was shown in Chapter 2, in the presence of the optomechanical frequency shift, the collective mode becomes the only non-degenerate mode of the system, and may decouple from he bath of other modes, increasing the quality factor of the collective mode. When measured on cavity resonance, the optomechanical frequency shift does not exist to pick the collective mode out of the N − 1 other degenerate modes.

6.7 The quantum–classical boundary: granularity revisited

It is commonly evoked that ~ sets the scale where quantum mechanics takes over the reigns of the classical world, and every description of the quantum–classical boundary 104

c e v o i o it l s i o n g p t ,

p k o o k p , t

n n io e at g !c a pli tiv am e

Figure 6.10: Spectrum of the cavity resonance. The side of the resonance which is elongated in frequency sweeps is also the side of resonance which exhibits a positive optical spring constant, and amplification. The edge of the resonance which ultimately becomes bistable exhibits cooling and a negative optical spring constant. contains a parameter, which is proportional to ~ that tunes this boundary. The granularity √ √ parameter, ∝ ~, is exactly this parameter. When → 0, (equivalent to ~ → 0) the optomechanical system is perfectly classical. The regime of > 1 is characterized by two statements. The impulse of a single photon displaces the collective mode by an amount greater than its zero-point rms momentum fluctuations. Similarly, collective displacement by an amount equal to its harmonic oscillator length shifts the resonance of the cavity by a half linewidth. When ' 1, only a single photon is needed to measure the collective position to the quantum limit. Nonlinearities in this regime are expected to be qualitatively different than in the non-granular regime. Imagine that at t = 0, with a detuning ∆ > κ, a single photon enters the cavity, and its impulse to the collective motion causes the cavity to shift out of resonance as the collective motion oscillates (i.e. the heating side of the line). After a time π/2ωz the cavity is further off-resonance and the likelihood that another photon will enter the cavity is reduced. At t = 3π/(2ωz) the cavity is nearest to resonance and it becomes more likely that the photon will enter the cavity (incidentally applying another kick to the collective motion). A central prediction from this picture is that the correlation of photons exiting the cavity be modulated by the trap frequency.

With ∆ca = −2π × 11.7 GHz, and Neff = 6900 we performed experiments (shown in Figure 6.11) with the granularity parameter = 0.67. The second order correlation function measures the conditional likelihood of detecting a second photon a time τ after a first photon. To obtain this correlation, I plotted a histogram of the time differences 105

1.2

1.0

0.8

n 0.6

0.4

0.2

0.0 -3 -20x10 -10 0 10 20 Time (ms)

3.0 ) 2.0 ho

2.5 1.5 2.0 ) τ 2 1.5 1.0

G ( al amplitude (Z/Z amplitude al

1.0 tr 0.5 0.5

0.0

0.0 spec ourier F 0 50 100 150 200 60 80 100 120 time (μs) frequency (kHz)

Figure 6.11: Nonlinearity near the granular regime. With Neff = 6900 and ∆ca = −2π×11.7 GHz, ωz ' 2π × 40 kHz, “wait and see” lineshapes were collected (top). The granularity parameter, = 0.67. The second order correlation function G2(τ) (lower left), calculated for data between 16.5 and 17 ms, (shown as a grey bar) exhibits clear modulation at a frequency of 82 kHz. The Fourier spectral amplitude, expressed in units of the collective motion (i.e. the collective motion necessary to give the measured amplitude) does not show a peak at 82 kHz, indicating that the collective mode is not simply oscillating. 106 between every detected photon, and normalized it to the mean photon flux. For a data trace of finite length T , time differences which are long occur less frequently. The correlation was normalized by a factor 1 − τ/T to account for this binning effect [127]. For regions of data which were near the cavity resonance a significant modulation of the correlation function was observed at a frequency of ∼ 2π × 82 kHz. Given the observed nmax, the data was taken at a setting where ∆/κ ∼ 0.2. The expected frequency p 2 2 of oscillationω ¯ ' ωz + 2κ ωz(∆/κ) ' 80 kHz agrees well with this value. A similar correlation function would also be obtained if the the collective mode was simply oscillating at frequencyω ¯, modulating the transmission at this frequency. Such a signal would also be resolved in the Fourier spectral amplitude of the detected photon over that data region. No clear peak is visible in the Fourier spectra indicating that the system was not simply oscillating. This tantalizing first glimpse of optomechanics in the quantum regime is a significant achievement. The ability to tune the force of a single photon with the atom-cavity detuning is a knob provided by atomic physics, yet is inaccessible to to traditional optomechanical systems. This ability will allow detailed studies of the quantum classical boundary, of squeezing, entanglement and precision measurement.

6.8 What? Who? Which way photons leave the cavity

I credit Tom Purdy with raising the paradox which I’ll state below. The question we are interested in answering is: given a detected photon ﬂux on the right side of a cavity, what is the intracavity intensity? The detected photon ﬂux (CPS) is related to the intracavity intensity as,

(CPS) n¯ = . (6.39) 2ηcavηκ

I have separated the eﬃciency of photo detection into two components, ηcav has to do with how photons get out of the cavity, and η is the eﬃciency of everything else. Assume for now that we have a loss-less, symmetric cavity.

• Situation #1: imagine that we manage to stick a photon into the cavity. This might be achieved by placing an atom inside the cavity and pumping it to an excited state from the side of the cavity. Now we ask the question, what is the probability that the 107

photon is detected on the right side of the cavity? Since the cavity is symmetric, the

photon has equal probability of leaving out of either mirror and ηcav = 1/2.

• Situation #2: now imagine that the cavity is probed exactly on resonance from the left side of the cavity. On resonance, the reﬂected power drops to zero and all the

photons are transmitted through the cavity. In this case, ηcav = 1. The reflected power drops to zero because light which reflects off the left mirror interferes deconstructively with light exiting the left cavity mirror.

In two different situations we get different efficiencies of detecting an intracavity photon2. The second situation is akin to a typical experiment, where the cavity resonance is probed in transmission, and the conclusion ηcav = 1 hinges on deconstructive interference between the intracavity light and the light reflected promptly off the left mirror. However, in the experiments we perform, atoms which are in the cavity interact to some varying degree with the intracavity intensity. To what degree does this interaction spoil the interference responsible for perfect transmission? Repeated quantum non-demolition measurements of photon number states have been performed, indicating that the intracavity intensity may be projected onto a Fock state through measurement [128]. Certainly if the intracavity field is projected onto a Fock state with undetermined phase, the light exiting the left side of the cavity interferes with the promptly reflected light with random phase, so on average half of the light is reflected (or we say, “it does not interfere”). All the light enters the cavity and then leaves through either mirror with equal probablitiy; ηcav = 1/2.

The question of whether ηcav = 1 or 1/2 depends on to what degree the collective motion of the atoms “measures” the intracavity field. If the single photon dipole force, 2 Fdp = ~kpNeffg0/∆ca is sufficient to displace the collective position by an amount larger p than the harmonic oscillator length Zho = ~/(2Mωz), then the atoms “measure” the

2It should be noted that after a survey of prominent “quantum opticians” including G. Milburn, S. Girvin and H. Mabuchi, only H. Mabuchi was sure enough to say, “I can state with 96% confidence [that it is always situation #1]”. So, while the paradox may be poorly posed, and photons always are detected at a rate proportional to the corresponding mirror’s transmission, the correct answer is at least not obvious to 2 out of 3 prominent specialists in quantum optics. 108 intracavity field sufficiently strongly to project it onto a Fock state.

Fdp 2 ≥ Zho, (6.40) Mωz 2κ ≥ 1. (6.41) ωz This inequality is satisﬁed for every experiment reported in this thesis, but only marginally.

For example, the measurement of quantum back action heating performed at ∆ca = −2π ×

100, 2κ/ωz = 4.4. It should therefore be possible to explore this hypothesis experimentally.

The real cavity used in our experiments ηcav is further reduced by losses in the mirrors. We therefore expect,

Tt Tt ηcav = = 0.136, ηcav = = 0.160. (6.42) situation 1 Tt + Tb + 2L situation 2 Tt + 2L This is a 16% eﬀect on the probability of detecting an intracavity photon and should be readily observable. By comparing the cavity transmission with varying numbers of atoms, the transmitted intensity should vary by 16%. 109

Chapter 7

Betatron motion in the ultracold atom storage ring

This chapter discusses the observation and study of betatron resonances in the ultracold atom storage ring. Portions of the work in this chapter were presented in the publication:

• K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Dispersion management using betatron resonances in an ultracold-atom storage ring, Phys. Rev. Lett. 96, 013202 (2006)

In route to building the Cavity QED apparatus, and conducting the experiments presented in chapters 2 - 6, we spent some time testing the mm–scale magnetic trap which was to be the workhorse of those experiments. In my version of the story, one day Dan told us that he thought we should be able to make a µm-scale ring by simply lowering the currents in our gradient coils slightly. Our job that afternoon was to go into lab and make this ring, condense atoms, and look in time-of-flight for persistent currents. It was a tall order for a Tuesday afternoon, but it seemed to be worth a try. Kevin and I fiddled with the magnetic fields for a few hours and after failing to see persistent currents on the first iterations of the experiment, lowered the currents in the gradient coils gradually to zero. The day’s experiments were further complicated by the fact that we had decided that our little ring experiments were so brilliant that they should be kept TOP SECRET. Especially that they should even be kept secret from the post–doctoral candidate who was 110 visiting the lab and participating in the experiment that day. He kept asking: “So, tell me again; what are you guys are trying to do?” — “Oh, we’re just fiddling with different settings.” He must of been sure that we were the two most brainless and inarticulate graduate students at Berkeley. “Hey Kevin”, I’d say, “how about we try increasing that setting we were working with and look in less time of flight for the you know what”. Eventually I broke down and told our visitor that we were looking for a ring that probably wasn’t there. Kevin gave me a searing look — it’s going to be all your fault when we get scooped. And then we found the ring — a slightly different ring than Dan had sent us after that afternoon, but a ring all the same. We spent a good half year conducting studies on this ring. We learned how to make Bose-condensates in a corner of the ring and eventually learning launch these matter waves into unterminated motion around the ring [129]. During a study of superradiance and propagation of atoms in the ring [130] I noticed something strange about the propagation dynamics of the recoiling pulse of cold atoms. Looking closer, I noted that the expansion of the pulse corresponded to a kinetic temperature below 100 pK, surely the lowest temperature ever observed. Many hypotheses, trying to understand what aspect of the superradiant light-atom interaction caused the atoms to become so cold, emerged. After a month of furious data and discussion, we finally concluded that superradiance played no role at all, the culprit, a well known accelerator physics headache, a “betatron” resonance had compressed the longitudinal velocity class hugely. This chapter begins, in that case, back in 1940, when D. Kerst constructed the first “betatron”, an electron induction accelerator and used it to accelerate electrons to an energy of 2.3 MeV. Years later, in considering the stability of circular orbits for particles, Kerst and Serber published a paper [131] analyzing particle oscillations in such a device. These particle oscillations occur in all particle accelerators and have come to be known as “betatron oscillations”[132]. A betatron resonance occurs when these oscillations are resonantly excited by static perturbations to the storage ring. The focus of particle accelerators and storage rings has been to push the energy of stored and accelerated particles to ever higher levels. On the sidelines, magnetic storage rings have been developed to store neutral atoms [133, 134, 129, 135] and molecules [136, 136] for purposes ranging from interferometry [137] to studies of low-energy collisions [138, 139]. The purposes of these low energy storage rings differs from the high energy demands of 111 particle physics and synchrotron radiation sources which suggests that familiar physics should be examined in new light. This chapter explores betatron resonances in the context of the ultracold atom storage ring [129]. In this ring 87Rb atoms were evaporatively cooled to form a Bose Einstein Condensates and subsequently accelerated to circulate in the ring at energies of roughly kB × 100µK per atom (or, in more incendiary language 100 peV per nucleon). At these circulation energies, a velocity spread across the matter wave of ∆v = 1.7 mm/s caused the matter wave to expand, filling a large portion of the storage ring after 20 revolutions. We found that, by launching the matter wave at specific angular velocities a portion of its kinetic energy could be coupled to oscillatory motion in the transverse degrees of freedom. Specifically slowing the front portion of the matter wave in this way, the velocity spread of the atomic beam was dramatically reduced. Betatron resonances are deleterious to the performance of high energy accelerators. When particles are accelerated to relativistic energies even the slightest coupling of the longitudinal kinetic energy into the transverse directions causes oscillations so severe that particles collide with apertures in the beam path. With ultracold gasses the initial dispersion and total kinetic energy very modest, the exchange of energy at the level of a few percent does not cause the matter wave to be lost. These cold gasses were used as a source of “bright” atomic wavepackets. Brightness refers to the velocity spread across a launched matter wave, in the case of these Rubidium condensates, the remaining mean field energy after decompression yielded a small velocity spread, and excellent source of atoms for testing properties of the ring. Here we refer to the launched matter wave as an atomic wave packet, a matter wave, an atomic beam, and as a pulse of atoms interchangeably.

7.1 Forming a storage ring for cold atoms

The details of the construction and operation of the ulracold atom storage ring are discussed in detail elsewhere [129, 67]. For readability, some background is provided here. The ﬁeld from the curvature and anti-bias coils, as shown in Figure 7.2, B is written in cylindrical coordinates [67], with ρ the radial coordinate,

B00 h ρ2 i B = B zˆ + z z2 − zˆ − zρρˆ . (7.1) 0 2 2

00 2 2 Bz = d Bz/dz is the axial ﬁeld curvature. We can see that the the ﬁeld B vanishes in the 112

z z

z y y

1 mm z y

Figure 7.1: Forming a circular magnetic storage ring for ultracold atoms. Four coaxial coils were used to generate the static and rotating fields used to make the storage ring. The field in the x–y plane formed by the curvature coils (top and bottom coils) is maximum at ρ = px2 + y2 = 0, and points in thez ˆ direction. Applying a bias field using the anti-bias coils (pair nearest z = 0) in the opposing direction cancels the field from the curvature coils at a radius ρ0 producing a ring of field zeros (×’s)in the x-y plane. Rapidly rotating the field zeros around the trapped atoms produced the time orbiting ring trap. 113

p 00 z = 0 plane at a radius ρ0 = 4B0/Bz . This field zero forms a attractive potential for 0 p 00 weak field seeking atoms. Expanding about this locus of minima, the gradient B = B0Bz . This Q-ring of magnetic field zeros is not a suitable container for ultracold atoms because “Majorana losses” cause atoms to be lost from the regions of small field. By applying a rotating field Br, Br = Br cos(ωrt)ˆz + sin(ωrt)ˆρ , (7.2)

Which rapidly rotates the magnetic ﬁeld zero around the location of the atoms Majorana losses can be eliminated. When the rotation rate ωr is fast compared to motional frequencies, the atoms “see” a time averaged magnetic ﬁeld,

02 B 2 2 |B| = Br + (ρ + z ). (7.3) 4Br The potential of the ring can then be written as, 1 U = U + mω2ρ2 + ω2z2. (7.4) 0 2 ρ z

p 02 Where, so far, ωz = ωρ = µB /(2mBr) is the same for both axial and radial directions. In practice, the rotating field is not identical in thez ˆ andρ ˆ directions, resulting in different trapping frequencies for both directions of confinement. Equation 7.4 describes a perfectly flat ring in the absence of other bias fields or inhomogenous forces. If things were this simple, this chapter would likely be about persistent currents and interferometry. Rather, (7.4) is only the zeroth order approximation to the potential landscape of the ring, and we should consider higher order corrections to the perfect ring geometry.

7.2 Betatron resonances

To develop a deeper understanding of these resonances, a classical treatment of betatron resonances which follows the treatment of [140] is presented in this section. The atomic beam in our experiments was derived from a BEC and was thus characterized by a macroscopic quantum wavefunction. Away from a resonance, its transverse velocity width was consistent with that of a minimum uncertainty quantum state [130]. Nevertheless, since the energy transferred to betatron motion is several hundred times larger than ~ωr,z, the relevant harmonic oscillator energy quanta, a purely classical treatment suﬃces to describe 114

top imaging plane

design z 1 mm side orbit imaging average plane beam ρο path θ

R side r imaging beam

atom pulse top imaging beam

Figure 7.2: An ultracold-atom storage ring. Axes are indicated with gravity along −zˆ. Top or side view absorption images of the atoms allow their motion to be studied. Dashed circles indicate the 270 µm radial extent of top-view images displayed in polar coordinates as “annular views” in ﬁgures 7.5 and 7.6. We refer to motion in r as radial, in z as axial, and in θ as azimuthal or longitudinal. The nominally circular storage ring contains radial δr (exaggerated for illustration) and axial δz beam-path errors and an azimuthal variation U(θ) in the potential. 115 the atomic motion near a resonance. We start with the Lagrangian for a particle circulating in the storage ring, m L = ρ˙2 + ρ2θ˙2 +z ˙2 + µB(ρ, θ, z) − mgz. (7.5) 2 Equations of motion are obtained as usual, d ∂L ∂L = , (7.6) dt ∂x˙ ∂x ∂B mρ¨ = mρθ˙2 + µ , (7.7) ∂ρ d ∂B mρ2θ˙ = µ , (7.8) dt ∂θ ∂B mz¨ = µ − mg. (7.9) ∂z

2 Assuming that the ring is perfect, ∂B/∂θ = 0 and angular momentum Lz = mρ θ˙ is conserved. It is useful to introduce the mean orbit radius R and work with deviations away

from that orbit. Note that R > ρ0, the design orbit for any nonzero angular momentum, since the “centrifugal force” displaces the orbit. We will work with deviations away from the mean orbit and set r = ρ − R.

L2 r −3 ∂B mr¨ = z 1 + + µ , (7.10) mR3 R ∂r ∂B mz¨ = µ . (7.11) ∂z We have ignored the −mg term since this constant force just shifts the location of the mean beam path downward slightly. It is natural and commonplace in storage ring analysis to use θ as a time variable, rather than carry around three position coordinates and time. To do this we replace all derivatives with derivatives with respect to θ. dr dr dr L = θ˙ = z , (7.12) dt dθ dθ mρ2 d2r L 2 d2r 2 L 2dr 2 = z − z . (7.13) dt2 mρ2 dθ2 ρ mρ2 dθ As discussed in previous sections, the TORT ring potential used for these experiments has the form, ∂B −µ = mω2(ρ − ρ ), (7.14) ∂r ρ 0 ∂B −µ = mω2(z − z ). (7.15) ∂z z 0 116

Further simpliﬁcations for our equations of motion can be obtained by noting that at radius ρ = R the force is zero, from (7.10),

L2 ∂B z 2 3 = −µ = mωρ(R − ρ0). (7.16) mR ∂r ρ0 This equality indicates that the centrifugal force is balanced by the magnetic potential at the mean orbit radius R. Pulling this all together,

L 2 d2r 2m L 2dr 2 L2 r −3 ∂B m z − z = z 1 + + µ . (7.17) mρ2 dθ2 ρ mρ2 dθ mR3 R ∂r

2 dr Now for the approximations. First we neglect the dθ term. Also, assuming that r R we’ll take the taylor expansion of the appropriate term. So far, the equation of motion for the radial coordinate has been reduced to:

L 2 d2r L2 r ∂B m z = z 1 − 3 + µ . (7.18) mρ2 dθ2 mR3 R ∂r

Utilizing (7.16), and writing the angluar frequency of the orbit Ω = θ,˙

d2r Ω2 = −3Ω2r − ω2r, (7.19) dθ2 ρ d2r = −ν2r. (7.20) dθ2 r

Equation 7.19 indicates how the curvature of the centrifugal force adds to the effective radial q 2 2 trap frequency. We define the effective trap frequency ωr(Ω) ≡ 3Ω + ωρ, and the “radial q 2 2 betatron tune”, νr ≡ ωr/Ω = 3 + ωρ/Ω . This takes care of the radial equation of motion For the z direction, we replace the time derivatives with the angular coordinate. This time we neglect the (dz/dθ)(dρ/dθ) term.

d2z L 2 d2z 2 L 2 dz dρ = z − z , (7.21) dt2 mρ2 dθ2 ρ mρ2 dθ dθ d2z ' Ω2 . (7.22) dθ2

Introducing the “axial betatron tune”, νz = ωz/Ω,

d2z = −ν2z. (7.23) dθ2 z 117

Equations 7.20 and 7.23 describe uncoupled oscillatory motion for the axial and transverse directions,

r(θ) = r0 cos(νrθ), (7.24)

z(θ) = r0 cos(νzθ). (7.25)

So far, nothing very interesting is going on, along the nominal beam path a particle can oscillate as it circulates at a constant angular velocity. In the radial direction, the frequency of motion is slightly different from what you would see from a linear waveguide; the curvature of the centrifugal force effectively increases the trap frequency. m 2 In general, we write a perturbation to the effective potential, U(r, z) = 2 νzz + 2 νrr of a perfect ring as,

X k l Upert = Aklqr z cos(qθ + φklq). (7.26) klq

We consider one type of “lowest order” error pertubation. Potential errors, where k + l = 0, denoted as 0th order resonances have already been neglected in our analysis when we conserved angular momentum for simplicity. “Beam path errors”, or 1st order resonances where k + l = 1 can be thought of as a local gradient perturbation to the ring which simply shifts the beam path. When either l or k = 0 energy is exchanged between a transverse and the longitudinal motion. These “synchro-betatron” resonances are the type discussed in this chapter. In general, the condition for resonance is [141],

kνr + lνz = q, (7.27) with q being an integer. For the case of a k = 1 synchro-betatron resonance, we can solve the equation of motion exactly,

d2r A + ν2r = cos qθ. (7.28) dθ2 r m A/m cos(qθ) r(θ) = − 2 2 + C1 cos(νrθ) + C2 sin(νθ). (7.29) q − νr A/m h i r(θ) = 2 2 cos(νrθ) − cos(qθ) , (7.30) q − νr with the initial condition that the particle’s radial coordinate is at rest and not displaced.

Note here that A = A10q is a “force” but has funny units of [mass] × [length] since we have parameterized time by θ. A is related to a real perturbative force as F = AΩ2. This equation 118 describes the resonant coupling of energy from longitudinal motion to radial motion. Even here, where we have made every simplifying assumption conceivable, the dynamics remain quite rich. Rewritten more procatively,

−2A/m νr − q νr + q r(θ) = 2 2 sin θ sin θ . (7.31) q − νr 2 2

Very near resonance, and for short propagation distances, (νr − q)θ/2 1, the radial oscillation at frequency (νr + q)/2 amplitude grows as, A θ. (7.32) m(q + νr)

7.3 Modeling betatron resonances

To model axial and radial betatron resonances we consider first order perturbations to the magnetic field of the ring. These “beam path errors” shift the magnetic minimum of the ring, allowing the potential of the ring to be written, m m U = ω2(ρ − ρ + A sin qθ)2 + ω2(z + B sin qθ)2 + mg(z + B sin qθ). (7.33) 2 ρ 0 2 z To get the full equations of motion as before we differentiate the Lagrangian L = T − U appropriately. m L = ρ˙2 + ρ2θ˙2 +z ˙2 2 m m − ω2(ρ − ρ + A sin qθ)2 − ω2(z + B sin qθ)2 − mg(z + B sin qθ), (7.34) 2 ρ 0 2 z

˙2 2 ρ¨ = ρθ − ωρ(ρ − ρ0 + A sin qθ), (7.35) ˙ 2 ¨ 2 2ρρ˙θ + ρ θ = −ωρ(ρ − ρ0 + A sin qθ)Aq cos qθ 2 − ωz (z + B sin qθ)Bq cos qθ − gBq cos qθ, (7.36) 2 z¨ = −mωz (z + B sin qθ) − g. (7.37)

These equations of motion can be numerically integrated, as shown in Figure 7.3, to study betatron motion.

7.4 Dispersion Management

In optical communications short pulses of light are used to communicate bits of data. In materials, these pulses of travel a group velocity which is less than the speed of 119 z) (H

π initial

Ω /

final final

Ωinitial /2π (Hz) Ωinitial /2π (Hz)

Figure 7.3: Computer simulation of a νz = 5 betatron resonance. Near a betatron resonance, a fraction of the total kinetic energy is transfered to radial oscillation, resulting in a slowing of particles which circulate as speciﬁc angular velocities. The input-output relationship (right) for the betatron resonance shows that particles which are launched near a “stopband” are slowed to a diﬀerent angular velocity. For this, (ωρ, ωz) = 2π × (69, 73) Hz, l = 5 and B = 2 µm.

light. Dispersion, or more properly group velocity dispersion, results from a group velocity which is also a function of the wave’s frequency. This causes different spectral components of a pulse to spread out and eventually overlap with neighboring pulses and interfere with the communication signal. For this reason, the management of dispersion, especially in optical fibers is very important for optical communications as it limits the length of a fiber through which a communication signal can be sent. Current fiber optic dispersion management is attained by using periodic modulation of the index of refraction of the fiber to alternate between positive and negative dispersion. Development of atom waveguides, outcouplers, beamsplitters, and a source of coherent matter waves has resulted in the atom based analogy to many optical phenomena. Atom optics are of significant interest for interferometry because the per-atom phase shifts may be exceedingly larger that for single photons. For instance an atom Sagnac gyroscope has a fundamental signal to noise limit which is 11 orders of magnitude greater than that for an optical gyroscope [137]. Most single mode atom sources require strong intra-particle interactions so dispersion occurs in atom pulses. In analogy to optic fiber-based dispersion management, periodic modulation has been used to control the dispersion of matter waves [142]. 120

ΔΩ final z) (H

π /2

final ΔΩ final Ω

ΔΩ initial

Ω initial /2π (Hz)

Figure 7.4: Basis of dispersion management with betatron resonances. For a range of initial angular velocities, ∆Ωinitial, the final spread of angular velocities ∆Ωfinal can be significantly reduced by coupling energy into the radial or axial directions. Because atoms are ultracold, only a very small amount of kinetic energy needs to be coupled to the transverse degrees of freedom to completely eliminate any dispersion.

In this chapter I describe how the exchange of energy between longitudinal and transverse motion at several betatron resonances was utilized to manage the dispersion of an atomic matter wave. For specific angular velocities of circulation the front of the atom pulse was slowed by the betatron resonance. This exchange dramatically reduced the longitudinal dispersion of the atomic beam. After BECs were formed in a tipped portion of the ring, the magnetic minimum was advanced by ∼ π/4 radians along the ring by applying a side bias field causing atoms to accelerate toward the new minimum. After ∼ 60 ms, the magnetic fields which controlled the tilt of the ring were ramped to a setting which made the ring as flat as possible [67]. Different launch velocities were obtained by applying bias fields of different magnitude, and 121

15 (a) νz =5 (b) (Hz)

π 1 mm /2

final 14 ν =5 Ω r

(c)

13 (d) 1 mm

13 14 15 /2π (Hz) Ωinitial

Figure 7.5: Stopbands for νr = 5 radial (lower Ωi) and the νz = 5 axial (higher Ωi) betatron resonances. Mean ﬁnal angular velocities, determined after 440 ms of propagation in the ring, are shown vs. Ωi. Near the resonances, the atomic beam divided into two portions (Fig. 7.7), and mean angular velocities of each are given. Simulation results for the axial resonance (solid line), and the relation Ωi = Ωf (dashed line) are shown. The (a) annular and (b) side views of the atoms launched at νr = 5 (radial resonance), and (c) annular and (d) side views of those at νz = 5 (axial resonance), show which transverse oscillation was enhanced at each resonance. The horizontal scale for (a) and (c) is chosen to give equal cloud lengths in annular- and side-view images. Here, (ωr, ωz) = 2π × (69, 73) Hz.

a linear relationship between the launch velocity and the applied bias ﬁeld was determined for settings far from integer tune parameters. For some launch velocities, the mean angular velocity of the beam was reduced from its initial velocity. This “dispersion relation”, shown in Figure 7.5, was a systematic way to study the strength of resonances in the ring. For varying initial velocities, the azimuthal position of the atomic beam was determined from absorption images after 400 and 500 ms of propagation, and its mean angular velocity determined from it’s angular advance. Figure 7.5 exhibits two clear stopbands, speciﬁc angular velocities which were

disallowed in the ring, each for the νr and νz = 5 resonances. The axial νz = 5 resonance was simulated by numerically integrating the equations of motion (7.35–7.37) for a single particle with a l = 5, B = 1.2 µm beam path error. As is suggested by Figure 7.4 the spread in angular velocities should vary signiﬁ- 122 cantly for beams which were launched at resonant or non resonant tune parameters. When launched at the axial tune νz = 4.2, the rms azimuthal width of the cloud grew steadily according to an rms azimuthal linear velocity spread of ∆v = 1.7 mm/s [129], equivalent −2 to an rms variation in the tune of ∆νz = 6 × 10 , or a longitudinal kinetic temperature 2 of T = m(∆v) /kB = 28 nK. At this rate of expansion, the propagating cloud ﬁlled a large portion of the storage ring after 20 revolutions.

When launched at an integer νz = 4.0 tune parameter, the a atomic beam evolved quite differently. As shown in Figure 7.6, about half of the circulating atoms were “caught” in the axial betatron resonance and were slowed to circulate at velocities below the stopband. This part of the atomic beam then circulated without dispersion so that a compact portion of the beam remained after 18 revolutions. After 250 ms, the width of this compact portion exhibited no further growth. To estimate the rms velocity spread of this part of the gas we consider the azimuthal extent of the beam at the earliest and latest times shown in Figure 7.6. From this we estimate the rms velocity spread of this portion of the beam to −3 be below 100 µm/s, reducing the range of tunes to ∆νz < 4 × 10 . This corresponds to a longitudinal kinetic temperature below 100 pK, the lowest kinetic temperature ever reported for an atomic beam. While the longitudinal kinetic temperature of the beam was significantly reduced when launched at a resonant tune parameter, this does not imply that it was cooled. “Tem- perature” is actually a misleading notion in this situation. Rather, a fraction of the kinetic energy was exchanged between the longitudinal and transverse directions in such a way as to eliminate the dispersion of the cloud. Figure 7.7 shows the effects of betatron resonance in more detail. Atomic wavepackets were launched at tune parameters near νz = 5 and then imaged after 640 ms of propagation. The data clearly reveal a stopband, a range of velocities which cannot be maintained in the storage ring. Atoms which circulate at angular velocities within the stopband are slowed by the betatron resonance to velocities which circulate just below the stopband. The increase in linear density of the atomic beam at the low velocity edge of the stopband is clearly shown. Resonances were identified as either axial or radial by directly observing the betatron motion. A radial resonance produces an atomic beam which oscillates radially with a phase which differs across the length (i.e. across different azimuthal velocities) of the cloud. Likewise, an axial resonance, as shown in Figure 7.8, produces oscillations in the 123

(a) laps=2 11 1.0 18

(b) laps=2 11 0.5 18

Angular rms width (rad) 2π

0.0 0 200 400 600 800 1000 1200 Time (ms)

Figure 7.6: Dispersion management of matter waves in a storage ring. Annular (top) views are shown after 2, 11, and 18 complete revolutions for initial mean axial tunes of (a) νz = 4.2 and (b) νz = 4.0. Slight bends in the cloud resulted from radial betatron motion excited during the injection sequence. (c) The rms width of all (X’s for νz = 4.2) or just the compact portion (circles for νz = 4.0) of the atomic beam is shown vs. propagation time, with data for νz = 4.0 limited to times when the compact and diﬀuse portions were separated. A linear ﬁt to data for νz = 4.2 (solid line), and a line joining earliest and latest data for νz = 4.0 (dashed line) are shown. Here (ωr, ωz) = 2π (48, 60) Hz.

axial direction but does not excite radial motion. Using either side or top imaging, resonances were identified as either axial or radial. The resonant tune parameter was calculated using the measured trap frequencies and angular velocities. For radial betatron resonances, q 2 2 the effective radial trap frequency, ωr(Ω) ≡ 3Ω + ωρ was measured by exciting betatron motion while an atomic beam was circulating at the relevant angular velocity. Axial trap frequencies were measured when the atomic beam was either at rest or circulating in the ring. These two measures agreed well. By directly observing betatron motion we also confirmed that energy was conserved. For instance, the 60 µm maximum amplitude of radial betatron motion observed in Fig. 7.5(c) corresponds to an energy of kB × 3.5 µK. Given the initial kinetic energy of kB × 62 µK for atoms launched at νr = 5, this motion should reduce the longitudinal velocity by 2.7%, in agreement with the measured magnitude of the stopband. The manipulation of matter waves through betatron resonances represents a novel approach to managing dispersion in a circular, multimode atomic waveguide. The usefulness of such dispersion management hinges on the ability to control the strength and resonant 124

15.2

15.0

14.8 (H

initial 14.6 Ω /2π

14.4

14.0 14.2 14.4 14.6 14.8 15.0 15.2

Ωfinal /2π (Hz)

Figure 7.7: Matter wave dispersion at an axial betatron resonance. The distribution of final azimuthal velocities is shown for beams with initial mean angular velocities Ωi/2π evenly spaced between 14.3 (bottom curve) and 15.4 Hz (top curve). These distributions were obtained from the radially-integrated column density in top view images taken after 640 ms of propagation. Atoms expelled from the stopband (gray shading) accumulated at its low-velocity edge. Since the stopband was narrower than the full initial range of velocities in the beam, only a portion of the beam was affected. Low-velocity atoms for Ωi/2π = 14.3 Hz were affected by the νr = 5 radial betatron resonance. Here (ωr, ωz) = 2π (70, 74) Hz. 125

Figure 7.8: Axial betatron motion. Side images of axial betatron motion for the νz = 5 resonance. Absorption images were taken after 503 ms (and every 3 ms thereafter)of propagation. As the atomic beam circulates around the ring it moves out of focus. 126

ν = 4 18 = 3 ν ν= 5

16 z) (H

π 14 /2 s e r Ω 12 ν = 7

ν = 8 10 ν = 6

50 55 60 65 70 75 80 85 Betatron frequency (Hz)

Figure 7.9: Tuning the resonant velocities of betatron resonances. Measured angular frequencies Ωres at the low velocity edge of stopbands observed at several radial (open circles) and axial (filled circles) betatron resonances are shown vs. measured betatron frequencies. velocities of such resonances. To vary the betatron frequency, or equivalently the resonant angular velocity, we adjusted the the trap frequencies of the storage ring. For various axial or radial trap frequencies we observed betatron resonances for tune parameters ν = 4 − 6 accessible with the storage ring (Fig. 7.9), which could be tuned to occur at any specific angular velocity. Aside from the utility of many betatron resonances, this demonstration of betatron resonances at nearly every harmonic perturbation of the ring indicates a complicated potential landscape of the ring [67]. It was unsurprising that the ring contained so many static perturbations, since it had been constructed by hand and contained many small electrical shorts and defects. The occurrence of so many high order (q > 2) perturbations may preclude flattening the ring to the required level for a condensate the fill the multiply connected geometry of the ring. To vary the strength of a specific resonance, we added to our magnetic trap a static radial-quadrupole magnetic field with its axis coinciding with that of the storage ring. This magnetic field added two types of errors to the ring: a q = 2 axial beam-path error, and a q = 4 potential error U(θ) which is proportional to the azimuthal field magnitude. (see Ref.

[129]) As shown in ﬁgure 7.9(b), the strength of the radial betatron resonance at νr = 4, assessed by the magnitude of the stopband, was adjusted by varying the magnitude of the 127

17.5

n = 1.2 17.0 n = 7.2 n = 13.4 (Hz) π 16.5 n = 19.4 /2 n = 24.2 final Ω 16.0

15.5

15.5 16.0 16.5 17.0 17.5 Ωinitial/2π (Hz)

Figure 7.10: The νr = 4 resonance for (ωr, ωz) = 2π (68, 74) Hz was characterized using a deliberately applied U(θ) = U4 sin 4θ azimuthal potential of varying strength, given in units of the harmonic oscillator quanta for the radial conﬁnement.

deliberately applied q = 4 modulation. The management of dispersion demonstrated in this chapter, and its deliberate manipulation shown in ﬁgures 7.10 and 7.9 is an unexpected twist on a familiar accelerator physics phenomena. Such dispersion management may improve measurements of rotation rates [143, 144, 145], fundamental constants [146], and other quantities by atom- interferometric schemes which are sensitive only to longitudinal velocities [147]. The resonant nature of betatron resonances may also be used to study properties of the ring itself. We have already noted the complicated potential landscape of the ring, deduced from the many orders of axial and radial betatron resonances which were observed. Only very small perturbations are needed to resonantly excite betatron motion. For example, the deliberately applied modulations in ﬁgures 7.10 and 7.11 were incremented in steps of 20 nK, equivalent to a force of 6 × 10−26 N, a fantastically small force. Given the 700 ms propagation time necessary for a measurement, this corresponds to a demonstrated √ force sensitivity of much better than 7 × 10−26N/ Hz. This force sensitivity suggests that we should look toward fundamental physics measurements which should be conducted with this device. 128

Figure 7.11: Manipulating the νr = 4 betatron resonance. Apllied q = 4 modulations of 24 nK, (left) and 81 nK (right) result in diﬀerent evolutions. Top images were taken after 100 ms of propagation, and bottom images after 700 ms. The radial excitation for the larger modulation is quite signiﬁcant. 129

Chapter 8

A proposal to test the gravitational inverse-square law at short distances

If there are two things which the reader should have learned from reading the last two chapters of this thesis, they might be that (1) quantum-limited measurements of position may be achieved for collections of atoms contained within a high finesse optical cavity, and (2) at certain resonant velocities, the motion of atoms in a circular storage ring may be very sensitive to static perturbations. We ask, what fundamental measurements may be conducted with both tools at our disposal? Gravity holds a special place in the study of physics. It dominates our our intuition of forces and mechanics, yet is the weakest of the four known forces, much weaker than even the “weak” force. Newton’s theory of gravity was the first unified theory of physics, connecting his famous falling apple to the orbits of planets, unifying physics on all accessible scales. Today, gravity, updated by Einstein’s theory of general relativity remains unincluded in the standard model of physics. Unifying gravitation with the quantum field theory which forms the standard model of particle physics is a central challenge in physics today, and may be accomplished by string theory or other extensions to the standard model. Some of these theories predict extra dimensions which are curled up at short distances, and would result in deviations to the inverse square law of the gravitational force. Excellent reviews of theoretical and experimental works in this area have been written [148, 149]. 130

Experiments which test the inverse square law of gravity at short length scales may discover, or at least constrain the existence of extra dimensions at short length scales. These experimental tests are quite difficult; because small objects have relatively small mass the gravitational force is easily swamped by electromagnetic forces. Current limits on deviations to the inverse square law at the shortest distances were set by experiments measuring the Casimir force between a sphere and a surface [150]. At slightly longer length scales limits have been set by measuring the resonant response of cantilever [151], torsion balance [152], and a torsional pendulum [148]. Here, the proposed measurement technique utilizes the response of a high-Q resonator to a weak, but resonant force. Equation 7.32 worked out the growth of the the radial betatron amplitude due to such a static perturbation. Exactly on resonance, νr = q, the amplitude of radial motion R grows in response to a force F as, F/Ω2 F R = θ = t. (8.1) 2qm 2mωr This simple result doesn’t actually require the contortions of equations used in Chapter 7 to discuss betatron motion; it may also be be obtained by simply considering the response of an oscillator which was kicked with an impulse F t/2. If this displacement A is detected at the quantum limit Zho, the force sensitivity is, r Z 2mω 2 mω F = ho r = ~ r . (8.2) min t t N 5 Plugging in numbers from Figure 7.10, ωr = 2π × 68 Hz, N = 10 , and t = 700 ms, we find −31 that Fmin = 7.2 × 10 N. Reasonable and demonstrated improvements can be envisioned by increasing the number of atoms to 106, reducing the trap frequency to 2π × 10 Hz, and −33 increasing the interrogation time to t = 20s, at which point, Fmin = 3.1 × 10 N. Another effect of betatron coupling to a periodic force F is a slowing of the mean orbital velocity, which for the betatron studies in Chapter 7 was an way to quantitatively study betatron resonances. Excitation of betatron motion of amplitude R slows the mean 2 2 2 2 orbital velocity by an amount δv ' ωr R /(2v) = ωr R q/(4πR). After a propagation 2 3 2 time t the position lags by an amount F t q/(16πm ωrR). Setting this equal to the limit

for a continuous measurement of position at the quantum limit for a free particle, Zfp = p ~τ/(2mN) [153], we have 1 1/2 F = p τ/(2mN)16πm2ω R (8.3) min t3q ~ r 131

Where τ is the measurement time. For the parameters of Figure 7.10 and τ = 100 µs, −29 Fmin = 2 × 10 N. However, for a measurement which takes a reasonable amount of time, orbital motion will smear out the position resolution. This limits the force sensitivity to,

4πmRω r2τ F = r (8.4) min q t3

−27 Again for our parameters in Figure 7.10, Fmin = 7 × 10 N. More optimistic parameters, −33 and using q = 100, R = 100 µm, set Fmin = 5 × 10 N. Gravity is unfortunately very weak, and in the scheme of things, Rubidium atoms are pretty light. The gravitational force between a 87Rb atom which is located on the

surface of of a sphere of some material with density ρs and radius rs and the sphere is, 4π F = Gmρ r . (8.5) grav 3 s s

4 3 −36 Considering a sphere of lead, ρs = 1.143×10 kg/m , with rs = 10 µm, Fgrav = 4.6×10 N. This force is much smaller than even our optimistic force sensitivity. However, it is possible that gravity is actually stronger at short distances and would actually be measurable. This would be great, but in the case that no gravitational force is measured, we might at least get the consolation prize of stating with certainty that the deviations to gravity are at least smaller than some amount. Deviations to the inverse square law of Newtonian gravity are most commonly expressed as a Yukawa type addition to Newton’s law, which describes the exchange of virtual bosonic particles.

4πGmρ r3 V = − s s 1 + αe−r/λ (8.6) 3r

By measuring the gravitational force at short distances, one places limits on the dimensionless strength α and the length scale λ. Diﬀerentiating 8.6, a bound on α(λ) from Fmin can be obtained (with rs = r). 4π h i F = Gmρ r 2 + α(2 − r/λ)e−r/λ , (8.7) min 3 s 3 F er/λ α ≤ min − 2 . (8.8) 4π Gmρsr 2 − r/λ In practice, circulating atoms cannot be placed exactly on the surface of the test masses, a reasonable separation s is necessary for electrostatic shielding between the test 132

Lamoreaux 97 Lamoreaux 97 3 μm 3 μm Chiaverini et. al. 03 Chiaverini et. al. 03 |α| |α| 10 μm 10 μm Long et. al. 04 Long et. al. 04

-33 -32 Fmin = 3 x 10 N Fmin = 3 x 10 N Hoyle et. al. 04 Hoyle et. al. 04

λ (m) λ (m)

Figure 8.1: Excluded regions for Yukawa additions to Newtonian gravity are shown. Given −33 −32 force sensitivity of Fmin = 3 × 10 N (left) and Fmin = 3 × 10 N (right) the possible excluded regions for test with 3 µm and 10 µm test masses are shown. In both cases the beam is speparated from the surface of the mass by 5 µm. Current excluded regions are shaded gray [148, 152, 151, 150]. 133 masses and the waveguide. Taking this into account, the Yukawa modified gravitational potential is, 4πGmρ(r − s)3 V = − 1 + αe−r/λ (8.9) 3r The theoretically achievable bounds on α(λ) were obtained from (8.9) in the same manner as (8.7) and (8.8) and plotted in Figure 8.1 for different force sensitivities. A cold atom– betatron resonance based measurement can potentially exclude values of α(λ) which are not currently excluded for Yukawa type additions to Newtonian gravity at short distances. These measurements could constrain theories of string dilation [154], gauge bosons, Gluon, and Strange modulus [155]. Implementation of such an experiment is still subject to many experimental un- knowns. However, a few key benefits of using betatron resonances to measure small forces are present. With ultracold atoms, measurement at the quantum limit is a real possibility, as has been demonstrated (albeit in a different context) in this thesis. It may be necessary to integrate a cavity into a storage ring apparatus to obtain quantum limited measurements, however, this cavity would not need attain the single atom strong coupling regime as mentioned in Chapter 5. Environmental vibrations may also result in excitation of betatron motion owing to the low frequency (10-50 Hz) of oscillation. However, because betatron resonances occur at specific angular velocities, an atomic beam which contains a range of angular velocities may be used to reject vibration induced common mode oscillations. Another ambitious advantage of using betatron resonances to measure deviations of the inverse square law is the possibility of performing quantum nondemolition measurements. This possibility arises from the fact that the amplitude of betatron motion is also coupled to the momentum of the circulating cloud. Momentum measurements can in principle be quantum nondemolition measurements, because for a free particle a disturbance of the position does not affect the momentum of the particle [43]. In the next few years as a second generation of ultracold atom storage ring comes online and quantum limited measurements are further explored in the E2 laboratory, the feasibility and focus of precision measurements with cold atoms will become clearer. Testing the limits of force sensitivity with betatron resonances using deliberately applied modulations to the ring potential will hopefully be among the first experiments performed with the new storage ring. Likewise, the detection small optical forces with the cold atoms-cavity system will be explored in parallel. 134

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