Quantum Limits on Measurement

Rob Schoelkopf Applied Physics Yale University

Gurus: Michel Devoret, Steve Girvin, Aash Clerk

And many discussions with D. Prober, K. Lehnert, D. Esteve, L. Kouwenhoven, B. Yurke, L. Levitov, K. Likharev, …

Thanks for slides: L. Kouwenhoven, K. Schwab, K. Lehnert,…

Noise and Quantum Measurement 1 R. Schoelkopf Overview of Lectures

Lecture 1: Equilibrium and Non-equilibrium Quantum Noise in Circuits Reference: “Quantum Fluctuations in Electrical Circuits,” M. Devoret Les Houches notes

Lecture 2: Quantum Spectrometers of Electrical Noise Reference: “Qubits as Spectrometers of Quantum Noise,” R. Schoelkopf et al., cond-mat/0210247 Lecture 3: Quantum Limits on Measurement References: “Amplifying Quantum Signals with the Single-Electron Transistor,” M. Devoret and RS, Nature 2000. “Quantum-limited Measurement and Information in Mesoscopic Detectors,” A.Clerk, S. Girvin, D. Stone PRB 2003. And see also upcoming RMP by Clerk, Girvin, Devoret, & RS Noise and Quantum Measurement 2 R. Schoelkopf Outline of Lecture 3 • Quantum measurement basics: The Heisenberg microscope No noiseless amplification / No wasted information

• General linear QND measurement of a qubit

• Circuit QED nondemolition measurement of a qubit Quantum limit? Experiments on dephasing and photon shot noise

• Voltage amplifiers: Classical treatment and effective circuit SET as a voltage amplifier MEMS experiments – Schwab, Lehnert

Noise and Quantum Measurement 3 R. Schoelkopf Heisenberg Microscope ∆p Measure position of free particle: ∆x ∆x = imprecision of msmt.

wavelength of probe photon: λ = hc / Eγ ∆p = backaction due to msmt. momentum “kick” due to photon: ∆pE= / c hc E ∆x∆p = ∼ h E c Only an issue if: 1) try to observe both x,p or 2) try to repeat measurements of x

Uncertainty principle: ∆x∆p ≥ /2

Noise and Quantum Measurement 4 R. Schoelkopf No Noiseless Amplification! Clerk & Girvin, Linear amplifier after Haus & Mullen, 1962 input output and Caves, 1982 mode mode a want: bG= a b †† † bG= a ⎡⎤aa,1† = ⎡bb,1⎤ = ⎣⎦ ⎣ ⎦ photon number gain, G extra †† c but then ⎡⎤bb,,= G⎡aa ⎤≠ 1 mode ⎣⎦⎣⎦ bG=+aG−1 c† bG††= a+−G1 c

†† † ⎣⎦⎡⎤bb,,= G⎣⎡aa ⎦⎤+−(G 1)⎣⎡c,c⎦⎤=1

Noise and Quantum Measurement 5 R. Schoelkopf No Noiseless Amplification! - II

input output mode mode bG=+aG−1 c† a b bG††=+aG−1 c

2 11 extra ()∆xa=+a††aa=n+ mode c in 22a

2 1 G G 1 ()∆=xbb††+bb={}a+c†,a†+c out 22 ⎛ 11⎞ = Gn⎜ ac++n+⎟ ⎝ 22⎠

amplified inputNoise vacuumand Quantum Measurementadded noise 6 R. Schoelkopf wasted No Wasted Information mode d output (e.g. Clerk, 2003) input mode mode a b bG=+aG−1cc† oshθ +dsinhθ extra () † mode bh= ..c c ⎡bb,1† ⎤ = G 1 ⎣ ⎦ 2 1 G ()∆=xb{},cb††={}a+coshθθ+dsinh,h.c. out 22

2 ⎛ 1122⎛⎞ ⎛1⎞⎞ ()∆=xGout ⎜ na ++cosh θθ⎜⎟nc ++sinh ⎜nd +⎟⎟ ⎝ 22⎝⎠ ⎝2⎠⎠

Noise “Excess”and Quantum Measurement noise above quantum limit7 R. Schoelkopf Two Manifestations of Quantum Limit Position meas. of a beam QND meas. of a qubit Mech. HO with SET/APC detector Circuit QED: Box + HO (Cleland et al.; Schwab et al.; Lehnert et al. ) (Yale )

Cg Cge Vds Cg

Vge

ω kT ≥ 1 N 2 T Γ ≥ m φ 2 min. noise energy of amplifier meas. induces dephasing

Noise and Quantum Measurement 8 R. Schoelkopf Linear QND Measurement of Qubit

ˆ Oˆ no transitions caused I G by measurement: A quantum ⎡HHˆˆ,0⎤ = H =− ω σˆ Hˆˆ= AIσˆ ⎣ Q 1 ⎦ nondemolition Qz2 01 1 z always some “demolition,” ⎡⎤HHˆˆ,0≠ in reality: ⎣⎦1 Universe e.g. spontaneous emission

ψσ==±1 if qz can measure repeatedly, no errors =↑ or ↓

σˆ z = 0 but if ψ q =+,o−=→ r← we get ±1 at random9 Linear QND Measurement - II

ˆ linear amplifier: Iˆ O G Otˆ() =−A dτ G t τσˆ (τ) ∫ ( ) z i t A ψψ==()td0(− τHˆ ψ0) ∫−∞ 1 i t ˆˆ ψ =+ψτ(0) dAψ(0) σˆ Iˆ H1 = AIσˆ z ∫−∞ z i ∞ ψψOdˆˆ=− τΘ()t−τψ(0)⎡⎤O,Hˆ (τ)ψ(0) ∫−∞ ⎣⎦1 i ∞ Otˆˆ() =− dτ Aστˆ ( )Θ(t−τ) ⎡O,Iˆ(τ)⎤ ∫−∞ z ⎣ ⎦ ⎡ ˆ ˆ ⎤ recognize Gt()=−iΘ()t ⎣O()t,I(0)⎦ ⎡ ˆ ˆ ⎤ input and output don’t but if G ≠ 0 ⎣Ot(),I(0)⎦ ≠ 0 commute, and have noise!10 Measurement Time

ˆ Integrate output: Iˆ O t G Mˆ ()td= ∫ τ Oˆ(τ ) 0 t A ↑↑Mˆ =dAτσGˆ ()τ=+AGt ˆˆ ∫0 z H1 = AIσˆ z ↓↓Mˆ =−AGt Distinguish when 2 ˆˆ 2 ()↑↑MM−↓↓ ()2AGt 4A2 2 ==2 t ~1 ()∆M StOOS /G

S 1 Measurement T = O Stronger coupling, time m GA224 faster measurement

Spectral density of output noise, referred to input 11 Dephasing by QND Measurement ˆ ˆ But It() also fluctuates! Iˆ O ˆˆ G H ()tA=−ωσ01 ˆˆzz/2+ σI()t

=−()ωδ01 + ω()t σˆ z /2 A so transition (Larmor) freq. fluctuates phase ˆ unperturbed ψσ()01==z ± fluctuates! 1 1 itφ () ψ ()01=+()+−1 ψ ()te= ()++11− 2 2 ttA fluctuations Gaussian φ()tt=+ωτdδω(τ)=+ωtdτIˆ()τ 01 ∫∫01 and rapid: 2200 22AA2 ()∆=φ ()∆I tS= t=Γt 22I φ spectral density of input dephasing rate Stronger coupling, faster dephasing! 12 Quantum Limit for QND Measurement

ˆ Oˆ Compare dephasing rate I G A and measurement time: S 1 T = O Measurement time: m GA224 2A2 Dephasing rate: Γ= S φ 2 I

2 SAOO12 SSIindependent of TSmIΓ=φ = GA22422 G22 coupling! and since ∼⎡ ˆ ˆ ⎤ 22 2 GO⎣ ()t,I(0)⎦ (∆∆OI)( )≥(G) Quantum 1 TmΓφ ≥ Measurement is dephasing Limit! 2 13 Measurement Dephasing – Quantum Dots A “which path” experiment in mesoscopics - Heiblum group, Weizmann 1998 A-B oscillations of ring tests coherence

QPC “detector” Gring

B-field

Quantum dot in a ring QPC current senses which way electrons go around ring, Visibility destroys fringes. QPC current E. Buks et al., Nature 391, 871 (1998) 14 “Circuit QED” – Box + Transmission Line Cavity 2g = vacuum Rabi freq. κ = cavity decay rate γ = “transverse” decay rate Strong Coupling = g > κ , γ out cm 2.5 λ ~ transmission L = line “cavity”

10 µm Cooper-pair box “atom” 10 GHz in

Theory: Blais et al., Phys. Rev. A 69, 062320 (2004) 15 Implementation of Oscillator on a Chip Superconducting transmission line 2 cm Si

Niobium films gap = mirror 6 GHz:

ω = 300mK nγ 1 @ 20 mK ω R even when nγ = 0 RMS voltage: V0 = ∼ 1Vµ 2CR 16 Energy Levels of Cooper Pair Box

E E H =−Coulomb σ x Josephson σ z 22 Tune σx with voltage: (Stark)

EECoulomb = 41Cg(n− )

Tune σz with Φ: (Zeeman) EE= cos πΦΦ/ Josephson Jbmax [ 0 ] 17 Box Coupled to Oscillator E Hˆ =− J σˆ box 2 z ˆ † HaHOR=+ω (1a/2) C ˆˆ⎛⎞g Heint =− ⎜⎟Vσˆ x ⎝⎠CΣ =−g()σσ−+aa† + Jaynes-Cummings LR ~ ½ nH; CR ~ ½ pF

1 eCg ω 112 g = R CVR 0 = ωR 24 22 CCΣ R ω V = R ∼ 1Vµ So for: CCg /0Σ = .1 0 2C R g ∼10 −100 MHz 18 The Chip for Circuit QED Wallraff et al., Nature 431, 162 (2004).

Nb

Si Al No wires attached Nb to qubit!

19 Dispersive QND Qubit Measurement

reverse of Nogues et al., 1999 (Ecole Normale) QND of single photon using Rydberg atoms!

20 A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 062320 (2004) Alternate View of the QND Measurement 22 ⎛⎞gg† ⎛⎞ Haeff ≈+⎜⎟ωrzσωa+⎜a+⎟σz ⎝⎠∆∆2⎝⎠

cavity freq. shift Lamb shift atom ac Stark shift vacuum ac Stark shift =×2n cavity pull

⎛⎞g2 ⎡⎤1 Ha≈+ω ††aωσ+2 aa+ eff r ⎜⎟az⎢⎥ 22⎝⎠∆ ⎣⎦ nIˆ ~ ˆ A 21 cQED Measurement and Backaction - Predictions Input = photon number in cavity Output = voltage outside cavity 2g 2 phase shift on transmission: θ = 0 κ∆ measurement rate: 1 ⎛⎞P (expt. still ~ 40 Γ= =22θ 22=θκn m 00⎜⎟ times worse) Tmr⎝⎠ω dephasing rate:

22⎛⎞P Γ=φ 22θ00⎜⎟=θκn ⎝⎠ωr quantum 2x limit, since half of information limit?: TmΓφ =1 wasted in reflected beam22 Microwave Setup for cQED Experiment n ~ 1−100 Transmit-side Receive-side

typical input power ~ 10-17 Watts

ndet ~40

23 Observing ac Stark Shift Measure absorption spectrum of CPB w/ continuous msmt. shift proportional to n

n =1 n = 40

Line broadened as qubit is dephased by photon shot noise24 Observing Backaction of Measurement 11⎛⎞g 2 ⎡⎤ Ha≈−ω ††aωσ+2 aa+ eff r ⎜⎟az⎢⎥ 22⎝⎠∆ ⎣⎦ n fluctuations in photon number

expt: Schuster et al., PRL 94, 123602 (2005). 25 Cavity QED - SET Analogy

e-

Cg Cge Vds

Vge

shot noise of SET photon shot noise current causes induces qubit dephasing backaction

26 Summary of Lecture 3

† • Quantum limit on measurement comes from ⎣⎡aa,1⎦⎤ = • Two equivalent manifestations of quantum limit: Min. noise ω 1 T ≥ Meas. induced T Γ ≥ temperature N 2k dephasing m φ 2 • Mesoscopic expts. can approach these limits: Sensitivity ~ 10-100 times limit obtained Dephasing due to measurement observed

• But true quantum limit not yet observed/tested!

• Future: back-action evasion, squeezing, quantum feedback, … 27 Equivalent Circuit of an Amplifier

SV ()ω

SI ()ω

“ficticious noise source” (V2/Hz) = output noise referred to input SV ()ω

2 SI ()ω a real noise (A /Hz) driven thru input terminals

here assume uncorrelated, though typically not!

Noise and Quantum Measurement 28 R. Schoelkopf Noise Temperature of an Amplifier Def’n (IEEE) : temperature of a load @ input which doubles the system’s output noise (assumes Rayleigh-Jeans) V (ω) sig SV

SI

2 tot ZZin S total noise at input: SSVV=+SI ZZin + S equate to Johnson tot noise of source: SkVN= 4RTe[Zs] for ZRin ==in Rs Zs TSNV= ( //RS+ SIRS) 4k

Noise and Quantum Measurement 29 TN depends onR. Schoelkopf source impedance Optimum Noise Temperature of Amplifier

TSNV= ( //RS+ SIRS) 4k N log T

log Rsource

Ropt = SV / SI TS= S/2kEk==T SS/2 NVopt I NNopt opt VI

EN is energy of signal that can be detected with SNR = 1

QM imposes minimum: EN ≥ ω /2 Noise and Quantum Measurement 30 R. Schoelkopf Noise of a Single Electron Transistor

charge advance, k

kN= ( 12+ N)/2 dk n Ie= ds dt island charge, n

nN= 12− N

Ideally, SET has only shot noise (T=0, ω

Fluctuations of k limit msmt. of response (Ids) Fluctuations of n cause island potential to change current flows thru gate @ ω ≠ 0 Noise and Quantum Measurement 31 R. Schoelkopf Properties of an SET Amplifier

V (ω) sig SV

SI

2 (1−α 2 )(1+α 2 ) ⎛ C ⎞ In limit of: S ()ω = eV R ⎜ Σ ⎟ indep. normal state, V 8α 2 ds Σ ⎜ C ⎟ ⎝ g ⎠ of ω T=0, no cotunneling, 2 2 (1−α 2 )eV ⎛ C ⎞ ⎛ eωR ⎞ ω << V/eR ds g Σ 2 SI ()ω = ⎜ ⎟ ⎜ ⎟ ~ ω 4 RΣ ⎝ CΣ ⎠ ⎝ Vds ⎠

M. Devoret and RS (2000), α = (2C gVg − e)/ CΣVds similar results by Schon et al, Averin, Korotkov Noise and Quantum Measurement 32 R. Schoelkopf Noise Energy of SET

Sequential Tunneling: (e.g. Devoret & RS, 2000)

4 2 π (1−α ) (1+α )RΣ E N ()ω = SV SI = ω 2α 2 RK 1 ∼ 8 at 16 MHz EN < 2ω Ropt ≈ 10 Ω ωCg Cotunneling limit: (e.g. Averin, Korotkov) EN → ω /2 Resonant Cooper-pair tunneling (DJQP): (e.g. Clerk) EN → ω /2 Experimentally: still factor of 10-100 from intrinsic shot noise limit

Noise and Quantum Measurement 33 R. Schoelkopf Other Amplifiers Near Quantum Limit Josephson parametric amplifier at 19 GHz

T = 0.45K ~ hν/2k Yurke et al.; N Movshovich et al., PRL 65, 1419 (1990)

SIS mixer at 95 GHz (heterodyne detection using quasiparticle nonlinearity)

Noise added = 0.6 photons Mears et al., APL 57, 2487 (1990)

Microwave SQUID amplifier at 500 MHz

TN = 50 mK ~ 2hν/k Muck, Kycia, and Clarke, APL 78, 967 (2001) No measurement of crossover, or backaction yet.

Noise and Quantum Measurement 34 R. Schoelkopf NEMS Oscillator Measured by SET – Schwab group

35 Sample

Beam Gate Silicon Nitride 8µm X 200nm X 100nm

fO = 19.7MHz Q ~ 30-45000

Single Electron Transistor

Al/AlxOy/Al Junctions 2K Charging Energy 70kΩ Resistance 70 MHz Bandwidth

Beam/SET Separation: 600nm 27aF Capacitance 36 Resonator Response

T= 30mk

Vg= 10V

0.50 Q = 54,00010 1 2 2 1 mωox = kBT 8 2 2 0.25 500 T=100mK

450 Vg= 10V ) 6 e) d a 400 Q=36,000

(r 0.00 π de (m / z) e 350 /H as 4 plitu 2 m e Ph µ

A 300

-0.25 2 250

Power ( 200

-0.50 0 150 19.674 19.675 19.676 Frequency (MHz) 100

19.668 19.670 19.672 19.674 19.676 19.678 19.680 Driven Response Frequency (MHz)

Thermal Response 37 Noise Power vs. Temperature

Saturates Below 100mK

Use Linear Data to Nth= 58 Calibrate Below 100mK

Lowest Mode Temp Measured: T=56mk

38 Noise Temperature

V =15V Noise Temperature g

TN = 15.6mK T=100mK

Energy Sensitivity

EN ≈ 17 ћω0 TN = 15.6mK

Position Sensitivity

∆x = 4.3∆xQL

Closest approach yet to uncertainty principle limit! 39 How far can we push this technique ?

Preamp noise floor 1000 √Sqq=10µe/√Hz Induced Charge: δQ= VgδCg = (CgVg/d) δx

QL Charge Sensitivity (forward 10 X ∆ coupling): X/ X (fm)

∆ 1/2 1/2 ∆ Sx = Sqq d/(CgVg) 100 Back-Action: 1/2 1/2 Sx = Svv CgVgQ/(kd)

1 20 1 3 10 30 V (Volts) Ideal Shot Beam Noise Limit Back-Action 40 Circuit Model

BEAM SET Total Noise Power = Gain x 2 Ids(q) [Sqq + Sthermal + Svv/|ωZin(ω) | ] Lm Sqq Svv 4 Cm Cg /Hz 2 e 3 Rm -9 Rj/2 2Cj 4kBTRm 2

1 Output Noise 10

0 9.366 9.367 9.368 9.369 9.370 9.371 Frequency (MHz) 2 2 Cm = Cg(CgVg /kd ) = 0.06 aF @ Vg=10V 2 2 2 Lm = 1/(Cgω )(kd /CgVg ) = 4500 H 2 2 Rm = 1/(QCgω)(kd /CgVg ) = 2.8 MΩ 41 Sensitivity Optimization

10000

2R = 75KΩ ) S j 1/2 qq = 2C =1.3fF 10 j Hz 0 / µ e K=1.7 N/m /H 5 1000 z 1 Q=1.5x10 /2 S h n 1/2 itivity (fm ot o S =2.2µe/Hz (shot noise) s i qq N ct S =100µe/Hz1/2 (preamp) oi A qq se k- 1/2 L ac Svv=1nV/Hz 100 im B it 1/2 (SqqSvv) ≈ 3h

Position Sen Standard Quantum Limit

10 0.01 0.1 1 10 50 Vg (Volt) 2 Rm=6.2 MΩ/Vg

1/2 Loading: Roptimum = (Svv/Sqq) / ω 2 2 0.5 = 47 MΩ ω = ω0(1- (CgVg /kd ) (Cg/2Cj)) -1 -1 2 2 42 Qeff = Q + (CgVg /kd )(Cg/2Cj)ω0(RjCj) Atomic Point Contact Displacement Detector: Lehnert group at JILA/CU

as in an STM

Infer postion from tunnel current

λ m Sensitive: δ≈x e ≈1.2×10−15 with 1 nA current Ne /Hτ z

Local: ideal for sub-micron objects 43 Atomic Point Contact Displacement Detector: Simple Noise Analysis Imprecision Backaction (shot noise limit) (momentum diffusion) 1/2 1/2 ⎛⎞21e ⎛⎞I ∆=x λ ∆=p ⎜⎟τ e ⎜⎟ 22e ⎝⎠I τ λe ⎝⎠ 1/2 1/2 ∆=xNλee()1 ∆=pN()e 2λe

Tunneling Counting Momentum Number length scale statistics per tunneling diffusion attempt Ideal quantum displacement amplifier ∆xp∆= B. Yurke PRL 1990, A. A. Clerk PRB 2004 2 44 Thermal Motion at 43 MHz Resonanace

•Zero-point motion:

ω0 δ=xZP 2kBsw 1/2 δ=xZP 100 am/Hz

δxT δx I = 28 δxT δxZP •Mechanical bandwidth

Bw ≈ 9 kHz ; Q ≈ 5000 δxI •Sensitivity to normal coordinate

45