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 R. Schoelkopf 1
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 : Quantum Spectrometers of Electrical Noise Reference: Qubits as Spectrometers of Quantum Noise, R. Schoelkopf et al., cond-mat/01047 Lecture 3: Quantum Limits on Measurement References: Amplifying Quantum Signals with the Single-Electron Transistor, M. Devoret and RS, Nature 000. Quantum-limited Measurement and Information in Mesoscopic Detectors, A.Clerk, S. Girvin, D. Stone PRB 003. And see also upcoming RMP by Clerk, Girvin, Devoret, & RS Noise and Quantum Measurement 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 R. Schoelkopf 3
Heisenberg Microscope Measure position of free particle: x = imprecision of msmt. wavelength of probe photon: p = backaction due to msmt. momentum kick due to photon: Only an issue if: x p = hc E E c h λ = hc / E γ p= E/ c 1) try to observe both x,p or ) try to repeat measurements of x p x Uncertainty principle: x p / Noise and Quantum Measurement R. Schoelkopf 4
input mode a aa =, 1 No Noiseless Amplification! Linear amplifier extra mode c output mode b bb, = 1 Clerk & Girvin, after Haus & Mullen, 196 and Caves, 198 want: b= G a b = G a photon number gain, G but then bb = b= G a+ G 1 c b = G a + G 1 c bb G aa ( G ) c c, G aa, 1, =, + 1, = 1 Noise and Quantum Measurement R. Schoelkopf 5
No Noiseless Amplification! - II input mode a output mode b b= G a+ G 1 c b = G a + G 1 c extra mode c 1 1 xin = aa + a a = na + ( ) G 1 ( x ) = bb + b b = { a+ c, a + c} out 1 G 1 1 = G na + + nc + amplified input vacuum Noise and Quantum Measurement R. Schoelkopf added noise 6
wasted mode input mode a d No Wasted Information output mode b Noise and Quantum Measurement R. Schoelkopf (e.g. Clerk, 003) ( θ θ ) b= G a+ G 1 c cosh + dsinh extra mode c b = hc.. bb, = 1 G 1 ( ) 1 { } G x, { cosh sinh,..} out = b b = a+ c θ + d θ hc ( ) 1 1 1 xout = G na + + cosh θ nc + + sinh θ nd + Excess noise above quantum limit 7
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 ) C g C ge V ds Cg V ge kt N min. noise energy of amplifier ω 1 TmΓ φ meas. induces dephasing Noise and Quantum Measurement R. Schoelkopf 8
Linear QND Measurement of Qubit Î G Ô no transitions caused by measurement: H Q A = ω ˆ 01σ z Hˆ ˆ ˆ 1 Aσ z I Hˆ, ˆ 0 Q H = = 1 quantum nondemolition in reality: ˆ, ˆ 0 H1 H Universe always some demolition, e.g. spontaneous emission if but if ψ q = σ =± 1 z = or ψ =+ =, or q can measure repeatedly, no errors we get ± 1 ˆ σ z = 0 at random9
recognize but if G 0 Linear QND Measurement - II linear amplifier: Î Ô G Ot ˆ() = A dτ G( t τ) ˆ σz ( τ) A i t ψ = ψ ( t = 0 ) dτ Hˆ 1 ψ(0) i t ψ = ψ(0) + dτ ψ(0) Aˆ σ ˆ zi Hˆ 1 = Aˆ σ ˆ z I ˆ i ψ Oψ = dτ Θ( t τ) ψ(0) Oˆ, Hˆ 1( τ) ψ(0) ˆ i Ot () = dτ A ˆ σ ( ) ( ) ˆ, ˆ z τ Θ t τ OI( τ) Gt () = Θ i () t Ot ˆ(), Iˆ (0) Ot ˆ(), Iˆ (0) 0 input and output don t commute, and have noise! 10
Î A Hˆ = Aσ Iˆ 1 ˆ z Distinguish when Measurement Time G Ô ( ˆ ˆ ) ( M ) Integrate output: t Mˆ () t = dτ Oˆ ( τ ) 0 t = 0 z =+ M ˆ dτ AG ˆ σ ( τ ) AGt ˆM = AGt M M ( AGt) 4A = = t St O SO / G ~1 Measurement time T m = SO 1 Stronger coupling, G 4A faster measurement Spectral density of output noise, referred to input 11
Dephasing by QND Measurement But It ˆ( ) also fluctuates! Î Ô G Hˆ() t = ωσˆ ˆ ˆ 01 z / + AσzI() t = ( ω01 + δω() t ) ˆ σz / A so transition (Larmor) freq. fluctuates phase ψ ( 0) = ˆ σ z =± 1 unperturbed fluctuates! 1 1 ψ ( 0) = ( + 1 + 1 ) () ( i () t ψ t = + 1 + e φ 1 ) t t A fluctuations Gaussian φ() t = ω ˆ 01t+ dτ δω( τ) = ω01t+ dτ I( τ) and rapid: 0 0 A A φ = I t = S It =Γφt ( ) ( ) spectral density of input Stronger coupling, faster dephasing! dephasing rate 1
Quantum Limit for QND Measurement Î Ô A G Measurement time: Dephasing rate: Compare dephasing rate and measurement time: S Tm = G Γ φ = 1 4A O A SI T m S 1 A S S O O I S I Γ φ = = G 4A G and since G Oˆ (), t Iˆ (0) Quantum Limit! independent of coupling! O I G ( )( ) ( ) Tm Γ φ 1 Measurement is dephasing 13
Measurement Dephasing Quantum Dots A which path experiment in mesoscopics - Heiblum group, Weizmann 1998 A-B oscillations of ring tests coherence QPC detector G ring B-field Quantum dot in a ring QPC current senses which way electrons go around ring, destroys fringes. E. Buks et al., Nature 391, 871 (1998) Visibility QPC current 14
Circuit QED Box + Transmission Line Cavity transmission line cavity L = λ ~.5 cm g = vacuum Rabi freq. κ = cavity decay rate γ = transverse decay rate Strong Coupling = g > κ, γ out 10 µm 10 GHz in Cooper-pair box atom Theory: Blais et al., Phys. Rev. A 69, 0630 (004) 15
Implementation of Oscillator on a Chip Superconducting transmission line cm Si Niobium films gap = mirror 6 GHz: ω = 300mK 1 @ 0 mk n γ RMS voltage: ωr V0 = 1µ V C R even when n γ = 0 16
Energy Levels of Cooper Pair Box E E Coulomb x Josephson z H = σ σ Tune σ x with voltage: (Stark) ( ) ECoulomb = 4EC ng 1 Tune σ z with Φ: (Zeeman) max [ ] E = E cos πφ / Φ Josephson J b 0 17
Box Coupled to Oscillator ˆ EJ H ˆ box = σ z ˆ HHO = ωr( a a+ 1/) ˆ Cg H ˆ ˆ int = e Vσ x CΣ = g σ a + σ a + ( ) L Jaynes-Cummings R ~ ½ nh; C R ~ ½ pf 1 1 CR V0 = ωr 4 g = 1 ecg ωr C C ωr V0 = 1µ V So for: Cg / C Σ = 0.1 C R g Σ 10 100 MHz R 18
The Chip for Circuit QED Wallraff et al., Nature 431, 16 (004). Nb Si Nb Al No wires attached to qubit! 19
Dispersive QND Qubit Measurement reverse of Nogues et al., 1999 (Ecole Normale) QND of single photon using Rydberg atoms! A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and RS, PRA 69, 0630 (004) 0
Alternate View of the QND Measurement H eff g g ωr σ + z a a+ ω a+ σz cavity freq. shift atom ac Stark shift = n cavity pull Lamb shift vacuum ac Stark shift g + + + r a 1 H ω a a ω a a σ eff z A ˆ ~ n Iˆ 1
cqed Measurement and Backaction - Predictions Input = photon number in cavity Output = voltage outside cavity phase shift on transmission: θ 0 g = κ measurement rate: 1 P Γ m = = θ 0 = θ0nκ Tm ωr dephasing rate: quantum limit?: P Γ φ = θ 0 = θ0nκ ωr Tm Γ = 1 φ (expt. still ~ 40 times worse) x limit, since half of information wasted in reflected beam
Microwave Setup for cqed Experiment Transmit-side Receive-side n ~ 1 100 typical input power ~ 10-17 Watts n det ~40 3
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 noise 4
Observing Backaction of Measurement 1 g 1 Heff ωra a ωa a a σz + + n fluctuations in photon number expt: Schuster et al., PRL 94, 1360 (005). 5
Cavity QED - SET Analogy e - C g C ge V ds V ge photon shot noise induces qubit dephasing shot noise of SET current causes backaction 6
Summary of Lecture 3 Quantum limit on measurement comes from Two equivalent manifestations of quantum limit: Min. noise temperature T N aa =, 1 ω Meas. induced 1 TmΓ k φ dephasing 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, 7
Equivalent Circuit of an Amplifier SV ( ω) S ( ω) I SV ( ω) ficticious noise source (V /Hz) = output noise referred to input S ( ω) I a real noise (A /Hz) driven thru input terminals here assume uncorrelated, though typically not! Noise and Quantum Measurement R. Schoelkopf 8
Noise Temperature of an Amplifier Def n (IEEE) : temperature of a load @ input which doubles the system s output noise (assumes Rayleigh-Jeans) V sig (ω) S V S I total noise at input: equate to Johnson noise of source: for Z = R R = Z in in s s Z Z S = S + S Z Z tot in S V V I in + S tot S = 4kT Re[ Z ] V N s = ( + ) T S / R S R /4k N V S I S T N depends on source impedance Noise and Quantum Measurement R. Schoelkopf 9
Optimum Noise Temperature of Amplifier ( ) T = S / R + S R /4k N V S I S log T N R opt = S T = S S /k N V I opt V / S I opt opt log R source E = kt = S S N N V I E N is energy of signal that can be detected with SNR = 1 QM imposes minimum: EN Noise and Quantum Measurement R. Schoelkopf ω / / 30
Noise of a Single Electron Transistor n charge advance, k ( ) k = N + N I ds 1 / dk = e dt island charge, n n= N1 N Ideally, SET has only shot noise (T=0, ω<v/er) Fluctuations of k limit msmt. of response (I ds ) Fluctuations of n cause island potential to change current flows thru gate @ ω 0 Noise and Quantum Measurement R. Schoelkopf 31
Properties of an SET Amplifier V sig (ω) S V S I In limit of: normal state, T=0, no cotunneling, S V ( ω ) ( )( 1 α 1+ α ) ev R C Σ = ds Σ 8α C g ( 1 α ) indep. of ω ω << V/eR ds g Σ S ( ω ) = ~ ω M. Devoret and RS (000), similar results by Schon et al, Averin, Korotkov I 4 ev R Σ C C Σ eωr Vds ( C ) gvg e CΣVds α = / Noise and Quantum Measurement R. Schoelkopf 3
E N ( ω ) Noise Energy of SET Sequential Tunneling: (e.g. Devoret & RS, 000) E N = S V < ω Cotunneling limit: S I E π = ( 4 ) ( 1 α 1 + α ) α R opt 1 ωc g R R 10 (e.g. Averin, Korotkov) ω / Resonant Cooper-pair tunneling (DJQP): N E N ω / Σ K 8 ω Ω (e.g. Clerk) at 16 MHz Experimentally: still factor of 10-100 from intrinsic shot noise limit Noise and Quantum Measurement R. Schoelkopf 33
Other Amplifiers Near Quantum Limit Josephson parametric amplifier at 19 GHz T N = 0.45K ~ hν/k Yurke et al.; 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, 487 (1990) Microwave SQUID amplifier at 500 MHz T N = 50 mk ~ hν/k Muck, Kycia, and Clarke, APL 78, 967 (001) No measurement of crossover, or backaction yet. Noise and Quantum Measurement R. Schoelkopf 34
NEMS Oscillator Measured by SET Schwab group 35
Sample Beam Silicon Nitride 8µm X 00nm X 100nm f O = 19.7MHz Q ~ 30-45000 Gate Single Electron Transistor Al/Al x O y /Al Junctions K Charging Energy 70kΩ Resistance 70 MHz Bandwidth Beam/SET Separation: 600nm 7aF Capacitance 36
Resonator Response Phase/π (rad) 0.50 0.5 0.00-0.5 T= 30mk V g = 10V Q = 54,000 10 8 6 4 Amplitude (me) Power (µe /Hz) 500 450 400 350 300 50 00 1 mωo x = 1 k B T T=100mK V g = 10V Q=36,000-0.50 19.674 19.675 19.676 0 150 Frequency (MHz) 100 Driven Response 19.668 19.670 19.67 19.674 19.676 19.678 19.680 Frequency (MHz) Thermal Response 37
Noise Power vs. Temperature Saturates Below 100mK N th = 58 Use Linear Data to Calibrate Below 100mK Lowest Mode Temp Measured: T=56mk 38
Noise Temperature Noise Temperature V g =15V T N = 15.6mK T=100mK Energy Sensitivity E N 17 ћω 0 T N = 15.6mK Position Sensitivity x = 4.3 x QL Closest approach yet to uncertainty principle limit! 39
How far can we push this technique? 1000 Preamp noise floor S qq =10µe/ Hz Induced Charge: δq = V g δc g = (C g V g /d) δx X (fm) 100 10 X/ X QL Charge Sensitivity (forward coupling): S x 1/ = S qq 1/ d/(c g V g ) Back-Action: S x 1/ = S vv 1/ C g V g Q/(kd) 0 1 3 10 30 V Beam (Volts) Ideal Shot Noise Limit 1 Back-Action 40
Circuit Model BEAM SET I ds (q) Total Noise Power = Gain x [S qq + S thermal + S vv / ωz in (ω) ] L m S vv S qq C m C g 4 R m 4k B TR m R j / C j Output Noise 10-9 e /Hz 3 1 C m = C g (C g V g /kd ) = 0.06 af @ V g =10V L = 1/(C m g ω )(kd /C g V g ) = 4500 H R m = 1/(QC g ω)(kd /C g V g ) =.8 MΩ 0 9.366 9.367 9.368 9.369 9.370 9.371 Frequency (MHz) 41
Sensitivity Optimization 10000 R j = 75KΩ C j =1.3fF K=1.7 N/m Q=1.5x10 5 S qq =.µe/hz 1/ (shot noise) S qq =100µe/Hz 1/ (preamp) S vv =1nV/Hz 1/ Position Sensitivity (fm/hz 1/ ) 1000 100 Shot Noise Limit Sqq =100µe/Hz1/ Back-Action Standard Quantum Limit (S qq S vv ) 1/ 3h 10 0.01 0.1 1 10 50 V g (Volt) R m =6. MΩ/V g Loading: ω = ω 0 (1- (C g V g /kd ) (C g /C j )) 0.5 Q eff -1 = Q -1 + (C g V g /kd )(C g /C j )ω 0 (R j C j ) R optimum = (S vv /S qq ) 1/ / ω = 47 MΩ 4
Atomic Point Contact Displacement Detector: Lehnert group at JILA/CU as in an STM Infer postion from tunnel current Sensitive: λe 15 m δx 1. 10 with 1 na current N / τ Hz e Local: ideal for sub-micron objects 43
Atomic Point Contact Displacement Detector: Simple Noise Analysis Imprecision (shot noise limit) x =λe x =λ e e 1 I τ ( 1 N ) e 1/ 1/ Backaction (momentum diffusion) I p = τ λe e p= ( Ne ) λ e 1/ 1/ Tunneling length scale Counting statistics Momentum per tunneling attempt Ideal quantum displacement amplifier B. Yurke PRL 1990, A. A. Clerk PRB 004 x p= Number diffusion 44
Thermal Motion at 43 MHz Resonanace Zero-point motion: δx T δx T δ x = ZP δ x = ZP δx δx I ZP ω0 kb s 100 am/hz = 8 w 1/ δx I Mechanical bandwidth Bw 9 khz ; Q 5000 Sensitivity to normal coordinate 45