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Quantum Limits on Measurement 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 ()∆=xbout b+bb={}a+c,a+c 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 ()∆=xbout {},cb={}a+coshθθ+dsinh,h.c. 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.
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