Quantum Spectrometers of Electrical Noise

Transcription

1 Quantum Spectrometers of Electrical Noise 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

2 Overview of Lectures Lecture : 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/0047 Lecture 3: Quantum Limits on Measurement References: Amplifyin Quantum Sinals with the Sinle-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 upcomin RMP by Clerk, Girvin, Devoret, & RS Noise and Quantum Measurement R. Schoelkopf

3 Outline of Lecture A spin or two-level system (TLS) as spectrometer The meanin of a two-sided spectral density The Cooper-pair box (CPB): an electrical TLS Usin the CPB to analyze quantum noise of an SET Majer and Turek, unpub. Other quantum analyzers: SIS junction: a continuum ede (DeBlock, Onac, & Kouwenhoven) Nanomechanical system: a harmonic oscillator (Schwab et al.; Lehnert et al.) Noise and Quantum Measurement R. Schoelkopf 3

4 The Electrical Enineer s Spectrum Analyzers Vt () FFT SV ( ω) ω = 0 ω Vt () tuned filter SV ( ω) ω = 0 ω Both measure only the symmetrized spectral density: S iωt SVV ( ω) = dt e V() t V(0) + V(0) V() t = S ( + ω) + S ( ω) VV VV Noise and Quantum Measurement R. Schoelkopf 4

5 The Quantum Mechanic s Analyzer a Spin Vt () B () x t B zˆ 0 = to manetometer Spins only absorb at ω = ω = µ B / 0 0 Larmor frequency Resolution = inverse of spin coherence time Able to measure both sides of a spectral density! Noise and Quantum Measurement R. Schoelkopf 5

6 Vt () Spin as Spectrum Analyzer - II B () x t B zˆ 0 H 0 H ω = 0 σ z = AV t σ () x ψ α () t = αe() t with initial condition: ψ (0) = i ψ() t = ψ(0) dτ H() t ψ(0) t 0 t ia ia αe = τ σ τ τ = τ τ iω0τ ( t) d e ( ) V( ) d e V( ) x 0 0 t t iω0( τ τ ) A τ τ τ τ V ω0 0 0 A A e = S ( ω ) Γ ( ) 0 e S + ω0 A pe() t = d d e V( ) V( ) = t S ( ) Γ V = t V Noise and Quantum Measurement R. Schoelkopf 6

7 Interpretation of Two-Sided Spectrum Γ e Γ ω 0 SV ω R e ω S V ω kt ω ( ) ( ) = / T 0 T = 0 absorption by spin emission by source Γ ω 0 SV ( ω ) 0 ω = kt emission by spin absorption by source 0 +ω 0 ω Γ S + V ( ω ) 0 Noise and Quantum Measurement R. Schoelkopf 7

8 Polarization of Spin and Noise Spectra Γ e Γ ω 0 Steady-state: dpe = pγ peγ dt dp = peγ p Γ dt p Γ = p Γ e Define polarization of spin: P = p p Thermal equilibrium: Requires particular asymmetry! e If noise is truly classical, p p p / p = e ω e V e and no polarization! 0 / kt S ω e S ω ω0 ( ) / kt + = ( ) 0 V 0 Noise and Quantum Measurement R. Schoelkopf 8

9 Polarization of Spin - II Define steady-state polarization: P SS d ( ω0) S( ω0) ( ω ) S( ω ) Γ Γ S + = = Γ +Γ S due to relative asymmetry of noise Define deviation from steady state: ( P() t ) dt ( ) Pt = Pt () Γ + Γ = () Γ A S 0 S 0 T ( ω ) ( ω ) [ ] Γ = = + + Pt () = Pt () PSS So relaxation rate (Γ ) due to total noise (and couplin) Noise and Quantum Measurement R. Schoelkopf 9

10 Ways to Characterize a Quantum Reservoir Fermi s Golden Rule Fluctuation- Dissipation Relation NMR Harmonic Oscillator Quantum Optics A Γ = S V ( ω ) E n Γ A Γ = S V ( + ω ) = Re ( Γ ) Γ [ Z( ω) ] A Γ Γ P ( Γ Γ ) = ω = = ( Γ +Γ ) Γ +Γ = ( Γ +Γ ) ω ( Γ Γ ) BEinstein = Γ T ω = = ( Γ Γ ) Q γ AEinstein = Γ Γ 0

11 V Cooper Pair Box as Two-Level System C C j q = enˆ ˆ ˆ Box H E = 4 E ( nˆ C V / e) el box = E el c σ x E c = e ( C + C ) j ~5 GHz n = V Enery n = 0 4E C (Buttiker 87; Bouchiat et al., 98) n = C V e /

12 V Cooper Pair Box as Two-Level System C C j q = enˆ J ˆ E E H ˆ ˆ box = el σ x σ z Box E = 4 E ( nˆ C V / e) el c E c = e ( C + C ) j ~5 GHz E J π = 5 GHz er 4 J 0 V Enery 0 + E J 4E C (Buttiker 87; Bouchiat et al., 98) n C V e = /

13 V Cooper Pair Box as Two-Level System C C j q = enˆ J ˆ E E H ˆ ˆ box = el σ x σ z Box E = 4 E ( nˆ C V / e) el c E c = e ( C + C ) j ~5 GHz E J π = 5 GHz er 4 J V Enery E J 4E C (Buttiker 87; Bouchiat et al., 98) n C V e = / 3

14 V Cooper Pair Box as Two-Level System C C j q = enˆ J ˆ E E H ˆ ˆ box = el σ x σ z Box E = 4 E ( nˆ C V / e) el c E c = e ( C + C ) j ~5 GHz E J π = 5 GHz er 4 J V Enery E J 4E C (Buttiker 87; Bouchiat et al., 98) n C V e = / 4

15 Cooper-pair Box Coupled to an SET Box SET Electrometer V C C c C e V ds Box V e SET Superconductin tunnel junction Cooper-pair Box Qubit Quantum spectrum analyzer or Noise and Quantum Measurement R. Schoelkopf SET Transistor Quantum state readout Nonequilibrium noise source 5

16 B eff a b c What Does SET Measure? ˆ el Measure box chare ˆ J H = E σ ˆ x E σ z n = ( σ / ) x + E el = 4 Ec( nˆ CV / e) Eel ẑ n a b c E J 0 CV ˆx 0 e B eff E J E J E el B eff E 0 a b c Ground state Excited state 0 C V e 6

17 Spectroscopy of Box Enery 0 with microwave 0Chare 0 Box Gate Chare (e) SV ( ω) peaks saturate ω to q=e Spectrum of oscillator Noise and Quantum Measurement R. Schoelkopf 7

18 Effects of Voltae Noise on Pseudo-Spin E J ẑ B eff θ E el ω = µb 0 eff ˆx δµ B = δe el cosθ δ E el = eδ V θ slow fluctuations of B T ϕ box e =Γ ϕ = SV box ω δµ B = δe el sinθ ( ) dephasin 0cos θ EJ sinθ = ω 0 resonant fluctuations of B T mix e =Γ = Noise and Quantum Measurement R. Schoelkopf S mix V box ω ( ) 0 transitions sin θ 8

19 Spontaneous Emission of Cooper-pair Box T = 0 R 50 Ω env V C Excited-state Box SV env H = AVσ x C H = E sinθδv e C x ( ) Σ σ = ec / C sinθ δv ( ω ) = 0 0 σ x S ( + ω ) = ω (50 Ω) V env 0 0 =Γ = S ( ) V 0 lifetime, T env + κ T = C / C % estimate: Σ e sin κ ω θ T 0. µ s Polarization = 00% Noise and Quantum Measurement R. Schoelkopf 9

20 Cooper-pair Box at Finite Temperature T > 0 R 50 Ω env V C Box ( ) ( ) ( ω ) ( ω ) 0 0 SV env 0 0 ( ) S ( + ω ) = ω R n + V env ( ω ) = ω Rn 0 0 P S + ω0 S ω0 n + n 0 tanh ω = = = S + + S n + kt T e ( n ) + κ 0 + =Γ Γ = ω R sin θ Noise and Quantum Measurement R. Schoelkopf 0

21 Excited-state Lifetime Measurement of Box follow peak heiht after turnin off microwaves Peak heiht (e) 0.3e S S n n n T =.3 µ s P~ with continuous measurement 0 time 0 µs 9 ( + ω0) 5 0 pairs/ Hz ( ω ) ~0 (@ 76 GHz) box box 0 C V e K. Lehnert et al., PRL 90, 0700 (003).

22 Couplin of SET Backaction to Box R env Box e - C C c SET V e e Environment relaxes box SET couples backaction to box Chare fluctuations on SET island with S Q ( ± ω ) 0 Noise and Quantum Measurement R. Schoelkopf

23 Double JQP Process in the SSET 0 JQP DJQP Ec / e Reflected power from RF-SET ~ Conductance of SSET 0 Γ qp 3

24 Quantum Shot Noise of DJQP* Process *Double Josephson-quasiparticle cycle: (A. Clerk et al. PRL (00)) lo S V SET noise spectrum ω ( ) on resonance off resonance Predicted box chare ω = 0 n ω = Excitation of box ω = + Relaxation of box 0 C V e 4

25 SET Determines Relaxation Time (T ) H. Majer and B. Turek, unpublished Calculated symmetric noise at 30 GHz T ~ µs low hih T < 00 ns Theory of quantum noise for DJQP A. Clerk et al., PRL (00) 5

26 What About Asymmetry in SET Noise? Measured reflected power Calculated 5 GHz asymmetric noise S Q ( + ω) S ( ω) Q Red: excites Blue: relaxes Γ qp Below resonance SET relaxes box 0 above resonance SET excites 6

27 Population Inversion Calculated asymmetric noise at 5 GHz Measured box chare neative SET excites Qubit positive SET Gate Chare (e) SET relaxes Qubit SET creates an neative effective spin temperature 7

28 Inversion: Theory and Experiment Numerical calculation includin stron couplin to SET + environmental relaxation (A. Clerk) 8

29 Other Quantum Spectrometers Delft Group Equivalent AC circuit Deblock, Onac, Gurevich, and Kouwenhoven, Science 30, 03 (003) 9

30 SIS detection principle Voltae biased SIS junction V SIS Hih-frequency detection based on Photon Assisted Tunnelin superconductor hν superconductor insulator Numerical simulations Density of States 30

31 Quasi-particle shot noise White noise fit S I as fit parameter Only emission part: S I (ω)=ei Resolution: 80 fa /Hz (3mK on a kω resistor) 3

32 Summary of Lecture Positive frequency noise = reservoir absorbs Neative frequency noise = reservoir emits A quantum system can act as a quantum spectrometer, able to measure both positive and neative frequency components of a non-classical noise source. Usin CPB (= electrical TLS) as quantum spectrometer Observed quantum noise (at > GHz) of SET backaction - SET controllin relaxation time of box - SET can create population inversion in box due to asymmetry of noise Noise and Quantum Measurement R. Schoelkopf 3

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34 Sinle Quantum Dot as Noise Detector Device picture Quantum Dot in CB reime: noise detector Quantum Point Contact: noise source Inverse picture: QPC as a chare detector Detector back-action on the studied quantum system 34 J.M.Elzerman et.al, Nature 430, 43 (004)

35 Sinle Quantum Dot as Noise Detector Transport throuh orbital states QPC transmission dependence κ δ δ κ ev dot V QPC =.7 mev V dot = 30 µev Temperature=00 mk δ = 45 mev ~ 60 GHz δ = 580 mev ~ 40 GHz Γ s = GHz Set of fittin parameters Γ es = 5.75 GHz Γ es = 4.05 GHz κ = κ = κ = κ =

36 Double Quantum Dot Detection Non-coherent limit ε»t C ; Γ i «Γ L, Γ R I inel = e/ħ (Γ L - + Γ i - + Γ R - ) - e/ħ Γ i - P(ε) probability for enery exchane with the enviroment Circuit transimpedance S V (ω) = Z(ω) S I (ω) Z(ω) = κ R K R. Auado and L.P. Kouwenhoven, PRL 84, 986 (000) 36