Overview of LecturesOutline of Lecture 2The Electrical Engineer’s Spectrum AnalyzersThe Quantum Mechanic’s Analyzer – a SpinSpin as Spectrum Analyzer - IIInterpretation of Two-Sided SpectrumPolarization of Spin and Noise SpectraPolarization of Spin - IIWays to Characterize a Quantum ReservoirCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper Pair Box as Two-Level SystemCooper-pair Box Coupled to an SETSpectroscopyQubit Coupled to Ohmic EnvironmentCharging Diagram of SET ElectrometerQuantum Shot Noise of DJQP* ProcessPredicted Effects of DJQP on Box ChargeSET Determines Relaxation Time (T1)What About Asymmetry in SET Noise?Population InversionInversion: Theory and ExperimentOther Quantum Spectrometers – Delft GroupSIS detection principleQuasi-particle shot noiseSingle Quantum Dot as Noise DetectorSingle Quantum Dot as Noise DetectorDouble Quantum Dot DetectionQuantum Spectrometers of Electrical NoiseRob SchoelkopfApplied PhysicsYale UniversityGurus:Michel Devoret, Steve Girvin, Aash ClerkAnd 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 MeasurementR. Schoelkopf1Overview of LecturesLecture 1: Equilibrium and Non-equilibrium Quantum Noisein CircuitsReference: “Quantum Fluctuations in Electrical Circuits,”M. Devoret Les Houches notesLecture 2: Quantum Spectrometers of Electrical NoiseReference: “Qubits as Spectrometers of Quantum Noise,”R. Schoelkopf et al., cond-mat/0210247Lecture 3: Quantum Limits on MeasurementReferences: “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, & RSNoise and Quantum MeasurementR. Schoelkopf2Outline of Lecture 2• A spin or two-level system (TLS) as spectrometer• The meaning of a two-sided spectral density• The Cooper-pair box (CPB): an electrical TLS• Using the CPB to analyze quantum noise of an SETMajer and Turek, unpub.• Other quantum analyzers:• SIS junction: a continuum edge (DeBlock, Onac, & Kouwenhoven)• Nanomechanical system: a harmonic oscillator(Schwab et al.; Lehnert et al.)Noise and Quantum MeasurementR. Schoelkopf3The Electrical Engineer’s Spectrum AnalyzersFFT()VSωω0ω=()Vt()VSωω0ω=tunedfilter()VtBoth measure only the symmetrized spectral density:( ) () (0) (0) ()SitVVS dte VtV V Vtωω∞−∞=+∫() ()VV VVSSωω=++ −Noise and Quantum MeasurementR. Schoelkopf4The Quantum Mechanic’s Analyzer – a SpinNoise and Quantum MeasurementR. Schoelkopf5tomagnetometer0ˆBz()xBt()Vt=Spins only absorb at Larmor frequency01 0/gBωωµ== Resolution = inverse of spin coherence timeAble to measure both sides of a spectral density!Spin as Spectrum Analyzer - II0ˆBz()xBt()Vt0102zHωσ=−1()xHAV tσ=()()gettαψα⎛⎞=⎜⎟⎝⎠10() (0) () (0)titdHtψψ τψ=−∫with initial condition:(0) gψ=0100() () () ()ttiexiA iAtdegV deVωτατστ τ τ τ=− =−∫∫01 1 222()12 1 2 012200() ( ) ( ) ( )ttie VAApt d d e V V t Sωττττττω−−==−∫∫()0122VgeSAω→Γ−=()0122VegSAω→Γ+=Noise and Quantum MeasurementR. Schoelkopf6Interpretation of Two-Sided Spectrum0T≠0T=01ω+01ω−absorption by spinemission by spin↑Γ↓Γge01ω()/2R1VkTSeωωω−=−kTω=()VSω0ωabsorption by sourceemission by source()01VSω↓Γ+∝()01VSω↑Γ−∝Noise and Quantum MeasurementR. Schoelkopf7Polarization of Spin and Noise Spectra↑Γ↓Γge01ωegegegdpppdtdpppdt↑↓↓↑=Γ− Γ=Γ− Γgepp↑↓Γ=ΓSteady-state:gePpp=−Define polarization of spin:If noise is truly classical, gepp≡and no polarization!01//kTegpp eω−=Thermal equilibrium:01/01 01() ()kTVVSeSωωω+=−Requires particular asymmetry!Noise and Quantum MeasurementR. Schoelkopf8Polarization of Spin - IIFor general case, define steady-state polarization()( )()()01 0101 01SSSSPSSωωωω↓↑↓↑Γ−Γ+−−==Γ+Γ + + −due to relative asymmetry of noise()()1()() ()dPtPt Ptdt↓↑∆=−∆ Γ + Γ = −∆ Γ()()[]210101211 ASSTωωΓ= = + + −() ()SSPt Pt P∆= −Define deviation from steady state:So relaxation rate (Γ1) due to total noise (and coupling)Noise and Quantum MeasurementR. Schoelkopf9Ways to Characterize a Quantum ReservoirFermi’s Golden RuleFluctuation-Dissipation RelationNMRQuantum Optics()2VASω↑⎛⎞Γ=−⎜⎟⎝⎠()2VASω↓⎛⎞Γ=+⎜⎟⎝⎠2()nγ↑↓↑Γ=Γ−Γ[]2()Re ( )ZAωω↓↑Γ−Γ=P↓↑↓↑Γ−Γ=Γ+Γ()11T−↑↓=Γ+ΓEinsteinA↓↑=Γ−ΓEinsteinB↑=Γ10Cooper Pair Box as Two-Level System11/gggnCV e=2~5 2( )jcgEeGHzCC=+4( /)ˆcgel gEnCV eE=−CgBox Vg2ˆqen=ˆˆ2box xelEHσ=CjVg10n=1n=4CEEnergy(Buttiker ’87; Bouchiat et al., 98)Cooper Pair Box as Two-Level Systemˆˆˆ22el Jbox x zHEEσσ=−2~5 2( )jcgEeGHzCC=+4( /)ˆcgel gEnCV eE=−12/gggnCV e=CgBox Vg2ˆqen=ˆˆ2box xelEHσ=Cj25 GHz4JJeREπ∆=≈Vg4CE1EJ01−01+Energy(Buttiker ’87; Bouchiat et al., 98)Cooper Pair Box as Two-Level Systemˆˆˆ22el Jbox x zHEEσσ=−2~5 2( )jcgEeGHzCC=+4( /)ˆcgel gEnCV eE=−13/gggnCV e=CgBox Vg2ˆqen=ˆˆ2box xelEHσ=Cj25 GHz4JJeREπ∆=≈Vg1EJ4CE↓↑Energy(Buttiker ’87; Bouchiat et al., 98)Cooper Pair Box as Two-Level Systemˆˆˆ22el Jbox x zHEEσσ=−2~5 2( )jcgEeGHzCC=+4( /)ˆcgel gEnCV eE=−14/gggnCV e=CgBox Vg2ˆqen=ˆˆ2box xelEHσ=Cj25 GHz4JJeREπ∆=≈Vg1EJ4CE↓↑Energy(Buttiker ’87; Bouchiat et al., 98)Noise and Quantum MeasurementR. Schoelkopf15Effects of Voltage Noise on Pseudo-SpinθsinelBE⊥=δµδ θcoselBEδµδ θ=slow fluctuations of Bdephasingresonant fluctuations ofB⊥transitions()22011sinboxmix VmixeSTωθ⎛⎞=Γ =⎜⎟⎝⎠01 effBωµ=()2210cosboxVeSTϕϕωθ⎛⎞=Γ = →⎜⎟⎝⎠2el boxeEV=δδeffBˆz2JEθ01sinJEθω=2elEˆxSpontaneous Emission of Cooper-pair Box1xHAVσ=Noise and Quantum MeasurementR. Schoelkopf16Box 50 envR ≈ΩgC()12sin/singCgxggxCHE VeeC C Vθδ σθδσΣ==Vg01()0envVSω−=0T =01 01()2 (50 )envVSωω+=×ΩExcited-statelifetime, T1()2201121sinenvVeSTκωθ↓=Γ = +⎛⎞⎜⎟⎝⎠10.1 1 sTµ≈−/12%gCCκΣ= ∼estimate:Polarization = 100%Cooper-pair Box at Finite Temperature0T >Noise and Quantum