# Quantum Spectrometers of Electrical Noise

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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 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 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 R Schoelkopf 2 Outline 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 SET Majer 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 Measurement R Schoelkopf 3 The Electrical Engineer s Spectrum Analyzers SV V t FFT tuned filter V t 0 SV 0 Both measure only the symmetrized spectral density S dt ei t V t V 0 V 0 V t S VV SVV SVV Noise and Quantum Measurement R Schoelkopf 4 The Quantum Mechanic s Analyzer a Spin B0 z V t Bx t to magnetometer Spins only absorb at 01 g B0 Larmor frequency Resolution inverse of spin coherence time Able to measure both sides of a spectral density Noise and Quantum Measurement R Schoelkopf 5 Spin as Spectrum Analyzer II Bx t V t B0 z H0 01 g t t e z 2 H1 AV t x with initial condition 0 g iA e t t 0 t d e x g V pe t 2 t d H t 0 1 0 0 A2 t i iA t i 01 d e V 0 t i 01 1 2 d d e V 1 V 2 t 1 2 0 g e A 0 2 2 SV 01 e g Noise and Quantum Measurement R Schoelkopf A2 2 A2 2 SV 01 SV 01 6 Interpretation of Two Sided Spectrum 2 R SV 1 e kT e 01 SV g absorption by spin emission by source 01 kT 0 SV 01 T 0 T 0 emission by spin absorption by source 01 SV 01 Noise and Quantum Measurement R Schoelkopf 7 Polarization of Spin and Noise Spectra e 01 g Steady state Define polarization of spin P pg pe Thermal equilibrium dpe pg pe dt dpg pe pg dt pg pe If noise is truly classical pg pe and no polarization pe pg e 01 kT Requires particular asymmetry SV 01 e Noise and Quantum Measurement R Schoelkopf 01 kT SV 01 8 Polarization of Spin II For general case define steady state polarization S 01 S 01 PSS S 01 S 01 due to relative asymmetry of noise Define deviation from steady state P t P t PSS d P t P t P t 1 dt 2 1 A 1 2 S 01 S 01 T1 So relaxation rate 1 due to total noise and coupling Noise and Quantum Measurement R Schoelkopf 9 Ways to Characterize a Quantum Reservoir Fermi s Golden Rule FluctuationDissipation Relation NMR Quantum Optics 2 A SV 2 n T1 BEinstein 2 A SV Re Z A2 1 P AEinstein 10 Cooper Pair Box as Two Level System Eel q 2en x H box 2 Cg Box Vg Eel 4 Ec n C gVg e Cj e2 Ec 5 GHz 2 C g C j Vg Energy n 1 Buttiker 87 Bouchiat et al 98 n 0 4 EC 1 ng C11gVg e Cooper Pair Box as Two Level System EJ Eel q 2en H box x z 2 Cg Box Vg 2 Eel 4 Ec n C gVg e Cj e2 Ec 5 GHz 2 C g C j EJ 2 4 e RJ 5 GHz Vg Energy 0 1 Buttiker 87 Bouchiat et al 98 EJ 4 EC 0 1 1 ng C12gVg e Cooper Pair Box as Two Level System EJ Eel q 2en H box x z 2 Cg Box Vg Eel 4 Ec n C gVg e Cj Energy e2 Ec 5 GHz 2 C g C j Vg 2 Buttiker 87 Bouchiat et al 98 EJ 2 4 e RJ 5 GHz EJ 4 EC 1 ng C13gVg e Cooper Pair Box as Two Level System EJ Eel q 2en H box x z 2 Cg Box Vg Eel 4 Ec n C gVg e Cj Energy e2 Ec 5 GHz 2 C g C j Vg 2 Buttiker 87 Bouchiat et al 98 EJ 2 4 e RJ 5 GHz EJ 4 EC 1 ng C14gVg e Effects of Voltage Noise on Pseudo Spin z B eff EJ 2 Eel 2e Vbox Eel 2 01 Beff sin B Eel sin B Eel cos EJ 01 x slow fluctuations of B dephasing 2 1 e SVbox 0 cos 2 T resonant fluctuations of B transitions 2 1 e mix SVbox 01 sin 2 Tmix Noise and Quantum Measurement R Schoelkopf 15 Spontaneous Emission of Cooper pair Box Renv 50 Cg Box Vg SVenv 01 0 T 0 Excited state lifetime T1 H1 AV x Cg H1 2 EC sin Vg x e e C g C sin Vg x SVenv 01 2 1 T1 2 e 01 50 2 SVenv 01 sin 2 C g C 12 estimate T1 0 1 1 s Polarization 100 Noise and Quantum Measurement R Schoelkopf 16 Cooper pair Box at Finite Temperature T 0 Renv 50 Cg Box Vg SVenv 01 2 SVenv 01 2 01R n 01R n 1 S 01 S 01 n 1 n 01 tanh P 2 n 1 S 01 S 01 2kT 1 T1 2 e 2 2 01R sin 2 2 n 1 Noise and Quantum Measurement R Schoelkopf 17 Cooper pair Box Coupled to an SET Box SET Electrometer Cg Vg Box Cc Cge Vds Vge SET Superconducting tunnel junction Cooper pair Box SET Transistor Qubit Quantum state readout or Nonequilibrium Quantum spectrum analyzer Noise and Quantum Measurementnoise source R Schoelkopf 18 Cooper pair Box as Quasi spin 1 2 Measure charge E el EJ H z x 2 2 C gVg 1 Eel 4 Ec 2 2e a z Beff z b Eel 2 EJ 2 z Beff x EJ 2 n Z 1 2 Ground state 1 n c EJ 2 x Beff a b Excited state 0 E c C gVg 1 0 5 2e 0 x Eel 2 a 0 b c 1 C gVg 2e 19 Energy Spectroscopy 0 1 2 0 1 Box Gate Charge e 2 Charge 2 with microwave 0 SV Spectrum of …