Shot Noise and the Non-Equilibrium FDT 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, 1
Outline Shot noise is quantum noise Shot noise of a tunnel junction Measurements of shot noise testing the non-eq. FDT Quantum shot noise measuring the frequency dependence of shot noise Experiments on the zero point noise in circuits Shot noise and the nonequilibrium FDT (time permitting) 2
Fundamental Noise Sources Johnson-Nyquist Noise S I 4 k BT ( f ) = R 2 A Hz Frequency-independent Temperature-dependent Used for thermometry Shot Noise 2 S ( f) = 2eI I A Hz Frequency-independent Temperature independent 3
Shot Noise Classically what s up here? n D I qdn Incident current of particles Barrier w/ finite trans. probability Poisson-distributed fluctuations white noise with SI = 2qI 4
Shot Noise is Quantum Noise Einstein, 1909: Energy fluctuations of thermal radiation Zur gegenwartigen Stand des Strahlungsproblems, Phys. Zs. 10 185 (1909) 2 3 2 π c 2 ( E) = ωρ( ω) + ρ ( ω) Vdω 2 ω particle term = shot noise! wave term first appearance of wave-particle complementarity? Can show that particle term is a consequence of (see Milloni, The Quantum Vacuum, Academic Press, 1994) 2 n = a aa a a a = + a ( a a 1) a a a = aaaa + aa aa 2 2 n = n + n 2 2 2 aa =, 1 ( / kt 1) 1 n n e ω = = + ( ) 1 n P = n / n + 1 n n aaaa = nn ( 1) P= 2n 2 n 5
Conduction in Tunnel Junctions I V Fermi functions Assume: Tunneling amplitudes and D.O.S. independent of energy Fermi distribution of electrons G I L R = fl(1 fr) de e G I R L = fr(1 fl) de e Difference gives current: I = I I = GV L R R L Conductance (G) is constant 6
Non-Equilibrium Noise of a Tunnel Junction (Zero-frequency limit) Sum gives noise: S ( f) = 2 e( I + I ) I L R R L ev SI ( f) = 2eIcoth 2 kt B I = V / R *D. Rogovin and D.J. Scalpino, Ann Phys. 86,1 (1974) 7
Non-Equilibrium Fluctuation Dissipation Theorem 2eI Shot Noise Transition Region ev~k B T 4k B TJohnson Noise R ev SI ( f) = 2 ev / Rcoth 2 kt B 8
Noise Measurement of a Tunnel Junction P SEM 5µ Al-Al 2 O 3 -Al Junction Measure symmetrized noise spectrum at ω < kt 9
Seeing is Believing δ P P = 1 B τ High bandwidth measurements of noise B 8 ~10 z, H τ = 1 second δ P P = 10 4 10
Test of Nonequilibrium FDT Agreement over four decades in temperature To 4 digits of precision L. Spietz et al., Science 300, 1929 (2003) 11
Self-Calibration Technique P(V) = Gain( S I Amp +S I (V,T) ) P(V) 2eI G 4kT B Amplifier + SI R 2 kt B / e V 12
Comparison to Secondary Thermometers 13
Two-sided Shot Noise Spectrum S (Quantum, non-equilibrium FDT) ω + ev /R ω ev /R ( ) ( ω ) ( ) ( ) I ω = + + ev / kt ev / kt 1 e 1 e SI ω ( ) ( ω ) V ev / 2 ω /R ev / ei ω = 0 T = 0 ω Aguado & Kouwenhoven, PRL 84, 1986 (2000). 14
Finite Frequency Shot Noise Symmetrized Noise: S = S( + ω) + S( ω) sym don t add powers! Shot noise Quantum noise 15
Measurement of Shot Noise Spectrum Theory Expt. Schoelkopf et al., PRL 78, 3370 (1998) 16
Shot Noise at 10 mk and 450 MHz hν / kt = 2 L. Spietz, in prep. 17
With An Ideal Amplifier and T=0 S I 2eI SI = 2hν G Quantum noise from source SI = 2hν G Quantum noise added by amplifier ev=-hν ev=hν V 18
Summary Lecture 1 Quantizing an oscillator leads to quantum fluctuations present even at zero temperature. This noise has built in correlations that make it very different from any type of classical fluctuations, and these cannot be represented by a traditional spectral density- requires a two-sided spectral density. Quantum systems coupled to a non-classical noise source can distinguish classical and quantum noise, and allow us to measure the full density next lecture! 19
Additional material on Johnson noise follows 20
Nyquist s Derivation of Johnson s Noise 21
Connection Between Johnson Noise and Blackbody Radiation* *R. H. Dicke, Rev. of Sci. Instrum. 17, 268 (1946) Coaxial-Line Antenna Resistor Temperature T R Enclosure Walls Black Temperature T bb T R =T bb Johnson Noise Power Blackbody Radiation Power 22
Connection Between Johnson Noise and Blackbody Radiation in Rayleigh-Jeans Limit Coaxial-Line Antenna Resistor Temperature T R S T h ν /k I R B Enclosure Walls Black = 4 k T/ R Temperature T bb B P k T B Johnson B R Rayleigh-Jeans limit P k T B bb-rayleigh-jeans B bb 23
Radiative Cooling of a Resistor? T > 0 P = ktb T=0 kt Total radiated power: ktbmax = kt h 2 k Total conductance: Gphoton = dp/ dt = T h More correct: hν P = hn ν ( ν) dν = dν h / kt e ν 1 0 0 One quantum of thermal conductance per electromagnetic mode Schmidt, Cleland, and Schoelkopf, PRL 93, 045901 (2004) 24
Resistor as Ideal Square Law Detector E γ ~ / H n E kt γ γ T=0 C e Single photon heats one resistor to T H = E/ C e If no other thermal conductances, cools entirely by radiation! Photon number gain is large!: Where s the nonlinearity? n ~ E / kt γ γ H 25