Lectures II, III: Quantum Noise Leonid Levitov (MIT) Boulder 2005 Summer School |Background on electron shot noise exp/th | Scattering matrix approach (intro) | Noise in mesoscopic systems: current partition, binomial statistics Counting statistics: • | Generating function for counting statistics | Passive current detector; Keldysh partition function representation Tunneling • | Odd vs even moments, nonequilibrium FDT theorem | Third moment S3 Driven many-body systems • | Many particle problem one particle problem (the determinant formula) | Coherent electron pumping! | Phase-sensitive noise, `Mach-Zender effect’, orthogonality catastrophe | Coherent many-body states | noise-minimizing current pulses Noise Introduction Fluctuating current I(t) Correlation function G2(τ) = I(t)I(t + τ) (time average, stationary flow) Temporal correlations due to quantum statistics and/or source Electrons counted in a flow, integrity preserved, without being pulled out of a many-body system, no single-electron resolution yet | ensemble average picture (but, turnstiles) i!τ Noise spectrum S(!) = 1 e− S(τ)dτ, −∞ 2 where S(τ) = I(t)I(t + τ) = I(t)I(t + τ) I hh R ii − L Levitov, Boulder 2005 Summer School Quantum Noise 1 Electron transport Coherent elastic scattering (mesoscopic systems, point contacts, etc.) | scattering matrix approach Interactions (nanotubes, quantum wires, QHE edge states) | Luttinger liquid theory, QHE fractional charge theories Quantum systems driven out of equilibrium (quantum dots, pumps, turnstiles, qubits) Current autocorrelation function: 1 G (τ) = (I(t)I(t + τ) + I(t + τ)I(t)) 2 h2 i (Note: no normal ordering, electrons counted without being destroyed) If you're an electron, L Levitov, Boulder 2005 Summer School Quantum Noise 2 Mesoscopic transport (crash course) A quintessential example (point contact): 1d single channel QM scattering ~2 on a barrier, 00 + U(x) = ψ. Scattering states in asymptotic form: −2m Functions u (x) Functions v (x) ikx k k e e−ikx 1/2 ikx t e 1/2 −ikx 1/2 −ikx t e ir e ir1/2 eikx ^ Express electric current through (x) = k a^kuk(x) + bkvk(x) : ^ e + i(k k0)x k+k0 + j(x) = 2m( (x)@x (x) + h:c:) = e k;k e − 2m (x) k(x) − P0 k0 k + k + P p 0 i(k k0)x ak t i rt ak ^j(x) = e e − +0 (x 0) 2m bk iprt r 1 bk Xk;k0 0 − − Time-averaged current (at eV EF only energies near EF contribute): + + dk j(x) = evF k t ak ak + (r 1) bk bk = evF t 2π~ [nL() nR()] = h i h i −2 h i − (et=h) [f( eV ) f()] d = e tV −P − h R R e2 Ohm's law: I = gV (IR = V ) with conductance g = 1=R = h t (Landauer) L Levitov, Boulder 2005 Summer School Quantum Noise 3 Multiterminal system, reservoirs, scattering states pt ipr Single channel S-matrix: S = (optical beam splitter) ipr pt Scattering manifest in transport | quantization of g in point contacts of adjustable width (many parallel channels which open one by one as a function of Vgate) 2e2 g = h n tn ConductancePquantum: 2 1 2e =h = 1=13 kΩ− adapted from van Wees et al. (1991) L Levitov, Boulder 2005 Summer School Quantum Noise 4 Electron beam partitioning e e e e e Incident electrons transmitted/reflected with probabilities t, r = 1 t; − At T = 0, bias voltage eV = µ µ , transport only at µ < < µ : L − L R L (i) Fully filled Fermi at < µL; µR; (ii) All empty states at > µL; µR. 2 µL d e Mean current I = 2eNL R = e t ~ = ~ tV (two spin channels) ! µR 2π Noiseless source (zero temperature)R | binomial statistics with the number of attempts during time τ: N = (µ µ )τ=2π~ = eV τ=h τ L − R 2 Transmitted charge Q = 2eNτ τ = (2e =h)V τ | agrees with microscopic calculation! h i L Levitov, Boulder 2005 Summer School Quantum Noise 5 Mesoscopic noise Find noise power spectrum for a single channel conductor (no spin): i!t S! = ^j(t)^j(0) + ^j(0)^j(t) e− dt hh ii + a t iprt a ^ i(k )t k k j(t) = evF e− − k0 +0 k;k0 b p b k i rt r 1 k „ 0 « „− − « „ «2 P a+ p 2 k t i rt ak S!=0 = k;k (evF ) δ(k k ) +0 0 − 0 b iprt r 1 bk »„ k0 « „− − « „ «– = e2Pv t(a+a b+b ) +DDiprt(a+b b+a ) [h:c:] EE k F hh k k − k k k k − k k × ii AveragingP withˆ the help of Wick's theorem, obtain˜ 2 S = e d t2(n (1 n ) + n (1 n )) + rt(n (1 n ) + n (1 n )) 0 h L − L R − R L − R R − L For reservoirsR ˆ at equilibrium, with n () = f( 1eV ), have ˜ L;R ∓ 2 e2 2 eV gkT eV kT; thermal noise; S0 = h t kT + rteV coth 2kT = 2 (re geV eV kT; shot noise Note: S (eV kT ) = re2I | Shottky noise, suppressed by r = 1 t (Lesovik '89) 0 − L Levitov, Boulder 2005 Summer School Quantum Noise 6 Noise due to electron beam partitioning e e e e e Incident electrons with µR < < µL transmitted/reflected with probabilities t, r = 1 t (No transport at < µ ; µ and > µ ; µ ) − L R L R Noiseless source (zero temperature) | binomial statistics with the number of attempts during time τ: N = (µ µ )τ=2π~ = eV τ=h τ L − R Probability of m out of Nτ electrons to be transmitted: m m N m m P = C t r − (C = N!=m!(N m)! | binomial coefficients) m N N − N N Mean value: m = 0 mPm = t@t(t + r) = tN Variance: δm2 = m2 m2 = (t@ )2(t + r)N m2 = rtN P − t − Variance = (1 t) Mean − × | agrees with microscopic calculation! L Levitov, Boulder 2005 Summer School Quantum Noise 7 Noise in a point contact, experiment 2e2 Shot noise summed over channels, S0 = n h tn(1 tn) | minima on QPC conductance plateaus (adapted from Reznikov et−al. '95) P L Levitov, Boulder 2005 Summer School Quantum Noise 8 Example from optics: photon beam splitter with noiseless source Consider n identical photons, in a number state n , incident on a beam splitter. j i a pt ipr a in = out b ipr pt b in out n = 1 (a+ )n 0 j i pn! in j i = 1 (pta+ + iprb+ )n 0 pn! out out j i 1 m=n n m m m=2 (n m)=2 + m + n m = i − C t r − (a ) (b ) − 0 pn! m=0 n out out j i m=Pn n m m 1=2 m=2 (n m)=2 = i − (C ) t r − n; n m m=0 n j − i m m n m ProbabilitP y to transmit m out of n photons is Pm = Cn t r − | binomial statistics Coherent, but noiseless, source classical beam partitioning ! L Levitov, Boulder 2005 Summer School Quantum Noise 9 Shot noise highlights (for experts) Noise suppression relative to Scottky noise S = eI (tunneling current). In a point • 2 contact S = (1 t)eI (Lesovik '89, Khlus '87) 2 − Multiterminal, multichannel generalization; Relation to the random matrix theory; • Universal 1=3 reduction in mesoscopic conductors (Buttik¨ er '90, Beenakker '92) Measured in a point contact (Reznikov '95, Glattli '96) • Measured in a mesoscopic wire (Steinbach, Martinis, Devoret '96, Schoelkopf '97) • Fractional charge noise in QHE (Kane, Fisher '94, de Picciotto, Reznikov '97, Glattli • '97 (ν = 1=3), Reznikov '99(ν = 2=5)) Phase-sensitive (photon-assisted) noise (Schoelkopf '98, Glattli '02) • Noise in NS structures, charge doubling (Kozhevnikov, Schoelkopf, Prober '00) • Luttinger liquid, nanotubes (Yamamoto, '03) • QHE system; Kondo quantum dots; Noise near 0:7e2=h structure in QPC (Glattli • '04) Third moment S measurement (Reulet, Prober '03, Reznikov '04) • 3 L Levitov, Boulder 2005 Summer School Quantum Noise 10 Experimental issues, briefly Actually measured is not electric current but EM field. Photons detached• from matter, transmitted by 1 m, amplified, and detected; ∼ Matter-to-field conversion harmless if there is no backaction; • Device + leads + environment. Engineer the circuit so that the interesting• noise dominates (e.g. a tunnel junction or point contact of high impedance) Detune from 1=f noise • Limitations due to heating and detection sensitivity • L Levitov, Boulder 2005 Summer School Quantum Noise 11 Full counting statistics L Levitov, Boulder 2005 Summer School Quantum Noise 12 Counting statistics generating function Probability distribution P cumulants m = nk n ! k hh ii m1 = n, the mean value; 2 2 2 m2 = δn = n n , the variance; 3 − 3 m3 = δn = (n n) , the skewness; ... − iλk Generating function χ(λ) = n e Pk (defined by Fourier transform), P m ln [χ(λ)] = k (iλ)k k! kX>0 While Pn is more easy to measure, χ(λ) is more easy to calculate! The advantage of χ(λ) over Pn similar to partition function in a `grand canonical ensemble' approach n n N n Ex I: Binomial distribution, Pn = CN p (1 p) − with N the number of attempts, p the success probability χ(λ)−= (peiλ + 1 p)N ! − n¯n n¯ iλ Ex II: Poisson distribution, P = e− , χ(λ) = exp(n¯(e 1)) n n! − L Levitov, Boulder 2005 Summer School Quantum Noise 13 Counting statistics, microscopic formula I WANTED: A microscopic expression for the generating function iλq χ(λ) = q P (q)e for a generic many-body system Spin 1=2Pcoupled to current: = (p aσ ; q) Hel;spin Hel − 3 λ Counting field a = 2x^δ(x x0) measures current through cross-section x = x0 − − λ Ex.