Lectures II, III: 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)

iωτ Noise spectrum S(ω) = ∞ 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, ψ00 + U(x)ψ = 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

ˆ Express electric current through ψ(x) = k aˆkuk(x) + bkvk(x) : ˆ e + i(k k0)x k+k0 + j(x) = 2m( ψ (x)∂xψ(x) + h.c.) = e k,k e − 2m ψ (x)ψk(x) − P0 k0 k + k + P √ 0 i(k k0)x ak t i rt ak ˆj(x) = e e − +0 (x 0) 2m bk i√rt 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 √t i√r Single channel S-matrix: S = (optical beam splitter) i√r √t   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): iωt Sω = ˆj(t)ˆj(0) + ˆj(0)ˆj(t) e− dt hh ii + a t i√rt a ˆ i(k  )t k k j(t) = evF e− − k0 +0 k,k0 b √ b k i rt r 1 k „ 0 « „− − « „ «2 P a+ √ 2 k t i rt ak Sω=0 = k,k (evF ) δ(k k ) +0 0 − 0 b i√rt r 1 bk »„ k0 « „− − « „ «– = e2Pv t(a+a b+b ) +DDi√rt(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. | i

a √t i√r a in = out b i√r √t b  in    out n = 1 (a+ )n 0 | i √n! in | i = 1 (√ta+ + i√rb+ )n 0 √n! out out | i 1 m=n n m m m/2 (n m)/2 + m + n m = i − C t r − (a ) (b ) − 0 √n! m=0 n out out | i m=Pn n m m 1/2 m/2 (n m)/2  = i − (C ) t r − n, n m m=0 n | − 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 = 2xˆδ(x x0) measures current through cross-section x = x0 − − λ Ex. Coupling to classical current = 2σ3I(t); time evolution of spin: iθ(t)/2 H iθ(t) = e− , = e /2 | ↑i | ↑i | ↓i | ↓i t Spin precesses in the XY plane, precession angle θ(t) = λ 0 I(t0)dt0 measures time-dependent transmitted charge x=x R

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θ(t) Disclaimer: Our goal is to clarify microscopic picture of current fluctuations, not to describe realistic measurement

L Levitov, Boulder 2005 Summer School Quantum Noise 14 Counting statistics, microscopic formula II

In QM current is an operator and [I(t), I(t0)] = 0 Need a more careful analysis! 6 → Spin evolution (ensemble-averaged):

(0) i λt i λt (0) i t i t ρ e H− e− H elρ ρ(t) = e− H ρ0e H el = i t i ↑↑ t (0) h (0) i ↑↓ h i e Hλ e− H λ ρ ρ h − iel ! ... = Tr (...ρ ) ↓↑ ↓↓ h iel el el Examine the classical current case: spin precesses by θn = λn for n transmitted particles.

(0) iθn (0) (0) iλn (0) ρ e− ρ ρ e ρ ρ(t) = P ↑↑ ↑↓ = ↑↑ h i ↑↓ n n iθn (0) (0) iλn (0) (0) e ρ ρ ! e− ρ ρ ! P ↓↑ ↓↓ h i ↓↑ ↓↓ Identify eiθ(t) = eiλq with χ(λ), the generating function h i h i

L Levitov, Boulder 2005 Summer School Quantum Noise 15 Main result: χ(λ) is given by Keldysh partition function

χ(λ) = TK exp i ˆλ(t0)dt0 − C0,t H ! D I E with the counting field λ(t) = λ antisymmetric on the forward and backward parts of the Keldysh contour C [0 t 0] 0,t ≡ → → Properties:

1. Normalization: q Pq = 1, since χ(λ = 0) = 1 2. P = e iqλχ(λ)dλ 0 q − P2π 3. Charge quantization:≥ χ(λ) is 2π-periodic in λ (for noninteracting particles)R Features: — Describes not just spin 1/2 but a wide class of passive charge detectors, such as heavy particle = p2/2M λf(t)q at large M (no recoil); — Minimal backaction,H measurement− affects only forward scattering: [ , σz] = 0; —H Good for generic many-body system

L Levitov, Boulder 2005 Summer School Quantum Noise 16 Compare: Larmor clock for QM collision Nuclear reaction, tunneling, resonance scattering, etc. — How long does it take?

Add a fictitious spin 1/2 to particle: U(x) U = U(x) + 1ω(x)σ → eff 2 z with fictitious field ω(x) nonzero in the spatial region of interest.

Potential barrier Resonance scattering

Spin precesses about the Z axis during collision: precession angle measures time. Analysis similar to passive detector (one particle!) yields

1 iωτ χ(ω) = Tr(S−ωSωρ) = e− P (τ)dτ − Z with P (τ) interpreted as probability to spend time τ in the region of interest

 0+iγ/2 ω ∆ iγ ω+∆ iγ Resonance scattering: S() = − gives χ(ω) = − − − with  0 iγ/2 ω ∆+iγ × ω+∆+iγ detuning ∆ = 2(  ). Obtain positive− −or negative probabilities!− (Caution) − 0

L Levitov, Boulder 2005 Summer School Quantum Noise 17 Variety of topics

Tunneling problem. S2, S3 Nonequilibrium FDT theorem. Relation with Glaub• er theory of photocounting. Driven many-body systems. For noninteracting particles (fermions or b•osons) χ(λ) can be expressed through time-dependent one-particle S- matrix. Pumps, coherent current pulses, photon-assisted noise next lecture → Mesoscopic noise in normal and superconducting systems (Nazarov, Nagaev)• Mesoscopic photon sources (Beenakker) • Entangled EPR states, counting statistics (Fazio) • Spin current noise (Lamacraft) • Backaction of spin 1/2 counter (Muzykantsky) • Role of environment (Kindermann) • /entropy (Callan & Wilczek, Vidal, Kitaev) • Orthogonality catastrophe, Fermi-edge singularity •

L Levitov, Boulder 2005 Summer School Quantum Noise 18 Counting statistics of tunneling current

Focus on the tunneling problem (generic interacting system). Tunneling Hamiltonian ˆ = ˆ + ˆ + Vˆ H H1 H2 where ˆ and Vˆ = Jˆ + Jˆ ) describe leads and tunneling coupling). The H1,2 12 21 counting field λ(t) is added to the phase of the tunneling operators Jˆ12, Jˆ21 as i λ(t) i λ(t) Vˆλ = e2 Jˆ12(t) + e−2 Jˆ21(t) with λ0

ieV t Transform the bias voltage into a phase factor, Jˆ12 Jˆ12e− , Jˆ Jˆ eieV t. In the interaction representation, write → 21 → 21

ˆ χ(λ) = TK exp i Vλ(t0)(t0)dt0 − C0,t ! D I E using cumulant expansion, as a sum of linked cluster diagrams.

L Levitov, Boulder 2005 Summer School Quantum Noise 19 The lowest order in the tunneling coupling Jˆ12, Jˆ21 is given by linked clusters of order two. Obtain χ(λ) = eW (λ), where

1 ˆ ˆ W (λ) = TKVλ(t0)(t0)Vλ(t00)(t00) dt0dt00 −2 C I I 0,tD E More explicitly,

iλ iλ W (λ) = (e 1)N1 2(t) + (e− 1)N2 1(t) − → − → t t t t N1 2 = Jˆ21(t0)Jˆ12(t00) dt0dt00 N2 1 = Jˆ12(t0)Jˆ21(t00) dt0dt00 → h i → h i Z0 Z0 Z0 Z0 with Nj k(t) = njkt the mean particle number transmitted between the contacts→in a time t (cf. ). Resulting counting statistics is bi-directional Poissonian:

iλ iλ χ(λ) = exp (e 1)N1 2(t) + (e− 1)N2 1(t) − → − → True in any interacting system, in the tunneling regime. 

L Levitov, Boulder 2005 Summer School Quantum Noise 20 Nonequilibrium Fluctuation-Dissipation theorem (iλ)k δqk ∞ The cumulants are generated as ln χ(λ) = k=1 k! hh k ii with q0 the q0 tunneling charge. Obtain P

k k (n12 n21)t, k odd δq = q0 − hh ii ((n12 + n21)t, k even

Setting k = 1, 2, relate n12 n21 with the time-averaged current and the low frequency noise power: 

n n = I/q , n + n = S /q2. 12 − 21 0 12 21 2 0 Relate the second and the first correlator:

2 S2 = δq /t = (N1 2 +N2 1)/(N1 2 N2 1)q0I = coth(eV/2kBT )q0I hh ii → → → − →

Nyquist for eV < kBT , Schottky for eV > kBT . A universal relation — holds for any I V characteristic (linear responce not required, cf. FDT in equilibrium) −

L Levitov, Boulder 2005 Summer School Quantum Noise 21 Application in metrology

L Levitov, Boulder 2005 Summer School Quantum Noise 22 Generalized Schottky formula for S3

3 Relate the third cumulant δq3 δq δq = S t with δq = It. hh ii ≡ − 3 h i Obtain a Schottky-like relation for the third correlator spectral power S3:  S δq3 /t = q2I 3 ≡ hh ii 0 — independent of the mean/variance ratio (n n )/(n + n ). 12 − 21 12 21 Since N1 2/N2 1 n12/n21 = exp(eV/kBT ) (detailed balance), the → 2 → ≡ relation S3 = q0I holds at any voltage/temperature ratio.

Good for using shot noise to determine particle charge in Luttinger liquids and fractional QHE (heating limitation: S2 = q0I requires eV > kBT ). A possibility to measure tunneling quasiparticle charge at temperatures k T eV B ≥

L Levitov, Boulder 2005 Summer School Quantum Noise 23 The 3-rd correlator in terms of the counting distribution profile: q = It = mean; δq2 = S t = variance; δq3 = S t = skewness h i hh ii 2 hh ii 3

−1 10

−2 10

−3 10

Counting probability −4 10 N =20, N =0 12 21 Gaussian

−5 10 0 5 10 15 20 25 30 35 40 Transmitted charge q/e*

The third moment determines skewness of the distribution P (q) profile. This is illustrated by a bi-directional Poissonian distribution and a Gaussian with the same mean and variance. For S3 > 0 the peak tails are stretched more to the right than to the left.

L Levitov, Boulder 2005 Summer School Quantum Noise 24 Measurement of S3 in a Tunnel Junction:

First experiment, low impedance (50 Ohm) tunnel junction (B. Reulet, J. Senzier, and D.E. Prober, Phys. Rev. Lett. 91, 196601 (2003));

High impedance junction (M. Reznikov et. al. ’04):

6 10 3 4 0 ) ) 22

counts −10 /Hz) /Hz/Hz 2 2 33 A 2 A A Voltage 0 −20 −10 0 10 −27 −46−46

(10 1

( 10 ( 10 −2

(2) 1 (3)(3) S counts counts SS −4 0.8 Voltage binbin numbernumber 0 −6 0 5 10 15 20 −20 −10 0 10 20 I (10−9 A) I (10−9 A)

Transmitted charge distribution and its moments S2 and S3. Poisson 2 behavior S3 = e I confirmed.

L Levitov, Boulder 2005 Summer School Quantum Noise 25 Tunneling summary:

1) The counting statistics of tunneling current is bi-directional Poissonian, universally and independently of the character of interactions and thus of the form of I V dependence. − 2) Nonequilibrium FDT relation S2 = coth(eV/2kBT )q0I

2 3) Shottky-like relation for the third correlator S3 = q0I at both large and small eV/kBT .

L Levitov, Boulder 2005 Summer School Quantum Noise 26 Counting statistics of a driven many-body system Time-dependent external field and scattering (noninteracting fermions) We obtain the generating function in the form of a functional determinant in the single-particle Hilbert space:

χ(λ) = det 1ˆ + n(t, t0) Tˆ (t) 1ˆ λ −    λ λi j ˆ i 4 i 4 Tλ(t) = S† λ(t)Sλ(t), Sλ(t)ij = e S(t)ije− − i i(t t0) with reservoirs density matrix n(t, t0)T =0 = = e− − , a 2π(t t0+iδ) <0 time-dependent S-matrix S(t) and separate λ fo−r each channel i P 2 3

1 4 ... 5

Time-dependent field (voltage V (t), etc.) included in Sλ(t). No time delay: S(t, t0) S(t)δ(t t0) (instant scattering apprx. — nonessential) ' −

L Levitov, Boulder 2005 Summer School Quantum Noise 27 Simple and not-so-simple facts from matrix alebra Useful relations between 2-nd quantized and single-particle operators:

N A Γ(A) = A a+a −→ ij i j i,j=1 X (mapping of matrices N N 2N 2N ). × −→ × Tr eΓ(A) = det 1 + eA β — fermion partition function Z = Tr e− H with  β = Γ(A)) Note: For A = 0 obtain 2N = 2N − H

Tr eΓ(A)eΓ(B) = det 1 + eAeB    Tr eΓ(A)eΓ(B)eΓ(C) = det 1 + eAeBeC ...   Proven using Baker-Hausdorff series for ln(eXeY ) (commutato r algebra for X, Y the same as for Γ(X), Γ(Y ))

L Levitov, Boulder 2005 Summer School Quantum Noise 28 Deriving the determinant formula I

Write c.s. generating function as

i λt i λt χ(λ) = Tr ρele H− e− H  1 β with λ = Γ(h λ), ρ = e− H0. Obtain H  Z 1 1 βh0 ih λt ihλt βh0 − βh0 ih λt ihλt χ = Z− det 1 + e− e − e− = det 1 + e− 1 + e− e − e− “ ” »“ ” “ ”– 1 Finally, with nˆ = 1 + eβh0 − , have  ih t ih t χ(λ) = det 1 nˆ + neˆ λ e− λ − − 

L Levitov, Boulder 2005 Summer School Quantum Noise 29 Deriving the determinant formula II Relate forward-and-backward evolution in time with scattering operator: t -1 S1 S0 ψ

^ -ih τ e 1

-1 S0 ψ ^ ih τ e 0 ψ x

ih λt ihλt 1 Thus t0 e − e− t = S−λ(t)Sλ(t)δ(t t0) with th |a wavepacket| airriving− at scatterer−at time t. | i 1 χ(λ) = det 1 nˆ + nSˆ −λSλ − − A single-particle quantity — Fermi-statistics accounted for by det! 1 (Generalize to bosons: det(1 + ...) det(1 ...)− ) −→ −

L Levitov, Boulder 2005 Summer School Quantum Noise 30 A more intuitive approach: DC transport

For elastic scattering different energies contribute independently:

d χ(λ) = χ (λ) , i.e. χ(λ) = exp t ln χ (λ) ,   2π~  Y  Z  (quasiclassical d = ddt/2π~). V 2 3

1 4 ... 5

Sum over all multiparticle processes:

i 2(λi +...+λi λj ... λj ) χ(λ) = e 1 k− 1− − k Pi ,...,i j ,...,j , 1 k | 1 k i ,...,i ,j ,...,j 1 Xk 1 k

L Levitov, Boulder 2005 Summer School Quantum Noise 31 with the rate of k particles transit i , ..., i j , ..., j is given by 1 k −→ 1 k 2 j1,...,jk Pi ,...,i j ,...,j = Si ,...,i (1 ni()) ni() . 1 k | 1 k 1 k − i=iα i=iα Y6 Y

with Sj1,...,jk antisymmetrized product of k single particle amplitudes. i1,...,ik This is equal to our determinant (Proven by reverse engineering) Positive probabilities!

L Levitov, Boulder 2005 Summer School Quantum Noise 32 Point contact (beam splitter): 2 2 matrices ×

2 i (λ λ ) 2 χ (λ) = (1 n )(1 n ) + ( S + e2 2− 1 S )n (1 n )  − 1 − 2 | 11| | 21| 1 − 2 2 i (λ λ ) 2 2 + ( S + e2 1− 2 S )n (1 n ) + det S n n , (1) | 22| | 12| 2 − 1 | | 1 2 with n () = f( 1eV ) and S is unitary: S 2 + S 2 = 1, det S = 1. 1,2 ∓ 2 ij | 1i| | 2i| | | iλ iλ Simplify: χ(λ) = 1 + t(e 1)n1(1 n2) + t(e− 1)n2(1 n1) Here t = S 2 = S 2 is the transmission− co−efficient and λ−= λ λ−. | 21| | 12| 2 − 1 At T = 0, since nF () = 0 or 1, for V > 0 have

iλ 1 e t + 1 t ,  < 2eV ; χ(λ) = − | | 1 ,  > 1eV ( | | 2

iλ N(τ) Full counting statistics: χτ (λ) = (e p + 1 p) — binomial, with the number of attempts N(τ) = (eV/h)τ. −

iλ iλ Similar at V < 0, with e e− (DC current sign reversal). −→

L Levitov, Boulder 2005 Summer School Quantum Noise 33 Note: the noninteger number of attempts is an artifact of a quasiclassical calculation. More careful analysis gives a narrow distribution PN of the number of attempts peaked at N = N(τ), and the generating function as a weighted sum N PN χN (λ). The peak width is a sublinear function of the 2 measurement time τ (in fact, δN T =0 ln τ), the statistics still binomial, to leading orderPin t. ∝

L Levitov, Boulder 2005 Summer School Quantum Noise 34 Strategies for handling the determinant

Two strategies: 1) For periodic S(t), in the frequency representation nˆ is diagonal, n(ω) = f(ω), while S(t) has matrix elements Sω0,ω with discrete frequency change ω0 ω = nΩ, with Ω the pumping frequency. In this method the energy axis−is divided into intervals nΩ < ω < (n + 1)Ω, and each interval is treated as a separate conduction channel with time-independent S-matrix

Sω0,ω. 2) The determinant can also be analyzed directly in the time domain:

1 ∂ ln χ(λ) = Tr (1 + n(T 1))− ∂ T λ λ − λ λ h i 1 i 1 with Tλ(t, t0) = S−λ(t)Sλ(t)δ(t t0), n(t, t0) = 2π(t t0 + iδ)− . − − −

The problem of inverting the integral operator R = 1 + n(Tλ 1) is the so-called Riemann-Hilbert problem (matrix generalization of Wiener-Hopf,− well-studied, exact and approximate solutions can be constructed)

L Levitov, Boulder 2005 Summer School Quantum Noise 35 Phase-sensitive noise Charge flow induced by voltage pulse V (t) in a point contact A pulse V (t)corresponds to a step ∆θ in the forward scattering phase:

e iθ(t)√t i√r e t S(t) = − θ(t) = V (t )dt i√r eiθ(t)√t ~ 0 0   Z−∞ The mean transmitted charge q = te∆θ/2π — independent of pulse shape In contrast, the variance δq2 exhibits complex dependence:

iθ12 t2 2 2 1 e e δq = 2t(1 t)e − dt dt θ = V (t0)dt0 − (t t )2 1 2 12 ~ Z Z 1 − 2 Zt1

2 Gives δq (1 cos ∆θ) ln(tmax/t0) + const — periodic in the pulse area and log-divergent∝ − (t ~/k T ) max ∼ B Interpret the log as orthogonality catastrophe: long-lasting change of scatterer at ∆θ = 2πn causes infinite number of soft particle-hole excitations 6 Shot noise is phase-sensitive — Mach-Zender effect in electron noise

L Levitov, Boulder 2005 Summer School Quantum Noise 36 Minimizing unhappiness

t2 e 1 exp i V (t0)dt0 − t1 ~ Find noise-minimizing pulse shapes: 2 dt1dt2 min “(tR1 t2) ” → — an interesting variational problem, solved by pulses− of integer area 2πn: RR ~ 2τ V (t) = i (τ > 0) e (t t )2 + τ 2 i i=1...n i i X − Lorentzian pulses (overlapping or nonoverlapping)

Degeneracy: δq2 = e2t(1 t)n, the same for all t , τ − i i

L Levitov, Boulder 2005 Summer School Quantum Noise 37 Coherent time-dependent many-body states

Variance of transmitted charge δq2 = e2t(1 t)n — independent of pulse − parameters ti, τi, same value as for binomial distribution with the number of attempts n Binomial counting statistics from the functional determinant (exactly solvable Riemann-Hilbert problem):

n χ(λ) = teiλ + 1 t −  Interpretation: pulses independent attempts to transmit charge ' Coherent current pulses: Noise reduced as much as the beam splitter partition noise permits Similarity to coherent states (QM uncertainty minimized) Many-body objects which behave like songle particles. Fully entangled?

L Levitov, Boulder 2005 Summer School Quantum Noise 38 Measurement of phase sensitive noise Instead of a train of pulses (which is difficult to realize) used a combination of DC and AC voltage, V = VDC + VAC cos Ωt — oscillations in noise power (Bessel functions), while DC current is ohmic:

∂S0 2 = e3 t (1 t ) J 2(V /~Ω)θ(eV n~Ω) m − m n AC DC − ∂VDC π m ! n X X Observed in mesoscopic wires (Schoelkopf ’98), point contacts (Glattli ’02)

L Levitov, Boulder 2005 Summer School Quantum Noise 39 Case studies

(i) Voltage pulses of different signs (D Ivanov, H-W Lee, and LL, PRB 56, 6839 (1997)):

h 2τ 2τ V (t) = 1 2 e (t t )2 + τ 2 − (t t )2 + τ 2  − 1 1 − 2 2  give rise to the counting distribution

2 iλ iλ z1∗ z2 χ(λ) = 1 2F + F (e + e− ) , F = t(1 t) − − − z1 z2 −

2 with z1,2 = t1,2 + iτ1,2. The quantity A = ... is a measure of pulses’ overlap in time: A = 0 (full overlap), A = 1 (no| |overlap). Note: χ(λ) factorizes for nonoverlapping pulses

(ii) Two-channel model of electron pump ((D Ivanov and LL, JETP Lett 58, 461 (1993)):

iΩτ iΩτ r t0 B + be− A¯ + ae¯ S(τ) = iΩτ ¯ iΩτ , ≡ t r0 A + ae− B¯ be    − − 

L Levitov, Boulder 2005 Summer School Quantum Noise 40 which is unitary provided A 2 + a 2 + B 2 + b 2 = 1, Aa¯ + B¯b = 0 (time-independent parameters).| | | | | | | |

For T = 0 and µL = µR the charge distribution for m pumping cycles is described by

iλ iλ m χ(λ) = 1 + p (e 1) + p (e− 1) 1 − 2 −  with p = a 4/( a 2 + b 2) and p = b 4/( a 2 + b 2): at each pumping 1 | | | | | | 2 | | | | | | cycle an electron is pumped in one direction with probability p1, or in the opposite direction with probability p2, or no charge is pumped with probability 1 p p . − 1 − 2 Also can be solved at µ = µ : more complicated statistics. L 6 R

L Levitov, Boulder 2005 Summer School Quantum Noise 41 Counting statistics of a charge pump DC current from an AC-driven open quantum dot. (Exp: Marcus group ’99)

The time-averaged pumped current is a purely geometric property of the path in the S-matrix parameter space, insensitive to path parameterization (Brouwer ’98, Buttiker’94). Noise dependence on the pumping cycle? Bounds on the ratio noise/current?

Here, consider generic but small path V1(t), V2(t):

V1

V2

L Levitov, Boulder 2005 Summer School Quantum Noise 42 Counting statistics for a generic pumping cycle Focus on the weak pumping regime, a small loop in the matrix parameter A(t) (0) space: S(t) = e S with perturbation A(t) (antihermitian, trA†A 1) and S(0) is the S-matrix in the absence of pumping.  ˆ ˆ Expand ln χ(λ) = det 1 + n(t, t0) S† λ(t)Sλ(t) 1 in A(t): − −    1 2 2 1 2 ln χ = tr nˆ A λ + Aλ 2A λAλ tr(nBˆ λ) 2 − − − − 2  iλσ iλσ with Aλ(t) = e 4 3A(t)e− 4 3, Bλ(t) = Aλ(t) A λ(t) 2 − − Using nˆT =0 = nˆT =0, separate a commutator:

1 1 2 2 2 ln χ(λ) = tr (nˆ[Aλ, A λ]) + tr nˆ Bλ tr(nBˆ λ) 2 − 2 −   The commutator is regularized as the Schwinger anomaly (splitting points, t0, t00 = t /2), which gives  1 n(t0, t00)tr (A λ(t00)Aλ(t0) Aλ(t00)A λ(t0)) dt 2 − − − I

L Levitov, Boulder 2005 Summer School Quantum Noise 43 Average over small  (insert additional integrals over t0, t00, or just replace 1 A (t) (A (t) + A (t0)), etc.) In the limit  0, obtain λ → 2 λ λ → i (ln χ)1 = tr (A λ∂tAλ Aλ∂tA λ) dt 8π − − − I The second term of ln χ is rewritten as

2 1 tr (Bλ(t) Bλ(t0)) (ln χ) = − dtdt0 2 4(2π)2 (t t )2 I I − 0

Now, combine (ln χ)1 with (ln χ)2, and simplify

L Levitov, Boulder 2005 Summer School Quantum Noise 44 Bi-directional Poissonian statistics

Convenient decomposition A = a0 + z + z†, such that [σ3, a0] = 0, [σ , z] = 2z, σ , z† = 2z†, gives 3 − 3

  iλσ iλσ iλ iλ A e− 4 3Ae 4 3 = a + e 2 z† + e− 2 z λ ≡ 0 iλ iλ B = e 2 e− 2 W, W z† z λ − ≡ −   In this representation,

2 sin λ (1 cos λ) tr (W (t) W (t0)) ln χ = tr ([σ , W ] ∂ W ) dt + − − dtdt0 8π 3 t 2(2π)2 (t t )2 I I I − 0 The first term is identical to the Brouwer result (invariant under reparameterization and has a purely geometric character), the second term describes noise. iλ iλ Note the λ-dependence: u(e 1) + v(e− 1) 2P statistics − − → iλ iλ χ(λ) = exp u(e 1) + v(e− 1) − −

L Levitov, Boulder 2005 Summer School  Quantum Noise 45 Prove that 1) u, v 0 for generic pumping cycle W (t); ≥ 2) u = 0 or v = 0 for special paths W (t) holomorphic in the upper/lower half-plane of complex time t; Then the ratio current/noise = (u v)(u + v) is maximal or minimal 1 − (equal q0− per cycle), when u = 0 or v = 0. The counting statistics in this caseis pure poissonian.

L Levitov, Boulder 2005 Summer School Quantum Noise 46 Example of a single channel system (two leads) driven by V1(t) = a1 cos(Ωt + θ), V2(t) = a2 cos(Ωt). The pumping cycle:

0 z(t) W (t) = , z(t) = z1V1(t)+z2V2(t), (z1,2 system parameters) z∗(t) 0   −1 1

0.5

0

−0.5

1.5 1 −1 0.5 −1.5 0 −1 −0.5 −0.5 Current to noise ratio, I/J, in the units of e 0 0.5 −1 Re w Im w 1 1.5 −1.5

Current to noise ratio, I/J = q 1(u v)/(u + v), as a function of the driving signal parameters for a single channel pump. 0− − iθ The two harmonic signals driving the system are characterized by relative amplitude and phase, w = (V1/V2)e . Maximum and minimum, as a function of w, are I/J = q 1.  0−

L Levitov, Boulder 2005 Summer School Quantum Noise 47 Pump noise summary:

1) Pumping noise is super-poissonian; counting statistics is double- poissonian; 2) The current/noise ratio can be maximized by varying the pumping cycle (relative amplitude or phase of the driving signals); 3) Extremal cycles correspond to poissonian counting statistics with a universal ratio 1 current/noise = q−  0 (another generalization of the Schottky formula).

L Levitov, Boulder 2005 Summer School Quantum Noise 48