Quantum shot noise in a tunnel junction Toward the dynamical control of tunneling processes

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1 Quantum shot noise in a tunnel junction Toward the dynamical control of tunneling processes Laboratoire de Physique des Solides, Université Paris Sud, UMR8502, Orsay, France Perspectives in Quantum Thermoelectricity, time-dependence, correlations & measurements

2 Charge transfer in a quantum conductor Is it possible to measure and/or control the coherent charge transfer in a conductor? Quantum conductor = h=ev electronic wave packet charge I(t) amp meter - Measurement of the current fluctuations gives the electronic wavepacket width τ = h/ev - Control of charge transfer with a classical voltage source

3 Single-particle excitations generated by voltage pulses Are there time-dependent voltage pulses which create single-particle excitations from a degenerate Fermi sea in a mesoscopic conductor?

4 Single-particle excitations generated by voltage pulses Are there time-dependent voltage pulses which create single-particle excitations from a degenerate Fermi sea in a mesoscopic conductor? It has been proven for voltage pulse with unit flux and a Lorentzian shape. Á Z T 0 V (t)dt = N h e V (t) = X n V t+nt 2 L. Levitov et al. (96), I. Keeling et al. (06), F. Hassler et al. (07)

5 Single-particle excitations generated by voltage pulses Are there time-dependent voltage pulses which create single-particle excitations from a degenerate Fermi sea in a mesoscopic conductor? It has been proven for voltage pulse with unit flux and a Lorentzian shape. - Dynamical control of a non-equilibrium electronic distribution function - Generation and manipulation of coherent excitations in mesoscopic conductors Fundamental questions about the nature of these elementary excitations E. Bocquillon et al. (Science 2013), J. Dubois et al. (Nature 2013)

6 Contents 1 Current noise measurement to probe electronic excitations Noise and photon-assisted noise in a tunnel junction Excess noise and electronic excitations 2 Photon-assisted noise under bi-harmonic excitation Lorentzian pulses vs. bi-harmonic excitation Exprimental setup Dynamical control of a non-equilibrium electronic distribution function Coherent scattering state and shot noise minimization 3 Outlook: power fluctuations in an ac-driven conductor Unlike current noise, energy noise reveals e-h correlations Case of Lorentzian pulses

7 Contents 1 Current noise measurement to probe electronic excitations Noise and photon-assisted noise in a tunnel junction Excess noise and electronic excitations 2 Photon-assisted noise under bi-harmonic excitation Lorentzian pulses vs. bi-harmonic excitation Exprimental setup Dynamical control of a non-equilibrium electronic distribution function Coherent scattering state and shot noise minimization 3 Outlook: power fluctuations in an ac-driven conductor Unlike current noise, energy noise reveals e-h correlations Case of Lorentzian pulses

8 Current fluctuations in a tunnel junction AFM picture (Lafe Spietz) - Tunnel junction = the simplest quantum conductor contact (L) contact (R) N e T n e N e τ = 1/Δf ev dc I = e Δf n e

9 Current fluctuations in a tunnel junction AFM picture (Lafe Spietz) - Tunnel junction = the simplest quantum conductor contact (L) contact (R) N e T n e N e τ = 1/Δf ev dc - Charge granularity - A toy model for a single conduction channel I = e Δf n e Probability of transferring n e electronsfor N e incoming electrons : P Ne n e = N e n e T n e(1 T) N e n e - The barrier (transmission T 1) induces partition noise. At zero temperature, the noise spectral density at low frequency for a dc voltage bias is given by: S 2 V dc = Δn 2 /Δf = e I n e

10 Current fluctuations in a tunnel junction AFM picture (Lafe Spietz) - Time dependent voltage bias: contact (L) contact (R) N e h N e T n e N e τ = 1/Δf - Probability of transferring n e electrons and n h holes: P Ne +N e h n e = N e + N e h n e P Ne h n h = N e h n h ev dc + ev ac The excess noise S 2 = S 2 V dc +V - S ac 2 V is related to the number of e-h dc pairs created by the time dependent excitation: S 2 N e h T n e(1 T) N e+n e h n e T n h(1 T) N e h n h I = e Δf (n e n h )

11 Current fluctuations in a tunnel junction Wave-particle duality is hidden in the probability per unit time - Fermi golden rule for a dc voltage bias gives: Γ ± = e h dε T(ε) f(ε ± ev/2) 1 f(ε ev/2) - Average charge current : I = e Γ + Γ

12 Current fluctuations in a tunnel junction S 2 /Ghν Wave-particle duality is hidden in the probability per unit time - Fermi golden rule for a dc voltage bias gives: Γ ± = e dε T(ε) f(ε ± ev/2) 1 f(ε ev/2) h - Average charge current : I = e Γ + Γ - Noise spectral density for a dc biased tunnel junction: S 2 Vdc = e 2 ev Γ + + Γ = e I coth 2k B T e 35 T e = 25 mk V dc f integrated time 1/ f 1ns ev dc /hν

13 Current fluctuations in a tunnel junction S 2 /Ghν Wave-particle duality is hidden in the probability per unit time - Fermi golden rule for a dc voltage bias gives: Γ ± = e h dε T(ε) f(ε ± ev/2) 1 f(ε ev/2) - Average charge current : I = e Γ + Γ - Noise spectral density for a dc biased tunnel junction: S 2 Vdc = e 2 ev Γ + + Γ = e I coth 2k B T e 35 T e = 25 mk Equilibrium noise (thermal fluctuations) 30 S 2 Vdc = 2Gk B T e Shot noise (discreteness of charge) S 2 Vdc = e I ev dc /hν

14 Photon-assisted noise in a tunnel junction The tunnel junction is biased with a time-dependent voltage: V t = V dc + V ac (t) - Fermi golden rule for an arbitrary periodic excitation gives: Γ ± = e h dε T(ε) f(ε ± ev dc /2) 1 f(ε ev dc /2) where f ε is an out of equilibrium distribution function: f ε = + n= c 2 n f ε + nhν exp i t and c n the Fourier coeff. of evac ħ 0 - Photon-assisted noise spectral density: S 2 Vdc +V ac t = + n= c n 2 S 2 (eq) evdc + nhν Where S 2 (eq) hν = G hνcoth hν 2k B T is the Johnson-Nyquist equilibrium noise

15 Photon-assisted noise in a tunnel junction S 2 /Ghν The tunnel junction is biased with a time-dependent voltage: V t = V dc + V ac (t) V dc + V ac (t) f Photon-assisted shot noise S 2 V dc +V ac (t) ev dc /hν probes the excitations created by pulses at frequency ν. The excess noise ΔS 2 = S 2 Vdc +V ac (t) S 2 Vdc at V dc = N e hν/e is related to the probability p k (N e ) (related to V ac ) to create k e-h pairs at zero temperature (hν k B T e ) S 2 (Nhº) = 2Ghº X k p k (N e ) / N e h M. Vanevic et al. (07)(08)(12)

16 Contents 1 Current noise measurement to probe electronic excitations Noise and photon-assisted noise in a tunnel junction Excess noise and electronic excitations 2 Photon-assisted noise under bi-harmonic excitation Lorentzian pulses vs. bi-harmonic excitation Exprimental setup Dynamical control of a non-equilibrium electronic distribution function Coherent scattering state and shot noise minimization 3 Outlook: power fluctuations in an ac-driven conductor Unlike current noise, energy noise reveals e-h correlations Case of Lorentzian pulses

17 Current noise measurement to probe electronic excitations - is minimal at unit-flux: Excess noise for different pulse shapes Á Z T 0 V (t)dt = N h e R. Schoelkopf et al. (98)(00)

18 Current noise measurement to probe electronic excitations - is minimal at unit-flux: Á Z T 0 Excess noise for different pulse shapes V (t)dt = N h e - reaches zero for Lorentzian pulses

19 Current noise measurement to probe electronic excitations - is minimal at unit-flux: Á Z T 0 Excess noise for different pulse shapes V (t)dt = N h e - reaches zero for Lorentzian pulses - Technical constrains: It is experimentally difficult to generate Lorentzian pulses with precise shape and high repetition rate ( ). The impedance of a single mode conductor induces a limitation of the detection bandwidth. bi-harmonic pulses to mimic Lorentzian pulses with width

20 Voltage pulses on a tunnel junction - Al/Al 2 O 3 /Al tunnel junction: channels transmission - For a N unit-flux Lorentzian pulse, the statistics of transferred charge corresponds to a coherent scattering state without e-h excitations : jãi = (2 ) X V (t) = ¹h 2= 1=2 e ²=¹h j1 ² i hhq n ii e 1 + (t= ) 2 = N R K ²>² F e n R - For N unit-flux (bi-)harmonic pulses, the coherent scattering state is surrounded by a ("small") cloud of e-h excitations :

21 Experimental setup dilution refrigerator sample holder 1 cm

22 Experimental setup V (t) = V dc + V ac1 cos (2¼ºt) + V ac2 cos (4¼ºt + ')

23 Experimental setup (calibration) V (t) = V dc + V ac1 cos (2¼ºt) + V ac2 cos (4¼ºt + ')

24 Dynamical control of a non-equilibrium electronic distribution function The non-equilibrium distribution is closely related to the derivative of the noise f ε = + n= c n 2 f ε + nhν Shot noise spectroscopy gives the non equilibrium distribution function, in the limit f ε F + ξ R S 2 (ev dc ) ξ=evdc Possibility to dynamically control and engineer the energy distribution function of the electrons in a mesoscopic conductor (e.g. dynamically control the amplitude of the critical current of a SNS tunnel junction).

25 Measurement of photon-assisted process probabilities

26 Coherent scattering state and shot noise minimization For a bi-harmonic excitation V (t) = V dc + V ac1 cos (2¼ºt) + V ac2 cos (4¼ºt + ') The total probability to create e-h pairs at zero temperature is: P X p k = S 2 2Ghº k V dc = 5hν/e and JG et al. (13), J. Dubois et al. (13)

27 Contents 1 Current noise measurement to probe electronic excitations Noise and photon-assisted noise in a tunnel junction Excess noise and electronic excitations 2 Photon-assisted noise under bi-harmonic excitation Lorentzian pulses vs. bi-harmonic excitation Exprimental setup Dynamical control of a non-equilibrium electronic distribution function Coherent scattering state and shot noise minimization 3 Outlook: power fluctuations in an ac-driven conductor Unlike current noise, energy noise reveals e-h correlations Case of Lorentzian pulses

28 Outlook: power fluctuations in an ac-driven conductor Motivated by the fluctuations relations, we could look at the dissipated work in the conductor and define the power operator provided by the source. P t = V(t) I L (t) where I L (t) is the current operator described in the scattering theory approach - For a conductor with energy-indep. transmission D, power fluctuations are given by: Δ P 2 = D(1 D) h e 2 h D2 + n= + n= c n 2 (nhν + ev dc ) 3 coth nhν + ev dc 2k B T v n 2 nhν coth - For transparent barrieres, an interference term appears in addition to the usual transport term nhν 2k B T v n = ν 0 1/ν + V t e i2πnνt dt F. Battista et al. (PRB 2014)

29 Outlook: power fluctuations in an ac-driven conductor For Lorentzian pulses: V (t) = X n V t+nt 2 Fluctuations of power yields an interference contribution related to the intrinsic energy spread of single particle excitation: Δ P 2 = D(1 D) h 0 1/ν ev t 3 dt + h τ 2 D 2 ν 2 ψ = 2τ ε>ε F e τε/ħ 1 ε F. Battista et al. (PRB 2014)

30 Outlook - Take home message The photon-assisted noise shows the possibility to dynamically control and engineer the energy distribution function of the electrons in a mesoscopic conductor. It is the first step towards the creation of the "electronic coherent states". - Outlook ψ = 2τ ε>ε F e τε/ħ 1 ε Is this source equivalent to single electron source based on a quantum dot (LPA experiment)? Study of Crooks and Jarzynski fluctuation relations for the tunneling current. How high order cumulant of power fluctuations have to be symmetrized? P + (W) P ( W) = exp W F k B T with W = V t I t dt

31 References [1] D. A. Ivanov, H. W. Lee and L. S. Levitov, Phys. Rev. B 56, 6839 (1997) [2] J. Keeling, I. Klich and L. S. Levitov, Phys. Rev. Lett. 97, (2006) [3] C. Grenier, R. Herve, E. Bocquillon, F. D. Parmentier, B. Placais, J.-M. Berroir, G. Feve, and P. Degiovanni, New J. Phys. 13, (2011) [4] M. Vanevic, Y. V. Nazarov, andw. Belzig, Phys. Rev. B 78, (2008) [5] M. Vanevic and W. Belzig, Phys. Rev. B 86, (R) (2012) [6] J. Dubois et al., Nature 502, 659 (2013) [7] J.G and B.R, Phys. Rev. B 87, (2013) Thank you!

32 Coherence of the electronic wave packet Collaboration with B. Reulet, Sherbrook University