Elementary charge-transfer processes in mesoscopic conductors

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Elementary charge-transfer processes in mesoscopic conductors

Mihajlo Vanević,1_{Yuli V. Nazarov,}2_{and Wolfgang Belzig}3

1_{Departement Physik, Universität Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland} 2_{Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands}

3_{Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany}

共Received 22 September 2008; published 8 December 2008兲

We determine charge-transfer statistics in a quantum conductor driven by a time-dependent voltage and identify the elementary transport processes. At zero temperature unidirectional and bidirectional single-charge transfers occur. The unidirectional processes involve electrons injected from the source terminal due to excess dc bias voltage. The bidirectional processes involve electron-hole pairs created by time-dependent voltage bias. This interpretation is further supported by the charge-transfer statistics in a multiterminal beam-splitter geom-etry in which injected electrons and holes can be partitioned into different outgoing terminals. The probabilities of elementary processes can be probed by noise measurements: the unidirectional processes set the dc noise level, while bidirectional ones give rise to the excess noise. For ac voltage drive, the noise oscillates with increasing the driving amplitude. The decomposition of the noise into the contributions of elementary pro-cesses reveals the origin of these oscillations: the number of electron-hole pairs generated per cycle increases with increasing the amplitude. The decomposition of the noise into elementary processes is studied for different time-dependent voltages. The method we use is also suitable for systematic calculation of higher-order current correlators at finite temperature. We obtain current noise power and the third cumulant in the presence of time-dependent voltage drive. The charge-transfer statistics at finite temperature can be interpreted in terms of multiple-charge transfers with probabilities which depend on energy and temperature.

DOI:10.1103/PhysRevB.78.245308 PACS number共s兲: 73.23.⫺b, 72.70.⫹m, 72.10.Bg, 05.40.⫺a I. INTRODUCTION

The charge transmitted through a mesoscopic conductor during a fixed time interval fluctuates because of charge dis-creteness and stochastic nature of transport. The objective of the statistical theory of quantum transport, full counting sta-tistics, is to completely characterize the probability distribu-tion of transferred charge. The field of full counting statistics has attracted significant attention because it provides the most detailed information on charge transfer accessible in the measurements of the average current, current noise power,1 and higher-order current correlations.2–4

The problem of full counting statistics has been addressed first by Levitov and Lesovik5,6 _{for the case of dc biased} multiterminal junctions and subsequently generalized to a time-dependent voltage bias.7,8 _{In addition to the scattering} theory of Levitov and Lesovik5,6_{共see also Ref.}₉_{兲, the} theo-retical approaches to full counting statistics include the so-called stochastic path-integral approach10,11 _and _the quantum-mechanical theory based on an extension of the Keldysh-Green’s-function technique.12–16 _Although equiva-lent, these theories provide different methods to access the charge-transfer statistics. Using the extended Keldysh-Green’s-function technique, the theory of full counting sta-tistics of a general quantum-mechanical observable has been formulated,17,18 _{including the effects of a detector back} action.19_{The problem of quantum measurement in the} con-text of full counting statistics of noncommuting spin observ-ables has also been analyzed.20,21 _{The discretization}22,23 _in space of the Keldysh-Green’s functions results in a quantum circuit theory24–26 _{which greatly simplifies the calculations.} The circuit theory can describe junctions with different con-nectors and leads, as well as multiterminal circuits.27

The goal of evaluation of full counting statistics in meso-scopic conductors has essentially been accomplished: the aforementioned theories provide methods and techniques to calculate the cumulant generating function of transferred charge from which the probability distribution can be ob-tained. However, the interpretation of the resulting statistics is often not straightforward. This is because the total statis-tics are composed of many electrons injected toward conduc-tor, each exhibiting different chaotic scattering and multiple reflections before entering an outgoing terminal. At finite temperatures, the charge flow is bidirectional with both in-jection and absorption of electrons from terminals. The par-ticle correlations due to Pauli principle, interactions, and those induced by the superconducting and/or ferromagnetic proximity effect also affect the charge-transfer statistics.

To understand the statistical properties of collective charge transfer, one has to find the independent elementary processes constituting the statistics. The independent pro-cesses can be revealed by decomposition of the total cumu-lant generating function into a sum of simpler ones. The physical interpretation obtained this way goes beyond infor-mation contained in the average current, noise, and finite-order cumulants and pertains to all transport measurements. The elementary processes in a dc biased normal junction are single-electron transfers, which are independent at differ-ent energies. At finite temperatures, the electrons are ferred in both directions with correlated left and right trans-fers. At low temperatures only unidirectional electron transfers remain, with direction set by polarity of applied voltage. The elementary processes in a superconductor- nor-mal contact are single- and double-charge transfers.24,28 _At low temperatures and voltages below the superconducting gap only double-charge Andreev processes remain, while above the gap normal single-charge transfers dominate. The

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elementary charge-transfer processes between superconduct-ors with dc bias applied consist of multiple-charge transfers due to multiple Andreev reflections.29–32 _{The interpretation} of elementary charge-transfer processes between supercon-ductors with constant phase difference is more subtle. In this case charge transfers acquire formally negative probabilities and the proper physical interpretation has to include dynam-ics of a detector.13,17,18

In contrast to dc voltage bias, time-dependent voltage drive mixes electron states of different energies which in combination with the Pauli principle leads to a nontrivial charge-transfer statistics. The signatures of this statistics have been studied first through the noise in an ac-driven junction 共photon-assisted noise兲.33,34 _{The photon-assisted} noise at low temperatures is a piecewise linear function of a dc voltage offset with kinks corresponding to integer mul-tiples of the driving frequency and slopes which depend on the amplitude and shape of the ac component. This depen-dence has been observed experimentally in normal coherent conductors35,36 _{and diffusive normal-metal–superconductor} junctions.37 _{The photon-assisted noise at small driving} am-plitudes is due to an electron-hole pair which is created by ac drive共with a low probability per voltage cycle兲 and injected toward the scatterer.38–40_{However, to obtain the elementary} processes, the knowledge of full charge-transfer statistics is needed.

The elementary processes in the presence of time-dependent drive have been identified in Refs.41–43 for the special choice of a driving voltage. The authors have studied Lorentzian voltage pulses of the same sign carrying integer numbers of charge quanta. In this case the elementary pro-cesses are electrons injected toward the scatterer without cre-ation of the electron-hole pairs. The charge transfer is unidi-rectional with binomial statistics set by the effective dc voltage. This result is independent of the relative position of the pulses, their duration, and overlap. The many-body quan-tum state generated by these pulses has been obtained recently.43An alternative way to inject single electrons free of electron-hole pairs is by using time-dependent shifts of a resonant level in a quantum dot.44–48

The elementary charge-transfer processes for arbitrary time-dependent voltage drive have been identified in Ref.49. At low temperatures these processes are single-charge trans-fers which originate from electrons and electron-hole pairs injected toward the scatterer. The electrons are injected due to excess dc voltage applied and give a binomial contribution to the total cumulant generating function. The electron-hole pairs are created by the ac component of the voltage drive. The probabilities of pair creations per voltage cycle depend on the details of the ac drive. For the special choice of opti-mal Lorentzian voltage pulses no electron-hole pairs are cre-ated. In general, however, an ac drive does create electron-hole pairs, with more and more pairs created per voltage cycle as the amplitude of the drive increases. A geometric interpretation of elementary processes is studied in Ref. 50. For time-dependent scatterer, the constraints imposed on charge-transfer statistics restrict the allowed charge transfers.51

In this paper we present a comprehensive study of el-ementary processes in a generic mesoscopic conductor in

different physical regimes, as depicted in Fig. 1. We first obtain the cumulant generating function for a coherent voltage-driven quantum contact at finite temperatures, using both the formalism of Levitov and Lesovik5,6_{and the circuit} theory of quantum transport 共Sec. II兲. We then identify the elementary charge-transfer processes at zero temperature 共Sec. III A兲. Decomposition of the noise into contributions of elementary processes for different voltage signals is studied in Sec. III B.

For ac voltage drive, the differential noise oscillates as the amplitude of the drive increases due to new electron-hole pairs being created per voltage cycle. The number of electron-hole pairs per cycle becomes large for driving am-plitudes much larger than frequency. In this case the statistics in the leading order reduces to uncorrelated electron and hole transfers and depends on an effective voltage only, indepen-dent of the details of the ac drive. The elementary processes can be probed also for both ac and dc voltages present, e.g., in the regime in which the ac component of the drive is kept fixed and the dc offset changes. In this case both the elec-trons and created electron-hole pairs are injected toward the junction. The electron-hole pairs give rise to the excess noise with respect to the noise level set by dc voltage offset.

The notion of elementary processes, being a matter of interpretation of the full charge-transfer statistics, provides physically plausible and intuitive description of transport in terms of statistically independent events. Such a description persists in all transport measurements. For example, the pic-ture of electrons and electron-hole pairs injected toward the FIG. 1. The elementary processes shown in different regimes. 共a兲 For a dc bias voltage V¯ and at arbitrary temperature Te, the

electrons are transferred independently at different energies. The probability of transfer is given by occupation numbers f1,2共E兲 of the

leads and transmissions兵T_p其 of the junction. The transport occurs when the incoming electronic state is occupied, the outgoing state is empty, and the electron is transferred across the junction.共b兲 On the other hand, an ac voltage drive⌬V共t兲 mixes the electron states of different energies. At zero temperature it creates the electron-hole pairs with probabilities pk, in addition to electrons injected due to

excess dc voltage. A created electron-hole pair contributes to trans-port only if one particle from a pair is transmitted and the other is reflected, which occurs with probability T_pR_p共R_p being reflection coefficient兲. 共c兲 Finally, at finite temperature and in the presence of a time-dependent voltage drive, the statistics can be interpreted in terms of multiple n-charge transfers with probabilities qn.

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scatterer, which we have inferred from a study of a two-terminal junction, is further supported by the evaluation of the full counting statistics in a beam-splitter geometry in which the injected particles can be separated into different outgoing terminals. These processes can be probed by cur-rent cross correlations共Sec. III C兲.

The method we use is also suitable for systematic calcu-lation of the higher-order current correlators at finite

tem-perature. The current noise power and the third cumulant in

the presence of time-dependent voltage drive are obtained in Sec. IV. The interpretation of the full counting statistics and the elementary processes are fundamentally different at finite temperature. For a periodic drive, the elementary processes are transfers of multiple integer charge quanta with prob-abilities which depend on energy and temperature共Sec. V兲.

II. CUMULANT GENERATING FUNCTION

The system we consider is a generic two-terminal meso-scopic conductor characterized by a set of transmission ei-genvalues兵Tp其. For a large number of channels, the transport

properties are universal and independent of microscopic de-tails of geometry of the junction and positions of impurities. We can neglect the energy dependence of transmission ei-genvalues if the electron dwell time is small with respect to time scales set by the inverse temperature and applied volt-age. The cumulant generating function of the transferred charge is given by determinant formula5,41,52

S共␹兲 = ln det关1 + f共S†_⌳ ␹ †_S_⌳ ␹− 1兲兴, 共1兲 where f =

冉

⬘

−E⬙兲,

where fi共E兲=共e共E−eVi兲/Te+ 1兲−1, Viis the voltage applied, and

Te is the electronic temperature. In the presence of

time-dependent drive the occupation numbers fi共E

⬘

,E

⬙

兲 are not

diagonal in energy indices and do not commute. The voltage drive V共t兲 can be incorporated via the gauge transformation in time representation f1→UfV=0U† with U共t

⬘

, t

⬙

兲

= exp关−i兰0 t⬘

eV共t兲dt兴␦共t

⬘

− t

⬙

兲, where we assume the convolu-tion over internal time indices.

Equation共1兲 can be simplified using polar decomposition

,

共4兲 where 2s takes into account the spin degree of freedom.

Since the operators in the first 共the second兲 row commute, the determinant can be calculated blockwise. Using Eq.共A1兲

given in Appendix we obtain

S共␹兲 = 2s

兺

p

Tr_Eln关1 + 共1 − f1兲f2Tp共ei␹− 1兲

+ f1共1 − f2兲Tp共e−i␹− 1兲兴共5a兲

=2s

兺

p Tr_Eln关1 + f₂共1 − f₁兲Tp共e i_␹ − 1兲 +共1 − f2兲f1Tp共e−i␹− 1兲兴. 共5b兲

Here we used the matrix identity ln det共M兲=Tr ln共M兲 and the trace Tr_E is taken in energy indices. The logarithm is taken assuming convolution over internal energy indices, e.g., 共f₁f₂兲_E_⬘_E_⬙=共2␲兲−1_兰dE

1f1共E

⬘

,E1兲f2共E1,E

⬙

兲.

In the following we show that the same result is obtained within the circuit theory of mesoscopic transport.24–26_{In this} case the system is represented by discrete circuit elements as depicted in Fig. 2. The cumulant generating function of the transferred charge is given by16,22

S共␹兲 =

兺

p Tr ln

冋

1ˇ +Tp 2

冉

兵Gˇ1共␹兲,Gˇ2共0兲其 2 − 1ˇ

冊

册

, 共6兲 where Gˇi共0兲 =

冉

1 2hi 0 − 1

冊

共7兲

are the quasiclassical Keldysh-Green’s functions of the ter-minals with hi= 1 − 2fi. The counting field is incorporated

FIG. 2. The schematic circuit-theory representation of a two-terminal junction. The left and right two-terminals are characterized by the quasiclassical Keldysh-Green’s functions Gˇ_1,2共0兲. The counting field␹ is related to the charges which enter the left terminal through the cross section indicated by dashed line. It can be incorporated into the corresponding Keldysh-Green’s function via gauge trans-formation given by Eq.共8兲.

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through the gauge transformation of the Keldysh-Green’s function of the left terminal,

Gˇ1共␹兲 = e−i␹␶ˇ1/2Gˇ1共0兲ei␹␶ˇ1/2, 共8兲 where

␶

ˇ₁=

冉

0 1

1 0

冊

共9兲

in Keldysh space. The logarithm and the trace in Eq.共6兲 are

taken assuming the convolution both in Keldysh and in en-ergy indices.

Equation共6兲 can be simplified by using matrix

represen-tation in Keldysh space. After rewriting Tr ln共¯兲=ln det共¯兲 and taking the determinant by blocks using Eq. 共A1兲, the

result coincides with Eq.共5兲. This proves the equivalence of

the circuit-theory expression forS共␹兲关Eq. 共6兲兴 and the

Levi-tov determinant formula 关Eq. 共1兲兴 for a coherent

two-terminal scatterer. The circuit-theory approach becomes ad-vantageous in the case of several scatterers in series23,54 _or for a multiterminal mesoscopic conductor with large number of conduction channels.27 It gives the charge-transfer statis-tics in terms of the scattering properties of individual ele-ments, thus circumventing a nontrivial task of obtaining the scattering matrix of the composite system and averaging over the phase shifts.

Equation 共5兲 is valid for a dc bias applied and

energy-dependent transmission probabilities. In this case the occu-pation numbers f1,2 are diagonal in energy and commute with each other and with Tp共E兲. The trace over energy

re-duces simply to the integration and we obtain

S共␹兲 =t0

␲

兺

p

冕

dE ln兵1 + 关1 − f1共E兲兴f2共E兲Tp共ei␹− 1兲

+ f₁共E兲关1 − f₂共E兲兴Tp共e−i␹− 1兲其. 共10兲

Here t₀ is the measurement time which is the largest time scale in the problem, much larger than the characteristic time scale on which the current fluctuations are correlated.55_The form ofS共␹兲 reveals that the elementary charge-transfer pro-cesses are single-electron transfers to the left and to the right. The electron transfers at different energies and in different channels are independent. The term 关1− f1共E兲兴f2共E兲Tpei␹ in

Eq. 共10兲 describes the electron transfer from the right to the

left lead at energyE in the channel p. The probability of this process is proportional to the probability f₂共E兲 that the state in the right lead is occupied, the probability 1 − f1共E兲 that the state in the left lead is empty, and the probability Tp共E兲 of

transfer across the scatterer. A similar analysis holds for the electron transfer from left to right. The left and right trans-fers are correlated becauseS共␹兲 is not a sum of the left- and right-transfer generating functions.

Equation共5兲 is also valid for the time-dependent voltage

applied and energy-independent transmission probabilities. In this case f_1,2共E

⬘

,E⬙兲 do not commute with each other. 关However, the order of f1and f2can be exchanged as shown by Eqs. 共5a兲 and 共5b兲.兴 The logarithm has to be calculated

with the matrix structure of f1,2in energy indices taken into

account because time-dependent drive mixes the electron states with different energies.

A few remarks on applicability of our approach are in order. The circuit theory we have used applies to instanta-neous scattering at the junction, with the frequency␻of the bias voltage much smaller than the inverse dwell time␶d

−1 . It also applies for a dot sandwiched between two terminals and

␻ much smaller than the inverse RC time␶_RC−1 of the charge relaxation on the dot. In a typical experimental situation

␶RCⰆ␶d,56,57with␻limited by the dwell time␻Ⰶ␶d−1.

How-ever, as shown in Ref.39, the photon-assisted noise SI共␻兲 in

the leading order共eV₀/␻兲2_{Ⰶ1 in driving amplitude V} 0does not depend on␶d: in a weakly driven junction the frequency

range is set by the RC time, ␻Ⰶ␶RC−1. The dwell time

be-comes important for larger voltage amplitudes. For eV₀/␻ ⲏ1 a maximum in ⳵SI/⳵␻ develops at the frequency ␻

⬃␶d−1 because the photon-assisted noise SI共␻兲 probes the

electronic distribution function f共E;t兲=兰d␶f共t+␶/2,t

−␶/2兲eiE␶_{which relaxes on a time scale given by} _␶

d.56,57At

frequencies␻⬃␶d −1

, the differential noise⳵SI/⳵V0as a func-tion of V₀is increased with respect to the case ␻Ⰶ␶d−1. The

increase is of the order of 20% with the distances between maxima and minima only weakly affected. This is consistent with the experiments of Schoelkopf et al.36_{and Kozhevnikov}

et al.37 _{which can be described by assuming the} instanta-neous scattering even though the frequency of the ac signal applied is comparable or larger than the inverse dwell time. III. ELEMENTARY CHARGE-TRANSFER PROCESSES AT

T_e= 0

A. Decomposition of the cumulant generating function into elementary processes

In this section we study elementary charge-transfer pro-cesses in a two-terminal junction with constant transmission eigenvalues 兵Tp其 and time-dependent voltage V共t兲 applied.

The cumulant generating function S共␹兲 is given by Eq. 共5兲.

To identify the elementary processes we diagonalize the op-erator under the logarithm in energy indices. As a first step, we rewrite Eq. 共5b兲 in terms of h˜⬅h1and h⬅h2,

S共␹兲 = 2

兺

p

ein␻t. 共13兲 Here V¯ =共1/␶兲兰V共t兲dt is the dc voltage offset and ⌬V共t兲 = V共t兲−V¯ is the ac voltage component. The coefficients an

n ln关1 + Tn共e−i␬␹− 1兲兴. 共16兲

Here ML= t0␻1/␲, MR= t0共␻−␻1兲/␲, and␻1= eV¯ −eV¯/␻␻, where x is the largest integer less than or equal x. The coefficient ␬=⫾1 in Eq. 共16兲 is related to direction of the

charge transfer, with ␬= 1共␬= −1兲 for eV¯⬎0 共eV¯⬍0兲. The total measurement time t0 is much larger than the period ␶ and the characteristic time scale on which the current fluc-tuations are correlated.

The parameters pkL,Rin Eq.共15兲 depend on the details of

the time-dependent voltage drive. They are given by p_kL,R = sin2共␣kL,R/2兲, where e⫾i␣kL共R兲are the eigenvalues of共hh˜兲nm

calculated forE苸共0,␻1兲关E苸共␻1,␻兲兴. The eigenvalues are obtained using a finite-dimensional matrix 共hh˜兲nm, with the

cutoff in indices n and m being much larger than the charac-teristic scale on which兩an兩 vanish.

Equations共15兲 and 共16兲 give the charge-transfer statistics

in a two-terminal junction driven by a time-dependent volt-age. The result is valid in the zero-temperature limit, with no thermal excitations present. The probability distribution of the number of charges N transferred within measurement time is given by P共N兲=共2␲兲−1_兰

−␲

␲ _d_␹_exp关S共_␹_兲−iN_␹_{兴. The} cumulants of N are given byCn=关⳵i␹

n_S共_␹_兲兴

␹=0and are related to the higher-order current correlators at zero frequency.17,18

For example, the average current, the current noise power, and the third cumulant of current fluctuations are given by

I =共e/t0兲⳵i␹S兩␹=0, SI=共e2/t0兲⳵i␹ 2_S兩 ␹=0, and CI=共e3/t0兲⳵i␹ 3_S兩 ␹=0, respectively.59

The elementary charge-transfer processes can be identi-fied from the form ofS, similarly as in Refs.60and61. The result is depicted schematically in Fig. 3. The S₂共␹兲 de-scribes unidirectional single-electron transfers due to the ex-cess dc bias voltage V¯ applied. The number of attempts for an electron to traverse the junction within measurement time

t₀ is given by t₀兩eV¯兩/␲共we take into account both spin ori-entations兲. The transfer events in different channels are inde-pendent. The term Tne−i␬␹ in Eq. 共16兲 describes a

single-electron transfer with probability Tn in nth channel. The

unidirectional charge-transfer processes give contributions to the average current and higher-order cumulants.

The S1L共␹兲 and S1R共␹兲 describe bidirectional charge-transfer processes. Different bidirectional processes are la-beled by k in Eq. 共15兲. These processes represent

electron-hole pairs created in the source terminal by the time-dependent voltage drive and injected toward the scatterer. The probability of such an electron-hole pair creation is given by pk and depends on the details of the ac voltage

component⌬V共t兲. The charge transfer in nth channel occurs if one particle 共e.g., electron兲 is transmitted and the hole is reflected or vice versa. The probability for the whole process is given by TnRnpk. The bidirectional processes contribute to

the noise and higher-order even cumulants. However, they give no contributions to the average current and higher-order odd cumulants because electrons and holes are transmitted with the same probability.

The two types of bidirectional processesS1L andS1R dif-fer in the number of attempts ML,R and have different

prob-abilities pkL,R. The ML,Rdepend on the number eV¯ /␻of

uni-directional attempts per period per spin. The simplest statistics is obtained for an integer value of eV¯ /␻ for which

S1Lvanishes, in agreement with Ref.7.

B. Comparison of different time-dependent voltages The elementary processes at zero temperature can be probed by noise measurements. In what follows we compare FIG. 3. The elementary charge-transfer processes in a voltage-driven junction. The unidirectional processes represent single elec-trons injected toward the contact due to excess dc offset voltage V¯ applied. The bidirectional processes represent electron-hole pairs created by time-dependent voltage drive and injected toward the contact. The probabilities pk of pair creations depend on the ac

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the elementary processes and the noise generated by different time-dependent voltages. We focus on standard periodic volt-age signals such as cosine, square, triangle, and sawtooth. We also present results for Lorentzian voltage pulses which provide the simplest charge-transfer statistics for certain am-plitudes. The applied voltage is characterized by coefficients

an given by Eq. 共13兲. These coefficients are calculated

ex-plicitly in Appendix, Sec. B for the driving voltages of inter-est.

We first consider ac voltage drive with no dc bias applied,

V

¯ =0. In this case only the bidirectional processes of R-type

remain, S共␹兲=S1R共␹兲. The number of attempts during the measurement time is given by M = 2t0/␶which corresponds to a single attempt per voltage cycle per spin. The current noise power is given by

SI= 2e2_␻ ␲

冉

兺

p TpRp

冊冉

兺

k pk

冊

, 共17兲

with the probabilities pkof electron-hole pair creations

ob-tained from the eigenvalues of 共hh˜兲nm, as discussed in Sec.

III A.

The probabilities pk for the harmonic voltage drive V共t兲

= V₀cos共␻t兲 are shown in Fig. 4共a兲. As the amplitude V₀ increases the probabilities pk also increase and new pairs

start to enter the transport. This results in the oscillatory change in the slope of SI as a function of V0. The decompo-sition of the differential noise ⳵SI/⳵V0 into elementary pro-cesses is shown in Fig. 4共b兲. The differential noise ⳵SI/⳵V0 for different time-dependent voltages is shown in Fig.5 for comparison.

The bidirectional processes with the unit probability pk

= 1 represent electron-hole pairs which are created and in-jected toward the scatterer in each voltage cycle. In this case the electron and hole transfers are statistically independent. This can be seen from the corresponding cumulant generat-ing function which reduces to Snk=Snk

e

+S_nkh , where S_nke,h共␹兲 =共2t0/␶兲ln关1+Tn共e⫿i␹− 1兲兴 describe electron and hole

trans-port. The electron-hole pairs which are created with the prob-ability 0⬍pk⬍1 result in correlated electron and hole

trans-fers. The corresponding cumulant generating function is given bySnk共␹兲=共2t0/␶兲ln关1+TnRnpk共ei␹+ e−i␹− 2兲兴 and

can-not be partitioned into independent electron and hole contri-butions.

The interpretation of the shot noise in terms of electron-hole pair excitations has been studied previously in Ref. 38

in the regime of low-amplitude harmonic driving eV₀/␻ Ⰶ1. In this case only one electron-hole pair is excited per period with probability p1⬇共eV0/2␻兲2Ⰶ1. Remarkably, Fig.

4shows that the single-electron-hole pair is excited not only for small amplitudes but also for amplitudes comparable or even larger than the drive frequency. This extended range of validity can be covered by taking into account the higher-order terms in the expression for probability p1 ⬇兺n=1⬁ n关Jn共eV0/␻兲兴2, where Jn共x兲 denote Bessel functions

共cf. Sec. IV兲. The first three terms approximate the exact p1 FIG. 4. The probabilities pkof electron-hole pair creations for

harmonic drive V共t兲=V0cos共␻t兲 as a function of the amplitude V0

关panel 共a兲兴. With increasing V0more and more pairs are created per

voltage cycle. The decomposition of the differential noise ⳵SI/⳵共eV0兲共solid line兲 into contributions of elementary processes

共dashed lines兲 is shown in panel 共b兲.

FIG. 5. The differential noise ⳵S_I/⳵共eV₀兲 as a function of the amplitude V0for different ac driving voltages 共V¯=0兲 is shown in

top panel: square 共dashed-dotted兲, cosine 共solid兲, triangle 共dotted兲, sawtooth 共dashed兲, and Lorentzian pulses 共short-dotted line兲. The width of Lorentzian pulses is␶_L= 0.1␶. The oscillations are due to elementary processes which are created as the voltage amplitude increases 共cf. Fig. 4兲. The decomposition of the differential noise into contributions of elementary processes is shown in lower panels.

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shown in Fig.4to accuracy better than 0.3% for eV₀/␻ⱗ2. For large driving amplitudes the charge-transfer statistics does not depend on the details of time-dependent drive8,42 and can be characterized by an effective voltage Veff =␶−1_兰

␶dt兩V共t兲兩共here we assume V¯=0兲. In this case there are

N =兩eVeff兩/2␻Ⰷ1 processes with pk= 1. The charge-transfer

statistics in the leading order in N consists of Ne,h= 2Nt0/␶ uncorrelated electrons and holes injected toward the contact in each transport channel during measurement time: S共␹兲 = Ne兺nln关1+Tn共e−i␹− 1兲兴+Nh兺nln关1+Tn共ei␹− 1兲兴. This result

can be interpreted by comparison with the generating func-tion of a dc bias given by Eq.共16兲. The electrons 共holes兲 are

injected during time intervals␶e共␶h=␶−␶e兲 per voltage cycle

in which eV共t兲⬎0 关eV共t兲⬍0兴. For a large number of injected particles, the time-dependent drive in these intervals can be replaced by effective dc voltages V_eff共e兲 and V_eff共h兲. The number of attempts is given by Ni=共t0/␲兲共␶i/␶兲兩eVeff共i兲兩共i=e,h兲, in agreement with Eq.共16兲. The noise generated is the sum of

independent electron and hole contributions SI

=共兩e3_V

eff兩/␲兲兺nTnRn. This explains the asymptotic behavior

of differential noise at large driving amplitudes shown in Figs.4and5.

In the following we consider time-dependent voltage drive with a nonzero dc offset V¯ ⫽0 which creates both directional and bidirectional elementary processes. The uni-directional processes generate dc noise which is given by

Sdc=共e2/␲兲共兺pTpRp兲eV¯. The bidirectional processes given

by Eq. 共15兲 generate the excess noise SI− Sdc =共e2_/t 0兲⳵i2_␹S1兩␹=0, SI− Sdc= 2e2␻ ␲

冉

兺

p TpRp

冊

兺

k 关v¯pkL+共1 − v¯兲pkR兴, 共18兲

where¯ = eVv ¯ /␻−eV¯/␻ is the fractional part of eV¯/␻. For

v

¯⫽0,1 there are two types of bidirectional processes

共la-beled by L and R兲 with different numbers of attempts and different probabilities of electron-hole pair creations.

We study bidirectional processes first in the regime V₀ = V¯ in which the ac amplitude and the dc offset increase simultaneously. The excess noise SI− Sdc for different time-dependent voltages is shown in Fig.6共a兲for comparison. The probabilities pkL共R兲as a function of voltage are shown in Fig.

6共b兲 for the cosine voltage drive. The decomposition of the excess noise into contributions of elementary processes is shown in Fig.6共c兲. For the cosine voltage drive there are two bidirectional processes共one of L type and another of R type兲 which are excited per period. The L-type processes transform continuously into the R-type ones at integer values of eV¯ /␻, while the R-type processes disappear. The steplike evolution of the probabilities of R-type processes as a function of volt-age does not introduce discontinuities in the current noise because the corresponding number of attempts vanishes. In-stead, the interplay between L and R processes results in kinks and the local minima at integer values of eV¯ /␻, as shown in Fig. 6共a兲. The L-type 共R-type兲 processes give the dominant contributions as eV¯ /␻ approaches the integer

val-ues from the left 共right兲 because of the number of attempts which is proportional tov¯ 共1−v¯兲.

The Lorentzian voltage drive is special because it pro-vides the simplest one-particle charge-transfer statistics. This is achieved for impulses carrying an integer number of charge quanta eV0/␻= N at offset voltages V¯ ⱖV0. In this case the Lorentzian pulses do not create electron-hole pairs and only single-particle processes given by Eq. 共16兲 remain.

The charge-transfer statistics in each transport channel is

ex-actly binomial.41,42_{The noise is reduced to the minimal noise} level Sdc of the effective dc bias, as shown in Fig. 6共a兲. A formal reason for this is the vanishing of the coefficients

an= 0 for n⬍−N 关cf. Eqs. 共29兲 and 共A9兲兴. A many-body state

which is created by optimal Lorentzian pulses has been ob-tained by Keeling et al.43 _recently.

Now we focus on regime in which the ac component ⌬V共t兲 of the drive is fixed and the dc offset V¯ changes. At low temperatures, the shot noise SI is a piecewise linear

function of the dc voltage offset V¯ with kinks corresponding to integer multiples of the driving frequency eV¯ /␻= N and slopes which depend on the shape and the amplitude of the ac voltage component.33,34 _{The differential noise} _⳵_S

I/⳵共eV¯兲

FIG. 6. The excess noise SI− Sdcis shown in panel共a兲 for

dif-ferent time-dependent bias voltages: square共dashed兲, cosine 共short-dotted兲, triangle 共dashed-dotted兲, sawtooth 共dotted兲, and Lorentzian pulses共solid line兲. The width of Lorentzian pulses is␶_L= 0.1␶. The amplitude and the dc offset increase simultaneously, V0= V¯ . The

probabilities p_kL 共dotted lines兲 and p_kR 共dashed lines兲 of bidirec-tional elementary processes are shown in panel 共b兲 for the cosine voltage drive V共t兲=V¯+V¯ cos共␻t兲. Only one elementary process of the L type and one of the R type can be created per period共k=1兲. The decomposition of the excess noise SI− Sdcinto contributions of

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consists of a series of steps, as shown in Fig.7共a兲. The piece-wise linear dependence of noise can be understood in terms of probabilities of elementary processes.7_{For the ac} compo-nent fixed, the probabilities pkRand pkLof electron-hole pair

creations are piecewise constant as a function of V¯ and can be relabeled by pkR共V¯兲⬅pk共N兲 and pkL共V¯兲⬅pk共N+1兲 where N

=eV¯/␻. This results in the piecewise linear dependence of the excess noise given by Eq.共18兲 as a function of V¯. For the

dc offset in the interval m − 1ⱕeV¯/␻⬍m with m integer, the excess noise is a linear combination of processes pk共m−1兲and

p_k共m兲共k=1,2,...兲 with the number of attempts proportional to

1 −v¯ and v¯, respectively. For mⱕeV¯/␻⬍m+1 a different set

of processes pk共m兲and pk共m+1兲contributes and the excess noise

changes slope. The excess noise for harmonic drive is shown in Fig. 7共b兲共solid line兲 with decomposition into contribu-tions of elementary processes 共dotted lines兲. The probabili-ties of elementary processes are shown in the inset.

C. Beam-splitter geometry

Here we study a multiterminal beam-splitter geometry de-picted in Fig. 8. The source terminal is biased with a

time-dependent periodic voltage V共t兲 and through a mesoscopic conductor attached to several outgoing terminals. The con-ductor is characterized by a set of transmission eigenvalues 兵Tp其 and the outgoing leads by conductances gi. We are

in-terested in the limit in which the outgoing leads play a role of a detector and only weakly perturb the charge transfer across the conductor. This is achieved for the conductance

g_⌺⬅兺igito the outgoing leads much larger than the

conduc-tance g⬅共e2_/_␲_兲兺

pTp of a conductor. In this case the

par-ticles which traverse the conductor enter the outgoing termi-nals with negligible back reflection into the source terminal. A similar setup with spin-selective outgoing contacts has been used in Ref.61to reveal singlet electron states.

The cumulant generating function is calculated using the circuit theory similarly as in Sec. II for the two-terminal case. In contrast to Sec. II, here we assign the counting fields

␹i to the outgoing terminals. Green’s function Gˇ 共0兲 of the

source terminal and Green’s functions Gˇi共␹i兲 of the outgoing

ones are given by

Gˇ 共0兲 =

冉

1 2h˜ 0 − 1

冊

共19a兲 and Gˇi共␹i兲 = e−i␹i␶ˇ1/2

冉

1 2h 0 − 1

冊

e i␹_i␶ˇ₁/2 . 共19b兲

Here h˜ and h are the matrices in energy indices defined in Sec. III A. Green’s function Gˇcof the internal node is given

by matrix current conservation and normalization condition

Gˇ_c2= 1. In the limit g_⌺Ⰷg, the central node is strongly coupled to the outgoing terminals and the Gˇccan be obtained

in the lowest order with the terminal Gˇ unattached. For sim-plicity we assume tunnel couplings to the outgoing terminals. In this case matrix current conservation reduces to22,24

冋

兺

_i giGˇi共␹i兲,Gˇc

册

= 0. 共20兲

In the following we work in the low-temperature limit in which h2= h˜2= 1. We seek for the solution in the form Gˇc

= pc·␶ˇ where␶ˇ =共␶ˇ1,␶ˇ2,␶ˇ3兲 is the vector of Pauli matrices in Keldysh space. From Eqs. 共19兲 and 共20兲 we obtain pc

FIG. 7. The differential noise ⳵SI/⳵共eV¯兲 as a function of dc

offset V¯ is shown in panel 共a兲 for different time-dependent bias voltages: cosine 共solid兲, square 共dashed兲, and sawtooth 共dashed-dotted line兲. The ac amplitude of the drive is kept constant,

eV0/␻=5. The excess noise SI− Sdcas a function of V¯ 共solid line兲

and the decomposition into contributions of elementary processes 共dotted lines兲 are shown for the cosine drive in panel 共b兲关cf. Eq. 共18兲兴. The probabilities of elementary processes are shown in the inset.

FIG. 8. Schematic representation of a multiterminal beam split-ter. The source terminal is biased with a time-dependent voltage

V共t兲 with respect to the outgoing terminals. The total conductance g_⌺=兺_ig_iof the outgoing leads is assumed to be much larger than the conductance g =共e2/␲兲兺pTpof the source contact.

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=共h,ihc−s,ihs+c兲 where c=兺i˜gicos共␹i兲, s=兺i˜gisin共␹i兲,

and g˜i= gi/g_⌺. The cumulant generating function of the

charge transferred is given by

S共兵␹其兲 =

兺

p Tr ln

冋

1ˇ +Tp 2

冉

兵Gˇ共0兲,Gˇc共兵␹其兲其 2 − 1ˇ

冊

册

, 共21兲 where summation over the internal Keldysh and energy indi-ces is assumed. The diagonalization of 兵Gˇ,Gˇc其E⬘E⬙ is

per-formed along the lines in Ref.49.

For the cumulant generating function we obtain S共兵␹其兲 =S1L+S1R+S2 whereS1L,Rare the contributions of bidirec-tional processes and S2 is the contribution of the unidirec-tional ones. The unidirecunidirec-tional processes are described by

S2共兵␹其兲 = t0eV¯ ␲

兺

n ln

˜i˜gj共ei␹ie−i␹j+ e−i␹iei␹j− 2兲

冊

册

. 共23兲

Here k labels the bidirectional processes, n labels transport channels, i and j label the outgoing terminals, and ␣= L , R. The number of attempts M_␣and the probabilities pk␣are the

same as in Sec. III A.

Equations 共22兲 and 共23兲 give the statistics of the charge

transfer in a multiterminal beam splitter at low temperatures in the presence of a periodic time-dependent drive at the source terminal. TheS1共兵␹其兲 and S2共兵␹其兲 have a direct physi-cal interpretation. The unidirectional processes, which are described by S₂, are the single-electron transfers across the structure due to the dc offset voltage V¯ applied to the source terminal. The term Tn˜giei␹iin Eq.共22兲 represents the process

in which an electron in the nth transport channel traverses the conductor with probability Tn and enters the outgoing

terminal i with probability g˜i= gi/g⌺.

On the other hand, the bidirectional processes represent the electron-hole pairs created in the source terminal and injected toward the conductor. The probabilities of such ex-citations are given by pk␣. The interpretation of the cumulant

generating function given by Eq.共23兲 can be obtained from a

simple counting argument. The term pkTnRn˜giei␹iin Eq.共23兲

represents the process in which electron-hole excitation is created, hole is reflected, and electron is transmitted into the outgoing terminal i. Similarly, the term pkTnRn˜gie−i␹i

repre-sents the process in which the electron is reflected and the hole is transmitted. Finally, the term pkTn

2_˜_g

i˜gjei␹ie−i␹j 共i⫽ j兲

represents the process in which both particles are transmitted with electron entering terminal i and hole entering terminal j. Charge-transfer statistics and current correlations can be obtained using the cumulant generating functionS共兵␹其兲. For

example, the current cross correlation between different ter-minals i and j is given by Sij=共e2/t0兲⳵2i␹_i,i␹_jS兩␹=0,

Sij= − 2e2␻ ␲

冉

兺

n Tn 2

冊

g ˜i˜gj

兺

k 关v¯pkL+共1 − v¯兲pkR兴. 共24兲

We find that only bidirectional processes in which both par-ticles are transferred共one into the terminal i and another into the terminal j兲 give the contributions to the cross correlation

Sij. This has been obtained previously by Rychkov et al.38in

the special case of harmonic drive in the limit of small driv-ing amplitudes共V¯=0 and eV0/␻Ⰶ1兲.

The cross correlation Sij depends on bidirectional

pro-cesses and is proportional to the excess noise SI− Sdc of a two-terminal junction关Eq. 共18兲兴. In a two-terminal setup, the

excess noise is just a small correction to Sdc generated by unidirectional processes for a bias voltage with V¯ ⯝V0 共Fig.

6兲. The contribution of Sdc component can be reduced by measuring current cross correlations between different out-going terminals in the beam-splitter geometry with negligible backscattering.

IV. CUMULANTS AT TeÅ 0

The full counting statistics and the corresponding elemen-tary transport processes obtained in Sec. III are valid descrip-tion in the low-temperature limit only, in which electron-hole pairs are created by the applied voltage and no thermally excited pairs exist. Formally, the diagonalization of the op-erator in Eq.共11兲 in energy indices, which is needed to

de-duce the elementary processes, is based on the involution property of h operators: h2_{= h}˜2_{= 1. This property no longer} holds at finite temperatures which are comparable to the ap-plied voltage. Nevertheless, the method we use enables the efficient and systematic analytic calculation of the higher-order cumulants at finite temperatures.

The cumulants can be obtained directly from Eq.共11兲 by

expansion in the counting field to the certain order before taking the trace. The trace of a finite number of terms can be taken in the original basis in which h and h˜ are defined. In the following we illustrate the approach by calculation of the average current, the current noise power, and the third cumu-lant at finite temperatures. From Eq.共11兲 we obtain

k,m

akak+mⴱ h共E

⬘

兲2␲␦共E

⬙

−E⬘− m␻兲, 共26a兲

h

˜共E

⬘

,E

⬙

兲 =

兺

k,m

akak+mⴱ h共E

⬘

− k␻− eV¯ 兲2␲␦共E

⬙

−E

⬘

− m␻兲.

共26b兲 Here h共E兲=tanh共E/2Te兲 and we used the properties of 兵an其

given by Eq.共14兲. After integration over energy in Eq. 共25兲

we obtain the average current I =共e2/␲兲共兺pTp兲V¯. The current

noise power and the third cumulant are given by

SI= e2 ␲

冋

2Te

p Tp共1 − Tp 2_{兲 + 3}

兺

p Tp 2_{共1 − T} p兲 ⫻

兺

n=−⬁ ⬁ 兩an兩2

冋

2Tecoth

冉

eV¯ + n␻ 2Te

冊

−共eV¯ + n␻兲coth2

冉

eV¯ + n␻ 2Te

冊

册

冎

, 共28兲

respectively. For a dc voltage bias an=␦n,0and the noise and

the third cumulant reduce to SI= eFI and CI= e2F3I at zero temperature. Here F =关兺pTp共1−Tp兲兴/共兺pTp兲 is the Fano

fac-tor and F3=关兺pTp共1−Tp兲共1−2Tp兲兴/共兺pTp兲.

The average current is linear in dc voltage offset, which is consistent with the initial assumption of energy-independent transmission eigenvalues and instant scattering at the con-tact. The result for the current noise power 关Eq. 共27兲兴

de-scribes the photon-assisted noise for arbitrary periodic volt-age drive. The coefficients an for harmonic drive V共t兲=V¯

+ V0cos共␻t兲 are given by the Bessel functions, an

= Jn共eV0/␻兲, and Eq. 共27兲 reduces to the previous results obtained by Lesovik and Levitov33 _{and Pedersen and} Büttiker34 _{共see also Ref.} ₁_{兲. The accurate noise} measure-ments at finite temperatures in the presence of the harmonic driving are performed in Ref. 35. The results are in agree-ment with Eq.共27兲.

In the following we discuss the low- and high-temperature limits of SIand CI. At high temperatures TeⰇ兩eV¯兩, n0␻, with

n0␻being the characteristic energy scale on which兩an₀兩

van-ish, the current noise power reduces to the thermal equilib-rium value SI= 2TeG, which is just a manifestation of the

fluctuation-dissipation theorem. The third cumulant is in this regime proportional to the average current, CI= e2FI. At high

temperatures SIand CIcarry no information on the details of

the time-dependent voltage drive.

At low temperatures TeⰆ兩eV¯兩, n0␻, the current noise power reduces to SI= e2 ␲

冉

兺

p TpRp

冊

兺

n=−⬁ ⬁ 兩eV¯ + n␻兩兩an兩2. 共29兲

The differential noise ⳵SI/⳵V¯ is a piecewise constant

N−0 =2e 3 ␲

冉

兺

p TpRp

冊

兩a−N兩2. 共30兲 All steps add up to 共2e3_/_␲_兲兺

pTpRp because of 兺n兩an兩2= 1

关see Fig.7共a兲兴.62_{The steps in}_⳵_S

I/⳵V¯ have been measured for

harmonic drive in normal36 _{and normal-superconductor}37 junctions. In the superconducting state they appear at integer values of 2eV¯ /␻, which can be interpreted as a signature of the elementary charge transport processes in units of eⴱ= 2e. The effective charge is doubled in the superconducting state due to the Andreev processes. We point out that for a general voltage drive, certain steps at integer values of eⴱV¯ /␻= N may vanish if the corresponding coefficient a−N= 0. For ex-ample, for a square-shaped drive with integer amplitude

eⴱV₀/␻= m, the steps at eⴱV¯ /␻= m + 2k 共k⫽0兲 vanish 关cf. Eq. 共A4兲 and Fig.7共a兲兴.

At low temperatures, the third cumulant reduces to CI

= e2_F

3I. Unlike the current noise power, the third cumulant at low temperatures does not depend on the ac component of the voltage drive. This is because the bidirectional processes, created by the ac voltage component, do not contribute to odd-order cumulants at low temperatures关recall Eq. 共15兲兴.

We conclude this section by comparison of two formulas for the current noise power at zero temperature. For simplic-ity we take V¯ =0. Equation 共29兲 for the current noise power

reduces to SI= 2e2␻ ␲

冉

兺

p TpRp

冊

冉

兺

n=1 ⬁ n兩an兩2

冊

. 共31兲

Here we used Eq.共14兲 to restrict the summation to the

posi-tive n only. On the other hand, the current noise power is also given by Eq. 共17兲. Regardless the similar form, the

physical content of these two equations is very different. Both equations give the same result for SIas a consequence

of the invariance of trace. However, Eq. 共17兲 has been

ob-tained by taking the trace of the operator in Eq.共11兲 in the

basis in which it is diagonal. The cumulant generating func-tion given by Eq. 共15兲 is decomposed into contributions of

elementary and statistically independent processes. The terms proportional to pk which appear in Eq. 共17兲 are the

contributions of these processes to the noise. Equation 共31兲

has been obtained by taking the trace of the operator in Eq. 共11兲 in a basis in which it is not diagonal. Although the end

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n兩an兩2 which appear in Eq.共31兲 have no direct physical

in-terpretation.

In the limit of small-amplitude voltage drive, eV0/␻Ⰶ1, only one electron-hole pair is excited per period with prob-ability p₁. Comparing Eqs. 共17兲 and 共31兲 we obtain that in

this case p1=兺n=1⬁ n兩an兩2. In fact, as shown in Fig.4 for

har-monic drive, the assumption of small amplitudes can be re-laxed to the amplitudes comparable or even larger than the drive frequency. The accuracy of this approximation depends on the actual voltage drive共cf. Figs.4and5兲.

V. CHARGE-TRANSFER PROCESSES AT TeÅ 0

Here we study the effect of finite temperature on elemen-tary charge-transfer processes. For a periodic voltage drive, the cumulant generating function S共␹兲 in Eq. 共11兲 can be

recast in the following form:

S共␹兲 =t0 ␲

兺

p

冕

0 ␻ dE ln

冉

1 +

兺

n=−⬁ ⬁ qn共ein␹− 1兲

冊

. 共32兲

Here qnein␹ describes the process in which n charges are

transmitted through the scatterer with probability qn. The

charge transfer can occur in either direction depending on the sign of n. To obtain the probabilities qn we diagonalize the

operator under the logarithm in Eq.共11兲 numerically,

multi-ply all the eigenvalues, and take the inverse Fourier transfor-mation with respect to␹. For simplicity we focus on ac volt-age drive with no dc offset. For a given ac drive, the probabilities qndepend on energy, temperature, and

transmis-sion, qn⬅qn共E/␻, Te/␻; T兲. Transport properties of the

junc-tion enter through summajunc-tion over the transmission eigen-channels 兵Tp其.

Equation 共32兲 allows us to study the crossover from

ac-driven fluctuations at zero temperature to thermal fluctua-tions at temperatures much larger than the voltage drive. The important difference between the two regimes is that ac drive mixes electron states of different energies in contrast to ther-mal fluctuations which are diagonal in energy. As we have seen, this is also reflected in the charge-transfer statistics. The statistics for an ac drive 共at zero temperature兲 can be interpreted in terms of electron-hole pairs after mapping the problem into energy interval set by the driving frequency␻. The mixing of energy states is taken into account by diago-nalization of the remaining matrix structure in energy. On the other hand, the statistics in the thermal limit is composed of electron transfers共in either direction兲 which are independent at different energies. In the crossover regime there is still some mixing of different energy states present and we use the probabilities qnin Eq.共32兲 with energy mapped in the␻

interval. These different physical situations are depicted schematically in Fig.1.

The probabilities qnare shown in Fig.9for cosine voltage

drive V共t兲=V0cos共␻t兲 and a transport channel of transmis-sion Tp= 0.5. We first discuss thermal limit of probabilities qn

shown by solid lines in Figs.9共a兲and9共b兲. For␻ⰇTe, the

mapping into interval␻is irrelevant since it is larger than the energy scale of thermal fluctuations. In this case the prob-abilities reduce to q_⫾1= f共1− f兲Tp and qn= 0 otherwise, in

accordance with Eq. 共10兲关cf. Fig. 9共b兲兴. As the mapping

interval becomes comparable or smaller than the tempera-ture, the higher-order “bands” qn 共n= ⫾1, ⫾2,...兲 start to

appear. For ␻ⱗ2Te the probabilities qn are independent of

energy 关Fig.9共a兲兴 and are given by

qn=

冕

−␲

␲ _d_␹ 2␲e

S˜共␹兲−in␹ _共33兲

with S˜共␹兲=−4共Te/␻兲arcsin2关

冑

Tpsin共␹/2兲兴. Here S˜共␹兲 is

re-lated to the cumulant generating function S共␹兲 in thermal equilibrium by S共␹兲=共2t0/␶兲兺pS˜共␹兲. The latter is obtained

from Eq. 共10兲 after energy integration.8

We emphasize again that the picture of multiple-charge transfers in thermal equilibrium limit is solely due to map-ping into the energy interval ␻ in Eq. 共32兲. Although such

mapping is not needed because thermal fluctuations are di-agonal in energy, we nevertheless perform it here to provide a limit for ac-driven processes as the temperature increases. At temperatures much larger than ac drive there is no mixing of different energy states and the statistics in the full energy FIG. 9. The probabilities q_⫾n of multiple-charge transfers for harmonic voltage drive V共t兲=V₀cos共␻t兲 and transmission T_p= 0.5. The temperature dependence of qnis shown in panel共a兲 for

differ-ent driving amplitudes. The energy dependence of qn at low

tem-peratures is indicated by plots for E/␻=0 共top line兲 and E/␻=0.5 共bottom line兲. The energy dependence of qnis shown in the thermal

limit 共b兲 eV₀/␻=0 and for 共c兲 eV₀/␻=1. At zero temperature q_⫾1 do not depend on energy and are given by q_⫾1= T共1 − T兲p₁共eV₀/␻兲共cf. Fig.4兲. As the temperature increases, multiple-charge transfers start to enter the transport. The limit of thermal fluctuations is reached at temperatures larger than the driving am-plitude. It can be recast again into single-charge transfers indepen-dent in the full energy range.

(12)

range reduces to single-electron transfers which are indepen-dent at different energies 关Eq. 共10兲兴.

Now we focus on the ac-driven processes and the cross-over region between ac and thermal fluctuations as the tem-perature increases. For simplicity, we consider driving am-plitudes eV0/␻= 1 and eV0/␻= 2. In this case q⫾1共Tp兲

= Tp共1−Tp兲p1共eV0/␻兲 at zero temperature and do not depend on energy. These single-charge transfers originate from an electron-hole pair which is created with probability p1 per voltage cycle 共Fig.4兲. As the temperature increases, the

in-terplay of thermal and ac excitations introduces a nontrivial energy dependence of probabilities q_⫾1shown in Fig.9共c兲. It also modifies the dependence on transmission eigenvalues with q_⫾1共Tp兲 no longer being proportional to Tp共1−Tp兲关cf.

Eq.共33兲兴. As the temperature increases further, the

multiple-charge transfers come into play as shown in Fig. 9共a兲. At temperatures larger than the driving amplitude the limit of thermal transport is reached with statistics which can be re-cast again in the form of single-charge transfers, independent at different energies.

VI. CONCLUSION

We have studied charge-transfer statistics in a generic me-soscopic contact driven by a time-dependent voltage. We have obtained the analytic form of the cumulant generating function at zero temperature and identified the elementary charge-transfer processes. The unidirectional processes rep-resent electrons which are injected from the source terminal due to excess dc bias voltage. The bidirectional processes represent electron-hole pairs which are created by the time-dependent voltage bias and injected toward the contact. This interpretation is consistent with the charge-transfer statistics in a multiterminal beam-splitter geometry in which electrons and holes can be partitioned into different outgoing termi-nals.

The elementary charge-transfer processes can be probed by current noise power and higher-order current correlators. The bidirectional processes contribute to the noise and higher-order even cumulants of the transferred charge at low temperature and give no contributions to the average current and higher-order odd cumulants. For an ac voltage with no dc offset, the noise is entirely due to bidirectional processes. The individual processes can be identified from the oscilla-tions of the differential noise⳵SI/⳵V0as the amplitude V0of the drive increases.

A time-dependent voltage drive with a nonzero dc offset generates both unidirectional and bidirectional processes. The bidirectional processes give rise to the excess noise with respect to dc noise level which is set by the unidirectional processes. The excess noise can be probed by measuring current cross correlations between different outgoing termi-nals in the beam-splitter geometry. The cross correlations are only due to processes in which the incoming electron-hole pair is split and the particles enter different outgoing termi-nals.

The method we use enables the systematic calculation of the higher-order current correlators at finite temperatures by expansion of the cumulant generating function in the

count-ing field. We have obtained the current noise power and the third cumulant for arbitrary periodic voltage applied.

We have also studied the effect of finite temperature on elementary charge-transfer processes. In the limits of low 共high兲 temperatures the single-charge transfers occur in ei-ther direction due to ac excited共thermally excited兲 electron-hole pairs. However, the nature of the two is very different: the ac drive mixes the electron states of different energies, while thermal fluctuations are diagonal in energy. In the crossover region there is still some mixing of different en-ergy states present. Such an interplay of thermal and ac ex-citations can be interpreted in terms of multiple-charge trans-fers with probabilities dependent on energy and temperature.

ACKNOWLEDGMENTS

This work has been supported by the German Research Foundation 共DFG兲 through Grants No. SFB 513 and No. SFB 767 and the Swiss National Science Foundation.

APPENDIX

1. Determinants of block matrices

Let A, B, C, and D be the quadratic matrices of the same size. Then the following equalities hold:

det

冉

A B C D

冊

=

冦

det共AD − BC兲, 关C,D兴 = 0 det共DA − BC兲, 关B,D兴 = 0 det共DA − CB兲, 关A,B兴 = 0 det共AD − CB兲, 关A,C兴 = 0.

冧

共A1兲

In the case in which more than two blocks commute with each other, the corresponding determinants on the right-hand side of Eq.共A1兲 coincide.

2. Coefficients a_n

Here we calculate coefficients 兵an其 given by Eq. 共13兲 for

different time-dependent voltages. For the cosine voltage drive V共t兲=V¯+V0cos共␻t兲, the coefficients an can be

calcu-lated using Jacobi-Anger expansion,63

eiz sin共␪兲=

兺

n=−⬁ ⬁

Jn共z兲ein␪, 共A2兲

where Jn are the Bessel functions of the first kind. From Eq.

共13兲 we obtain

an= Jn共eV0/␻兲. 共A3兲

The square voltage drive is given by⌬V共t兲=V0 for 0⬍t ⬍␶/2 and ⌬V共t兲=−V0for␶/2⬍t⬍␶. For noninteger values of eV₀/␻, the coefficients anare given by

an= 2 ␲ eV0 ␻ sin关共n − eV0/␻兲␲/2兴 n2−共eV0/␻兲2 ei共n−eV0/␻兲␲/2_. 共A4兲

For integer values of eV₀/␻, the coefficients anare obtained