Stochastic Path Integral Formulation of Full Counting Statistics

Abstract

We derive a stochastic path integral representation of counting statistics in semi-classical systems. The formalism is introduced on the simple case of a single chaotic cavity with two quantum point contacts, and then further generalized to find the propagator for charge distributions with an arbitrary number of counting fields and generalized charges. The counting statistics is given by the saddle point approximation to the path integral, and fluctuations around the saddle point are suppressed in the semi-classical approximation. We use this approach to derive the current cumulants of a chaotic cavity in the hot-electron regime.PACS numbers: 72.70.+m, 76.36.Kv Noise properties of electrical conductors are interesting because they reveal additional information beyond linear response [1]. In the pioneering work of Levitov and Lesovik [2], the optics concept of full counting statistics (FCS) for photons was introduced for electrons in the context of mesoscopic physics. FCS gives the probability of counting a certain number of particles at a measurement apparatus in a certain amount of time and finds not only conductance and shot noise, but all higher current cumulants as well. Several methods have been used in finding this quantity. Originally, quantum mechanical methods based on scattering theory [2,3], the Keldysh approach put forth by Nazarov [4] or sigma-models [5] have been advanced and have been successfully applied to a number of problems among which we mention only multiterminal structures [6], normal-superconducting samples [7], combined photon/electron statistics [8], and conductors which are current (instead of voltage) biased [9].A quantum mechanical treatment of transport shows that the leading contribution to current cumulants is of the order of the channel number N . For many conductors or circuits of interest, this leading order is a semi-classical quantity [10]. Weak localization or universal conductance fluctuations provide only a small correction of order 1. Clearly, it is desirable to have a purely semi-classical theory to calculate semi-classical results. To provide such a derivation of FCS is the main purpose of this work.That a purely classical theory should be developed was realized by de Jong [11] who put forth a discussion for problems which can be described with the help of master equations. A more general approach, leading to a set of rules for a cascade of higher order cumulants, was recently invented by Nagaev [12] and applied to chaotic cavities [13]. The work presented here aims at providing a foundation for the cascade approach by deriving a functional integral from which FCS, but also dynamical quantities such as correlation functions, can be obtained.The approach provided here applies to an arbitrary mesoscopic network. Its semi-classical nature does not allow the treatment of weak localization corrections nor is it applicable to macroscopic quantum effects like the 25 50Full counting statistics of a chaotic cavity at zero temperature: Comparison between hot (thick lines) and...

Current fluctuations in rough superconducting tunnel junctions

“…We will be interested in the full counting statistics (FCS) of charge transfer through the junction, i.e., the probability distribution P t0 (N ) for N charge quanta to be transmitted within measurement time t 0 [31,32,33]. In addition to the noise characteristic, proportional to the second cumulant, this distribution will also supply us with higher, non-Gaussian cumulants such as they occur in RTN.…”

Fluctuation theorem for heat transport probed by a thermal probe electrode

“…44 In the present paper we adopt a simpler approximation which captures the relevant physics. [15][16][17][18][19][20][21] This approach relies on the existence of two distinct time scales. The faster one pertains to the traveling time of each electron through the conductor and the subsequent relaxation in any of the electrodes.…”

“…(23), we adopt the saddle-point approximation, 15 for which δS/δ(iλ P (t)) = δS/δE(t) = 0, and consequentlẏ…”

“…For example, the rapid flow of electrons in and out of a voltage-probing electrode results in much slower charge fluctuations there, thus allowing for a stochastic path-integration of the CGF of the full setup (e.g., a three-terminal one) over all configurations of the probe charge, to obtain the reduced CGF of the physical setup (e.g., a two-terminal one). [15][16][17][18] A similar treatment has been carried out for the stochastic temperature and chemical potential fluctuations in an overheated metallic island. [19][20][21] However, to the best of our knowledge there are no studies of the fluctuation theorem (FT) [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] in systems coupled to thermal probes.…”

“…In order to map the three-terminal junction of Fig. 1 onto the effective two-terminal one we adopt the stochastic path-integral formalism, [15][16][17][18] originally devised for describing electric conduction through a chaotic cavity. We analyze the FT pertaining to the resulting effective two-terminal setup.…”