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...

W e consi dera m esoscopi c m easuri ng devi ce w hose conductance i ssensi ti ve to the state ofa tw o-l evel system . T he detector i sdescri bed w i th the hel p ofi ts scatteri ng m atri x. Itsel em entscan be used to cal cul ate the rel axati on and decoherence ti m e ofthe system ,and determ i ne the characteri sti c ti m e for a rel i abl e m easurem ent. W e deri ve condi ti ons needed for an e ci ent rati o of decoherence and m easurem ent ti m e. To i l l ustrate the theory w e di scuss the di stri buti on functi on ofthe e ci ency of an ensem bl e ofopen chaoti c cavi ti es.PA C S num bers: 03. 65. Ta,03. 67. Lx M esoscopi c physi cs i s evol vi ng toward a stage w here the understandi ng ofthe m easurem ent process becom es ofi ncreasi ng i m portance.O fi nterestare detectorsw hi ch al l ow a fastdeterm i nati on ofthe state ofthe system but at the sam e ti m e l eave the coherence of the m easured system as una ected as possi bl e. T hese are con i cti ng requi rem ents:Fori nstancea tunnelcontacti san e ci ent butsl ow detector.T herefore,the questi on ari sesw hether i ti spossi bl eto devel op detectorsw hi ch areboth fastand e ci ent. To answer thi s questi on we i nvesti gate m esoscopi c m ul ti channelconductors and anal yze thei r speed and e ci ency. T he e ci ency ofa detector i s determ i ned by the rati o ofthe m easurem ent ti m e and the decoherence ti m e ofthe m easured system .T hee ectofa detectoron a phasecoherentm esoscopi c system hasbeen el egantl y dem onstrated i n recentexperi m ents [ 1{3] . T heoreti caldi scussi ons addressed di erent aspectsofweak m easurem enti n m esoscopi csystem s:the rel ati on to scatteri ng theory [ 1,4,5]and screeni ng [ 5] ,the m easurem ent ti m e and i nteracti ons [ 6]and the rel ati on between detector noi se and decoherence rate [ 7,8] . T he ti m e evol uti on ofsystem and detector has been studi ed usi ng a m aster equati on approach [ 9,10] . R e ned cal cul ati ons consi der the condi ti onalevol uti on ofthe system dependi ng on the outcom e of the m easurem ent [ 11,12] . Tunnelcontactsand si ngl eel ectron transi storshavebeen i denti ed ascandi datesfore ci entm easurem entdevi ces [ 13] . B oth the m easurem ent ti m e and the decoherence rate depend on the scatteri ng m atri x ofthe conductor(detector). C onsequentl y both of these quanti ti es depend on the sam pl e speci c geom etry and i m puri ty di stri buti on of the detector. It i s therefore necessary to i nvesti gate the di stri buti on of the quanti ti es of i nterest (m easurem entti m e,decoherence rate and e ci ency)ofensem bl es ofm acroscopi cal l y i denti caldetectors. H ere we focus on bal l i sti cdetectorsforw hi ch ensem bl em em bersdi eronl y i n thei r geom etry.T he m odelwe consi der i s show n i n Fi g. 1. It consi sts ofa doubl e dot(D D )thatpl aysthe rol eofthe system .It i s an e ecti ve two-l evelsystem : T he topm ostel ectron i n theD D can ei theroccupy theupperorthel owerdot.T he detector i s a m esoscopi c two-term i nalconductor (M C ): I...

The modified optimal path method and non-adiabatic II-order transitions in noisy perturbed dynamical systems AIP Conf. Proc. 502, 54 (2000) We investigate the statistics of fluctuations in a classical stochastic network of nodes joined by connectors. The nodes carry generalized charge that may be randomly transferred from one node to another. Our goal is to find the time evolution of the probability distribution of charges in the network. The building blocks of our theoretical approach are (1) known probability distributions for the connector currents, (2) physical constraints such as local charge conservation, and (3) a time scale separation between the slow charge dynamics of the nodes and the fast current fluctuations of the connectors. We integrate out fast current fluctuations and derive a stochastic path integral representation of the evolution operator for the slow charges. The statistics of charge fluctuations may be found from the saddlepoint approximation of the action. Once the probability distributions on the discrete network have been studied, the continuum limit is taken to obtain a statistical field theory. We find a correspondence between the diffusive field theory and a Langevin equation with Gaussian noise sources, leading nevertheless to nontrivial fluctuation statistics. To complete our theory, we demonstrate that the cascade diagrammatics, recently introduced by Nagaev, naturally follows from the stochastic path integral. By generalizing the principle of minimal correlations, we extend the diagrammatics to calculate current correlation functions for an arbitrary network. One primary application of this formalism is that of full counting statistics (FCS), the motivation for why it was developed in the first place. We stress however, that the formalism is suitable for general classical stochastic problems as an alternative approach to the traditional master equation or Doi-Peliti technique. The formalism is illustrated with several examples: Both instantaneous and time averaged charge fluctuation statistics in a mesoscopic chaotic cavity, as well as the FCS and new results for a generalized diffusive wire.

We propose a simple semiclassical method for calculating higher-order cumulants of current in multichannel mesoscopic conductors. To demonstrate its efficiency, we calculate the third and fourth cumulants of current for a chaotic cavity with multichannel leads of arbitrary transparency and compare the results with ensemble-averaged quantum-mechanical quantities. We also explain the discrepancy between the quantum-mechanical results and previous semiclassical calculations.Comment: 7 pages, 4 figure

A method to calculate the statistics of energy exchange between quantum systems is presented. The generating function of this statistics is expressed through a Keldysh path integral. The method is first applied to the problem of heat dissipation from a biased mesoscopic conductor into the adjacent reservoirs. We then consider energy dissipation in an electrical circuit around a mesoscopic conductor. We derive the conditions under which measurements of the fluctuations of heat dissipation can be used to investigate higher order cumulants of the charge counting statistics of a mesoscopic conductor. 72.70.+m, During the second decade of mesoscopic physics there has been an increasing interest in thermal phenomena. Several experiments succeeded in measuring properties of heat transport in mesoscopic samples: Experimental tools based on the Coulomb blockade to measure accurately local temperatures in mesoscopic samples have been established [1,2]. It has been demonstrated that the heat conductance through one-dimensional phonon modes of a micro bridge is quantized at low temperatures [3]. The universal heat conductance per mode is given by π 2 k 2 T /3h. The same universal heat quantum is also found for carriers other than bosons (see [4] and Refs. therein). Thermopower was experimentally investigated for a quantum dot in the Coulomb blockade regime [5,6] and for multiwalled carbon nanotubes [7]. The Andreev reflection process is ineffective for heat transport and has been employed to measure thermopower in Andreev interferometers. The thermopower depends on the magnetic flux and shows geometry dependent time inversion properties [8].All these experiments have in common that they investigate mean properties averaged over time. In electrical transport, however, noise measurements have become a very useful tool to study properties of non-equilibrium systems which are hidden in measurements of the mean current (for a review see [9]). A recent milestone was the first successful experimental investigation of higher order current correlators by Reulet et al. [10]. It revealed an unexpected temperature dependence that was explained by accounting for an external measurement circuit [11]. The calculation of average currents and current fluctuations can be unified within the concept of full counting statistics that has been introduced into mesoscopic physics a decade ago. It was shown by Levitov and Lesovik that coherent charge transfer through a two-terminal conductor can be seen as probabilistic process defined through a set of transmission probabilities [12]. There- * Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge MA 02139, USA.after different works investigated the statistics of other measurable quantities in mesoscopic structures such as voltage [13], momentum [14], and charge inside a mesoscopic volume [15].In this publication we address the statistics of fluctuations of energy exchange. We investigate the statistics of energy flow into a subvolume of a quantum system. In the fi...