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arXiv:0801.0661 · Physical Review Letters

Counting Statistics of Non-Markovian Quantum Stochastic Processes

Christian Flindt, Tomáš Novotný, Alessandro Braggio, Maura Sassetti, Antti‐Pekka Jauho

Published Apr 17, 2008• 205 citations
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Abstract

We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.

DOI
10.1103/physrevlett.100.150601

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We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.
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We have recently developed methods for calculating the statistics of a single counted quantity in a stochastic transport process described by a non‐Markovian generalized master equation. Here, we expand on these ideas and extend our formalism to several counted quantities, allowing us to study cross‐correlations between different variables. As an illustrative example, we consider charge transport through a double quantum dot capacitively coupled to a quantum point contact. We calculate the cross‐correlations between the current through the double quantum dot and the current through the quantum point contact, and demonstrate the sensitivity of this cross‐correlator to measurement quantum back‐action.
Based on our recently developed quantum transport theory in term of an exact master equation, the corresponding particle-number resolved (nnn-resolved) master equation and the related shot noise spectrum formalism covering the full frequency range are constructed. We demonstrate that the noise spectra of transport current through single quantum dot and double quantum dots show characteristic steps and/or peak-dips in different tunneling frequency regimes through tuning the applied bias voltage and/or gate voltage at low temperatures. The peak-dips crossing the tunneling resonant frequencies is a combination effect of non-Markovian and cotunneling processes. These voltage-dependent tunneling resonance characteristics can be utilized to effectively modulate the internal Rabi resonance signature in the noise spectrum.
We consider stochastic and open quantum systems with a finite number of states, where a stochastic transition between two specific states is monitored by a detector.The long-time counting statistics of the observed realizations of the transition, parametrized by cumulants, is the only available information about the system.We present an analytical method for reconstructing generators of the time evolution of the system compatible with the observations.The practicality of the reconstruction method is demonstrated by the examples of a laser-driven atom and the kinetics of enzyme-catalyzed reactions.Moreover, we propose cumulant-based criteria for testing the non-classicality and non-Markovianity of the time evolution, and lower bounds for the system dimension.Our analytical results rely on the close connection between the cumulants of the counting statistics and the characteristic polynomial of the generator, which takes the role of a cumulant generating function.
In this thesis, certain aspects of time-dependent quantum transport properties in nanoscale structures are investigated using the non-equilibrium Green’s function(NEGF). For the transient behaviors of the electric current and heat current after a quantum quench, the exact analytical solution beyond the wide band limit (WBL) is obtained for the first time which paves a way to simulate realistic physical systems. The time-dependent thermoelectric effects are evaluated based on this general solution and it is shown that the time-dependent thermopower is significantly enhanced in the transient regime. By taking initial correlation into account, the combined Caroli and Cini scheme in quantum transport is proposed and its exact solution beyond WBL is derived for the first time as well. Furthermore, besides the average electric current, a current conserving and gauge invariant formalism for the total electric current operator is established at equilibrium state with the help of the electric displacement current operator formulated by taking the Coulomb interaction into account at the operator level. By means of the NEGF and the scattering matrix theory (SMT), a current conserving theory for equilibrium noise power at finite frequencies is obtained which solves one of the fundamental problems in quantum transport. A generalized equilibrium fluctuation-dissipation theorem is presented. Moreover, the entanglement entropy \nfluctuation and its distribution are investigated by introducing the entanglement entropy operator. The essential feature of the entanglement entropy generated by quantum transport is that it is a stochastic quantity whose fluctuation and distribution can be obtained from the generating function (GF). Combining the NEGF with the Grassmann algebras, it is found that the GF of entanglement \nentropy can be expressed in terms of the lesser Green’s function of the subsystem. The general relation connecting the full counting statistics (FCS) of charge transfer with the GF of entanglement entropy is found which indicates that, from the experimental point of view, the entanglement entropy fluctuation can be measured indirectly. As an illustration of the general theoretical framework of entanglement entropy fluctuation, the quantum point contact (QPC) system as well as the quantum dot (QD) system are studied. It is shown that for a QPC under dc bias, the entanglement entropy is maximized and fluctuationless when the transmission coefficient T is one-half. For the quantum dot system, at short times, a universal scaling relation among transient peak values of cumulants of entanglement entropy is shown.