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arXiv:2207.01943 · Journal of Physics Condensed Matter

Non-Markovian master equation for quantum transport of fermionic carriers

Дмитрий Н. Максимов, S. V. Aksenov, Andrey R. Kolovsky

Published Oct 13, 2023• 5 citations
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Abstract

We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix (SPDM) in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the SPDM in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.

DOI
10.1088/1361-648x/ad0351

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We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix (SPDM) in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the SPDM in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.
We consider quantum transport of non-interacting particles in a tight-binding chain coupled to Markovian reservoirs at its boundaries and subject to uniform on-site dephasing. Using the masterequation approach, we derived the exact analytic solution for the mean current along the chain. This analytic solution is obtained in the framework of single-particle quantum mechanics and, thus, equally applies for transport of non-interacting bosons and fermions. In the absence of phase damping, the results are extended to the complete many-body solution for bosonic problem in the form of anti-normally ordered characteristic function. This function allows us to calculate the distribution function for the current at each bond in the chain as well as the inter-bond current correlation functions.
We analyze the quantum state of fermionic carriers in a transport channel attached to a particle reservoir. The analysis is done from first principles by considering microscopic models of the reservoir and transport channel. In the case of infinite effective temperature of the reservoir we demonstrate a full agreement between the results of straightforward numerical simulations of the system dynamics and the solution of the master equation on the single-particle density matrix of the carriers in the channel. This allows us to predict the quantum state of carriers in the case where the transport channel connects two reservoirs with different chemical potentials.
In this review article, we present a non-equilibrium quantum transport theory for transient electron dynamics in nanodevices based on exact Master equation derived with the path integral method in the fermion coherent-state representation. Applying the exact Master equation to nanodevices, we also establish the connection of the reduced density matrix and the transient quantum transport current with the Keldysh nonequilibrium Green functions. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account. In applications, we utilize the theory to specific quantum transport systems, a variety of quantum decoherence and quantum transport phenomena involving the non-Markovian memory effect are investigated in both transient and stationary scenarios at arbitrary initial temperatures of the contacts.
We build an exact framework to evaluate heat, energy, and particle transport between Gaussian reservoirs mediated by a quadratic quantum system. By combining full counting statistics with newly developed non-Markovian master equation approaches, we introduce an effective master equation whose solution can be used to generate arbitrary moments of the heat statistics for any number of reservoirs. This theory applies equally to fermionic and bosonic systems, holds at arbitrarily strong coupling, and resolves out-of-equilibrium transient dynamics determined by the system's initial state. In the steady-state, weak-coupling limit, we recover results analogous to those of the well-known Landauer-Büttiker formalism. We conclude our discussion by demonstrating an application of the method to a prototypical fermionic system. Our results uncover a regime of transient negative heat conductance contingent upon the initial system preparation, providing a clear signature of non-trivial out-of-equilibrium dynamics.
We derive a non-Markovian master equation for the evolution of a class of open quantum systems consisting of quadratic fermionic systems coupled to wideband reservoirs. This is done by providing an explicit correspondence between master equations and nonequilibrium Green's function approaches. Our findings permit us to address non-Markovian regimes characterized by negative decoherence rates and to characterize the dynamics with respect to a recently proposed measurement of ``non-Markovianity.'' We study the real-time dynamics and the steady-state solution of two illustrative models: a tight-binding electronic model and XY\mathit{XY}XY spin chains. The rich set of nonequilibrium steady-state phases encountered extends previous results to the non-Markovian regime.
We consider a fermionic quantum system exchanging particles with an environment at a fixed temperature and study its reduced evolution by means of a Redfield-I equation with time-dependent (non-Markovian) coefficients. We find that the description can be efficiently reduced to a standard-form Redfield-II equation, however, with a obeying a universal law. At early times, after the system and environment start in a product state, the appears to be very high, yet eventually it settles down towards the true environment value. In this way, we obtain a time-local master equation, offering high accuracy at all times and preserving the crucial properties of the density matrix. It includes non-Markovian relaxation processes beyond the secular approximation and time-averaging methods and can be further applied to various types of Gorini-Kossakowski-Sudarshan-Lindblad equations. We derive the theory from first principles and discuss its application using a simple example of a single quantum dot.