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arXiv:1405.3370 · Journal of Physics A Mathematical and Theoretical

Superoperator coupled cluster method for nonequilibrium density matrix

Alan A. Dzhioev, Daniel S. Kosov

Published Dec 4, 2014• 26 citations
arXivHTMLPDF

Abstract

We develop a superoperator coupled cluster method for nonequilibrium open many-body quantum systems described by the Lindblad master equation. The method is universal and applicable to systems of interacting fermions, bosons or their mixtures. We present a general theory and consider its application to the problem of quantum transport through the system with electron–phonon correlations. The results are assessed against the perturbation theory and nonequilibrium configuration interaction theory calculations.

DOI
10.1088/1751-8113/48/1/015004

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We develop a superoperator coupled cluster method for nonequilibrium open many-body quantum systems described by the Lindblad master equation. The method is universal and applicable to systems of interacting fermions, bosons or their mixtures. We present a general theory and consider its application to the problem of quantum transport through the system with electron–phonon correlations. The results are assessed against the perturbation theory and nonequilibrium configuration interaction theory calculations.
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We study the dynamics of interacting open quantum subsystems. Interactions merge these subsystems into a unified system. It is often believed that the Lindblad superoperator of the unified system is the sum of the Lindblad superoperators of the separate subsystems. Such an approach, however, results in a violation of the second law of thermodynamics. To avoid a cumbersome direct derivation of the correct superoperator from first principles, we develop a perturbation theory based on using Lindblad superoperators of separate subsystems, which are assumed to be known. Using interacting two-level systems as an example, we show that, starting with a certain order of the perturbation theory, the second law of thermodynamics holds. We demonstrate that the theory developed can be applied to the problems of quantum transport.
We present an alternative form of master equation, applicable to the analysis of nonequilibrium dynamics of fermionic open quantum systems. The formalism considers a general scenario, composed by a multipartite quantum system in contact with several reservoirs, each one with a specific chemical potential and in thermal equilibrium. With the help of Jordan-Wigner transformation, we perform a fermion-to-qubit mapping to derive a set of Lindblad superoperators that can be straightforwardly used on a wide range of physical setups. To illustrate our approach, we explore the effect of a charge sensor, acting as a probe, over the dynamics of electrons on coupled quantum molecules. The probe consists of a quantum dot attached to source and drain leads, that allows a current flow. The dynamics of populations, entanglement degree, and purity show how the probe is behind the sudden deaths and rebirths of entanglement, at short times. Then, the evolution leads the system to an asymptotic state being a statistical mixture. Those are signatures that the probe induces dephasing, a process that destroys the coherence of the quantum system.
We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R\mathcal{R}R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R\mathcal{R}R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.
We develop a method of calculating the nonequilibrium steady state (NESS) of an open quantum system that is weakly coupled to reservoirs in different equilibrium states. We describe the system using a Redfield-type quantum master equation (QME). We decompose the Redfield QME into a Lindblad-type QME and the remaining part R\mathcal{R}R. Regarding the steady state of the Lindblad QME as the unperturbed solution, we perform a perturbative calculation with respect to R\mathcal{R}R to obtain the NESS of the Redfield QME. The NESS thus determined is exact up to the first order in the system-reservoir coupling strength (pump/loss rate), which is the same as the order of validity of the QME. An advantage of the proposed method in numerical computation is its applicability to systems larger than those in methods of directly solving the original Redfield QME. We apply the method to a noninteracting fermion system to obtain an analytical expression of the NESS density matrix. We also numerically demonstrate the method in a nonequilibrium quantum spin chain.
Abstract We consider the transition from the description of a closed quantum system consisting of an open quantum system and a reservoir to the description of the open quantum system alone by eliminating the reservoir degrees of freedom by averaging over them. An approach based on the Lindblad master equation for the density matrix is used. A general scheme is developed for deriving the Lindblad superoperator that emerges after averaging the von Neumann equation over the reservoir degrees of freedom. This scheme is illustrated with the cases of radiation of a two-level atom into free space and the dynamics of the transition of a two-level atom from a pure state to a mixed state due to interaction with a dephasing reservoir. Special attention is paid to the open system consisting of several subsystems each of which independently interacts with the reservoir. In the case of noninteracting subsystems, the density matrix is a tensor product of the subsystem density matrices, and the Lindblad superoperator of the system is a sum of Lindblad superoperators of those subsystems. The interaction between the subsystems results not only in the emergence of the corresponding term in the Hamiltonian of the combined system but also in the nonadditivity of the Lindblad superoperators. This is often overlooked in modern literature, possibly leading, as is shown in this methodological note, to serious errors; for example, the second law of thermodynamics could be violated.