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Physical Review A

Perturbation theory for Lindblad superoperators for interacting open quantum systems

Vladislav Yu. Shishkov, Evgenii S. Andrianov, Александр Александрович Пухов, A. P. Vinogradov, Alexander A. Lisyansky

Published Sep 8, 2020• 18 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.102.032207

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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.
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.
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.
Abstract If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator’s jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system’s bare Hamiltonian.
Master equations under appropriate assumptions are efficient tools for the study of open quantum systems. For many-body systems, subsystems of which locally couple to thermal baths and weakly interact with each other, the local approach provides a more convenient description than the global approach. However, these local master equations are believed to generate inconsistencies with the laws of thermodynamics when intersubsystem interactions exist. Here we develop an alternative local master equation by virtue of similar approximations used in deriving the traditional Gorini-Kossakowski-Lindblad-Sudarshan master equation. In particular, we stick to using eigenstates of each subsystem to construct quantum jump operators, and the secular approximation is also employed to modify the intersubsystem interactions. Our results show that violations of thermodynamic laws will be avoided after correcting intersubsystem interactions. Finally, we study a two-qubit heat transfer model and this further shows the validity of our modified master equation.
We develop a general theory describing the thermodynamical behavior of open quantum systems coupled to thermal baths beyond perturbation theory. Our approach is based on the exact time-local quantum master equation for the reduced open system states, and on a principle of minimal dissipation. This principle leads to a unique prescription for the decomposition of the master equation into a Hamiltonian part representing coherent time evolution and a dissipator part describing dissipation and decoherence. Employing this decomposition we demonstrate how to define work, heat and entropy production, formulate the first and second law of thermodynamics, and establish the connection between violations of the second law and quantum non-Markovianity.
If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniqueley specified by requiring the dissipator's jump operators to be traceless. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.