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Proceedings of the Steklov Institute of Mathematics

Finding Stationary Solutions of the Lindblad Equation by Analyzing the Entropy Production Functional

Anton Sergeevich Trushechkin

Published May 1, 2018• 11 citations
Publisher

Abstract

A necessary and sufficient condition is derived for a density operator to be a stationary solution for a certain class of Lindblad equations in the theory of open quantum systems. This condition is based on the properties of a functional that in some cases corresponds to entropy production. Examples are given where this condition is used to find stationary solutions.

DOI
10.1134/s008154381804020x

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A necessary and sufficient condition is derived for a density operator to be a stationary solution for a certain class of Lindblad equations in the theory of open quantum systems. This condition is based on the properties of a functional that in some cases corresponds to entropy production. Examples are given where this condition is used to find stationary solutions.
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In the framework of the Lindblad axiomatic approach for open quantum systems various models for heavy-ion reactions are considered. Assuming the existence of a stationary solution for the Lindblad equation in Gibbs form, the relations between the friction and diffusion coefficients are derived. In this case the evolution operator depends on temperature. Models and Lindblad's evolution operators, which are constructed with the generators of SO(3) and SO(4) algebras, are considered. A master equation for the density operator is derived for the case of a general potential and its connection with similar approaches in nuclear physics is discussed.
The celebrated Lindblad equation governs the nonunitary time evolution of density operators used in the description of open quantum systems. It is usually derived from the von Neumann equation for a large system, at given physical conditions, when a small subsystem is explicitly singled out and the rest of the system acts as an environment whose degrees of freedom are traced out. In the specific case of a subsystem with variable particle number, the equilibrium density operator is given by the well-known grand canonical Gibbs state. Consequently, solving the Lindblad equation in this case should automatically yield, without any additional assumptions, the corresponding density operator in the limiting case of statistical equilibrium. Current studies of the Lindblad equation with varying particle number assume, however, the grand canonical Gibbs state a priori: the chemical potential is externally imposed rather than derived from first principles, and hence the corresponding density operator is not obtained as a natural solution of the equation. In this work, we investigate the compatibility of grand canonical statistical mechanics with the derivation of the Lindblad equation. We propose an alternative and complementary approach to the current literature that consists in using a generalized system Hamiltonian which includes a term μN. In a previous paper, this empirically well-known term has been formally derived from the von Neumann equation for the specific case of equilibrium. Including μN in the system Hamiltonian leads to a modified Lindblad equation which yields the grand canonical state as a natural solution, meaning that all the quantities involved are obtained from the physics of the system without any external assumptions.
Abstract The Lindblad equation describes the time evolution of a density matrix of a quantum mechanical system. Stationary solutions are obtained by time-averaging the solution, which will in general depend on the initial state. We provide an analytical expression for the steady states of the Lindblad equation using the quantum jump unraveling, a version of an ergodic theorem, and the stationary probabilities of the corresponding discrete-time Markov chains. Our result is valid when the number of states appearing the in quantum trajectory is finite. The classical case of a Markov jump-process is recovered as a special case, and differences between the two are discussed.