Zeroth law of thermodynamics for thermalized open quantum systems having integrals of motion

TL;DR

This paper investigates the dynamics of open quantum systems with conserved quantities, showing that stationary states can be found by analyzing invariant subspaces without explicitly identifying constants of motion.

Contribution

It introduces a method to determine stationary states through invariant subspaces, bypassing the need to find constants of motion, thus simplifying analysis of open quantum systems.

Findings

Invariant subspaces correspond to basis of constants of motion

Stationary states are weighted sums of subspace states

Method allows direct construction of invariant subspaces

Abstract

We study the evolution of an open quantum system described by a dynamical semigroup having the Lindblad superoperator as a generator. This generator may have an eigenfunction with a unity eigenvalue, referred to as a constant of motion (COM). An open quantum system has a unique stationary state if and only if it has no COMs. A system with multiple stationary states has a basis of COMs; any COM of the system is a linear combination of the basis COMs. The basis divides the space of system states into subspaces. Each subspace has its own stationary state, and any stationary state of the system is a linear combination of these states. Usually, neither the basis of COMs nor even the number of COMs is known. We demonstrate that finding the stationary state of the system does not require looking for the COMs. Instead, one can construct a set of invariant subspaces. If the system evolution begins from one of these subspaces, the system will remain in it, arriving at a stationary state independent of evolution in other subspaces. We suggest a direct way of finding the invariant subspaces by studying the evolution of the system. We show that the sets of invariant subspaces and subspaces generated by the basis of COMs are equivalent. A stationary state of the system is a weighted sum of stationary states in each invariant subspace; the weighted factors are determined by the initial state of the system.