Explicit expressions for stationary states of the Lindblad equation for a finite state space

TL;DR

This paper derives explicit analytical expressions for the stationary states of the Lindblad equation in finite quantum systems, connecting quantum dynamics with classical Markov processes.

Contribution

It introduces a novel analytical method for finding steady states of the Lindblad equation using quantum jump unraveling and Markov chain stationary probabilities.

Findings

Provides explicit formulas for quantum steady states

Shows the classical Markov process as a special case

Discusses differences between quantum and classical stationary states

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 discret-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.