Stationary states of boundary driven quantum systems: some exact results

TL;DR

This paper analyzes the stationary states of boundary-driven quantum systems, showing that under certain conditions, the stationary density matrix must be a product state, and discusses the uniqueness and properties of these states.

Contribution

The paper provides exact results characterizing stationary states of boundary-driven quantum systems with ergodic dissipators, including conditions for their uniqueness and form.

Findings

Stationary states are product states under ergodic boundary dissipation.

Gibbs states are not stationary unless there is no interaction.

Criteria for the uniqueness of stationary states are established.

Abstract

We study finite-dimensional open quantum systems whose density matrix evolves via a Lindbladian, $\dot{\rho}=-i[H,\rho]+{\mathcal D}\rho$. Here $H$ is the Hamiltonian of the isolated system and ${\mathcal D}$ is the dissipator. We consider the case where the system consists of two parts, the "boundary'' $A$ and the ``bulk'' $B$, and ${\mathcal D}$ acts only on $A$, so ${\mathcal D}={\mathcal D}_A\otimes{\mathcal I}_B$, where ${\mathcal D}_A$ acts only on part $A$, while ${\mathcal I}_B$ is the identity superoperator on part $B$. Let ${\mathcal D}_A$ be ergodic, so ${\mathcal D}_A\hat{\rho}_A=0$ only for one unique density matrix $\hat{\rho}_A$. We show that any stationary density matrix $\bar{\rho}$ on the full system which commutes with $H$ must be of the product form $\bar{\rho}=\hat{\rho}_A\otimes\rho_B$ for some $\rho_B$. This rules out finding any ${\mathcal D}_A$ that has the Gibbs measure $\rho_\beta\sim e^{-\beta H}$ as a stationary state with $\beta\neq 0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\bar{\rho}$ for systems with interactions between $A$ and $B$. Related results for non-ergodic cases are also discussed.