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arXiv:2408.06887 · Physical Review A

Stationary states of boundary-driven quantum systems: Some exact results

Eric A. Carlen, David A. Huse, Joel L. Lebowitz

Published Jan 14, 2025• 1 citation
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Abstract

We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here $H$ is the Hamiltonian of the 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 $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where ${\mathcal{I}}_{B}$ is the identity superoperator on part $B$. Let ${\mathcal{D}}_{A}$ be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with $H$ must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any ${\mathcal{D}}_{A}$ that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts $A$ and $B$. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between $A$ and $B$. Related results for nonergodic cases are also discussed.

DOI
10.1103/physreva.111.012210

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