Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior
Authors
Eric A. Carlen
David A. Huse
Joel L. Lebowitz
Abstract
We study composite open quantum systems with a finite-dimensional state space ${\mathcal H}_A\otimes {\mathcal H}_B$ governed by a Lindblad equation $ρ'(t) = {\mathcal L}_γρ(t)$ where ${\mathcal L}_γρ= -i[H,ρ] + γ{\mathcal D} ρ$, and ${\mathcal D}$ is a dissipator ${\mathcal D}_A\otimes I$ acting non-trivially only on part $A$ of the system, which can be thought of as the boundary, and $γ$ is a parameter. It is known that the dynamics simplifies for large $γ$: after a time of order $γ^{-1}$, $ρ(t)$ is well approximated for times small compared to $γ^2$ by $π_A\otimes R(t)$ where $π_A$ is a steady state of ${\mathcal D}_A$, and $R(t)$ is a solution of $\frac{\rm d}{{\rm d}t}R(t) = {\mathcal L}_{P,γ}R(t)$ where ${\mathcal L}_{P,γ} R := -i[H_P,R] + γ^{-1} {\mathcal D}_P R$ with $H_P$ being a Hamiltonian on ${\mathcal H}_B$ and ${\mathcal D}_P$ being a Lindblad generator over ${\mathcal H}_B$. We prove this assuming only that ${\mathcal D}_A$ is ergodic and gapped. In order to better control the long time behavior, and study the steady states $\barρ_γ$, we introduce a third Lindblad generator ${\mathcal D}_P^\sharp$ that does not involve $γ$, but still closely related to ${\mathcal L}_γ$. We show that if ${\mathcal D}_P^\sharp$ is ergodic and gapped, then so is ${\mathcal L}_γ$ for all large $γ$, and if $\barρ_γ$ denotes the unique steady state for ${\mathcal L}_γ$, then $\lim_{γ\to\infty}\barρ_γ= π_A\otimes \bar R$ where $\bar R$ is the unique steady state for ${\mathcal D}_P^\sharp$. We show that there is a convergent expansion $\barρ_γ= π_A\otimes\bar R +γ^{-1} \sum_{k=0}^\infty γ^{-k} \bar n_k$ where, defining $\bar n_{-1} := π_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.
