Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior

We study composite open quantum systems with a finite-dimensional state space ${\mathcal H}_A\otimes {\mathcal H}_B$ governed by a Lindblad equation $\rho'(t) = {\mathcal L}_\gamma \rho(t)$ where ${\mathcal L}_\gamma\rho = -i[H,\rho] + \gamma {\mathcal D} \rho$, 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 $\gamma$ is a parameter. It is known that the dynamics simplifies for large $\gamma$: after a time of order $\gamma^{-1}$, $\rho(t)$ is well approximated for times small compared to $\gamma^2$ by $\pi_A\otimes R(t)$ where $\pi_A$ is a steady state of ${\mathcal D}_A$, and $R(t)$ is a solution of $\frac{{\rm d}}{{\rm d}t}R(t) = {\mathcal L}_{P,\gamma}R(t)$ where ${\mathcal L}_{P,\gamma} R := -i[H_P,R] + \gamma^{-1} {\mathcal D}_P R$ 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 $\bar\rho_\gamma$, we introduce a third Lindblad generator ${\mathcal D}_P^\sharp$ that does not involve $\gamma$, but still closely related to ${\mathcal L}_\gamma$. We show that if ${\mathcal D}_P^\sharp$ is ergodic and gapped, then so is ${\mathcal L}_\gamma$ for all large $\gamma$, and if $\bar\rho_\gamma$ denotes the unique steady state for ${\mathcal L}_\gamma$, then $\lim_{\gamma\to\infty}\bar\rho_\gamma = \pi_A\otimes \bar R$ where $\bar R$ is the unique steady state for ${\mathcal D}_P^\sharp$. We show that there is a convergent expansion $\bar\rho_\gamma = \pi_A\otimes\bar R +\gamma^{-1} \sum_{k=0}^\infty \gamma^{-k} \bar n_k$ where, defining $\bar n_{-1} := \pi_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.