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}_Aotimes {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}_Aotimes 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 γ² by π_Aotimes 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,γR(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→inftybarρ_γ= π_Aotimes 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_Aotimesbar R +γ^-1 sum_k=0^infty γ^-k bar n_k where, defining bar n_-1 := π_Aotimesbar R, {mathcal D} bar n_k = -i\[H,bar n_k-1\] for all kgeq 0.