Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior
Eric A. Carlen, David A. Huse, Joel L. Lebowitz
TL;DR
This work analyzes boundary‑driven open quantum systems in the finite‑dimensional Zeno regime, showing rapid projection onto a steady‑state manifold $oldsymbol{ ext M}=ig\{oldsymbol{ ext π}_Aoldsymbol{ extotimes}Rig brace$ and deriving an effective bulk dynamics governed by $oldsymbol{ ext L}_{P,oldsymbol{ ext γ}}=oldsymbol{ ext K}_P+oldsymbol{ ext γ}^{-1}oldsymbol{ ext D}_P$. A key contribution is the introduction of Davies’ averaging $oldsymbol{ ext D}_P^ sharp$, which yields a γ‑independent generator that governs long‑time behavior and steady states; under ergodicity and gap assumptions this links the true steady state to the unique steady state of $oldsymbol{ ext D}_P^ sharp$. The paper establishes a convergent Hilbert expansion for the steady state, showing $ar ho_oldsymbol{ ext γ}=oldsymbol{ ext π}_Aoldsymbol{ extotimes}ar R+oldsymbol{ ext γ}^{-1}ar n_0+oldsymbol{ ext γ}^{-2}ar n_1+\cdots$, with solvability controlled by $H$ and the projected generators. A hydrodynamic‑limit perspective ties the coherent bulk dynamics to a hierarchy of effective equations, suggesting a quantum analogue of Euler, Navier–Stokes, and Hilbert expansions for boundary‑driven systems. These results enable systematic computation of non‑equilibrium steady states and quantify the decoupling of boundary dissipation from bulk dynamics in the Zeno regime.
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 $k\geq 0$.