MCS-MP Seminar: Eric Carlen

Speaker: Eric Carlen (Rutgers)

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

Abstract: We study composite open quantum systems with a finite-dimensional state space ${\cal H}_{AB} = {\cal H}_A\otimes {\cal H}_B$
governed by a Lindblad equation
$\rho'(t) = {\cal L}_\gamma \rho(t)$
where ${\cal L}_\gamma\rho = -i[H,\rho] + \gamma {\cal D} \rho$.
Here, $H$ is a Hamiltonian on ${\cal H}_{AB}$ while ${\cal D}$ is a dissipator ${\cal D}_A\otimes I$ 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 as the Zeno limit, $\gamma \to \infty$, is approached: after a initial time of order
$\gamma^{-1}$, $\rho(t)$ is well approximated by $\pi_A\otimes R(t)$ where $\pi_A$ is a density matrix on ${\cal H}_A$ such that ${\cal D}_A\pi_A =0$, and $R(t) := {\rm Tr}_A[\rho(t)]$.
We study the long-time behavior of $R(t)$ near the Zeno limit, and apply this to steady states for the full system. We construct a new Lindbladian
dissipator ${\cal D}_P^\sharp$ acting on density matrices on ${\cal H}_B$
and show that if this dissipator is gapped and ergodic, the so is ${\cal L}_\gamma$.
In this case, if $\bar\rho_\gamma$ denotes the
unique steady state for ${\cal L}_\gamma$, then
$\lim_{\gamma\to\infty}\bar\rho_\gamma = \pi_A\otimes \bar R$ where $\bar R$ is the unique solution of
\[
{\cal D}_P^\sharp R =0\ . \eqno(*)
\]
We prove there is a trace norm convergent expansion
${\displaystyle
\bar\rho_\gamma = \pi_A\otimes\bar R +\gamma^{-1} \sum_{k=0}^\infty \gamma^{-k} \bar n_k}$
where, defining $\bar n_{-1} := \pi_A\otimes\bar R$, ${\cal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$.
Using properties of ${\cal D}_P$ and ${\cal D}_P^\sharp$, we show that this system of equations has a unique solution,
and prove convergence. Thus $(*)$ is the equation that selects the starting point of a convergent expansion for $\bar\rho_\gamma$
from among the infinitely many states of the form $\pi_A\otimes R$.
This is joint work with David Huse and Joel Lebowitz; see arXiv 2512.12825.