Fugu-MT 論文翻訳(概要): Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior

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

arxiv url: http://arxiv.org/abs/2512.12825v1
Date: Sun, 14 Dec 2025 20:13:54 GMT
ステータス: 翻訳完了
システム内更新日: 2025-12-16 17:54:56.461302
Title: Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior
Title（参考訳）: ゼノ極限近傍の境界駆動量子系:定常状態と長時間の挙動
Authors: Eric A. Carlen, David A. Huse, Joel L. Lebowitz,
Abstract要約: 有限次元状態空間 $mathcal H_Aotimes 数学 H_B$ はリンドブラッド方程式 $'(t) = 数学 L_(t)$ によって支配される。もし$mathcal D_Psharp$がエルゴディックでギャップがあるなら、$mathcal L_$ for all large $$、$bar_$が$mathcal L_$のユニークな定常状態であることを示す。
参考スコア（独自算出の注目度）: 0.0
License: http://creativecommons.org/licenses/by/4.0/
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$.
Abstract（参考訳）: 有限次元状態空間 ${\mathcal H}_A\otimes {\mathcal H}_B$ をリンドブラッド方程式 $ρ'(t) = {\mathcal L}_γρ(t)$ ここで${\mathcal L}_γρ= -i[H,ρ] + γ{\mathcal D} ρ$ とし、${\mathcal D}$ を Dissipator ${\mathcal D}_A\otimes I$ をシステムの部分 $A$ に非自明に作用させる。 a a time of order $γ^{-1}$, $ρ(t)$ is well approximated for times than $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}R:=-i[H_P,R] + γ^{-1} {\mathcal D}R==-i[H_P,R]+\mathcal DP$ $H_P$ a Hamilton on ${\mathcal D}_B}${\mathcal D}R(t)$ $R(t)$ ${\mathcal D}R(t)$$ ${\mathcal D}D}_B}$$D(t)$ の解である。このことは、${\mathcal D}_A$ がエルゴード的でギャップがあるという仮定で証明する。長時間の挙動をよりよく制御し、定常状態 $\barρ_γ$ を研究するために、第3のリンドブラッド生成器 ${\mathcal D}_P^\sharp$ を導入する。すると、${\mathcal D}_P^\sharp$がエルゴード的かつギャップ的であれば、${\mathcal L}_γ$はすべての大きな$γ$に対して${\mathcal L}_γ$であり、$\barρ_γ$は${\mathcal L}_γ$に対して一意的な定常状態を表すならば、$\lim_{γ\to\infty}\barρ_γ= π_A\otimes \bar R$である。収束展開 $\barρ_γ = π_A\otimes\bar R +γ^{-1} \sum_{k=0}^\infty γ^{-k} \bar n_k$ ここで、$\bar n_{-1} := π_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$ が定義される。

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

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