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arXiv:2512.12825 · Journal of Statistical Physics

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

Eric A. Carlen, David A. Huse, Joel L. Lebowitz

quant-phmath-phSubmitted Dec 14, 2025• Announced Dec 16, 2025• 0 citations
v1 · Dec 14, 2025v2 · Jan 6, 2026v3 · Feb 4, 2026
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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 $k\geq 0$.

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We investigate the time evolution of an open quantum system described by a Lindblad master equation with dissipation acting only on a part of the degrees of freedom H0{\mathcal{H}}_{0}H0​ of the system, and targeting a unique dark state in H0{\mathcal{H}}_{0}H0​. We show that, in the Zeno limit of large dissipation, the density matrix of the system traced over the dissipative subspace H0{\mathcal{H}}_{0}H0​, evolves according to another Lindblad dynamics, with renormalized effective Hamiltonian and weak effective dissipation. This behavior is explicitly checked in the case of Heisenberg spin chains with one or both boundary spins strongly coupled to a magnetic reservoir. Moreover, the populations of the eigenstates of the renormalized effective Hamiltonian evolve in time according to a classical Markov dynamics. As a direct application of this result, we propose a computationally efficient exact method to evaluate the nonequilibrium steady state of a general system in the limit of strong dissipation.
Markovian open quantum systems are well modeled by the Lindblad Master Equation (ME) ddtρt=Lρt\frac{\mathrm{d}}{\mathrm{d} t} ρ_t = \mathcal{L} ρ_tdtd​ρt​=Lρt​, where L\mathcal{L}L is a linear (super-)operator and ρtρ_tρt​ is the system state, a positive matrix. When designing or characterizing a quantum system, one is usually interested in the steady state ρ∞ρ_\inftyρ∞​ (such that Lρ∞=0\mathcal{L} ρ_\infty = 0Lρ∞​=0), the first few excited states, and trajectories t↦ρtt\mapsto ρ_tt↦ρt​. In finite dimension, ρtρ_tρt​ is an n×nn\times nn×n matrix, L\mathcal{L}L thus typically costs n4n^4n4 to store explicitly as a dense matrix, and O(n6)O(n^6)O(n6) to diagonalize or invert exactly, making standard linear algebraic techniques expensive for large systems. However, L\mathcal{L}L usually costs only O(n3)O(n^3)O(n3) to apply. This makes iterative methods appealing, but they do not work without a good preconditioner. In this article, our main observation is that a part of the Lindblad equation, corresponding to the so-called no-jump evolution S\mathcal{S}S, can be inverted efficiently. Using this inverse map, we introduce an auxiliary completely positive trace-preserving (CPTP) map ΦΦΦ whose fixed point is directly related to ρ∞ρ_\inftyρ∞​, all the other eigenvalues having smaller magnitude. The map ΦΦΦ is thus well suited to iterative methods, and ρ∞ρ_\inftyρ∞​ can be found in a few Arnoldi iterations. Using the same inverse map S−1\mathcal{S}^{-1}S−1 as preconditioner, we compute the low-lying spectrum efficiently via shift-invert Arnoldi, and, as a proof of concept, build an implicit time integrator that is competitive on stiff systems in the low-precision regime. For the steady-state and low excited states problems, our methods scale like O(n3)O(n^3)O(n3) per iteration and offer state-of-the-art performance on CPU and GPU.
We study finite-dimensional open quantum systems in contact with macroscopic equilibrium systems at their boundaries such that the system density matrix $\ensuremath{\rho}$ evolves via a Lindbladian, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}=\ensuremath{-}i[H,\ensuremath{\rho}]+\mathcal{D}\ensuremath{\rho}$. Here HHH is the Hamiltonian of the system and D\mathcal{D}D is the dissipator. We consider the case where the system consists of two parts, the ``boundary'' AAA and the ``bulk'' BBB, and D\mathcal{D}D acts only on AAA, so $\mathcal{D}={\mathcal{D}}_{A}\ensuremath{\bigotimes}{\mathcal{I}}_{B}$, where IB{\mathcal{I}}_{B}IB​ is the identity superoperator on part BBB. Let DA{\mathcal{D}}_{A}DA​ be ergodic, so ${\mathcal{D}}_{A}{\ensuremath{\pi}}_{A}=0$ only for one unique density matrix ${\ensuremath{\pi}}_{A}$. We show that any stationary density matrix $\overline{\ensuremath{\rho}}$ on the full system which commutes with HHH must be of the product form $\overline{\ensuremath{\rho}}={\ensuremath{\pi}}_{A}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{B}$ for some ${\ensuremath{\rho}}_{B}$. This rules out finding any DA{\mathcal{D}}_{A}DA​ that has the Gibbs measure ${\ensuremath{\rho}}_{\ensuremath{\beta}}={e}^{\ensuremath{-}\ensuremath{\beta}H}/Z(\ensuremath{\beta})$ as a stationary state with $\ensuremath{\beta}\ensuremath{\ne}0$, unless there is no interaction between parts AAA and BBB. We give criteria for the uniqueness of the stationary state $\overline{\ensuremath{\rho}}$ for systems with interactions between AAA and BBB. Related results for nonergodic cases are also discussed.
Understanding the transport properties of quantum many-body systems is a central challenge in condensed matter and statistical physics. Theoretical studies usually rely on two main approaches: Dynamics of linear-response functions in closed systems and boundary-driven dynamics governed by Markovian master equations for open systems. While the equivalence of their dynamical behavior has been explored in recent studies, a systematic comparison of the transport coefficients obtained from these two classes of approaches remains a largely open question. Here, we address this gap by comparing and contrasting the dc diffusion constant Ddc\mathcal{D}_{\text{dc}}Ddc​ according to the two approaches, focusing on the specific example of magnetization transport in the spin-1/21/21/2 XXZ chain. Using exact numerical simulations for finite system sizes, we find (i) a clear mismatch between the two Ddc\mathcal{D}_{\text{dc}}Ddc​ and (ii) a strong dependence of Ddc\mathcal{D}_{\text{dc}}Ddc​ on the system-bath coupling strength for the open system, where neither (i) nor (ii) tend to vanish in the thermodynamic limit. These findings suggest limitations of the open-system approach to transport coefficients. To gain insight into the origin of (i) and (ii), we go beyond Ddc\mathcal{D}_{\text{dc}}Ddc​ and analyze the full time dependence of the diffusion coefficient D(t)D(t)D(t) in the open system. In this way, we find that both (i) and (ii) vanish up to a finite time scale. While this time scale gradually increases with system size and tends to be macroscopic in the thermodynamic limit, this increase is still slow compared to the increase of the time to reach the steady state, where (i) and (ii) do not vanish. This observation can be seen as a wrong, yet unavoidable order of limits of long times first and large system sizes afterwards.
We present a rigorous and comprehensive classification of the asymptotic behavior of time-dependent Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations under the assumption of Hermitian jump operators. Our results apply to a broad class of GKSL equations whose time dependence is assumed to be recurrent, including time-independent, periodic, quasiperiodic, and certain classes of random time dependence. Our main contributions are twofold: first, we establish a criterion for the uniqueness of steady states. The criterion is formulated in terms of the algebra generated by the GKSL generators and provides a necessary and sufficient condition when the generators are analytic functions of time. We demonstrate the utility of our criterion through prototypical examples, including quantum many-body spin chains. Second, we extend the concept of strong symmetry for time-dependent GKSL equations by introducing two distinct forms, strong symmetry in the Schrödinger picture and that in the interaction picture, and completely classify the asymptotic dynamics with them. More concretely, we rigorously uncover that the strong symmetry in the interaction picture is responsible for non-trivial time-dependent steady states, such as coherent oscillations, whereas that in the Schrödinger picture controls the existence of time-independent steady states. This classification not only encompasses established mechanisms underlying non-trivial oscillatory steady states, such as strong dynamical symmetry and Floquet dynamical symmetry, but also reveals symmetry-predicted, time-dependent asymptotic dynamics in a novel class of open quantum systems. Our framework thus provides a rigorous foundation for controlling dissipative quantum systems in a time-dependent manner.
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