Theory of Steady States for Lindblad Equations beyond Time-Independence: Classification, Uniqueness and Symmetry

TL;DR

The paper develops a rigorous framework to classify the asymptotic behavior of time-quasiperiodic GKSL equations with Hermitian jump operators, introducing two distinct strong-symmetry notions in the Schrödinger and interaction pictures and proving a necessary-and-sufficient criterion for steady-state uniqueness based on the algebra generated by the GKSL generators.It shows that the interaction-picture symmetry $\mathcal{C}^{\mathrm{Int}}$ governs time-dependent steady states, while the Schrödinger-picture symmetry $\mathcal{C}^{\mathrm{Sch}}$ controls time-independent ones, enabling a four-class classification of long-time dynamics under quasiperiodic driving and linking to known mechanisms like strong dynamical and Floquet dynamical symmetry.The authors provide practical criteria based on the algebra $\mathcal{A}_{t}^{\mathrm{ad}}$ and demonstrate applications to prototypical systems such as two-level atoms and quantum spin chains, including time-quasiperiodic and Floquet settings, thereby establishing a rigorous foundation for engineering dissipative, time-dependent quantum states.Overall, the framework broadens the understanding of open quantum system dynamics under time-dependent driving and offers a principled path to design and control steady-state behavior via symmetry considerations.

Abstract

We present a rigorous and comprehensive classification of the asymptotic behavior of time-quasiperiodic Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations under the assumption of Hermitian jump operators. 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.

Figures (3)

• Figure 1: Schematic illustration of our main results. We present a novel framework to analyze the steady-state structure of open quantum systems governed by time-dependent Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equations with Hermitian jump operators with rigorous analysis. (a) If the algebra $\mathcal{A}^\mathrm{ad}_{t_0}$ defined in \ref{['eq:aad']} coincides with the full operator algebra $\mathcal{B}(\mathcal{H})$ for some $t_0$, then every state relaxes to the completely mixed state $\mathbb{I}/d$ for GKSL generators that are time-periodic or, more generally, quasiperiodic (Theorem \ref{['thm:main2']}). Conversely, if $\mathcal{A}^\mathrm{ad}_{t_0}\neq \mathcal{B}(\mathcal{H})$ for all $t_0$, we prove that the steady state is not unique under the assumption that the GKSL generators are analytic at all times. (b) For time-dependent GKSL equations, we define strong symmetries characterized by two symmetry algebras: $\mathcal{C}^\mathrm{Sch}$ and $\mathcal{C}^\mathrm{Int}$ defined in \ref{['eq:def_cst']} and \ref{['eq:def_cint']}, respectively. Since $\mathcal{C}^\mathrm{Sch} \subseteq \mathcal{C}^\mathrm{Int}$ holds, the following four possibilities arise: $\mathcal{C}^\mathrm{Sch} = \{ c\mathbb{I} \mid c \in \mathbb{C} \}$ or $\mathcal{C}^\mathrm{Sch} \neq \{ c\mathbb{I} \mid c \in \mathbb{C} \}$, and $\mathcal{C}^\mathrm{Int} \setminus \mathcal{C}^\mathrm{Sch} = \emptyset$ or $\mathcal{C}^\mathrm{Int} \setminus \mathcal{C}^\mathrm{Sch} \neq \emptyset$. The table summarizes the possible steady states for each case. For example, (iv) corresponds to the existence of time-dependent steady states without non-trivial time-independent ones and represents a class unique to time-dependent GKSL equations. Note that the completely mixed state $\mathbb{I}/d$ is always a trivial steady state. See Definition \ref{['def:steady_state']} for the precise meaning of the classification and Sec. \ref{['sec:summary_classification']} for an explanation.
• Figure 2: Schematics of $\mathcal{L}_t$ that satisfies Condition \ref{['con:quasiperiodic']}. Although $\mathcal{L}_t$ is actually a superoperator, it is depicted here as if it were a real-valued function. (top) The spectral norm of $\mathcal{L}_t$ is bounded by $M$ for any $t$. Moreover, for any $\varepsilon>0$, $\delta>0$, and $t_0\geq0$, one can take an infinite sequence of mutually disjoint time intervals $[t_n,t_n+\delta]$$(n=0,1,\ldots)$. (bottom left) Plots of $\mathcal{L}_{t_n+t}$$(0\leq t\leq \delta)$. (bottom right) Plot of the errors $\mathcal{L}_{t_n+t} - \mathcal{L}_{t_0+t}$. For any $n$ and $0 \leq t \leq \delta$, the spectral norms of them are bounded by $\varepsilon$.
• Figure 3: Numerical simulations of the dynamics of a dissipative Hubbard model under periodic (a--c) and quasiperiodic (d--f) driving. (a,d) Trajectories of the time evolution of the expectation values $\langle S_1^x\rangle$ and $\langle S_1^y\rangle$. (b,e) Time evolution of $\langle S_1^y\rangle$. (c,f) Magnitude of the Fourier spectra corresponding to (b) and (e). The dynamics under the quasiperiodic driving leads to a much larger number of peaks than that under the periodic driving. The Fourier analysis is performed using data in the time interval $t\in[0,500]$, with a window function $\exp\!\left[-(t-300)^2/100^2\right]$. The parameters are $L=4$, $J=U=\kappa_j=1$, and $\varepsilon=\{-0.786,\,0.657,\,-0.133,\,-0.176\}$. For periodic (quasiperiodic) driving, we set $B=\omega=\pi$ ($B_1=B_2=\omega_1=\pi$, and $\omega_2=\phi\pi$), where $\phi=\frac{1+\sqrt{5}}{2}$. The initial state is a random pure state with fixed particle number $N=4$.

Theorems & Definitions (27)

• Definition 1
• Example 1
• Example 2
• Example 3
• Example 4: Exponentially decaying dephasing
• Theorem 2
• Example 5: Multi-frequency drives
• Example 6: Fibonacci drives
• proof : Proof of Theorem \ref{['thm:main1']}
• Lemma 4