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Dynamical Mode Decomposition for Infinite-Dimensional Open Quantum Systems; Liouvillian Spectral Analysis and Parameter Estimation

譲 加藤, Yuzuru Kato, 裕也 中尾, Hiroya Nakao

Published Jan 1, 2024• 0 citations
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Dynamical Mode Decomposition for Infinite-Dimensional Open Quantum Systems; Liouvillian Spectral Analysis and Parameter Estimation

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Dynamic mode decomposition (DMD) is a data-driven method for the estimation, prediction, and control of complex dynamical systems, which has gained much attention in the fields of nonlinear dynamics and fluid mechanics. A DMD method for quantum spin systems, described by a linear dynamical system of a finite-dimensional set of observables, has been proposed recently. In this study, we propose two DMD methods applicable to infinite-dimensional open quantum systems, which use time-series data obtained by quantum state tomography. First, we propose a kernel DMD method for a data-driven spectral analysis of the Liouville superoperator. Second, we propose a method for the parameter estimation of the Liouville superoperator, which incorporates prior knowledge of the model structure into DMD. The proposed methods can accurately reconstruct the system dynamics and show that DMD frameworks can be applicable to infinite-dimensional open quantum systems.

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Dynamic mode decomposition (DMD) is a data-driven method for the estimation, prediction, and control of complex dynamical systems, which has gained much attention in the fields of nonlinear dynamics and fluid mechanics. A DMD method for quantum spin systems, described by a linear dynamical system of a finite-dimensional set of observables, has been proposed recently. In this study, we propose two DMD methods applicable to infinite-dimensional open quantum systems, which use time-series data obtained by quantum state tomography. First, we propose a kernel DMD method for a data-driven spectral analysis of the Liouville superoperator. Second, we propose a method for the parameter estimation of the Liouville superoperator, which incorporates prior knowledge of the model structure into DMD. The proposed methods can accurately reconstruct the system dynamics and show that DMD frameworks can be applicable to infinite-dimensional open quantum systems.
Equivalent approaches to determine eigenfrequencies of the Liouvillians of open quantum systems are discussed using the solution of the Heisenberg-Langevin equations and the corresponding equations for operator moments. A simple damped two-level atom is analyzed to demonstrate the equivalence of both approaches. The suggested method is used to reveal the structure as well as eigenfrequencies of the dynamics matrices of the corresponding equations of motion and their degeneracies for interacting bosonic modes described by general quadratic Hamiltonians. Quantum Liouvillian exceptional and diabolical points and their degeneracies are explicitly discussed for the case of two modes. Quantum hybrid diabolical exceptional points (inherited, genuine, and induced) and hidden exceptional points, which are not recognized directly in amplitude spectra, are observed. The presented approach via the Heisenberg-Langevin equations paves the general way to a detailed analysis of quantum exceptional and diabolical points in infinitely dimensional open quantum systems.
We investigate how quasiperiodic spatial structure enables protocol-induced acceleration in open quantum systems by analyzing the Pontus-Mpemba effect in one-dimensional chains subject to Markovian dephasing. The dynamics are governed by a Lindblad superoperator that drives all initial states toward a maximally mixed infinite-temperature steady state, isolating dynamical mechanisms from static equilibrium properties. Considering two representative quasiperiodic models, namely a tight-binding chain with a mosaic potential and its extension with power-law long-range hopping, we show that a properly engineered two-step protocol, in which the system is first steered to a finite temperature intermediate state, yields a strictly shorter overall relaxation time than direct evolution from the same initial configuration. This protocol-induced acceleration persists for both initially localized and extended eigenstates and remains robust in the presence of long-range hopping. A Liouvillian spectral analysis reveals that the mechanism originates from a redistribution of spectral weight that suppresses overlap with the slowest decay modes, rather than from any modification of the decay spectrum itself. Our results establish quasiperiodic chains as a controlled setting for engineering relaxation pathways through Liouvillian spectral structure.