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arXiv:2512.09007

An ETH-ansatz-based environmental-branch approach to master equation

Wen-ge Wang

quant-phSubmitted Dec 9, 2025• Announced Dec 10, 2025• 0 citations
v1 · Dec 9, 2025v2 · Apr 24, 2026
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Abstract

In this paper, a method for deriving master equation is developed for a generic small quantum system, which is locally coupled to an environment as a many-body quantum chaotic system that satisfies the eigenstate thermalization hypothesis ansatz, resorting to neither the Born approximation nor the Markov approximation. The total system undergoes Schrödinger evolution, under an initial condition in which the environmental branches possess no correlation with the interaction Hamiltonian. Derivation of the master equation is based on piecewise usage of a second-order expansion of a formal expression, which is derived for the evolution of the environmental branches. Approximations used in the derivation are mainly based on dynamic properties of the environment.

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18 pages

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In this paper, a method for deriving master equation is developed for a generic small quantum system, which is locally coupled to an environment as a many-body quantum chaotic system that satisfies the eigenstate thermalization hypothesis ansatz, resorting to neither the Born approximation nor the Markov approximation. The total system undergoes Schrödinger evolution, under an initial condition in which the environmental branches possess no correlation with the interaction Hamiltonian. Derivation of the master equation is based on piecewise usage of a second-order expansion of a formal expression, which is derived for the evolution of the environmental branches. Approximations used in the derivation are mainly based on dynamic properties of the environment.
We use the eigenstate thermalization hypothesis to derive a quantum master equation for a system that is weakly coupled to a chaotic finite-sized bath prepared in a pure state. We show that the emergence of Markovianity is controlled by the spectral function of the ETH and that local detailed balance emerges in the Markovian regime for a broad class of pure bath states. We numerically verify this result by comparing the master equation to dynamics computed using exact diagonalization of a chaotic Hamiltonian. We also compare the master equation to exact dynamics for an integrable bath and find that at finite size they strongly disagree. Our work puts forward eigenstate thermalization as a foundation for open quantum systems theory, thus extending it beyond ensemble bath preparations to chaotic many-body environments in generic pure states.
We study the long-time average of the reduced density matrix (RDM) of a central system that is locally coupled to a large environment, under a Schr\"odinger evolution of the total system. We consider a class of interaction Hamiltonian, whose environmental part satisfies the so-called eigenstate thermalization hypothesis ansatz with a constant diagonal part in the energy region concerned. Relations among elements of the averaged RDM are derived. When steady states of the central system exist, these relations imply the existence of a preferred basis, which is given by the eigenbasis of a renormalized self-Hamiltonian that includes certain averaged impact of the system-environment interaction. Numerical simulations performed for a qubit coupled to a defect Ising chain confirm the analytical predictions.
The Markovian dynamics of open quantum many-body systems are typically governed by the Lindblad master equation, yet obtaining the reduced density matrix for bosonic and fermionic systems remains a formidable numerical challenge due to the exponential growth of the Hilbert space. Here, we introduce a computationally efficient framework for constructing the reduced density matrix by modeling the environment as a copy of the primary system with a monotonically decaying coupling that enforces unidirectional energy flow. Our method directly yields exact solutions that rigorously satisfy the Lindblad master equation for both bosonic and fermionic cases, bypassing the need for costly Liouvillian diagonalization. We validate our approach through applications to paradigmatic models, demonstrating accurate reproduction of dissipative dynamics across a broad parameter regime. This work provides a straightforward and powerful tool for simulating Markovian open-system evolution, with immediate applicability to quantum transport and control problems in many-body physics.
We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.
The Eigenstate Thermalization Hypothesis (ETH) has played a major role in understanding thermodynamic phenomena in closed quantum systems. However, its connection to the timescale of thermalization for open system dynamics has remained elusive. This paper establishes a rigorous link between ETH and fast thermalization to the global Gibbs state. Specifically, we demonstrate fast thermalization for a system coupled weakly to a bath of quasi-free Fermions that we refresh periodically. To describe the joint evolution, we derive a finite-time version of Davies' generator with explicit error bounds and resource estimates. Our approach exploits a critical feature of ETH: operators in the energy basis can be modeled by independent random matrices in a near-diagonal band. This gives quantum expanders at nearby eigenstates of the Hamiltonian and reduces the problem to a one-dimensional classical random walk on the energy eigenstates. Our results explain finite-time thermalization in chaotic open quantum systems.
Exact and approximate master equations were derived by the projection operator method for the reduced statistical operator of a multi-level quantum system with finite number N of quantum eigenstates interacting with arbitrary external classical fields and dissipative environment simultaneously. It was shown that the structure of these equations can be simplified significantly if the free Hamiltonian driven dynamics of an arbitrary quantum multi-level system under the influence of the external driving fields as well as its Markovian and non-Markovian evolution, stipulated by the interaction with the environment, are described in terms of the SU(N) algebra representation. As a consequence, efficient numerical methods can be developed and employed to analyze these master equations for real problems in various fields of theoretical and applied physics. It was also shown that literally the same master equations hold not only for the reduced density operator but also for arbitrary nonequilibrium multi-time correlation functions as well under the only assumption that the system and the environment are uncorrelated at some initial moment of time. A calculational scheme was proposed to account for these lost correlations in a regular perturbative way, thus providing additional computable terms to the correspondent master equations for the correlation functions.