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

TL;DR

The paper addresses how to derive a master equation for a small quantum system S coupled to a chaotic environment E that satisfies the ETH, without invoking the Born or Markov approximations. It introduces an environmental-branch representation of the RDM, partitions time into short intervals, and uses a second-order expansion together with ETH-based filtering to identify the dominant contributions that yield a Lindblad-form generator. The main result is a concrete master equation with a dissipator determined by a diagonal ETH slope, under conditions of weak correlation between initial environmental states and the interaction, and chaotic environmental dynamics that suppress branch correlations. The approach provides a dynamic route to open-system behavior beyond standard bath models and is validated against pure dephasing scenarios and random-matrix theory predictions when the interval scale $\tau$ is chosen appropriately, highlighting its potential for non-Markovian dynamics and realistic chaotic environments.

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.