Fermi's golden rule, the origin and breakdown of Markovian master equations, and the relationship between oscillator baths and the random matrix model

TL;DR

This paper elucidates how a thermal environment modeled by harmonic oscillators can be understood as a cascade of Fermi's golden rule structures, clarifying the connection to Markovian master equations and contrasting it with random matrix models.

Contribution

It demonstrates that the harmonic oscillator bath decomposes into a cascade of quasi-continuum couplings, providing a physical derivation of the Markovian master equation from Fermi's golden rule.

Findings

The oscillator bath can be decomposed into a cascade of golden rule structures.

The cascade structure is essential for the bath to produce state-independent transition rates.

Differences between oscillator baths and RMT models are attributed to the cascade structure.

Abstract

Fermi's golden rule applies to a situation in which a single quantum state $|\psi\rangle$ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of $|\psi\rangle$. Here we show that the coupling of a quantum system to the standard model of a thermal environment, a bath of harmonic oscillators, can be decomposed into a "cascade" made up of the quasi-continuum coupling structures of Fermi's golden rule. This clarifies the connection between the physics of the golden rule and that of a thermal bath, and provides a non-rigorous but physically intuitive derivation of the Markovian master equation directly from the former. The exact solution to the Hamiltonian of the golden rule, known as the Bixon-Jortner model, generalized for an asymmetric spectrum, provides a window on how the evolution induced by the bath deviates from the master equation as one moves outside the Markovian regime. Our analysis also reveals the relationship between the oscillator bath and the "random matrix model" (RMT) of a thermal bath. We show that the cascade structure is the one essential difference between the two models, and the lack of it prevents the RMT from generating transition rates that are independent of the initial state of the system. We suggest that the cascade structure is one of the generic elements of thermalizing many-body systems.