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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 $|\psi\rangle$. Herewe show that the coupling of a quantum system to the standard model of athermal environment, a bath of harmonic oscillators, can be decomposed into a"cascade" made up of the quasi-continuum coupling structures of Fermi's goldenrule. This clarifies the connection between the physics of the golden rule andthat of a thermal bath, and provides a non-rigorous but physically intuitivederivation of the Markovian master equation directly from the former. The exactsolution to the Hamiltonian of the golden rule, known as the Bixon-Jortnermodel, generalized for an asymmetric spectrum, provides a window on how theevolution induced by the bath deviates from the master equation as one movesoutside the Markovian regime. Our analysis also reveals the relationshipbetween the oscillator bath and the "random matrix model" (RMT) of a thermalbath. We show that the cascade structure is the one essential differencebetween the two models, and the lack of it prevents the RMT from generatingtransition rates that are independent of the initial state of the system. Wesuggest that the cascade structure is one of the generic elements ofthermalizing many-body systems.

journal of physics. a, mathematical and theoretical

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