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

Abstract

Fermi's golden rule applies to a situation in which a single quantum state |ψ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of |ψ . 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.

Accurate Lindblad-form master equation for weakly damped quantum systems across all regimes

“…It has already been established that the spectral density must be sufficiently flat in order that the damping induced by the bath be exponential 32,53 . We can use system identification, which we used above to find the master equation for the V-system, to determine the number of dynamical variables required to reproduce the evolution of the open system as the slope of the spectral density is increased.…”

“…These two Lindblad master equations are obtained from the Bloch-Redfield master equation by making the secular (rotating-wave) approximation. However, no Lindblad master equation has been obtained for the neardegenerate regime 31,32 . Thus to simulate systems in which two or more distinct transitions are separated by less than a few linewidths, one must resort to the Bloch-Redfield (B-R) master equation 33,34 .…”

“…Jeske et al 40 also noted that when transitions are close enough that they share the same value of the spectral density, the B-R equation reduces to the degenerate master equation, which is a key element in our analysis here. These recent works raise an interesting question: other authors have assumed that the near-degenerate regime is non-Markovian, due to the apparent lack of a Lindblad-form master equation in that regime 31,32 . If the dynamics of weakly-damped systems is indeed Markovian in the near-degenerate regime, then it is not unreasonable to suggest that there may be a completely positive Markovian master equation that accurately describes it.…”

Ability of Markovian master equations to model quantum computers and other systems under broadband control

“…We now consider what happens when the transition is driven with a drive frequency of ω d = ω 0 and a fixed Rabi frequency of Ω = ω 0 /32, while the transition frequency is modulated by a function with a given bandwidth and RMS amplitude. To this end we choose ω(t) = ω c + N k=−N c k cos(νk + ω m ) + s k sin(νk + ω m ) (27) in which N is an integer. Here ω c is the "center" (also the time-averaged) value of the transition frequency, ω m is the center frequency of the modulation, and ω w = N ν/2 is the modulation bandwidth.…”

The Leggett–Garg inequalities and the relative entropy of coherence in the Bixon–Jortner model

“…These kicks are due to the couplings between the single state and the quasi-continuum background states. The model is discussed often in the field of quantum optics because its Hamiltonian can be diagonalized exactly [8,9,10].…”