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Journal of Physics A Mathematical and General

Generalisation of Fermi's golden rule for the strong bound-free transitions

R J Kuklinski, Kazimierz Rza̧żewski

Published Oct 7, 1989• 0 citations
Publisher

Abstract

A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.

DOI
10.1088/0305-4470/22/19/011

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A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.
Fermi's golden rule applies in the limit where an initial quantum state is weakly coupled to a continuum of other final states overlapping its energy. Here we investigate what happens away from this limit, where the set of final states is discrete, with a nonzero mean level spacing; this question arises in a number of recently investigated many-body systems. For different symmetry classes, we analytically and/or numerically calculate the universal crossovers in the average decay of the initial state as the level spacing is varied, with the golden rule emerging in the limit of a continuum. Among the corrections to the exponential decay of the initial state given by Fermi's golden rule is the appearance of the spectral form factor in the longtime regime for small but nonzero level spacing.
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.
Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.
The authors show that the population trapping phenomenon found by Deng and Eberly (1986, 1988) for free-free transitions between structureless continua occurs under much more general conditions. They solve the time-dependent models exactly and show for asymptotically long times that population trapping can occur for two continua with energy-dependent matrix elements, and for an infinite number of structureless continua with coupled bands. Furthermore, they show that the trapping persists for two continua coupled by a noisy laser. The amount of trapping depends on the initial conditions.