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arXiv:quant-ph/0609011

Decay of discrete state resonantly coupled to a continuum of finite width. (arXiv:quant-ph/0609011v6 UPDATED)

Eugene Kogan

Published Feb 18, 2019• 0 citations
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Abstract

A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong (Fano-Anderson model), is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory. The deviations from the exponential decay law, predicted by the Fermi's Golden Rule, are discussed. We also study analytic structure of the Green's function (GF) for the model. We analyze the GF poles, branch points and Riemann surface, and show how the Fermi's Golden Rule, valid in perturbative regime for not to large time, appears in this context. The knowledge of analytic structure of the GF in frequency representation opens opportunities for obtaining easy for numerical calculations formulas for the GF in time representation, alternative to those using the spectral density.

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A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong (Fano-Anderson model), is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory. The deviations from the exponential decay law, predicted by the Fermi's Golden Rule, are discussed. We also study analytic structure of the Green's function (GF) for the model. We analyze the GF poles, branch points and Riemann surface, and show how the Fermi's Golden Rule, valid in perturbative regime for not to large time, appears in this context. The knowledge of analytic structure of the GF in frequency representation opens opportunities for obtaining easy for numerical calculations formulas for the GF in time representation, alternative to those using the spectral density.
A simple quantum mechanical model consisting of a discrete level resonantly coupled to a continuum of finite width, where the coupling can be varied from perturbative to strong, is considered. The particle is initially localized at the discrete level, and the time dependence of the amplitude to find the particle at the discrete level is calculated without resorting to perturbation theory and using only elementary methods. A simple and convenient, both for qualitative analysis and for numerical calculations, formula for the amplitude as a function of time is obtained.
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.
A simple quantum-mechanical model consisting of a two-level system coupled to a continuum, where the interaction can be varied from perturbative to strong, is examined for what can be learned about quantum mechanical decay using elementary methods. The decay spectrum and time dependence of the amplitudes can be calculated in this model, without resorting to perturbation theory. The spectrum is not precisely Lorentzian nor is the decay precisely exponential, features that appear to be shared by all quantum-mechanical models of decaying systems. The conditions for exponential decay and some of the reasons for deviation from this at both short and long times are discussed.
Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.
We study a class of Friedrichs Hamiltonians, that is, operators describing an excited state coupled to a bath through a rank-one perturbation, in the case where the bath Hamiltonian has purely discrete spectrum. We consider sequences of such Hamiltonians for which the spectral measures associated to the coupling functions by the bath Hamiltonians converge weakly to a limiting measure that is absolutely continuous with a Hölder continuous density near the excited energy. Under this assumption, we show that the survival probability of the excited state decays in an approximately exponential manner on suitable time scales and under favourable conditions, with a decay rate given by Fermi's golden rule for the limiting measure. The error is estimated in terms of the coupling strength and the Lévy distance between the discrete spectral measures and the limiting measure. As an application, we treat a two-level atom coupled to a massless bosonic field confined to a large cavity in the rotating-wave approximation.
The decay of isolated quantum systems is studied. In particular, a specific model for the radiative decay of an excited atomic state is considered. The atom is taken to have two levels and is described by a current distribution $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$. Both the case in which the modes of the electromagnetic field are continuous and the case in which they are discrete are considered. Conditions on $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$ are found under which the decay law for the excited state is approximately exponential with a lifetime given by Fermi's golden rule. The current should be well localized, not too strong, and not too singular. The $2p\ensuremath{\rightarrow}1s$ decay in hydrogen is examined in detail. It is also found that the existence of an approximately exponential decay law does not depend upon the ability to extend the resolvent operator of the system to a second Riemann sheet. The case in which the electromagnetic field is quantized in a cavity illustrates this point.