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Physical Review

Theory of Sequential Decays

Alfred S. Goldhaber, Kenneth Watson

Published Aug 25, 1967• 18 citations
Publisher

Abstract

The theory of sequential decays of an unstable system is studied. Examples include the sequential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate levels, and the decay of an unstable particle into other unstable particles. To describe these phenomena, a factorization of the Green's function is introduced. This leads to a simple, and intuitively obvious, description of sequential decays. It also makes possible an assessment of the accuracy of this description.

DOI
10.1103/physrev.160.1151

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The theory of sequential decays of an unstable system is studied. Examples include the sequential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate levels, and the decay of an unstable particle into other unstable particles. To describe these phenomena, a factorization of the Green's function is introduced. This leads to a simple, and intuitively obvious, description of sequential decays. It also makes possible an assessment of the accuracy of this description.
The theory of se~uential decays of an unstable system is studied.Examples include the se~uential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate leveis} and the decay of an unstable particle into other unstable particles.To describe these phenomena} a factorization of the Green's function is introduced.This leads to a simple} and intuitively obvious} description of se~uential decays.It also makes possible an assessment of the accuracy of this description.
A theory of sequential decay of an unstable system is presented using the projection-operator algebra suggested by the recent projection-operator approach to reaction theory. The formulation is an attempt to provide an alternative to perturbation theory. While exact, the theory is cast in a phenomenological form that is suggestive in its interpretation and may be extended so as to apply to sequential transitions in general. Examples treated include the transition between unstable states of an atom induced by an intense electromagnetic field as well as a problem in resonance fluorescence. Higher-order corrections to the more familiar theory are a natural consequence of the description of sequential decay and sequential transitions presented here.
Detailed descriptions of one- and two-photon ionization are presented. We go beyond conventional perturbation theory to investigate the dynamics of both processes for a broad range of field intensities. The general theory of unstable decaying states is applied to the problem of one-photon ionization (the photoeffect). It is found that a non-trivial dependence of the transition dipole moment on the continuum state energy can lead to corrections to exponential decay, and even oscillations in the bound state population. As examples we consider the photoionization of hydrogen and atoms with autoionizing states. Only in the absence of autoionizing states corrections to exponential decay remain inconsequential for any experimentally relevant intensity. Our description of two-photon ionization is formulated in the Heisenberg picture. Besides the usual two-level atom projection operators we introduce explicitly the operators referring to the atomic continuum. By eliminating from the equations of motion the terms related to the electromagnetic self field as well as the terms referring to the continuum, transverse and longitudinal damping corresponding to spontaneous emission and photoionization is obtained. The use of field correlation functions allows us to account for the partial coherence of the applied field. We develop a technique to extract from the equations of motion a unique two-photon ionization rate. Such a characterization of the ionization process is not only instructive because of its simplicity, it is also in many cases completely satisfactory. The resulting general expression for the two-photon ionization rate is the central result of our treatment. From it we recover several familiar results in different limiting cases. In addition, our general rate expression provides the first description of the complete hierarchy of saturation effects that one should expect. We present a number of examples where we calculate the two-photon ionization rate as a function of the detuning, and where we emphasize the saturation and field correlation effects.
It is well known, both theoretically and experimentally, that the survival probability for an unstable quantum state, formed at t = 0 , is not a simple exponential function, even if the latter is a good approximation for intermediate times. Typically, unstable quantum states/particles can decay in more than a single decay channel. In this work, the general expression for the probability that an unstable state decays into a certain i -th channel between the initial time t = 0 and an arbitrary t > 0 is provided, both for nonrelativistic quantum states and for relativistic particles . These partial decay probabilities are also not exponential and their ratio turns out to be not a simple constant, as it would be in the exponential limit. Quite remarkably, these deviations may last relatively long, thus making them potentially interesting in applications. Thus, multichannel decays represent a promising and yet unexplored framework to search for deviations from the exponential decay law in quantum mechanical systems, such as quantum tunneling, and in the context of particle decays .
Abstract A quantum model of the scattering of a particle via absorption and re-emission at the origin is described. The model is capable of exhibiting an arbitrary number of resonances, and the associated unstable states are discussed. The emphasis is on obtaining exact results. It is found that the exponential decay law is of limited application, and that in all cases of non-zero coupling the decay goes ultimately as t−3. Finally, it is pointed out that the treatment of unstable states in this type of model depends only on a knowledge of the phase shift as a function of radial momentum, and a possible phenomenological method is suggested.