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Bulletin de la Classe des sciences

Construction of a physical representation for sequential decay

Françoise Hénin

Published Jan 1, 1982• 0 citations
Publisher

Abstract

We consider a three level system (| 2 >, | 1 >, | 0>) where levels I 2 > — I 1 > interact through one kind of bosons and levels | 1 > — | 0 > interact through another kind of bosons. Only real transitions are included in the hamiltonian. This simple model allows the discussion of sequential decay in a non perturbative way and leads to an interpretation of line shape as a consequence of the existence of two boson processes. In this paper, we show how to construct the elements of the evolution operator describing the cascade decay in a physical representation.

DOI
10.3406/barb.1982.57261

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We consider a three level system (| 2 >, | 1 >, | 0>) where levels I 2 > — I 1 > interact through one kind of bosons and levels | 1 > — | 0 > interact through another kind of bosons. Only real transitions are included in the hamiltonian. This simple model allows the discussion of sequential decay in a non perturbative way and leads to an interpretation of line shape as a consequence of the existence of two boson processes. In this paper, we show how to construct the elements of the evolution operator describing the cascade decay in a physical representation.
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.
An exactly soluble three-level model for a system of fermions with Hamiltonian formed from generators of the group SU(3) is studied. A basis for the representation of which the ground state is a member is constructed. It is demonstrated that in this representation, the generators can each be expanded in a series as functions of a pair of ``kinematical'' boson operators; the series are uniquely determined to satisfy the operator algebra and the invariants of the representation by a method of Marumori. It is seen that the lowest anharmonic approximation to the Hamiltonian and other operators yields excellent numerical agreement with exact results for all regimes of interaction strength considered. An alternative description of the system in terms of a dynamically more meaningful boson, called the ``physical'' boson, is shown to be appropriate for relatively weak coupling where one has a near harmonic spectrum.
Previous work of the authors on a three-level quantum system is extended to allow the radiation field in interaction with the system to have a continuous spectrum of possible frequencies. The limiting procedure involved in the passage from discrete to continuous spectra is complicated by the need to express sums over different discrete spectra as integrals with well-behaved limits. Exact expressions are found, for a spontaneous emission problem, for the time evolution of the probabilities that the system be in (each) one of its three states and representative calculations of these probabilities are presented. The emergence of irreversible behavior upon constructing the thermodynamic limit of the problem is plainly seen and this demonstration permits for the first time a discussion of the relative effectiveness of competing decay channels in three-level quantum systems without imposing any ad hoc assumptions (such as exponential decay). Rather, the actual form of the decay emerges as a consequence of the structure and parameters of the Hamiltonian defining the model, and hence one can examine the variety of circumstances in which the evolution can reasonably be described as exponential. The results obtained should be of great use in clarifying certain outstanding conceptual problems in radiation physics, particularly those which deal with the universality of exponential decay in three- (and two-) level quantum systems in interaction with a radiation field and the conditions under which nonexponential or even nonergodic behavior can emerge in such dissipative quantum systems.
The problem of giving a theoretical description of sequential decays is reconsidered. It is shown that the usual quantum description of such processes cannot be reconciled with the classical one. This fact suggests a more critical analysis of the actual physical situation occurring in such processes. By taking into consideration the remarkable successes which have been obtained in the description of nonsequential decays by considering the unavoidable interactions of the decaying system with the measuring apparatuses, an analogous formalism is developed for the problem under consideration. The description of the process is given in the language of the density operator. The evolution equation for such an operator has the standard form of the dynamical equations in the theory of quantum dynamical semigroups. Also in the present case, as for the nonsequential decay, the more realistic description which takes into account the actual set-up of a decay experiment turns out to yield results which agree completely with the classical ones. Some interesting problems arising within the present formalism are discussed.
After the introduction of the two-level model, we add a third level to the quantum mechanical system to discuss some of the effects that cannot be described with only two states. The main feature is the exchange of order between states and transitions, which may proceed through populations or coherences. We concentrate our attention on the latter possibility. Phenomenological introduction Model atoms In the preceding section, we discussed the interaction between light and two-level atoms – probably one of the most popular physical models. The basis of this popularity is its intuitively simple interpretation combined with the posibility of explaining a wide range of physical phenomena. An interesting aspect of the two-level system is that, although its dynamics are formally equivalent to those of a classical angular momentum, it can explain many aspects of quantum mechanics. Once these aspects are understood, it is tempting to look further into the behaviour of real systems, trying to find patterns inconsistent with the predictions of the two-level model. This section discusses aspects of the interaction between matter and radiation that are incompatible with the two-level model. We do not consider specific atomic systems or attempt a complete analysis of the dynamics that a three-level system can exhibit. Instead we select a number of phenomena that play an important role in the discussion of those physical systems that form the subject of the subsequent sections.
A simple systematic procedure to construct the three-level unitary and orthogonal algebras which appear in the bosonic three -level pairing models and for arbitrary choice of level degeneracies is presented. We draw attention to the existence of the three types of dynamical symmetry. The new analytical expressions of the energies are derived. The results presented show very clearly the duality between orthogonal algebras and the quasi-spin algebras for the three-level system with pairing interactions. The seniority selection rules of the transition operators in three level boson system are given. Illustrative calculations are carried out for U 21 algebra.