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Journal of Physics A Mathematical and General

Resolvent operator theory of sequential quantum processes

James D. Cresser, Bryan J. Dalton

Published Mar 1, 1980• 6 citations
Publisher

Abstract

The Mower sequential decay theory of quantum processes has been extended in order formally to remove certain spurious poles in the matrix elements of the resolvent operator, and to recast the results into a more symmetrical general form. A special choice of intermediate manifolds of states leads to a further simplification. An illustrative application of the theory is given.

DOI
10.1088/0305-4470/13/3/013

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The Mower sequential decay theory of quantum processes has been extended in order formally to remove certain spurious poles in the matrix elements of the resolvent operator, and to recast the results into a more symmetrical general form. A special choice of intermediate manifolds of states leads to a further simplification. An illustrative application of the theory is given.
The theory of sequential quantum processes has been extended to Liouville space via the use of non-Hermitian projection operators in order to treat the evolution of the quantum density operator and to enable physically important matrix elements of the density operator to be calculated. The formal relationship of master equation methods to the theory of sequential quantum processes is established, and a new set of coupled master equations is derived. Special choices of projection operators lead to further simplification of the results. The Markoff approximation is also examined.
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.
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.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
Resonances and decaying states have been associated with first-order poles of the S-matrix and represented by exponentially decaying Gamow kets using the Rigged Hilbert Space formulation of quantum mechanics. In analogy, higher-order S- matrix poles, represented by a mixture of Gamow-Jordan vectors, are associated with higher-order decaying states. The exact Golden Rule for decaying states associated to a higher-order pole of the S-matrix is calculated. These higher-order decaying states operators also obey the exponential decay law, but they have a more complicated linewidth than the Breit-Wigner energy distribution. This linewidth depends on the interaction V. To test whether these higher-order decaying state operators represent possible physical states, we discuss their linewidth for various assumptions about the matrix elements of V.
The well-known magnetic or optical Bloch equations are extended to the case of unstable systems such as, e.g., spontaneously decaying particles and molecules undergoing chemical reactions or decaying from excited states back to states other than those involved in resonance. The modified equations are derived from a general theory of irreversible processes where quantum Markovian master equations and the corresponding completely positive dynamical semigroups dictate the time evolution of density operators. The results are of particular importance in those situations where decay occurs level-selectively and all relaxation and decay constants are of comparable order of magnitude. The obtained generalized Bloch equations are of direct applicability to problems of time-resolved magnetic or optical resonance spectroscopy.