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Physical Review A

Soluble quantum map with continuous spectrum

Kazimierz Rza̧ewski, Jan Mostowski

Published May 1, 1987• 9 citations
Publisher

Abstract

We present a soluble model of an atomic system driven by a train of \ensuremath{\delta}-function impulses. The atomic Hamiltonian describes an atom with one bound state and one continuum representing an autoionizing resonance. The model is soluble or requires very simple numerics. The results depend strongly on the shape of the resonance. Population trapping is discovered for a Gaussian-shaped continuum. Corrections coming from the counterrotating terms are discussed.

DOI
10.1103/physreva.35.4414

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We present a soluble model of an atomic system driven by a train of \ensuremath{\delta}-function impulses. The atomic Hamiltonian describes an atom with one bound state and one continuum representing an autoionizing resonance. The model is soluble or requires very simple numerics. The results depend strongly on the shape of the resonance. Population trapping is discovered for a Gaussian-shaped continuum. Corrections coming from the counterrotating terms are discussed.
Resonances in quantum mechanics are commonly introduced as quasi-bound states embedded in the continuum, a perspective that can be conceptually challenging due to the abstract nature of continuum states. In this work, we discuss an alternative approach that avoids an explicit treatment of the continuum by formulating the problem in terms of discrete quantum states. Our discussion is based on the stabilization method, in which the system is confined to a finite region such that the continuum is replaced by a discrete energy spectrum. Resonances then appear as characteristic features in the energy levels under variation of the confining box size, providing an intuitive interpretation in terms of a two-level system while remaining closely connected to standard quantum mechanics curriculum. We review the method, derive selected results, and discuss practical strategies for extracting resonance parameters from stabilization diagrams. In addition to established fitting procedures, we introduce a novel approach based on the analysis of spatial localization of resonant states, which enables a robust identification of resonance properties. The approach is illustrated using both attractive and repulsive delta-shell potentials, which serve as simple and instructive model systems amenable to analytical treatment.
An exactly soluble model is used to describe transitions, induced by two strong laser beams, proceeding from the initial discrete state to the continuum via two channels. In the first channel the transition occurs via an intermediate, near-resonant discrete state located close to the continuum edge. In the second channel the intermediate state belongs to the continuum. Population trapping occurs due to an interference between the two channels. The model can be used to describe the photodetachment of electrons from negative ions or the photodissociation of molecules.
Considering a gas of independent fermions in the presence of an attractive localized potential, one can show that the properties of the system as a whole are smooth (analytic) functions of the strength parameter of the potential, even at those values where new single-particle bound states appear. Thus for the system as a whole, the transition from ``unbound'' to ``bound'' states is continuous and the concept of a bound state cannot be made precise. This is illustrated here for a simple mathematically soluble model—noninteracting spinless fermions moving in the presence of a delta-function potential in one dimension. Some related physical ideas are also presented.
Studies the coupling of N discrete levels to a common continuum. The discrete levels, which can include both genuine bound states and autoionisation states, are assumed to be coupled to the continuum and to each other in such a way as to permit a rotating wave approximation in the Schrodinger equation. A general solution for the model is derived with the aid of Laplace transforms and the Weisskopf-Wigner approximation. The author discusses general properties of the discrete level populations and the electron spectrum like population trapping, zeros, crossings, resonances and confluences. As examples the author treats cases where two or three discrete levels interact with a continuum.