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Journal of Physics A Mathematical and Theoretical

A two-level atom coupled to a two-dimensional supersymmetric and shape-invariant system: models

A N F Aleixo, A. B. Balantekin

Published Mar 20, 2007• 10 citations
Publisher

Abstract

A class of bound-state problems which represents the coupling of a two-level atom with a two-dimensional supersymmetric system involving two shape-invariant potentials is introduced. We study two models with different coupling Hamiltonians.
DOI
10.1088/1751-8113/40/14/011

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A class of shape-invariant bound-state problems which represent transitions in a two-level system introduced earlier are generalized to include arbitrary energy splittings between the two levels. We show that the coupled-channel Hamiltonians obtained correspond to the generalization of the non-resonant Jaynes-Cummings Hamiltonian, widely used in quantized theories of laser. In this general context, we determine the eigenstates, eigenvalues, the time evolution matrix and the population inversion matrix factor.
The quantum dynamics of a two-level system coupled to a shape-invariant potential is investigated. Shape-invariant potentials share an integrability condition called shape invariance which identifies an underlying algebraic structure, in general infinite dimensional, that transforms the potential parameters such as strength, range and diffuseness. We determine the time-evolution operator, the density operator of the system, and obtain the temporal behaviour of various dynamical variables by considering either pure or mixed initial states of the system, constructed with the generalized coherent state of the shape-invariant coupling potential. We consider specific examples of shape-invariant coupling potentials (harmonic oscillator, Pöschl–Teller and self-similar potentials). The results obtained for all dynamical variables exhibit rapid oscillations which periodically collapse and regenerate in different ways depending on the coupling potential nature.