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Journal of Physics A Mathematical and General

General sets of coherent states and the Jaynes$ndash$Cummings model

Mohammed Daoud A, Véronique Hussin

Published Aug 13, 2002• 38 citations
Publisher

Abstract

General sets of coherent states are constructed for quantum systems admitting a nondegenerate infinite discrete energy spectrum. They are eigenstates of an annihilation operator and satisfy the usual properties of standard coherent states. The application of such a construction to the quantum optics Jaynes–Cummings model leads to a new understanding of the properties of this model.

DOI
10.1088/0305-4470/35/34/310

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General sets of coherent states and the JaynesndashndashndashCummings model
General sets of coherent states are constructed for quantum systems admitting a nondegenerate infinite discrete energy spectrum. They are eigenstates of an annihilation operator and satisfy the usual properties of standard coherent states. The application of such a construction to the quantum optics Jaynes–Cummings model leads to a new understanding of the properties of this model.
Generalized and Gaussian coherent states constructed for quantum system with degeneracies in the energy spectrum are compared with respect to some minimal definitions and fundamental properties they have to satisfy. The generalized coherent states must be eigenstates of a certain annihilation operator that has to be properly defined in the presence of degeneracies. The Gaussian coherent states are, in the particular harmonic oscillator case, an approximation of the generalized coherent states and so the localizability in phase space of the particle in those states is very good. For other quantum systems, this last property serves as a definition of those Gaussian coherent states. The example of a particle in a two-dimensional square box is thus revisited having in mind the preceding definitions of generalized and Gaussian coherent states and also the preservation of the important property known as the resolution of the identity operator.
In the representation of dressed states, which diagonalizes the Hamiltonian in the Jaynes-Cummings model, the Bose operator for excitation annihilation Â, the spin operator Σ^_\hat \Sigma \_Σ^_ , and the corresponding Hermitian-conjugate operators are found. Using these operators, correlated coherent states were constructed and statistical properties of a field mode and an atom in these states are studied. Nonclassical field properties, such as the deviation of the probability distribution for the number of photons from the Poissonian distribution, its time dependence, and the squeezing of quadrature field components are found. For small interaction coefficients k, all these characteristics of the model vary weakly in time in the resonance case.