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Physica Scripta

Path integral for the generalized Jaynes-Cummings model

Tahar Boudjedaa, Abdelhafid Bounames, Khireddine Nouicer, L. Chétouani, Théophile F. Hammann

Published Sep 1, 1996• 14 citations
Publisher

Abstract

We use the coherent state path integral and a fermion model for the spin to solve a non-linear bosonic generalization of the Jaynes-Cummings model. The propagators are given explicitly as perturbation series. These are summed up exactly for special cases and the corresponding atomic population inversion as well as energy spectrum and wave functions are deduced.

DOI
10.1088/0031-8949/54/3/001

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We use the coherent state path integral and a fermion model for the spin to solve a non-linear bosonic generalization of the Jaynes-Cummings model. The propagators are given explicitly as perturbation series. These are summed up exactly for special cases and the corresponding atomic population inversion as well as energy spectrum and wave functions are deduced.
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We present a general qubit-boson interaction Hamiltonian that describes the Jaynes-Cummings model and its extensions as a single Hamiltonian class. Our model includes non-linear processes for both the free qubit and boson field as well as non-linear, multi-boson excitation exchange between them. It shows an underlying algebra with supersymmetric quantum mechanics features allowing an operator based diagonalization that simplifies the calculations of observables. As a practical example, we show the evolution of the population inversion and the boson quadratures for an initial state consisting of the qubit in the ground state interacting with a coherent field for a selection of cases covering the standard Jaynes-Cummings model and some of its extensions including Stark shift, Kerr-like, intensity dependent coupling, multi-boson exchange and algebraic deformations.
A two-parameter (p, q) deformation of the Jaynes-Cummings model is obtained using a recently developed (p, q)-deformed oscillator. In the rotating wave approximation (RWA) the dynamical symmetry of the model is the quantum superalgebra u p,q (1 mod 1). The partition function of the model is obtained as a path integral over generalized Perelomov coherent states corresponding to the quantum algebra U p.q (1 mod 1). The complete spectrum of the model is extracted for both the case when the coupling constants are Grassman valued and when they are c-number valued. It is noted that a relaxation of the RWA extends the dynamical symmetry to the quantum superalgebra osp p,q (2 mod 2).