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Journal of Physics A Mathematical and General

A two-parameter deformation of the Jaynes-Cummings model: path integral representation

R. Chakrabarti, R. Jagannathan

Published Dec 7, 1992• 12 citations
Publisher

Abstract

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).

DOI
10.1088/0305-4470/25/23/033

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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).
The q-deformations of the quantum harmonic oscillator are used for to describe the generalized Jaynes-Cummings model (JCM) by using the q-analog of the Holstein-Primakoff realization of the su(1,1). The corresponding dynamical symmetry is described by a quantum algebra. The q-analogs of the Barut-Girardello and the Perelomov coherent states are introduced and the expectation value of ${\mathrm{\ensuremath{\sigma}}}_{3}$ is calculated. The periodic revivals of the generalized JCM are destroyed for increasing deformation parameter q. The deformed original JCM in the rotating-wave approximation can be described by the u(1\ensuremath{\Vert}1)q{)}_{\mathit{q}})q​, while its relaxation extends the dynamical algebra to the osp(2\ensuremath{\Vert}2)q{)}_{\mathit{q}})q​ quantum superalgebra.
We introduce a parafermionic version of the Jaynes–Cummings Hamiltonian, by coupling k Fock parafermions (nilpotent of order F ) to a 1D harmonic oscillator, representing the interaction with a single mode of the electromagnetic field. We argue that for k = 1 and F ⩽ 3 there is no difference between Fock parafermions and quantum spins . We also derive a semiclassical approximation of the canonical partition function of the model by assuming ℏ to be small in the regime of the large enough total number of excitations n , where the dimension of the Hilbert space of the problem becomes constant as a function of n . We observe in this case an interesting behaviour of the average of the bosonic number operator showing a single crossover between regimes with different integer values of this observable. These features persist when we generalize the parafermionic Hamiltonian by deforming the bosonic oscillator with a generic function Φ( x ); the q -deformed bosonic oscillator corresponds to a specific choice of the deformation function Φ. In this particular case, we observe, at most k ( F − 1), crossovers in the behaviour of the mean bosonic number operator, suggesting a phenomenology of superradiance similar to the k -atoms Jaynes–Cummings model.
We show how, by resorting to a one-fermion realization of the Pauli operators, the Hamiltonian of the Jaynes-Cummings (JC) model can be identified as an element of the superalgebra u(1|1), which plays the role of a dynamical algebra. The extension of this notion to osp(2|2) allows adding both virtual and real two-photon processes to the JC Hamiltonian. The exact diagonalization problem is tackled here in the special case when the coupling constants of the fermionic terms of the "dressed" JC Hamiltonian are assumed to nilpotent Grassman-Banach numbers.