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Journal of Physics A Mathematical and Theoretical

Then-level,n− 1-mode Jaynes–Cummings model: spectrum and eigenvectors

T. Skrypnyk

Published Jan 21, 2013• 8 citations
Publisher

Abstract

We explicitly find eigenvectors and eigenvalues of the general n -level, n − 1-mode Jaynes–Cummings model in Λ-configuration possessing arbitrary detuning parameters Δ i , and arbitrary interaction strengths g i , . In the case of equal interaction strengths g i = g j , , we compare the obtained answers with those obtained by the algebraic Bethe ansatz method (Skrypnyk 2008 J. Phys. A: Math. Theor. 41 475202; 2009 J. Math. Phys. 50 103523).

DOI
10.1088/1751-8113/46/5/052001

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We explicitly find eigenvectors and eigenvalues of the general n -level, n − 1-mode Jaynes–Cummings model in Λ-configuration possessing arbitrary detuning parameters Δ i , and arbitrary interaction strengths g i , . In the case of equal interaction strengths g i = g j , , we compare the obtained answers with those obtained by the algebraic Bethe ansatz method (Skrypnyk 2008 J. Phys. A: Math. Theor. 41 475202; 2009 J. Math. Phys. 50 103523).
Abstract Using the quantum group technique we construct a one-parametric family of integrable modifications of the n -level, n − 1 mode Tavis–Cummings Hamiltonian possessing an additional Stark-type term. We show that in the ‘quasiclassical’ limit the constructed Hamiltonian transforms into the integrable Hamiltonian of the quantum n -level, n − 1 mode Tavis–Cummings model with the equal interaction strengths considered in Skrypnyk (2008 J. Phys. A: Math. Theor . 41 475202, 2009 J. Math. Phys . 50 103523). We diagonalize the constructed ‘modified’ Tavis–Cummings Hamiltonian and its second order integrals of motion using the nested Bethe ansatz.
A generalized three-level Jaynes - Cummings model (JCM) which includes various ordinary JCMs is shown explicitly to have an SU(3) structure: the Hamiltonian can be treated as a linear function of the generators of the SU(3) group. Based on this algebraic structure, the exact algebraic solutions of the Schrödinger equation, as well as eigenvalues and eigenstates of the Hamiltonian, are obtained by an algebraic method. Thus the three-level JCM is completely solved algebraically. The SU(N) structure of the N-level JCM is also constructed explicitly and can be solved by the same method.
The unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.
The Jaynes-Cummings model describing the interaction of a single linearly- polarized mode of the quantized electromagnetic field with an isolated two- level atom is generalized to the case of atomic levels degenerate in the projections of the angular momenta on the quantization axis, which is a usual case in the experiments. This generalization, like the original model, obtains the explicit solution. The model is applied to calculate the dependence of the atomic level populations on the angle between the polarization of cavity field mode and that of the laser excitation pulse in the experiment with one-atom micromaser.
Using the technique of classical r -matrices and quantum Lax operators we construct the most general form of the quantum integrable ‘two-level, one-mode’ spin-boson Jaynes–Cummings–Dicke-type model. We consider in detail two subclasses of this model and the corresponding examples associated with concrete classical r -matrices with spectral parameters. We show that the so-called ‘diagonal’ in the root basis r -matrices produces integrable Jaynes–Cummings–Dicke-type models with the ‘rotating wave approximation’, while all other types of classical r -matrices produce integrable JCD models without the rotating wave approximation. We construct an explicit example of an integrable Hermitian Jaynes–Cummings–Dicke-type Hamiltonian containing both ‘rotating’ and ‘counter-rotating’ terms. In the case of the diagonal r -matrices we find the spectrum of the corresponding quantum Hamiltonians using the algebraic Bethe ansatz.