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Journal of Physics A Mathematical and Theoretical

Modified n -level, n − 1-mode Tavis–Cummings model and algebraic Bethe ansatz

T. Skrypnyk

Published Oct 19, 2017• 3 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/aa94af

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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.
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Abstract This chapter extends the algebraic Bethe ansatz to the quantum Tavis–Cummings model, an N atom generalization of the Jaynes–Cummings model to describe the strong interaction between light and quantum matter. In the case of the quantum Tavis–Cum- mings model there is no underlying vertex model to suggest the constituent building blocks of the algebraic Bethe ansatz approach, e.g.like the L-matrix or ultimately the transfer matrix. The algebraic Bethe ansatz is then first applied to the Tavis–Cummings Hamiltonian with an added Stark term using a conjecture for the transfer matrix. The original Tavis–Cummings model and its algebraic Bethe ansatz are obtained in the limit of vanishing Stark term, which requires considerable care.
We consider quantum integrable systems associated with non-skew-symmetric sl (2)-valued classical r -matrices. For a special class of such r -matrices we construct a one-parametric family of integrable modifications of the ‘two-level one-mode’ Jaynes–Cummings–Dicke Hamiltonians, with non-uniform coupling constants, containing additional Kerr- and Stark-type nonlinearities. We also construct a family of integrable Bose–Hubbard-type dimers and a family of integrable models that unifies Bose–Hubbard- and Jaynes–Cummings–Dicke-type models and may be called a ‘two-level, two-mode’ Jaynes–Cummings–Dicke model or a ‘spin generalization’ of a Bose–Hubbard dimer. We diagonalize the constructed models with the help of the algebraic Bethe ansatz technique in any irreducible representation of the sl (2) ⊕ N spin algebra.
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.