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Journal of Geometry and Physics

Quantum integrable models of interacting bosons and classical r -matrices with spectral parameters

T. Skrypnyk

Published Jul 13, 2015• 8 citations
Publisher
DOI
10.1016/j.geomphys.2015.07.015

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arXiv:2410.19035•Theoretical and Mathematical Physics•Feb 1, 2025•1 citation
Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.
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.