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In particular, the main problem of the Bogoliubov transformation comes when, with the parameters HT ht and HT ht , one tries to verify if the transformation is unitary. In fact, if one computes the commutators between HT ht and HT ht , an expression similar to Eq. (3) is obtained. [Extracted from the article]

We study the Tavis-Cummings model with three modes of oscillation by using four different algebraic methods: the Bogoliubov transformation, the normal-mode operators, and the tilting transformation of the SU(1, 1) and SU(2) groups. The algebraic method based on the Bogoliubov transformation and the normal-mode operators lets us obtain the energy spectrum and eigenfunctions of a particular case of the Tavis-Cummings model, while with the tilting transformation we are able to solve the most general case of this Hamiltonian. Finally, we compute some expectation values of this problem by means of the SU(1, 1) and SU(2) group theory. [ABSTRACT FROM AUTHOR]

We introduce two generalizations of the Jaynes–Cummings (JC) model for two modes of oscillation. The first model is formed by two Jaynes–Cummings interactions, while the second model is written as a simultaneous Jaynes–Cummings and Anti-Jaynes–Cummings (AJC) interactions. We study some of its properties and obtain the energy spectrum and eigenfunctions of these models by using the tilting transformation and the Perelomov number coherent states of the two-dimensional harmonic oscillator. Moreover, as physical applications, we connect these new models with two important and novel problems: The relativistic non-degenerate parametric amplifier and the relativistic problem of two coupled oscillators. [ABSTRACT FROM AUTHOR]