On the Bethe ansatz for the Jaynes–Cummings–Gaudin model

Abstract

Abstract. We investigate the quantum Jaynes-Cummings model -a particular case of the Gaudin model with one of the spins being infinite. Starting from the Bethe equations we derive Baxter's equation and from it a closed set of equations for the eigenvalues of the commuting Hamiltonians. A scalar product in the separated variables representation is found for which the commuting Hamiltonians are Hermitian. In the semi classical limit the Bethe roots accumulate on very specific curves in the complex plane. We give the equation of these curves. They build up a system of cuts modeling the spectral curve as a two sheeted cover of the complex plane. Finally, we extend some of these results to the XXX Heisenberg spin chain.

Eigenvalue-based method and form-factor determinant representations for integrable XXZ Richardson-Gaudin models

“…It was already noted by Babelon and Talalaev that similar equations could be found for the XXX Heisenberg spin 155102-10 chain [22], so the question naturally arises if this method can be generalized to other Bethe ansatz equations obtained from the quantum inverse scattering method [49]. The extension of the proposed determinant expressions to degenerate models will be the subject of future research.…”

“…The equations for the Dicke model were independently presented by Babelon and Talalaev [22]. In this method, an alternative set of equations is derived in terms of variables related to the eigenvalues of the constants of motion.…”

Algebraic Bethe ansätze and eigenvalue-based determinants for Dicke–Jaynes–Cummings–Gaudin quantum integrable models

“…While many efforts have been made over time to deal with these equations directly [13][14][15][16], the rewriting of the Bethe equations as an ensemble of N quadratic equations [7,17,18] has recently greatly simplified the numerical treatment of such systems [19][20][21][22]. It has indeed been shown that the Bethe equations for models with…”

Топологическое разложение модели $\beta$-ансамбля и квантовая алгебраическая геометрия в рамках секторного подхода

“…Это условие "нулевой монодромии" автоматически выполняется для всех форм, происходящих из уравнения Шредингера. Оно эквива-лентно уравнениям анзаца Бете, которым удовлетворяют s i , и аналогично свойству, имеющему место в модели Годена [18].…”