Algebraic Bethe Ans\"atze and eigenvalue-based determinants for Dicke-Jaynes-Cummings-Gaudin quantum integrable models

TL;DR

This paper introduces an alternative algebraic Bethe ansatz formulation for Dicke-Jaynes-Cummings-Gaudin models, enabling simplified determinant expressions for eigenstates and overlaps, which could improve numerical analysis in quantum optics.

Contribution

It develops a new Bethe ansatz approach for these models and expresses overlaps and form factors as determinants depending only on eigenvalues.

Findings

Eigenstates expressed as determinants of eigenvalue-dependent matrices

Overlaps and form factors written as partition functions with domain wall boundary conditions

Simplified quadratic Bethe equations facilitate numerical computations

Abstract

In this work, we construct an alternative formulation to the traditional Algebraic Bethe ansatz for quantum integrable models derived from a generalised rational Gaudin algebra realised in terms of a collection of spins 1/2 coupled to a single bosonic mode. The ensemble of resulting models which we call Dicke-Jaynes-Cummings- Gaudin models are particularly relevant for the description of light-matter interaction in the context of quantum optics. Having two distinct ways to write any eigenstate of these models we then combine them in order to write overlaps and form factors of local operators in terms of partition functions with domain wall boundary conditions. We also demonstrate that they can all be written in terms of determinants of matrices whose entries only depend on the eigenvalues of the conserved charges. Since these eigenvalues obey a much simpler set of quadratic Bethe equations, the resulting expressions could then offer important simplifications for the numerical treatment of these models.