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

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.

26 pages, 0 figure, published version

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• Eigenvalue-based determinants for scalar products and form factors in Richardson-Gaudin integrable models coupled to a bosonic mode
• Completeness of the Bethe states for the rational, spin-1/2 Richardson--Gaudin system
• Exact solution of the p+ip Hamiltonian revisited: duality relations in the hole-pair picture
• Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields
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• Multifractality in the interacting disordered Tavis-Cummings model
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• Determinant representation of the domain-wall boundary condition partition function of a Richardson-Gaudin model containing one arbitrary spin
• Quantum-classical duality for Gaudin magnets with boundary
• Steady-states of out-of-equlibrium inhomogeneous Richardson-Gaudin quantum integrable models in quantum optics
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary