Common framework and quadratic Bethe equations for rational Gaudin magnets in arbitrarily oriented magnetic fields

in SciPost Physics (2017)

In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure ... [more ▼]

In this work we demonstrate a simple way to implement the quantum inverse scattering method to find eigenstates of spin-1/2 XXX Gaudin magnets in an arbitrarily oriented magnetic field. The procedure differs vastly from the most natural approach which would be to simply orient the spin quantisation axis in the same direction as the magnetic field through an appropriate rotation. Instead, we define a modified realisation of the rational Gaudin algebra and use the quantum inverse scattering method which allows us, within a slightly modified imple- mentation, to build an algebraic Bethe ansatz using the same unrotated reference state (pseudovacuum) for any external field. This common framework allows us to easily write determinant expressions for certain scalar products which would be highly non-trivial in the rotated system approach. [less ▲]

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

in Journal of Physics. A, Mathematical and Theoretical (2014), 47

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 ... [more ▼]

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. [less ▲]