Hugo Tschirhart and Alexandre Faribault, Algebraic Bethe ansätze and eigenvalue-based determinants for Dicke–Jaynes–Cummings–Gaudin quantum integrable models, Journal of Physics A: Mathematical and Theoretical 47, 405204 (2014)
In this work, we construct an alternative formulation to the traditional algebraic Bethe ansätze for quantum integrable models derived from a generalized rational Gaudin algebra realized 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.
Groenendijk, Solofo and Calzona, Alessio and Tschirhart, Hugo and Idrisov, Edvin G. and Schmidt, Thomas L., Parafermion braiding in fractional quantum Hall edge states with a finite chemical potential, Phys. Rev. B 100, 205424 (2019)
Tschirhart, Hugo and Ong, Ernest T. S. and Sengupta, Pinaki and Schmidt, Thomas L., Phase diagram of spin-1 chains with Dzyaloshinskii-Moriya interaction, Phys. Rev. B 100, 195111 (2019)
Hugo Tschirhart and Thierry Platini and Alexandre Faribault, Steady-states of out-of-equlibrium inhomogeneous Richardson–Gaudin quantum integrable models in quantum optics, Journal of Statistical Mechanics: Theory and Experiment 2018, 083102 (2018)
In this work we present numerical results for physical quantities in the steady-state obtained after a variety of product-states initial conditions are evolved unitarily, driven by the dynamics of quantum integrable models of the rational (XXX) Richardson–Gaudin family, which includes notably Tavis–Cummings models. The problem of interest here is one where a completely inhomogeneous ensemble of two-level systems (spins-1/2) are coupled to a single bosonic mode. The long-time averaged magnetisation along the z-axis as well as the bosonic occupation are evaluated in the diagonal ensemble by performing the complete sum over the full Hilbert space for small system sizes. These numerically exact results are independent of any particular choice of Hamiltonian and therefore describe general results valid for any member of this class of quantum integrable models built out of the same underlying conserved quantities. The collection of numerical results obtained can be qualitatively understood by a relaxation process for which, at infinitely strong coupling, every initial state will relax to a common state where each spin is in a maximally coherent superposition of its and states, i.e. they are in-plane polarised, and consequently the bosonic mode is also in a maximally coherent superposition of different occupation number states. This bosonic coherence being a feature of a superradiant state, we shall loosely use the term superradiant steady-state to describe it. A finite value of the coupling between the spins and the bosonic mode then leads to a long-time limit steady-state whose properties are qualitatively captured by a simple ‘dynamical’ vision in which the coupling strength plays the role of a time at which this ‘relaxation process’ towards the common strong coupling superradiant steady-state is interrupted.
Alexandre Faribault and Hugo Tschirhart and Nicolas Muller, Determinant representation of the domain-wall boundary condition partition function of a Richardson–Gaudin model containing one arbitrary spin, Journal of Physics A: Mathematical and Theoretical 49, 185202 (2016)
In this work we present a determinant expression for the domain-wall boundary condition partition function of rational (XXX) Richardson–Gaudin models which, in addition to spins , contains one arbitrarily large spin S. The proposed determinant representation is written in terms of a set of variables which, from previous work, are known to define eigenstates of the quantum integrable models belonging to this class as solutions to quadratic Bethe equations. Such a determinant can be useful numerically since systems of quadratic equations are much simpler to solve than the usual highly nonlinear Bethe equations. It can therefore offer significant gains in stability and computation speed.