Steady-states of out-of-equlibrium inhomogeneous Richardson–Gaudin quantum integrable models in quantum optics

Abstract

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.