Chaos and integrability in experimentally accessible all-to-all spin models

In recent years, models of disordered fermions with random, all-to-all couplings have emerged as prime candidates for studying the limit of strong chaos in quantum mechanical systems. However, such models are prohibitively difficult to realize experimentally. By contrast, spin models with random, all-to-all couplings can be engineered in the context of cavity QED and could provide an opportunity to probe strongly interacting, disordered physics in the laboratory. We show that the class of models most naturally realized in this system has the unusual property of possessing two integrable points in its phase diagram. We construct the integrals of motion explicitly and propose a method to directly measure their characteristics in the experiment. This scheme raises the possibility of tuning the system between classical and quantum physics on the one hand, by varying the effective spin per site, and between integrable and chaotic physics on the other, by varying the effective cavity interactions.