Thermal state structure in the Tavis-Cummings model and rapid simulations in mesoscopic quantum ensembles

Hybrid quantum systems consisting of a collection of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math> spin-<b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mn>1</b:mn><b:mo>/</b:mo><b:mn>2</b:mn></b:mrow></b:math> particles uniformly interacting with an electromagnetic field, such as one confined in a cavity, are important for the development of quantum information processors and will be useful for metrology, as well as tests of collective behavior. Such systems are often modeled by the Tavis-Cummings model, and having an accurate understanding of the thermal behaviors of this system is needed to understand their behavior in realistic environments. We quantitatively show in this work that the Dicke subspace approximation is at times invoked too readily. Specifically, we show that there is a temperature above which the degeneracies in the system become dominant and the Dicke subspace is minimally populated. This transition occurs at a lower temperature than previously considered. In such a temperature regime, the key constants of the motion are the total excitation count between the spin system and cavity and the collective angular momentum of the spin system. These enable perturbative expansions for thermal properties in terms of the energy shifts of dressed states, called Lamb shifts herein. They enable efficient numeric methods that scale in terms of the size of the spin system. Notably the runtime to obtain certain parameters of the system scales as <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>O</c:mi><c:mo>(</c:mo><c:msqrt><c:mi>N</c:mi></c:msqrt><c:mo>)</c:mo></c:mrow></c:math>, and is thus highly efficient. These provide methods for approximating, and bounding, properties of these systems as well as characterizing the dominant population regions, including under perturbative noise. In the regime of stronger spin-spin coupling, the perturbations outweigh the expansion series terms, and inefficient methods must likely be employed, removing the computational efficiency of simulating such systems. The results in this work can also be used for related systems such as coupled-cavity arrays, cavity-mediated coupling of collective spin ensembles, and collective spin systems.