Critical Behavior of Light and Matter in Two-Dimensional Pöschl-Teller Potentials

Ideal two-dimensional homogeneous Bose gases do not undergo true Bose–Einstein condensation (BEC) at finite temperature or density. Nonetheless, critical behavior can emerge in the presence of boundaries softening or inter-particle interactions, the former being cleanly exemplified by a two-dimensional Pöschl–Teller potential, as discussed here. As we will illustrate, such a boundary softening crucially modifies the density of states of a rigid box, thus allowing for the emergence of finite-temperature and density BEC. We further discuss the critical behavior of thermodynamic properties and of the density correlation function as BEC is approached. In particular, we find that the compressibility diverges logarithmically and that the correlation length diverges with exponent $\nu = 1/2$, with logarithmic corrections. Remarkably, the finite-size transition disclosed in a photon gas confined in two dimensional optical microcavities can be reproduced within this analysis. Our results provide a unified framework for understanding condensation in confined two-dimensional quantum gases of light and matter.