Lee-Yang theory of the superradiant phase transition in the open Dicke model

The Dicke model describes an ensemble of two-level atoms that are coupled to a confined light mode of an optical cavity. Above a critical coupling, the cavity becomes macroscopically occupied, and the system enters the superradiant phase. This phase transition can be observed by detecting the photons that are emitted from the cavity; however, it only becomes apparent in the limit of long observation times, while actual experiments are of a finite duration. To circumvent this problem, we here make use of recent advances in Lee-Yang theories of phase transitions to show that the superradiant phase transition can be inferred from the factorial cumulants of the photon emission statistics obtained during a finite measurement time. Specifically, from the factorial cumulants, we can determine the complex singularities of generating functions that describe the photon emission statistics, and by extrapolating their positions to the long-time limit, one can detect the superradiant phase transition. We also show that the convergence points determine the tails of the large-deviation statistics of the photon current. Our work demonstrates how phase transitions in the Dicke model and in other quantum many-body systems can be detected from measurements of a finite duration.