  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.
