  We theoretically investigate the critical properties of a single
driven-dissipative nonlinear photon mode. In a well-defined thermodynamical
limit of large excitation numbers, the exact quantum solution describes a
first-order phase transition in the regime where semiclassical theory predicts
optical bistability. We study the behavior of the complex spectral gap
associated with the Liouvillian superoperator of the corresponding master
equation. We show that in this limit the Liouvillian gap vanishes exponentially
and that the bimodality of the photon Wigner function disappears. The
connection between the considered thermodynamical limit of large photon numbers
for the single-mode cavity and the thermodynamical limit of many cavities for a
driven-dissipative Bose-Hubbard system is discussed.
