  Superfluidity and Bose-Einstein condensation are usually considered as two
closely related phenomena. Indeed, in most macroscopic quantum systems, like
liquid helium, ultracold atomic Bose gases, and exciton-polaritons,
condensation and superfluidity occur in parallel. In photon Bose-Einstein
condensates realized in the dye microcavity system, thermalization does not
occur by direct interaction of the condensate particles as in the above
described systems, i.e. photon-photon interactions, but by absorption and
re-emission processes on the dye molecules, which act as a heat reservoir.
Currently, there is no experimental evidence for superfluidity in the dye
microcavity system, though effective photon interactions have been observed
from thermo-optic effects in the dye medium. In this work, we theoretically
investigate the implications of effective thermo-optic photon interactions, a
temporally delayed and spatially non-local effect, on the photon condensate,
and derive the resulting Bogoliubov excitation spectrum. The calculations
suggest a linear photon dispersion at low momenta, fulfilling the Landau's
criterion of superfluidity . We envision that the temporally delayed and
long-range nature of the thermo-optic photon interaction offer perspectives for
novel quantum fluid phenomena.
