  Two dimensional photon gases trapped in dye-filled microcavities can undergo
thermalization and nearly ideal equilibrium Bose-Einstein condensation.
However, they are inherently driven-dissipative systems that can exhibit an
intricate interplay between the thermalizing influence of the environment given
by the dye solution and the pump and loss processes driving the system out of
equilibrium. We show that this interplay gives rise to a robust mechanism for
two-mode emission, when the system is driven by an off-centered pump beam.
Namely, after the system starts lasing in the dominantly pumped excited mode,
in a second transition a photon condensate is formed in the ground mode, when
the pump power is increased further. This effect is a consequence of the
redistribution of excited dye molecules via the lasing mode in combination with
thermalization. We propose to exploit this effect for engineering controlled
two-mode emission and demonstrate that by tailoring the transverse potential
landscape for the photons, the threshold pump power can be tuned by orders of
magnitude.
