  We introduce an auxiliary-particle field theory to treat the non-Markovian
dynamics of driven-dissipative quantum systems of the Jaynes-Cummings type. It
assigns an individual quantum field to each reservoir state and provides an
analytic, faithful representation of the coupled system-bath dynamics. We apply
the method to a driven-dissipative photon Bose-Einstein condensate (BEC)
coupled to a reservoir of dye molecules with electronic and vibronic
excitations. The complete phase diagram of this system exhibits a hidden,
non-Hermitian phase transition separating temporally oscillating from
biexponentially decaying photon density correlations within the BEC. On one
hand, this provides a qualitative distinction of the thermal photon BEC from a
laser. On the other hand, it shows that one may continuously tune from the BEC
to the lasing phase by circumventing a critical point. This auxiliary-particle
method is generally applicable to the dynamics of open, non-Markovian quantum
systems.
