  We study systematically the role of temperature in the optical response of
doped two-dimensional semiconductors. By making use of a finite-temperature
Fermi-polaron theory, we reveal a crossover from a quantum-degenerate regime
with well-defined polaron quasiparticles to an incoherent regime at high
temperature or low doping where the lowest energy "attractive" polaron
quasiparticle is destroyed, becoming subsumed into a broad trion-hole
continuum. We demonstrate that the crossover is accompanied by significant
qualitative changes in both absorption and photoluminescence. In particular,
with increasing temperature (or decreasing doping), the emission profile of the
attractive branch evolves from a symmetric Lorentzian to an asymmetric peak
with an exponential tail involving trions and recoil electrons at finite
momentum. We discuss the effect of temperature on the coupling to light for
structures embedded into a microcavity, and we show that there can exist
well-defined polariton quasiparticles even when the exciton-polaron
quasiparticle has been destroyed, where the transition from weak to strong
light-matter coupling can be explained in terms of the polaron linewidths and
spectral weights.
