  Strong Coulomb interactions in atomically thin semiconductors like monolayer
WSe$_2$ induce not only tightly bound excitons, but also make their optical
properties very sensible to doping. By utilizing a microscopic theory based on
the excitonic Heisenberg equations of motion, we systematically determine the
influence of doping on the excitonic linewidth, lineshift, and oscillator
strength. We calculate trion resonances and demonstrate that the Coulomb
coupling of excitons to the trionic continuum generates a non-Markovian
interference, which, due to a time retardation, builds up a phase responsible
for asymmetric exciton line shapes and increased excitonic linewidths. Our
calculated doping dependence of exciton and trion linewidths, lineshifts, and
oscillator strengths explains recent experiments. The gained insights provide
the microscopic origin of the optical fingerprint of doped atomically thin
semiconductors.
