  Using the composite boson (coboson) many-body formalism, we determine under
which conditions "trion-polariton" can exist. Dipolar attraction can bind an
exciton and an electron into a trion having an energy well separated from the
exciton energy. Yet, the existence of long-lived "trion-polariton" is a priori
implausible not only because the photon-trion coupling, which scales as the
inverse of the sample volume, is vanishingly small, but mostly because this
coupling is intrinsically "weak". Here, we show that a moderately dense Fermi
sea renders its observation possible: on the pro side, the Fermi sea overcomes
the weak coupling by pinning the photon to its momentum through Pauli blocking,
it also overcomes the dramatically poor photon-trion coupling by providing a
volume-linear trion subspace to which the photon is coherently coupled. On the
con side, the Fermi sea broadens the photon-trion resonance due to the
fermionic nature of trions and electrons, it also weakens the trion binding by
blocking electronic states relevant for trion formation. As a result, the
proper way to observe this novel polariton is to use doped semiconductor having
long-lived electronic states, highly-bound trion and Fermi energy as large as a
fraction of the trion binding energy.
