  We present a semi-analytic and asymptotically exact solution to the problem
of phonon-induced decoherence in a quantum dot-microcavity system. Particular
emphasis is placed on the linear polarization and optical absorption, but the
approach presented herein may be straightforwardly adapted to address any
elements of the exciton-cavity density matrix. At its core, the approach
combines Trotter's decomposition theorem with the linked cluster expansion. The
effects of the exciton-cavity and exciton-phonon couplings are taken into
account on equal footing, thereby providing access to regimes of comparable
polaron and polariton timescales. We show that the optical decoherence is
realized by real phonon-assisted transitions between different polariton states
of the quantum dot-cavity system, and that the polariton line broadening is
well-described by Fermi's golden rule in the polariton frame. We also provide
purely analytic approximations which accurately describe the system dynamics in
the limit of longer polariton timescales.
