  We report on the anharmonic signatures in dissipative polaritons' stationary
energy distribution and thermodynamics under external periodic driving. First,
we introduce a dynamic model for the dissipative anharmonic Jaynes-Cummings
polariton with a generic time-periodic interaction representing modulations of
the polariton's energy due to an external force or field. We characterize the
stationary state in terms of the exciton, phonon, and interaction energy
dependence on the phonon anharmonicity, exciton-phonon coupling strength, and
intensity and form of the external field-polariton coupling. Our model also
captures the quantum thermodynamics of the driven polariton, which we analyze
in connection with the irreversible heat, maximum power, and efficiency of the
process. We find considerable differences in energy distribution and
thermodynamics between harmonic, moderate, and strongly anharmonic polaritons.
Moreover, comparing the external modulations to the phonon and exciton energy,
we conclude that the former enhances the polariton's energy storage capacity
and is occasionally limited by interference effects and energy saturation at
the exciton.
