  Recent experimental advances in Positronium (Ps) physics have made it
possible to produce dense Ps ensembles in which Ps-Ps interactions may occur,
leading to the production of Ps$_2$ molecules and paving the way to the
realization of a Ps Bose-Einstein Condensate (BEC). In order to achieve this
latter goal it would be advantageous to develop new methods to measure Ps
densities in real-time. Here we describe a possible approach to do this using
polaritonic methods: using realistic experimental parameters we demonstrate
that a dense Ps gas can be strongly coupled to the photonic field of a
distributed Bragg reflector microcavity. In this strongly coupled regime, the
optical spectrum of the system is composed of two hybrid positronium-polariton
resonances separated by the vacuum Rabi splitting, which is proportional to the
square root of the Ps density. Given that polaritons can be created on a
sub-cycle timescale, a spectroscopic measurement of the vacuum Rabi splitting
could be used as an ultra-fast Ps density measurement in regimes relevant to Ps
BEC formation. Moreover, we show how positronium-polaritons could potentially
enter the ultrastrong light-matter coupling regime, introducing a radically
novel platform to explore its non-perturbative phenomenology.
