Massive phonons and gravitational dynamics in a photon-fluid model

We theoretically investigate the excitation dynamics in a photon-fluid with both local and nonlocal interactions. We show that the interplay between locality and an infinite-range nonlocality gives rise to a gapped Bogoliubov spectrum of elementary excitations which, at lower momenta, correspond to massive particles (phonons) with a relativistic energy-momentum relation. In this regime and in the presence of an inhomogeneous flow the density fluctuations are governed by the massive Klein-Gordon equation on the acoustic metric and thus propagate as massive scalar fields on a curved spacetime. We finally demonstrate that in the non-relativistic limit the phonon modes behave as self-gravitating quantum particles with an effective Schr\"{o}dinger-Newton dynamics, although with a finite-range gravitational interaction and a non-zero cosmological constant. Our photon-fluid represents a viable alternative to BEC models for "emergent-gravity" scenarios and offers a promising setting for analogue simulations of quantum gravity phenomenology and semiclassical gravity.