  Various experimental platforms have proven to be valid testbeds for the
implementation of nondipolar light-matter interactions, where atomic systems
and confined modes interact via two-photon couplings. Here, we study a damped
quantum harmonic oscillator interacting with $N$ two-level systems via a
two-photon coupling in the so-called bad-cavity limit, in the presence of
finite-temperature baths and coherent and incoherent drivings. We have
succeeded in applying a recently developed adiabatic elimination technique to
derive an effective master equation for the two-level systems, presenting two
fundamental differences compared to the case of a dipolar interaction: an
enhancement of the two-level systems spontaneous-like emission rate, including
a thermal contribution and a quadratic term in the coherent driving, and an
increment of the effective temperature perceived by the two-level systems.
These differences give rise to striking effects in the two-level systems
dynamics, including a faster generation of steady-state coherence and a richer
dependence on temperature of the collective effects, which can be made stronger
at higher temperature.
