  A system of ${N_{osc}}$ charged oscillators interacting with the
electromagnetic field, spatially confined in a 3D lattice of sub-wavelength
dimension, can condense into a superradiant coherent state if appropriate
density and frequency conditions are met. In this state, the common frequency
$\omega$ of the oscillators and the plasma frequency $\omega_p$ of the charges
are combined into a frequency $\omega'=\sqrt{\omega^2+\omega_p^2}$ that is
off-shell with respect to the wavelength of the photon modes involved,
preventing them from propagating outside the material. Unlike other atomic
cavity systems, the frequency $\omega$ in this case is not determined by the
cavity itself but is defined by the periodic electrostatic potential that
confines the charged particles in the lattice. Additionally, the
electromagnetic modes involved have wave vectors distributed in all spatial
directions, resulting in a significant increase in coupling. The analytical
study of this system can be carried out in the limit of large ${N_{osc}}$ by
searching for an approximation of the ground state via suitable coherent trial
states. Alternatively, numerical simulations can be employed for smaller
${N_{osc}}$. In the numerical approach, it is possible to go beyond the
Rotating Wave Approximation (RWA) and introduce a dissipation term for the
photon modes. This dissipation term can account for the ohmic quench in a metal
and also consider photon losses at the boundary of the material. By utilizing
numerical solutions and Monte Carlo simulations, the presence of condensation
has been confirmed, and an energy gap of a few electron volts (eV) per particle
has been observed in typical metal crystals with protons bound to tetrahedral
or octahedral sites.
