Abstract
A system of Nosc 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 ω of the oscillators and the plasma frequency ωp of the charges are combined into a frequency ω′=ω2+ωp2 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 ω 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 Nosc by searching for an approximation of the ground state via suitable coherent trial states. Alternatively, numerical simulations can be employed for smaller Nosc. 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.