Numerical Simulations Unveil Superradiant Coherence in a Lattice of Charged Quantum Oscillators

TL;DR

This paper demonstrates through numerical simulations that a lattice of charged quantum oscillators can form a superradiant coherent state with a significant energy gap, influenced by the electrostatic confinement and electromagnetic mode distribution.

Contribution

It introduces a novel superradiant phase in a lattice of charged oscillators, highlighting the role of electrostatic confinement and off-shell frequencies in coherence formation.

Findings

Confirmation of condensation via numerical simulations.

Observation of an energy gap of a few eV per particle.

Enhanced coupling due to all-direction electromagnetic modes.

Abstract

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.