Synthetic dimensions in optical cavities and their analogies to two-dimensional materials

Abstract

A method to produce synthetic dimensions in an optical cavity is presented. The system is obtained by coupling two one-dimensional optical cavities by means of a moving mirror. Its translation generates a new synthetic dimension resulting in a transmittance pattern that corresponds to the energy dispersion of electrons in a two-dimensional lattice: in this case a strained triangular lattice. We elucidate the analogy between these two systems by relating the distribution of transmittances of the optical cavity to the density of states of the two-dimensional lattice and the Bragg diffraction modes to the Van Hove singularities. Our mapping makes the coupled optical cavity a simulator for lattice Hamiltonians found in solid-state physics, providing an easier alternative platform to access some of their properties, as for example, the electronic conductivity, which is found here using the Boltzmann formula. Moreover, it is proved that the truly synthetic behavior appears only when the two cavities have an irrational length ratio as modes resonant on one cavity are never resonant in the other. Mathematically, wave phase differences act independently as they are given by a function that behaves as a pseudorandom number generator. This provides an elegant way to study synthetic dimensions without the need to move the central mirror, i.e., in a single, fixed geometry device. Thus, our work establishes a bridge between the fields of optical cavities, synthetic dimensions, and two-dimensional materials.