Dirac equation on a square waveguide lattice with site-dependent coupling strengths and the gravitational Aharonov-Bohm effect

The main objective of this work is to present a theoretical proposal for an implementation of the $(2 + 1)$-dimensional Dirac equation in classical gravitational and electromagnetic backgrounds in a two-dimensional waveguide array. For this, a framework for achieving site-dependent effective coupling constants in two-dimensional waveguide arrays is developed. Implementability of the Dirac equation under the proposed scheme puts minor demands on gauge and spacetime backgrounds; however, a wide array of physical spacetimes, such as all vacuum and static solutions, prove to be implementable. As an interesting and instructive example, we discuss a tabletop realization of the gravitational Aharonov-Bohm effect: After devising a thought experiment in which signatures of the gravitational Aharonov-Bohm effect could be detected, we briefly discuss how the analogue of such a setting can in principle be implemented using the proposed waveguide setup.