Massive Klein--Gordon equation from a BEC-based analogue spacetime

TL;DR

This paper demonstrates how two-component Bose-Einstein condensates can simulate the massive Klein-Gordon equation in curved spacetime, advancing analogue gravity models to include particle mass and bi-metric structures.

Contribution

It introduces a method to decouple phonon modes in BECs, enabling simulation of massive particles and bi-metric spacetimes in analogue gravity experiments.

Findings

Decoupling of phonon modes is achievable under specific conditions.

Massive phonon modes exhibit relativistic dispersion relations.

Both phonon modes can be tuned to travel at the same speed, satisfying the Einstein equivalence principle.

Abstract

We extend the "analogue spacetime" programme by investigating a condensed-matter system that is in principle capable of simulating the massive Klein--Gordon equation in curved spacetime. Since many elementary particles have mass, this is an essential step in building realistic analogue models, and a first step towards simulating quantum gravity phenomenology. Specifically, we consider the class of two-component BECs subject to laser-induced transitions between the components. This system exhibits a complicated spectrum of normal mode excitations, which can be viewed as two interacting phonon modes that exhibit the phenomenon of "refringence". We study the conditions required to make these two phonon modes decouple. Once decoupled, the two distinct phonons generically couple to distinct effective spacetimes, representing a bi-metric model, with one of the modes acquiring a mass. In the eikonal limit the massive mode exhibits the dispersion relation of a massive relativistic particle: omega = sqrt[omega_0^2 + c^2 k^2], plus curved-space modifications. Furthermore, it is possible to tune the system so that both modes can be arranged to travel at the same speed, in which case the two phonon excitations couple to the same effective metric. From the analogue spacetime perspective this situation corresponds to the Einstein equivalence principle being satisfied.