We propose tests of the quantum nature of gravity using the gravitational
self-interaction of photons in a cavity. We demonstrate that this interaction
results in a number of quantum gravitational signatures in the quantum state of
the light that cannot be reproduced by any classical theory of gravity. We
rigorously assess these effects using quantum parameter estimation theory, and
discuss simple measurement schemes that optimally extract their signatures.
Crucially, the proposed tests are fundamentally relativistic, and probe aspects
of the gravitational field that are inaccessible to experiments with massive
particles, thus providing a new avenue for understanding the quantum nature of
gravity.
