  Several phenomenological approaches to quantum gravity predict the existence
of a minimal measurable length and/or a maximum measurable momentum near the
Planck scale. When embedded into the framework of quantum mechanics, such
constraints induce a modification of the canonical commutation relations and
thus a generalization of the Heisenberg uncertainty relations, commonly
referred to as generalized uncertainty principle (GUP). Different models of
quantum gravity imply different forms of the GUP. For instance, in the
framework of string theory the GUP is quadratic in the momentum operator, while
in the context of doubly special relativity it includes an additional linear
dependence. Among the possible physical consequences, it was recently shown
that the quadratic GUP induces a universal decoherence mechanism, provided one
assumes a foamy structure of quantum spacetime close to the Planck length.
Along this line, in the present work we investigate the gravitational
decoherence associated to the linear-quadratic GUP and we compare it with the
one associated to the quadratic GUP. We find that, despite their similarities,
the two generalizations of the Heisenberg uncertainty principle yield
decoherence times that are completely uncorrelated and significantly distinct.
Motivated by this result, we introduce a theoretical and experimental scheme
based on cavity optomechanics to measure the different time evolution of
nonlocal quantum correlations corresponding to the two aforementioned
decoherence mechanisms. We find that the deviation between the two predictions
occurs on time scales that are macroscopic and thus potentially amenable to
experimental verification. This scenario provides a possible setting to
discriminate between different forms of the GUP and therefore different models
of quantum gravity.
