In the past 20 years, the field of Quantum Gravity Phenomenology has blossomed (see [A. Camelia, Living Rev.Rel. 16 (2013) 5] for a review). Its main aim is to connect quantum gravity ideas with experiments and observations in the hope to shed some light on the fundamental structure of spacetime, or at least give theoreticians some pointers.

Presently, all various approaches to Quantum Gravity are far from being complete and several questions remain to be solved. Nevertheless, they have inspired well-defined phenomenological models aimed to probe specific features that the fundamental theory might have. Among the most notable examples of phenomenological avenues being explored we can list

- Violations of Lorentz Invariance (and of other exact symmetries)
- Deformation of Special Relativity
- Space-time foam
- Deformed field dynamics
- Generalized Uncertainty principle
- Quantum Cosmology

### Optomechanical Tests of Non-Local theories

Non-local field theories, i.e., field theories in which the kinetic terms has infinitely many space-time derivatives, are ubiquitous in Quantum Gravity. An example is given by Causal set theory (see the Causal set section).

In [Belenchia et al., Phys. Rev. Lett. 116, 161303 (2016) & Phys. Rev. D 95, 026012 (2017)], we obtained a non-local Schrödinger equation by taking the non-relativistic limit of a generic non-local Klein-Gordon operator. We then introduced a harmonic potential and studied the evolution of a harmonic oscillator with this modified dynamics.

From the results for the variances of position and momentum, we see that the non-locality introduces a spontaneous squeezing effect which could be a possible smoking gun for near future optomechanical experiments, allowing us to cast constraints on the non-locality scale orders of magnitude greater than the ones obtainable with high-energy experiments like LHC.

### Relativistic Quantum Information and Non-Locality

While optomechanics offers an unprecedented test-bed for a test of non-locality, other low-energy quantum systems are of interest for investigating the features of non-local field theories. A prominent example is Unruh-DeWitt detectors, largely employed in the field of Relativistic Quantum Information (RQI). An Unruh-DeWitt detector is a two-level system which couples to a quantum field, and can have a multitude of physical implementations (like a qubit can).

In [Belenchia *et al.*, Phys. Rev. D 94, 061902(R) (2016)] we studied the response

of an inertial Unruh-DeWitt detector coupled to a non-local scalar field (with the non-analytic kinetic term characteristic of Causal set theory). The response of the detector turns out to be deformed with respect to the local case and the deformation is only polynomially suppressed by the non-locality scale. This opens for the possibility that such deformation could be strongly bounded in apt experiments once the model is extended to physical fields (like the electromagnetic one).

RQI is a relatively recent research field. Born in the attempt to generalize quantum information protocols and concepts in the case in which relativistic effects cannot be ignored, it has grown in a multidisciplinary field attracting experts from both the theoretical and experimental communities.

In [Belenchia *et al., *Phys. Rev. D 96, 116006 (2017)] we investigated the transmission of information capabilities of a non-local field. In contrast to a local massless scalar field in 4 dimensions, a non-local scalar field (with non-analytic kinetic operator) does posses Green functions with support not limited to the light-cone. This, in turn, implies that information can be transmitted through the field between two agents even when they are timelike, an impossible feat with a local massless field.