  In this paper, we study the phenomenon of quantum interference in the
presence of external gravitational fields described by alternative theories of
gravity. We analyze both non-relativistic and relativistic effects induced by
the underlying curved background on a superposed quantum system. In the
non-relativistic regime, it is possible to come across a gravitational
counterpart of the Bohm-Aharonov effect, which results in a phase shift
proportional to the derivative of the modified Newtonian potential. On the
other hand, beyond the Newtonian approximation, the relativistic nature of
gravity plays a crucial r\^ole. Indeed, the existence of a gravitational time
dilation between the two arms of the interferometer causes a loss of coherence
that is in principle observable in quantum interference patterns. We work in
the context of generalized quadratic theories of gravity to compare their
physical predictions with the analogous outcomes in general relativity. In so
doing, we show that the decoherence rate strongly depends on the gravitational
model under investigation, which means that this approach turns out to be a
promising test bench to probe and discriminate among all the extensions of
Einstein's theory in future experiments.
