The control of quantum systems requires the ability to change and read-out
the phase of a system. The non-commutativity of canonical conjugate operators
can induce phases on quantum systems, which can be employed for implementing
phase gates and for precision measurements. Here we study the phase acquired by
a radiation field after its radiation pressure interaction with a mechanical
oscillator, and compare the classical and quantum contributions. The classical
description can reproduce the nonlinearity induced by the mechanical oscillator
and the loss of correlations between mechanics and optical field at certain
interaction times. Such features alone are therefore insufficient for probing
the quantum nature of the interaction. Our results thus isolate genuine quantum
contributions of the optomechanical interaction that could be probed in current
experiments.