  In cavity optomechanics, single photon detection of Raman scattered photons
can be a useful tool for observing nonclassical features of both radiation and
motion. While this tool has been employed in experiments with pulsed laser
driving of a cavity mode, it has not been readily accessible to experiments
with continuous and constant laser driving. To address this, we present a study
of a standard optomechanical system where the cavity mode is continuously
driven at two different frequencies, and where sideband photons are detected by
single photon detectors after frequency filtering the output from the cavity
mode around its resonance frequency. We first derive the normalized second
order coherence associated with the detected photons, and show that it contains
signatures of the quantum nature of the mechanical mode which would be absent
with only single-tone driving. To identify model-independent nonclassical
features, we derive two inequalities for the sideband photon statistics that
should be valid in any classical model of the system. We show that these
inequalities are violated in the proposed setup. This is provided that the
average phonon occupation number of the mechanical mode is sufficiently small,
which in principle can be achieved through sideband cooling intrinsic to the
setup. Violation of the first inequality means that there is no well-defined
probability distribution of the Glauber-Sudarshan type for the cavity mode. In
contrast, a violation of the second inequality means that there is no joint
probability distribution for the cavity mode at two times separated by a finite
interval, which originates from the noncommutativity of the motional
quadratures of the mechanical mode. The proposed setup thus employs a
mechanical oscillator in order to generate a steady-state source of
nonclassical radiation.
