  Two-photon coherent states are one of the main building pillars of non-linear
and quantum optics. It is the basis for the generation of minimum-uncertainty
quantum states and entangled photon pairs, applications not obtainable from
standard coherent states or one-photon lasers. Here we describe a
fully-resonant optomechanical parametric amplifier involving a polariton
condensate in a trap lattice quadratically coupled to mechanical modes. The
quadratic coupling derives from non-resonant virtual transitions to extended
discrete excited states induced by the optomechanical coupling. Non-resonant
continuous wave (cw) laser excitation leads to striking experimental
consequences, including the emergence of optomechanically induced inter-site
parametric oscillations and inter-site tunneling of polaritons at discrete
inter-trap detunings corresponding to sums of energies of the two involved
mechanical oscillations (20 and 60 GHz confined vibrations). We show that the
coherent mechanical oscillations correspond to parametric resonances with
threshold condition different to that of standard linear optomechanical
self-oscillation. The associated Arnold tongues display a complex scenario of
states within the instability region. The observed new phenomena can have
applications for the generation of entangled phonon pairs, squeezed mechanical
states relevant in sensing and quantum computation, and for the bidirectional
frequency conversion of signals in a technologically relevant range.
