Optomechanical parametric oscillation of a quantum light-fluid lattice

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.