Deterministic synthesis of mechanical NOON states in ultrastrong optomechanics

We propose a protocol for the deterministic preparation of entangled NOON mechanical states. The system is constituted by two identical, optically coupled optomechanical systems. The protocol consists of two steps. In the first, one of the two optical resonators is excited by a resonant external π-like Gaussian optical pulse. When the optical excitation coherently partly transfers to the second cavity, the second step starts. It consists of sending simultaneously two additional π-like Gaussian optical pulses, one at each optical resonator, with specific frequencies. In the optomechanical ultrastrong coupling regime, when the coupling strength becomes a significant fraction of the mechanical frequency, we show that NOON mechanical states with quite high Fock states can be deterministically obtained. The operating range of this protocol is carefully analyzed. Calculations have been carried out taking into account the presence of decoherence, thermal noise, and imperfect cooling.