Quantum walks on a circle with optomechanical systems

We propose an implementation of a quantum walk on a circle on an optomechanical system which is composed of an optical cavity coupled to a mechanical resonator. The position of a coherent state in the phase space for the radiation mode is used to encode the position of the walker, while the mechanical mode plays the role of a quantum coin. We investigate the situation for a two-sided coin by considering low excitation of the mechanical resonator, where it can be appropriately described by a two-level system. The shift operator is implemented via the coupling to the mechanical resonator. The dynamics of the system is obtained by applying Suzuki-Trotter decomposition. By adjusting the mechanical resonator frequency, its evolution is driven by a Hadamard-like operator, necessary in the evolution.
We numerically show that the system displays a typical behavior of quantum walks, namely, the probability distribution evolves ballistically and the standard deviation of the phase distribution is linearly proportional to the number of steps. The transition from quantum to classical walk due to noise effects on the coin is investigated by introducing decoherence via the phase damping channel on the coin space.