Asymptotics of a quantum random walk driven by an optical cavity

We investigate a novel quantum random walk (QRW) model, of possible use in quantum algorithm implementation, that achieves a quadratically faster diffusion rate compared to its classical counterpart. We evaluate its asymptotic behaviour expressed in the form of a limit probability distribution of a double-horn shape. Questions of robustness and control of that limit distribution are addressed by introducing a quantum optical cavity in which a resonant Jaynes-Cummings type of interaction between the quantum walk coin system realized in the form of a two-level atom and a laser field is taking place. Driving the optical cavity by means of the coin-field interaction time and the initial quantum coin state, we determine two types of modification of the asymptotic behaviour of the QRW. In the first one the limit distribution is robustly reproduced up to a scaling, while in the second one the quantum features of the walk, exemplified by an enhanced diffusion rate, are washed out and Gaussian asymptotics prevail. Verification of these findings in an experimental set-up that involves two quantum optical cavities that implement the driven QRW and its quantum to classical transition is discussed