  We investigate a novel quantum random walk (QRW) model, possibly useful in
quantum algorithm implementation, that achieves a quadratically faster
diffusion rate compared to its classical counterpart.
  We evaluate its asymptotic behavior 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 behavior 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 enhanced diffusion
rate, are washed out and Gaussian asymptotics prevail. Verification of these
findings in an experimental setup that involves two quantum optical cavities
that implement the driven QRW and its quantum to classical transition is
discussed.
