Quantum random walk in periodic potential on a line

We investigated the discrete-time quantum random walks on a line in periodic potential. The probability distribution with periodic potential is more complex compared to the normal quantum walks, and the standard deviation $\sigma$ has interesting behaviors for different period $q$ and parameter $\theta$. We studied the behavior of standard deviation with variation in walk steps, period, and $\theta$. The standard deviation increases approximately linearly with $\theta$ and decreases with $1\mathrm{/}q$ for $\theta \in (0,\pi \mathrm{/}4)$, and increases approximately linearly with $1\mathrm{/}q$ for $\theta \in [\pi \mathrm{/}4,\pi \mathrm{/}2)$. When $q=2$, the standard deviation is lazy for $\theta \in [\pi \mathrm{/}4+n\pi ,3\pi \mathrm{/}4+n\pi ],n\in Z$.