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 $σ$ has interesting behaviors for different period $q$ and parameter $θ$. We studied the behavior of standard deviation with variation in walk steps, period, and $θ$. The standard deviation increases approximately linearly with $θ$ and decreases with $1/q$ for $θ\in(0,π/4)$, and increases approximately linearly with $1/q$ for $θ\in[π/4,π/2)$. When $q=2$, the standard deviation is lazy for $θ\in[π/4+nπ,3π/4+nπ],n\in Z$.

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