Quantum Random Walk in Periodic Potential on a Line

  • We investigate 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 study 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∈(0,π/4), and increases approximately linearly with 1/q for θπ/4,π/2). For θ∈(π/4, 3π/4), the transmission is larger than the reflection, with sensibility the diffussion will be accelerated. However, when q=2, the standard deviations are nearly the same, and when q>2, the standard deviation will decrease.
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