Abstract
We consider a discrete time quantum walker in one dimension, where at each step, the step length ℓ is chosen from a distribution P(ℓ) ∝ ℓ^-δ -1 with ℓ ≤ ℓ_max. We evaluate the probability f(x,t) that the walker is at position x at time t and its first two moments. As expected, the disorder effectively localizes the walk even for large values of δ. Asymptotically, x² ∝ t^3/2 and x ∝ t^1/2 independent of δ and ℓ, both finite. The scaled distribution f(x,t)t^1/2 plotted versus x/t^1/2 shows a data collapse for x/t < α(δ,ℓ_max) ∼ O(1) indicating the existence of a universal scaling function. The scaling function is shown to have a crossover behaviour at δ = δ^* ≈ 4.0 beyond which the results are independent of ℓ_max. We also calculate the von Neumann entropy of entanglement which gives a larger asymptotic value compared to the quantum walk with unique step length even for large δ, with negligible dependence on the initial condition.