Controlling and reversing the transition from classical diffusive to quantum ballistic transport in a quantum walk by driving the coin

We show that the standard quantum-walk quantum-to-classical transition, characterized by ballistic-to-diffusive spreading of the walker's position, can be controlled by externally modulating the coin state. We illustrate this by showing an oscillation between classical diffusive and quantum ballistic spreading using numerical and asymptotically exact closed-form solutions, and we prove that the walker is in a controllable incoherent mixture of classical and quantum walks with a reversible quantum-to-classical transition.

Figure 1

(a) Numerically evaluated variance of the walker's position over time $t$ with initial state $\left|0\right.〉(\left|+\right.〉+i\left|-\right.〉)/\sqrt{2}$: unitary evolution (blue solid line), driving with $\kappa \left(t\right)=\cos t/10$ (red dotted line), and random walk (black solid line). Position distribution $P$ at (b) $t=22$, (c) $t=32$, and (d) $t=48$.

Figure 2

Numerically evaluated variance of the walkers position over time $t$ with the same initial state as in Fig. 1, but driving with (a) $\kappa \left(t\right)=\cos t/2$ and (b) $\kappa \left(t\right)=\cos 3t/10$.

Figure 3

(a) The sawtooth driving function $\kappa \left(t\right)$ and (b) the corresponding numerically evaluated variance of the walker's position as a function of time $t$.

Figure 4

Numerically evaluated entropy $H$ of the walker over time $t$ with initial state $\left|0\right.〉(\left|+\right.〉+i\left|-\right.〉)/\sqrt{2}$: unitary evolution (blue dashed line), driving with $\kappa \left(t\right)=\cos (t/10)$ (red dotted line), and random walk (black solid line).

Figure 5

Walker's position distributions $P_{\mathrm{w}}(x,t)$ for even values of $x$ and (a) $t=90$ and (b) $t=94$ with initial state $\left|0\right.〉(\left|+\right.〉+i\left|-\right.〉)/\sqrt{2}$ and driven-coin function $\kappa \left(t\right)=\cos (t/10)$ for precise numerical simulations (dots) and asymptotic expression (solid line).

Figure 6

(a) The MID and (b) QD for a QW on a line after $t=100$ steps for undriven (blue line), driven with $\kappa \left(t\right)=\cos (t/10)$ (red line), and random-walk (black line) cases with initial state $\left|0\right.〉(\left|+\right.〉+i\left|-\right.〉)/\sqrt{2}$.