Relativistic quantum walks

By pursuing the deep relation between the one-dimensional Dirac equation and quantum walks, the physical role of quantum interference in the latter is explained. It is shown that the time evolution of the probability density of a quantum walker, initially localized on a lattice, is directly analogous to relativistic wave-packet spreading. Analytic wave-packet solutions reveal a striking connection between the discrete and continuous-time quantum walks.

Figure 1

(a) Nonrelativistic $(a=5)$ and (b) relativistic $(a=0.5)$ solutions of the one-dimensional Dirac equation. The probability density $\rho (x,t)={\Psi }^{†}(x,t)\Psi (x,t)$ is shown at time $t=0$ (dotted lines, arbitrary units) and at $t=50$ (solid lines). Other parameters are the mass $m=1$ and the speed of light $c=1$.

Figure 2

(a) Nonrelativistic $(\alpha =22)$ and (b) relativistic $(\alpha =2.2)$ solutions of the one-dimensional discrete-time quantum walk (DTQW). The probability density $\rho (n,\tau )={\psi }^{†}(n,\tau )\psi (n,\tau )$ is shown at time $\tau =0$ (dotted lines, arbitrary units) and at $\tau =225$ (dots), along with the Bessel-function approximation (15) (solid lines). Other parameters are the rotation angle $\theta =3\pi ∕7$, the mass $m=\tan \theta \simeq 4.38$, and the speed of light $c=\cos \theta \simeq 0.22$. To compare with the Dirac solution, the parameters were chosen such that $c\tau \simeq 50$ and $\alpha ∕m=\alpha ∕\tan \theta \simeq a$.

Figure 3

The entanglement, in ebits, of the Dirac solution (solid), and the discrete-time quantum walk with $\theta =3\pi ∕7$ (dashed), as a function of the scaling parameter $a=\alpha ∕\tan \theta$. The entanglement measure is the spinor entropy [18], using a base-2 logarithm.

Figure 4

(a) Nonrelativistic $(\alpha =22)$ and (b) relativistic $(\alpha =2.2)$ solutions of the one-dimensional continuous-time quantum walk (CTQW). The probability density $\rho (n,t)={\psi }^{†}(n,t)\psi (n,t)$ is shown at time $t=0$ (dotted lines, arbitrary units) and at $t=225$ (solid lines). The remaining parameter is $c=2\gamma =\cos (3\pi ∕7)\simeq 0.22$.