Dirac quantum cellular automaton in one dimension: $\mathit{Zitterbewegung}$ and scattering from potential

We study the dynamical behavior of a quantum cellular automaton which reproduces the Dirac dynamics in the limit of small wave vectors and masses. We present analytical evaluations along with computer simulations, showing that the automaton exhibits typical Dirac dynamical features, such as the Zitterbewegung and, considering the scattering from potential, the so-called Klein paradox. The motivation is to show concretely how pure processing of quantum information can lead to particle mechanics as an emergent feature, an issue that has been the focus of solid-state, optical, and atomic-physics quantum simulators.

Figure 1

The Dirac automaton dispersion relation in Eq. (6) for four values of the mass: $m=0.1$, 0.2, 0.4, and 0.8.

Figure 2

Plots of $z\left(k\right)$ (left) and $\omega \left(k\right)/\pi$ (right) related to the oscillation amplitude and frequency of the position expectation value in Eq. (13). In both cases the plots are reported for different values of the mass ($m=0.1$, 0.2, 0.4, and 0.8, from top to bottom (left) and from bottom to top (right).

Figure 3

Automaton evolution of a state as in Eq. (17) showing the Zitterbewegung of the position expectation value. Top: $m=0.15$, $c_{+}=1/\sqrt{2}$, $c_{-}=i/\sqrt{2}$, $k_0=0$, and $\sigma ={40}^{-1}$. The calculated shift and oscillation frequency are, respectively, $\langle \psi |X\left(0\right)+Z_X\left(0\right)|\psi \rangle =3.2$ and $\omega \left(0\right)/\pi =0.05$, according to the simulation. Middle: $m=0.15$, $c_{+}=1/\sqrt{2}$, $c_{-}=1/\sqrt{2}$, $k_0=0$, $\sigma ={40}^{-1}$. The calculated shift and oscillation frequency are 0 and $0.13$, respectively. Bottom: $m=0.13$, $c_{+}=\sqrt{2/3}$, $c_{-}=1/\sqrt{3}$, $k_0={10}^{-2}\pi$, $\sigma ={40}^{-1}$. In this case the particle and antiparticle contribution are not balanced and the average position drift velocity is thus $\langle {\psi }_{+}\left|V\right|{\psi }_{+}\rangle +\langle {\psi }_{-}\left|V\right|{\psi }_{-}\rangle =\left(\right|c_{+}{|}^2-|c_{-}{|}^2)v\left(k_0\right)=0.08$, corresponding to an average position $x_{\psi }^{+}\left(800\right)+x_{\psi }^{-}\left(800\right)=464$ [see Eq. (16)]. Note that for $t\to \infty$ the term $2\mathrm{Re}[\langle {\psi }_{+}|Z_X\left(t\right)|{\psi }_{-}\rangle$, which is responsible for the oscillation, goes to 0.

Figure 4

Schematic of the potential.

Figure 5

Group velocity of the transmitted wave packet as a function of the potential barrier height $\varphi$ and of the momentum $k$ of the incident particle state. From top left to bottom right the transmitted group velocity is depicted for values of the mass $m=0.1$, 0.2, 0.4, and 0.8.

Figure 6

Reflection coefficient as a function of the potential barrier height $\varphi$ and of the momentum $k$ of the incident particle state. From top left to bottom right the reflection coefficient is depicted for values of the mass $m=0.1$, 0.2, 0.4, and 0.8.

Figure 7

Reflection coefficient for $m=0.4$ and momentum of the incident particle $k_0=2$ as a function of the potential barrier height $\varphi$ (section of plots in Figs. 5 and 6 for $m=0.4$, $k_0=2$).

Figure 8

Simulations of the Dirac automaton evolution with a square potential barrier. Here the automaton mass is $m=0.2$, while the barrier turns on at $x=140$. In the simulation the incident state is a smooth state of the form $|\psi \left(0\right)\rangle =\int dk\sqrt{2\pi }{g}_{k_0}\left(k\right){|+\rangle }_k$ peaked around the positive energy eigenstate ${|+\rangle }_{k_0}$ with $k_0=2$ and with $g_{k_0}$ a Gaussian having width $\sigma ={15}^{-1}$. The incident group velocity is $v\left(k_0\right)=0.90$. The simulation is run for four increasing values of the potential $\varphi$. Top left: Potential barrier height $\varphi =1.42$, reflection coefficient $R=0.25$, velocity of the transmitted particle $v\left(k_0^{\prime }\right)=0.63$. Top right: $\varphi =1.55$, $R=0.75$, $v\left(k_0^{\prime }\right)=0.1$. Bottom left: $\varphi =2$, $R=0.1$, $v\left(k_0^{\prime }\right)=0$. Bottom right: $\varphi =2.4$, $R=0.50$, $v\left(k_0^{\prime }\right)=0.33$.