Zitterbewegung for ultracold atoms in the merging of Dirac points

We investigate Zitterbewegung (ZB) of the ultracold atoms in a tunable honeycomb optical lattice. By tuning the parameters of the lattice one can realize the merging of two Dirac points. The process of two Dirac points merging into a hybrid point implies a topological phase transition from a semimetallic to a band insulator phase. After merging, it presents a linear-quadratic dispersion relation which is linear in one direction and quadric in the other orthogonal direction. We show that ZB occurs isotropically before merging but occurs only in the direction of the linear dispersion after merging. Furthermore, we obtain that ZB is exponentially damping for a negative energy gap but is a stable oscillation for a positive energy gap. The frequency and amplitude of ZB can be controlled in a detectable range, so the phenomena can be observable in the ultracold atomic experiments.

Figure 1

Numerically calculated probability distributions, ${\left|\Psi (x,y)\right|}^2$, at times $t=0$ (top), 6 (middle), 12 (bottom) with $\Delta =-2,-0.8,-0.2,0,0.2$ [from panels (a) to (e)]. The corresponding $\Delta$ are marked. The width of the initial wave packet $d=2$. Throughout $t$ is in units of $m^{*}/\left(\hslash k_y^2\right)$, the gap parameter $\Delta$ is in units of $\left(\hslash k_y^2\right)/m^{*}$, and $x$ and $y$ are in units of $\left(p_{y}t\right)/m^{*}$.

Figure 2

Analytical (lines) and numerically (symbols) calculated $\tau$ vs $\Delta$ in the process of merging. The left inset shows the average values $\overline{y}\left(t\right)$ change over time with different $\Delta$ before merging and the right one shows that after merging. The values of $\Delta$ are marked. The width of the initial wave packet $d=2$. Throughout $t$ and $\tau$ are in units of $m^{*}/\left(\hslash k_y^2\right)$, gap parameter $\Delta$ is in units of $\left(\hslash k_y^2\right)/m^{*}$, and $y\left(t\right)$ is in units of $\left(p_{y}t\right)/m^{*}$.