Observing Zitterbewegung with Ultracold Atoms

We propose an optical lattice scheme which would permit the experimental observation of (ZB) with ultracold, neutral atoms. A four-level tripod variant of the setup for stimulated Raman adiabatic passage (STIRAP) has previously been proposed for generating non-Abelian gauge fields. Dirac-like Hamiltonians, which exhibit ZB, are simple examples of such non-Abelian gauge fields; we show how a variety of them can arise, and how ZB can be observed, in a tripod system. We predict that the ZB should occur at experimentally accessible frequencies and amplitudes.

Figure 1

(a) Schematic geometry of tripod laser beams yielding the Rabi frequencies of Eqs. (1, 2, 3). (b) Tripod STIRAP scheme with Rabi frequencies as defined in Eqs. (1, 2, 3).

Figure 2

Energy surfaces in momentum space for ${}^{85}\mathrm{Rb}$ with $k_0=2\pi /820{\mathrm{nm}}^{-1}$ and $\epsilon ={\epsilon }_R$. A conical intersection occurs at $(k_y,k_z)=(k_0,0)$; a circuit of this conical intersection in $\overrightarrow{k}$ space gives a Berry phase.

Figure 3

ZB for an atom with zero momentum spread (dashed), and for a momentum spread corresponding to the velocity spread of a cloud initially in a trap of frequency 112 Hz (solid). ZB oscillations for finite momentum spread damp out over time, but persist over several periods.

Figure 4

Spin separation and ZB for $\epsilon =0(0.2)1$, as indicated on the right color bar. For each value of $\epsilon$ the dashed trajectory corresponds to the atom’s mean position in spin-up, while the solid trajectory corresponds to the mean position of the atom in spin-down; open and closed circles indicate the respective ends of these trajectories. In the Abelian cases, $\epsilon =0$ and 1, the trajectories are straight lines; the trajectory for $\epsilon =0$ is the vertical line $y=0$.

Figure 5

Time-evolved probability distribution in spin-up for an initial Gaussian wave packet ($\epsilon ={\epsilon }_R$, $k_{0}d=5$, $\hslash k_0^{2}t/m=10$). As seen in Fig. 4, the wave packet separates by internal state as it jitters; the black dot indicates the mean position, which has drifted to the right. The mixing between internal states gives rise to tails on the wave packet, resulting in ZB. The ZB damps as the internal states separate.