Observation of Zitterbewegung in a spin-orbit-coupled Bose-Einstein condensate

Spin-orbit-coupled ultracold atoms provide an intriguing new avenue for the study of rich spin dynamics in superfluids. In this Rapid Communication, we observe Zitterbewegung, the simultaneous velocity (thus position) and spin oscillations, of neutral atoms between two spin-orbit-coupled bands in a Bose-Einstein condensate (BEC) through sudden quantum quenches of the Hamiltonian. The observed Zitterbewegung oscillations are perfect on a short time scale but gradually damp out on a long time scale, followed by sudden and strong heating of the BEC. As an application, we also demonstrate how Zitterbewegung oscillations can be exploited to populate the upper spin-orbit band and observe a subsequent dipole motion. Our experimental results are corroborated by a theoretical and numerical analysis and showcase the great flexibility that ultracold atoms provide for investigating rich spin dynamics in superfluids.

Figure 1

(a) Experimental configuration for the creation of spin-orbit coupling in the $F=1$ manifold of a ${}^{87}$Rb BEC. (b) Typical band structure before (dashed black line) and after (solid red line) quenching the system by jumping the detuning $\delta$. The wave packets symbolically show the wave function directly after the quench. The short-time dynamics are dominated by ZB oscillations. (c) Similar to (b) but for a jump of the relative phase between the two Raman beams. The band structure is unaltered by the phase jump.

Figure 2

(a) Experimental observation of the ZB oscillation of $〈v_x〉$ for $\Omega =2.5E_r$ and $\delta$ jumped from $6.42E_r$ to $-6.42E_r$. Experimental data (open circles) are shown overlain with results of numerical simulations of the G-P equation (black line) and analytic prediction based on effective two-band Hamiltonian (dashed red line). Experimental (black dots) and numerical (solid black line) quasimomenta are shown in top part of plot. (b) $〈v_x〉$ for numerical simulation of experimental parameters with [solid line, same as (a)] and without (dashed line) interactions. (c) Experimental image taken at $t=325$ $\mu$s showing the Stern-Gerlach separation and 2$\hslash$k${}_r$ photon momentum separation of the bare states. (d) Experimental (open circles) and numerical results (solid circles) for ZB oscillation frequency vs $\delta$ after quench. The dashed line shows the band splitting calculated from the effective two-band model for $\Omega =2.5E_r$. The experimental error bars are determined from fit uncertainties, shot-to-shot variations of $k_x$, and calibration uncertainty of $\Omega$.

Figure 3

BEC loaded to $\Omega =1.8E_r$ and $\delta =1.6E_r$, followed by jump of $\delta$ to $-1.6E_r$. (a) Experimentally observed quasimomentum for the component of the BEC near $k_x=k_r$. The gray region indicates the onset of heating. (b) Corresponding experimental (open circles) and numerical results (solid line) for spin composition of the BEC. Insets in (a) and (b) show images of the BEC at $t=8$ ms and $t=9$ ms (i.e., just before and after the onset of heating). The vertical line indicates zero kinetic momentum. (c) Numerical simulations showing the real and momentum space composition for 7, 10, and 11.5 ms.

Figure 4

Loading of the upper band via two phase jumps separated by 40 $\mu$s where $\Omega =3.5E_r$ and $\delta =1.6E_r$. (a) Experimentally observed ZB oscillation (open circles) after a single phase jump and corresponding numerical simulations (solid line) for experimental conditions. (b) The time sequence. (c) and (d) Experimentally observed quasimomentum and spin polarization, respectively. Insets are experimental images taken during the evolution, where the vertical line indicates zero kinetic momentum.