Collective Dipole Oscillations of a Spin-Orbit Coupled Bose-Einstein Condensate

In this Letter, we present an experimental study of the collective dipole oscillation of a spin-orbit coupled Bose-Einstein condensate in a harmonic trap. The dynamics of the center-of-mass dipole oscillation is studied in a broad parameter region as a function of spin-orbit coupling parameters as well as the oscillation amplitude. The anharmonic properties beyond the effective-mass approximation are revealed, such as the amplitude-dependent frequency and finite oscillation frequency at a place with a divergent effective mass. These anharmonic behaviors agree quantitatively with variational wave-function calculations. Moreover, we experimentally demonstrate a unique feature of the spin-orbit coupled system predicted by a sum-rule approach, stating that spin polarization susceptibility—a static physical quantity—can be measured via the dynamics of dipole oscillation. The divergence of polarization susceptibility is observed at the quantum phase transition that separates the magnetic nonzero-momentum condensate from the nonmagnetic zero-momentum phase. The good agreement between the experimental and theoretical results provides a benchmark for recently developed theoretical approaches.

Figure 1

(a) Experimental layout. The field ${\mathbf{B}}_{\mathrm{bias}}$ is along the $z$ direction and the Raman lasers propagate in the $x\mathrm{\text{−}}y$ plane. (b) Raman coupling scheme within the $F=1$ manifold. (c) Single-particle phase diagram in the $\Omega \mathrm{\text{−}}\delta$ plane. The dispersion spectrum $\mathcal{E}(k_x)$ has two (one) local minima in the blue (grey) regime. The experiments are along the paths $P_1$, $P_2$, and $P_3$, with $\mathcal{E}(k_x)$ in the insets. (d) Experimental time sequence. (e–f) Dipole oscillation with $\Omega =3.3E_r$ and $\delta =1E_r$ along the $\widehat{x}(e)$ and $\widehat{z}$ (f) directions. For comparison, the oscillations without SO coupling are displayed as grey dashed curves.

Figure 2

(a) Oscillation period $T_x$ along path $P_1$ ($\delta =1E_r$) versus $\Omega$. The black triangles are without SO coupling, and the blue line denotes the estimated trap frequency. The purple squares are with SO coupling for relatively small (below $0.2k_r$) oscillation amplitude, while the green circles are for large (about $0.6k_r$) amplitude. The inset is for $T_z$ versus $\Omega$. (b) $T_x$ versus oscillation amplitude with $(\Omega =4.8,\delta =1.0)E_r$. The black squares with error bars are experimental data. (c) and (d) $T_x$ versus $\Omega$ along path $P_2$ and path $P_3$. In all plots, the red lines are the theory curves obtained by solving Eq. (2).

Figure 3

Magnetization oscillation for $\delta =1E_r$ and $\Omega =4.8E_r$. (a) Spin-resolved TOF images for various holding times $t_h$. (b) Quasimomentum $k_x$ versus $t_h$. (c) Polarization $\mathcal{M}$ versus $t_h$. The red circles are directly measured, while the black squares are deduced from (b) (see text for details).

Figure 4

Amplitude ratio of spin and momentum oscillation $A_{\sigma }/(A_k/k_r)$ versus $\Omega$, for magnetic phase (a) and nonmagnetic phase (b). The inset is for the spin polarization susceptibility $E_r\chi$ deduced from this ratio via Eq. (4). The red solid line is from the solution of Eq. (3).