Measurement of collective excitations in a spin-orbit-coupled Bose-Einstein condensate

We measure the collective excitation spectrum of a spin-orbit-coupled Bose-Einstein condensate using Bragg spectroscopy. The spin-orbit coupling is generated by Raman dressing of atomic hyperfine states. When the Raman detuning is reduced, mode softening at a finite momentum is revealed, which provides insight into a supersolid-like phase transition. We find that for the parameters of our system, this softening stops at a finite excitation gap and is symmetric under a sign change of the Raman detuning. Finally, using a moving barrier that is swept through the BEC, we also show the effect of the collective excitation on the fluid dynamics.

Figure 1

(a) BdG spectrum (blue solid line) and single-particle dispersion (red dotted line) of a spin-orbit-coupled BEC for a nonlinear coefficient of $g=0.186$, a Raman detuning of $0.28E_{\text{Raman}}$ ($\delta =2\pi \times 500$ Hz) and a Raman coupling strength of $2.5E_{\text{Raman}}$. (b) Schematic of mode softening with decreasing Raman detuning. The lines correspond to a Raman detuning of $1E_{\text{Raman}}$, $0.5E_{\text{Raman}}$, and 0 from top to bottom.

Figure 2

Roton-like minimum softening and energetic instability for decreasing Raman detuning $\delta$ in a miscible regime ($g_{11}=g_{22}=0.186,g_{12}=0.08$) with $\Omega =2.5E_{\text{Raman}}$. The lines correspond to the Raman detuning of $0.6E_{\text{Raman}}$, $0.4E_{\text{Raman}}$, $0.2E_{\text{Raman}}$ and 0, from top to bottom.

Figure 3

Bragg spectroscopy for BECs without spin-orbit coupling. (a) Schematic of the Bragg beam geometry. (b) Experimental image belonging to the blue data set (squared symbols) of (c), taken near the peak of the resonance. The fainter cloud on the left shows the Bragg scattered atoms. (c) Red dots: measured Bragg spectrum for a trapped BEC. Blue squares: spectrum obtained when the Bragg pulse is applied to a BEC after a 3 ms expansion time. The blue data are averages over four measurements; typical error bars are shown near the peak. The red data are from single measurements. The lines are Gaussian fits to the data.

Figure 4

(a) Bragg spectrum for a spin-orbit-coupled BEC, measured for $\hslash \Omega =3.5E_{\text{Raman}}$ and $\delta =2\pi \times 500$ Hz. Each point is an average over four measurement. (b) Schematic of the transitions corresponding to the three peaks in the spectrum.

Figure 5

Mode softening. (a) Position of Bragg peaks as a function of Raman detuning. Each point is an average over four data runs. These data were taken for $\hslash \Omega =3.5E_{\text{Raman}}$. Vertical error bars in (a) are on the order of the symbol size. The data quality in the uppermost branch is impacted by the smallness of the spin overlap between the initial and final states. (b) Zoomed-in view of the data for peak $\beta$. The lines in (a) and (b) represent the result of theoretical calculations.

Figure 6

(a) Spin composition of a spin-orbit-coupled BEC after a repulsive light sheet has been swept through it. The shaded rectangle indicates the region where the roton minimum starts to disappear. The roton does not exist to the right of this region. The data were taken for $\hslash \Omega =2.5E_{\text{Raman}}$, leading to very obvious roton-like structures for low Raman detunings as shown in Fig. 1. (b) Sweep direction of the light sheet for the two data sets. Drawing is not to scale.