Negative-Mass Hydrodynamics in a Spin-Orbit–Coupled Bose-Einstein Condensate

A negative effective mass can be realized in quantum systems by engineering the dispersion relation. A powerful method is provided by spin-orbit coupling, which is currently at the center of intense research efforts. Here we measure an expanding spin-orbit coupled Bose-Einstein condensate whose dispersion features a region of negative effective mass. We observe a range of dynamical phenomena, including the breaking of parity and of Galilean covariance, dynamical instabilities, and self-trapping. The experimental findings are reproduced by a single-band Gross-Pitaevskii simulation, demonstrating that the emerging features—shock waves, soliton trains, self-trapping, etc.—originate from a modified dispersion. Our work also sheds new light on related phenomena in optical lattices, where the underlying periodic structure often complicates their interpretation.

Figure 1

(a) Schematic representation of the 1D expansion of a SOC BEC. The asymmetry of the dispersion relation (solid curve) causes an asymmetric expansion of the condensate due to the variation of the effective mass. The dashed lines indicate the effective mass, and the shaded area indicates the region of negative effective mass. The parameters used for calculating the dispersion are $\mathrm{\Omega }=2.5E_R$ and $\delta =1.36E_R$. The color gradient in the dispersion shows the spin polarization of the state. (b) Experimental TOF images of the effectively 1D expanding SOC BEC for expansion times of 0, 10, and 14 ms.

Figure 2

Anisotropic expansion of a BEC along the direction of spin-orbit coupling ($x$ axis) with the Raman coupling strength $\mathrm{\Omega }=2.5E_R$, and various detunings of (a) $\delta =2.71E_R$, (b) $\delta =1.36E_R$, and (c) $\delta =0.54E_R$, from left to right, respectively. The first row shows the experimental integrated cross section of the condensate (dotted red curves) overlaid with results from the GPE simulation (solid black curves). The second row plots the location of the 20% and 80% quantiles with respect to the initial cloud center. The shaded regions present the GPE simulations, while the data points (and dotted lines to guide the eye) are the experimental measurements smoothed slightly by averaging the nearest 3 times. Error estimates are the standard deviation of 5 samples and are comparable with about twice the camera pixel resolution. The insets show the dispersion relation, with the region of negative effective mass lightly shaded. The bottom two rows show the evolution of the expanding condensate from the experiment (upper) and corresponding single-band axially symmetric 3D GPE simulations (lower). The dashed white lines depict the quantiles from the plot above. All experimental data presented here are from in situ imaging.

Figure 3

Zoom of a 1D simulation matching the lower middle frame of Fig. 2 showing the total density $n=n_{\uparrow }+n_{\downarrow }$ as a function of time in the region where the dynamic instability first appears. Dashed lines are the three group velocities $v_g={E_{-}}^{\prime }(k)$ at the quasimomenta $k$ where the inverse effective mass $m_{*}^{-1}={E_{-}}^{\prime \prime }(k)$ first becomes negative (steepest line), the point of maximum $m_{*}^{-1}$ (middle), and the point where the $m_{*}^{-1}$ returns to positive (least steep line). Red points demonstrate where the local quasimomentum lies in the negative-mass region. Note that, as described in the text, the pileup initially contains many density fluctuations, but sharpens as solitons and phonons “radiate” energy away from the wall.