Spin Rotations in a Bose-Einstein Condensate Driven by Counterflow and Spin-Independent Interactions

We observe spin rotations caused by atomic collisions in a nonequilibrium Bose-condensed gas of ${}^{87}\mathrm{Rb}$. Reflection from a pseudomagnetic barrier creates counterflow in which forward- and backward-propagating matter waves have partly transverse spin directions. Even though inter-atomic interaction strengths are state independent, the indistinguishability of parallel spins leads to spin dynamics. A local magnetodynamic model, which captures the salient features of the observed spin textures, highlights an essential connection between four-wave mixing and collisional spin rotation. The observed phenomenon is commonly thought not to occur in Bose condensates; our observations and model clarify the nature of these effective-magnetic spin rotations.

Figure 1

Experimental configuration. (a) Illustration of the quasi-1D scattering configuration for the two-component BEC. (b) Bloch sphere representation of the magnetization of the BEC and the pseudomagnetic field $\overrightarrow{B}$ within the barrier. (c) Spin rotation, $S_x$ (orange), $S_y$ (green), and $S_z$ (magenta), during collision of right-going (dashed curves) and left-going (solid curves) atomic density (black), generated in reflection from a barrier that also acts as a magnetic field. Initially, at $t=t_0$, the spin of the right-going wave packet is polarized along $|+x⟩$. Later, reflection from the barrier generates a left-going wave packet whose spin, not parallel with that of the right-going wave, rotates about the right-going spin (backaction on the right-going spin is not illustrated for simplicity). After complete reflection, at $t=t_f$, a spin texture is evident across the left-going density.

Figure 2

Observed spin textures. $S_x$ (left), $S_y$ (middle), and $S_z$ (right) spatial profiles for the reflected atomic cloud for various ${\mathrm{\Omega }}_B$ (denoted by color). The atomic cloud, with mean incident energy of 1.67(5) kHz and rms energy width of 0.30(3) kHz, scattered from a barrier, located at the origin, with peak energy of 2.25(9) kHz. (a)–(c) Measured spin profiles with the bands representing 1 standard error in either direction from the average profile of the dataset. (d)–(f) Spin profiles predicted by GP simulations of the experiment. The shaded regions are bounded by simulations including 1 standard error in the measured barrier height and velocity width. The dashed line shows the predicted spin profile without atomic interactions for ${\mathrm{\Omega }}_B/2\pi =170\mathrm{Hz}$. (g)–(i) Predictions of the LMD model at ${\mathrm{\Omega }}_B/2\pi =170\mathrm{Hz}$ for the average reflected velocity of this dataset.

Figure 3

Net rotation angles of the reflected cloud. Angles of rotation, ${\theta }_y$ (top) and ${\theta }_z$ (bottom), of the reflected atomic cloud for incident wave packets well below the barrier height (red) and near the barrier height (blue). The peak energies of the barriers are 2.25(9) and 1.72(7) kHz in the two cases, while the incident energy of the cloud was 1.61(6) kHz with an rms width of 0.29(3) kHz. Markers represent the measured angles of rotation. Color-coded bands show GP simulations of the experiment, with the shaded regions bounded as in Fig. 2, while the dashed lines indicate SE simulations for the average parameters. The solid curves depict LMD calculations for the average reflected velocity in each scenario.

Figure 4

Effect of spin rotation on transmission. Transmission for the data presented in Fig. 3, color coded as before. Markers display data with statistical error bars, shaded bands illustrate GP simulations bounded as in previous plots, and the dashed lines depict SE simulations for the average experimental parameters. The solid lines indicate LMD calculations averaged over the 0.29 kHz rms energy width of the incident cloud. (Inset) Transmission versus velocity for instances with ${\mathrm{\Omega }}_B/2\pi$ of 0 (black), 1 (blue), and 2 kHz (orange). Dashed lines depict SE simulations, while solid lines represent GP simulations. In the inset, the simulated energy width is 0.18 kHz, narrower than in the experiment to demonstrate the trends clearly.