Spin current generation and relaxation in a quenched spin-orbit-coupled Bose-Einstein condensate

Understanding the effects of spin-orbit coupling (SOC) and many-body interactions on spin transport is important in condensed matter physics and spintronics. This topic has been intensively studied for spin carriers such as electrons but barely explored for charge-neutral bosonic quasiparticles (including their condensates), which hold promises for coherent spin transport over macroscopic distances. Here, we explore the effects of synthetic SOC (induced by optical Raman coupling) and atomic interactions on the spin transport in an atomic Bose-Einstein condensate (BEC), where the spin-dipole mode (SDM, actuated by quenching the Raman coupling) of two interacting spin components constitutes an alternating spin current. We experimentally observe that SOC significantly enhances the SDM damping while reducing the thermalization (the reduction of the condensate fraction). We also observe generation of BEC collective excitations such as shape oscillations. Our theory reveals that the SOC-modified interference, immiscibility, and interaction between the spin components can play crucial roles in spin transport.

Combined TOF images vs t hold for two dressed spin components of a SO coupled BEC with equal populations in |↑ and |↓ (ΩF = 1 Er, δR = δ (ΩF , ε)) undergoing in-phase dipole oscillations, showing very little damping (1/Q < 0.05) with negligible thermalization. Each slice in the image shown is a TOF image at a given t hold , but compressed along the horizontal direction. The time step between successive image slices is 0.5 ms. The figure shows |↓ in red and |↑ in blue plotted in the lower and upper panels, respectively. To further verify the excitation of the m = 0 quadrupole mode in the dressed case, we intentionally changed the trap frequencies to ω z ∼ 2π × (21 ± 3) Hz and ω x ∼ ω y ∼ 2π × (144 ± 10) Hz, and measured the aspect ratio of the condensate as a function of t hold at Ω F = 1.3 E r ( Supplementary Fig. 3a, with select TOF images shown in Supplementary Fig. 3b) with all the other experimental parameters similar to Fig. 5f in the main text. The data after the dashed line (t hold ∼ 2τ damp ) is fitted to a damped sinusoidal function. The extracted aspect ratio oscillation frequency is around 34 Hz, again consistent with the prediction f m = √ 2.5ω z /(2π) ∼ 33 Hz for the m = 0 quadrupole mode. This confirms the excitation of the m = 0 quadrupole mode after the SDM is damped out in the dressed case. Er with trap frequencies ωz ∼ 2π × (21 ± 3) Hz and ωx ∼ ωy ∼ 2π × (144 ± 10) Hz used in this measurement, the observed aspect ratio oscillation frequency is around 34 Hz, consistent with the expected m = 0 quadrupole mode frequency fm=0 = √ 2.5ωz/(2π) ∼ 33 Hz. This further verifies the excitation of the m = 0 quadrupole mode. The oscillation frequency is obtained using a damped sinusoidal function to fit the data following the SDM is damped out (when t hold ∼ 2τ damp as indicated by the dashed line). The representative error bars are standard deviation of at least three measurements. (b) Select TOF images are typically the average of a few repetitive measurements.

Supplementary Note 3: Control Simulations, Phase of BEC Wavefunctions in SDM, and Movies
Effect of immiscibility on SDM. We have used GPE simulations for the bare case with intentionally modified interactions to study the effect of immisciblity on the SDM. Supplementary Fig. 4 shows the damping of the relative momentum k spin of the SDM for 5 cases without and with modified interactions. Case 1 is the original bare case without modification of interactions, with the intraspecies and interspecies interaction parameters g ii and g ij (i, j =↑, ↓ and i = j) given by Eqs. (18,19) in the main text, respectively. Case 2 corresponds to the same intraspecies interaction parameterg ii = g ii and a modified interspecies interaction parameterg ij = 1.5g ij . Case 3 corresponds tõ g ii = 1.5g ii andg ij = 1.5g ij . Case 4 corresponds tog ii = 1.5g ii andg ij = g ij . Case 5 corresponds tog ii = 1.8g ii and g ij = g ij . Such modification of interactions is done by immediately increasing the interaction g-parameters to the desired values as soon as Ω is changed from Ω I to Ω F . Among all the cases, only case 2 is immiscible and we observe that case 2 possesses the strongest damping, thus suggesting that immiscibility is particularly effective to enhance the damping of the SDM. This is further supported by the observation that case 4 and case 5 have similar damping which is less than the original bare case (case 1), presumably because these two cases are more miscible than case 1. We have also calculated and listed the immiscibility metric η (defined in Eq. (13) in the main text) in Supplementary  Fig. 4 for the various cases. Note that simply increasing all the interaction g-parameters without notably changing η can also enhance the SDM damping, as suggested by the observation that the damping in case 3 (η = −0.0045, miscible) is stronger than that in case 1 (η = −0.0045, miscible) but is not as prominent as in case 2 (η = 1.2341, immiscible).  For each case, the corresponding immisciblity metric η = (g ↑↓ 2 −g ↑↑g ↓↓ )/g ↑↑ 2 is calculated. The corresponding simulated SDM for each case is shown in Supplementary Fig. 4 above.
Effect of interference on the relative motion between two colliding BECs. To investigate the effect of interference on the relative motion between two colliding BECs, we have performed another set of control GPE simulations, in which two (bare) BECs are initially in a double well trap such that they are separated in real space by a potential barrier. Then, we change the double well trap to a single harmonic potential by suddenly removing the potential barrier at t hold = 0, allowing the two BECs to collide and oscillate against each other in the y direction. We conduct the following simulations: case 1, the two BECs initially in the double well are in the same spin state (called the single spin case), with only one interaction parameter g = 4π 2 m 100a 0 . Case 2, two BECs initially in the double well have orthogonal spin states (↓ and ↑) with g ↑↑ = g ↓↓ = g ↑↓ = g (called the two spin case; here all the interaction g-parameters are set to be the same to focus on the effect of interference. The cases where the interaction g-parameters are varied differently and the effect of immiscibility are also studied separately). The damping of the relative motion in case 1 is characterized by the t hold -dependent effective width (W eff,y , shown in Supplementary Fig. 5a) of the two BECs oscillating against each other in the y direction, where W eff,y = y 2 ( y 2 is the expectation value of y 2 and is calculated using the whole wavefunction of the two BECs). In this case, we find that the relative motion almost damps out after t hold = 30 ms (when we can no longer observe any relative motion between two BECs, which have merged into one BEC; the relatively undamped remnant oscillations in the data after ∼ 30 ms reflect the breathing of width of this merged BEC. See Supplementary Movie 1). On the other hand, in case 2 we observe prominent damping only after t hold = 60 ms ( Supplementary Fig. 5b). By comparing case 1 with case 2, we avoid the effect of immiscibility and investigate the effect purely due to the interference on damping. This suggests that the interference between the two colliding BECs can enhance the damping of the relative motion. In addition, in the two spin case we have modified the interaction parameters similar to the cases in Supplementary  Fig. 4. These results also suggest that immiscibility is particularly effective to enhance the damping of SDM.
Effect of turning off interactions on the relative motion between two colliding BECs. To further investigate the role of interactions on the relative motion between two colliding BECs, we have performed three control GPE simulations where all the interaction parameters are set to zero (i.e. g = g ↑↑ = g ↓↓ = g ↑↓ = 0): (1) the bare case SDM.
(2) the single spin case and the two spin case with two BECs initially in a double well as described in the previous section. (3) the dressed case SDM at Ω F = 1.3 E r . The results of these cases are shown respectively in Supplementary Figs. 6, 7, and 8. In all these non-interacting cases, we find that the relative motion between the two colliding BECs has no noticeable damping within the time of simulation. This suggests that interactions are essential for the damping mechanisms studied in this work. Spatial modulation in the phase of BEC wavefunctions in SDM. Supplementary Fig. 9 is an example showing the spatial modulation in the phase of BEC wavefunctions (Eq. (16) in the main text) at t hold = 7.2 ms during SDM for the bare case and the dressed case at Ω F = 1.3 E r (snapshots taken from Supplementary Movie 3 and Supplementary Movie 6 below). We notice much less spatial variation in the gradient of the phase in the bare case than in the dressed case at Ω F = 1.3 E r , suggesting that LC KE in the bare case is generally smaller than that in the dressed case at this time ( Here, the momentum-space (in the k x -k y plane) 2D density distributions (obtained by the integration over k z , where k x(y,z) is the mechanical momentum in the x(y, z) direction) of different bare spin components (separated vertically from each other for better visualization) are the Fourier transform of the real-space 2D densities (as those shown in Fig. 5 in the main text). The momentum-space and real-space 1D atomic densities in the y direction (SOC direction) are obtained by integrating the momentum-space and real-space 2D densities over k x and x, respectively. In addition, the snapshot shown in Supplementary Fig. 9 for comparing the phase in the cases of Ω F = 0 and Ω F = 1.3 E r is taken from Supplementary Movie 3 and Supplementary Movie 6.