Experimental realization of a non-magnetic one-way spin switch

Controlling magnetism through non-magnetic means is highly desirable for future electronic devices, as such means typically have ultra-low power requirements and can provide coherent control. In recent years, great experimental progress has been made in the field of electrical manipulation of magnetism in numerous material systems. These studies generally do not consider the directionality of the applied non-magnetic potentials and/or magnetism switching. Here, we theoretically conceive and experimentally demonstrate a non-magnetic one-way spin switch device using a spin-orbit coupled Bose–Einstein condensate subjected to a moving spin-independent repulsive dipole potential. The physical foundation of this unidirectional device is based on the breakdown of Galilean invariance in the presence of spin-orbit coupling. Such a one-way spin switch opens an avenue for designing quantum devices with unique functionalities and may facilitate further experimental investigations of other one-way spintronic and atomtronic devices.


Supplementary Note 2
In Fig. 4a of the main text, we experimentally observe a spin flip in the region of slow, right moving barrier velocities and the spin polarization has a smooth dependence on velocity below a crossover velocity around 9 mm s −1 . The non-interacting case provided in Fig. 5 of the main text shows that the spin polarization (transmission) exhibits a discontinuous jump from σ z = −1 to + 1 (T = 0 to 1) near this crossover velocity. The crossover velocities where σ z changes from negative to positive values are similar in both cases, indicating that the origin of the spin-flip process can be understood by analyzing the single-particle dynamics (i.e. the non-interacting case).
There is no known analytic solution for a single particle reflecting from a stationary Gaussian potential, like the one used in this work. However, using the potential V b defined in Methods, we find that a potential with a similar real-space profile (inset of Supplementary Fig. 4) can numerically reproduce the spin polarization results in a noninteracting system ( Supplementary Fig. 4). Scattering from this potential is analytically tractable and is characterized by a single dimensionless parameter ν = w 2 b U b /2 1 > 1/8. This leads to a transmission coefficient T , in the absence of SO coupling, given by [1] where is the single-particle energy. The resulting crossover speed v co for the transition from total reflection to total transmission is v co = −2.85v r ≈ −16.6 mm s −1 , relevant for a left sweeping barrier in our system. For a barrier moving to the right, GPE numerics show that the crossover velocity, given the experimental SO coupling parameters, is reduced to v co = 1.6v r ≈ 9.3 mm s −1 . These values are obtained using the non-interacting GPE model (g = 0). To understand where the difference between these crossover velocities for a left verse a right moving barrier originates from, we consider the co-moving frame with respect to the barrier. In this frame, the BEC moves at −v b and is scattered by a static barrier. As a first approximation, consider a single particle scattered by a given barrier in the absence of SO coupling. Depending on the dynamical energy (set by the velocity) of the particle, it will either be transmitted or reflected. For the Gaussian barrier implemented in the experiment, the dynamical energy needed for the transition between reflection and transmission is E 0 co ∼ 8.4E r (corresponding to a crossover velocity v 0 co ∼ 2.9v r ∼ 16.8 mm s −1 ). Now consider a moving barrier in the presence of SO coupling. The BEC is initially loaded into SO coupling with quasi-momentum q i and detuning δ in the lab frame. In the co-moving frame, as discussed in the main text, the quasimomentum becomes q i,cm = q i ± |v b | vr k r and δ cm = δ ± 4|v b | vr E r for a left (+) or right (-) moving barrier. The band structure in the co-moving frame is depicted in Fig. 5a and b, where in (a) the atoms are moving with positive velocity and in (b) the atoms are moving with negative velocity. The dynamics here are mainly characterized by the lowest band in the presence of strong SO coupling (Ω =1.53E r , as used in the experiment, is considered to be strong in this context). As a result, the effective dynamical energy for a left (L) or right (R) moving barrier is given by The crossover velocity for either direction is analytically obtained by solving E L,R D = E 0 co . Based on the parameters used in the experiment, we find v L co 16.6 mm s −1 and v R co 9.3 mm s −1 , consistent with GPE numerical simulations and experimental results quoted in the main text.
From this discussion, it is shown that the crossover velocities are primarily determined by the SO coupling parameters and the barrier shape. Although other systematic parameters like trapping frequencies and the atomic number could reduce the effective interatomic interaction strength, and thus improve the spin switch efficiency, these parameters have less of an effect on the crossover velocities. In this sense, a finely tuned system is not required to implement such a unidirectional spin switch.
In the theory developed so far, the reflection channel R ↑ is assumed to resided on the upper band of the SO- coupled dispersion (as depicted in Fig. 5b of the main text). This is typically only true for positive barrier velocities significantly larger than a minimum critical value. For barrier velocities lower than this critical speed, atoms are unable to overcome the momentum-space barrier due to a small Doppler shift and low energy imparted from the barrier, as shown in Supplementary Fig. 5c. The resonant coupling channel, R ↑ , therefore remains open and can exist on the lower branch of the SO-coupled dispersion. As the barrier speed is increased, the momentum-space barrier flattens and more energy is imparted on the system by the barrier, causing the R ↑ coupling channel to close, and the R ↓ coupling channel to subsequently open, resulting in a spin-flip. For the given parameters of the experiment, the lower critical velocity is found to be v b ≈ 1.1 mm s −1 (Supplementary Fig. 5d), consistent with GPE simulation results.

Supplementary Note 3
Based on our scattering analysis, a barrier with large potential height and a wide profile is desired for spin switching. This generates a large ν parameter (see Supplementary Note 2) and leads to a sharp transition between total reflection and total transmission, preventing an undesired mixture of reflection and transmission channels for a range of velocities. We note that even in the presence of interactions, the transition region from reflection to transmission is relatively narrow due to the large ν parameter. In addition, the barrier width w b may also affect how fast atoms can be driven while still following the lower band, due to different Landau-Zener tunneling rates to a higher band for a given Supplementary Figure 6: Barrier width dependence. Spin polarization with respect to the barrier width in the single particle regime for a barrier traveling at v b = +20 mm s −1 . Insets show corresponding momentum-space profiles of final states. The spin flip process at this speed is accomplished solely through the T ↓ channel.
sweeping speed. In general, when using a very wide barrier (or the "adiabatic" limit), we expect to observe a vanishing tunneling rate. Consequently, the spin-flip transmission channel (T ↓ ) should be completely suppressed for sufficiently large w b .
To confirm this intuitive picture, we perform GPE simulations in the non-interacting regime over a range of barrier widths for a fixed barrier velocity, v b = +20 mm s −1 . Results are shown in Supplementary Fig. 6 where T ↓ is completely suppressed for w b > 7 µm. In the narrow barrier limit, atoms flipped into the |↓ state (blue peaks in the insets of Supplementary Fig. 6) have a negative final momentum and T ↓ is observable. Recall that the barrier used in experiments has a Gaussian width in the relevant direction of 11 µm. This behavior is in agreement with predictions from the resonant condition.
In conclusion, a reasonably wide barrier leads to a sharp transition in the transmission coefficient T and, with the help of the avoided band crossing in the case of strong SO coupling, suppresses the T ↓ channel. As a result, a good one-way spin switch is expected under these experimental parameters.