Mean-Field Dynamics of Spin-Orbit Coupled Bose-Einstein Condensates

Spin-orbit coupling (SOC), the interaction between the spin and momentum of a quantum particle, is crucial for many important condensed matter phenomena. The recent experimental realization of SOC in neutral bosonic cold atoms provides a new and ideal platform for investigating spin-orbit coupled quantum many-body physics. In this Letter, we derive a generic Gross-Pitaevskii equation as the starting point for the study of many-body dynamics in spin-orbit coupled Bose-Einstein condensates. We show that different laser setups for realizing the same SOC may lead to different mean-field dynamics. Various ground state phases (stripe, phase separation, etc.) of the condensate are found in different parameter regions. A new oscillation period induced by the SOC, similar to the Zitterbewegung oscillation, is found in the center-of-mass motion of the condensate.

Figure 1

A tripod scheme for implementing Rashba spin-orbit coupling. (a) The atom-laser coupling. (b) The lowest three energy levels composed of hyperfine ground states $|1⟩$, $|2⟩$, $|3⟩$. $|D_1⟩$ and $|D_2⟩$ are degenerate dark states. $\Omega =\sqrt{|{\Omega }_1^2|+|{\Omega }_2^2|+|{\Omega }_3^2|}$. (c)-(d) Two different laser configurations [8, 16] for the tripod scheme.

Figure 2

Condensate wave function $\Phi$ in the simple system. (a)-(b) Density $|{\Phi }_{\uparrow }{|}^2$ (a) and phase arg $({\Phi }_{\uparrow })$ (b) in the TF phase. $C_1=10$, $C_2=6$, $\gamma =10$. (c)-(d) In the stripe phase. $C_1=6$, $C_2=10$, $\gamma =10$. (e)-(f) Density $|{\Phi }_{\uparrow }{|}^2$ and $|{\Phi }_{\downarrow }{|}^2$ with $C_1=10$ $C_2=6$, $\gamma =1$. (g)-(h) Density $|{\Phi }_{\uparrow }{|}^2$ and $|{\Phi }_{\downarrow }{|}^2$ with $C_1=C_2=0$ $\gamma =10$.

Figure 3

The center-of-mass motion. Red lines [upper lines in (a), (c), lower lines in (b), (d)]: $⟨x⟩$; Blue lines [lower lines in (a), (c), upper lines in (b), (d)]: $⟨y⟩$. (a)-(b) The simple system with the corresponding condensate wave function in Figs. 2a, 2b. The shift of the center of the harmonic trap $D=0.8$. The shifts are along the $\widehat{\mathbf{x}}$ (a) and $\widehat{\mathbf{y}}$ (b) directions, respectively. (c)-(d) The same c.m. motion as (a)-(b) but for the complex system. Parameters are the same as that in (a)-(b) except that new terms $C_3$, $C_4$, $C_5$ are included in the condensate and c.m. motion calculations.

Figure 4

Plot of the c.m. oscillation period $T$ with respect to the SOC strength $\gamma$ for the simple system in the TF phase. Circles are from the numerical stimulation of the G-P equation (3). The lines are from Eq. (5).