Quench dynamics of a Bose-Einstein condensate under synthetic spin-orbit coupling

We study the quench dynamics of a Bose-Einstein condensate under a Raman-assisted synthetic spin-orbit coupling. To model the dynamical process, we adopt a self-consistent Bogoliubov approach, which is equivalent to applying the time-dependent Bogoliubov–de Gennes equations. We investigate the dynamics of the condensate fraction as well as the momentum distribution of the Bose gas following a sudden change of system parameters. Typically, the system evolves into a steady state in the long-time limit, which features an oscillating momentum distribution and a stationary condensate fraction. We investigate how different quench parameters such as the inter- and intraspecies interactions and the spin-orbit-coupling parameters affect the condensate fraction in the steady state. Furthermore, we find that the time average of the oscillatory momentum distribution in the long-time limit can be described by a generalized Gibbs ensemble with two branches of momentum-dependent Gibbs temperatures. Our study is relevant to the experimental investigation of dynamical processes in a spin-orbit-coupled Bose-Einstein condensate.

Figure 1

(a) Time evolution of the condensate fraction $n_0/n$ as a function of time $t$, following a sudden change of interaction from a common initial scattering length of $k_r{a}_i=0.01$ to different final scattering lengths $a_f$. (b) Comparison of the steady-state condensate fraction with that of equilibrium states. The solid black line is the long-time condensate fraction $n_0/n$ as a function of the final interaction strength $k_r{a}_f$, with initial interaction strength $k_r{a}_i=0.01$. The dashed red line is the condensate fraction in the equilibrium state under the postquench parameters. Here, the time of evolution $t{E}_r=0.6$. In (a) and (b), $\mathrm{\Omega }/E_r=6,\delta =0$.

Figure 2

Time evolution of the condensate fraction after a sudden change of the interspecies interaction. The scattering length $a_i$ associated with the intraspecies interaction $\left(g_1=g_2=g=4\pi {\hslash }^2{a}_i/m\right)$ is fixed at $k_r{a}_i=0.01$, while the scattering length associated with the interspecies interaction $g_{12}$ is changed from $k_r{a}_i=0.01$ to $k_r{a}_f=0.6$. For the solid blue line, $\mathrm{\Omega }/E_r=6$; for the dashed red line, $\mathrm{\Omega }/E_r=2$. Here, $\delta =0$.

Figure 3

Time evolution of the condensate depletion $n_0/n$ following an SOC parameter switch from ${\mathrm{\Omega }}_i$ to ${\mathrm{\Omega }}_f$. Parameters for different quenches are as follows: ${\mathrm{\Omega }}_i/E_r=2,{\mathrm{\Omega }}_f/E_r=6$ (dashed red line); ${\mathrm{\Omega }}_i/E_r=6,{\mathrm{\Omega }}_f/E_r=2$ (solid blue line); ${\mathrm{\Omega }}_i/E_r=2,{\mathrm{\Omega }}_f/E_r=3$ (dash-dotted black line). Here, $g_1=g_2=g_{12}=4\pi {\hslash }^2{a}_s/m$, where the interaction strength is fixed at $k_r{a}_s=0.16$.

Figure 4

(a) Momentum distribution of the Bose gas at different times after a change of interactions from an initial scattering length of $k_r{a}_i=0.01$ to $k_r{a}_f=0.2$. The thick solid red line represents the momentum distribution given by the Gibbs ensemble. (b) Two branches of the momentum-dependent Gibbs temperature. Here, $\mathrm{\Omega }/E_r=6,\delta =0$.

Figure 5

(a) Momentum distribution of the Bose gas at different times after a change of Rabi frequency from ${\mathrm{\Omega }}_i/E_r=2$ to ${\mathrm{\Omega }}_f/E_r=6$. The thick solid red line is the momentum distribution given by the Gibbs ensemble. (b) Two branches of the momentum-dependent Gibbs temperature. Here, we have assumed SU(2)-invariant interactions with $k_r{a}_s=0.16$ and $\delta =0$.