Nonlinear dynamics of a spin-orbit-coupled Bose-Einstein condensate

Single-particle dynamics of a spin-orbit-coupled Bose-Einstein condensate has recently been investigated in experiments that explore the physics of Landau-Zener tunneling and of the Zitterbewegung. In this paper, we study the influence of a nonlinearity due to interactions on these dynamics and show that the dispersion relation develops interesting loop structures. The atoms can move along the nonlinear dispersion curve in the presence of a weak acceleration force, and we show that the loops lead to straightforward nonlinear Landau-Zener tunneling. However, we find that for the Zitterbewegung, induced by a sudden quench in the spin-orbit-coupling parameters, the nonlinear dispersion is irrelevant.

Figure 1

Nonlinear dispersion relation $\mu \left(k\right)$ for $g_{11}=g_{22}=g,\mathrm{\Omega }=0.3$, and $\gamma =1$. (a) Comparison of the nonlinear dispersion with equal nonlinear coefficients $g_{12}=g=1.5$ (blue lines) with the linear dispersion $g_{12}=g=0$ (green lines). (b) Cusps can be seen to develop in the upper branch (blue lines) for $g=1.5,g_{12}=1.2$ and in the lower branch (red lines) for $g=1.2,g_{12}=1.5$. (c) Loops appear in the upper branch, and their sizes change with different nonlinear coefficients. The red lines are for $g=1.5$ and $g_{12}=0.8$, and the blue lines are for $g=1.5$ and $g_{12}=0$. (d) Loops exist in the lower branch, the red lines are for $g=0.8$ and $g_{12}=1.5$, and the blue lines are for $g=0$ and $g_{12}=1.5$. In (a)–(d), $\delta =0$. (e) Loop structure with a finite detuning $\delta =-1$ for $g=0$ and $g_{12}=1.5$. The vertical dashed lines in (c)–(e) label the locations of the edges and the center of the loop structures.

Figure 2

Nonlinear Landau-Zener dynamics in the presence of the loop structures. The acceleration force is given by $F=0.0001$, and the other parameters are $\mathrm{\Omega }=0.3,\gamma =1$, and $\delta =0$. The right panels show the nonlinear dispersion relations, and the left panels display the dynamics for different initial states. (a) Loop in the lower band. The initial state corresponding to the thick gray lines in the left plot is prepared in the upper band (square in the right plot), and the initial state for the thin lines is in the lower band (the lower circle in the right plot). $g_{11}=g_{22}=g=0$ and $g_{12}=1.5$. (b) Loop in the upper band. The initial state for the thick lines in the left plot corresponds to the circle in the right plot, and that for the thin lines corresponds the square in the right plot. $g_{11}=g_{22}=g=1.5$ and $g_{12}=0.8$.

Figure 3

Nonlinear dynamics induced by the quenching of the Rabi frequency from $-\mathrm{\Omega }$ to $\mathrm{\Omega }$ for different values of the nonlinear coefficients and $\gamma =0.6$ and $\delta =0.2$. In the right-hand side panels, the quench dynamics is shown by the evolution of the population difference $|{\mathrm{\Phi }}_1{|}^2-{\left|{\mathrm{\Phi }}_2\right|}^2$. In the left-hand side panels, the nonlinear dispersion $\mu \left(k\right)$ (thin back lines) and the collective excitations $\omega \left(q\right)$ (thick green lines) are displayed together. The open circles are eigenvalues for a fixed quasimomentum, and the solid circles are the excitations for fixed quasimomenta of excitations. (a) Nonlinear dynamics in the presence of a loop structure in the upper band. $\mathrm{\Omega }=0.4,g_{11}=g_{22}=g=1.5$, and $g_{12}=0.2$. The values of the open circles are $\mu =(1.62,1.38,1.06,0.65)$ from top to bottom. The value of the solid circle is $\mu =1.47$. The oscillation period is $T=11.67$. (b) Nonlinear dynamics in the absence of loop structure. $\mathrm{\Omega }=1,g=1.5$, and $g_{12}=0.8$. The open circles represent $\mu =(0.65,1.75)$, and the location of the solid circle is at $\mu =1.95$. The oscillation period is $T=7.26$. (c) Effective linear dynamics with the parameters $\mathrm{\Omega }=1$ and $g=g_{12}=1.5$. The open and solid circles overlap $\mu =(2.11,0.97)$ from top to bottom. The oscillation period is $T=5.52$, which exactly matches with the energy difference between the top and the bottom circles.