Dynamics of spin-nematic bright solitary waves in spin-tensor-momentum coupled Bose-Einstein condensates

We explore the dynamics of bright solitary waves in one-dimensional spin-tensor-momentum coupled spin-1 Bose-Einstein condensates. Our results show that the solitary waves have nonzero spin and nematicity simultaneously, hence we call them spin-nematic solitary waves. The ground-state solitary waves carry a finite momentum and display the zero-energy modes arising from symmetry breaking. In the motion of the solitary wave, the spin-tensor-momentum coupling induces the flip of the spin-nematic vector on a unit Bloch sphere, which in turn produces a force to result in the oscillatory motion of the solitary wave. The effects of a harmonic trap are also analyzed. These findings reveal the unique properties of solitary waves affected by the spin-tensor-momentum coupling.

Figure 1

Single-particle energy-band structures in the momentum space. (a) ${\mathrm{\Omega }}_R/k_R^2=4$. (b) ${\mathrm{\Omega }}_R/k_R^2=0.1$.

Figure 2

(a) and (b) The parameters $\theta , k$, and $Q_{x,y,z}$ in the ground state for different ${\mathrm{\Omega }}_R/k_R^2$ by fixing $g_n/k_R=-1$. (c) and (d) Real and imaginary parts of the ground-state wave functions at ${\mathrm{\Omega }}_R/k_R^2=1$. The lines (solid line, dashed line, dash-dotted line) and the circles represent the results obtained by the GP simulations and variational calculations, respectively.

Figure 3

(a) Frequency ${\omega }_{\pm }$ obtained by the variational approach as a function of ${\mathrm{\Omega }}_R/k_R^2$. (b) Bogoliubov excitation frequency $\omega$ for ${\mathrm{\Omega }}_R/k_R^2=1$ and $g_n/k_R=-1$, where the two blue circles represent the frequency ${\omega }_{\pm }$ in Eq. (22) and the red points are obtained by the BdG equation.

Figure 4

Time evolution of the initially population-balanced solitary wave with ${\theta }_0=arccos(\sqrt{3}/3)$ and ${\phi }_0=0$. (a) and (b) Time evolution of the spin-nematic vector for momentum $k=4{\mathrm{\Omega }}_R/k_R$. In panel (a), the solid red line, the dashed blue line, and the dash-dotted green line represent the numerical results of the GP simulations, and the circles are the results of the variational analysis. In panel (b), the blue arrow represents the spin-nematic vector at $t=0$, the colored line is the trajectory of the spin-nematic vector, and the colorbar is the corresponding time. (c) and (d) Time evolution of the density for the initial momentum $k=0$ obtained by the GP simulations. The solid black lines are the evolutions of the center of mass given by the variational analysis. The other parameters are ${\mathrm{\Omega }}_R=0.5, g_n=-10$, and $k_R=0.1$ in panels (a)–(d). (e) and (f) Oscillation period $T$ of the center of mass as a function of ${\mathrm{\Omega }}_R/k_R^2$ for $g_n/k_R=-10$. The momentum in panels (e) and (f) are $k/k_R=0$ and $k/k_R=0.2$, respectively. The solid black lines and the dashed red lines in panels (e) and (f) represent the periods given by the GP simulations and the variational analysis, respectively.

Figure 5

Properties of the ground-state solitary waves for $g_s/k_R=-0.1$ [panels (a) and (b)] and $g_s/k_R=0.1$ [panels (c) and (d)], where the strength of the spin-independent interaction is $g_n/k_R=-1$. Panels (a) and (c) show $\theta , k$, and $z_c$ for different ${\mathrm{\Omega }}_R/k_R^2$. Panels (b) and (d) show the norms of the ground-state wave functions for ${\mathrm{\Omega }}_R/k_R^2=1$. In panels (a)–(d), the lines (solid line, dashed line, dash-dotted line) and circles represent the results obtained by the GP simulations and the variational calculations, respectively. Panels (e) and (f) show the Bogoliubov excitation frequencies (red points) given by the BdG equations for $g_s=-0.1$ and $g_s=0.1$, respectively, where ${\mathrm{\Omega }}_R/k_R^2=1$. The blue crosses represent the four excitation frequencies ${\omega }_{1,\pm }$ and ${\omega }_{2,\pm }$ given by the variational analysis.

Figure 6

Time evolution of the initially population-balanced solitary wave with ${\theta }_0=arccos(\sqrt{3}/3), {\phi }_0=0$, and $k_0=1$. (a) Momentum. (b)–(e) Spin-nematic vector. (f) and (g) Norms of the wave functions obtained by the GP simulations. (h) Center of mass. The parameters are $g_n=-10, g_s=-1, {\mathrm{\Omega }}_R=0.5$, and $k_R=0.2$. In panel (e), the blue arrow represents the spin-nematic vector at $t=0$, the colored line is the trajectory of the spin-nematic vector, and the colorbar is the corresponding time. In panels (a)–(d) and (h), the solid black lines represent the results given by the GP simulations, and the dash-dotted red lines and the red circles are the variational results.

Figure 7

(a) and (b) Parameters $\theta , k$, and $z_c$ in the harmonically trapped ground-state solitary wave for different ${\mathrm{\Omega }}_R/k_R^2$. The solid red line, the dashed blue line, and the dash-dotted green line are the results given by the GP simulations, and the red, blue, and green lines with solid circles are the results of the variational analysis. (b) Parameters $\theta$ and $k$ in the stationary solitary wave with $z_c=0$ for different ${\mathrm{\Omega }}_R/k_R^2$, which are obtained by solving the time-independent equation (13). (c) and (d) The excitation frequencies ${\omega }_{1,\pm }$ and ${\omega }_{2,\pm }$ of the stationary solitary wave with $z_c=0$ for different ${\mathrm{\Omega }}_R/k_R^2$, which are obtained by the linear stability analysis of Eq. (13). In panels (a)–(d), the other system parameters are $g_n/k_R=-10, g_s/k_R=1$, and $\gamma /k_R^2=0.2$.

Figure 8

Time evolution of the initially population-balanced solitary wave in the harmonic trap with $\gamma =0.1$. (a)–(c) Time evolutions of the densities and the center of mass for $g_s=0$, where the parameters are $g_n=-10, {\mathrm{\Omega }}_R=0.5$, and $k_R=0.1$. (d) Evolution of the center of mass for $g_s=-0.1$. The other parameters are the same as those in panels (a)–(c). Panels (a) and (b) are given by the GP simulations, and the solid black lines and the red circles in panels (c) and (d) represent the results of the GP simulations and the variational analysis, respectively.