Gap solitons in spin-orbit-coupled Bose-Einstein condensates in optical lattices

While different ways to realize spin-orbit coupling in Bose-Einstein condensates exist, not all are currently experimentally implementable. Here we present a detailed study of gap solitons in a Bose-Einstein condensate with experimentally realizable spin-orbit coupling and discuss two cases relating to a spin-dependent parity symmetry. In the parity symmetric case, two families of fundamental gap solitons in the second linear energy gap are demonstrated with the opposite sign of the parity, with one family having single-humped densities and the other double-humped ones. In the case of broken parity symmetry, the fundamental modes manifest spin polarization. Both families possess an opposite sign of the spin polarization.

Figure 1

Linear Bloch spectrum and Bloch waves at the symmetry points of the spectrum. (a) The linear spectrum for zero detuning. The Bloch waves in the lowest two bands are shown in (c)–(f), corresponding to the points indicated in (a). (b) The linear spectrum with a finite detuning. The corresponding Bloch waves are plotted in (g)–(j). Thick (red) lines are for ${\varphi }_1,$ while thin (blue) lines are for ${\varphi }_2$. The gray shadows represent $-cos\left(2z\right)<0$. The unit of $z$ is $1/k_{\text{lat}}$. Note that we only show the lowest four bands.

Figure 2

Linear Bloch spectrum. Shaded regions represent the bands and white ones the gaps. (a) and (b) Spectrum as a function of the Rabi frequency $\Omega$ for $\delta =0$ and $\delta =0.5E_{\mathrm{lat}},$ respectively. The blue vertical lines in (a) identify the values that will be used in Figs. 3 and 6, while the ones in (b) indicate parameters that relate to results shown in Figs. 8 and 9. (c) and (d) Spectrum as a function of the detuning $\delta$ for $\Omega =8E_{\mathrm{lat}}$ and $\Omega =2E_{\mathrm{lat}}$. The horizontal lines in (c) and (d) mark the values relating to results shown in Figs. 10 and 11, respectively.

Figure 3

Gap solitons in the first and second gaps in the $\left(N,\mu \right)$ plane for $\delta =0$ and $\Omega =8E_{\mathrm{lat}}$. Shaded areas correspond to the linear bands. The soliton profiles at the labeled points in the first gap are shown in Fig. 4 and the ones in the second gap are shown in Fig. 5.

Figure 4

Soliton profiles at the points indicated in the first gap in Fig. 3. The grayed areas correspond to $-cos\left(2z\right)<0$. The thick (red) lines are for ${\varphi }_1$ and the thin (blue) lines are for ${\varphi }_2$.

Figure 5

Time evolution of gap solitons in the first gap shown in Fig. 3. The initial wave functions are the gap soliton solutions with 10% Gaussian distributed noise added on. (a) The out-of-phase mode of Fig. 4. (b) The in-phase mode shown in Fig. 4. Only the density distribution of the first component, $|{\varphi }_1{|}^2,$ is shown.

Figure 6

Soliton profiles at the points indicated in the second gap in Fig. 3. (a) The first family of the fundamental mode; (b) the second family.

Figure 7

Time evolution of the second family in the second gap as shown in Fig. 3. 10% Gaussian distributed noise was added to the initial wave function to trigger potential instabilities. (a) An unstable evolution at $\mu =-3.75$ and (b) a stable evolution at $\mu =-3.7$. Only the density distribution of the first component, $|{\varphi }_1{|}^2,$ is shown. Note the different scale of $t$ in (a) and (b).

Figure 8

Fundamental families in the second gap for $\delta =0$ and $\Omega =2E_{\mathrm{lat}}$. Shaded areas represent the linear bands. (a) The dotted line corresponds to the first family and the solid line to the second family. The profiles of the labeled points are shown in (b)–(d). Since $|{\varphi }_1{\left(z\right)|}^2={\left|{\varphi }_2\left(z\right)\right|}^2,$ only the densities of ${\varphi }_1$ are shown and the thick (red) lines correspond to the first family, while the thin (blue) lines correspond to the second family.

Figure 9

Eigenspectrum and eigenstates of a spin-orbit-coupled single-particle system in a harmonic trap with the Hamiltonian given in Eq. (12). (a) Lowest two eigenenergies. The dotted line is the ground-state energy while the solid line is the energy of the first excited state. (b)–(d) Density distributions of corresponding eigenstates as indicated in (a). Since $|{\varphi }_1{\left(z\right)|}^2={\left|{\varphi }_2\left(z\right)\right|}^2$ we only show one component and the dotted lines are the densities of the ground state and solid lines are the ones of the first excited state. Here $\gamma =1.96$ and the unit of $z$ is $\sqrt{\hslash /m{\omega }_z}$.

Figure 10

Fundamental gap solitons in the first and second gap for $\delta =0.5E_{\mathrm{lat}}$ and $\Omega =8E_{\mathrm{lat}}$. (a) Soliton families in the $\left(N,\mu \right)$ plane. The dotted lines correspond to the first family and solid line to the second family. The shaded regions are the linear Bloch bands and in the embedded figures the spin polarization of corresponding families are shown. The soliton profiles at the labeled points are shown in (b) and (c), where the thick (red) lines are for ${\varphi }_1$ and the thin (blue) lines are for ${\varphi }_2$.

Figure 11

Fundamental gap solitons in the second gap for $\delta =0.5E_{\mathrm{lat}}$ and $\Omega =2E_{\mathrm{lat}}$. (a) Soliton families and (b) spin polarization. In both plots the dotted line corresponds to the first family and solid line to the second family. The shaded areas represent the linear Bloch bands. The soliton profiles of labeled points in (a) are shown in (c) and (d). The thick (red) lines are for ${\varphi }_1$ and the thin (blue) lines are for ${\varphi }_2$. The gray shadows indicate $-cos\left(2z\right)<0$.

Figure 12

Behavior of the first and second fundamental families in the second gap with respect to detuning for fixed chemical potential $\mu =-4E_l,$ i.e., along the horizontal line in Fig. 2 with $\Omega =8E_l$. (a) Soliton families in the $\left(N,\delta \right)$ plane and (b) spin polarization. In (a) and (b) the dotted lines correspond to the first family and the solid lines to the second family. The density distributions of the labeled points in (a) are shown in (c)–(h). The thick (red) lines are for ${\varphi }_1$ and the thin (blue) lines are for ${\varphi }_2$. The shaded areas represent $-cos\left(2z\right)<0$.

Figure 13

Behavior of the first and second fundamental families in the second gap with respect to the detuning for fixed chemical potential $\mu =-3{\mathrm{E}}_l$ along the horizontal line in Fig. 2. Here $\Omega =2{\mathrm{E}}_l$.