Split and overlapped binary solitons in optical lattices

We analyze the energetic and dynamical properties of bright-bright (BB) soliton pairs in a binary mixture of Bose-Einstein condensates subjected to the action of a combined optical lattice, acting as an external potential for the first species while modulating the intraspecies coupling constant of the second. In particular, we use a variational approach and direct numerical integrations to investigate the existence and stability of bright-bright (BB) solitons in which the two species are either spatially separated (split soliton) or located at the same optical lattice site (overlapped soliton). The dependence of these solitons on the interspecies interaction parameter is explicitly investigated. For repulsive interspecies interaction we show the existence of a series of critical values at which transitions from an initially overlapped soliton to split solitons occur. For attractive interspecies interaction only single direct transitions from split to overlapped BB solitons are found. The possibility to use split solitons for indirect measurements of scattering lengths is also suggested.

Figure 1

(Left panel) Dependence of the minimum energy of the system, within ansatz (9), vs the number of atoms $N_1$, with $N_1+N_2=3$. (Right panel) Difference between chemical potentials ${\mathrm{\Delta }}_{\mu }={\mu }_1-{\mu }_2$ vs $N_1$. The parameters are fixed to ${\gamma }_1={\gamma }_2=-1,V_{01}=-0.5$, and $V_{02}=-0.25$. Energies and chemical potentials are in units of $2E_r$.

Figure 2

Existence curves in the ${\mu }_i-N$ plane $\left(N\equiv N_1=N_2\right)$ for BB solitons of the GPEs with attractive (${\gamma }_{12}=-0.5$, left) and repulsive (${\gamma }_{12}=0.1$, right) interspecies interaction. Other parameters are fixed as ${\gamma }_1={\gamma }_2=-1,V_{01}=-0.5,V_{02}=-0.25$. Squares and triangles denote numerical results obtained by self-consistent diagonalization of Eq. (8) and refer to the first and the second component, respectively. Dashed and dot-dashed lines represent the corresponding results obtained from the minimization of the energy functional. Chemical potentials in units of $2E_r$.

Figure 3

Profiles of BB solitons with $N_1=N_2=N$ corresponding to the $N=1$ points on the curves of Fig. 2 for attractive (${\gamma }_{12}=-0.5$, left panel) and repulsive (${\gamma }_{12}=0.1$, right panel) interspecies interaction. Dashed and dot-dashed (blue) lines and continuous and dotted (red) lines refer to first and second component, respectively. In both panels continuous and dashed lines refer to numerical solutions of Eq. (4), while dotted and dot-dashed lines to variational analysis. Position in units of $k_L^{-1}$.

Figure 4

Numerical minimization of the pseudopotential $\mathrm{\Pi }$ [see (22)] with soliton peaks around $(x_{01},x_{02})=(0,0)$ (top panels), $(x_{01},x_{02})=(\pi ,0)$ (central panels), and $(x_{01},x_{02})=(\pi ,-\pi )$ (bottom panels). Plots in the left column represent the variation of the optimal soliton widths $a_1$ (solid blue lines) and $a_2$ (dashed red lines), in units of $k_L^{-1}$, with ${\gamma }_{12}$. Plots in the right column show the behavior of one optimal width with respect to the other. The jumps observed in the central and bottom lines (right column) are due to the disappearance of local minima with $\mathrm{\Delta }x\simeq \pi$ and $\mathrm{\Delta }x\simeq 2\pi$ as ${\gamma }_{12}$ decreases under a critical value. Minima found below the critical ${\gamma }_{12}$ coincide with the values in the top plots and are thus not shown in the right column.

Figure 5

(Left panel) Peak positions $x_{01}$ (solid blue line) and $\left(x_{02}\right)$ (dotted red line), in units of $k_L^{-1}$, of optimal BB profiles, obtained by searching the local minimum around $(x_{01},x_{02})=(\pi ,-\pi )$, for different values of ${\gamma }_{12}$. The sudden jump towards $(\pi ,\pi )$ is due to the disappearance of the local minimum (see also Fig. 4, bottom panels). (Right panel) Local minimum value of the effective potential energy in Eq. (22), in units of $2E_r$.

Figure 6

Time evolution of density profiles for overlapped solitons in the first (solid blue lines) and second (dashed red lines) species for ${\gamma }_{12}=-0.25$ (left panel) and ${\gamma }_{12}=-1$ (right panel). The results are obtained by direct numerical integration. Position in units of $k_L^{-1}$. Time in units of $\hslash /\left(2E_r\right)$.

Figure 7

Time evolution of density profiles for split solitons in the first (solid blue lines) and second (dashed red lines) species for ${\gamma }_{12}=-0.25$ (left panel) and ${\gamma }_{12}=-0.38$ (right panel). In the latter case, numerical integration shows the instability of the soliton pair. Position in units of $k_L^{-1}$. Time in units of $\hslash /\left(2E_r\right)$

Figure 8

Dependence of chemical potentials (left panel) and distance between peaks (right panel) on repulsive interspecies interactions of a BB soliton with $N_1=N_2=1$. Other parameters are fixed as ${\gamma }_1={\gamma }_2=-1,V_{01}=-0.5,V_{02}=-0.25$. Top (blue) and bottom (red) curves in the left panel refer to chemical potentials of first and second component, respectively, while vertical lines separate ${\gamma }_{12}$ regions for existence of BB solitons in $B{B}_0$ (overlapped), $B{B}_1$, and $B{B}_2$ (split). Chemical potential in units of $2E_r$, distance in units of $k_L^{-1}$.

Figure 9

Split BB solitons inside the regions $n=1$ and $n=2$ of the left panel of Fig. 8, at ${\gamma }_{12}=0.5$ (top panel) and ${\gamma }_{12}=1.0$ (bottom panel). The (blue) profiles centered at $x=0$ and the (red) ones centered at $x<0$ refer to first and second component, respectively, while the sinusoidal (black) lines shows the periodicity of the optical lattice. Position in units of $k_L^{-1}$.

Figure 10

Dependence on attractive interspecies interaction of chemical potentials (left panel) and distance between density centers, $x_1,x_2$, (right panel) of a BB soliton with $N_1=N_2=1$. Bottom (blue) and top (red) curves in the left panel refer to the first and the second component, respectively, while dotted and continuous lines (left panel) and square and triangle symbols (right panel) refer to BB solitons with centers initially separated by $\pi$ and by $2\pi$, respectively. Other parameters are fixed as ${\gamma }_1={\gamma }_2=-1,V_{01}=-0.5,V_{02}=-0.25$. Chemical potential in units of $2E_r$, distance in units of $k_L^{-1}$.