Multiple dark-bright solitons in atomic Bose-Einstein condensates

Motivated by recent experimental results, we present a systematic theoretical analysis of dark-bright-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component Bose-Einstein condensates. We study analytically the interactions between two dark-bright solitons in a homogeneous condensate and then extend our considerations to the presence of the trap. We illustrate the existence of robust stationary dark-bright-soliton “molecules,” composed of two or more solitons, which are formed due to the competition of the interaction forces between the dark- and bright-soliton components and the trap force. Our analysis is based on an effective equation of motion, derived for the distance between two dark-bright solitons. This equation provides equilibrium positions and characteristic oscillation frequencies of the solitons, which are found to be in good agreement with the eigenfrequencies of the anomalous modes of the system.

Figure 1

Experimental images indicating DB-soliton clustering in a two-component BEC. The upper cloud in each image [and the red (gray) curve in the inset] shows atoms in the $|2,-2\rangle$ state, while the lower cloud (black curve) shows atoms in the $|1,-1\rangle$ state. Prior to imaging, the two components are overlapped in the trap for 5 s. Insets show integrated cross sections of the boxed regions.

Figure 2

The top panel depicts the stationary solution for a single DB soliton for ${\mu }_1=3/2$, ${\mu }_2=1$, and $\Omega =0.1$. The bright (dark) components are shown by the dashed green (solid blue) line. The middle (bottom) panel shows the normalized imaginary part ${\lambda }_i/\Omega$ of the eigenvalues for the single DB soliton as a function of ${\mu }_1$ (${\mu }_2$) for ${\mu }_2=1$ (${\mu }_1=3/2$). The red dashed line depicts the analytical prediction of Ref. [16] for the DB-soliton oscillation frequency [cf. Eq. (17)], providing an excellent approximation to the anomalous mode eigenfrequency.

Figure 3

(top) A stationary DB-soliton pair: the solid blue line denotes the two-dark-soliton state (recall that each dark soliton is associated with a zero crossing), while the dashed green line denotes the respective two-bright-soliton state. The chemical potentials are ${\mu }_1=3/2$ and ${\mu }_2=1$. (middle) The equilibrium center of mass $x_0$ as a function of the chemical potential ${\mu }_2$ (for ${\mu }_1=3/2$). Red stars denote the analytical prediction of Eq. (23), while blue circles denote the numerically obtained soliton center $x_0$. (bottom) The oscillation frequency for the out-of-phase motion of the DB-soliton pair as a function of ${\mu }_2$ (for ${\mu }_1=3/2$). Red stars depict the analytical result for ${\omega }_0$ [cf. Eq. (26)], while blue circles depict the corresponding numerically obtained imaginary eigenvalue ${\lambda }_i$ (for the out-of-phase soliton motion) of the excitation spectrum.

Figure 4

The left and right columns correspond, respectively, to an in-phase and an out-of-phase dark-bright-soliton pair in a harmonic trap with $\Omega =0.1$. The top row of panels depicts the profiles of the DB-soliton pairs (solid blue lines and dashed green lines correspond, respectively, to the dark and bright components) and the trapping potential (dashed-dotted red line). The middle row of panels depicts the spectral plane $({\lambda }_r,{\lambda }_i)$ rescaled by the trap frequency $\Omega$. The bottom row of panels depicts the numerical (small red stars) and the analytical (blue circles) results for the equilibrium distance between the solitons as a function of ${\mu }_2$; the theoretical prediction is based on Eq. (27).

Figure 5

The left and right columns of panels correspond, respectively, to an in-phase and an out-of-phase dark-bright-soliton pair in a harmonic trap with $\Omega =0.1$. Shown are the imaginary (top row of panels) and the real (bottom row of panels) parts of the eigenvalues as functions of ${\mu }_2$ for ${\mu }_1=3/2$. In the top panels, the theoretical predictions for the eigenfrequencies of the anomalous modes of the system, pertaining to the in-phase (${\omega }_2$) and out-of-phase (${\omega }_1$) oscillations of the DB solitons [see Eqs. (28) and (29)], are depicted by dashed red lines. Notice that collisions of modes (eigenvalue crossings) observed in the top panels indicate the emergence of instability windows observed in the bottom panels. The instabilities are of the Hamiltonian-Hopf type and result in the emergence of eigenvalue quartets.

Figure 6

The left and right columns of panels correspond, respectively, to in-phase and out-of-phase three-DB-soliton configurations. The first row of panels depicts the respective stationary states for ${\mu }_1=3/2$, ${\mu }_2=1$, and $\Omega =0.1$; solid blue lines depict the dark-soliton components, dashed green lines depict the bright ones, and the dash-dotted red line shows the harmonic trap. The second row of panels depicts the spectral planes for the above stationary states, and the third and fourth rows are equivalent to those of Fig. 5, but for the three-DB-soliton configurations.