Solitons in tunnel-coupled repulsive and attractive condensates

We study solitons in the condensate trapped in a double-well potential with far-separated wells, when the $s$-wave scattering length has different signs in the two parts of the condensate. By employing the coupled-mode approximation it is shown that there are unusual stable bright solitons in the condensate, with the larger share of atoms being gathered in the repulsive part. Such unusual solitons derive their stability from the quantum tunneling and correspond to the strong coupling between the parts of the condensate. The ground state of the system, however, corresponds to weak coupling between the condensate parts, with the larger share of atoms being gathered in the attractive part of the condensate.

Figure 1

Illustration of the double well: the parabolic trap potential in the $y$ direction modified by a Gaussian barrier. The (horizontal) solid lines indicate the numerically computed eigenvalues in the combined potential. The dashed lines show the bottom energies in the two wells of the trap. We use dimensionless units.

Figure 2

The ground- and first excited-state wave functions for the double-well trap in Fig. 1 (dashed lines) and the wave functions localized in the complementary wells (solid lines) constructed as indicated in the text. All units are dimensionless.

Figure 3

Behavior of the number of atoms in the two wells of the trap vs the chemical potential with variation of the zero-point energy difference between the wells [all quantities pertain to the dimensionless system (13)]. Here the dotted lines correspond to $N_u$, the dashed lines to $N_v$, and the solid ones to the total number of atoms $N=N_u+N_v$. In panel (d) only the part of the picture about the local maximum is shown [the full picture is similar to that in panel (c)].

Figure 4

Unstable (1) and stable (2) soliton solutions to the dimensionless system (13) with the same total number of atoms which correspond to the two stars in panel (d) of Fig. 3.

Figure 5

The energy vs the total number of atoms [for the dimensionless system (13)] for two values of the zero-point energy difference: $\epsilon =0$ (the dashed line) and $\epsilon =-\kappa$ (the solid line). The solid line has a very narrow twist without self-intersection, which occupies the interval $8<N∕\sqrt{\kappa }<12$.

Figure 6

Bifurcation of the soliton solutions for two values of the zero-point energy difference: $\epsilon =-5.30\kappa$ [panel (a)] and ${\epsilon }_{\text{cr}}=-5.45\kappa$ [panel (b)]. The dimensionless units of system (13) are used. Here $a=0.001$. In both panels: the dotted lines correspond to $N_u$, the dashed lines to $N_v$, and the solid ones to the total number of atoms.