Spontaneous symmetry breaking of gap solitons and phase transitions in double-well traps

We study stationary states of a two-dimensional (2D) Bose-Einstein condensate with both attractive and repulsive nonlinearities in a combination of a double-square-well potential in one direction and a perpendicular optical lattice. We look for dual-core solitons in this configuration, focusing on their symmetry-breaking bifurcations. For attractive interactions, without the lattice, a similar analysis was performed [M. Matuszewski et al., Phys. Rev. A 75, 063621 (2007)], where subcritical bifurcation transforming antisymmetric gap solitons into asymmetric ones was found. Here we focus on the effect of an optical lattice and the so created gap solitons. We discover that a phase transition occurs when the lattice depth increases and the additional dimension becomes strongly suppressed. The bifurcation type changes from subcritical (typical for a 2D system, with hysteresis) to supercritical (typical for a one-dimensional system). An additional advantage of the lattice is that gap solitons exist even for repulsive interactions. In this case we also discover bifurcation of a supercritical type. The analysis is based on a variational approximation, which is surprisingly well verified by numerical results.

Figure 1

The shape of the quasi-one-dimensional double-well potential, $U\left(x,y\right)$. The wiggles indicate quasi-1D lattice along $y$.

Figure 2

(Color online) A set of numerically found bifurcation diagrams in the model with attraction, showing degree of a asymmetry of dual-core soliton, $\nu$, as a function of the soliton’s total norm, $N$, see Eqs. (9). The diagrams pertain to fixed values of parameters of the transverse double-well configuration, $L=D=1$, $U_0=-0.7$ (attractive case), while the strength of the longitudinal optical-lattice potential gradually increases. One can check from the analysis of Eq. (17) that the turning points are at $\nu =0$ and $\pm N/\sqrt{3}$. One can clearly see that the supercritical bifurcation will turn into subcritical bifurcation with increase of the optical latttice strength. It can be interpreted as a phase transition.

Figure 3

(Color online) A set of bifurcation diagrams for gap solitons in the model with repulsive nonlinearity, for $L=D=1$, $U_0=4$, and a set of different values of the optical-lattice strength, $\rho$.

Figure 4

(Color online) The top and bottom panels demonstrate examples of cross-section profiles, along $y=0$, of stable asymmetric and symmetric gap solitons, as obtained from a numerical solution to Eq. (3) and predicted by the variational approximation (dashed and continuous lines, respectively). Parameters of the double-well potential are $L=D=1$, $U_0=-0.7$ ($\sigma =s=1$, attractive case), and $\rho =1$. Norms of the asymmetric and symmetric solitons are, respectively, $N=0.52$ and $N=0.28$. The asymmetry parameter for the former soliton, see Eqs. (9), is $\nu =0.34$.

Figure 5

(Color online) Two-dimensional version of the situation pictured in Fig. 4. Only numerical results following from Eq. (3) are shown. The insets illustrate solitons—the density scale is represented by the intensity of the shading. In the background the horizontal stripes represent double-well structure and vertical modulation visualize optical-lattice potential. Norms of the (a) asymmetric and (b) symmetric solitons are, respectively, $N=0.52$ and $N=0.28$. The asymmetry parameter equal to $\nu =0.34$ is in agreement with the variational predictions of Fig. 2.