Bose-Einstein condensates in a ring-shaped trap with a nonlinear double-well potential

We develop the mean-field theory for Bose-Einstein condensates in a one-dimensional ring with two types of nonlinear double-well potentials, based on a pair of $\delta$ functions or Gaussian of a finite width, placed at diametrically opposite points. By analyzing the ground states (GSs) in these cases, we find a qualitative difference between them. With the Gaussian profile, the GS always undergoes the phase transition from the symmetric shape to an asymmetric one at a critical value of the norm. In contrast, the symmetry-breaking transition does not happen with the $\delta$-functional profile of the nonlinearity, and the GS always remains symmetric. In addition, the numerical analysis for the Gaussian profile demonstrates that the type of symmetry-breaking transition depends on the width of the nonlinear-potential well.

Figure 1

A sketch of the system. The two gray dots represent the positions of the nonlinear potential, around which the interaction strength is modulated. Values of azimuthal coordinate $\theta$ are indicated as defined in the text.

Figure 2

Examples of the different types of solutions at $\mu =-0.1$.

Figure 3

The norm of the (a) asymmetric and (b) symmetric states as functions of the chemical potential. The asymmetric state exists at $\left|\mu \right|>0.0879$.

Figure 4

The time evolution of the wave function in direct simulations of the model with the $\delta$ functions modeled by the narrow Gaussians. From top to bottom, the initial state is given by the symmetric solution at $\left|\mu \right|=0.1$, the symmetric solution at $\left|\mu \right|=1$, and the asymmetric one at $\left|\mu \right|=1$. Only the top solution is stable.

Figure 5

The (a) energy and (b) chemical potential of the symmetric and asymmetric states as functions of the norm. (c) Imbalance $\Theta$ for the asymmetric state as a function of the norm. All the quantities plotted have been rescaled to be dimensionless.

Figure 6

(a) Energies of the symmetric and asymmetric states as functions of the norm in the model with the Gaussians of width $a=0.1$. (b) The chemical potential of the ground state. (c) The population imbalance of the ground state.

Figure 7

Same as Fig. 6, but for the width of the Gaussian $a=0.3$.

Figure 8

The numerically found critical norm $N_c$ as a function of the width of the nonlinearity-modulation Gaussian, $a$. The symmetry-breaking phase transition of the first kind occurs at $a⩽a_{\mathrm{crit}}\simeq 0.28$ (dashed line), while the phase transition of the second kind occurs at $a>a_{\mathrm{crit}}$ (solid line).

Figure 9

(a) The jump of the population imbalance $\Delta \Theta$ of the ground state at the critical point, $N=N_c$, as a function of width $a$. At $a\lesssim 0.28$, the order phase transition of the first kind occurs at $N=N_c$. (b) The jump of the chemical potential, $\Delta \mu ={\mu }_{\mathrm{s}}-{\mu }_{\mathrm{as}}$, between the symmetric and asymmetric states at the transition point, $N=N_c$, for different values of the Gaussian's width $a$.