Model of a $\mathcal{PT}$-symmetric Bose-Einstein condensate in a $\delta$-function double-well potential

The observation of $\mathcal{PT}$ symmetry in a coupled optical waveguide system that involves a complex refractive index has been demonstrated impressively in the experiment by Rüter et al. [Nat. Phys. 6, 192 (2010)]. This is, however, only an optical analog of a quantum system, and it would be highly desirable to observe the manifestation of $\mathcal{PT}$ symmetry and the resulting properties also in a real, experimentally accessible, quantum system. Following a suggestion by Klaiman et al. [Phys. Rev. Lett. 101, 080402 (2008)], we investigate a $\mathcal{PT}$-symmetric arrangement of a Bose-Einstein condensate in a double-well potential, where in one well cold atoms are injected while in the other particles are extracted from the condensate. We investigate, in particular, the effects of the nonlinearity in the Gross-Pitaevskii equation on the $\mathcal{PT}$ properties of the condensate. To study these effects we analyze a simple one-dimensional model system in which the condensate is placed into two $\mathcal{PT}$-symmetric $\delta$-function traps. The analysis will serve as a useful guide for studies of the behavior of Bose-Einstein condensates in realistic $\mathcal{PT}$-symmetric double wells.

Figure 1

Eigenvalues $\kappa$ of the nonlinear equation (1) as functions of the size of the loss-gain parameter $\gamma$ for $a=2.2$. The value of the nonlinearity parameter chosen is $g=1$. The case $g=0$ is drawn (dashed lines) for comparison. A branch of complex conjugate eigenvalues appears which bifurcates from the ground-state branch before the critical value of $\gamma$ where the branches of the real eigenvalues coalesce in an exceptional point. There a pair of complex eigenvalues only emerges after an analytical continuation of the nonlinearity in the Gross-Pitaevskii equation. All quantities are plotted dimensionless.

Figure 2

Real and imaginary parts and moduli of the eigenstates of the nonlinear Hamiltonian in Eq. (1) for $g=1$, $a=2.2$, and $\gamma =0.35$, (a) ground state and (b) excited state. The wave functions are $\mathcal{PT}$ symmetric, and the moduli are symmetric functions, producing the $\mathcal{PT}$ symmetry of the total nonlinear Hamiltonian. All quantities are plotted dimensionless.

Figure 3

Real and imaginary parts and moduli of the eigenstates of the nonlinear Hamiltonian in Eq. (1) for $g=1$, $a=2.2$, and $\gamma =0.5$, (a) solution with imaginary part of $\kappa >0$ and (b) imaginary part of $\kappa <0$. The $\mathcal{PT}$ symmetry is broken, the moduli are not symmetric functions, and the $\mathcal{PT}$ symmetry also of the nonlinear Hamiltonian is broken. All quantities are plotted dimensionless.