Dipolar Bose-Einstein condensates in a $\mathcal{PT}$-symmetric double-well potential

We investigate dipolar Bose-Einstein condensates in a complex external double-well potential that features combined parity and time-reversal symmetry. On the basis of the Gross-Pitaevskii equation we study the effects of the long-ranged anisotropic dipole-dipole interaction on ground and excited states by the use of a time-dependent variational approach. We show that the property of a similar nondipolar condensate of possessing real energy eigenvalues in certain parameter ranges is preserved despite the inclusion of this nonlinear interaction. Furthermore, we present states that break the $\mathcal{PT}$ symmetry and investigate the stability of the distinct stationary solutions. In our dynamical simulations we reveal a complex stabilization mechanism for $\mathcal{PT}$-symmetric as well as for $\mathcal{PT}$-broken states which are, in principle, unstable with respect to small perturbations.

Figure 1

Visualization of the $\mathcal{PT}$-symmetric double-well potential in the repulsive configuration. The arrows indicate the direction of the dipoles. Isosurfaces show the shape of the real part $V_{\mathrm{ext}}^{\mathrm{r}}$ of the external potential, where $l$ is the interwell spacing as given in the text. The contour lines show a slice cut through the center of the imaginary part $V_{\mathrm{ext}}^i$ of the external potential, with the corresponding color bars on the left.

Figure 2

Mean-field energy as a function of the gain-loss parameter $\Gamma$ at different values of the nonlinearity $Na$ for the nondipolar case. The upper panel shows the real and the lower one the imaginary part. For $Na=-0.0022$ the tangent bifurcation is marked by a dot labeled $T$ and the pitchfork bifurcation by a dot labeled $P$. For $Na=0$ the two bifurcations merge in the singular point $E$. $E_{\mathrm{mf}}$ and $\Gamma$ are given in the units introduced in Sec. 1.

Figure 3

(a) Mean-field energy of the stationary states in the repulsive configuration as a function of the gain-loss parameter $\Gamma$ for different values of $Na$. The upper panel shows the real and the lower one the imaginary part. The dipole strength is set to $N{a}_{\mathrm{dd}}=0.3$. For $Na=-0.03$ the arrows in (a) point at the lines of the states for which in (b) $S_{0,1}$, (c) $S_{0,2}$, (d) $S_{1,1}$, and (e) $S_{1,2}$ the absorption images are shown at $\Gamma =0.2$. The field of view is $1\times 1$ in the units given in the text. In (a) the two tangent bifurcations $T_0$ and $T_1$ are labeled with red dots.

Figure 4

Mean-field energy as a function of the gain-loss parameter $\Gamma$, where the upper panel shows the real and the lower one the imaginary part. The dipole strength is set to $N{a}_{\mathrm{dd}}=0.3$. The points $P$ and $P^{\prime }$ denote pitchfork bifurcations for different values of the scattering length. At the point $E$ the pitchfork bifurcation has merged with the tangent bifurcation $T_0$ (see Fig. 3) of the $\mathcal{PT}$-symmetric states. Note that for $Na=-0.0415$ the $\mathcal{PT}$-broken state for values of $\Gamma <P^{\prime }$ has almost the same real part of the energy as the $\mathcal{PT}$-symmetric one and lies on top of that in the upper panel.

Figure 5

Eigenvalues $\Lambda ={\Lambda }^{\mathrm{r}}+i{\Lambda }^{\mathrm{i}}$ of the Jacobian (10) as functions of the gain-loss parameter $\Gamma$ with $Na=-0.038$ and $N{a}_{\mathrm{dd}}=0.3$. (a)–(d) show the states $S_{0,1}$, $S_{0,2}$, $S_{1,1}$, and $S_{1,2}$, respectively. In (a) all eigenvalues are imaginary with vanishing real parts implying stable fixed points. The upper state, shown in (b) is stable up to the pitchfork bifurcation $P$. The $\mathcal{PT}$-broken states $S_{1,\alpha }$ ($\alpha =1,2$) shown in (c) and (d) are unstable in the whole range of $\Gamma$. In (d) the pitchfork bifurcation $P$ can be seen, where for values of $\Gamma <{\Gamma }_P$ only the state $S_{0,2}$ shown in (b) survives.

Figure 6

Real-time evolution of the state $S_{0,2}$ for $N{a}_{\mathrm{dd}}=0.3$, $Na=-0.038$, and $\Gamma =0.2$. No additional perturbations other than numerical fluctuations have been added to the initial state. (a) The upper panel shows the populations $I^k=\langle g^k|g^k\rangle$ ($k=1,2$) of the left and the right wells. The lower panel shows the overall norm of the wave function. In (b) absorption images during the first oscillation are plotted with the parameters given in Fig. 3. It can be seen that during the oscillation the wave function takes the shape of the different $\mathcal{PT}$-broken states. The time $t$ is given in units $m{l}^2/\hslash$ as introduced in Sec. I.

Figure 7

Real-time evolution of the state $S_{1,2}$ for $N{a}_{\mathrm{dd}}=0.3$ and $Na=-0.038$. In contrast to Fig. 6, here $\Gamma =0.25$ is slightly above the value ${\Gamma }_P$ of the pitchfork bifurcation [see Fig. 5], i.e., in the unstable regime. Yet the nonlinear coupling dynamically stabilizes the state for several oscillations until the condensates finally collapses. The time $t$ is given in units $m{l}^2/\hslash$ as introduced in Sec. I.