Cusp bifurcation in the eigenvalue spectrum of $\mathcal{PT}-\text{symmetric}$ Bose-Einstein condensates

A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interactions as long as the particle number and therefore the interaction strength does not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that, in fact, there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.

Figure 1

The bifurcation scenario of the $\mathcal{PT}$-symmetric stationary states of the three-dimensional double-well potential at strong attractive interactions. The chemical potential of the stationary solutions is shown as a function of the interaction strength $Na$ for different values of the gain-loss parameter $\gamma$. For $\gamma =0$ (black dashed line) two tangent bifurcations T1 and T2 are present. The states $|\mathrm{A}\rangle$ and $|\mathrm{D}\rangle$ emerge from the bifurcation T1 whereas T2 gives birth to the states $|\mathrm{B}\rangle$ and $|\mathrm{C}\rangle$. For small values of $\gamma$ (solid lines) a third bifurcation T3 appears at which the two states $|\mathrm{B}\rangle$ and $|\mathrm{C}\rangle$ bifurcate which connects the so-far-independent branches of the states $|\mathrm{C}\rangle$ and $|\mathrm{D}\rangle$. For higher values of $\gamma$ (dotted lines) the bifurcations T2 and T3 vanish, as does the intermediate state $|\mathrm{C}\rangle$; thus, the branches of the states $|\mathrm{B}\rangle$ and $|\mathrm{D}\rangle$ merge.

Figure 2

Trajectories of the two tangent bifurcations T2 and T3 in the $\gamma \text{−}Na$-parameter space. In the shaded area all three states involved in the two bifurcations are present while in the other areas only one of these states exists. For increasing values of the gain-loss parameter $\gamma$, the bifurcation T2 is strongly shifted to lower nonlinearity parameters $Na$. At ${\gamma }_{\mathrm{c}}\approx 0.000766$ the two bifurcations coincide and the state $|\mathrm{C}\rangle$ vanishes while the states $|\mathrm{B}\rangle$ and $|\mathrm{D}\rangle$ merge.

Figure 3

Bicomplex bifurcation scenario showing the ${\mu }_1$ and ${\mu }_j$ component of the chemical potential as a function of the nonlinearity parameter $Na$ for different values of the gain-loss parameter $\gamma$. The $i$ and $k$ components vanish for all states discussed; thus, ${\mu }^{*}=\mu$ holds and all states are $\mathcal{PT}$ symmetric. The states with a purely real chemical potential, i.e., ${\mu }_j=0$, are still present (solid lines). In addition, a pair of bicomplex states with $\mu ={\mu }_1\pm j{\mu }_j$ emerges at the tangent bifurcations T1, T2, and T3 (dotted lines). For $\gamma >{\gamma }_{\mathrm{c}}$, the bifurcations T2 and T3 have vanished, leading to a bicomplex branch that is completely independent from the remaining scenario. Independent of the parameters $\gamma$ and $Na$ there are four stationary solutions.

Figure 4

Comparison of the cusp bifurcation occurring in the eigenvalue spectrum (left panels) and in the normal form (18) (right panels). For parameter values smaller than the critical value ($\gamma <{\gamma }_{\mathrm{c}}$, $\rho <{\rho }_{\mathrm{c}}$) the tangent bifurcations T2 and T3 are separated. They coalesce at the critical parameter value ($\gamma ={\gamma }_{\mathrm{c}}$, $\rho ={\rho }_{\mathrm{c}}$) and vanish for $\gamma >{\gamma }_{\mathrm{c}}$, $\rho >{\rho }_{\mathrm{c}}$. Real branches are drawn as solid lines and (bi)complex branches as dotted lines.