Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive $1/r$ interaction

The extended Gross-Pitaevskii equation for the Bose-Einstein condensation of gases with attractive $1/r$ interaction has a second solution which is born together with the ground state in a tangent bifurcation. At the bifurcation point both states coalesce, i.e., the energies and the wave functions are identical. We investigate the bifurcation point in the context of exceptional points, a phenomenon known for linear non-Hermitian Hamiltonians. We point out that the mean field energy, the chemical potential, and the wave functions show the same behavior as an exceptional point in a linear, nonsymmetric system. The analysis of the analytically continued Gross-Pitaevskii equation reveals complex waves at negative scattering lengths below the tangent bifurcation. These solutions are interpreted as a decay of the condensate caused by an absorbing potential.

Figure 1

(Color online) Chemical potentials of the two solutions. The results of the variational and the numerical exact calculations are shown. Two solutions emerge in a tangent bifurcation at a critical value of the parameter $N^{2}a/a_u$. In the variational approximation the critical value is $N^{2}a/a_u=-3\pi /8$, whereas the numerical exact solutions emerge at $N^{2}a/a_u=-1.0251$ [10]. The “atomic” units introduced in Ref. [10] are used.

Figure 2

(Color online) Eigenvalues of the two-dimensional model defined in Eq. (20) with $c=1$ for a circle around the exceptional point. (a) Circle in the parameter space with radius $\varrho ={10}^{-3}$. The starting point $\lambda \left(0\right)$ is marked with an open square and the direction of progression is indicated with the arrow. (b) Path of the eigenvalues $e_{1,2}$ for the parameter values on the circle. The eigenvalues which belong to the first parameter value $\lambda \left(0\right)$ are labeled with open squares and the arrows point in the direction of progression.

Figure 3

(Color online) Phase of the product $p_{12}$ for a circle in the parameter space of type (24) with radius $\varrho ={10}^{-3}$ around the exceptional point. (a) Symmetric matrix with $c=0$. The phase changes by $\pi$ indicating the change in sign of one of the eigenvectors from Eq. (26). (b) Nonsymmetric matrix with $c=1$. There is no change in sign as noted in Eq. (27).

Figure 4

(Color online) Chemical potential and mean field energy for a circle in the complex $N^{2}a/a_u$ parameter space. The first row shows the parameter space circle of type (31) with $\varrho ={10}^{-3}$ used for the variational (a) and the numerical exact (b) calculations. The starting point is marked with an open square and the direction of progression is indicated with an arrow. In the second row, the chemical potential $\epsilon /N^2$ for the variational (c) and numerical (d) solutions is plotted. The variational (e) and numerical (f) mean field energies $E/N^3$ are presented in the third row. Each of the two solutions is drawn with a different line. The results obtained for the first parameter value on the circle are marked with an open square and the arrows point in the direction of progression. A permutation of the two solutions is present for the mean field energy as well as for the chemical potential indicating the existence of an exceptional point. All energies are given in units of the “Rydberg energy” $E_u$ mentioned in Sec. 2.

Figure 5

(Color online) Phase $\text{arg}\left(O_{12}\right)$ of the overlap integral $O_{12}$ defined in Eq. (34). The path in the parameter space $N^{2}a/a_u$ is a circle with the critical value as center point and $\varrho ={10}^{-3}$. The angle on the circle is denoted by $\phi$. Both the variational and the numerical results show that the phase of $O_{12}$ returns to its initial value at the end of the circle and that the change in sign from Eq. (26) is not present as expected (see text).

Figure 6

(Color online) Imaginary part of the chemical potential for real $N^{2}a/a_u$ below the bifurcation point obtained with the variational and the numerical calculations. Both the continuation of the ground state and the continuation of the excited state are plotted. One is the complex conjugate of the other.