Exceptional Points in Atomic Spectra

We report the existence of exceptional points for the hydrogen atom in crossed magnetic and electric fields in numerical calculations. The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities. A possibility for the observation of exceptional points in an experiment with atoms is proposed.

Figure 1

(a) Circle in parameter space $\lambda$ with the exceptional point ${\lambda }_c=i$ as center point for the simple model (1). (b) Eigenvalues ${ϵ}_{1,2}$ calculated for the parameter values from (a) indicated by squares and diamonds, respectively. In this special case, each of the two eigenvalues traverses a semicircle. In (a), (b), the filled symbols represent the first parameter value ${\lambda }_0$ and the corresponding eigenvalues ${ϵ}_{1,2}({\lambda }_0)$, respectively. The arrows point in the direction of progression.

Figure 2

(a) Paths of the two eigenvalues which degenerate at the exceptional point in the complex energy plane. The squares and the diamonds represent one of the eigenvalues, respectively. Each point of one eigenvalue belongs to a different set of parameters. The path in the field strength parameter space is a circle defined in Eq. (4) with $a=0.01$ (see inset). The first set of parameters and the corresponding eigenvalues are represented by filled symbols. The arrows indicate the direction of progression. The cross marks the position of the exceptional point in parameter space and the corresponding complex energy of the degenerate resonances. (b) Resonances in the complex energy plane for the parameter values $\gamma =0.00572$ and $f=0.000256$ (atomic units) at the exceptional point. The position of the two degenerate eigenvalues is marked with an arrow.

Figure 3

Phase of the complex matrix element $p$ defined in Eq. (5). It changes its value by $\pi$ during the traversal of the circle in parameter space. The result corresponds to the expected sign change.

Figure 4

(a) Photoionization cross section of the energy range in which two eigenvalues connected with an exceptional point appear. The cross section is shown for eight different pairs of parameter values $\gamma$ and $f$ being located on a circle around the exceptional point. All values in atomic units. (b) Complex energy eigenvalues extracted from the cross sections with the harmonic inversion method. Each eigenvalue is drawn with a different symbol. Altogether, 16 pairs of parameter values were used. In (a), every second cross section is shown. The signature of a branch point singularity connected with the two eigenvalues labeled with squares and diamonds, respectively, is clearly visible.