Exceptional points in the spectra of atoms in external fields

We investigate exceptional points, which are branch-point singularities of two resonance eigenstates, in spectra of the hydrogen atom in crossed external electric and magnetic fields. A procedure to systematically search for exceptional points is presented, and their existence is proven. The properties of the branch-point singularities are discussed with effective low-dimensional matrix models, their relation with avoided level crossings is analyzed, and their influence on dipole matrix elements and the photoionization cross section is investigated. Furthermore, the rare case of a connection between three resonances almost forming a triple degeneracy in the form of a cubic-root branch point is discussed.

Figure 1

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

Figure 2

(Color online) Paths of the two eigenvalues (represented by squares and diamonds) which degenerate at the exceptional point labeled 14 in Table in the complex-energy plane [40]. 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. (14) with $\delta =0.01$ (see inset). The initial set of parameters and the corresponding eigenvalues are represented by filled starting points. The arrows indicate the direction of progression. The filled triangle marks the position of the exceptional point in the parameter space and the corresponding complex energy of the degenerate resonances.

Figure 3

(Color online) Paths of the two eigenvalues considered in Fig. 2 for different radii of parameter-space circle (14), where the center $\left({\gamma }_0,f_0\right)=\left({\gamma }^{\left(\text{EP}\right)},f^{\left(\text{EP}\right)}\right)$ was always chosen to be exactly the exceptional point. (a) $\delta =3\times {10}^{-2}$; (b) $\delta ={10}^{-2}$; (c) $\delta ={10}^{-4}$. The position of the resonances at $\left({\gamma }^{\left(\text{EP}\right)},f^{\left(\text{EP}\right)}\right)$ is marked by a triangle in each figure. The squares and diamonds represent the exact quantum resonances. The lines represent the eigenvalues of the two-dimensional matrix model [Eq. (15)], whose coefficients have been fitted to the numerical results of the exact quantum calculations.

Figure 4

(Color online) (a) A circle in the parameter space (labeled 1) is chosen such that it passes through the parameter set $\left({\gamma }_1,f_1\right)$ at which one of the energy eigenvalues is identical with its position at the exceptional point. A straight line (labeled 2) connects the parameters of the exceptional point and the parameter set $\left({\gamma }_1,f_1\right)$. (b) Paths of the two numerically exact complex energies for the circle and the straight line described in (a). (c) Magnification of the path of the eigenvalue denoted by the red solid line in (b) for the parameter line labeled 2 in (a). The eigenvalue departs from the point of degeneracy (circles) and returns to its initial value (plus symbols).

Figure 5

(Color online) Squares $P^2$ of the dipole matrix elements between a wave function corresponding to a bound state and the two states evolving from the exceptional point labeled 12 in Table at field strengths according to form (14) with the constant angle $\phi =0.7$ and $\left({\gamma }_0,f_0\right)=\left({\gamma }^{\left(\text{EP}\right)},f^{\left(\text{EP}\right)}\right)$. $\delta$ is the distance from the critical values of the exceptional point. For both states the square modulus diverges in the form of a reciprocal square root of the distance. A fit of the data points to a function of the form $a/\sqrt{\delta }+b$, which is expected from Eq. (21), is shown by the solid red and dashed green lines. The real and the imaginary parts of the sum $P_1^2+P_2^2$ are also drawn.

Figure 6

(Color online) Photoionization cross section $\sigma \mathbf{\left(}\text{Re}\left(E^{\left(\text{EP}\right)}\right)\mathbf{\right)}$ with the bound-state energy $E_0=-0.125$ ($n=2$, $p$ orbital) as a function of the distance $\delta$ from the critical field values of the exceptional point. The numerical data (dots) converge linearly to a constant value as is expected from Eq. (25). Again, line (14) with constant angle $\phi =0.7$ and $\left({\gamma }_0,f_0\right)=\left({\gamma }^{\left(\text{EP}\right)},f^{\left(\text{EP}\right)}\right)$ is used.

Figure 7

(Color online) Illustration of a structure in the vicinity of two exceptional points in which three resonances are involved. Each of the resonances is denoted by a different symbol and color. The inset of (a) shows the three parameter-space loops (circles) and the position of the two exceptional points (up triangles). In (a) the parameter-space loop (indicated by dots) is chosen such that both exceptional points are located within the circular area. The resonance marked by down triangles forms a branch-point singularity with two further states at two different parameter values, which is shown by two additional parameter-space loops (small circles in the inset) in whose circular area only one of the two exceptional points is located [(b) and (c)]. The solid lines mark the eigenvalues of the three-dimensional model [Eqs. (28b, 28c, 28d)] fitted to the numerical data.

Figure 8

(Color online) Real part of the complex energy as a function of the one-dimensional parameter $\alpha$, $\gamma =0.80516\times {10}^{-2}\alpha$, $f=0.32169\times {10}^{-3}\alpha$, defined in Eqs. (29a, 29b). The blue circle marks an exceptional point. Avoided crossings (marked by blue squares) appear, which are related to exceptional points. Only resonances with $\left|\text{Im}\left(E\right)\right|<0.0005$ are drawn so as to not overload the figure. The black frame marks the region investigated in detail in Fig. 9a to demonstrate how an avoided crossing leads to an exceptional point if the parameters are varied.

Figure 9

(Color online) (a) Detailed illustration of the region marked by the black frame in Fig. 8. Three avoided energy crossings are present. In addition, the imaginary part is shown, which exhibits two crossings (arrows 1 and 3) and one avoided crossing (arrow 2). (b) The same region is shown for the field strengths varied on the line $\gamma =0.65378\times {10}^{-2}{\alpha }^{\prime }$, $f=0.28213\times {10}^{-3}{\alpha }^{\prime }$ given by Eqs. (30a, 30b). One of the three exceptional points is located on this line and the corresponding avoided energy crossing marked by arrow 3 has been transformed into a branch-point singularity, where both the real and imaginary parts are identical. The other two encounters show at least one avoided crossing in the real or in the imaginary part.