Classical dynamics and localization of resonances in the high-energy region of the hydrogen atom in crossed fields

When superimposing the potentials of external fields on the Coulomb potential of the hydrogen atom, a saddle point (called the Stark saddle point) appears. For energies slightly above the saddle point energy, one can find classical orbits that are located in the vicinity of this point. We follow those so-called quasi-Penning orbits to high energies and field strengths, observing structural changes and uncovering their bifurcation behavior. By plotting the stability behavior of those orbits against energy and field strength, the appearance of a stability apex is reported. A cusp bifurcation, located in the vicinity of the apex, will be investigated in detail. In this cusp bifurcation, another orbit of similar shape is found. This orbit becomes completely stable in the observed region of positive energy, i.e., in a region of parameter space, where the Kepler-like orbits located around the nucleus are already unstable. By quantum mechanically exact calculations, we prove the existence of signatures in quantum spectra belonging to those orbits. Husimi distributions are used to compare quantum-Poincaré sections with the extension of the classical torus structure around the orbits. Since periodic orbit theory predicts that each classical periodic orbit contributes an oscillating term to photoabsorption spectra, we finally give an estimation for future experiments, which could verify the existence of the stable orbits.

Figure 1

By encircling an $\mathrm{EP}2$, an interchange of two resonances can be observed. Parts (a) and (b) show the real and imaginary part of the two fixed points of Eq. (13), depending on the angle $\phi$ in $\mu =e^{i\phi }$. (c),(d) When encircling the cusp point along $(a,b)=(-5cos\phi ,-5sin\phi ),\phi \in [0,2\pi )$, one of the fixed points of Eq. (11) can be transformed into another one [marked by the red solid line starting in the upper left corner of (c)], without passing through a bifurcation. A second encircling passes both tangent bifurcations. Since the fixed points coincide in those bifurcations, it is not possible to determine the permutation behavior. An $\mathrm{EP}2$ or an $\mathrm{EP}3$ behavior can be observed accordingly. The angle $\phi$ and the position $x$ are given in dimensionless units [see Eqs. (11) and (13)].

Figure 2

(a) Shape of the orbit $B_1$, evolving from the elliptically shaped quasi-Penning resonances, at different energies and a fixed value of $F=0.6$. The orbit is localized in the $z=0$ plane. The position of the Stark saddle point on the left-hand side and the nucleus at the origin are marked by diamonds. At $E\approx 0.356$ the right point of intersection with the $x$ axis coincides with the nucleus. (b) Shape of the orbits $B_1,B_2,D_1$ and $K_1$ at $F=0.5$. The positions $x$ and $y$ are given in scaled atomic units [see Eq. (2)].

Figure 3

Stability of $B_1$ (a) and $B_2$ (b) in dependence on the scaled energy and the scaled field strength. Along the lines marking the borders of different stability regions, bifurcations occur. The stability within those regions is displayed by two upper indices. The first one refers to the stability ($s$) or instability ($u$) of the orbit in the direction perpendicular to the magnetic field, the second one to the stability in the direction parallel to the magnetic field. According to the different types of stability, the regions are additionally dyed by colors, shadings, and grayscales. One can clearly observe the stability apex for $B_1$ ending at $F\approx 0.5,E\approx 0.5$, i.e., a point in the $E-F$ space at which two border lines of the stability regions $B_1^{us}-B_1^{uu}$ meet, which was first uncovered by Flöthmann [38]. The dashed line on the left-hand side in (a) marks the energy of the saddle point (SP) as a function of the field strength. Specific points in parameter space are displayed by ${\mathbf{S}}_i=(F_i,E_i)$. The field strength $F$ and the energy $E$ are given in scaled atomic units [see Eq. (2)].

Figure 4

Stability of $B_1$ in dependence on the scaled energy and the scaled field strength. Part (a) is an enlargement of the rectangular area marked by dashed lines in Fig. 3. Part (b) is an enlargement of the equivalently marked area in (a). The different resolutions allow for a comparison with Fig. 6 and provide better insight in the relevant region around $F\approx 0.5,E\approx 0.6$. In the dark red area, two different versions of $B_1$ exist, which are indicated by $B_{1a}$ and $B_{1b}$. The field strength $F$ and the energy $E$ are given in scaled atomic units [see Eq. (2)].

Figure 5

Real (a) and imaginary part (b) of one of the intersection points of the orbits with the $x$ axis when encircling the cusp point ${\mathbf{S}}_{\mathrm{CP}B}$. The angle $\phi$ parametrizes the circle around ${\mathbf{S}}_{\mathrm{CP}B}$. $C{X}_1$ and $C{X}_2$ are the two complex orbits appearing. Just as in Figs. 1 and 1, it is not possible to determine the permutation behavior due to the coincidence of two orbits in a tangent bifurcation, respectively. The angle $\phi$ and the values of $x_0$ are given in dimensionless and scaled atomic units, respectively [see Eq. (2)].

Figure 6

(a) Stability of $B_2$ in dependence on the scaled energy and the scaled field strength. The observed stability island is triangular in shape, indicated by $B_2^{ss}$, and it ends in ${\mathbf{S}}_{z,B}$. Part (b) is an enlargement of the equivalently marked area in (a). The field strength $F$ and the energy $E$ are given in scaled atomic units [see Eq. (2)].

Figure 7

Stability of $D_1$ in dependence on the scaled energy and the scaled field strength. A very thin region marked $D_1^{ss}$ of complete stability can be observed. A cusp bifurcation occurs around the cusp point ${\mathbf{S}}_{\mathrm{CP}D}$. Part (b) is an enlargement of the equivalently marked area in (a). The field strength $F$ and the energy $E$ are given in scaled atomic units [see Eq. (2)].

Figure 8

Comparison between quantum-mechanical and semiclassical results at $B=4.26\times {10}^{-3}\left(\widehat{=}1000\mathrm{T}\right)$. For further information, see Sec. 5. Best agreements are obtained for $\lambda =1$ and $m_{\perp }=0,m_{\parallel }=0$ in the case of the first resonance series (a) and $m_{\perp }=0,m_{\parallel }=2$ in the case of the second resonance series (b). The parameter range shown is almost the same as in Fig. 6. All values are given in atomic units.

Figure 9

Comparison between semiclassical and quantum-mechanical results for the first resonance series at $B=1.70\times {10}^{-4}\left(\widehat{=}40\mathrm{T}\right)$. The resonance widths are evidently smaller but still do not vanish within the stability island, which is marked by dashed lines. For further information, see Sec. 5. All values are given in atomic units.

Figure 10

Probability density $\varrho \left(\mathbf{r}\right)$ of a resonance of the first series at $E=(1.95\times {10}^{-3})-(4.14\times {10}^{-5}i),F=4.76\times {10}^{-6}$, and $B=1.70\times {10}^{-4}$. One can observe an agreement with the course of the classical orbit (continuous line) in (a) and the restriction to the $z=0$ plane in (b). All values are given in atomic units.

Figure 11

Probability density $\varrho \left(\mathbf{r}\right)$ of a resonance of the second series at $E=(2.14\times {10}^{-3})-(5.87\times {10}^{-5}i),F=4.73\times {10}^{-6}$, and $B=1.70\times {10}^{-4}$. The differences to the classical orbit (continuous line) are significantly larger. All values are given in atomic units.

Figure 12

Probability density $|4U^2{\Psi }^{*}\left(\mathbf{U}\right)\Psi \left(\mathbf{U}\right)|$ of the resonance of Fig. 10 in regularized coordinates. Without the limitation of $U_2\ge 0$, the probability density shows a second symmetry relative to the line $U_2=0$. The lower part $(U_2\le 0)$ corresponds to $\alpha =2\pi$ and is not included in the calculations of the quantum-Poincaré sections. The classical orbit intersects the $U_1=0$ line twice but with a different sign of the velocity $V_1$ in the $U_1$ direction. All values are given in atomic units.

Figure 13

Comparison between quantum-Poincaré sections [shown by color bar (grayscales)] and classical torus structure. The dashed lines display the borders of the classically permitted region. Since the Hamiltonian is a quadratic function of the momentum, there are two possibilities to choose $P_1$ when calculating the section. Both solutions differ from each other in the sign of the velocity $V_1$ in the $U_1$ direction. Therefore, agreements with only one part of the torus structure can be achieved at any time. The color bar is chosen equal to compare both results. We set $\sigma$ to the constant value $\sigma =2$. All values are given in atomic units.