Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits

A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori and thereby allows one to characterize the periodic orbits by a set of winding numbers. With this knowledge, we construct the action variables as functions of the frequency ratios and carry out a semiclassical torus quantization. The semiclassical energy levels thus obtained agree well with exact quantum calculations.

Figure 1

(Color online) Periodic orbits at $\tilde{E}=-1.4$ and $\tilde{F}=0.5$. The FPOs $S^{\pm }$ and their repetitions are shown with black circles. (a) The 2-torus POs $T_2^p$ (red plus symbols) and $T_2^n$ (green triangles). (b) The $T_2^p$ and their 3-torus partners $T_3^p$ (blue crosses). (c) The $T_2^n$ and their 3-torus partners $T_3^n$ (magenta diamonds).

Figure 2

(Color online) The two planar FPOs $S^{+}$ (green) and $S^{-}$ (blue) at $\tilde{E}=-1.5$ (a) and $\tilde{E}=-1.4$ (b), where $S^{-}$ is already strongly deformed. The position of the nucleus is marked by a dot.

Figure 3

(Color online) Periodic orbits in the Poincaré surface of section $\tilde{y}=0$ for the planar subsystem, $\tilde{E}=-1.5$, $\tilde{F}=0.5$. Nonfundamental POs are labeled by their winding ratios.

Figure 4

The hierarchy of $N$-torus POs in the crossed-fields hydrogen atom at $\tilde{E}=-1.5$, $\tilde{F}=0.5$ (and at $\tilde{E}=-1.4$, $\tilde{F}=0.5$, except for the collapse on $S^{-}$). The two families of 3-torus POs $T_3^p$ and $T_3^n$ collapse onto their 2-torus partners $T_2^p$ and $T_2^n$ as ${\tilde{I}}_3\to 0$. The $T_2$ in turn collapse onto the FPOs $S^{+}$ and $S^{-}$. For a correct description of the collapse the appropriate coordinate system must be used: ${\tilde{I}}_1\to 0$ for $S^{-}$ and ${\tilde{I}}_1^{\prime }\to 0$ for $S^{+}$. The coordinate change is described by the topological invariants $M^p$ and $M^n$.

Figure 5

(Color online) The pattern of primitive and repeated POs for the first four series of 2-torus POs derived theoretically. The first series $w_1=1$ contains only primitive POs. The second series $w_1=2$ shows an alternating sequence of primitive POs (odd $w_2$) and repetitions of the POs in the first series (even $w_2$). (It cannot be derived from the multiplication principle if there is an additional primitive PO at the lower or upper end of the second series.) In the third series there are blocks of two primitive POs ($w_2$ indivisible by 3) separated by threefold repetitions of the first series POs ($w_2$ divisible by 3). The fourth series contains primitive POs for odd values of $w_2$, fourfold repetitions of the first series for positions with $w_2$ divisible by fourfold and twofold repetitions of the second series for $w_2$ divisible by 2 but not by 4. The scheme can be extended to arbitrarily large values of $w_1$.

Figure 6

(Color online) The pattern of primitive and repeated POs for the first four series of the $T_2^p$ (a) and the $T_2^n$ (b). POs in the first series and their repetitions are shown with red circles, POs in the second series and their repetitions with green diamonds and POs from the third and fourth series in blue squares and black triangles, respectively. The sequence of primitive and repeated POs within the individual series should be compared to the pattern shown in Fig. 5, which was derived from the multiplication principle. The numerical results agree perfectly with the theoretical predictions.

Figure 7

(Color online) Fourier spectra (absolute values of Fourier coefficients) of $\tilde{x}\left(\tilde{t}\right)$ for some POs of the $T_2^p$ family. Orbits are labeled by winding numbers calculated in two different coordinate systems (see text). The right-hand column shows the orbits in configuration space and the position of the nucleus.

Figure 8

(Color online) Fourier expansions of $\tilde{x}\left(\tilde{t}\right)$ (red) and $\tilde{z}\left(\tilde{t}\right)$ (green) for POs from the $T_2^n$ family. Peaks not associated with winding numbers are located at integer linear combinations thereof [37].

Figure 9

(Color online) Fourier spectra of $\tilde{x}\left(\tilde{t}\right)$ (red) and $\tilde{z}\left(\tilde{t}\right)$ (green). The second winding number is given in the unprimed coordinate system $w_2^0=2$. (a) A 2-torus PO in the $T_2^n$ family and (b) its 3-torus partner from $T_3^n$. (c) A 2-torus PO from the $T_2^p$ family and (d) its 3-torus partner from $T_3^p$.

Figure 10

(Color online) Elliptic (green) and hyperbolic (blue) POs with winding numbers 1:2 (a) in the $T_2^p$ family and (b) in the $T_2^n$ family.

Figure 11

(Color online) The actions ${\tilde{I}}_1$ and ${\tilde{I}}_2$ for the $T_2^p$ (red plus symbols) and $T_2^n$ (green triangles). (a) Actions at $\tilde{E}=-1.5$, $\tilde{F}=0.5$ in the unprimed action-angle coordinate system with $w_2^0=2$. The winding numbers are suitable to describe the approach to $S^{-}$, where ${\tilde{I}}_1$ tends to zero. The stability angles of the FPO $S^{+}$ have been transformed using Eq. (17). (b) The same data in the primed coordinate system with $w_2^0=4$. The situation around the FPOs is reversed. (c) Actions at $\tilde{E}=-1.4$, $\tilde{F}=0.5$ (unprimed coordinates). The $T_2^{p,n}$ do not collapse onto $S^{-}$ due to the ionizing region around that FPO. (d) Actions at $\tilde{E}=-1.4$, $\tilde{F}=0.5$ (primed coordinates).

Figure 12

(Color online) (a) Frequency map for $N$-torus POs at $\tilde{E}=-1.5$, $\tilde{F}=0.5$. The 3-torus POs $T_3^p$ (blue crosses) and $T_3^n$ (magenta diamonds) are bounded by the corresponding 2-torus POs $T_2^p$ (red plus symbols) and $T_2^n$ (green triangles). The 2-torus POs in turn are bounded by the FPOs $S^{\pm }$ (black dots). (b) The $T_3^p$ at $\tilde{E}=-1.4$, $\tilde{F}=0.5$ with some of the most prominent resonance lines highlighted. They intersect in the points $P_i=(\frac{1}{i},\frac{1}{i})$.

Figure 13

(Color online) Action variables for the $T_3^p$ family at $\tilde{E}=-1.4$, $\tilde{F}=0.5$. At the lower boundary, where the $T_3^p$ are bounded by the $T_2^p$, ${\tilde{I}}_3$ tends to zero while ${\tilde{I}}_{1,2}$ tend toward the values obtained from the 2-torus POs. For clarity the dotted line $\beta$ from Fig. 12b has been subtracted on the vertical axis.

Figure 14

(Color online) (a) The EBK spectrum as obtained from the quantization of the $T_3^p$ (upper half, red) at $\tilde{E}=-1.4$ and $\tilde{F}=0.5$ compared to the exact quantum-mechanical spectrum (lower half, green). The first three $n_2$ manifolds are labeled with ($n_1$, $n_2$, $n_3$). (b) Spectra at $\tilde{E}=-1.5$, $\tilde{F}=0.5$.

Figure 15

(Color online) The quantization of the $T_3^n$ family (blue) gives the two leftmost peaks in the $n_2=10$ and $n_1+n_3=10$ ($n=10$, $q=1$) subseries showing a good agreement with the exact quantum spectrum (green). The other peaks result from the quantization of the $T_3^p$ (red). The slight discrepancies of the peaks on the right-hand side are due to numerical inaccuracies in the interpolation procedure.