Nondispersive wave packets in planar helium

Two-electron wave packets found in driven planar helium follow a classical periodic orbit without spreading and are localized in nonlinear resonance islands of the classical phase space. The general mechanism that produces these nondispersive wave packets is the near-resonant coupling between highly correlated, asymmetric, and doubly excited states of the atom with an external periodic electromagnetic field. We provide a full characterization of these two-electron nondispersive wave packets found in the Floquet spectrum of driven planar helium. This includes the characteristic energy spectrum, relatively long lifetimes, and the identification of the resonance states of helium that are responsible for their formation. This is achieved with the help of an efficient quantum treatment of driven planar helium which combines Floquet theory, complex rotation, and the representation of the Hamiltonian in a set of four coupled harmonic oscillators.

Figure 1

Schematic block structure of the Hamiltonian matrices $\mathbf{A}$ (left) of the eigenvalue problem (27) and $\tilde{\mathbf{A}}$ (right) of the eigenvalue problem (31).

Figure 2

Phase space of the outer electron in the collinear frozen planet configuration.

Figure 3

Left: Conditional probability density (${\theta }_{12}=0$) of the lowest ${}^{1}S$ frozen planet states of the $N=4$ series in planar helium. For the ground FPS, the maximum of the probability density is localized near the equilibrium position $x_{\min }$ of the outer electron in the classical configuration [Eq. (33)]. For the excited FPS, the maximum of the probability density along the inner electron axis $r_2$ remains at the same position while the maximum along the outer electron axis $r_1$ increases with $n_{\mathrm{F}}$. Right: Projections in configuration space for the electronic density of the ${}^{1}S, N=4$ frozen planet states. In each plot, the left part corresponds to the inner electron density for a fixed position of the outer electron, while the right part shows the outer electron density for a fixed position of the inner electron. The value for the position of the fixed electron is given by the maximum of the corresponding conditional probability density displayed in the left panels.

Figure 4

Projections in phase space for the electronic density of the ${}^{1}S$ frozen planet states of the $N=6$ series in planar helium. For the ground FPS ${}^{1}S{\mathcal{F}}_1^6$ the maximum of the probability density is localized at the equilibrium position of the outer electron in the classical FPC shown in Fig. 2, while for the excited FPS the maximum of the probability density is localized along periodic orbits with higher energy of the classical configuration.

Figure 5

Phase space of the driven frozen planet configuration for fixed field frequency $\omega =0.2{(N-0.5)}^{-3}$ a.u. and three different field amplitudes: $F=0$ a.u. (left), $F=0.001{(N-0.5)}^{-4}$ a.u. (center), and $F=0.005{(N-0.5)}^{-4}$ a.u. (right).

Figure 6

Phase space of the driven frozen planet configuration for field amplitude $F=0.005{(N-0.5)}^{-4}$ a.u., frequency $\omega =0.2{(N-0.5)}^{-3}$ a.u., and variable driving field phases $\omega t$.

Figure 7

Complex energy spectrum of the unperturbed helium atom for triplet states in the energy region below the sixth ionization threshold, obtained after diagonalization of (17) for $n_{\mathrm{base}}=250$ and several angular momenta.

Figure 8

Husimi distributions of the triplet wave packets ${\mathcal{W}}_1^6, {\mathcal{W}}_2^6$, and ${\mathcal{W}}_3^6$ obtained for field parameters $F=5.5\times {10}^{-6}$ a.u. and $\omega =0.0012$ a.u., depicted for different phases of the driving field. For comparison, we show the classical phase-space structure obtained for the same field parameters (bottom).

Figure 9

Projections in configuration space of the electronic density of the same wave packets presented in Fig. 8. In each plot, the left part shows the inner electron density, while the right part shows the outer electron density.

Figure 10

Overlaps of the wave packet ${\mathcal{W}}_2^6$ with the states of the atomic basis organized according to angular momentum $L$ (top) and the Floquet number $k$ (bottom). Driving field parameters: $F=5.5\times {10}^{-6}$ a.u. and $\omega =0.0012$ a.u.

Figure 11

Behavior of the energies $E$ (left) and decay rates $\mathrm{\Gamma }$ (right) as a function of the field amplitude $F$, for driving frequency $\omega =0.0012$ a.u. Green, blue, and red circles identify the wave packets ${\mathcal{W}}_1^6, {\mathcal{W}}_2^6$, and ${\mathcal{W}}_3^6$, respectively. For field amplitudes $F>5.7\times {10}^{-6}$ a.u., there are no Floquet eigenstates with the largest overlap with the FPS ${}^{3}S{\mathcal{F}}_1^6$.

Figure 12

Overlaps between the triplet wave packet ${\mathcal{W}}_2^6$ and the elements in the basis. The left plot shows the overlaps as a function of the field amplitude $F$ for fixed frequency $\omega =0.0012$ a.u., while in the right plot we present the overlaps as a function of the field frequency $\omega$ for a fixed amplitude $F=5.5\times {10}^{-6}$ a.u. The green regions identify the field amplitude (left) and frequency (right) intervals for which the Husimi distribution of the wave packet ${\mathcal{W}}_2^6$ follows the classical trajectory as in Fig. 8.

Figure 13

Left plots display the overlaps between the wave packets obtained with the few level model and the atomic states in Table 4. In the right plot, we present the scheme of the essential states involved in the formation of the NDWP ${\mathcal{W}}_2^6, {\mathcal{W}}_3^6$, and ${\mathcal{W}}_4^6$. The dashed lines are separated by $\omega$, with energies (in atomic units) indicated in parentheses. ${}^{3}S$ (${}^{3}P$) states are highlighted in blue (orange).

Figure 14

Comparison between the unperturbed states ${}^{3}P{\mathcal{F}}_1^6$ and ${}^{3}P{\mathcal{G}}_1^6$. The left plots show the projections of the electronic density in phase space, while the projections of the electronic density in configuration space are depicted as a function of the distances $r_1$ and $r_2$ (for ${\theta }_{12}=0$) in the center plots, and for the $x$ and $y$ coordinates (as in Fig. 3) in the right plots.

Figure 15

Husimi distributions of the “hyperbolic” (green) and “elliptic” (red) wave packets ${\mathcal{W}}_2^6$ for electromagnetic driving at frequency $\omega =0.0012$ a.u. and amplitude $F=5.5\times {10}^{-6}$ a.u., depicted for different phases of the driving field.

Figure 16

Energy of the “hyperbolic” (green) and “elliptic” (red) wave packets ${\mathcal{W}}_2^6$ obtained with the few level model (top). Overlaps between the hyperbolic wave packet ${\mathcal{W}}_2^6$ and the elements in the atomic basis (bottom). The results are presented for variable field amplitude with fixed frequency $\omega =0.0012$ a.u. (left) and variable field frequency with fixed amplitude $F=5.5\times {10}^{-6}$ a.u. (right).

Figure 17

Contour plot of the Husimi distribution of the triplet elliptic wave packets ${\mathcal{W}}_2^N$, obtained by diagonalization of Eq. (31) for the field parameters in Table 6, at different phases of the driving field.