Ab initio quantum approach to planar helium under periodic driving

We present a method for the accurate quantum treatment of the planar three-body Coulomb problem under electromagnetic driving. Our ab initio approach combines Floquet theory, complex dilation, and the representation of the Hamiltonian in suitably chosen coordinates, without adjustable parameters. The resulting complex-symmetric sparse banded generalized eigenvalue problem of rather high dimension is solved using advanced techniques of parallel programming. This theoretical and numerical machinery is employed to provide a complete description of the bound and of the doubly excited spectrum of the field-free two-dimensional (2D) helium atom. For the driven atom, we focus on the near resonantly driven frozen planet configuration, and give evidence for the existence of nondispersive two-electron wave packets.

Figure 1

Dependence of the ground-state energy of the 2D helium atom $(\gamma =1)$ in atomic units, on the dilation parameter $\alpha$, for $n_{\mathrm{base}}=60$ (a), and ground-state energy in a.u. as a function of the strength $\gamma$ of the electron-electron interaction (b). In (b), the dashed line is obtained from first-order perturbation theory, while the solid line represents our numerical results. The inset in (a) zooms into the apparent plateau of $E$ vs $\alpha$, in the vicinity of the actual minimum value of $E$ at $\alpha \simeq 0.36$.

Figure 2

(Color online) Complex energy spectrum of the rotated 2D helium Hamiltonian (2), for singlet states, ${\Pi }_x=+1$, and $\mid l\mid =0$. The data were obtained by several runs of the Lanczos algorithm, choosing the shift parameter [34, 62] close to the ionization thresholds $I_N$, $N=1\cdots 5$. Due to the truncation of the basis, $I_N^{\mathrm{eff}}<I_N$ [66]. In all cases $\theta =0.3$, while the parameters $\alpha$ and $n_{\mathrm{base}}$ have to be readjusted to obtain optimal convergence in the different spectral ranges. For the eigenvalues above the sixth ionization series we used $n_{\mathrm{base}}=300$, what implies a basis size of $76076$.

Figure 3

Energies (a) and decay rates (b) of the resonances of the 2D three-body problem, as a function of $\gamma$ for singlet exchange symmetry, ${\Pi }_x=+1$, and $\mid l\mid =0$, close to the 4th helium ionization threshold $(I_4=-8∕49\text{a.u.})$. The lowest resonance $(\circ )$ of the 5th Rydberg series exhibits several avoided crossings before crossing the 4th ionization threshold (indicated by the long dashed line), with associated dramatic enhancements of its decay rate at the crossings (note the logarithmic scale of the right-hand plot).

Figure 4

Energy (a) and decay rate (b), in atomic units, of the lowest resonance $N=6,n=6$ $(•)$ of the 6th Ryberg series for singlet exchange symmetry, ${\Pi }_x=+1$, and $\mid l\mid =0,$ under variation of the electron-electron interaction $\gamma$. This resonance exhibits a clear avoided crossing with one of the $N=4,n=9$ resonances $(\circ )$, at $\gamma \simeq 0.7$, where its decay rate is dramatically reduced.

Figure 5

Contour plots of the Husimi representation of the $n_F=1$ (b), $n_F=2$ (c), and $n_F=3$ (d) triplet frozen planet states (FPS) of the sixth series, compared with the classical phase space of the collinear frozen planet configuration (FPC) in (a) [16, 73]. The Husimi functions show perfect phase space localization: while the fundamental state (a) is localized at the minimum of the effective potential, at 2.6 scaled units [20, 75], the excited states are localized along frozen planet trajectories with higher energy, with outer turning points (at the maxima of the Husimi densities) at $4.2$ (c) and 5.5 (d) scaled units.

Figure 6

Conditional probability densities (with a fixed interelectronic angle ${\theta }_{12}=0$) of the first four triplet frozen planet states (FPS) of the sixth series $(N=6)$: (a) $n_F=1$; (b) $n_F=2$; (c) $n_F=3$; (d) $n_F=4$. The maxima of the electronic densities in the radial distance $r_2$ of the inner electron remain invariant, while the most likely position of the outer electron (at the outer turning point of its classical orbit) increases with $n_F$. The number of nodes in the radial coordinate $r_1$ is precisely given by $n_F-1$.

Figure 7

Contour plot of the Husimi distribution (A13) of two different wave-packet triplet eigenstates (top and middle) along a $N=6$ frozen planet trajectory of 2D helium, under electromagnetic driving at frequency $\omega =0.0012\text{a.u.}$ and amplitude $F=5.5\times {10}^{-6}\text{a.u.}$, projected onto the phase space component spanned by $x_1$ and $p_1$ (the position and momentum of the outer electron). For comparison, also the classical phase space structure of the restricted collinear problem [20] is shown (bottom), for the same driving field parameter. Clearly, the electronic density is associated with the chaotic phase space region, and is localized around the hyperbolic fix point of the 1:1 resonance.

Figure 8

Complex-valued Floquet spectrum of Eq. (17), for triplet states, within one Floquet zone of width $\omega$. The real parts of the resonance poles (crosses) correspond to the energies, the imaginary parts to half the decay rates of the atomic resonance states in the field. The state highlighted by a black spot and an arrow at $\mathrm{Re}\left(E\right)=-0.0731723\text{a.u.}$ exhibits the largest overlap with the third excited FPS of the $N=6$ series [see Fig. 5d]. $F=5.5\times {10}^{-6}\text{a.u.}$, $\omega =0.0012\text{a.u.}$, $n_{\mathrm{base}}=200$, $7$ photon blocks, $k_{\min }=-2$, $k_{\max }=4$, $l_{\max }=3$, $n_{\mathrm{tot}}=521795$.

Figure 9

Contour plot of the Husimi phase space distribution (A13) of the wave-packet singlet eigenstate (top) along a $N=6$ frozen planet trajectory of 2D helium, under electromagnetic driving at frequency $\omega =0.0012\text{a.u.}$ and amplitude $F=5.5\times {10}^{-6}\text{a.u.}$ The (restricted) phase space is spanned by the outer electron’s position $x_1$ and momentum $p_{x_1}$ along the driving field polarization axis. For comparison, also the classical phase space structure of the restricted collinear problem [20] is shown (bottom), for the same driving field parameters. The electronic density remains localized around the elliptic fix point of the driving-induced 1:1 resonance.

Figure 10

Electronic density of the $N=6$ triplet wave packet (Fig. 7) in configuration space, for different phases of the driving field: $\omega t=0$ [(a1), (b1), (c1)], $\omega t=\pi ∕2$ [(a2), (b2), (c2)], $\omega t=\pi$ [(a3), (b3), (c3)]. (a1)–(a3) show the electronic density as a function of the distances $r_1$ and $r_2$ of the electrons from the nucleus, for fixed interelectronic angle ${\theta }_{12}=0$. (b1)–(b3) depict the electronic density of the inner electron, when the outer electron is fixed at a distance $r_1\simeq 165\text{a.u.}$ along the $x$ axis—its outer turning point in (a1)–(a3). In (c1)–(c3) the probability density of the outer electron is shown, for the inner electron fixed at $r_2\simeq 23\text{a.u.}$ along the $x$ axis.

Figure 11

Electronic density of the $N=6$ triplet wave packet (Figs. 7, 10) in configuration space, for different phases of the driving field: $\omega t=0$ [(a1), (b1)], $\omega t=\pi ∕2$ [(a2), (b2)], $\omega t=\pi$ [(a3), (b3)]. (a1)–(a3) shows the electronic density of the outer electron for the inner electron fixed at a distance $r_2=23\text{a.u.}$ on the diagonal $y=x$. (b1)–(b3) represents the electronic density of the outer electron when the inner electron is delocalized over a circle with radius $r_2=23\text{a.u.}$ The $x$ profiles of (b1)–(b3) are shown in (c1)–(c3).