Atomic Siegert states in an electric field: Transverse momentum distribution of the ionized electrons

The Siegert states of atoms in a static uniform electric field, defined as the solutions to the stationary Schrödinger equation satisfying the regularity and outgoing-wave boundary conditions, are discussed. An efficient method to calculate not only the complex energy eigenvalue, but also the eigenfunction for a general class of one-electron atomic potentials is introduced. An exact expression for the transverse momentum distribution of the ionized electrons in terms of the Siegert eigenfunction in the asymptotic region is derived. The method is illustrated by calculations of the energy, ionization width, and transverse momentum distribution as functions of the electric field for several lowest states of H, outer $p$ shells of Ne, Ar, Kr, and Xe, and the active electron in ${\mathrm{H}}^{-}$. We also discuss the ionization of Ar by the pulse of a unidirectional time-dependent electric field, which illustrates the role of the Siegert states in the recently developed adiabatic theory of ionization of atoms by intense laser pulses [O. I. Tolstikhin et al., Phys. Rev. A 81, 033415 (2010)].

Figure 1

The energy $\mathcal{E}$ and width $\Gamma$ for four lowest Siegert states of H labeled by parabolic quantum numbers $(n_{\xi },n_{\eta },m)$. Solid lines: exact results. Dashed lines: the second-order perturbation theory [21], for $\mathcal{E}$, and semiclassical asymptotics [21, 24], for $\Gamma$.

Figure 2

The transverse momentum distributions for the same four states of H as shown in Fig. 1. The normalization factor $\mathcal{N}$ is defined by Eq. (45). Thick dashed lines: the weak-field limit results from Eq. (43).

Figure 3

The energy $\mathcal{E}$ and width $\Gamma$ for the Siegert states of Ne$(2p)$, Ar$(3p)$, Kr$(4p)$, and Xe$(5p)$ in the single-active-electron approximation with the model potential defined by Eqs. (46) and (47), see Refs. [57, 58]. The results labeled by ${\text{Kr}}^{\prime }(4p)$ are obtained with slightly modified parameters of the potential.

Figure 4

The transverse momentum distributions for the same four atoms as shown in Fig. 3. The normalization factor $\mathcal{N}$ is defined by Eq. (45). Thick dashed lines: the weak-field limit results from Eq. (41).

Figure 5

The asymptotic coefficients $f_{\nu }$ for Ar$(3p)$.

Figure 6

The energy $\mathcal{E}$ and width $\Gamma$ for the Siegert state of ${\mathrm{H}}^{-}$ in the single-active-electron approximation with the model potential defined by Eq. (48).

Figure 7

The transverse momentum distributions for ${\mathrm{H}}^{-}$. The normalization factor $\mathcal{N}$ is defined by Eq. (45). Thick dashed line: the weak-field limit results from Eq. (41).

Figure 8

(a) The photoelectron momentum distribution $P(k_{\perp },k_z)$ for Ar$(3p)$ generated by a half-cycle pulse (49) with $F_0=0.414$ and $T=45.48$ obtained by solving the time-dependent Schrödinger equation. (b) The corresponding normalized photoelectron momentum distribution $\mathrm{P̃}(k_{\perp },k_z)$ defined by Eq. (51). (c) The normalized photoelectron momentum distribution in the adiabatic approximation obtained from Eqs. (51) and (53).