Dynamical core polarization of two-active-electron systems in strong laser fields Zengxiu Zhao∗ and Jianmin Yuan Department of Physics, National University of Defense Technology, Changsha 410073, P. R. China (Dated: July 25, 2018) The ionization of two-active-electron systems by intense laser fields is investigated theoretically. In comparison with time-dependent Hartree-Fock and exact two electron simulation, we show that the ionization rate is overestimated in SAE approximation. A modified single-active-electron model is formulated by taking into account of the dynamical core polarization. Applying the new approach to Ca atoms, it is found that the polarization of the core can be considered instantaneous and the large polarizability of the cation suppresses the ionization by 50% while the photoelectron cut-off energy increases slightly. The existed tunneling ionization formulation can be corrected analytically by considering core polarization. PACS numbers: 33.80.Rv, 42.50.Hz, 42.65.Re Various of non-perturbative phenomena occurring dur- time-varying. In the case of absence of resonant exci- ing atom-laser interactions are started with single ion- tation, the polarization is instantaneously following the ization, e.g., above threshold ionization (ATI) and high laser field. One therefore expects that ionization rates harmonic generation (HHG). Although they have been from single-active electron theory needed to be corrected successfully interpreted by the rescattering model based by taking the dynamical core-polarization (DCP) into on single active electron (SAE) approximation (see Re- account [12]. Recently we have incorporated the DCP views e.g., [1, 2]), detailed examination showed that mul- into simulations [13] successfully interpreting the exper- tielectron effects are embedded in the photon and elec- imentally measured alignment-dependent ionization rate tron spectra [3–9]. It is found that high-order harmonic of CO molecules [14]. In this work, we further inves- generation (HHG) from molecules records interference of tigate the effects of DCP on the photoelectron spectra different channels suggesting more than one molecular of alkali-earth atoms that have two strongly correlated orbitals are involved [3] and electron rearrangement is valence electrons. In particular, we benchmark the var- occuring [4], which is certainly beyond the scope of the ious related theories by comparison with exact solution SAE theory. On the other hand, two-electron events such of the time-dependent Schr¨odinger equation (TDSE) for as non-sequential double ionization can not be explained a model hydrogen molecule with both electrons moving either without considering the electron-electron interac- in one dimension. tion [10]. It is thus desirable to examine in details the We start with the SAE approximation and then take multielectron effects occurring in the ionization of atomic into account of the multielectron symmetry [15–18] and systems beyond SAE. the core-polarization induced by laser fields. For a N- The single ionization of atoms in strong laser fields e− system interacting with laser fields, the valence elec- can be pictured as tunneling of one electron through the trons will be strongly perturbed compared to the inner barrier formed by the atomic potential and the laser- electrons. After the liberation of one electron, the ion atom dipole interaction. The Keldysh parameter mea- becomes tighter bounded giving rise to higher secondary sures the ratio of tunneling time to the optical period, ionization potential. Therefore the SAE approximation 2 is usually adopted assuming the ionic core is frozen. The γ = Ip/2Up, where Ip = κ /2 is the ionization po- 2 2 effective TDSE for the active electron in a laser field takes tentialp and Up = E /4ω is the ponderomotive energy of a free electron in a laser field of strength E and fre- the form of 2 quency of ω. When γ < 1, tunnel ionization occurs so ∂ ∇ arXiv:1312.4055v1 [physics.atom-ph] 14 Dec 2013 i ψ = [− + V + V ]ψ. (1) rapid that the electric field can be considered as a static ∂t 2 n L field at each instant. The so-called adiabatic approxi- where VL = ~r · E~ is the interaction of the active elec- mation is the root of Ammosov-Delone-Krainov (ADK) tron with the external laser field E~ and Vn is the ef- -like theories [11] for obtaining ionization rates. Based on fective potential from the frozen core (atomic units are this picture, the rate is mainly determined by the unit- 3 3 used throughout unless indicated otherwise). One of the less quantity κ /E with κ representing the atomic field approaches to obtain the effective potential is approxi- strength at the classical radius of the electron motion. mating the Hartree-Fock potential in the local density It is obvious that the adiabatic approximation will approximation, that gives the correct asymptotic behav- 1 break down if the atomic potential acting on the tun- ior of Vn →− r as the active electron is detached from the neling electron is varing sooner than the tunneling time. atomic system. The initial wave function can be taken For more than one electron systems, the core can be po- as the the Hartree-Fock orbital of the valence electron. larized by the laser fields, hence the atomic potential is We will refer to this treatment as the SAE theory. 2 The SAE theory assumes the electrons can be distin- the Coulombic field of the outer electron as well as the guished as the active electron and the core electrons. Al- permanent dipole moment. though the static (both Coulombic and exchange) poten- Different from the theories presented above, the time- tials from the core electrons are taken into account, the dependent Hartree-Fock theory in principle takes all elec- antisymmetrization of the total wave function due to the trons into account based on the mean-field approxima- Pauli exclusion principle is disregarded in the dynamics tion. We limit ourselves to the case of two valence elec- driven by external fields. It can be partly remedied by trons that forms a singlet state and keep the other N-2 requiring the wavefunction ψ(~r, t) orthogonal to the oc- electrons frozen. Restricting the two electrons occupying cupied orbitals during the time propagation, therefore for the same orbital, and using the effective potential from many-electron systems, the occupied orbitals by the core the other N-2 electrons which forms the closed-shell core, electrons limit the configuration space that the active we have the following nonlinear equation, electron can occupy. We refer this treatment as SAE+O ∂ ∇2 1 theory. i ψ = [− + VN 2 + ψ| |ψ + VL]ψ. (4) ∂t 2 − r12 Another shortcoming of SAE theory is that it fails when the dynamic response of the core electrons comes where VN 2 is the effective potential from the core con- − into play, such as for systems that have more than one stituted by the other N-2 electrons, which has asymptotic 2 weakly bounded electrons. The interplay between elec- behavior as − r . This method will be referred as time- trons would lead to complex multielectron effects includ- dependent restricted Hartree (TDRH) method. The re- ing multiorbital (multichannel) and multipole effects [19]. pulsive Coulomb potential from the other valence elec- Here we focus on the effect of the adiabatic evolution tron is evaluated at each time as or polarization of the ionic core induced by the exter- 1 2 1 nal laser field. Within the adiabatic approximation, it is ψ| |ψ = d~r2|Ψ(~r2,t)| (5) r12 Z |~r − ~r2| possible to derive an effective Hamiltonian of the active electron which takes into account of the laser-induced which includes the induced polarization from the inter- core polarization [20–22]. We give a brief description in action of the laser field with the other valence electron. the follows. Here we have made a crude assumption that the two Denoting the polarizability tensor of the atomic core valence electrons have the same time-dependent orbital. as βˆ+, the induced dipole moment is given by d~ = βˆ+E~ , Note that if the potential in Eq. 5 is evaluated with the where E~ is the external laser field. For symmetric atomic initial field-free Hartree-Fock valence orbital, we again core, the polarizability is uniform in all direction, the obtain the TDSE given in Eq. 1. induced dipole moment is parallel to the external field Now we apply those theories to the ionization of alkali- earth atom Ca by a laser field at wavelength of 1600 nm, and the potential due to laser-induced core polarization 13 2 is given by [20, 22] intensity of 1 × 10 W/cm . The laser pulse has a du- ration of 15fs with a Gaussian envelop. The Ca atom 2 2 6 2 6 2 β+E~ · ~r has a configuration of 1s 2s 2p 3s 3p 4s with two va- V = − . (2) cp r3 lence electrons outside a closed-shell. The hartree-Fock calculation gives ionization potential of 0.1955 a.u.. The When the active electron is close to the atomic core, the polarizability of Ca is found of 154 a.u. After obtain- form of polarization potential is not valid because of the ing the effective potential from HF calculation using the electron screening. Therefore the polarization potential local density approximation, we perform the SAE calcu- is cut to zero below r0 that is estimated from the atomic lation and obtain the similar ionization energy at 0.1947 3 polarizability (≈ r0) [13, 20]. It can be seen that the a.u. The ponderomotive energy Up is about 3.08 times magnitude of the potential from the polarized core is pro- of photon energy, and the Keldysh parameter is close to portional to the strength of the external electric field.