Dynamical core polarization of two-active-electron systems in strong laser fields
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec 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 thatthe ionization rate is overestimated in SAE approximation. A modified single-active-electron modelis formulated by taking into account of the dynamical core polarization. Applying the new approachto Ca atoms, it is found that the polarization of the core can be considered instantaneous and thelarge polarizability of the cation suppresses the ionization by 50% while the photoelectron cut-offenergy increases slightly. The existed tunneling ionization formulation can be corrected analyticallyby considering core polarization.
PACS numbers: 33.80.Rv, 42.50.Hz, 42.65.Re
Various of non-perturbative phenomena occurring dur-ing atom-laser interactions are started with single ion-ization, e.g., above threshold ionization (ATI) and highharmonic generation (HHG). Although they have beensuccessfully interpreted by the rescattering model basedon single active electron (SAE) approximation (see Re-views e.g., [1, 2]), detailed examination showed that mul-tielectron effects are embedded in the photon and elec-tron spectra [3–9]. It is found that high-order harmonicgeneration (HHG) from molecules records interference ofdifferent channels suggesting more than one molecularorbitals are involved [3] and electron rearrangement isoccuring [4], which is certainly beyond the scope of theSAE theory. On the other hand, two-electron events suchas non-sequential double ionization can not be explainedeither without considering the electron-electron interac-tion [10]. It is thus desirable to examine in details themultielectron effects occurring in the ionization of atomicsystems beyond SAE.The single ionization of atoms in strong laser fieldscan be pictured as tunneling of one electron through thebarrier formed by the atomic potential and the laser-atom dipole interaction. The Keldysh parameter mea-sures the ratio of tunneling time to the optical period, γ = p I p / U p , where I p = κ / U p = E / ω is the ponderomotive energyof a free electron in a laser field of strength E and fre-quency of ω . When γ <
1, tunnel ionization occurs sorapid that the electric field can be considered as a staticfield at each instant. The so-called adiabatic approxi-mation is the root of Ammosov-Delone-Krainov (ADK)-like theories [11] for obtaining ionization rates. Based onthis picture, the rate is mainly determined by the unit-less quantity κ /E with κ representing the atomic fieldstrength at the classical radius of the electron motion.It is obvious that the adiabatic approximation willbreak down if the atomic potential acting on the tun-neling electron is varing sooner than the tunneling time.For more than one electron systems, the core can be po-larized by the laser fields, hence the atomic potential is time-varying. In the case of absence of resonant exci-tation, the polarization is instantaneously following thelaser field. One therefore expects that ionization ratesfrom single-active electron theory needed to be correctedby taking the dynamical core-polarization (DCP) intoaccount [12]. Recently we have incorporated the DCPinto simulations [13] successfully interpreting the exper-imentally measured alignment-dependent ionization rateof CO molecules [14]. In this work, we further inves-tigate the effects of DCP on the photoelectron spectraof alkali-earth atoms that have two strongly correlatedvalence electrons. In particular, we benchmark the var-ious related theories by comparison with exact solutionof the time-dependent Schr¨odinger equation (TDSE) fora model hydrogen molecule with both electrons movingin one dimension.We start with the SAE approximation and then takeinto account of the multielectron symmetry [15–18] andthe core-polarization induced by laser fields. For a N-e − system interacting with laser fields, the valence elec-trons will be strongly perturbed compared to the innerelectrons. After the liberation of one electron, the ionbecomes tighter bounded giving rise to higher secondaryionization potential. Therefore the SAE approximationis usually adopted assuming the ionic core is frozen. Theeffective TDSE for the active electron in a laser field takesthe form of i ∂∂t ψ = [ − ∇ V n + V L ] ψ. (1)where V L = ~r · ~E is the interaction of the active elec-tron with the external laser field ~E and V n is the ef-fective potential from the frozen core (atomic units areused throughout unless indicated otherwise). One of theapproaches to obtain the effective potential is approxi-mating the Hartree-Fock potential in the local densityapproximation, that gives the correct asymptotic behav-ior of V n → − r as the active electron is detached from theatomic system. The initial wave function can be takenas the the Hartree-Fock orbital of the valence electron.We will refer to this treatment as the SAE theory.The SAE theory assumes the electrons can be distin-guished as the active electron and the core electrons. Al-though the static (both Coulombic and exchange) poten-tials from the core electrons are taken into account, theantisymmetrization of the total wave function due to thePauli exclusion principle is disregarded in the dynamicsdriven by external fields. It can be partly remedied byrequiring the wavefunction ψ ( ~r, t ) orthogonal to the oc-cupied orbitals during the time propagation, therefore formany-electron systems, the occupied orbitals by the coreelectrons limit the configuration space that the activeelectron can occupy. We refer this treatment as SAE+Otheory.Another shortcoming of SAE theory is that it failswhen the dynamic response of the core electrons comesinto play, such as for systems that have more than oneweakly bounded electrons. The interplay between elec-trons would lead to complex multielectron effects includ-ing multiorbital (multichannel) and multipole effects [19].Here we focus on the effect of the adiabatic evolutionor polarization of the ionic core induced by the exter-nal laser field. Within the adiabatic approximation, it ispossible to derive an effective Hamiltonian of the activeelectron which takes into account of the laser-inducedcore polarization [20–22]. We give a brief description inthe follows.Denoting the polarizability tensor of the atomic coreas ˆ β + , the induced dipole moment is given by ~d = ˆ β + ~E ,where ~E is the external laser field. For symmetric atomiccore, the polarizability is uniform in all direction, theinduced dipole moment is parallel to the external fieldand the potential due to laser-induced core polarizationis given by [20, 22] V cp = − β + ~E · ~rr . (2)When the active electron is close to the atomic core, theform of polarization potential is not valid because of theelectron screening. Therefore the polarization potentialis cut to zero below r that is estimated from the atomicpolarizability ( ≈ r ) [13, 20]. It can be seen that themagnitude of the potential from the polarized core is pro-portional to the strength of the external electric field. Instrong field regime, it is comparable or larger than the in-teraction of the active electron with the permanent dipolemoment of the atomic core if it exists.The effective TDSE for the active electron turns into i ∂∂t ψ = [ − ∇ V n + V cp + V L ] ψ. (3)The method of direct propagating Eq. 3 will be namedas SAE+CP. Similar to the SAE theory discussed pre-viously, the initial wave function is taken as the theHartree-Fock orbital of the valence electron. Note in thistheory, we neglect the polarization of the core induced by the Coulombic field of the outer electron as well as thepermanent dipole moment.Different from the theories presented above, the time-dependent Hartree-Fock theory in principle takes all elec-trons into account based on the mean-field approxima-tion. We limit ourselves to the case of two valence elec-trons that forms a singlet state and keep the other N-2electrons frozen. Restricting the two electrons occupyingthe same orbital, and using the effective potential fromthe other N-2 electrons which forms the closed-shell core,we have the following nonlinear equation, i ∂∂t ψ = [ − ∇ V N − + (cid:28) ψ | r | ψ (cid:29) + V L ] ψ. (4)where V N − is the effective potential from the core con-stituted by the other N-2 electrons, which has asymptoticbehavior as − r . This method will be referred as time-dependent restricted Hartree (TDRH) method. The re-pulsive Coulomb potential from the other valence elec-tron is evaluated at each time as (cid:28) ψ | r | ψ (cid:29) = Z d ~r | Ψ( ~r , t ) | | ~r − ~r | (5)which includes the induced polarization from the inter-action of the laser field with the other valence electron.Here we have made a crude assumption that the twovalence electrons have the same time-dependent orbital.Note that if the potential in Eq. 5 is evaluated with theinitial field-free Hartree-Fock valence orbital, we againobtain the TDSE given in Eq. 1.Now we apply those theories to the ionization of alkali-earth atom Ca by a laser field at wavelength of 1600 nm,intensity of 1 × W/cm . The laser pulse has a du-ration of 15fs with a Gaussian envelop. The Ca atomhas a configuration of 1 s s p s p s with two va-lence electrons outside a closed-shell. The hartree-Fockcalculation gives ionization potential of 0.1955 a.u.. Thepolarizability of Ca is found of 154 a.u. After obtain-ing the effective potential from HF calculation using thelocal density approximation, we perform the SAE calcu-lation and obtain the similar ionization energy at 0.1947a.u. The ponderomotive energy U p is about 3.08 timesof photon energy, and the Keldysh parameter is close to1, therefore tunneling ionization dominates.The equations of motion, Eq. (1, 3, and 4) are solvedrespectively in a spherical box of 1600 a.u. with 20 par-tial waves using pseudospectral grid and split-operatorpropagation method [23, 24]. The photoelectron spectraare obtained by projecting the final wavefunction to thecontinuum states P E = X l =1 |h φ El | ψ ( T f ) i| (6)where φ El is the energy normalized continuum state ofgiven energy E and angular momentum l . The single-electron ionization probability is defined as p i = R P E dE, Photoelectron energy ( ω) -6 -5 -4 -3 -2 -1 I on i za ti on p r ob a b ilit y SAESAE+OSAE+CPTDRH10U p p FIG. 1: Photoelectron spectra of Ca atom calculated in var-ious approximations. The arrows indict the maximum energyof directly escaped and rescattered electron, with U p as theponderomotive energy. and the total ionization probability of the system is givenby P I = 1 − (1 − p i ) , where independent particle approx-imation is applied.In Fig. 1, we present the photoelectron energy spectrafor Ca calculated with the various theories. All calcula-tions capture the main features of above threshold ioniza-tion that there exists two main peaks located at 2 U p and10 U p . The 2 U p peak corresponds to those electrons di-rectly escape from the atom after tunnelling through thebarrier formed by the atomic potential and the dipole in-teraction with the laser field. The 10 U p is the maximumenergy that the tunnnelling electron can gain after rescat-tered backward by the atomic core [25]. In the SAE+Ocalculation, we keep the outer electron wavefunction or-thogonal to the other occupied orbitals, which impliesthat the core can not be penetrated. However the spec-tra intensity shows only a little increasing compared tothe SAE calculation.When the core is polarized by the laser fields, it takesmore energy for the electron to move from inside thecore to the outer region. However once it tunnels outthe barrier, the polarization potential repels the electronaway resulting slightly increase of the cut-off energy asdemonstrated by the comparison of SAE results with theSAE+CP results multiplied by 2 shown in Fig. 1. We seethat the core polarization causes the suppression of ion-ization rates and marginally increases the maximum en-ergy that electrons can gain. The reason for the latter liesin the rescattering electron dynamics [25, 26]. Based onthe classical trajectory analysis, the laser field is close tozero when the electron collides the atomic core with max-imum kinetic energy. The corresponding instantaneouspolarization of the atomic core is small thus it makeslittle impact on the cut-off energy of the photoelectron.On the other hand, it can be expected that the harmonicspectra can be modified by the core-polarization due todifferent recombination instants and the resulted polar-ization as shown in [12]. Note that we use the available polarizability of the neu-tral atom found in database. The polarizability of theCa + is about 20% smaller which makes small change tothe ionization ratio. We also perform a different check bycalculating the time-dependent induced dipole momentfrom propagating the Ca + in the same laser field andthen use it in the simulation of the outer electron dynam-ics. The results show almost no difference suggesting thepolarization is indeed instantaneous, therefore the phaselag due to different time response of the outer electronand the atomic core to the laser field makes no effect inthe present study. However, this might be not true whenresonant excitation is present which is beyond the scopeof the present study. To further justify the considerationof the core polarization, TDRH calculation is performedas well. The resulted spectra shown in Fig. 1 is very closeto that computed from SAE+CP calculation indicatingthat the core polarization can not be neglected for thelaser intensity we considered.The suppression of ionization rate due to the core po-larization can be understood within the tunneling pic-ture. The potential barrier along the laser polarizationdirection is given by V ( z ) = V n ( z ) − Ez + V cp (7)where V cp = β + E/z . In the standard theory of tun-nel ionization, the core polarization is not taken intoaccount, and the potential barrier takes the form of V ( z ) = V n ( z ) − Ez . According to the ADK model [11],the tunnel ionization rate when disregarding the core po-larization can be estimated by w ≈ exp (cid:20) − Z z z p κ + 2 V ( z ) dz (cid:21) (8)where κ is related to the ionization potential I p by I p = κ /
2, and z and z are the inner and outer turningpoints respectively. When the core polarization is takeninto account, the correction of the ionization rate can beapproximated by [21, 27–29] R c = w/w ≈ exp (cid:20) − Z z z V cp κ dz (cid:21) (9)The potential barrier of Ca atom in the laser field isplotted in Fig. 2. The polarization potential is cut tozero below r = 5 a.u. that is estimated from the atomicpolarizability ( ≈ r ) and is consistent with the size ofCa atom. Taking z = r as the inner turning point.The outer turning point is determined by I p /E . At thelaser intensity of 10 W/cm , the outer turning point is12 a.u. from the nucleus. According to Eq. 9, we foundthe correction factor is 0.386, while the numerical simu-lation shows that total photoelectron spectra intensity inSAE+CP calculation is about 52% of that obtained fromSAE calculation in good agreement with the analyticalcorrection. -20 -15 -10 -5 0 5 10 15 20 25 30 z (a.u.) -0.5-0.4-0.3-0.2-0.100.1 P o t e n ti a l ( a . u . ) V n (z)V n (z)-EzV n (z)+V p (z)-Ez FIG. 2: Illustration of the effective potentials for the cases oflaser-free (solid line), laser field included (dashed lines) andcore-polarization considered (dotted lines).
Laser intensity (I ) -3 -2 -1 I on i za ti on p r ob a b ilit y TAESAESAE+CPTDRH 4 6 8 10 12 14
Laser intensity (I ) -0.6-0.30.00.30.6 R e l a ti v e r a ti o FIG. 3: Ionization probability of a model Hydrogen moleculecalculated from exact two-electron calculation (TAE, filledsquares), one-electron TDSE using the effective potential(SAE, dotted lines with open circles), modified SAE calcu-lation by considering the core-polarization (SAE+CP, solidline with open squares) and TDRH (solid line with diamonds).The inset displays the relative ratios comparing to the TAEcalculation from the other three theories.
In order to further check the validity of our theory, weperform exact two-active-electron (TAE) calculation fora model hydrogen molecule with both electrons moving inone dimension. The soft-core Coulomb potential has theform of | V ( x ) | = √ ǫ + x with ǫ N = 0 . ǫ e = 1 . I ( I = 10 W/cm ), the ionization probabil-ity obtained from TDRH calculation is higher than thatfrom the exact TAE calculation. The failure of TDRH atlow laser intensity might be caused by two facts. Firstly,the HF ground state energy differs from the exact twoelectron energy by the correlation energy (0.025 a.u.).When the ionization rate is less than the correlation en- ergy, it has been shown that the correlation of the twoelectrons has profound effects [31]. The threshold laserintensity at which the correlation can not be neglectedmight be estimated from that the dipole interaction en-ergy at the classical radius is in the order of the correla-tion energy. In the present studied system, the averageddistance h r i ≈ . . I . For laser intensities much higher,the difference of the ionization energy and the initial wavefunction in the restricted Hartree-Fock theory from theexact calculation can be neglected due to the strong in-teraction with laser filed. Secondly, however, the twoelectrons are treated as equivalent and independent par-ticles in TDRH which is not appropriate during ioniza-tion when the two electrons in fact can be distinguishedas the inner and the outer electron. After the removal ofthe first electron, it will be very difficult to further ionizethe ionic core. Therefore the total ionization probabilitydefined by P I = 1 − (1 − p i ) is not proper anymore andneeds to be replaced by p i neglecting the ionization ofsecond electron.For higher laser intensity, the correlation plays less rolein the ionization process. When the ionization rate ismuch larger than the correlation energy, the difference ofthe inner and outer electron orbital is not the main causeany more. At the laser intensity of 5 I , the TDRH givesthe same ionization probability to the TAE calculation.However, as laser intensity keeps increasing, the ratesfrom TDRH becomes smaller than the TAE results. Thereason lies in the assumption that the two electron areequivalent and are occupying the same time-dependentorbital. When ionization probability is large, less normfrom the orbital is bounded. The nuclear charge is lessshielded and the potential on the electrons become un-physical due to this self-interaction in the TDRH the-ory such that further ionizing is incorrectly suppressed.Therefore TDRH theory fails in both low and high laserintensities.As shown in Fig 3, the rates from the SAE calculationare larger than the TAE results for the laser intensityconsidered. When introducing the core-polarization po-tential, we see that the deviation is reduced to less than20% for laser intensity < I . Noted for laser intensityabove 8 I , the ionization mechanism switches from tun-nel ionization to over the barrier ionization (OTB). Inthe OTB regime, the removal of one electron is so rapid,there is no time for the two electrons exchanging energysuch that SAE theory works better as shown in the insetof Fig 3.In conclusion, we have shown that it is necessaryto consider core polarization in strong field ionizationof multielectron atoms. By incorporating the polariza-tion potential into theory the tunneling ionization the-ory can be improved within the frame of SAE. Compar-ing with exact two-electron model calculations and time-dependent restricted Hartree-Fock calculation, we showthat single-active electron approximation overestimatesthe ionization rate due to the additional barrier from thedipole potential. The maximum photoelectron energy isfound increased slightly by the core-polarization.This work is supported by the National Basic Re-search Program of China (973 Program) under grantno 2013CB922203, the Major Research plan of NationalNSF of China (Grant No. 91121017) and the NSF ofChina (Grants No. 11374366, No. 11104352). Z. X.acknowledges Dr. Xiaojun Liu for helpful discussions. ∗ Electronic address: [email protected][1] M. Protopapas, C. H. Keitel, and P. L. Knight, Rep.Prog. Phys. , 389 (1997).[2] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545(2000).[3] O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich,D. Villenenve, P. Corkum, and Y. Ivanov, Nature ,972 (2009).[4] Y. Mairesse, et al., Phys. Rev. Lett. , 213601 (2010).[5] M. Spanner, J. Mikosch, A. Gijsbertsen, A. E. Bo-guslavskiy, and A. Stolow, New Journal of Physics ,093001 (2011).[6] A. D. Shiner et al., Nat. Phys. , 464 (2011).[7] A. E. Boguslavskiy et al., Science , 136 (2012).[8] B. Bergues et al., Nat. 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