arXiv:1312.4055v1 [physics.atom-ph] 14 Dec 2013 ue h ai ftneigtm oteotclperiod, optical the to time mea- tunneling laser- parameter of γ the Keldysh ratio and The the potential sures interaction. atomic the dipole the through atom electron by one formed of tunneling barrier as pictured be can the details in examine SAE. atomic of beyond to ionization systems desirable the in thus occurring is effects multielectron It interac- [10]. electron-electron tion the explained considering be without not can either the ionization such of events double two-electron scope non-sequential hand, as the other the is beyond On certainly rearrangement theory. SAE is electron which and molecular [4], [3] occuring one involved than are more orbitals suggesting harmonic of channels interference high-order records different that molecules elec- found from and is (HHG) It photon generation the [3–9]. in spectra embedded tron mul- are that showed effects Re- examination tielectron (see detailed 2]), approximation [1, (SAE) e.g., views electron based active model single been rescattering on have the by they interpreted high Although successfully and (HHG). (ATI) ion- generation ionization single harmonic threshold with above started e.g., are ization, interactions atom-laser ing fafe lcrni ae edo strength of field laser a in electron free a of unyof quency eta and tential aie ytelsrfils ec h tmcptnilis potential atomic the po- hence be fields, can laser core the the tun- by systems, time. the electron larized tunneling one on the than than acting more sooner For potential varing is atomic electron the neling if down break motion. electron the of radius classical the at strength hspcue h aei anydtrie yteunit- the by determined mainly is quantity less rate on the Based picture, rates. ionization this approxi- (ADK) obtaining Ammosov-Delone-Krainov for adiabatic [11] of theories so-called -like root The static the a is as instant. mation considered each be at can field field electric the that rapid h igeinzto faosi toglsrfields laser strong in atoms of ionization single The dur- occurring phenomena non-perturbative of Various ti biu htteaibtcapoiainwill approximation adiabatic the that obvious is It = yaia oeplrzto ftoatv-lcrnsyste two-active-electron of polarization core Dynamical p I p / ω 2 eateto hsc,Ntoa nvriyo ees Tech Defense of University National Physics, of Department U U When . ASnmes 38.v 25.z 42.65.Re 42.50.Hz, 33.80.Rv, numbers: PACS b polarization. can ionizati core core ionizati tunneling considering the existed the by of The suppresses slightly. polarization cation increases the the energy that of found po polarizability is core large it dynamical atoms, the of Ca account to into approximation. taking SAE by in formulated overestimated t is is exact rate and ionization Hartree-Fock the time-dependent with comparison In κ p p h oiaino w-cieeeto ytm yitnel intense by systems two-active-electron of ionization The where , 3 /E = E with 2 < γ / 4 I ω p κ 2 3 ,tne oiainocr so occurs ionization tunnel 1, = stepneooieenergy ponderomotive the is ersnigteaoi field atomic the representing κ 2 / steinzto po- ionization the is 2 egi Zhao Zengxiu E Dtd uy2,2018) 25, July (Dated: n fre- and ∗ n ini Yuan Jianmin and aigteHrreFc oeta ntelcldensity behav- asymptotic local of correct ior the the gives approxi- in that is potential approximation, Hartree-Fock potential effective the the mating the of obtain One to otherwise). approaches indicated unless throughout used etv oeta rmthe from potential fective ewl ee oti ramn steSEtheory. SAE the electron. as valence treatment this the taken to of be refer orbital will can Hartree-Fock We function the wave the initial as The system. atomic h oeplrzto nue ylsrfils o N- a For fields. laser by and e [15–18] induced symmetry core-polarization multielectron the the of account into moving electrons both dimension. with one in molecule hydrogen model solution for a (TDSE) exact Schr¨odinger equation with time-dependent var- comparison the by of the benchmark theories correlated we related particular, strongly ious spectra inves- In two photoelectron further have electrons. the we that valence on work, atoms DCP alkali-earth this of of In effects the [14]. tigate molecules CO rate of ionization DCP exper- alignment-dependent the the measured interpreting imentally incorporated successfully have [13] into we simulations (DCP) into Recently core-polarization [12]. dynamical account rates the corrected ionization be taking to that by needed expects theory electron therefore exci- single-active One from the resonant following of field. instantaneously absence is laser of polarization case the the tation, In time-varying. ffcieTS o h cieeeto nalsrfil takes field of laser a The form in the frozen. electron active is the core for ionic TDSE effective the approximation assuming SAE adopted ion usually the is the Therefore secondary higher electron, to potential. one rise ionization giving of inner bounded the liberation tighter to the becomes compared After perturbed strongly electrons. be will trons where rnwt h xenllsrfield laser external the with tron − esatwt h A prxmto n hntake then and approximation SAE the with start We nfruaincnb orce analytically corrected be can formulation on ytmitrcigwt ae ed,tevlneelec- valence the fields, laser with interacting system nb 0 hl h hteeto cut-off photoelectron the while 50% by on V V n oie igeatv-lcrnmodel single-active-electron modified A srfilsi netgtdtheoretically. investigated is fields aser ooy hnsa407,P .China R. P. 410073, Changsha nology, L aiain pligtenwapproach new the Applying larization. − → oeeto iuain eso that show we simulation, electron wo = osdrdisatnosadthe and instantaneous considered e ~ r 1 r i · ∂t steatv lcrni eahdfo the from detached is electron active the as ∂ E ~ ψ steitrcino h cieelec- active the of interaction the is si toglsrfields laser strong in ms [ = − ∇ 2 frozen 2 + V n oe(tmcuisare units (atomic core + E ~ V L and ] ψ. V n steef- the is (1) 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. In 1, therefore tunneling ionization dominates. strong field regime, it is comparable or larger than the in- The equations of motion, Eq. (1, 3, and 4) are solved teraction of the active electron with the permanent dipole respectively in a spherical box of 1600 a.u. with 20 par- moment of the atomic core if it exists. tial waves using pseudospectral grid and split-operator The effective TDSE for the active electron turns into propagation method [23, 24]. The photoelectron spectra

2 are obtained by projecting the final wavefunction to the ∂ ∇ continuum states i ψ = [− + Vn + Vcp + VL]ψ. (3) ∂t 2 20 2 The method of direct propagating Eq. 3 will be named PE = |hφEl|ψ(Tf )i| (6) as SAE+CP. Similar to the SAE theory discussed pre- Xl=1 viously, the initial wave function is taken as the the where φEl is the energy normalized continuum state of Hartree-Fock orbital of the valence electron. Note in this given energy E and angular momentum l. The single- theory, we neglect the polarization of the core induced by electron ionization probability is defined as pi = PE dE, R 3

-1 10 2U Note that we use the available polarizability of the neu- p SAE SAE+O tral atom found in database. The polarizability of the -2 10 SAE+CP + TDRH Ca is about 20% smaller which makes small change to the ionization ratio. We also perform a different check by -3 10 calculating the time-dependent induced dipole moment from propagating the Ca+ in the same laser field and 10U -4 p 10 then use it in the simulation of the outer electron dynam-

Ionization probability ics. The results show almost no difference suggesting the

-5 10 polarization is indeed instantaneous, therefore the phase lag due to different time response of the outer electron

-6 10 0 5 10 15 20 25 30 35 40 45 and the atomic core to the laser field makes no effect in Photoelectron energy (ω) the present study. However, this might be not true when resonant excitation is present which is beyond the scope FIG. 1: Photoelectron spectra of Ca atom calculated in var- of the present study. To further justify the consideration ious approximations. The arrows indict the maximum energy of the core polarization, TDRH calculation is performed of directly escaped and rescattered electron, with Up as the ponderomotive energy. as well. The resulted spectra shown in Fig. 1 is very close to that computed from SAE+CP calculation indicating that the core polarization can not be neglected for the and the total ionization probability of the system is given laser intensity we considered. 2 by PI =1−(1−pi) , where independent particle approx- The suppression of ionization rate due to the core po- imation is applied. larization can be understood within the tunneling pic- In Fig. 1, we present the photoelectron energy spectra ture. The potential barrier along the laser polarization for Ca calculated with the various theories. All calcula- direction is given by tions capture the main features of above threshold ioniza- V (z)= Vn(z) − Ez + Vcp (7) tion that there exists two main peaks located at 2Up and + 2 10Up. The 2Up peak corresponds to those electrons di- where Vcp = β E/z . In the standard theory of tun- rectly escape from the atom after tunnelling through the nel ionization, the core polarization is not taken into barrier formed by the atomic potential and the dipole in- account, and the potential barrier takes the form of teraction with the laser field. The 10Up is the maximum V0(z)= Vn(z) − Ez. According to the ADK model [11], energy that the tunnnelling electron can gain after rescat- the tunnel ionization rate when disregarding the core po- tered backward by the atomic core [25]. In the SAE+O larization can be estimated by calculation, we keep the outer electron wavefunction or- z1 2 thogonal to the other occupied orbitals, which implies w0 ≈ exp −2 κ +2V0(z)dz (8)   that the core can not be penetrated. However the spec- Zz0 p tra intensity shows only a little increasing compared to where κ is related to the ionization potential Ip by the SAE calculation. / Ip = κ 2, and z0 and z1 are the inner and outer turning When the core is polarized by the laser fields, it takes points respectively. When the core polarization is taken more energy for the electron to move from inside the into account, the correction of the ionization rate can be core to the outer region. However once it tunnels out approximated by [21, 27–29] the barrier, the polarization potential repels the electron z1 away resulting slightly increase of the cut-off energy as Vcp Rc = w/w0 ≈ exp −2 dz (9) demonstrated by the comparison of SAE results with the  Zz0 κ  SAE+CP results multiplied by 2 shown in Fig. 1. We see that the core polarization causes the suppression of ion- The potential barrier of Ca atom in the laser field is ization rates and marginally increases the maximum en- plotted in Fig. 2. The polarization potential is cut to ergy that electrons can gain. The reason for the latter lies zero below r0 = 5 a.u. that is estimated from the atomic 3 in the rescattering electron dynamics [25, 26]. Based on polarizability (≈ r0) and is consistent with the size of the classical trajectory analysis, the laser field is close to Ca atom. Taking z0 = r0 as the inner turning point. zero when the electron collides the atomic core with max- The outer turning point is determined by Ip/E. At the imum kinetic energy. The corresponding instantaneous laser intensity of 1013W/cm2, the outer turning point is polarization of the atomic core is small thus it makes 12 a.u. from the nucleus. According to Eq. 9, we found little impact on the cut-off energy of the photoelectron. the correction factor is 0.386, while the numerical simu- On the other hand, it can be expected that the harmonic lation shows that total photoelectron spectra intensity in spectra can be modified by the core-polarization due to SAE+CP calculation is about 52% of that obtained from different recombination instants and the resulted polar- SAE calculation in good agreement with the analytical ization as shown in [12]. correction. 4

ergy, it has been shown that the correlation of the two V (z) 0.1 n electrons has profound effects [31]. The threshold laser Vn(z)-Ez V (z)+V (z)-Ez n p intensity at which the correlation can not be neglected 0 might be estimated from that the dipole interaction en- -0.1 ergy at the classical radius is in the order of the correla- tion energy. In the present studied system, the averaged -0.2

Potential (a.u.) distance hri ≈ 1.2 a.u. such that the threshold laser in- -0.3 tensity is roughly 0.2I0. For laser intensities much higher,

-0.4 the difference of the ionization energy and the initial wave function in the restricted Hartree-Fock theory from the -0.5 -20 -15 -10 -5 0 5 10 15 20 25 30 exact calculation can be neglected due to the strong in- z (a.u.) teraction with laser filed. Secondly, however, the two electrons are treated as equivalent and independent par- FIG. 2: Illustration of the effective potentials for the cases of laser-free (solid line), laser field included (dashed lines) and ticles in TDRH which is not appropriate during ioniza- core-polarization considered (dotted lines). tion when the two electrons in fact can be distinguished as the inner and the outer electron. After the removal of the first electron, it will be very difficult to further ionize TAE SAE the ionic core. Therefore the total ionization probability SAE+CP TDRH 2 -1 defined by P =1 − (1 − p ) is not proper anymore and 10 I i needs to be replaced by pi neglecting the ionization of second electron. 0.6 For higher laser intensity, the correlation plays less role -2 0.3 10 in the ionization process. When the ionization rate is 0.0 much larger than the correlation energy, the difference of Ionization probability -0.3 Relative ratio the inner and outer electron orbital is not the main cause -0.6 -3 4 6 8 10 12 14 any more. At the laser intensity of 5I0, the TDRH gives 10 Laser intensity (I ) 0 the same ionization probability to the TAE calculation. 3 4 5 6 7 8 9 10 However, as laser intensity keeps increasing, the rates Laser intensity (I0) from TDRH becomes smaller than the TAE results. The FIG. 3: Ionization probability of a model Hydrogen molecule reason lies in the assumption that the two electron are calculated from exact two-electron calculation (TAE, filled equivalent and are occupying the same time-dependent squares), one-electron TDSE using the effective potential orbital. When ionization probability is large, less norm (SAE, dotted lines with open circles), modified SAE calcu- from the orbital is bounded. The nuclear charge is less lation by considering the core-polarization (SAE+CP, solid shielded and the potential on the electrons become un- line with open squares) and TDRH (solid line with diamonds). The inset displays the relative ratios comparing to the TAE physical due to this self-interaction in the TDRH the- calculation from the other three theories. ory such that further ionizing is incorrectly suppressed. Therefore TDRH theory fails in both low and high laser intensities. In order to further check the validity of our theory, we As shown in Fig 3, the rates from the SAE calculation perform exact two-active-electron (TAE) calculation for are larger than the TAE results for the laser intensity a model hydrogen molecule with both electrons moving in considered. When introducing the core-polarization po- one dimension. The soft-core Coulomb potential has the tential, we see that the deviation is reduced to less than form of |V (x)| = 1 with ǫ = 0.7 and ǫ = 1.2375 √ǫ+x2 N e 20% for laser intensity < 8I0. Noted for laser intensity for the electron-nucleus and electron-electron interaction above 8I0, the ionization mechanism switches from tun- [30]. The energies of the neutral and the cation are found nel ionization to over the barrier ionization (OTB). In of -2.2085 a.u. and -1.4103 a.u. respectively. The laser the OTB regime, the removal of one electron is so rapid, pulse has a duration of 2 cycles with wavelength of 1200 there is no time for the two electrons exchanging energy nm. The calculated ionization probability from the the- such that SAE theory works better as shown in the inset ories above are shown in Fig. 3. For laser intensities less of Fig 3. 14 2 than 5I0 (I0 = 10 W/cm ), the ionization probabil- In conclusion, we have shown that it is necessary ity obtained from TDRH calculation is higher than that to consider core polarization in strong field ionization from the exact TAE calculation. The failure of TDRH at of multielectron atoms. By incorporating the polariza- low laser intensity might be caused by two facts. Firstly, tion potential into theory the tunneling ionization the- the HF ground state energy differs from the exact two ory can be improved within the frame of SAE. Compar- electron energy by the correlation energy (0.025 a.u.). ing with exact two-electron model calculations and time- When the ionization rate is less than the correlation en- dependent restricted Hartree-Fock calculation, we show 5 that single-active electron approximation overestimates Kulander, Phys. Rev. Lett. 69, 2642 (1992). the ionization rate due to the additional barrier from the [11] M. V. Ammosov, N. B. Delone, and V. P. Krainov, Sov. dipole potential. The maximum photoelectron energy is Phys. JETP 64, 1191 (1986). 10 found increased slightly by the core-polarization. [12] G. Jordan and A. Scrinzi, New Journal of Physics , 025035 (2008). This work is supported by the National Basic Re- [13] B. Zhang, J. Yuan, and Z. Zhao, Phys. Rev. Lett. 111, search Program of China (973 Program) under grant 163001 (2013). no 2013CB922203, the Major Research plan of National [14] J. Wu et al., Phys. Rev. Lett. 108, 183001 (2012). NSF of China (Grant No. 91121017) and the NSF of [15] S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Vil- China (Grants No. 11374366, No. 11104352). Z. X. leneuve, Phys. Rev. Lett. 97, 123003 (2006). acknowledges Dr. Xiaojun Liu for helpful discussions. [16] R. Santra and A. Gordon, Physical Review Letters 96, 073906 (2006). [17] Z. Zhao, J. Yuan, and T. Brabec, Phys. Rev. A 76, 031404 (2007). [18] Z. Zhao, J. Wu, and J. Yuan, J. Phys. B 41, 155601 ∗ (2008). Electronic address: [email protected] [19] S. Pabst, L. Greenman, D. A. Mazziotti, and R. Santra, [1] M. Protopapas, C. H. Keitel, and P. L. Knight, Rep. Phys. Rev. A 85, 023411 (2012). Prog. Phys. 60, 389 (1997). 9 72 [20] I. B. Bersuker, Opt. Spectrosc. , 363 (1960). [2] T. Brabec and F. Krausz, Rev. Mod. Phys. , 545 [21] T. Brabec, M. Cˆot´e, P. Boulanger, and L. Ramunno, (2000). Phys. Rev. Lett. 95, 073001 (2005). [3] O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, [22] V. L. Sukhorukov, I. D. Petrov, M. Sch¨afer, F. Merkt, D. Villenenve, P. Corkum, and Y. Ivanov, Nature 460, M.-W. Ruf, and H. Hotop, J. Phys. B 45, 092001 (2012). 972 (2009). 217 104 [23] X. M. Tong and S. I. Chu, Chem. Phys. , 119 (1997). [4] Y. Mairesse, et al., Phys. Rev. Lett. , 213601 (2010). [24] X. Chu and S. I. Chu, Phys. Rev. A 63, 013414 (2000). [5] M. Spanner, J. Mikosch, A. Gijsbertsen, A. E. Bo- 13 [25] G. G. Paulus, W. Becker, W. Nicklich, and H. Walther, guslavskiy, and A. Stolow, New Journal of Physics , J. Phys. B 27, L703 (1994). 093001 (2011). [26] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993). [6] A. D. Shiner et al., Nat. Phys. 7, 464 (2011). [27] Z. X. Zhao and T. Brabec, J. Phys. B 39, L345 (2006). [7] A. E. Boguslavskiy et al., Science 335, 136 (2012). 54 3 [28] Z. Zhao and T. Brabec, J. Modern. Optics , 981 (2007). [8] B. Bergues et al., Nat. Comm. , 813 (2012). 82 [9] S. Pabst and R. Santra, Phys. Rev. Lett. 111, 233005 [29] B. Zhang and Z. Zhao, Phys. Rev. A , 035401 (2010). [30] J. Zhao and Z. Zhao, Chin. Phys. Lett. 27, 063301 (2010). (2013). 32 [10] D. N. Fittinghoff, P. R. Bolton, B. Chang, and K. C. [31] A. J. Tolley, J. Phys. B , 3449 (1999).