The field-induced double ionization of a model two-electron quantum system is investigated by direct numerical integration of the nonstationary Schroedinger equation. The contribution of different processes to a formation of doubly charged ions is analyzed.
©2001 Optical Society of America
The advance in the creation of the high-intensity laser sources allows for the investigation of the atomic and molecular dynamics in the strong field region, where the perturbation theory does not appear to be valid. A lot of new interesting features of quantum systems interacting with a strong laser field were discovered: the phenomenon of atomic stabilization, above threshold ionization and dissociation, high harmonic generation and other [1–3].
In a series of experiments [4–7] in that the ionization yields produced by a laser pulse in He were measured, surprisingly large signal of double ionization has been observed. The double ionization yield appeared to be much larger than that predicted by the single active electron model. The additional double ionization yield found can be attributed to a nonsequential process and is observed as a characteristic “knee” or “shoulder” structure on the curve of intensity dependence of the ion yield.
Many theoretical and experimental studies are aimed to clarify the physical nature of such nonsequential ionization in a strong laser field. Several distinct mechanisms are supposed to contribute to appearance of doubly charged ions: classical rescattering or recollision , “shake-off’ mechanism  and the energy-exchange mechanism due to electron-electron correlation [9,10], though the latest could be understand as recollision process in more general sence.
In this paper the results of numerical simulations of the dynamics of two-electron system in a strong laser field are presented and the contribution of different channels to the double ionization process is analyzed.
The exact numerical simulations are performed by direct numerical integration of the nonstationary Schrödinger equation (NSE):
Here H0 is a field-free Hamiltonian of the system, the laser frequency ω corresponds to the photon energy of 2 eV and the laser pulse envelope ε(t) is chosen in a smoothed trapezoidal form with the turn-on time t f=2T and the plateau duration t p=5T (T is the duration of one optical cycle). The method of the numerical solution of the NSE is similar to that explained in .
The field-free Hamiltonian can be presented in the form:
Here α is a smoothing parameter chosen to be equal to 0.92 A.
For z=1 such a Hamiltonian describes the one-dimensional model of a negative hydrogen ion H - with the only bound state characterized by the binding energy Eb =-1.1 eV. The energy needed for detachment of the second electron I2 is equal to 11.46 eV. If z=2, this Hamiltonian corresponds to the 1D model of a two-electron atom with following first and second ionization potentials: I1 =12.95 eV and I2 =24.75 eV. Because of the value of the first ionization potential we shall call such a system a Xe-like atom.
Using the time-dependent wave function of the system found from the nonstationary Schrödinger equation the two-dimensional electron-density distribution can be obtained at any instant of time. At such pictures the single electron ionization is characterized by the so-called cross-distribution, while wave-packets of double ionization are found to appear in I–IV quadrants. Using such space pictures, the probability of non-ionization Wb can be calculated by integration inside the square area 14*14 Å near the nucleus: Wb = dx 2 dx 1 |ψ(x1 x2t) 2 . The probability of single ionization W1 is calculated by integration over the cross area:
And then the probability of double ionization is determined as:
W2 =1-W1 -Wb
The main result of our simulations is the probability of a single and double ionization calculated in dependence on laser intensity. For the Xe-like atom such curves are presented at Fig. 1 and are characterized by a non-monotonous behaviour in the low-intensity region. Quite similar double ionization signal with a few peaks was obtained by Eberly and co-workers . In our opinion such a non-monotonous structure can be attributed to the appearance and the vanishing of the multiphoton resonances between the initial and one of the single-electron excited bound states of the system.
In contrast, the double-ionization curve obtained for the H - (Fig.2, solid curve) is characterized by a rather monotonous behaviour, since for this system there is the only bound state and no resonances can be found.
It should be emphasized that the double ionization observed in the low intensity region can not be interpreted as a result of purely classical rescattering process . Indeed, the threshold for the (e,2e) scattering-like process appears to be approximately 1014 W/cm2 for H - and 2.1014 W/cm2 for the Xe-like atom. As for resonance peaks of double ionization, they are rather pronounced still in the vicinity of the threshold intensity. Since the rescattering cross section is known to be negligibly small at the threshold the found peaks do not seem to be attributed to the purely recollision process.
Besides possible multiphoton resonances, the energy exchange between electrons due to electron-electron correlations is found to result in the double-ionization process. This fact is confirmed by results obtained using the passive electron model. According to this model one of the electrons is assumed not to interact with the electromagnetic field. In this case the nonstationary Schrödinger equation can be written by following way:
So, the ejection of the second electron can occur because of the e-e interaction only. At Fig.3 the space density distribution calculated by passive model is presented for Xe –like atom for different instants of time. The correlated energy exchange results in the formation of the cross. Although one of the electrons is passive, the picture becomes rather symmetrical. By the end of the pulse it is difficult to distinguish active electron from the passive one, because of the energy exchange caused by e-e correlations. It should be noticed that it is the energy - exchange mechanism that results in the double ionization although the intensity is above the rescattring threshold.
Similar results are obtained for H -. So, we can conclude that in the low intensity region two important mechanisms of double ionization are found: possible multiphoton resonances and the energy exchange between electrons due to the e-e correlations.
In the high intensity region the results of the passive electron model can be easily understood in terms of classical rescattering process, which is illustrated by dynamics of the mean coordinates of electrons calculated in such a way:
The data are presented at Fig. 4. It can be seen, that the oscillations of the passive electron appear when the active electron approaches the nucleus and the ejection of the second electron is shown to occur due to e→2e scattering-like process. On the space distribution picture the rescattering process is characterized by the density wave-packets that are formed from the cross domain rather than from the zero point.
However the results of the exact calculations do not seem to be correctly explained by the the rescattering mechanism only. For high laser intensities the electron density distribution is characterized not only by the recollision wave-packets formed from the cross domain, but also by some jets that are coming from the zero-point (Fig.5). In our opinion, the formation of jets should be attributed to the simultaneous ejection of both electrons due to the direct action of the laser field on each electron. It can be seen that both electrons move in the same direction, but they remove each other because of the Coulomb repulsion. In dependence on the phase of laser field the electron wave packets are formed on the one or another side of the nucleus. Thus, the collective ejection of two electrons is seen to occur.
This process is also confirmed by direct comparison between the results obtained by exact calculations and by the passive electron model. In the high intensity region (3.17Up >I2 ) the double ionization yield is found to exceed approximately in one order of magnitude the results of passive model both for the H - and for the Xe-like atom. For H - both curves are presented at Fig.2. Thus, the direct action of the electromagnetic field on both electrons is proved to be very significant for the ionization process.
The laser-induced single and double ionization of a model two-electron system is studied by means of direct numerical integration of nonstationary Schrödinger equation. The contribution of different channels to the double ionization is analyzed.
It is found that
1) At rather low laser intensities both a correlated energy exchange between electrons and possible multiphoton resonances are proved to play a role in ionization process.
2) The rescattering process is shown to take place at rather high intensities but does not appear to be the only mechanism of double ionization. This conclusion seems to correlate with results of calculation of double ionization of He using a recollision model . The recollision calculations are qualitatively in agreement with experiment, but quantitative difference by a factor of 3 is found.
3) The direct action of the laser field on both electrons is found to be very significant and collective ejection of two electrons is confirmed to be a very important channel of double ionization.
This study was supported by Russian Foundation for Basic Research, grants 00-02-16046, 00-15-96554 and INTAS, grant 99-1495
References and links
1. M. Gavrila, Atoms in intense laser field (Academic, New-York, 1992)
2. N.B. Delone and V.P. Krainov, Multiphoton processes in atoms (Springer, Berlin, 1994) [CrossRef]
3. M.V. Fedorov, Atomic and free electrons in a strong light field (World Scientific, Singapore, 1997)
4. S. Augst, D. Strickland, D.D. Meyerhofer, S.L. Chin, and J.H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys.Rev.Lett. 63, 2212–2215, (1989). [CrossRef] [PubMed]
5. D.N. Fittingoff, P.R. Bolton, B. Chang, and K.C. Kulander, “Observation of non-sequential double ionization of helium with optical tunneling,” Phys.Rev.Lett. 69, 2642–2645, (1992). [CrossRef]
6. D.N. Fittingoff, P.R. Bolton, B. Chang, and K.C Kulander, “Polarization dependence of tunneling ionization of helium and neon by 120-fs pulses at 614 nm,” Phys.Rev.A 49, 2174, (1994). [CrossRef]
9. A. Becker and F.H.M. Faisal, “Mechanism of laser-induced double ionization of helium,” J.Phys.B 29, L197–L202, (1996). [CrossRef]
10. M.Y Kuchiev., “Adiabatic mechanism of multiply charged ion production by a laser field through ATI states of an atom,” Phys.Lett.A. 212, 77, (1996). [CrossRef]
11. E.A. Volkova, A.M. Popov, and O.V. Tikhonova, “Double-electron ionization of a quantum system in a laser field: rescattering and interparticle correlations,” JETP 91, 706–712, (2000). [CrossRef]
12. W.-C. Liu, J.H Eberly, S.L. Haan, and R. Grobe, “Correlation effects in two-electron model atoms in intense laser fields,” Phys.Rev.Lett. 83, 520–523, (1999). [CrossRef]
13. H.W. van der Hart and K. Burnett, “Recollision model for double ionization of atoms in strong laser fields,” Phys.Rev.A 62, 013407(10), (2000). [CrossRef]