Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Miladinovi, T. B.; Petrovi, V. M. Quasiclassical approach to tunnel ionization in the non relativistic and relativistic regimes Revista Mexicana de Física, vol. 60, núm. 4, julio-agosto, 2014, pp. 290-295 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57031049004 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative RESEARCH Revista Mexicana de F´ısica 60 (2014) 290–295 JULY-AUGUST 2014 Quasiclassical approach to tunnel ionization in the non relativistic and relativistic regimes T. B. Miladinovic´ and V. M. Petrovic´ Department of Physics, Faculty of Science, Kragujevac University, Kragujevac, Serbia, e-mail:[email protected] Received 6 January 2014; accepted 29 May 2014 Non relativistic and relativistic transition rates were observed for the deep tunnel regime for the case of a linear polarized laser field. The contribution of the initial momentum of an ejected photoelectron, the ponderomotive potential and the linear Stark shift for the Ar atom and its ions were taken into account. It was shown that, in both regimes, these processes have an influence on the transition rate behavior. The curves obtained for the transition rate show that with increasing laser filed intensities and ion charge the influence of relativistic effects increases. Keywords: Tunnel ionization; transition rate. PACS: 32.80 - t; 32.80 Fb 1. introduction laser field this field influences the electron’s binding poten- tial, perturbs it and makes it much higher than the unper- Studying of the photoionization of atomic and molecular sys- turbed value. There are at least two reasons for this increase: tems is one of the important ways to investigate the charac- the linear Stark effect and the ponderomotive potential. teristic properties of the interaction of these systems with an If an atom is placed in an external electric field, its energy electromagnetic field. This provides many fundamental in- levels are altered. This phenomenon is essential and known sights into light-matter interactions and because of that it has as the Stark effect. As the field increases, the states spread been the subject of many theoretical and experimental inves- out in energy and therefore have linear Stark effects even in tigations over the years. As a result, many theories were de- lower intensity fields. fined as the theoretical framework of these processes. Every- The ponderomotive potential is caused by the quiver mo- thing started in 1964 with the Keldysh theory [1] that showed, tion of an electron in an atom. The ponderomotive potential for the first time, that the tunnel effect and the multiphoton is directly proportional to the laser field intensity, meaning ionization of atoms are two limiting cases of photoioniza- that its significance increases with increasing laser intensity. tion. The theory is quasi classical meaning that an atom is Therefore, the description of ionization must take into ac- described quantum-mechanically whereas the laser field is count both effects. treated classically. Keldysh introduced the Keldysh param- eter of adiabaticity, determined through the field strength F , the frequency ! of the external field and the binding energy 2. Corrections of the ionization potential E γ = !(2E )1=2=F i of the electron in the atom, i (in the The field free ionization potential of an atom can change un- atomic units) [1]. Tunneling ionization is the limiting case of der laser irradiation. We observed two effects which change a photoionization process, when the dimensionless Keldysh the ionization potential and, at the same time, the tunnel- γ ¿ 1 parameter . ing rate in non relativistic and relativistic domains of laser Tunneling ionization of atoms occurs in an intense laser field intensities. In recent years (recently) modern laser sys- field when the potential barrier of an atom is deformed in tems are able to generate laser pulses with peak intensities a way that bound electrons can tunnel through this barrier up to » 1022 Wcm¡2. This progress in laser physics and and “escape” from an atom easily. Tunneling plays a cen- the laser technique enables numerous experiments opening tral role in the interaction of matter with intense laser pulses. a window to the physical phenomena occurring in the rela- For γ ¿ 1 the multiphoton dynamics dominates. Soon after, tivistic domain. They became significant for intensities up to Perelomov, Popov and Terent’ev developed the new PPT the- » 1018 Wcm¡2. We considered the deep tunneling regime, ory [2] to calculate the ionization rate for hydrogen like atoms where the Keldysh parameter γ · 0:2. According to Reiss in a linearly and circularly polarized laser field. Twenty years [4] the γ ! 0 limit also requires relativistic treatment. later Ammosov, Delone and Krainov derived the ADK the- An electron is always oscillating around its nucleus, but ory [3] for the case of tunneling ionization for arbitrary com- we did not take into account this motion in our analysis. plex atoms and atomic ions. This is still one of the commonly We observed only electron motion in an oscillating electric used models for calculating the ionization rate. field. The physical picture of this motion is mathematically The ionization process is directly determined by the elec- described by the ponderomotive potential which represents tron binding energy. When an atom is placed in an intense the time average kinetic energy of the electron oscillating QUASICLASSICAL APPROACH TO TUNNEL IONIZATION IN THE NON RELATIVISTIC AND RELATIVISTIC REGIMES 291 in a laser field. We have an electron in a varying external Equation 7 indicates that under the influence of the laser laser field and with a kinetic energy large enough so New- field, ignoring the rapid oscillatory term, the movement of the tonian equations of classical physics ~rÄ = F~(~r; t) could be slow-varying term is driven by an effective force: used (here and below, the atomic system of units is used ~ 1 e = m = ~ = 1). F(~r; t) is the electric field vector at f~ = ¡ r(F~ 2) (8) the position of the electron. The field intensity should be 4!2 large along the beam axis, and should fall smoothly to zero at ~ some distance from the axis. F~(~r; t) can be decomposed into f is normally referred to as the ponderomotive force which a fast oscillation term modulated by a slowly varying ampli- points along the gradient of the intensity. If the laser field is tude F~ (~r). For a linearly polarized field: non-time-varying, this force is a conservative one, and can be associated with a potential given by: ~rÄ = F~ (~r) cos !t: (1) F 2 U = ; (9) As the light frequency from the optical range is very big, p 4!2 we can picture our electron wiggling slightly around the cen- tral position at the optical frequency. This picture is mathe- or, in a standard system of units: matically described if one adds a slow drift term, and a slow e2F 2 amplitude modulation, to the standard rapidly oscillating si- Up = 2 : (10) nusoidal solution of this problem: 4me! ~r(t) = ~®(t) + ¯~(t) cos !t; (2) From Eq. 9 follows that with increasing of the laser field intensity the influence of the ponderomotive potential on the where ~®(t) is the coordinate of the position of the electron field free ionization potential Ei becomes larger and it be- and ¯~(t) the amplitude of the wiggling motion. The temporal comes more significant. Increase of the laser field intensity variations of ®; ¯ take place in a time interval much longer also leads to the relativistic domain and for these intensities ~ than one optical cycle. That means we can expand F (~r) in a the relativistic ponderomotivep potential may be written in the rel 4 2 2 series in ¯ and keep the first two terms in the expansion: following form Up = c + 2c Up ¡ c [5], where c is the speed of light and U is the nonrelativistic ponderomotive po- ~ ~ p F (~®(t) + ¯(t) cos !t) tential (see the Eq. 9). To account for the ponderomotive po- ~ ~ ~ tential we replaced the ground bound state Ei with the shifted = F (~®) + (¯ ¢ r)F (~®) cos !t + ¢ ¢ ¢ : (3) 2 energy Eief = Ei + Up = Ei + F¡p=4! for the nonrelativistic¢ rel rel 4 2 2 Substituting Eq. (2) and Eq. (3) into Eq. (1) we obtain: and Eief = Ei + Up = Ei + c + 2c Up ¡ c for the relativistic domain. ~®Ä ¡ !2¯~(t) cos !t = [F~ (~®(t)) As we already mentioned when an atom is placed in an ~ ~ external electrical field its energy levels undergo a splitting + (¯ ¢ r)F (~®(t)) cos !t] cos !t: (4) proportional to the field intensity. This effect is known as On the right hand side of Eq. 4 we have the term which in- the linear Stark effect and unlike the weak fields where it cludes cos2 !t, a factor very rapidly oscillating at frequency may be neglected in a strong laser field that is not the case. !. After averaging over the rapid oscillation at frequency ! Using the perturbation theory approximation the shifted ion- ization potential can be represented in the following form will be equivalent in gain to the term on the left hand side 2 which does not contain the cos !t factor, so we obtain: Eief = Ei + ESt = Ei + ®F =4 [6], where ESt is the shift caused by the linear Stark effect and ® is the static polariz- ZT 1 ability of the atom.