TRANSITION RATES DEPENDENCE ON THE TURNING POINT

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1 31 Kragujevac J. Sc. 7 (5) TRANSITION RATES DEPENDENCE ON THE TURNING POINT V. M. Rstć, M. M. Radulovć and J. M. Stevanovć Faculty of Scence, Kragujevac Unversty, R. Domanovća 1, 34 Kragujevac, Serba and Montenegro (Receved March 1, 5) ABSTRACT. In the case of the short-range potental [1, 5, 1, 11], the estmaton of the transton rate of onzaton of atoms s defned takng nto account Keldysh approxmaton, whch states that ths knd of potental does not affect the energy of the fnal states f of the ejected electron n the laser feld, because electron s far enough from the nucleus. When the Coulomb potental s taken nto account, t can be treated as a perturbaton to the energy of the fnal state [1, 1]. So there are three dfferent transton rates, one obtaned for short range potental, the other for the ADKtheory and thrd for ADK result wth turnng pont corrected wth the Coulomb nteracton. If 1 17 plotted for the felds from 1 W cm to 1 W cm the three varants gve behavor whch were predcted many tmes theoretcally, but stll wth rather poor expermental support, see [11]. Yet result for short range potental s not realstc at all for greater felds and t s gven here only to be compared wth more realstc results shown n Fgs and 3. Indeed n Fg 3 one has corrected result of ADK-theory, whch ncludes the turnng pont calculated wth the Coulomb correcton, and gves the saturaton effect after 1 13 W/cm, dfferng from the ADK case whch gves ths effect after 1 14 W/cm. Whch of two s correct only experment can decde. 1. INTRODUCTION In the late XX Century tunnelng onzaton (Keldysh s [6] parameter γ 1) of atoms and ons by strong low-frequency laser felds has become the subject of ntensve research. In that perod, one of the most utlzed theores was Ammosov-Delone-Kranov (ADK) theory [1], whch uses the concept of a quas-statonary external feld ejectng valence electrons va tunnelng. Recently [8, 1, 13], ths theory has been extended to the barrer-suppresson onzaton of complex atoms and atomc ons,.e. to the case of super-strong felds. Before that, the ADK-theory has been used to descrbe onzaton whch occurs when the laser ntenstes, n experments, were up to 1 1 W/cm (atomc feld s approxmately 1 16 W/cm ). Nowdays, when we are dealng wth feld strengths of the order of atomc and even hgher, the ADK theory had to be adjusted to the stuaton. Usng Landau-Dykhne adabatc approxmaton [1-13] the ADK-theory [1] starts wth the transton ampltude between ntal and fnal states ( Ef > E on real axes) Af = exp (t) dt ωf (1) t1

2 3 where ωf s frequency of the transton from - ntal to f - fnal state n the presence of the external feld, and s the complex turnng pont n the tme plane. The complex turnng pont s obtaned from equaton whch s classcally forbdden [11]: ω ( ) =. () f Also, the transton rate f s gven by expresson [1, 11] f f { } W = A = exp ImδS(). (3) In the case of the short-range potental [11, 1], the estmaton of the transton rate of onzaton of atoms usng (3) s performed takng nto account the Keldysh approxmaton, whch states that ths knd of potental does not affect the energy of the fnal state f of the ejected electron n the electromagnetc feld, because the electron s far enough from the nucleus. When the Coulomb potental s taken nto account, t can be treated as a perturbaton to the energy of the fnal state [1, 1]. Yet, orgnally [1, 1], the Coulomb potental n ths knd of estmaton was not ncluded nto calculatng the turnng pont from (). Ths was done n [7], but only for the felds below the atomc feld (up to1 14 W cm ). Now we are extendng our calculaton that ncluded the Coulomb correcton nto the estmatng the turnng pont to the felds that are much stronger (up to1 17 W cm ), whch s justfed by the results of [1, 13]. That results n the shft of the poston of the turnng pont. So the paper s comparng the results of the nfluence of that shft on the transton rate for atoms n superstrong low frequency laser felds, wth the results obtaned for the same rate n the "pure" ADK-theory.. CALCULATING THE COMPLEX TURNING POINT, BY TAKING INTO ACCOUNT THE COULOMB INTERACTION The method for calculatng the transton rate usng the Landau-Dykhne adabatc approxmaton s gven n [1, 11]. It begns wth the equaton (),.e., f ( ) ( ) E =E, (4) where E (), E f () are the ntal and fnal energy, respectvely, n the external electromagnetc feld, and s the complex tme, related to the turnng pont. Because external feld F s much smaller than the atomc feld F at (and that s so even n the case of superstrong felds W/cm s the hghest value we are takng nto account, see explanaton n the paragraph after equaton 19 - because as shown n [8], and n ths paper, after ntenstes of laser felds of 1 14 W/cm the atoms are onzed so that Z~1, thus producng the atomc feld much stronger then external), we wll take n consderaton nfluence of external feld only on fnal state E f, whle assumng that ntal state s non-perturbed (see [6]). In the case of lnear feld polarzaton E( ) = E, where E s the onzaton potental of the ground state of valence electron, and energy of fnal state E f s gven as: 1 F Z Ef ( ) = p- snω -, ω η( ) where the last term n the above expresson s due to Coulomb nteracton. From (4), t follows:

3 33 1 F Z p snω = E. (5) ω η ( ) As Coulomb term n (5) s small compared to other terms, we wll be usng teraton. In approxmaton of zero order, only the external electrc feld s taken nto account dz = Fcosω, dt whch, after ntegraton and rememberng that z= η (we are usng parabolc coordnates, as n [11]), gves F η() = E (1 cosω). (6) ω Frst term n (6) was chosen n such way as to make t possble that, at the ntal tme t =, energy of the electron equals atomc energy E. By neglectng a second term n (6), as was done earler n ADK theory [1], one has η( ) = E, where s the turnng pont n the zero-order approxmaton p+ E =. (7) F But, f we form a power seres of cosne: cosω 1 ω, expresson (6) becomes η() E F = +. (8) By followng an teraton procedure, we put nstead of and obtan p + E η( ) =. (9) F Now, let us go back to expresson (5): because external feld has a low-frequency ( ω << ωat ), we are allowed to expand sne n power seres, snω ω. Thus we get followng expresson 1 Z p F = E 1. (1) η() E If, n expresson (1), we substtute η( ), because of teraton, we wll obtan F Z =. (11) p F E 1 p +E E As Coulomb correcton under the root of the above expresson s small compared wth onzaton potental E there follows another expandng: 1 x 1 x, whch gves 1 F Z p F E = 1 p, +E E.e. p E Z F = + 1, F F E p +E and, fnally, an expresson for Coulomb-correcton-ncluded turnng pont, already obtaned n [7]

4 34 p+ E Z = F (p +E ) E. (1) 3. RATES OF TUNNELING IONIZATION OF ATOMS IN VARIUOS APPROACHES The transton rate n the adabatc approxmaton of Landau-Dykhne n the case of an atom n an external electromagnetc feld s gven by expresson (3), where S( ) s tme dependent part of the classcal acton, defned as: () = { f () () } S E t E t dt, or, usng Keldysh approxmaton: 1 Ssr ( ) = ( p-ft ) + E dt For the case of a short-range potental, the transton rate (3) s: ( E ) 3/ Wsr = exp. (13) 3F Includng the Coulomb nteracton [1] nto the tme-dependant part of the acton, wll lead to the followng expresson for energy of the fnal state: 1 n 1 E(t) f (p Ft) n η(t) = *, (14) where η(t) = E + Ft, n beng one of parabolc quantum numbers that defne a gven state, and n * = Z E s an effectve prncpal quantum number all these follow from our expressng the Coulomb nteracton n parabolc coordnates. And so, the Coulomb nteracton gves the followng part of the acton δs (n + 1) C = E η(t) Z whch we shall dvde nto two parts [7] C a ta δs = δs + δs = + ta dt. (15). (16) where t a represents tme related to arbtrary turnng pont ηa = ra, that we are allowed to use because, at ths dstance from atom, the nfluence of atomc resdue on ejectng electron s already small, and the external feld can stll be neglected. Now, n a regon η< ηa, whch corresponds to tme t < ta, quantum effects are very strong, so t should be treated n a completely dfferent manner than regon η> ηa, whch corresponds to tme t > ta. Correspondng gan to the acton by δ S can be obtaned usng semclasscal approxmaton, and by takng nto account that, for t < ta, the wave functon can be treated as unperturbed atomc functon

5 35 n* 1 Ze ImδS = ln E * t a n. Analogous gan to the acton by δ Sa can be obtaned by ntegratng: δs (n + 1) a = E ta After substtutng a turnng pont η(), and a few elementary transformatons, we have E n + 1 F ta δsa = ln. E ta F As E / F >> t а, we can neglect t a n the denomnator of above fracton, and shall nclude expresson (7) for the turnng pont n the zero-order approxmaton. Takng nto * account that ln = π /, usng a fact n = n 1, t follows from (16): max * 4Ze ImδS c = (n 1)ln E * Fn. * Rememberng that E = Z / n and ncludng part of acton due to already mentoned shortrange potental, the transton rate s now gven as: η(t) * 3 n 1 3 4Z e Z WADK = exp *4 *3. (17) Fn 3Fn At the end, we have to examne the nfluence of turnng pont on a pre-exponent obtaned n ADK theory. For that purpose, we shall nclude an expresson for Coulombcorrected turnng pont (1) n formulae for δ Sa ; after rather cumbersome but pretty much straghtforward procedure, we obtan expresson for magnary part of acton: dt. * * n 1 ZF Z F Fn ImδSa = ln , (18) (p + E ) E (p + E ) 4 Ze E Fnally, for the mproved transton rate one has ( S sr beng the part of the acton due to the short-range potental, and S c the part of the acton due to Coulomb potental): ( δ ) ( δ ) W = exp Im S exp Im S = CADK sr C n* 1 4Z e 1 Z = exp Fn ZF Z F 3 Fn 3 3 * 4 * (p +E 3 ) E (p +E ). (19) Here we denoted by W CADK the transton rate obtaned n ADK-theory whch was corrected by ntroducng Coulomb nteracton nto estmaton of the turnng pont. In

6 36 expresson (19) for transton rate W CADK, the second ratonal term n the parentheses s a correcton due to the Coulomb nteracton whch was obtaned n [7]. For the felds up to 1 1 W cm, the correcton s small and could be neglected (for nstance, n the case of 1 potassum onzaton n the laser feld of 1 W cm [5], t s , [7]), but for 14 greater felds (n [7] the felds were used up to 1 Wcm < I at ~1 16 W/cm, because at that tme the ADK-theory was not extended to the felds that are greater than the atomc [1, 13]) ths correcton gans n amount (for nstance, 1 14 W cm gves , and 17 1 W cm gves ). If plotted for the felds from 1 1 W/cm to 1 W cm (because for 1 W cm and hgher felds relatvstc effects become predomnant [9], and, also, t s extremely dffcult to obtan so strong laser pulses as contnuous, and for hgher ntenstes that s even mpossble, for the moment), transton rate (17) gves behavor whch were predcted many tmes theoretcally, but wth rather poor expermental support yet, see [13]. One has, after strong 14 ncrease of the transton rate of onzaton of atoms, untl the feld ntenstes of 1 W cm, 14 rapd decreasng at ntenstes of order 1 W cm (whch can be explaned by the onzaton va tunnelng effect of all avalable electrons n the last orbts that, n the cases of alkal metals or noble gases, gve nuclear charge of the resduum Z 1, so ADK- theory s stll 15 vald [8, 1]), then a saturaton at very low level of transton rate for felds from 1 W cm 17 to 1 W cm - see Fg. Because Z 1 the Keldysh approxmaton s stll vald [1], and ADK-theory can be used to descrbe onzaton of electrons from deeper orbts. Yet result for short range potental s not realstc at all for greater felds and t s gven here only to be compared wth more realstc results shown n Fgs. and 3. Indeed n Fg. 3 one has corrected result of ADK-theory, whch ncludes the turnng pont calculated wth the Coulomb correcton (therefore transton rate has an ndex CADK: W CADK ), and gves the saturaton after 1 13 W/cm, and Fg. gves ths saturaton only after ntenstes of 1 14 W/cm. Whch of two s correct only experment can decde. [W/cm ] Fgure 1.W sr plotted vs. ntensty of the feld [W/cm ]

7 37 [W/cm ] Fgure.W ADK plotted vs. ntensty of feld [W/cm ] Fgure 3.W CADK plotted vs. ntensty of feld 4. CONCLUSION For short-range potental, estmaton of transton rate of onzaton of atoms was made, based on assumptons of Keldysh approxmaton [6], that short-range potental does not affect energy of the fnal state of ejected electron, when t leaves the atom. Coulomb potental s then treated as perturbaton of fnal state energy thus obtanng the ADK-theory. But not when calculatng the turnng pont. Ths was done n [7], though only for felds wth ntenstes below those of the atomc feld. But as the ADK-theory was recently extended to the case of superstrong felds [1,13], our calculatons now nclude extenson of potental 17 range up to 1 W cm, ths leads to the shft of poston of the turnng pont, whch then nfluences transton rates for atoms n the low-frequency electromagnetc feld of superstrong lasers. Fnally, we can conclude that, for the felds whose ntensty vary from 1 1 W/cm 17 to 1 Wcm, transton rates gven by (17 and 19) show behavor whch was predcted many

8 38 tmes theoretcally (but wth rather poor expermental support so far, see [11]),.e. n Fgs. and 3 (not n Fg. 1 whch s not showng the realstc case, and s gven here only for comparaton wth the other two, whch are more realstc) the sudden decrease s shown at 14 laser-feld ntenstes of order 1 W cm (but even after ntenstes of order 1 13 W/cm, n the case of the ADK-theory wth the turnng pont corrected to the Coulomb nteracton, see formula 19 and Fg. 3), and then a saturaton at a very low level of transton rate for felds from 1 W cm to 1 W cm, whch s, as mentoned above, all applcable only to multcharged ons wth the on charge larger or equal to ten and wth at least one electron left n a bound state (Z beng 1 on both Fgs. and 3). ACKNOWLEDGEMENTS Ths work was supported n part by the Mnstry of Scence and Ecology, Republc of Serba (Project 147). References [1] AMMOSOV, M.V., DELONE, N.B., KRAINOV, V.P., Sov. Phys. JETP 64, 1191 (1986). [] CHIN, S.L., YERGEAU, F., LAVIGNE, P., J. Phys. B 18, L13 (1985). [3] YERGEAU, F., CHIN, S.L., LAVIGNE, P., J. Phys. B, 73 (1987). [4] XIONG, W., YERGEAU, F., CHIN, S.L., LAVIGNE, P., J. Phys. B 1, L159 (1988). [5] XIONG, W., CHIN, S.L.,, Sov. Phys. JETP 7, 68 (1991). [6] KELDYSH, L.V., Sov. Phys. JETP, 137 (1964). [7] RISTIĆ, V.M., RADULOVIĆ, M.M., KRAINOV, V.P., Laser Physcs 4, 98 (1998). [8] KRAINOV, V.P., J. Opt. Soc. Am. B 14, 45 (1997). [9] MILOŠEVIĆ, N., KRAINOV, V.P., BRABEC, T., Phys. Rev. Lett. 89, 1931 (). [1] RISTIĆ, V.M., The Foundaton and Extenson of Some Aspect of ADK-Theory, Ph.D. Thess, Moscow, Kragujevac, 199. [11] LANDAU, L.D., LIFSHITZ, E.M., Quantum Mechancs, Oxford: Pergamon, [1] DELONE, N.B., KRAINOV, V.P., n Atoms n Strong Lght Felds, Berln: Sprnger, [13] DELONE, N.B., KRAINOV, V.P., n Multphoton Processes n Atoms, Berln: Sprnger,.