TREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA

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1 41 Kragujevac J. Sc. 5 (00) TREATMENT OF THE TURNING POINT IN ADK-THEORY INCLUDING NON-ZERO INITIAL MOMENTA Vladmr M. Rstć a and Tjana Premovć b a Faculty o Scence, Department o Physcs, Kragujevac Unversty, P.O. Box 60, 4000 Kragujevac, Serba and Montenegro e-mal: rstc@kg.ac.yu b Faculty o Scence, Department o Physcs, Prstna Unversty, Kosovska Mtrovca, Kosovo, Serba and Montenegro (Receved March 1, 00) ABSTRACT. The ADK-theory usually has been used to descrbe the onzaton whch occurs when the ntal momentum o the ejected electrons s zero. There has been one try to nclude a nonzero ntal momentum o electrons [7], but t dd not draw much attenton at the moment, as ths eect had not had great mpact on the results o the theory because the laser ntenstes n the experments were gvng the elds up to 10 1 Wcm - (atomc eld s approxmately Wcm - ). Now, wth ADK-theory dealng wth eld strengths o the order o atomc, and even hgher [6, 8], ths eect may be expected to gan n ts mportance, so we shall consder the correctons to transton rate n ADK-theory up to the terms wth the eld strength to mnus th as n [9], but wth nonzero ntal momentum, whch s the rst tme ever to nclude ths eect n so hgh orders o correctons. 1. ITRODUCTION In last two decades o XX Century tunnelng onzaton (Keldysh's [1] parameter γ << 1) o atoms and ons by a strong low-requency electromagnetc eld has become the subject o ntensve research. The theory whch was one o the most propulsve ones n the eld was the Ammosov-Delone-Kranov (ADK) theory [-5]. Ths theory uses the concept o a quas-statonary external eld whch s ejectng valence electrons va tunnelng. It has been, at the end o ths perod [6], extended to the barrer-suppresson onzaton o complex atoms and atomc ons,.e. to the case o super-strong elds. But the ADK-theory usually has been used to descrbe the onzaton whch occurs when the ntal momentum o the ejected electrons s zero. There has been one try to nclude a nonzero ntal momentum o electrons [7], but t dd not draw much attenton at the moment, as ths eect had not had great mpact on the results o the theory because the laser

2 4 ntenstes n the experments were gvng the elds up to 10 1 Wcm - (atomc eld s approxmately Wcm - ). Now, wth ADK-theory dealng wth eld strengths o the order o atomc, and even hgher [6, 8], ths eect could have been expected to gan n ts mportance, so we consdered the correctons to transton rate n ADK-theory up to the terms wth the eld strength to mnus th as n [9], but wth nonzero ntal momentum, whch s the rst tme ever to nclude ths eect n so hgh orders o correctons.. TRANSITION RATE IN GENERAL As already mentoned we are dealng wth tunnelng onzaton (γ << 1, [1, 1]). In ths case the transton rate s gven by the expresson [11, 1]: τ [E (t) E (t)]dt 0 W = e e S( τ). (1) Here S(τ) s the classcal acton and E (t) and E (t) are energes o the ntal and nal states as unctons o the tme parameter t. The condton γ << 1 mples that the requency ω o the external (laser) eld s much smaller than atomc requency ω << E (atomc unts system: e = m e = h = 1 s used throughout ths paper), whch justes usng the adabatc approxmaton n obtanng (1). Upper lmt n the ntegral n (1) s the complex turnng pont obtaned rom relaton Here τ E ( τ ) = E ( τ). () s beng complex, as t beng obtaned rom equaton () corresponds to "turnng pont" o Classcal Mechancs, but ths knd o transton s classcaly orbdden whch produces the complexty o the roots o equaton (). E ( τ) = E In the case o lnear eld polarzaton the Keldysh approxmaton [1] gves, where E s the onzaton potental o the ground state o valence electron and r 1 r 1 r 1 r F E ( τ) = p A( ) = p sn ωt + τ c, where F r s the eld strength vector and p r s the ω momentum o an ejected electron. As the transton rate s equal to (see Equaton 1 and [11,1])

3 4 W = e ImS( τ), () one has to calculate the magnary part o the ntegral τ 1 r F S( τ ) = p snωτ + E dt. (4) ω 0 I we take nto account that the probablty o onzaton s the bgest the vector p r were r drected along the eld vector F [see 10,1], and havng n mnd that the lower lmt o ntegraton s real, we have p F pf F Im S = Im + E + τ + cosωτ sn ωτ. (5) 4ω ω 8ω Usually here momentum p was consdered zero, to make the calculatons smpler, but as n realty there should be some nonzero dstrbuton o electron momenta, we shall now take nto account ths correcton.. THE CORRECTIONS TO THE TRANSITION RATE DUE TO NON-ZERO MOMENTA OF EJECTED ELECTRONS Explct calculatons o Equaton () gve us p + κ sn ωτ = ω α, F or, (6) ωτ = arcsn α, where κ = E. As requency ω s very small n the case o the low requency eld, we can consder the newly ntroduced parameter α a small quantty, and expand the above expresson n the ollowng manner

4 44 5 α α ωτ α + +. (7) 6 40 Also, n equaton (5), we may expand n powers cosne and sne under the same condtons, and obtan ω τ ω τ ω τ cosωτ sn ωτ ωτ ω τ + ω τ ω τ. (8) we obtan Ater smoewhat lenthy, but straghtorward calculatons, substtutng (7) and (8) nto (5) κ p κ κ ω κ p κ p κ ω ImS( τ ) = + 5 F F , (9) F where we have collected all the terms correspondng to eld power up to -5, because t was shown earler [10] that these are the only relevant correctons to the basc actor κ, whch F n the expresson or probablty gves the well known short range potental term / (E ) / F [1]. The rst and thrd term o expresson (9) could be combned to gve [10]: κ F 5 κ ω κ κ ω κ γ = 1 = , (10) F F F F κω where we used the act that Keldysh s parameter s gven by [1]: γ =. So, snce γ << 1 F we can neglect the second term n Eq. (10). As t can be shown that atomc eld strength s proportonal to κ [1], or very strong elds one should not take nto account none o the terms n parenthess beore F -5, thus the only term o any relevance wth nonzero momentum beng

5 45 p κ ω p γ K1 = =, (11) 6F 6ω because the small parameters γ and ω may cancel each other and leave some space or nluence o nonzero ntal momentum o the electron, whch s not new [10], but t can now gve greater mpact on probablty w, because the elds are much stronger. And the nluence o the eld on ntal momentum can be seen rom the expresson or the nal energy E o the electron, n a long laser pulse approxmaton: E = p / + F / 4 ω. (11) So, nally, p / = E F / 4ω, (1) we see that the nluence o the eld strngth on the ntal momentum o the electron s consderable, and some eects n the measured probabltes could be expected. 4. CONCLUSION Fnally, we can conclude that ncludng non-zero ntal momenta o ejected electrons n the case o onzaton o atoms and ons by the strong laser elds to the hgher order o correcton then beore [10], gves the results that are already obtaned n earler works [7,10], but as they were estmated or the strong elds, and now superstrong elds strengths [6] can be used ths should nluence ther mact on the electron ejecton probabltes. But exact estmatons we leave or uture work.

6 46 Reerences [1] L.V. Keldysh, Ionzaton n the Feld o a Strong Electromagnetc Wave, Sov. Phys. JETP 0 (1965), [] M. V. Ammosov, N. B. Delone, and V. P. Kranov, Tunnel Ionzaton o Complex Atoms and Atomc Ions by an Alternatng Electromagnetc Feld, Sov. Phys. JETP 64 (1986), [] F. Yergeau, S. L. Chn, and P. Lavgne, Multple Ionzaton o Rare-gas Atoms by an Intense CO Laser (10 14 Wcm - ), J. Phys. B: At. Mol. Phys. 0 (1987), [4] W. Xong, F. Yergeau, S. L. Chn, and P. Lavgne, Multphoton Ionzaton o Rare Gases by a CO Laser Spectroscopy, J. Phys. B: At. Mol. Phys. 1 (1988), L159-L164. [5] W. Xong and S.L. Chn, Tunnel Ionzaton o Potassum and Xenon Atoms n the Feld o an Intense CO Laser, Sov. Phys. JETP 7 (1991), [6] V. P. Kranov, Ionzaton Rates and Energy and Angular Dstrbutons at the Barrersupresson Ionzaton o Complex Atoms and Atomc Ions, J.Opt. Soc. Am. B 14 (1997), [7] V. P. Kranov and V. M. Rstć, Energy Spectra o Electrons n Tunnel Ionzaton o Atoms and Ions by a Strong Low-requency Electromagnetc Feld, Sov. Phys. JETP 101 (199), [8] N. Mloševć, V. P. Kranov, and T. Brabec, Semclasscal Drac Theory o Tunnel Ionzaton, Phys.Rev.Lett. Vol. 89, No. 9 (00), [9] V.M. Rstć and M.M. Radulovć, About the Behavor o Correctons to the Transton Rate or Tunnel Ionzaton o Atoms and Ions by a Strong Low-requency Electromagnetc Feld, Coll.Sc.Pap.Fac.Sc. Kragujevac 0 (1998) [10] V. P. Kranov and V. M. Rstć, Knetc Energy Dstrbuton n Tunnel Ionzaton o Atoms and Ions by Strong Low-requency Electromagnetc Feld, Coll.Sc.Pap.Fac.Sc. Kragujevac 1 (1991), [11] L.D. Landau, E.M. Lshtz, Quantum Mechancs, rd ed., Pergamon Press, Oxord (1997). [1] N.B. Delone, V.P. Kranov, Atoms n Strong Lght Felds, Sprnger-Verlag, Berln (1984).