ISSN 10637761, Journal of Experimental and Theoretical Physics, 2009, Vol. 108, No. 6, pp. 947–962. © Pleiades Publishing, Inc., 2009. Original Russian Text © S.B. Popruzhenko, V.D. Mur, V.S. Popov, D. Bauer, 2009, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2009, Vol. 135, No. 6, pp. 1092–1108. ATOMS, MOLECULES, OPTICS

Multiphoton Ionization of Atoms and Ions by HighIntensity XRay Lasers S. B. Popruzhenkoa, b, V. D. Mura, V. S. Popovc, and D. Bauerb a Moscow Institute of Engineering Physics, Kashirskoe sh. 31, Moscow, 115409 Russia b MaxPlank Institut für Kernphysik, Postfach 103980, 69029, Heidelberg, Germany c Institute for Theoretical and Experimental Physics, ul. Bol’shaya Cheremushkinskaya 25, Moscow, 117218 Russia email: [email protected] Received December 31, 2008

Abstract—Coulomb corrections to the action function and rate of multiphoton ionization of atoms and ions in a strong linearly polarized electromagnetic field are calculated for high values of the Keldysh adiabaticity parameter. The Coulomb corrections significantly increase the ionization rate for atoms (by several orders of magnitude). An interpolation formula proposed for ionization rate is valid for arbitrary values of the adiaba ticity parameter. The high accuracy of the formula is confirmed by comparison with the results of numerical calculations. The general case of elliptic polarization of laser radiation is also considered. PACS numbers: 32.80.Rm, 32.80.Fb, 03.65.Sq DOI: 10.1134/S1063776109060053

1. INTRODUCTION been mainly associated with experiments using high power optical and IR lasers with a wavelength on the Considerable advances have been made in recent 1 years in the physics of the interaction of highintensity order of a micrometer or longer. Accordingly, theo laser fields with matter in connection with the devel retical investigations were also associated mainly with opment and commissioning of powerful sources of low frequencies of the laser field. It should be recalled coherent radiation in the UV and Xray wavelength that electromagnetic field in the theory of multipho ranges. Such sources are based on freeelectron lasers. ton ionization is assumed to be a lowfrequency field if A unique device of this type (FLASH) operates at parameter γ, introduced by Keldysh [15], is much present at the DESY laboratory (Hamburg, Germany) smaller than unity. Ionization in this case is known as [1, 2]. In 2002, the FLASH device generated electro tunneltype ionization. The opposite limit (γ 1) is magnetic radiation pulses at a wavelength of about usually referred to as multiphoton ionization. When 100 nm (photon energy ω ≈ 13 eV) with a duration on atoms and positive ions are ionized by the field of a 13 2 the order of 100 fs and an intensity of up to 10 W/cm highintensity infrared optical laser, the Keldysh [3]. Since 2007, experiments with radiation pulses of a parameter is usually on the order of unity or smaller. wavelength of 13 nm (ω ≈ 93 eV), a duration of about 16 2 The development of highpower Xray lasers extends 10 fs, and an intensity of up to 10 W/cm have been the experimental potential to the range of high fre carried out [4]. In the nearest future, transition to a wavelength of 6 nm is planned and use will be made of quencies and small wavelengths. Typical values of the Keldysh parameter in experiments [4, 7] amounted to the third and fifth harmonics of the fundamental fre γ ≈ quency (the latter harmonic corresponds to a wave 30–100, which corresponds to the multiphoton length of 1.2 nm) [2]. Numerous experiments on the ionization mode. Until recently, much less attention interaction of highintensity coherent rf radiation with has been given to the development of the theory in this atoms, molecules, multiply charged ions, nanostruc range of parameters. In this study, we analyze mul tures, and solids were performed at the DESY labora tiphoton ionization of atoms and positive ions in a tory in 2002–2007 using freeelectron lasers (see [4–9] strong highfrequency (rf) field (i.e., for γ 1) and and the literature cited therein). derive analytic expressions for the total ionization probability per unit time. The ionization rate calcu One of the fundamental nonlinear processes lated using these formulas is compared to the result of induced by a highintensity laser field is multiphoton ionization of atoms, which was observed for the first exact numerical calculations. time in 1965 [10] (on the modern state of the art, see 1 Among powerful lasers, the most widely used are titanium–sap reviews [11–14]). Until recently, the physics of inter phire, neodymium, and CO2 lasers with wavelengths of 0.79– action of highintensity laser fields with matter has 0.80, 1.06, and 10.6 µm, respectively.

947 948 POPRUZHENKO et al.

In analyzing multiphoton ionization of atoms by It should be emphasized that ionization potential I highintensity laser radiation, the Keldysh theory [15], and asymptotic coefficient Cκ in expression (2) corre also known as the strong field approximation [16], is spond to the atomic state for which the ionization widely used. In this approximation, the electromagnetic probability is calculated rather than to the state in a field of a wave is taken into account exactly, and the Cou shortrange potential. Finally, the Keldysh parameter lomb interaction of an emitted electron with the atomic is defined as core is disregarded, which makes it possible to derive 2mIω 1 convenient analytic formulas for the ionization probabil γ == , (4) ity and momentum spectra of photoelectrons (see [17, e 2K0F 18] and the literature in [12]). For ionization of negative ω ions, the strong field approximation leads to good quan where K0 = I/ is the multiquantum parameter of the titative agreement with the results of exact numerical cal process. In the Keldysh theory, it was assumed that culations and experimental data (see, for example, [19] K 1, F 1, (5) and the literature therein). At the same time, in the case 0 of neutral atoms and positive ions, the Coulomb interac here, parameter γ can assume arbitrary values. Hence tion suppresses the potential barrier through which an forth, we will use atomic units: = e = m = 1. electron tunnels, which considerably increases the ion Expression (1) for the Coulomb correction to the ization probability. For example, in a constant electric ionization rate is applicable not only in the case of a field of strength , ionization rate w for the s state of an constant field, but also in the tunnel mode (γ < 1) for a atom differs from analogous quantity wsr for an energy plane electromagnetic wave with an arbitrary polariza level in the shortrange well [12, 20, 21] (with the same tion. Usually, this correction increases the tunnel ion ionization potential I) by a factor of [22] ization probability by several orders of magnitude (this () effect was reliably established in experiments; see, for (), ω ≡ () w QC = 0 Q0 = example, [24]). At present, the expressions for tunnel wsr() (1) ionization rate taking into account the Coulomb cor ⎛⎞22κ3 2n* ⎛⎞2 2n* rection (and also known as the Ammosov–Delone– 1, = ⎝⎠ = ⎝⎠ Krainov formulas [25]; see [26]) are widely used for me F calibrating the laser pulse intensity. where In this study, we obtain the Coulomb correction ω κ2 QC( , ) in the opposite limit of high frequency, () 2 ⎛⎞2 ch γ 1, in the simplest case of ionization of the s state wsr = Cκ exp⎝⎠– . (2) m ch 3 of an atom or a positive ion by a linearly polarized field (ρ = 0). In this case, the formulas for tunnel 2κ3 Here, k = 2mI / and ch = /me are the character ionization probability are completely inapplicable istic momentum and the electric field of the atom (for the even for obtaining qualitative estimates. For exam ≡ 2 5 4 16 2 ground state of the hydrogen atom, ch a = m e / = ple, for an intensity on the order of 10 V/cm used 9 2 2 –1/2 5.14 × 10 V/cm), n* = e m/ k = (I/IH) is the in experiment [4], successive tunnel ionization effective principal quantum number of the level (I = leads to the formation of ions up to Xe8+–Xe9+ H 21+ me4/22 = 13.6 eV is the ionization potential of the instead of observed Xe ions, and the tunnel ion ization probability for the Xe10+ ion for the given hydrogen atom), m and –e are the electron mass and –24 charge, and is the charge of the atomic core ( = 1 for intensity amounts to a value on the order of 10 for a pulse with a duration of about 10 fs. In our calcu neutral atoms and = 0 for negative ions H–, Na–, etc.). lations, we use the imaginary time method (ITM), Ratio F = / ch is often referred to as the reduced field, in which subbarrier trajectories satisfying the classi ⎛⎞I –3/2 cal equation of motion, but in imaginary time, are F = 0.169 , (3) 2⎝⎠ considered for describing quantum tunneling 1 + ρ IH through a barrier [27, 28]. In the derivation of the where ρ is the ellipticity of radiation (–1 ≤ ρ ≤ 1; ρ = 0 Coulomb correction, we will use the Kapitza and ±1 corresponds to the linear and circular polariza method [29, 30] for describing the averaged motion tions of the wave, respectively), and is the intensity of a particle in a rapidly oscillating field. The appli measured in units of 1015 W/cm2. Finally, C is the cation of the ITM for analyzing Coulomb effects in κ the spectral–angular distributions of photoelec dimensionless asymptotic coefficient of the atomic 2 trons in a highintensity laser field is considered in wavefunction at distances r /κ from the nucleus. detail in [31]. The article has the following structure. In Section 2, 2 For the ground states of neutral atoms and singly charged ions, the method for calculating the Coulomb correction in these coefficients are numerically close to unity [12]; for exam a linearly polarized rf field is formulated and an ple, Cκ = 1, 0.95, and 1.04 for the H, K, and Na atoms, respec – tively, and Cκ = 1.15 for the H ion. A simple approximate for explicit expression is derived for this correction. In mula for coefficients Cκ was proposed by Hartree [23]. Section 3, the results are analyzed and compared with

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 949 the results of numerical calculations. In Section 4, the The trajectory begins at a complex instant ts(p) that general case of elliptically polarized field is consid can be determined from the equation ered. In Section 5, the applicability conditions for the 2 derived analytic formulas are formulated. Appendices ⎛⎞1 2 p + A()t + κ = 0. (9) contain the description of the Kapitza averaging method ⎝⎠c s applied to the photoionization problem, details of calcu lations for a linearly polarized field, and a brief descrip The same equation also defines the initial complex tion of numerical methods used for controlling the accu velocity v0(ts) = p + A(ts)/c. The trajectory itself has the racy of the analytic formula derived here. Preliminary form results of this study were announced in [32]. 1 v ()t = p + A()t , 0 c (10) 2. COULOMB CORRECTION IN A LINEARLY () ()() () POLARIZED FIELD r0 t = X0 ++p tt– s G t – G ts .

2.1. Imaginary Time Method in the Ionization Problem Quantity X0 has the meaning of the initial point of sub The probability of ionization of a level with a bind barrier motion, X0(p) = r0(t = ts(p)). Usually, we set ing energy of –I by an electromagnetic field with X0 = 0; in this case, the trajectory in real time for a nonzero value of momentum p has a constant imagi amplitude 0 and characteristic frequency ω in the ITM is controlled (with an exponential error) by the nary part [28]. imaginary part of a function of truncated action calcu To calculate the total ionization probability with an lated along the trajectory on which an electron has a exponential accuracy, it is sufficient to find action (7) preset momentum p [18, 28]: along the trajectory corresponding to the most proba

w()0,,ω p ∝ exp[]–2ImW()0,,ω p . (6) ble value of finite momentum p0. We will henceforth consider a field linearly polarized along the x axis, For a particle bound by shortrange forces, the action and the trajectory are controlled by the field of the ()t = –0 cosωt; (11) electromagnetic wave. In a wave with vector potential · 4 A(t) and electric field (t) = –A (t)/c (c is the velocity in this case , p0 = 0. The corresponding trajectory is of light), truncated action W0 and Lagrange function onedimensional and directed along the polarization 3 vector: 0 have the form ∞ x ()τ = 0()coshτ – coshτ , () () 0 2 0 W0 = ∫ 0 – I dt + F0 p , ω t ()p s τ0 ≥ τ ≡ –iωt ≥ 0, (12) 1 2 () 0 = v0 – t r0, (7) x ()θ = 0()coshτ – cosθ , θω≡ t ≥ 0, 2 0 ω2 0 () ()⋅ t → ∞ ⋅ []() F0 p ==– v0 r0 ppts + G ts , tt= s while the stationary point corresponding to it has the 5 where form t ω ()τ 1 ts p0 = 0 = i 0, G()t = dt'A()t' c∫ ⎧γ, γ 1, (13) and the value of the function at point t ∞, with a τ0 = arcsinhγ ≈ ⎨ purely real contribution, which does not affect the ⎩ln()2γ , γ 1. probability, is omitted on the righthand side of the expression for F0(p). Subscripts “0” indicate that all Quantity quantities are evaluated taking into account the laser field alone; i.e., the trajectory is a solution to the New 4 To be more precise, the most probable energy of a photoelec ton equation disregarding the Coulomb interaction, tron is not necessarily zero, but it does not exceed the photon energy so that 0 ≤ p < 2ω and p /κ < 1/K 1. r·· v· (). (8) 0 0 0 0 ==0 – t 5 Here, we have chosen the trajectory along which the electron moves to the right (in the positive direction of the x axis) so that 3 These formulas are one of the main results of the ITM in the (τs) = –0coshτ0 < 0. A symmetric trajectory passing to the theory of ionization of atoms by an ac electric field. The deriva left of the atom begins at the instant ωts = π + iτ0 and makes the tion of these formulas can be found in [27, 28]. same contribution to the ionization probability.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 950 POPRUZHENKO et al. Another correction appears due to distortion of the ≡ ()θ 0()γ2 bx0 = 0 = 1+ – 1 electron trajectory by the Coulomb field. If correction ω2 r1(t) to the trajectory has been determined in a partic (14) ular way (e.g., numerically), the corresponding cor ⎧()I/ ()1 – γ2/4 , γ 1, ≈ ⎨ 0 rection to the action is –1 ⎩()2I/ω ()1 – γ , γ 1 ∞ ⋅ 1 2 () has the meaning of the tunnel barrier width. An elec WC1 = v0 v1 + v1 – t r1 dt + F1, (16) ∫ 2 tron begins moving in real time at this point. It should t be noted that if conditions (5) hold, the barrier width s is always much larger than the size of the bound state, where κb 1, which ensures the applicability of the semi t → ∞ classical approximation (ITM in our case). F1 = –()v0 ⋅ r1 + v1 ⋅ r0 + v1 ⋅ r1 . (17) tt= s To calculate correction (16), we first consider the 2.2. Coulomb Corrections part of the trajectory behind the barrier for which time is real. For γ 1, we can average over fast oscillations The Coulomb interaction between an emitted elec in the laser field and consider only the smooth motion tron and the atomic core can be taken into account in the Coulomb field [33]. In this approximation, the assuming that the trajectories and action introduced problem becomes conservative, and the energy con above are the zeroth approximation and calculating servation law gives the relation between momentum p|| corrections to these quantities [22]. Naturally, this at the exit from under the barrier (for the 1D trajectory method is applicable when the Coulomb field is small considered here, the momentum is directed along the in a certain sense as compared to the laser field. The smallness criterion will be formulated below (see also polarization vector of the wave) and momentum p0 = 0 [31, 33]). In this case, two types of corrections appear. at infinity. Details of calculations are given in Appen dix A, and the results have the form First, Coulomb interaction energy UC = –/r leads to the following correction to action: 2 p|| ===p + 2/b 2n*ωκ2µ. (18) ∞ ∞ 0 ()() dt WC0 ==–∫UC r0 t dt ∫. (15) This formula contains a small parameter 2() t t r0 t s s ω n* µ == 1, (19) 3 Since this correction explicitly contains a “small” κ 2K0 Coulomb interaction (it is proportional to charge ), the integral can be evaluated along a trajectory unper determining the contribution of the Coulomb field to turbed by the Coulomb field. At the lower limit, the the distortion of the electron trajectory. integral diverges logarithmically and requires match τ ≤ τ ing to the asymptotic form of the wavefunction of a In imaginary time (i.e., for 0 < 0), the Coulomb free atom. The matching procedure is described in correction to trajectory (12) can be determined from detail in [22, 31, 33] (see Section 4 below, in which the equation correction (15) is calculated in the case of elliptic polarization). For a linearly polarized field, such a cal x·· = – . (20) 1 2 culation leads to familiar result (1). It turns out that x0 correction (1) has the same form in the tunnel and multiphoton limits and is formally independent of fre The solution to this equation corresponding to quency altogether; however, this applies only to fields momentum (18) at instant ωt = 0 is obtained in 6 Appendix A (see also [33]): with linear polarization. This correction always ()τ () increases the ionization probability. In the tunnel x0 = b 1 – u , mode, the effect of increasing ionization probability can easily be interpreted since the Coulomb field low ()τ ⎛ µτ ers the barrier through which an electron tunnels. The x1 = ba⎝ + i 2 fact that expression (1) for the Coulomb correction is (21) “pulled” into the multiphoton region was established 1 2 ⎞ + µ τ – ln()1 – u + L ()u , for the first time in [22]. 2 2 ⎠ 6 This statement is correct for a strictly monochromatic field with ()ττ a constant temporal envelope. For short pulses, the correction is u = exp – 0 , a function of frequency even in the case of linear polarization [33]; see also Appendix D. where L2 is the Euler logarithm [34],

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 951

u ∞ 3. DISCUSSION OF RESULTS () k () ln 1 – t u AND COMPARISON WITH NUMERICAL L2 u ==–∫dt ∑ t k2 CALCULATIONS 0 k = 1 Coulomb correction (25) contains two factors, one 1 2 3 ⎧ u ++ u Ou(), u 0, of which, Q0, is independent of the adiabaticity ⎪ 4 parameter and the other, Q 0, tends to unity for = ⎨ 1 2 γ γ ⎪π 1 and Q1 1 for 1 as follows from (23). Thus, 1++()– u []…ln()1 – u – 1 , u 1. ⎩ 6 Coulomb correction Q1 obtained here effectively low ers the ionization probability in the multiphoton It can be seen that the correction to the trajectory in region; however, total correction (25) remains numer imaginary time is complexvalued. Realvalued con ically large: stant a vanishes for µ 0 and does not change the ≈ ()2n* γ result. QC 2eK0 1, 1. (26) The contribution from the correction to trajectory (21) The ionization rate for the s state in the shortrange to action (16) is calculated in Appendix B. The imagi potential is given by [12, 17, 18] nary part of the action is given by 2 κ 2 –3/2 1/2 2 β () κ wsr = CκK0 β()NN– µ τ π ∑ th ImWC1 = 0. (22) ≥ ω NNth (27) Substitution of this quantity into the exponent in rela ⎧⎫2g()γ tion (6) leads to the emergence of an additional factor × exp⎨⎬– – 2c1()NN– th , ⎩⎭3F Q ==exp()–2ImW () 2γ –2n*, γ 1. (23) 1 C1 where in the ionization probability. Simple, although not rig orous, analysis proves the correctness of asymptotic γ2 ()γ 3 ⎛⎞1 γ 1 + form (23) for Q1 and even makes it possible to refine it. g = 1⎝⎠+ acrsinh – , 2γ 2γ2 2γ In accordance with relation (18), p|| ≥ κ 2µ at the instant when the electron emerges from under the bar 2γ β = , (27') rier (the lower boundary corresponds to the photoion γ2 ization threshold). Such a momentum is required for 1 + the electron to overcome the Coulomb attraction dur x ing its motion above the barrier averaged over fast () ()2 2 (), 2 x ==∫dyxexp – + y x1F1 1 3/2; –x oscillations. On the other hand, the probability of extraction of the electron from the potential well 0 γ2 (shortrange potential) decreases upon an increase in is the Dawson function [35] and Nth = K0(1 + 1/2 ) is longitudinal momentum p|| [12, 18]: the threshold number of quanta required for ioniza tion of the level. ⎧⎫2 7 p|| The ionization rate of the state is given by wsr()p|| = wsr()0 exp⎨⎬–c1()γ , s ⎩⎭ω (24) wQ= Cwsr. (28) γ In the lowfrequency limit γ 1, correction Q1 1, c1()γ = arcsinh γ– . 2 and expression (28) is transformed into the familiar 1 + γ formula for the tunnel ionization rate for the s level in 2 2 2 a lowfrequency field [22, 25], Substituting p|| = 2µκ and µκ /ω = n* into this 2n* – 1 γ γ 2 2 ⎛⎞ ⎛⎞ expression and considering that c1( ) = ln(2 ) – 1+ κ 3F 2 2 w = 2Cκ ⎝⎠ exp⎝⎠– , (29) O(γ–2) for γ 1, we immediately arrive at relation (23). π F 3F Thus, the total Coulomb correction can be written which differs from relations (1) and (2) by a factor of in the form 3F/π . This additional factor [18] is connected with () , ω {}()γ QC 0 ==Q0Q1 Q0 exp –2n*c1 , (25) the time variation of the field (it should be recalled that which is valid for arbitrary values of adiabaticity ionization rate (2) corresponds to a static field). parameter γ. Consequently, although result (25) was obtained It should be noted that the imaginary part of func for γ 1, it ensures the correct static limit γ 0. In tion F1 (17) was not taken into account in [33], which 7 led to an erroneous result. Formula (25) derived above It should be noted that the ionization rate for an atom in [36] was also calculated using formula (28); probability wsr was deter and having the same meaning as formulas (14) and mined numerically for arbitrary values of parameter γ, but the (20) in [33] corrects this inaccuracy. Coulomb correction was calculated in tunnel limit (1).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 952 POPRUZHENKO et al.

20 10 5 2 γ (25) in a simpler form convenient for obtaining esti 0 mates: 2n* (a) ⎛⎞2 –1 –2n* Q ≈ ()12+ e γ , e = 2.718… (30) C ⎝⎠F −4 In the limit γ 1, correction (26) is independent of intensity and is numerically large. Far from the closure

w 8 of multiphoton ionization channels, the Dawson

log − 8 function in (27) is on the order of unity and the largest contribution to the sum comes from the first term; i.e., ionization occurs with absorption of the minimal pos −12 sible number of photons, Nmin . The expression for ionization rate in this case assumes the following sim ple form: 1012 1013 1014 2 I 2Nmin J, W/cm w ≈ AK(), n* F , 0 20 10 5 2 γ (31) 0 N + 2n* 2N + 2n*3/2– AK(), n* ≈ 22n*C2 e min K min . (b) 0 κ 0 Thus, a power law typical of perturbation theory of order N is reproduced, which is natural in the mul −4 min tiphoton mode [12, 15]. It is important, however, that we have also obtained an explicit value of coefficient w A(K0, n*), which is in good agreement with the results

log −8 of numerical calculations (curves in Fig. 1). This coef ficient is numerically large and is the product of large numbers; consequently, it is quite sensitive to even insignificant changes in parameters. Coulomb correc −12 tion (25) makes a significant contribution to the value of A. For parameters K0 ≈ 5–10 and n* ≈ 1–3 typical 1012 1013 1014 of ionization of atoms and ions in a strong field, the 2 2 10 J, W/cm value of QC varies in the interval 10 –10 . Factor Q1 Fig. 1. Ionization rate (in atomic units; 1 a.u. = 4.13 × responsible for the deviation from the tunnel limit is 16 –1 –9 –1 10 s ) calculated by formulas (28) (solid curve), (27) also of fundamental importance: Q1 ≈ 10 –10 . (dashed curve), and tunnel formula (29) (dotted curve), as It should be noted that a simple analytic expression well as determined numerically by the Floquet method (᭹) and from the solution of the timedependent Schrödinger for the ionization rate of atoms in the multiphoton equation (᭡) as a function of intensity for the 1s state of the mode was derived by Bersons in [37] for n* ≥ Nmin. For hydrogen atom in the field of a linearly polarized wave of K 1, this condition indicates that we are dealing λ 0 lengths = 400 nm (a) and 422 nm (b). Parameters K0 are with ionization of the upperlying Rydberg level and 4.39 and 4.63, respectively. Intensities of 1014 and that parameter (19) introduced above is large, µ > 1. 13 2 10 W/cm corresponds to field amplitudes 0 = Our results are applicable in the opposite limit µ < 1, 0.053a and 0.0169a. The Coulomb correction is QC ≈ K0 n* and, hence, do not ensure the limiting transi 6 × 102. tion to onephoton ionization, Nmin = 1. Conversely, the Bersons formula for Nmin = 1 is transformed into an intermediate range of γ ≈ 1, formula (25) for the the wellknown Kramers relation for the photoeffect Coulomb correction can be treated as an interpolation [38]. In addition, the ionization rate determined in formula. Since quantity QC is a monotonic function of [37] is averaged over the orbital quantum number l of the parameters, the accuracy of such an interpolation Rydberg level, while our result corresponds to l = 0. As a turns out to be quite high, which is confirmed by com consequence, the expression derived in [37] for a coef parison with the results of numerical calculations ficient analogous to A(K0, n*) differs substantially given below. Thus, expressions (25), (27), and (28) from relation (31). define the ionization probability of the s states of Tables 1 and 2 illustrate the effect of Coulomb cor atoms and positive ions for arbitrary values of the adi rections on the ionization rate of atoms and ions in the abaticity parameter (provided that ionization is of the field of IR and Xray lasers. multiphoton type; i.e., 1). It should also be noted K0 8 that exp[–2n*c (γ)] ≈ (1 + 2e–1γ)–2n* to a high degree The closure of a channel is a change in the minimal number of 1 photons required for ionization, Nmin = [K0] + 1, by unity ([K0] of accuracy, which allows us to represent correction is the integral part of number K0); i.e., Nth = Nmin.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 953

Table 1. Coulomb corrections for neutral atoms µγ Atom I, eV FK0 n* QC Q1 wsr w K 4.34 0.094 2.77 1.78 0.321 1.90 8.2(+3) 0.16 3.0(–5) 0.25 Na 5.14 0.073 3.29 1.63 0.248 2.06 7.5(+3) 0.15 1.1(–5) 8.3(–2) Li 5.39 0.068 3.48 1.59 0.228 2.12 7.1(+3) 0.15 6.4(–6) 4.6(–2) I 10.5 0.025 6.77 1.14 0.084 2.96 3.0(+3) 0.14 2.0(–11) 6.1(–8) Xe 12.1 0.020 7.81 1.06 0.068 3.18 2.4(+3) 0.14 3.9(–13) 9.4(–10) H 13.6 0.017 8.77 1.00 0.057 3.37 2.0(+3) 0.14 7.5(–15) 1.5(–11) He 24.6 0.007 15.8 0.74 0.023 4.53 7.2(+2) 0.16 5.1(–28) 3.7(–25)

Note: The table contains parameters F, K0, n*, µ, and γ, as well as Coulomb correction (25), correction Q1, ionization rate (27) from the shortrange potential with the same value of binding energy as in the corresponding atom, and analogous quantity (28) taking into account the Coulomb correction for neutral atoms in the field of a titanium–sapphire laser (ω ≈ 1.55 eV, λ ≈ 800 nm) for 13 2 b intensity J = 10 W/cm . Coulomb correction Q1 was taken in the form Q1(γ) = exp[–2n*c1(γ)]. Notation a(b) ≡ a × 10 is used. The ionization probability during the time of action of the pulse can be obtained by multiplying the reduced ionization rate by the pulse duration in atomic units: 100 fs ≈ 2.5 × 103 a.u. In the case of K, Na, and Li atoms for which the value of this quantity is greater than unity, saturation of single ionization occurs before termination of the pulse.

Table 2. Coulomb corrections for positive ions

Ion I, eV FK0 n* µγQC Q1 wsr w Xe2+ 32 0.015 2.52 1.96 0.388 13 2.7(+4) 1.3(–4) 1.2(–8) 3.5(–4) Xe3+ 47 0.0083 3.70 2.15 0.291 16 4.0(+5) 2.3(–5) 6.2(–12) 2.5(–6) Xe4+ 60 0.0057 4.72 2.38 0.252 18 5.1(+6) 4.1(–6) 3.5(–15) 1.8(–8) Xe5+ 72 0.0044 5.67 2.61 0.230 20 5.8(+7) 7.8(–7) 1.9(–18) 1.1(–10) Ar2+ 41 0.010 3.23 1.73 0.268 15 2.0(+4) 2.4(–4) 2.1(–11) 4.1(–7) Ar3+ 60 0.0058 4.72 1.91 0.202 18 2.3(+5) 4.9(–5) 4.5(–15) 1.1(–9) Note: The same as in Table 1 for positive ions in the field of an Xray laser with a photon energy ω ≈ 12.7 eV (λ ≈ 98 nm) and an intensity of 1014 W/cm2.

It can be seen from Table 1 that in the case of ion To estimate the accuracy of expression (25), we ization of atoms by the field of a titanium–sapphire compared the ionization rate calculated using formula laser, the adiabaticity parameter is not very large (γ ≈ (28) with the results of numerical calculations [32]. 9 Figure 1 shows the logarithms of the ionization rate for 2–3) for reasonable values of intensity. Accordingly, a hydrogen atom in the 1s state in field (11) of the sec the deviation of the Coulomb correction from its tun ond harmonic of a titanium–sapphire laser with a ω nel limit (1), characterized by quantity Q1, is insignif photon energy of = 3.10 eV and in the field with a slightly lower frequency (ω = 2.94 eV). Numerical icant (Q1 ≈ 0.1; see Fig. 2 below). The situation changes radically when we pass to multiply charged results were obtained by two different methods, briefly described in Appendix C. The ionization rate calcu ions of noble gases in the field of an Xray laser (see lated by the complex quasienergy (Floquet) method Table 2): the ionization rate remains accessible for is shown by dark circles. Triangles correspond to the observation for γ ≥ 20. Coulomb corrections increase results obtained from numerical solution of the time in absolute value mainly due to an increase in the dependent Schrödinger equation in the field of a short effective principal quantum number n* (this effect can pulse with a duration of ten field periods. The obvious be traced even in Table 1). As a result, Coulomb cor perfect matching of the results of two independent rections increase the ionization probability by 5–7 numerical calculations in Fig. 1 point to the reliability orders of magnitude or more (see Fig. 3 below). The of this method. deviation of the Coulomb correction from the tunnel For not very high values of the adiabaticity param limit is also significant. eter, the difference of Coulomb correction QC from Q0 is insignificant, which is demonstrated by the curves in 9 Parameter γ increases upon a decrease in intensity, but the ion ization probability simultaneously becomes so low that its analy Fig. 2 plotted for ionization of the hydrogen atom by sis is meaningless. It can be seen from Table 1 that in the case of the radiation field of a Nd laser with a wavelength of helium, this is observed even for intensity = 1013 W/cm2. 1064 nm. Here, the results of numerical calculation of

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 954 POPRUZHENKO et al.

43 2 γ 8040 20 10 5 γ

0 −8

−4 w w

log −12 − log 8

QC −12 −16 −16 0.4 1 4 1015 1016 1017 1018 J, 1013 W/cm2 J, W/cm2

Fig. 3. Same as in Figs. 1 and 2 for the Xe17+ ion (I = 434 eV, 4p0 shell) in the field of a Xray laser with wave Fig. 2. Same as in Fig. 1 for a Nd laser with a wavelength of length λ= 13.3 nm (ω ≈ 93 eV) [4]. Parameters K0, n*, 1064 nm. The thin solid line shows the ionization rate cal and µ are 4.67, 3.19, and 0.34, respectively; Coulomb cor culated taking into account correction Q alone. 8 0 rection QC ≈ 9 × 10 . the ionization rate by the Floquet method are even 1013 W/cm2 (i.e., under the conditions of experiment closer to the analytical curve plotted by taking into [7], in which charge states were observed up to Xe6+ account only correction Q0 as compared to the curve inclusive). In [39], multiphoton cross sections calculated using formula (28). N σ = w /j min , Figure 3 shows the ionization rate of the 4p0 state of Nmin Nmin the Xe17+ ion calculated using formulas (25), (27), and (28) and determined by numerical integration of the were determined numerically, where Nmin is the mini oneparticle timedependent Schrödinger equation mal number of photons required for ionization (intro for an electron in the effective potential in the electric duced above) and j = c2 /8πω is the photon flux den field of a wave (see Appendix C). Comparison of the 0 analytic and numerical results shows good quantitative sity. In experimental conditions [7], the flux density ≈ × 30 –2 –1 agreement even in the case of xenon, although param was j 5 10 cm s . Knowing the cross sections, eter µ = 0.34 is not very small in this case, and the we can easily calculate the corresponding probabilities bound state is characterized by a unit angular momen and compare them with analogous values calculated tum, while analytic results were obtained for the s using formula (31). Comparison of our results with the states. Deviations of analytic results from numerically probabilities calculated from the results of [39] dem determined ionization rates are mainly associated with onstrate satisfactory agreement: formula (31) gives 3+ ≈ × –10 4+ ≈ × –13 resonances disregarded in the given theory. The fact w4(Xe ) 4.2 10 a.u. and w5(Xe ) 4.0 10 a.u., that it is resonances that are responsible for the dis while the corresponding values of these quantities 3+ –9 crepancy between numerical results and the theory is determined in [39] are w4(Xe ) ≈ 2.7 × 10 a.u. and 4+ –12 confirmed by comparison of Figs. 1a and 1b. In the w5(Xe ) ≈ 8.3 × 10 a.u. The probabilities calculated former case, fourphoton resonance with one of the from formula (31) are lower than the numerical results Rydberg levels with the position controlled by the obtained in [39]. In our opinion, the systematic dis intensity due to dynamic Stark shift is observed in the crepancy is connected with the method of determining range of intensities (2–3) × 1013 W/cm2. In Fig. 1b, the cross sections from these numerical calculations: the wavelength that does not correspond to any specific authors of [39] used the complex absorbing potential laser is chosen so that there is no resonance. The coin operating at distances r > R0 = 4 a.u. from the nucleus. cidence of analytic and numerical results is almost As a result, the electrons populating the upperlying perfect. Rydberg levels were interpreted as ionized, which Apart from the numerical results considered above, effectively increased the ionization cross section. As we are aware of only one example of calculations of the shown in recent experiment [40], the probability of multiphoton ionization probability of positive ions in exciting an electron to a Rydberg state during mul the radiation field of a highintensity Xray laser, per tiphoton ionization by the field of a highintensity formed in [39] for ionization of xenon by the field with short pulse is not low at all. In the numerical calcula a photon energy of ω = 12.7 eV and an intensity of tions used here, electrons excited to Rydberg states are

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 955 not treated as ionized (see Appendix C), while analytic C(γ, ρ) results generally do not take into account the presence ρ = 0 of the bound excited state. 1.0 0.25

4. ELLIPTIC POLARIZATION 0.8 0.50 4.1. Monochromatic Field Let us now consider the correction to action (15) 0.75 0.6 and the corresponding correction Q0 to the ionization probability in the general case of elliptic polarization. The contribution to the function of action emerging 0.90 due to the Coulomb interaction energy and regular 0.4 0.95 ized by matching to the asymptotic form of the atomic 1.00 wavefunction has the form

τ 0.2 ∗ τ 01020304050 []κ ()τ d γ WC0 = – in*ln r0 * – i ∫ . (32) ω 2()τ 0 r0 Fig. 4. Factor C(γ, ρ) in Coulomb correction (33) as func τ τ τ τ tion of γ for various values of ellipticity ρ. Here, ∗ is the matching point such that 0 – ∗ 0 κ τ and r0( ∗) 1 [22, 33]. The subbarrier trajectory trajectory r0 can be found analytically: r0(τ) and the equation for the imaginary initial time τ0 correspond to = 0. Following [22], it is convenient ⎧ to write the formula for the Coulomb correction in the r ()τ = 0⎨()coshτ – coshτ e 0 2 0 x form ω ⎩ (35) 2n* ⎧⎫ ⎛⎞τ ⎫ 2 ()γρ, + iρ sinhτ – sinhτ e ⎬, Q0 = ⎨⎬C , (33) ⎝⎠τ 0 y ⎩⎭F 0 ⎭

2 where 2 2⎛⎞1 2 sinh τ 1 – ρ cothτ – = γ . (36) 0 ⎝⎠0 τ τ 0 C()γρ, = 0 2γ Numerical calculations based on Eqs. (33'), (35), and (36) shows that (Fig. 4) factor C(γ, ρ) decreases with ⎧⎫τ γ 0 (33') increasing , the increase being especially strong for a ⎪⎪⎛⎞1 1 × exp⎨⎬⎜⎟ – dτ nearly circular polarization. Substituting Eq. (35) into ∫ τ τ (33) and (33'), we arrive at the expression ⎪⎪⎝⎠ξ2()τ 0 – ⎩⎭0 0 Q = exp{}–2ImW 0 C0 (37) {}[]()τ ()γρ, and ξ0 = ωr0/κ. The latter expression was derived by = exp 2n* ln 2K0 0 + J , regularizing formula (32) and does not contain arbi where trary matching point τ∗, which is convenient for 1 ⎧ numerical calculations. It can easily be shown that γ ()τ []τ ()2 |ξ τ | τ τ τ τ J = ∫ds⎨ [ cosh 0 – cosh 0 1 – s 0( ) = 0 – + … for 0; consequently, singu ⎩ larities of the integrand in formula (33') are cancelled 0 (38) out and the integral always converges. It should be ⎫ ρ2()()τ []τ ()2 ]–1/2 1 noted that expressions (32), (33), and (33') are valid – 1 – s sinh 0 – sinh 0 1 – s – ⎬, s not only for an arbitrary polarization, but also for an ⎭ arbitrary (not necessarily monochromatic) pulse. s = 1 – τ/τ0. For ρ = 0, the integral can be evaluated For a monochromatic elliptically polarized field analytically, J = ln(2γ/τ0), which gives familiar expres sion (1) for the Coulomb correction in a linearly () ()ω ρ ω t = 0 ex cos t + ey sin t (34) polarized field.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 956 POPRUZHENKO et al.

J(γ, ρ) and Coulomb correction (37) assumes the form 2n* 10 Q0 ≈ ()2K0 exp()πn* 2lnγ . (42)

ρ = 0 8 4.2. Adiabatic Limit 0.5 If γ 1 (which is usually observed for optical and 6 IR lasers), all functions in expressions (35) and (38) 0.8 2 can be expanded into power series in τ, τ0, and γ . After simple but cumbersome calculations, we obtain [41] 4 2 1 2 1 ⎛⎞25 2 4 1.0 C()γ, ρ = 1 – ρ γ – 1 – ρ γ + … , (43) 18 45⎝⎠72 2 which matches the results in Fig. 4 for γ < 2 with an error on the order of 1%; consequently, in this range of parameters, we can use expansion (43) without evalu 0 12345 ating integral (38). Expression (43) can be generalized logγ to the relativistic case (i.e., for a level with a binding 2 Fig. 5. Plots of functions (38) (solid curves) and (41) (dot energy comparable to rest energy mc ). In this case, the ted curves) of angle γ for various values of ellipticity ρ. For coefficient of γ2 no longer vanishes for linear polariza ρ = 0.5, the curves are practically indistinguishable in the tion [41]. entire range of parameter γ; for lower values of ellipticity, the difference between the exact result and the curve depicting formula (41) is even smaller. 4.3. Plane Wave of Arbitrary Shape Above, we considered ionization by the field of monochromatic wave (11); however, formulas (33) In the multiphoton limit τ0 1, we have the asymptotic form and (33') are also valid in a more general case. Let us τ ()γρ, suppose that 0 ()t = 0ϕθ(), –∞ < θω≡ t < ∞, (44) ⎧ln()2γ/ 1 – ρ2 , 1 – ρ2 1/ln()2γ , (39) where function ϕ(θ) defines the shape of the pulse. To = ⎨ evaluate Coulomb factor C(γ, ρ), we can substitute ⎩ ()γ γ ρ2 ln2ln , = 1 into Eq. (33') the subbarrier trajectory ξ0(τ) for = 0, (asymptotic form τ0 is more complicated in a narrow which can be determined from the equations given in transition region in the vicinity of ρ = ±1). In this case, Appendix D. For γ 1, the ionization probability is expression (38) is simplified: controlled by the behavior of field (τ) in the vicinity of its maximum. Assuming that 1 ⎧ ≈ []τ2 ρ2()τ 2 1/2 ()()τ 2 a a J ∫ds⎨ 0 – 0 – 1 [ 1 – exp – 0s ϕθ() 2θ2 4θ4 … > ⎩ = 1 – ++ , a2 0, (45) 0 2! 4! (40) for θ 0, we obtain (see [42] for details) –1/2 ⎫ ρ2()()τ 2 ] 1 1 2 4 6 – 1 – s – exp – 0s – ⎬. C()γ = 1 ++()γa – a O()γ . (46) s ⎭ 216 4 2 For γ 1, we can write the following simple interpo It should be noted that this expansion begins with γ4 lation formula for integral (40): (and not with γ2), which, however, is a peculiarity of linear polarization (cf. formula (43)). As a rule, the 2 2 2 arcsinρ J()γρ, = τ– ρ ()τ – 1 coefficient of γ4 is numerically small (see examples in 0 0 ρ Appendix D); consequently, the Coulomb correction (41) πρ2 in domain γ ≤ 1 can be used in the same form (1) as in – – lnτ . 2 2 0 the case of a constant field. 2[]τ0()ρ1 – ρ + The curves plotted in Fig. 5 indicate that this formula 5. CONCLUSIONS for γ ≥ 10 ensures high accuracy for all values of ρ; for a polarization far from circular (ρ ≤ 0.5), expression The analytic formulas for Coulomb corrections (41) matches exact result (40) up to γ ≈ 1. have been derived by the method of perturbation the In the case of circular polarization, we have ory in the Coulomb field. It would be interesting to consider limitations imposed on the application of this π τ J ≈≈ 2 τ – lnτ , τ ln()γ 2lnγ , method. Correction x1( ) to the subbarrier trajectory 2 0 0 0 can be calculated in perturbation theory; conse

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 957 quently, the Coulomb field must be weaker than the ACKNOWLEDGMENTS laser field on the majority of the subbarrier trajectory. γ The authors are grateful to S.P. Goreslavskii, For 1, this condition can be reduced to the ine D.F. Zaretskii, B.M. Karnakov, M.V. Fedorov, and quality N.I. ShevtsovShilovskii for fruitful discussions and to γµ 1, (47) S.G. Pozdnyakov for his help in numerical calcula tions. 10 which is rather stringent. Indeed, it can be seen from This study was partly supported by the Russian Tables 1 and 2 that parameter (19) for the ground states Foundation for Basic Research (project nos. 0502 of atoms and for positive ions of noble gases varies in 04016–NNIO_a, and 070201116). the limits µ ≈ 0.05–0.2. In accordance with inequality (47), this means that the upper bound of the adiabatic ity parameter is γ ≈ 10. APPENDIX A However, Figs. 1 and 2 show that the applicability of result (25) is extended to the range of much larger Kapitza’s Method in the Ionization Problem values of γ. To establish a more exact criterion of appli In the case of ionization of an atom by the field of a cability, we must compare the Coulomb corrections to linearly polarized wave, the imaginary part of action action (i.e., the exponents of each factor in relation assumes the minimal possible values for 1D trajecto (25)) with the action accumulated in the laser field. In ries lying on the straight line directed along the polar the multiphoton mode, this action is [ln(2γ) – 1/2]. K0 ization of the wave. Let us consider such a 1D trajec As a result, we obtain a less stringent inequality instead tory in the field = – 2/ of the atomic core and of (47), UC e r electromagnetic wave (11), which satisfies the equa µ 1, (48) tion which does not contain γ, but suggests that γ 1. e2 mx·· = – + e cosωt (A.1) 2 0 Thus, if the condition K0 1 is satisfied, formulas x (25) and (28) are applicable for arbitrary values of adi γ (in this appendix, we return to dimensional , e and abaticity parameter and ensure a correct passage to m). In a rapidly oscillating field, we represent, in the limit of a weak field, in which perturbation theory accordance with the Kapitza method [29, 30], the of the lowest order in Nmin is valid. These results gen motion of an electron as a displacement along a eralize the familiar expressions [22] for the rate of smooth trajectory X(t) with small oscillations at fre multiphoton ionization of atoms and positive ions to quency ω about this trajectory: the case of a strong field of an arbitrary frequency. The e most natural range of application of the above theory xt()= Xt()+ 0 () 1 – cosωt . (A.2) corresponds to experiments on ionization of matter in mω2 the field of powerful Xray lasers; such experiments are being actively performed at present. The condition for the applicability of the averaging method has the form ωT 1, where T is the charac It should be noted that the inclusion of the Cou teristic time of classical motion in the Coulomb field lomb interaction in the Keldysh theory was considered [30] after emergence from under the barrier, in early publications [22, 45], but this problem had not 3/2 m 2π 2K been solved completely. Nikishov and Ritus developed T ==2πb 0. (A.3) a special diagram technique (exponentiation method e2 ω n* [45]) and derived the following relation in the case of circular polarization: Here, b = κ/mω is barrier width (14) and parameters K0 and n* are defined in expressions (1) and (4). Thus, w() , ω 2 –2n* the Kapitza method is applicable under the multipho Q (), ω ==0 ()F 1 + γ , (49) C () , ω ton ionization conditions: wsr 0 2K which is in qualitative agreement with formula (42) for ωT ≈µ0 = –1/2 1, (A.4) n* γ 1. On the other hand, the result w = w exp(c K ) sr 0 where dimensionless parameter µ is defined in (19). In described in [45] for linear polarization with a con γ stant c ~ 1 does not coincide with formula (25) and is this case, condition 1 is also assumed to be satis strange in our opinion: this expression for ω 0 is fied, which ensures the smallness of the amplitude of not transformed into familiar formula (1) for the Cou oscillations as compared to spatial scale b characteriz lomb correction for ionization by a constant field. ing the averaged motion of the electron: e –1 10 0 = γ 1. (A.5) It is well known, however, that semiclassical formulas are often 2 “pulled” beyond the range of their formal applicability [43, 44]. mω b

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 958 POPRUZHENKO et al.

Passing in (A.1) to dimensionless variables ξ = x/b and This equality was obtained taking into account the fact θ = ωt and averaging over period 2π/ω, we obtain that the value of τ0 ≈ ln(2γ) 1 for γ 1 so that 2π τ ≈ τ γ 2 cosh 0 sinh 0 = . Integrating relation (A.12) twice d ξ µ dθ ∂ ξ τ ξ ≡ ==– –, under initial conditions ( = 0) = 0 1 + a, 2 π ∫ –1 2 ∂ξ dθ 2 []ξγ– ()1 – cosθ ξ τ τ µ 0 (A.6) d /d ( = 0) = i 2 , we obtain correction (21) to the µ trajectory. Real constant a vanishes for µ = 0 and does = –. not affect the ionization probability (see Appendix B). ξ2 – 2ξ/γ ξ θ The motion corresponding to smooth coordinate ( ) APPENDIX B is conservative with an energy integral Calculation of the Correction to Action 1⎛⎞dξ 2 + ()ξ ==E const. (A.7) 2⎝⎠dθ We will describe here the derivation of formula (23). Let us suppose that We are interested in a solution corresponding to the () () () …, (B.1) ionization threshold (i.e., zero electron velocity at r t = r0 t ++r1 t infinity, E = 0); it is this solution that controls the where r0 is trajectory (10) disregarding the Coulomb extremal [12, 28] trajectory along which the ionization interaction and r1 is the firstorder correction in (or probability attains its maximal value. In this case, we in µ) to this trajectory; here, v = r· ≡ dr /dt. have i i i To calculate correction (16), we note that the first ()dξ/dθ 2 ξ2 – 2γ–1ξ = 2µ. (A.8) term in the integrand can be evaluated by parts: For γ 1 and ξ ≥ 1, potential can be reduced to the ∞ t → ∞ ∞ ⋅ ⋅ · ⋅ Coulomb potential and relation (A.8) gives ∫v0 v1dt = v0 r1 – ∫v0 r1dt. (B.2) 2/3 3/2 3 t tt= t ξξ= + 2µθ()– θ . (A.9) s s s 0 2 0 As a result, the integrated term in formula (B.2) can Here, ξ and θ = 0 are the dimensionless initial point cels out with the same term in function F1 having the 0 0 · and initial instant of motion in real time. Relation opposite sign. Since v0 = – (t), the integral on the (A.9) leads to the average velocity (drift momentum) righthand side of formula (B.2) also cancels out with of the electron emerging from under the barrier (18): the third term in formula (16). Analogous cancella · tions take place in derivation of the formula p|| ==mX()κθ = θ0 2µ/ξ0 ≈ κ 2µ, (A.10) t ξ ≈ µ ξ s since 0 1 (for = 0, we have 0 = 1). Thus, the point 1 2 2 ImW = []κ + p ()t dt, (B.3) of emergence from under the barrier is not the point at 2∫ which the particle comes to a halt even in the 1D case 0 in contrast to the conventional semiclassical approxi which is valid [28] in the case of shortrange poten mation [20]. tials. The ionization probability is affected by only the Passing to imaginary time τ = –iθ, we obtain the imaginary part of action; consequently, the contribu ξ τ boundary condition for subbarrier trajectory ( ): tion from the upper limit in function F1 is insignifi ξ()dξ/dτ 2 ==–2µ, τ 0, (A.11) cant. For the same reason, it is sufficient to evaluate the integral in relation (16) only with respect to imag which shows that this trajectory is not realvalued any ω τ ≤ τ ≤ τ ω τ inary time t = i , 0 0, ts = i 0. Thus, the imag longer in the presence of the Coulomb interaction. inary part of the function of action (16) is Consequently, for µ ≠ 0, the initial instant of motion ts τ is also naturally shifted from the imaginary time axis. 0 1 2 ImW = – v dτ + Im()r ⋅ v + r ⋅ v For motion in imaginary time, the laser field is not C1 2ω∫ 1 0 1 1 1 oscillating and the Kapitza method is inapplicable. ττ= 0 0 (B.4) For γµ 1 (see relation (47)), the Coulomb field is τ 0 smaller than the laser field on the majority of the sub 1 2 = – x· dτ + Im()x x + x x· barrier trajectory, so that the correction to the trajec 2ω∫ 1 0 1 1 1 tory can be found from Eq. (20), which assumes the 0 ττ= ξ τ 0 following form in variables and : (the trajectory is onedimensional). Retaining contri 2 d ξ1 µ butions on an order no higher than the first in param = , eter µ, we obtain from relation (21) dτ2 ξ2 0 (A.12) · 2 2 x1 ==2κ µ,Imx1()τ0 b 2µτ0, ξ ()τ 1()τ τ ≈ ()ττ (B.5) 0 = cosh 0 – cosh 1 – exp – 0 . · γ Rex()τ0 = κ2µ.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 959

Finally, from relation (12), we have x0(τ0) = 0. As a integration of the timedependent oneelectron result, we obtain from (B.4) Schrödinger equation in a given effective atomic κ2µ potential in the field of an electromagnetic wave with τ µτ κ µ ImWC1 = – 0 + b 2 0 2 , (B.6) a time envelope corresponding to a pulse with a length ω of ten periods: which leads precisely to (22). The real part of correc ()t = f()ωt sin ωt, tion (16) is infinitely large at the lower integration 0 limit, but this does not affect the ionization probabil ⎧θ/4π, 0 ≤ θ < 4π, ity. In particular, constant a in the Coulomb correction ⎪ to trajectory (21) does not affect this probability. f()θ = ⎨1, 4πθ≤≤16π, ⎪ ⎩()20πθ– /4π, 16πθ< ≤ 20π. APPENDIX C To evaluate the ionization probability, norm (t) Numerical Methods (integral of the probability density over the volume) The results of numerical calculations of the ioniza was determined in a sphere of large radius R after ter tion rate presented in Figs. 1–3 were obtained using mination of the laser pulse. Probability 1 – (∞) two different methods. The Floquet method (also divided by pulse duration τ gives the value of the ion known as the complex quasienergy method) is based ization rate. The same value can be estimated by eval on representation of the solution to the Schrödinger uating the time derivative of (t). It is important to equation for an atomic electron in the presence of a note that choosing a large radius of the sphere (R = monochromatic electromagnetic wave in the form of a 40 a.u. for the hydrogen atom), we exclude the inter Fourier series with coefficients (position functions) pretation of electrons excited to the Rydberg levels as that can be determined from the solution of an infi ionized electrons. The results of calculations show that nitely large system of ordinary differential equations. the fraction of such excited electrons is significant. The eigenvalues of this system are complex and known As compared to the Floquet method, the solution as quasienergies [46, 47]. The doubled imaginary part of the timedependent Schrödinger equation is more of the quasienergy with the minus sign is the ioniza convenient for a generalization to the case of a non tion probability for an atomic level per unit time. A hydrogenlike atom or ion. State 4p0 of a valence elec detailed description of the Floquet method applied to tron in the Xe17+ ion, which plays the role of the initial the problem of photoionization of atoms in a strong condition of the timedependent Schrödinger equa field is given, for example, in [48]. The same publica tion, was constructed as an eigenstate in the effective tion also contains an easily accessible STRELO potential β numerical code for computing the quasienergies of Ur()= – /r – ()54 – e– r/r, atomic levels in a monochromatic field of an arbitrary β polarization, which was used for obtaining our results where = 18, and parameter = 7.93 was chosen so represented in Figs. 1 and 2. For the hydrogen atom, that the 4p0 state possessed the known ionization the Floquet method is quite effective since the time for potential I = 434 eV. calculating quasienergy for a given state and fixed Evaluation of the ionization rate for an energy level parameters of the field (i.e., a single point in Figs. 1 by solving the timedependent Schrödinger equation and 2) using a mediumpower multiprocessor com requires a much longer time: the calculation of one puter is about 10 s. In a purely monochromatic field, point (triangle) in Figs. 1 and 2 on a multiprocessor multiphoton resonances between bound states with computer takes several hours. energies depending on the applied field due to the dynamic Stark shift play a significant role (see, for APPENDIX D example, Fig. 1a). With increasing field, the probabil ities of excitation of multiphoton resonances also Coulomb Factor C(γ) for a ArbitraryShape Pulse increases, which considerably complicates computa In the case of a linearly polarized electromagnetic tions. For this reason, we failed to calculate the line pulse of arbitrary shape (44), the subbarrier trajectory widths for the hydrogen atom using the Floquet ξ0(τ) for = 0 (i.e., disregarding the Coulomb inter method for intensities exceeding 1014 W/cm2. action) is determined from the equations An alternative approach to calculating the ioniza ·· –1 ξ0 = –γ ϕ()iτ , tion rate is based on direct numerical integration of the (D.1) Schrödinger equation for an atomic electron in the · · 2 ξ ()τ == ξ0()0 0, ξ0()τ =1. field of a finiteduration laser pulse. From the analytic 0 0 0 ξ ωκ–1 ≤ τ ≤ τ ϕ θ point of view, this approach boils down to solving the Here, 0 = x0(t), 0 0, ( )defines the Cauchy problem with the initial condition corre shape of the pulse, and the derivative with respect to sponding to an electron in a certain bound state. The imaginary time τ = –iθ is denoted by dot. The last results shown in Figs. 1 and 3 by triangles were condition in (D.1) defines the “initial instant” τ0 of obtained using the Qprop code [49] for numerical the subbarrier motion.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 960 POPRUZHENKO et al.

In some cases, these equations can be solved tion rate on the Keldysh parameter and the momen exactly. tum spectra of photoelectrons [50, 51] were calculated (a) For a monochromatic field ϕ(θ) = cosθ, we for various fields of type (44). obtain 1 ξ ()τ == ()coshτ – coshτ , τ arcsinγ (D.2) REFERENCES 0 γ 0 0 and the integral in Eq. (33') is evaluated analytically 1. J. Andruszkow, B. Aunte, V. Ayvazyan, N. Baboi, R.Bakker, V. Balakin, D. Barni, A. Bazhan, M. Ber and gives C(γ) = 1 for all values of γ so that Q1 = 2n* nard, A. Bosotti, J. C. Bourdon, W. Brefeld, R. Brink (2/F) is independent of γ (i.e., on frequency ω for a mann, S. Buhler, J.P. Carneiro, M. Castellano, P. Cas fixed amplitude 0). The latter, however, is a specific tro, L. Catani, S. Chel, Y. Cho, S. Choroba, E. R. Colby, feature of linear polarization and of the given shape of W. Decking, P. Den Hartog, M. Desmons, M. Dohlus, the field, which will be demonstrated in the following D. Edwards, H. T. Edwards, B. Faatz, J. Feldhaus, examples. M. Ferrario, M. J. Fitch, K. Flöttmann, M. Fouaidy, ϕ θ 2θ A. Gamp, T. Garvey, C. Gerth, M. Geitz, E. Gluskin, (b) For ( ) = 1/cosh (solitonlike pulse), we V. Gretchko, U. Hahn, W. H. Hartung, D. Hubert, have M. Hüning, R. Ischebek, M. Jablonka, J. M. Joly, 1 cosτ M. Juillard, T. Junquera, P. Jurkiewicz, A. Kabel, ξ ()τ == ln, τ arctanγ, (D.3) 0 γ cosτ 0 J. Kahl, H. Kaiser, T. Kamps, V. V. Katelev, J. L. Kirch 0 gessner, M. Körfer, L. Kravchuk, G. Kreps, J. Krzywin 1 4 6 ski, T. Lokajczyk, R. Lange, B. Leblond, M. Leenen, C()γ = 1 ++ γ O()γ . (D.4) 54 J. Lesrel, M. Liepe, A. Liero, T. Limberg, R. Lorenz, 2 –3/2 Lu Hui Hua, Lu Fu Hai, C. Magne, M. Maslov, (c) For a pulsed field ϕ(θ) = (1 + θ ) , we obtain G. Materlik, A. Matheisen, J. Menzel, P. Michelato, 1 2 2 W.D. Möller, A. Mosnier, U.C. Müller, O. Napoly, ξ ()τ = 1()– τ – 1 – τ , 0 γ 0 A. Novokhatski, M. Omeich, H. S. Padamsee, (D.5) C. Pagani, F. Peters, B. Petersen, P. Pierini, J. Pflüger, γ P. Piot, B. Phung Ngoc, L. Plucinski, D. Proch, τ0 = , 2 K. Rehlich, S. Reiche, D. Reschke, I. Reyzl, J. Rosen 1 + γ zweig, J. Rossbach, S. Roth, E. L. Saldin, W. Sandner, Z. Sanok, H. Schlarb, G. Schmidt, P. Schmüser, 1 4 2 6 C()γ = 1 ++ γ – γ … (D.6) J. R. Schneider, E. A. Schneidmiller, H.J. Schreiber, 6 15 S. Schreiber, P. Schütt, J. Sekutowicz, L. Serafini, 2 D. Sertore, S. Setzer, S. Simrock, B. Sonntag, B. Sparr, (d) In the above examples, coefficient c4 = a4 – a2 F. Stephan, V. A. Sytchev, S. Tazzari, F. Tazzioli, in expansion (46) is nonnegative, as well as for ϕ(θ) = M. Tigner, M. Timm, M. Tonutti, E. Trakhtenberg, 2 exp(–θ ), 1/coshθ, etc. However, c4 can also be R. Treusch, D. Trines, V. Verzilov, T. Vielitz, V. Vogel, smaller than zero in some cases. Let us suppose, for G. V. Walter, R. Wanzenberg, T. Weiland, H. Weise, example, that J. Weisend, M. Wendt, M. Werner, M. M. White, I.Will, S. Wolff, M. V. Yurkov, K. Zapfe, P. Zhogolev, and F. Zhou, Phys. Rev. Lett. 85, 3825 (2000). ⎧⎫1 2 ϕθ()= exp⎨⎬–1()θ– s cos()sθ ⎩⎭2 2. http://flash.desy.de (D.7) 3. V. Ayvazyan, N. Baboi, I. Bohnet, R. Brinkmann, 1 2 1 2 4 = 1 – θ ++()θ 3– 2 s …, M. Castellano, P. Castro, L. Catani, S. Choroba, 2 24 A.Cianchi, M. Dohlus, H. T. Edwards, B. Faatz, ∞ ≤ A. A. Fateev, J. Feldhaus, K. Flöttmann, A. Gamp, where – < s 1. Then, in accordance with relation T.Garvey, H. Genz, Ch. Gerth, V. Gretchko, B. Grig (46), we have oryan, U. Hahn, C. Hessler, K. Honkavaara, M. Hün 1 2 4 6 ing, R. Ischebeck, M. Jablonka, T. Kamps, M. Körfer, C()γ = 1 ++1()γ– s O()γ (D.8) 108 M. Krassilnikov, J. Krzywinski, M. Liepe, A. Liero, T.Limberg, H. Loos, M. Luong, C. Magne, J. Menzel, γ and c4 < 0 for s < –1. Thus, the Coulomb factor C( ) P. Michelato, M. Minty, U.C. Müller, D. Nölle, may either increase or decrease with increasing γ A. Novokhatski, C. Pagani, F. Peters, J. Pflüger, P. Piot, depending on the shape of the pulse. It should be L. Plucinski, K. Rehlich, I. Reyzl, A. Richter, J. Ross noted that for 0 < s ≤ 1, Eq. (D.7) corresponds to a bach, E. L. Saldin, W. Sandner, H. Schlarb, modulated pulse with a Gaussian envelope (this model G. Schmidt, P. Schmüser, J. R. Schneider, E. A. Schneid is often used in laser physics); for s ≤ 0, oscillations miller, H.J. Schreiber, S. Schreiber, D. Sertore, S. Set vanish and formula (D.7) describes a single pulse. zer, S. Simrock, R. Sobierajski, B. Sonntag, B. Steeg, F. Stephan, K. P. Sytchev, K. Tiedtke, M. Tonutti, In experiments, ultrashort pulses are used [14] with R. Treusch, D. Trines, D. Türke, V. Verzilov, R. Wan a duration of only a few field periods and with a shape zenberg, T. Weiland, H. Weise, M. Wendt, I. Will, far from an ideal sinusoid. In this connection, the S. Wolff, K. Wittenburg, M. V. Yurkov, and K. Zapfe, functions determining the dependence of the ioniza Phys. Rev. Lett. 88, 104 802 (2002).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 MULTIPHOTON IONIZATION OF ATOMS AND IONS 961

4. A. A. Sorokin, S. V. Bobashev, T. Feigl, K. Tiedtke, 22. A. M. Perelomov and V. S. Popov, Zh. Éksp. Teor. Fiz. H. Wabnitz, and M. Richter, Phys. Rev. Lett. 99, 52 (2), 514 (1967) [Sov. Phys. JETP 25 (2), 336 213002 (2007). (1967)]. 5. H. Wabnitz, L. Bittner, A. R. B. de Castro, R. Döhr 23. D. R. Hartree, Proc. Cambridge Philos. Soc. 24, 89 mann, P. Gürtler, T. Laarmann, W. Laasch, J. Schulz, (1927). A. Swiderski, K. von Haeften, T. Möller, B. Faatz, 24. S. L. Chin, C. Rolland, P. B. Corkum, and P. Kelly, A. Fateev, J. Feldhaus, C. Gerth, U. Hahn, E. Saldin, Phys. Rev. Lett. 61, 153 (1998). E. Schneidmiller, K. Sytchev, K. Tiedtke, R. Treusch, 25. M. V. Ammosov, N. B. Delone, and V. P. Kraіnov, Zh. and M. Yurkov, Nature (London) 420, 482 (2002). Éksp. Teor. Fiz. 91 (6), 2008 (1986) [Sov. Phys. JETP 6. T. Laarmann, A. R. B. de Castro, P. Gürtler, W. Laasch, 64 (6), 1191 (1986)]. J. Schulz, H. Wabnitz, and T. Möller, Phys. Rev. Lett. 26. V. S. Popov, Usp. Fiz. Nauk 169 (7), 819 (1999) 92, 143 401 (2004). [Phys.—Usp. 42 (7), 733 (1999)]. 7. H. Wabnitz, A. R. B. de Castro, P. Gürtler, T. Laar 27. V. S. Popov, V. P. Kuznetsov, and A. M. Perelomov, Zh. mann, W. Laasch, J. Schulz, and T. Möller, Phys. Rev. Éksp. Teor. Fiz. 53 (1), 331 (1967) [Sov. Phys. Usp. 26 Lett. 94, 023 001 (2005). (1), 222 (1967)]. 8. N. Stojanovic, D. von der Linde, K. Sokolowski 28. V. S. Popov, Preprint No. 1304, ITÉF (Institute of Tinten, U. Zastrau, F. Perner, E. Förster, R. Sobiera Theoretical and Experimental Physics, Moscow, 2000); jski, R. Nietubyc, M. Jurek, D. Klinger, J. Pelka, Yad. Fiz. 68 (4), 717 (2005) [Phys. At. Nucl. 68 (4), 686 J. Krzywinski, L. Juha, J. Cihelka, A. Velyhan, (2005)]. S. Koptyaev, V. Hájková, J. Chalupsky, J. Kuba, 29. P. L. Kapitsa, Zh. Éksp. Teor. Fiz. 21, 588 (1951); Usp. T.Tschentscher, S. Toleikis, S. Düsterer, and H. Redlin, Fiz. Nauk 44, 7 (1951). Appl. Phys. Lett. 89, 241 909 (2006). 30. L. D. Landau and E. M. Lifshitz, Course of Theoretical 9. S. W. Epp, J. R. Crespo LópezUrrutia, G. Brenner, Physics, Vol. 1: Mechanics (Nauka, Moscow, 1989; But V. Mäckel, P. H. Mokler, R. Treusch, M. Kuhlmann, terworth–Heinemann, Oxford, 1991). M. V. Yurkov, J. Feldhaus, J. R. Schneider, M. Well 31. S. V. Popruzhenko, G. G. Paulus, and D. Bauer, Phys. höfer, M. Martins, W. Wurth, and J. Ullrich, Phys. Rev. Rev. A: At., Mol., Opt. Phys. 77, 053409 (2008); Lett. 98, 183 001 (2007). S. V. Popruzhenko and D. Bauer, J. Mod. Opt. 55, 2573 10. G. S. Voronov and N. B. Delone, Zh. Éksp. Teor. Fiz., (2008). Pis’ma Red. 1 (2), 42 (1965) [JETP Lett. 1 (2), 66 32. S. V. Popruzhenko, V. D. Mur, V. S. Popov, and (1965)]. D. Bauer, Phys. Rev. Lett. 101, 193 003 (2008). 11. W. Becker, F. Grasbon, R. Kopold, D. B. Milosv evicv , 33. V. S. Popov, V. D. Mur, and S. V. Popruzhenko, Pis’ma G. G. Paulus, and H. Walther, Adv. At., Mol., Opt. Zh. Éksp. Teor. Fiz. 85 (5), 275 (2007) [JETP Lett. 85 Phys. 48, 35 (2002). (5), 223 (2007)]; S. V. Popruzhenko, V. D. Mur, and 12. V. S. Popov, Usp. Fiz. Nauk 174 (9), 921 (2004) V. S. Popov, Phys. Lett. A 372, 59 167 (2008). [Phys.—Usp. 47 (9), 855 (2004)]. 34. H. Bateman and A. Erdelyi, Higher Transcendental 13. A. Becker and F. H. M. Faisal, J. Phys. B: At., Mol. Functions (McGrawHill, New York, 1955; Nauka, Opt. Phys. 38, R1 (2005). Moscow, 1966), Vol. 1. v v 35. Handbook of Mathematical Functions with Formulas, 14. D. B. Milos evic , G. G. Paulus, D. Bauer, and W. Graphs, and Mathematical Tables, Ed. by M. Abramow Becker, J. Phys. B: At., Mol. Opt. Phys. 39, R203 itz and I. A. Stegun (Dover, New York, 1972; Nauka, (2006). Moscow, 1979). 15. L. V. Keldysh, Zh. Éksp. Teor. Fiz. 47, 1945 (1964) 36. A. Becker, L. Plaja, P. Moreno, M. Nurhuda, and [Sov. Phys. JETP 20, 1307 (1964)]. F. H. M. Faisal, Phys. Rev. A: At., Mol., Opt. Phys. 64, 16. F. H. M. Faisal, J. Phys. B: At. Mol. Phys. 6, L89 023408 (2001). (1973); H. R. Reiss, Phys. Rev. A: At., Mol., Opt. Phys. 37. I. Bersons, Phys. Lett. A 84, 364 (1981); I. Ya. Bersons, 22, 1786 (1980). Zh. Éksp. Teor. Fiz. 83 (4), 1276 (1982) [Sov. Phys. 17. A. I. Nikishov and V. I. Ritus, Zh. Éksp. Teor. Fiz. 50 JETP 56 (4), 731 (1982)]. (1), 255 (1966) [Sov. Phys. JETP 23 (1), 168 (1966)]. 38. H. A. Kramers, Philos. Mag. 46, 836 (1923). 18. A. M. Perelomov, V. S. Popov, and M. V. Terent’ev, Zh. 39. Robin Santra and Chris H. Greene, Phys. Rev. A: At., Éksp. Teor. Fiz. 50 (5), 1393 (1966) [Sov. Phys. JETP Mol., Opt. Phys. 70, 053 401 (2004). 23 (5), 924 (1966)]; Zh. Éksp. Teor. Fiz. 51 (1), 309 40. T. Nubbemeyer, K. Gorling, A. Saenz, U. Eichmann, (1966) [Sov. Phys. JETP 24 (1), 207 (1966)]. and W. Sandner, Phys. Rev. Lett. 101, 233 001 (2008). 19. N. L. Manakov, M. V. Frolov, B. Borca, and A. F. Sta 41.V. S. Popov, V. D. Mur, B. M. Karnakov, and race, J. Phys. B: At., Mol. Opt. Phys. 36, R49 (2003). S. G. Pozdnyakov, Phys. Lett. A 358, 21 (2006). 20. L. D. Landau and E. M. Lifshitz, Course of Theoretical 42. V. S. Popov, Doctoral Dissertation in Mathematical Physics, Vol. 3: Quantum Mechanics: NonRelativistic Physics (Institute of Theoretical and Experimental Theory (Nauka, Moscow, 1989; Butterworth–Heine Physics, Moscow, 1974). mann, Oxford, 1991). 43. R. E. Langer, Phys. Rev. 75, 1573 (1949). 21. Yu. N. Demkov and G. F. Drukarev, Zh. Éksp. Teor. 44. A. B. Migdal, Qualitative Methods in Quantum Theory Fiz. 47, 918 (1964) [Sov. Phys. JETP 20, 614 (1964)]. (Nauka, Moscow, 1975; Benjamin, London, 1977).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009 962 POPRUZHENKO et al.

45. A. I. Nikishov and V. I. Ritus, Zh. Éksp. Teor. Fiz. 52 49. D. Bauer and P. Koval, Comput. Phys. Commun. 174, (1), 223 (1967) [Sov. Phys. JETP 25 (1), 145 (1967)]. 396 (2006). 46. Ya. B. Zel’dovich Zh. Éksp. Teor. Fiz. 51 (5), 1492 50. V. S. Popov, Laser Phys. 10, 1033 (2000); V. S. Popov, (1966) [Sov. Phys. JETP 24 (5), 1006 (1966)]; Usp. Fiz. Pis’ma Zh. Éksp. Teor. Fiz. 73 (1), 3 (2001) [JETP Lett. Nauk 110, 139 (1973) [Sov. Phys.—Usp. 16, 427 73 (1), 1 (2001)]; Zh. Éksp. Teor. Fiz. 120 (2), 315 (1973)]. (2001) [JETP 93 (2), 278 (2001)]. 51. B. M. Karnakov, V. D. Mur, and V. S. Popov, Pis’ma Zh. 47. V. I. Ritus, Zh. Éksp. Teor. Fiz. 51 (5), 1544 (1966) Éksp. Teor. Fiz. 88 (7), 495 (2008) [JETP Lett. 88 (7), [Sov. Phys. JETP 24 (5), 1041 (1966)]. 423 (2008)]. 48. R. M. Potvliege, Comput. Phys. Commun. 114, 42 (1998). Translated by N. Wadhwa

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 108 No. 6 2009