INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 2245–2254

www.iop.org/Journals/jb

PII: S0953-4075(01)20703-2

Ionization of hydrogen atoms by intense vacuum ultraviolet radiation ´ 2 , Bernard Piraux3 , Robert Potvliege4 , Jarosław Bauer1 , Łukasz Plucinski 5,6 ´ 5,7 Mariusz Gajda and Jacek Krzywinski 1 Katedra Fizyki J¸ adrowej i Bezpiecze´nstwa Radiacyjnego, Uniwesytet Ł´odzki, ul. Pomorska 149/153, 90-236 Ł´od´z, Poland 2 Hamburg University, Institut f¨ ur Experimentalphysik, Notkestrasse 85, 20603 Hamburg, Germany 3 Laboratoire de Physique Atomique et Mol´ eculaire, Universit´e Catholique de Louvain, 2 Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium 4 Physics Department, Durham University, South Road, Durham DH1 3LE, UK 5 Instytut Fizyki, Polska Akademia Nauk, Aleja Lotnik´ ow 32/46, 02-668 Warsaw, Poland 6 Szkoła Nauk Scisłych, ´ Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland 7 Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22603 Hamburg, Germany

Received 10 January 2001, in final form 27 April 2001 Abstract We study the ionization of hydrogen atoms by an intense pulsed beam of photons with energies of 17 or 50 eV. The work is motivated by the demonstration of the free-electron laser (FEL) action at the DESY Laboratory. The parameters chosen for the incident field are in the regime accessible by the FEL. Ionization yields are obtained within three different approaches, namely the strong-field approximation, the Floquet method and a numerical solution of the timedependent Schr¨odinger equation. A marked stabilization effect for 50 eV photons is shown.

1. Introduction For more than a decade, great effort has been devoted to the investigation of the interaction of atoms with highly intense laser fields in the optical domain. A number of non-perturbative phenomena have been observed at these wavelengths, such as above threshold ionization (ATI), multielectron ionization, strong field stabilization, high-harmonic generation and Coulomb explosion of molecules and clusters [1]. The observation of self-amplified spontaneous emission at vacuum ultraviolet (VUV) wavelengths in a free-electron laser (FEL) has been reported recently [2]. This breakthrough opens the interesting possibility of an experimental study of the interaction of matter with a very intense electromagnetic field of much higher frequency than can presently be achieved. Lasing at high frequency has been demonstrated recently at the DESY FEL [2], in a machine which is still under development. Much enhanced performances are expected in the near future. The perspectives are very interesting. Simulations show [3, 4] that the maximum number of photons radiated during a pulse lasting a few hundred femtoseconds should be of the 0953-4075/01/112245+10$30.00

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photon energy [a.u.] Figure 1. The phase diagram; full line, Keldysh parameter γ = 1; long broken line, excursion parameter α0 = 1; short broken line, ponderomotive potential Up = ω.

order of 1013 –1014 for photon energies ranging from 20 to 200 eV. The focusing of the photon beam is limited by the performance of optical elements rather than by diffraction. Using stateof-the-art technology, one may expect to attain a minimum beam size of the order of a few hundred nanometres with a transmission efficiency of 10% [5]. Correspondingly, the expected maximum intensity is 1017 –1018 W cm−2 . By compressing the pulses by a factor 100 [6], one should be able to increase the maximum intensity to 1019 –1020 W cm−2 . After compression the pulse can be as short as a few femtoseconds. At the present stage of development, the FEL intensity fluctuates; however, work on intensity stabilization is now in progress [7]. Although it is only in the last few years that coherent radiation sources operating at intense high frequency have been realized, a number of theoretical studies of the ionization of atoms by such fields have already been published. The possibility of the stabilization of hydrogen atoms in short high-frequency laser pulses was established 10 years ago [8]. Floquet rates of ionization at high frequency have also been obtained [9]. Several authors have discussed the adiabatic following of the Floquet state in high-frequency pulses [10–16]. Results for one-dimensional models of multielectron atoms [17] and hydrogen atoms [18] exposed to the DESY FEL pulse have recently been reported. In this paper, we study the ionization of hydrogen atoms by an intense pulsed beam of photons with 17 or 50 eV energy and parameters within the ranges that are expected to be accessible by the DESY FEL. The atom is assumed to be initially in the ground state. Ionization yields are obtained using three different approaches, namely the strong-field approximation (SFA), the Floquet method and a numerical solution of the time-dependent Schr¨odinger equation (TDSE). These three approaches are briefly described in section 2. The results are given in section 3. In this introduction, we start by discussing the typical parameters which characterize the interaction of atoms with an intense FEL field. We limit our analysis to the simplest atomic system, namely a hydrogen atom initially in its ground state. In figure 1 we present a phase diagram in the two-dimensional parameter space of the angular frequency,

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ω, and the intensity, I , of the applied field. The lines shown in the diagram correspond to characteristic values of the most important parameters of the incident field, as discussed in the following paragraphs. In a strong external oscillating field the simplest approach to the ionization process is to neglect the atomic Coulomb potential compared to the interaction with the field. The role of the atomic potential is therefore limited to the ‘preparation’ of the initial state. The average kinetic energy of a classical electron oscillating in the field, i.e. the ponderomotive potential with all quantities expressed in atomic units is I . (1) 4ω2 Large values of this parameter, i.e., Up > ω, imply, in the low-frequency regime, a peak switching of the ATI spectrum—the ionized electron must absorb enough energy to overcome the atomic binding and to support the field-induced quiver motion [19]. (The region of Up > ω is located above the short broken line in figure 1.) The ponderomotive potential thus contributes to a shift of the ionization threshold. The maximal energy which an electron can acquire while leaving the atom is equal to about 3Up and this value gives the cutoff for frequencies of high-harmonic generation, i.e. the photon emission occurring when atomic electrons excited to high-energy states by the field recombine radiatively with the core. In the high-frequency regime, 2Up > ω implies strong-field adiabatic stabilization [9]. The effect of the magneticfield component of the incident field becomes too strong above Up ≈ cω/4 for the dipole approximation to remain valid and strong-field stabilization to persist [20–22]. The classical amplitude of the quiver motion, α0 , also known as the excursion parameter, is another important quantity. In atomic units it is √ I α0 = 2 . (2) ω If the excursion parameter is small compared to the Bohr radius (α0 < 1), the electron remains in the neighbourhood of the nucleus where it can absorb a substantial amount of energy. This absorption finally results in the ionization of the atom. On the other hand, if the amplitude of the quiver motion is large (α0 > 1), the electron spends most of the time far from the nucleus where it is unlikely to absorb energy. In this case the ionization can be substantially reduced. In figure 1, the long broken line corresponds to α0 = 1 and the region where α0 > 1 is located above the long broken line. In general, and depending on the frequency and the intensity, the ionization process can be viewed either as photon absorption in an oscillating field or as tunnelling (or over-the-barrier ionization). In the low-frequency regime, these two ionization mechanisms correspond to different values of the Keldysh parameter [23], γ , which is the ratio of the tunnelling time (i.e., the width of the barrier divided by the electron velocity) to the period of the field. One readily finds that
b γ = , (3) 2Up Up =

where b is the ionization potential, which for the hydrogen atom is equal to b ≡ 0.5 au. At relatively low intensities the Keldysh parameter is large (γ  1) and photon absorption then dominates over any tunnelling. At higher intensities, for which γ  1, the electron may be released through tunnelling or over-the-barrier ionization. In figure 1, this high-intensity regime is located above the solid line. However, photon absorption from the ground state of the Kramers–Henneberger potential is likely to be a more efficient mechanism of ionization than tunnelling and over-the-barrier ionization for a long, high-frequency pulse [24]. The Keldysh

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parameter is normally less than 1 in the strong-field stabilization regime (2Up > ω) when ω > b . It is worth noting that the order of the lines shown in figure 1 depends on the frequency of the applied field, since these lines cross each other. At wavelengths in the visible or the infrared spectra, tunnelling and over-the-barrier ionization (γ < 1) can be achieved only when both Up and α0 are large. In contrast, at shorter wavelengths it is possible to find intensities for which γ < 1 but both Up and α0 are small. For 50 eV photons, one passes successively from one regime to the other by increasing the field intensity. Figure 1 indicates that the coherent electromagnetic field provided by the FEL source can be expected to lead to photoionization in dynamical regimes as yet experimentally unexplored. 2. Theoretical approaches 2.1. Numerical solution of time-dependent Schr¨odinger equation The use of the TDSE is required for the exact analysis of the ionization of a hydrogen atom by a laser pulse. The equation reads   ∂ e i¯h (r , t) = H0 − A(t)p (r , t), (4) ∂t c where H0 is the atomic Hamiltonian, A(t) the vector potential of the field, E (t) = −(1/c)∂ A(t)/∂t, and r , p are the relative position and momentum of the electron–nucleus system. In equation (4) we have used the dipole approximation and the ‘velocity’ form of the interaction Hamiltonian. In order to describe the light pulses, we will assume the following form of the vector potential: A(t) = A0 f (t) ez sin(ωt).

(5)

In the above equation, A0 is the amplitude of the potential vector, ω the angular frequency of the laser light, ez a unit polarization vector (supposed for simplicity to be oriented in the z-direction) and the function f (t) describes the slowly varying envelope of the pulse. In our calculations we assume sin-square pulses, with f (t) = sin2 (π t/td ) where td is the total duration of the pulse. Equation (4) is solved numerically by expanding the atomic wavefunction in a basis of Coulomb–Sturmian functions. The close link between the Coulomb– Sturmian functions and the hydrogen radial functions makes this basis very convenient. The corresponding matrix elements of the atomic and the interaction Hamiltonians are easily obtained, and the resulting matrices are sparse. The details of the numerical method can be found in [25]. 2.2. Floquet method Our Floquet calculations assume that the atom remains at all times in a single dressed state, namely the state that develops adiabatically from the field-free ground state as the envelope function f (t) increases from 0 to 1. For a given value of f (t), this state is obtained by solving the TDSE for the stationary laser field of the vector potential amplitude A0 f (t), assuming that the solution has the Floquet form (r , t) = exp(−i t/¯h)F (r , t), where F (r , t) is periodic in time with period 2π/ω and satisfies radiation (Siegert) boundary conditions [26,27]. The quasienergy, , is then found as a complex eigenvalue of a system of time-independent differential equations equivalent to the TDSE. We solve this system numerically in the velocity gauge, using a basis of spherical harmonics and complex radial Coulomb–Sturmian functions; the calculation amounts to finding generalized eigenvalues and eigenvectors of a complex nonHermitian matrix [28]. The rate of ionization of the atom,  = −2Im /¯h, does not depend

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on the normalization of the wavefunction. As the ionization process is non-resonant, one may expect that the atom follows the dressed ground state adiabatically, and, correspondingly, that the probability that it is ionized by the pulse is   ∞  PFl = 1 − exp − [A0 f (t)] dt . (6) −∞

2.3. Strong-field approximation Approximate Floquet calculations can be done relatively easily, beyond perturbation theory (PT) and with tunnelling taken into account, within the approaches introduced by Keldysh [23], Faisal [29] and Reiss [30]. Reiss’ approach is also known as the SFA. We have compared its predictions to those of our fully numerical time-dependent and Floquet calculations. Within the SFA, ionization is described as a one-step transition from the initial ground state to a final Volkov state (the free-electron state in an oscillating field). The interaction of the electron with the core is thus neglected in the final state. The SFA is expected to become reliable when the ponderomotive potential exceeds the atomic binding energy (Up > b ). Unlike PT, the SFA improves as the laser field becomes stronger. This model gives the following expression for the total ionization rate of a hydrogen atom in a linearly polarized plane wave:  2   ∞    Up 2π 1    , SFA = ( d ( + )J k · α ; k ) (7) k n b n 0   h ¯ (2π)3 2¯hω n=n0

where k is the electron momentum, n = (n¯hω − b − Up ) is the electron final kinetic energy, and n0 is the smallest values of n for which n
0. Jn denotes the generalized Bessel function of order n and (k) the Fourier transform of the ground-state radial wavefunction. The integration is carried out over all angles of the final momentum of the electron. The total ionization rate is obtained as an infinite sum of partial rates corresponding to absorption of n0 , n0 + 1, . . . photons, respectively. The largest contribution to the sum (7) comes from the first two terms; the magnitude of the following terms indeed rapidly decreases with increasing n. For a constant radiation intensity a kinetic energy spectrum of outgoing photoelectrons consists of several ATI peaks of discrete energies separated by the photon energy. As in the fully numerical Floquet calculation, SFA is an ionization rate for a stationary field. The ionization yield at the end of a pulse can be obtained by replacing  by SFA in equation (6). Note that the position of the ATI peaks, n , also depends on the intensity. 3. Results In this section we compare results obtained using the three approaches already described. We apply them to the study of the ionization of an atom of hydrogen, initially in its ground state, by two different kinds of pulses: (1) pulses with photon energy h ¯ ω = 17 eV (0.625 au) and total ¯ ω = 50 eV (1.838 au) and total duration td = 40(2π/ω); and (2) pulses with photon energy h duration td = 30(2π/ω). These pulse durations are of the order of a few femtoseconds, which requires compressing the pulses presently generated by the DESY FEL by a factor of the order of 100. This compression, together with intensity stabilization, is expected to be feasible in the near future. For both frequencies we vary the pulse-peak intensity. The largest intensities considered are well within the limits of validity of the dipole approximation (4Up /cω = 0.2 for ω = 1.838 au at 7 × 1018 W cm−2 intensity). In figures 2 and 3 we compare the SFA ionization rates with the Floquet rates obtained entirely numerically. In general, for both frequencies, the Floquet ionization rates depend smoothly on the intensity. They reach a maximum value at I ≈ 1016 W cm−2 for ω = 17 eV

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and I ≈ 4 × 1017 W cm−2 for ω = 50 eV. These two maxima occur at values of the excursion parameter close to unity (at α0 ≈ 1.3 and 1.0, respectively), and at values of the intensity such that 2Up ≈ h ¯ ω [9] (2Up /¯hω ≈ 0.6 and 0.9, respectively). For larger intensities the ionization

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total ionization yield

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peak intensity [W/cm ] Figure 4. Ionization probability of a hydrogen atom interacting with the sin-square pulse, td = 30 (2π/ω) and the photon energy h ¯ ω = 50 eV: squares, PT; circles, TDSE; triangles, Floquet method.

rate decreases, which indicates strong-field stabilization. Let us stress that the maximal value of the ionization rate for 50 eV photons is about three times smaller than the corresponding maximal value for 17 eV photons. The difference suggests that the total ionization yield decreases with frequency. The corresponding ionization rates obtained within the SFA behave differently (see figures 2 and 3). At first glance one can observe a number of sharp maxima at high intensities. These jagged profiles arise from a succession of multiphoton thresholds. The smooth parts of the curves are, in fact, beyond the limit of applicability of the SFA. For certain values of intensity the index n0 in equation (7) changes by one; each jump results in a local minimum in the ionization rate. The SFA, however, neglects the non-ponderomotive ac Stark shift of the initial ground state. This shift is positive at VUV frequencies, contrary to the optical case, and cancels the effect of the ponderomotive potential. One can also observe from the figures that the ionization rate (in the region of applicability of the SFA model) exceeds by about one order of magnitude the ionization rate resulting from exact Floquet calculations. The TDSE approach leads directly to ionization probabilities and photoelectron spectra. These quantities are obtained by projecting the wavefunction, after the field is turned off, on the positive energy states of the atom. This method, contrary to the others, takes the pulsed-nature field into account from the start. In figure 4 we compare the total ionization rates resulting from the Floquet method and those obtained from the TDSE. We do not show the SFA results. The latter significantly overestimate the ionization yield and give, for the whole range of intensities considered here, an ionization probability practically equal to one. We present instead the results of first-order PT. It has been shown [18] that the perturbative approach works quite well in a wide range of intensities for the energy h ¯ ω = 17 eV. It is shown in figure 4 that the Floquet and TDSE approaches agree almost exactly. Both methods predict clear stabilization for high intensities, contrary to the PT and the SFA. The decrease in ionization probability, which in the optical regime is equal to a few percent, reaches about 20% here. PT agrees with

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energy [a.u.] Figure 5. Spectrum of outgoing electrons for the hydrogen atom interacting with the sinsquare pulse td = 30 (2π/ω) and the photon energy h ¯ ω = 50 eV and the peak intensity I = 2.5 × 1018 W cm−2 : full curve, SFA; broken curve, TDSE. 0

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energy [a.u.] Figure 6. Spectrum of outgoing electrons for the hydrogen atom interacting with the sinsquare pulse td = 30 (2π/ω) and the photon energy h ¯ ω = 50 eV and the peak intensity I = 2.5 × 1019 W cm−2 : full curve, SFA; broken curve, TDSE.

the calculated ionization yields up to saturation. Finally, the energy spectrum of the outgoing photoelectrons is presented in figures 5 and 6. We compare the ‘exact’ spectra, obtained through the time-dependent calculation, with the predictions of the SFA model. The TDSE spectrum shows a characteristic ATI structure,

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namely a number of peaks separated by the photon energy. Our results indicate that the position of the peaks depends only very weakly on the laser intensity. In particular, we do not observe any peak switching. This fact can be explained by the analysis of ac Stark shifts obtained with the help of the Floquet method. The dressed ground state is shifted appreciably towards lower binding energies, rather than towards higher binding energies across multiphoton thresholds: for an intensity of 1017 W cm−2 , the reduction in binding energy amounts to about 6 eV. The agreement of the electron spectra with those obtained within the SFA model is poor. As seen from the figure, the shape, position and width of the ATI peaks are very different. The pulse-peak intensities considered obey the applicability condition of the SFA very well, Up / b  1. But our calculations show that the ionization yield attains about 90% of its final value during switching off and on of the pulse when the intensity is too small for the SFA to apply. To enter the strong-field stabilization regime one should pass very quickly through the region where the excursion parameter is of the order of unity [8]. This, in turn, may compromise the adiabatic following of the dressed initial state. 4. Summary We have studied the interaction of an hydrogen atom, initially in its ground state, with an intense high-frequency laser pulse. We have presented the perspectives of development of the FEL source and have given the relevant ranges of the parameters characterizing the coherent VUV radiation which should soon be generated. We have compared three approaches commonly used at optical frequencies. The solution of the TDSE and the Floquet approach predict marked stabilization at high intensities. Both approaches agree very well for the pulse parameters considered here. The stabilization occurs in the barrier suppression regime (γ < 1) when both the excursion parameter and the ponderomotive energy become large (α0 > 1, 2Up > ω). The TDSE and Floquet results are in a clear contradiction with the prediction of the SFA. The latter is not accurate in the regimes of intensities and frequencies studied here. We have also obtained energy spectra of the emitted photoelectrons. These spectra are composed of several ATI peaks separated by the photon frequency. We have not observed any peak switching. We attribute this fact to the large and positive ac Stark shift of the initial ground state. Acknowledgment This work was supported by the KBN grant no 2P03B10313. References [1] Piraux B, L’Hullier A and Rz¸az˙ ewski K (ed) 1993 Super-Intense Laser-Atom Physics (NATO ASI Series) (New York: Plenum) [2] Andruszkow J et al 2000 Phys. Rev. Lett. 85 3825 [3] Brefeld W et al 1997 Nucl. Instrum. Methods A 393 119 [4] Saldin E L, Schneidmiller E A and Yurkov M V 1997 Nucl. Instrum. Methods A 393 157 [5] Schulte-Schrepping H Private communication [6] Pellegrini C 2000 Nucl. Instrum. Methods A 445 124 [7] Feldhaus J et al 1999 Inst. Phys. Conf. Ser. 159 553 [8] Kulander K C, Schafer K J and Krause J L 1991 Phys. Rev. Lett. 66 2601 [9] D¨orr M, Potvliege R M, Proulx D and Shakeshaft R 1991 Phys. Rev. A 43 3729 [10] D¨orr M, Latinne O and Joachain C J 1995 Phys. Rev. A 52 4289 [11] Vivirito R M A and Knight P L 1995 J. Phys. B: At. Mol. Opt. Phys. 28 4357 [12] Zakrzewski J and Delande D 1995 J. Phys. B: At. Mol. Opt. Phys. 28 L667 [13] Piraux B and Potvliege R M 1998 Phys. Rev. A 57 5009

2254 [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

J Bauer et al Patel A, Kylstra N J and Knight P L 1999 J. Phys. B: At. Mol. Opt. Phys. 32 5759 Day H C, Piraux B and Potvliege R M 2000 Phys. Rev. A 61 031402(R) D¨orr M and Potvliege R M 2000 J. Phys. B: At. Mol. Opt. Phys. 33 L233 Brewczyk M and Rz¸az˙ ewski K 1999 J. Phys. B: At. Mol. Opt. Phys. 32 L1 Gajda M, Krzywi´nski J, Pluci´nski Ł and Piraux B 2000 J. Phys. B: At. Mol. Opt. Phys. 33 1271 Eberly J H, Javanainen J and Rz¸az˙ ewski K 1991 Phys. Rep. 204 333–83 V´azques de Aldana J R and Roso L 1999 Opt. Express 5 144 Kylstra N J, Worthington R A, Patel A, Knight P L, V´azques de Aldana J R and Roso L 2000 Phys. Rev. Lett. 85 1835 Reiss H R 2000 Phys. Rev. A 63 013409 Keldysh L V 1964 Zh. Eksp. Teor. Fiz. 47 1945 (Engl. Transl. 1965 Sov. Phys.–JETP 20 1307) Gavrila M 1992 Atoms in intense laser fields Adv. At. Mol. Opt. Phys. (Suppl.) 1 435 Piraux B, Huens E, Bugacov A and Gajda M 1998 Phys. Rev. A 55 2132 Maquet A, Chu S I and Reinhardt W P 1985 Phys. Rev. A 27 2946 Potvliege R M and Shakeshaft R 1992 Atoms in intense laser fields Adv. At. Mol. Opt. Phys. (Suppl.) 1 435 Potvliege R M 1998 Comput. Phys. Commun. 114 42 Faisal F H M 1973 J. Phys. B: At. Mol. Phys. 6 L89 Reiss H R 1980 Phys. Rev. A 22 1786