Документ взят из кэша поисковой машины. Адрес оригинального документа : http://danp.sinp.msu.ru/LIIWM/JPhysB40_2007_F299Kab.pdf Дата изменения: Wed Oct 24 01:10:34 2007 Дата индексирования: Mon Oct 1 22:14:36 2012 Кодировка: Windows-1251

IOP PUBLISHING

JOURNA

L OF

PHYSICS B: ATO

MIC,

MOLECULAR

AND

OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 40 (2007) F299-F305

doi:10.1088/0953-4075/40/21/F02

FAST TRACK COMMUNICATION

An attosecond time-resolved study of strong-field atomic photoionization
A K Kazansky1,4 and N M Kabachnik2,3
1 2

Fock Institute of Physics, State University of Sankt Petersburg, Sankt Petersburg 198504, Russia ЕЕ Е Fakultat fur Physik, Universitat Bielefeld, D-33615 Bielefeld, Germany 3 Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia 4 Donostia International Physics Center, E-20018 San Sebastian/Donostia, Basque Country, Spain

Received 12 September 2007 Published 22 October 2007 Online at stacks.iop.org/JPhysB/40/F299 Abstract The time evolution of atomic photoionization by an intense few-cycle laser pulse is theoretically investigated. A possible modification of the recent attosecond tunnelling experiment in Uiberacker et al (2007 Nature 446 627) is proposed, which consists of measuring the yield of the doubly charged Li ions produced by the combined action of a strong few-cycle infrared pulse and an ultrashort (attosecond) extreme ultraviolet (XUV) pulse. We predict the results of such an experiment, based on the numerical solution of the time-dependent Е Schrodinger equation, which describes the atomic electron in a strong laser field. The influence of the XUV pulse is treated in the sudden approximation. It is shown that the dependence of the double ionization cross section on the time delay between the two pulses reflects the time evolution of the strong-field ionization. We demonstrate that even more detailed information can be gained from the forward-backward asymmetry of the photoelectron emission. (Some figures in this article are in colour only in the electronic version)

Recently, the first attosecond real-time observation of strong-field ionization of weakly bound excited atomic states has been reported [1]. In the experiment two short pulses, an attosecond extreme ultraviolet (XUV) pulse and a strong few-cycle laser near-infrared (NIR) pulse, were used in a kind of pump-probe experiment. In particular, the XUV pulse (pump) ionized and excited (shake up) the neon atom. The resulting excited ion was further ionized by the strong NIR field which played a role of analysing probe pulse. The yield of the doubly charged Ne ions was measured as a function of the time delay between the two pulses. In this way a time-resolved depletion of the transient excited states by the strong NIR field was studied. It was found that the depletion occurs in sharp steps lasting several hundred attoseconds. Moreover, the experiment shows that the ionization steps are preceded by dips in the yield curve. Interpretation of the experiment was given on the basis of the non-adiabatic tunnel
0953-4075/07/210299+07$30.00 ї 2007 IOP Publishing Ltd Printed in the UK F299


F300

Fast Track Communication

ionization theory [2]. The modelling described well the general features of the experiment (for details see supplementary information to the paper [1]). However, some details, such as the steepness of the ionization steps and the dips preceding them in the measured data, are not well reproduced by the model. As a proof-of-principle attosecond tunnelling experiment, the investigation of Ne double ionization is very important. However, its interpretation encounters serious inherent problems. At the attosecond XUV ionization many shake-up states can be populated. Thus, the yield curve of Ne2+ contains contributions from the strong-field ionization of several states, and it is not at all easy to separate the contribution of an individual state which contains information on the time development of the strong-field ionization. Besides, the excitation of the shake-up ionic states occurs in the presence of the strong NIR field which can significantly affect the population probability. This has led the authors of the paper [1]to the plausible suggestion that the origin of the small dips in the yield curve is connected with the influence of the strong laser field on the shake-up process. This question is still open. In this communication we propose a possible modification of the above-described experiment suitable for studying the time development of the strong-field ionization on the attosecond scale which is free from the above-discussed shortcomings. We suggest studying the strong-field ionization of Li atoms by measuring the yield of the doubly charged Li ions as a function of the time delay between the strong infrared (IR) pulse and the probing attosecond XUV pulse. The outer 2s electron in the Li atom has the binding energy of Eb = 5.39 eV. It will be effectively ionized already at moderate laser intensity of 1012 -1013 W cm-2 . On the other hand, the ionization potential of Li+ is very high, 75.64 eV, therefore double ionization of Li atoms by the IR field is practically impossible for such laser intensities. Double ionization by a weak XUV pulse is also very improbable (less than 1% of the total ionization yield [3]). Therefore, the doubly charged Li ions will be mainly produced by joint action of both fields. The process we are discussing may be considered as comprising three stages. At the first stage the Li atoms are `heated' by the strong IR field, and some of the electrons start to leave the parent atoms. Then the ultrashort (attosecond) XUV pulse arrives and ejects an inner shell 1s electron. Due to a sudden increase in the attractive potential, for those excited valence electrons, which are still close to the nucleus, the emission probability becomes considerably suppressed. However, the electrons, which are already far away (in the region where the atomic potential is negligible), will continue to be driven by the strong IR field and may be finally ejected. Thus, the double ionization will be completed. Roughly speaking, one can say that the XUV pulse interrupts the electron-emission process initiated by the laser pulse. In this sense, the XUV pulse can be considered as a probe pulse. The final probability of double ionization will depend on the time delay between the IR and the XUV pulses. By measuring the yield of the doubly charged ions as a function of the time delay between the two pulses, one can obtain information on the time evolution of the strong-field ionization. Note that in the case of Li, the initial state of the electron to be ionized is a well-defined ground state and therefore the accurate theoretical analysis of the experiment is more feasible than that of the first experiment [1]. In the following, we present a theoretical model of the above process which reveals the characteristic features similar to those which were observed in recent experiment on attosecond tunnelling [1]. The model is based on the numerical solution of the time-dependent Е Schrodinger equation (TDSE) and includes on equal footing the realistic atomic and ionic potentials and the strong electromagnetic field. We calculate the double-ionization probability for different intensities of the IR laser field under conditions which are very close to the recent experiment [1]. We consider a short (5 fs FWHM) laser pulse with the basic wavelength of 800 nm and with the intensity in the interval of I (0.5 - 2) Ч 1013 W cm2 . For such intensities, the Keldysh parameter [4]is = Eb /2Up (3 - 1.5) (Up is the ponderomotive


Fast Track Communication

F301

energy). Thus, we consider the intermediate regime between the multiphoton ( 1) and tunnelling ( 1) strong-field ionization. We suppose that the XUV pulse is extremely short TL /2. so that its duration (X ) is much smaller than the half period of the laser light, X For the considered laser wavelength this means that X < 100 as. This condition allows us to considerably simplify the computations since the sudden approximation can be used for a description of the XUV pulse effect. However, if in a real experiment the XUV pulse is not so short, one can use the theoretical approach similar to that developed in our paper [5]. Thus, the limitation introduced by the sudden approximation is not very important. For a description of the strong-field ionization of the Li atom, we solve the TDSE within the single-active electron approximation. By expanding the electron wavefunction in partial waves one obtains the system of non-stationary equations for the partial wavefunctions (atomic units are used throughout unless otherwise indicated): i ^ u (r, t ) = h( ) (r )u (r, t ) + r E L (t ) t
L
max

C ( , )u (r, t ).
=0

(1)

Here u (r, t ) is the radial part of the partial wavefunction of the active electron corresponding ^ to the orbital angular momentum , h( ) (r ) is the -dependent single-electron Hamiltonian, and the coefficients C ( , ) are the angular parts of the dipole matrix elements (see, for example, equation (3) in [6]). We suppose that initially, i.e. before the atom is irradiated by the electromagnetic pulses, the electron is in the atomic 2s state, u2s (r ). Therefore, the initial condition for the functions u (r, t ) can be written as u (r, t = 0) = 0 u2s (r ), where 0 is the Kronecker symbol. The single-electron Hamiltonian 1 2 ( +1) ^ h( ) (r ) = - + U (r ) + , 2 r 2 2r 2 contains an effective -dependent potential U (r ) which we have chosen as follows. (3) (2)

(i) If t td (td is the delay of the XUV pulse counted from the onset of the IR pulse at t = 0), then the potential of the atomic core is taken in the l-dependent form [7]: U (r ) = - 1+2 exp -a (1) r + a (2) r exp -a (3) r r- 2 r2 + d
22

,

(4)

where the parameters a (i ) (i = 1 - 3), and d have been fitted in such a way that the calculated energy positions of the low-energy levels of each symmetry ( = 0, 1, 2) reproduce the experimental data [8]. For the higher orbital momenta the potential was taken the same as for = 2. (ii) If t > td , then the potential of the ionic core, which consists of the nucleus and one 1s electron, is taken as a screened hydrogen-like potential: U (r ) = -[2 + exp(-6r)]/r - 3exp(-6r). (5)

Thus, in the spirit of the sudden approximation, at the moment t = td , the potential, which is felt by the valence electron, suddenly changes from (4)to(5). The last term in equation (1) describes the dipole interaction of the electron with the strong laser field EL (t ):
EL (t ) = E0L L (t ) cos(L t),

(6)


F302

Fast Track Communication

0.8

0.6

Probability

0.4

0.2

0 0

100

20 0 300 Delay time (a.u.)

400

Figure 1. The probability of double ionization as a function of the delay time (in atomic units) between the XUV and the IR pulses for different intensities of the IR pulse: 2 Ч 1013 W cm-2 (circles), 1 Ч 1013 W cm-2 (squares) and 0.5 Ч 1013 W cm-2 (diamonds). Symbols show the calculated results; the lines are drawn to guide the eyes. Dashed line shows the squared vector potential of the IR field in arbitrary units. The horizontal bars on the right show the asymptotic values of the probability for each of the intensities.

where E0L is the amplitude, L (t ) is the envelope and L is the carrier frequency of the IR pulse. We suppose that the laser field is linearly polarized along the z direction and has a cosine-type envelope: L (t ) = 0.5{cos[(t /L - 1)]+1}. (7) The pulse starts at t = 0, reaches its maximal value at t = L and the FWHM of its envelope is L . Since the laser field is strong, it mixes the partial states with different ; the coefficients C ( , ) in (1) describe this mixing. The system of equations (1) has been solved using the split-propagation algorithm with the Crank-Nicolson propagator [9, 10]. Up to 50 partial waves have been taken into account. Further details of the calculations as well as a description of the used algorithm can be found in [11]. In figure 1, the calculated probability of Li double ionization is presented as a function of the delay between the IR laser and the XUV pulses for several intensities of the laser. (In the presentation of the results we do not include the probability of 1s ionization by the XUV pulse which is independent of the delay time. For simplicity it is set equal to unity. Thus, we compute the probability of the process per one absorbed XUV photon. To obtain the cross section of double ionization, one should multiply the shown probability by the cross section of single ionization of the 1s electron by the XUV pulse.) In all cases the probability is increasing from some minimal value, corresponding to the strong-field ionization of the Li+ ion with a vacancy in an inner 1s shell which was produced instantaneously by the XUV pulse acting before the laser pulse. At small delay time, the probability is almost constant since the IR field is still weak so that before the XUV pulse the valence electron is not yet excited. However, at the delay td > 100 au (2.4 fs), the ionization probability quickly increases with the delay. At small laser intensity it increases almost monotonically. However, already at the intensity of 1 Ч 1013 W cm-2 the sharp steps with preceding minima appear. The first step is especially well pronounced. The sharp increase of the probability and the following minimum are correlated with the maxima and minima of the vector potential of the laser field at least for the first two strong peaks of the squared vector potential (shown by the dashed curve). The


Fast Track Communication

F303

0.5 0.4

Probability

0.3 0.2 0.1 0 0 100 200 300 Delay time (a.u.) 400

Figure 2. The probability of double ionization as a function of the delay time (in atomic units) between the XUV and the IR pulses for the IR pulse intensity I = 0.5 Ч 1013 W cm-2 (black circles). Separately shown are the contributions of ionization when electrons are emitted in the forward hemisphere (red triangles up) and in the backward hemisphere (blue triangles down). Symbols show the calculated results; the lines are drawn to guide the eyes. The IR field and its squared vector potential are shown by dot-dashed and dashed lines, respectively, in arbitrary units. The horizontal bars on the right show the asymptotic values of the corresponding probabilities.

steps become less pronounced at larger time delay. The reason is obvious: for a large delay the emitted electron wave packet has already propagated at large distance from the atom and the influence of the potential change due to the XUV ionization is not so strong as at smaller delays. The asymptotic values of the probability at large delays, when the XUV pulse comes long after the laser pulse has already terminated, are shown by the horizontal bars. They correspond to the single-ionization probability of the Li atom by the IR laser pulse. This probability remains less than unity and tends to the value about 0.75 as the intensity of the IR pulse further increases (not shown in the figure) which is known as the stabilization effect [12]. Note that at the two larger intensities shown in figure 1, the ionization is almost completed before the IR pulse achieves its maximum (t = 206 au or 5 fs). The calculated dependence of the double ionization cross section on the delay time between the two pulses shows the principle characteristic features, which have been observed in the experiment [1]. To avoid possible confusion we would stress that the visual similarity of the yield curves in figure 1 of this paper and in figure 4 of the experimental work [1] should be considered with some caution. In fact, the sequence of processes in the two papers is different and, correspondingly, the definitions of the delay time are also different. In [1] the state, which is ionized by the strong field, is created by the XUV pulse which shakes up an electron; therefore, the maximal ionization occurs when the laser pulse follows the XUV pulse. In the process considered currently, the laser pulse ionizes the existing electronic state until the XUV pulse arrives; therefore, the maximal ionization occurs when the XUV pulse follows the laser pulse. Nevertheless, the physical process, namely the strong-field ionization of a weakly bound atomic state, is essentially the same and the qualitative similarity of the results is not accidental but reflects the same time development of the field ionization process. In order to achieve better understanding of the time evolution of the ionization, we have also calculated the probability of ionization separately for the case when the electrons are emitted forward (in the forward hemisphere) and backward (in the backward hemisphere) with


F304

Fast Track Communication

0.6

Probability

0.4

0.2

0 0 100 200 300 Delay time (a.u.) 400

Figure 3. The same as in figure 2 but for the laser intensity I = 1 Ч 1013 Wcm

-2

.

0.8

0.6

Probability

0.4

0.2

0 0 100 200 300 Delay time (a.u.) 400

Figure 4. The same as in figure 2 but for the laser intensity I = 2 Ч 1013 Wcm

-2

.

respect to the direction of the IR pulse electric field at the envelope maximum (t = L ). The results are presented in figures 2-4 for the pulse intensities 0.5 Ч 1013 Wcm-2 , 1 Ч 1013 Wcm-2 and 2 Ч 1013 Wcm-2 , respectively. Consider first the lowest intensity (figure 2). One can see that the first step is mainly due to ionization events with emission of electrons in the backward direction (blue triangles pointing down). This peak is associated with the first strong maximum of the vector potential when the electric field is decreasing. In contrast, the second maximum is connected with the emission of electrons in the forward direction (red triangles pointing up). It is stronger since it is associated with the stronger peak of the vector potential, but at another phase of the field. The following maxima in both curves are less pronounced. This can be simply explained by the fact that the number of electrons which are still close to the atom quickly diminishes with the delay time. Note that the asymptotic value of the probability with backward emission is larger than that with the forward emission. Similar forward-backward asymmetry was found theoretically for strong-field ionization of hydrogen atoms by few-cycle pulses in [13], and was also observed experimentally for Xe atoms by Paulus et al [14]. One can see from the figure that the time variation of the strong-field ionization is much more pronounced when observed with the selection of electrons by the direction of emission.


Fast Track Communication

F305

When the intensity of the IR pulse increases (see figures 3 and 4 for the pulse intensities of 1 Ч 1013 Wcm-2 and 2 Ч 1013 Wcm-2 , respectively), the second maximum in the backward emission curve practically disappears. It is interesting that here, contrary to the previous case, the forward emission asymptotically dominates. Thus the forward-backward asymmetry of electron emission changes its sign depending on the field intensity. This phenomenon was already noted and discussed in paper [13]. At larger intensity (figure 4) the dominance of the forward emission becomes stronger. Besides, the first step appears already in both `forward' and `backward' curves. The first step in the sum curve becomes very strong and sharp. The probability of ionization is already close to the stabilization level, therefore the second step is much less pronounced. Comparing figures 3 and 4, one can see that the probability of the backward emission is almost independent of the IR laser intensity, while the probability of the electron emission in the forward direction changes drastically during the first half of the IR pulse. In conclusion, the suggested pump-probe experiment on double ionization of Li atoms seems to be feasible with modern experimental facilities. It allows one to obtain information on the strong-field ionization in the time domain. At the same time its interpretation is comparatively easy since the ionized state is the ground atomic state. Our model calculations show all the principal characteristic features that were observed in the first attosecond tunnelling experiment [1]. We have also found that the study of the time development of the strong-field ionization process with the separation of the forward and backward emission of slow electrons is more informative than only the double-charged ion yield and reveals a pronounced time structure. Acknowledgments We would like to thank J-M Rost, J Ullrich and A Dorn for stimulating discussions. We gratefully appreciate numerous helpful discussions with U Heinzmann. We acknowledge the financial support from Russian Foundation for Fundamental Researches via grant 06-0216289. NMK is grateful to Bielefeld University for hospitality and for financial support via SFB 613. References
[1] Uiberacker M et al 2007 Nature 446 627 [2] Yudin G L and Ivanov M Yu 2001 Phys. Rev. A 64 013409 [3] Huang M-T, Wehlitz R, Azuma Y, Pibida L, Sellin I A, Cooper J W, Koide M, Ishijima H and Nagata T 1999 Phys. Rev. A 59 3397 [4] Keldysh L V 1965 Sov. Phys.--JETP 20 1307 Keldysh L V 1964 J. Exp. Theor. Phys. 47 1945 (in Russian) [5] Kazansky A K and Kabachnik N M 2006 J. Phys. B: At. Mol. Opt. Phys. 39 5173 [6] Kazansky A K and Kabachnik N M 2007 J. Phys. B: At. Mol. Opt. Phys. 40 3413 [7] Bardsley J N 1974 Case studies At. Phys. 4 299 [8] http://physics.nist.gov/cgi-bin/ASD/energy1.pl [9] Crank J and Nicolson P 1947 Proc. Camb. Phil. Soc. 43 50 [10] Press W H, Flannery B P, Teulkovsky S A and Vetterling W T 1989 Numerical Recipes (Cambridge: Cambridge University Press) [11] Kazansky A K and Kabachnik N M 2007 J. Phys. B: At. Mol. Opt. Phys. 40 2163 [12] Popov A M, Tikhonova O V and Volkova E A 2003 J. Phys. B: At. Mol. Opt. Phys. 36 R125 [13] Chelkowski S and Bandrauk A D 2002 Phys. Rev. A 65 061802(R) [14] Paulus G G, Lindner F, Walther H, Baltuska A, Goulielmakis E, Lezius M and Krausz F 2003 Phys. Rev. Lett. 91 253004