Abstract
In the research area of stronglaserfield interactions and attosecond science^{1}, tunnelling of an electron through the barrier formed by the electric field of the laser and the atomic potential is typically assumed to be the initial key process that triggers subsequent dynamics^{1,2,3}. Here we use the attoclock technique^{4} to obtain experimental information about the electron tunnelling geometry (the natural coordinates of the tunnelling current flow) and exit point. We confirm vanishing tunnelling delay time, show the importance of the inclusion of Stark shifts^{5,6} and report on multielectron effects clearly identified by comparing results in argon and helium atoms. Our combined theory and experiment allows us to single out the geometry of the inherently onedimensional tunnelling problem, through an asymptotic separation of the full threedimensional problem. Our findings have implications for laser tunnel ionization in all atoms and in particular in larger molecular systems with correspondingly larger dipoles and polarizabilities.
Main
One of the most striking manifestations of the rules of quantum mechanics is the possibility for a particle to move from one side of a potential barrier to the other regardless of the energy height of that barrier. This includes the classically forbidden case, referred to as tunnelling, where the potential energy of the barrier is higher than the energy of the particle (Fig. 1a). In linearly polarized laser fields, electron tunnelling is expected to eventually lead to abovethreshold ionization, enhanced double ionization and coherent emission up to the Xray regime with highorder harmonic generation^{7,8,9,10}. Therefore, a detailed understanding of the tunnelling step is of paramount importance for attosecond science, including generation of attosecond pulses^{11,12} and attosecond measurement techniques^{4,13,14}.
The attoclock^{4} is an attosecond streaking technique^{13}. The rotating electric field vector of a closetocircularly polarized laser field gives the time reference, in a manner similar to the hands of a clock, and the time is measured by counting fractions of cycles with the exact angular position of the rotating electric field. In this way it is possible to obtain attosecond time resolution by employing a femtosecond pulse. The attoclock was used to set an upper limit to the tunnelling delay time during the tunnel ionization process in helium^{15}, and to measure the ionization times in double ionization of argon^{16,17}. For the attoclock, a very short fewfemtosecond pulse is used to both ionize an atom and to provide the time reference. The pulse duration is kept sufficiently short such that the ionization event is limited to within one optical cycle around the peak of the pulse. As a result of the closetocircular polarization, rescattering of the liberated electron with the parent ion is mostly suppressed. Assuming classical propagation of the liberated electron, the instant of ionization can be mapped to the angle of the final momentum of the electron in the polarization plane, measured with cold target recoil ion momentum spectroscopy^{18} (Fig. 2).
Here, we use the attoclock to measure the offset angle θ (defined in Fig. 3) that is directly related to the complex parent ion interaction and therefore extremely sensitive to the exact tunnel geometry. The attoclock cycle, the time zero (that is, the direction of the maximum laser field vector) and the exact time evolution of the streaking laser field are fully characterized independently (Supplementary Information). We minimize systematic errors in the angular streaking using both clockwise and anticlockwise polarized pulses (Fig. 3). The ionization event remains the same, but the clockwise and anticlockwise laser fields will streak the electron at the exit of the tunnel by equal amounts but in opposite directions.
In the experiments, we vary the peak intensity and therefore the Keldysh parameter^{19} from 0.5 to 1.1 and to 1.4 for helium and argon, respectively. Figure 4 shows the angular shift θ due to the interaction with the ionic potential during the angular streaking, as explained in Fig. 3. No significant intensity dependence of θ is observed for helium over the investigated intensity range (Fig. 4c), whereas argon exhibits a monotonic downwards trend of θ with increasing intensity (Fig. 4b).
Timedependent Schrödinger equation (TDSE) simulations^{20} (Fig. 4b) could be carried out only for three peak intensities without serious numerical problems (Supplementary Information). The TDSE results refer to a calculation employing a shorter pulse than the experimental one (a pulse of duration of three cycles as opposed to three cycles in fullwidth at halfmaximum of the intensity, employed in the experiment). Volume effects, not included in the TDSE results, would result in larger angular shifts compared with considering only the peak intensity. We use a semiclassical tunnelling model, TIPIS (tunnel ionization in parabolic coordinates with induced dipole and Stark shift; Fig. 4), that consists of an initial tunnelling step and subsequent classical propagation of the electron trajectory starting at the tunnel exit point (Fig. 1). The offset angle is especially sensitive to the ion–electron interaction and the tunnel exit point: the attraction to the ion at the beginning of the electron trajectory is reflected in an angular offset θ when compared with the laseronly trajectory (Figs 2b and 3). Without the forces from the parent ion the angular offset θ would be zero^{15,20}, and the final momentum would then be given by the integral of the electric field from the instant of ionization t_{0}; that is, the final momentum would follow the vector potential A(t_{0}) (ref. 21). Even without considering the effect of the ionic potential, any tunnelling delay time Δt_{D} will manifest itself as an angular offset, because instead of −A(t_{0}), the final momentum would then be −A(t_{0}−Δt_{D}). The offset angle is extremely sensitive to the tunnelling delay time, with one degree in θ corresponding to Δt_{D}≈5–10 as. The excellent agreement of our theory for both atoms and over a large intensity range below and above the Keldysh parameter γ=1 confirms zero tunnelling time within the experimental accuracy of 10 as.
Although the electron feels the full threedimensional (3D) potential (Fig. 1a), only the potential in the direction of the field is usually considered^{22}. We will refer to this model as the ‘fielddirection model’. Using that model, one can predict that overthebarrierionization (OBI) occurs for argon within the intensity range investigated in this work. The comparison between theory and experiment (Fig. 4b,c) shows that the calculation based on tunnel exit points obtained from the fielddirection model fails to reproduce the experimental trend in θ, the curves exhibiting a nonmonotonic dependence on laser intensity by having a pronounced local maximum close to the OBI intensity (Fig. 4b). The reason is that for the problem at hand the direction along the field is nonseparable.
A separation of the Schrödinger equation for the electron in the external field is possible for the pure Coulomb problem in parabolic coordinates ξ=r+z and η=r−z (see Fig. 1b). In that case, the resulting onedimensionality of the separated problem enables an analytical treatment of tunnelling^{23}. Here we consider multielectron systems, and show that a separation of the problem is still possible for relevant ξ and η values. In the inner region (Fig. 1b,c), we do not know the potential, but our procedure to determine the exit points is not sensitive to this lack of knowledge, as the region of space where the exit points are is separable in parabolic coordinates (Fig. 1c). Our procedure reveals the universal tunnelling geometry for atoms.
The complete derivation of our model is given in the Supplementary Information; here we provide only a sketch. In a static field F, we use the adiabatic approximation to find the effective potential^{24} for the outer electron, including the field and the multielectron effects^{5} (atomic units are used),
where α^{I} is the static polarizability of the ion. Then, we carry out an approximate separation of the oneelectron problem in a potential (1) in parabolic coordinates in the limit ξ/η≪1 (satisfied for the present experiment). The obtained effective potential along the η coordinate (the coordinate through which tunnelling occurs) is
where I_{p}(F)=I_{p}(0)+1/2(α^{N}−α^{I})F^{2} is the Starkshifted ionization potential^{6,25,26} and α^{N} is the static polarizability of the atom. We note that the procedure above is not just a change of coordinates, rather another separation procedure that defines a different tunnelling geometry. In the parabolically separated problem, the tunnelling geometry is a line along the parabolic η coordinate but it defines a whole region in the 3D space, whereas in the fielddirection model, the tunnelling geometry is a line in a 3D space. The exit points are determined by equating the potential (2) with m=0 and the energy term −I_{p}(F)/4. We note that the induced dipole (polarization) term is not included in the TDSE. The effect of the multielectron polarization term is small for the intensities at which the TDSE results could be obtained (Fig. 4b).
The tunnel exit points obtained from the separated problem are generally larger than those obtained in the fielddirection model^{26}, and the inclusion of the Stark shifts (larger binding energy) and induced dipole of the ion (larger barrier) pushes the exit points even farther away from the origin (Fig. 4a). The induced dipole and Stark shifts also lead to an increase of the OBI; however, the experimental parameters are such that in the parabolically separated problem, no OBI occurs (Fig. 4a). As a result of the very small polarizabilities of He and He^{+}, the Stark shifts and the induced dipole barrier modification are negligible, and it is the parabolically separated model that leads to a plateau in the θ dependence on intensity, rather than the weak monotonic increase produced by the fielddirection model (Fig. 4c). For the case of argon, with larger polarizabilities, the inclusion of Stark shifts, and even more, the multielectron effect through the increase of the barrier due to the induced dipole of the ion, becomes decisive for the decrease of θ with increasing intensity (Fig. 4b). The prediction for the trend in θ is insensitive to the exact form of the rate, the only requirement being that the rate increases with increasing intensity (Supplementary Information), and that gives further confidence about the calculated exit points in time and space.
The attoclock technique allowed us to gain new insights into laserinduced tunnelling, one of the paradigms of modern strongfield and ultrafast science. We have confirmed vanishingly small tunnelling time and, above all, revealed the natural geometry of laserinduced tunnelling in atoms, defined by the tunnelling current flow along one parabolic coordinate. Furthermore, our model showed and quantified the contributions of the Stark shifts of the boundstate energy levels and the multielectron effects describing the action of the induced dipole of the core electrons on the liberated electron. The extent to which the multielectron effects influence the tunnelling dynamics is system dependent as shown here by the difference in helium and argon. Argon is much easier to polarize than helium and is therefore affected more strongly.
The implications of our study are significant in particular to attosecond science and strongfield physics. First, the intensity range where the electron still emerges below the tunnel barrier is significantly extended towards higher intensities. This is important for new attosecond measurement concepts because the liberated electrons in the laserinduced tunnelling regime exhibit more precisely defined properties and theoretical tools suited for belowthebarrier ionization can be used. Second, the multielectron effects identified here will greatly affect further studies on larger molecules and on surfaces where much less is known. In particular, larger molecules are much more polarizable than the noble gas atoms studied here, and the effects reported here will be visible especially in experiments employing circularly or nearcircularly polarized laser pulses that isolate the ionization event. Third, attosecond measurements typically rely on streaking techniques that are highly sensitive to the parent ion interaction^{27,28}, which needs to be described precisely (taking additional force terms such as those identified in the present work) to draw conclusions on the time delays.
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Acknowledgements
This work was supported by the NCCR Quantum Photonics (NCCR QP) and NCCR Molecular Ultrafast Science and Technology (NCCR MUST) programmes, research instruments of the Swiss National Science Foundation (SNSF), ETH Research Grant ETH03 092, an SNSF equipment grant and the Danish Council for Independent Research, Natural Sciences.
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A.N.P., C.C., M.S. and U.K. carried out experiments and simulations. D.D., M.As. and L.B.M. developed the theory. All authors participated in the writing of the paper.
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Pfeiffer, A., Cirelli, C., Smolarski, M. et al. Attoclock reveals natural coordinates of the laserinduced tunnelling current flow in atoms. Nature Phys 8, 76–80 (2012). https://doi.org/10.1038/nphys2125
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DOI: https://doi.org/10.1038/nphys2125
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