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Atomic physics

Electrons get real

Nature volume 485, pages 313314 (17 May 2012) | Download Citation

Strong laser fields allow electrons to tunnel out of atoms. The response of such electrons to a second laser field supports the idea that they start tunnelling at a time defined by a complex number, but exit atoms at a 'real' time. See Letter p.343

Physicists are perfectly aware that the microscopic behaviour of electrons cannot be understood without the laws of quantum theory. Nevertheless, when scientists trace the dynamics of subatomic phenomena, they like to ask questions that are motivated by a classical, non-quantum perspective. In this spirit, Shafir et al.1 report on page 343 the exact times at which electrons 'exit' atoms that are irradiated by a short flash of laser light. The existence of such an exit time is seemingly counter-intuitive, given that electrons are described by wavefunctions that extend smoothly from the inside to the outside of atoms — part of the electron is always outside the atom. In the presence of a laser field, however, there is a continuous outward flow of electron density, which Shafir and colleagues have decomposed into different electron trajectories, assigning each trajectory an experimentally determined starting time.

The emission of electrons from atoms in Shafir and colleagues' experiments is a consequence of quantum tunnelling. The applied laser field changes the potential-energy profile experienced by the electron, forming a finite barrier that is impenetrable to classical Newtonian particles, but which can be tunnelled across by electrons. A similar process forms the basis of scanning tunnelling microscopy: electrons tunnel between the surface of the object under study and the tip of the microscope. Tunnelling occurs because electron wavefunctions encompass both sides of a potential barrier (Fig. 1a); so what is the meaning of an exit time?

Figure 1: Quantum tunnelling and high-harmonic generation.
Figure 1

a, The blue line depicts the potential-energy profile that binds an electron in a laser-irradiated atom. The atom's nucleus is at the profile's minimum. If the electron were a classical Newtonian particle, it could not enter the shaded 'forbidden' regions and would be trapped in the atom. But electrons are quantum-mechanical objects whose probability distributions in space are described by wavefunctions (Ψ, such as the one shown in green). Because wavefunctions extend through the right-hand forbidden area, electrons may tunnel out of the atom. b, In the phenomenon of high-harmonic generation, a laser field accelerates an electron (e) that has tunnelled out of an atom away from the resulting ion, then directs it back again. Recombination of the electron with the parent ion generates a high-energy photon (a high-harmonic emission). Shafir and colleagues' report1 suggests that high-harmonic emissions from helium atoms are described by 'quantum orbits'. This means that tunnelling proceeds in imaginary time (the imaginary part of time as defined by a complex number), but the electron moves as a classical particle in 'real' time once it has exited the atom. At the start of its real-time journey, the electron counter-intuitively moves towards the parent ion.

Before answering that question, one must realize that the authors did not detect electrons in their experiments. Instead, they recorded the light released on the return of an emitted electron to its parent ion. Light release occurs because the electric field in a laser pulse reverses direction periodically. This means that, about 1 femtosecond (10−15 seconds) after an electron has tunnelled out of an atom, the laser's force pushes it back towards the resulting ion (Fig. 1b). If the electron and ion recombine to form the same bound state that existed before ionization, then a photon is emitted2. Because the photon's frequency (and therefore its energy) is much higher than that of the incident laser light, the photon-forming process is called high-harmonic generation.

To resolve the electron emission in time, Shafir et al. perturbed electrons emitted from helium atoms with a second, weak probe field acting perpendicular to the main laser field. To picture the experiment, imagine a game in which people (the atoms) repeatedly and regularly throw balls (the electrons) vertically into the air and then catch them. Even without seeing the players, one can discern a successful capture by hearing them cheer “Yeah” (photon emission).

Now suppose you could blow a crosswind (the probe field) over the heads of the players. If the crosswind blows when the ball is in the air, it pushes the ball sideways. In this case, the ball lands some distance away from its starting point so that the player cannot catch it, and there is no cheer. By applying the crosswind at various times and then listening for the presence or absence of cheers, one can determine the precise throwing times (the ionization times).

In their experiments, Shafir et al. used a probe field that oscillated at twice the frequency of the main field, and monitored the intensity of the emitted harmonics as the researchers varied the temporal shift between the two fields (the time elapsing between a maximum of the main field and a maximum of the probe field). But for a robust analysis, they also needed to measure electron departure and return times independently, which meant that they had to take things further. In our analogous game, a crosswind whose direction alternates could affect the ball so much that it unexpectedly hits the player from the side, making him shout “Yikes” instead of “Yeah”. Similarly, Shafir et al. deduced the angle of electron return in their experiments by detecting specific photon emissions known as even-order harmonics. These emissions, whose frequencies are even multiples of the main laser's frequency, were generated when released electrons hit their parent ions 'from the side' — at an angle to the main field.

Of particular note, the authors found that every harmonic emission frequency has its own ionization time, all of which fall within a range of 200 attoseconds (1 attosecond is 10−18 s). The superposition of the many different associated electron trajectories forms a quantum-mechanical wave packet — a short 'pulse' of travelling wave activity — for emitted electrons. The observed ionization times1 are conceptually different from the extremely small tunnelling delay time (tens of attoseconds at most) reported in a previous study of helium atoms3, which measured the delay between the maximum of the applied oscillating laser field and the most likely time of electron departure.

One striking result is the excellent agreement of Shafir and colleagues' findings with a model of high-harmonic generation that is well known to atomic physicists — the quantum orbit model4. Put simply, the model states that an electron trajectory begins with negative kinetic energy at an instant of time defined by a complex number. Just the imaginary part of time changes for the electron as it tunnels through a potential barrier; time becomes real-valued only at the exit of the tunnel. This real time is the exit time measured by Shafir and colleagues. It is the time at which the electron starts to feel the effect of the probe field.

In molecules, high-harmonic generation often involves contributions from different 'channels' — that is, not only from the most weakly bound electrons of atoms, but also from more tightly bound ones5,6. Shafir et al.1 report that small differences in ionization times from two channels in carbon dioxide are, in principle, measurable. The characteristic differences are observable when the channels interfere nearly destructively. However, this scenario corresponds to a small range of the emitted spectrum, and generates a low number of photons. Determining reliable, channel-dependent ionization times will therefore be extremely challenging.

One limitation of Shafir and colleagues' study is that they measure only how ionization times and return times vary with harmonic frequency. But the absolute timing of ionization is of substantial interest too, because it is related to the tunnelling delay time and it may influence the absolute timing of the harmonic emission. It remains to be seen how well the authors' technique will work for mixtures of gases, in which differences between atomic species may complicate harmonic emission profiles. Extending the present study to such a case may reveal interference between emissions from different gases in the same way that different channels in the same molecule interfere with each other.

Accurate knowledge of the electron excursion times (the difference between return and ionization times) during high-harmonic generation is vital for our understanding of the many ultrafast experiments5,7 in which ionization triggers a dynamic process, and in which recombination of the resulting ion with the electron takes a snapshot of that process. Unlike the classical approach8 to ultrafast time-resolved chemistry, in which reactions are initiated using a 'pump' light pulse and a separate probe pulse is used to monitor reaction evolution, high-harmonic generation combines the pump and probe steps into just one shot. By facilitating the real-time observation of attosecond electron dynamics, this approach will increasingly compete with ultrafast spectroscopic methods in which molecules are directly probed by attosecond light pulses9.


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  1. Manfred Lein is at the Centre for Quantum Engineering and Space-Time Technology and at the Institute for Theoretical Physics, Leibniz Universität Hannover, 30167 Hanover, Germany.

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Correspondence to Manfred Lein.

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