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Electron tunneling time: Attoclock revisited

April 25, 2019

Quantum tunneling time is a highly debated topic – we explain why

This is the latest update on the electron tunneling time measured with the attoclock technique. Quantum tunneling time is a highly debated topic – we explain why. We discuss the attoclock technique to extract tunneling delays with regards to the typical approximations such as the dipole approximation, non-adiabatic effects, photoelectron momenta at the tunnel exit, electron correlation and exit coordinate. We can confirm that the He attoclock measurement is in agreement with finite tunneling time models. However, the adiabatic approximation gives the wrong field strength calibration which effectively increases the tunnel barrier width (Fig. 13). Unresolved is the issue of the starting time of the tunneling process. Some results indicate a starting time before the peak of the electric field which would increase the tunneling time shown in Fig. 12 and 13. Single active electron time-dependent Schrödinger equation (TDSE) calculations overlap with the non-adiabatic data (Fig. 6) - mostly within the error bars.

Quantum tunneling is a fundamental and ubiquitous effect that sparked a long-standing debate on the time duration of this process (1, 2). The main theoretical contenders, such as the Keldysh, Buttiker-Landauer, Eisenbud-Wigner (also known as Wigner-Smith), and Larmor time give contradictory answers.

The attoclock is a recently developed approach for the extraction of tunneling delay time in the context of strong field ionization (3, 4). Our most recent attoclock experimental measurements (5) found a finite tunneling time over a wide intensity range and therefore a large variation of tunnel barrier width. This result sparked a number of theoretical developments (6–9). Only two theoretical predictions are compatible within our experimental error:  the Larmor time, and the peak of the probability distribution of tunneling times constructed using a Feynman Path Integral (FPI) formulation.  The FPI theory matches the observed qualitative change in tunneling time over a wide intensity range, and predicts a broad tunneling time distribution with a long tail. The implication of such a probability distribution of tunneling times, as opposed to a distinct tunneling time, would imply that one must account for a significant, though bounded and measurable, uncertainty as to when the hole dynamics begin to evolve (5). The FPI theory also agrees well when we take into account all non-adiabatic corrections. Another independent attoclock experiment (10) recently found finite tunneling delay times as well.
While the experiments mostly seem to agree that quantum tunneling does not happen instantaneously, there is no consensus yet on the theoretical side (6–9). However this topic is important not only to the interpretation of time-resolved studies in attosecond physics, but also in the treatment of many experimental schemes in the AMO community which are based on a semiclassical view of strong-field ionization (11–13).

Fig. Most probable trajectory: The colour scale shows the classical trajectory Monte Carlo (CTMC) simulation real space (left) or momentum space (right) probability density after the laser pulse has passed. The orange line traces a single classical trajectory (SCT). The targetwas helium, irradiated by a laser field with the following parameters:  = 0.89 (indicated as the green solid polarization ellipse), λ = 735 nm, pulse duration FWHM 9 fs, I = 2.5 · 1014 W/cm2. The influence of the ion Coulomb force on the electron during the propagation is included. A SCT initiated with the most probable initial conditions traces the highest probability dencity of the wave packet (movie version, mp4-file at the top of the linked page).

In this latest invited review article we we have reviewed why tunneling time is such a highly debated theoretical concept in quantum mechanics and why both statements “cannot be measured because time is not an operator” and “just follow the peak of the wavepacket” do not resolve this issue. Following the peak of a wavepacket, for example, can be tricky and often misleading. In contrast to a light pulse, an electron wavepacket disperses even in vacuum. Since the propagation of the peak of the wavepacket is defined by the group delay, almost any group delay can be measured during propagation in combination with an appropriate energy-dependent transmission filter. In fact strong-field ionization in the dipole approximation (i.e. tunnel ionization) is much faster than the group delay of the electron wavepacket (i.e. Wigner delay).

We reviewed the recent theoretical and experimental developments in the attoclock approach to extract tunneling delays with regards to the typical approximations such as the dipole approximation, non-adiabatic effects, photoelectron momenta at the tunnel exit, electron correlation and exit coordinate. We reviewed the initial conditions of semiclassical models. These describe the photoelectron wavepacket at the tunnel exit, and how their choice affects the delays extractd from the attoclock experiment (6, 7, 9, 14–16). Furthermore non-adiabatic effects and their interplay with the field strength calibration of strong-field ionization experimental data are discussed (15, 17). Another section present the role of multi-electron effects (18–20) and the dipole approximation (21). More detailed discussions with further references are presented in the review article.
(1)     Landsman, A.S.; Keller, U. Physics Reports 2015, 547, 1–24.
(2)     Landauer, R. Nature 1989, 341 (6243), 567–568.
(3)     Eckle, P.; Smolarski, M.; Schlup, P.; et al. Nature Physics 2008, 4 (7), 565–570.
(4)     Eckle, P.; Pfeiffer, A.N.; Cirelli, C.; et al. Science 2008, 322 (5907), 1525–1529.
(5)     Landsman, A.S.; Weger, M.; Maurer, J.; et al. Optica 2014, 1 (5), 343.
(6)     Zimmermann, T.; Mishra, S.; Doran, B.R.; et al. Physical Review Letters 2016, 116 (23), 233603.
(7)     Ni, H.; Saalmann, U.; Rost, J.M. Physical Review Letters 2016, 117 (2), 023002.
(8)     Teeny, N.; Yakaboylu, E.; Bauke, H.; et al. Physical Review Letters 2016, 116 (6), 063003.
(9)     Torlina, L.; Morales, F.; Kaushal, J.; et al. Nature Physics 2015, 11 (6), 503–508.
(10)   Camus, N.; Yakaboylu, E.; Fechner, L.; et al. Physical Review Letters 2017, 119 (2), 023201
(11)   Meckel, M.; Comtois, D.; Zeidler, D.; et al. Science 2008, 320 (5882), 1478–1482.
(12)   Lin, C.D.; Le, A.T.; Chen, Z.; et al. Journal of Physics B: At. Mol. Opt. Phys. 2010, 43 (12), 122001.
(13)   Bruner, B.D.; Soifer, H.; Shafir, D.; et al. Journal of Physics B: At. Mol. Opt. Phys. 2015, 48 (17), 174006.
(14)   Ivanov, I.A.; Kheifets, A.S. Physical Review A 2014, 89 (2), 021402.
(15)   Boge, R.; Cirelli, C.; Landsman, A.S.; et al. Phys. Rev. Lett. 2013, 111 (10), 103003.
(16)   Klaiber, M.; Hatsagortsyan, K.Z.; Keitel, C.H. Physical Review Letters 2015, 114 (8), 083001.
(17)   Hofmann, C.; Zimmermann, T.; Zielinski, A.; et al. New Journal of Physics 2016, 18 (4), 043011.
(18)   Pfeiffer, A. N.; Cirelli C.; Smolarski M.; et al. Nature Physics 2012, 8, 76
(19)   Emmanouilidou, A.; Chen, A.; Hofmann, C.; et al. Journal of Physics B: At. Mol. Opt. Phys. 2015, 48 (24),
(20)   Majety, V.P.; Scrinzi, A. Journal of Modern Optics 2017, 1–5.
(21)   Ludwig, A.; Maurer, J.; Mayer, B. W.; et al. Physical Review Letters 2014, 113, 243001

Reference: Hofmann, C., Landsman, A.S., and Keller, U. (2019). Attoclock revisited on electron tunnelling time. J Modern Opt 66, 1052-1070. (

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