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Solid State Communications

Is the momentum of a tunneling electron real or imaginary? An alternative description of the tunneling process

Pierre Guéret

Published Aug 1, 1991• 2 citations
Publisher
DOI
10.1016/0038-1098(91)90493-f

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A plausible description of traversal time was given, both in classically allowed and forbidden regions, through a path-integral solution of the telegrapher's equation. This analysis was applied to a simulation based on microwave propagation in a waveguide considered as a one-dimensional system. An extension of the analysis has been performed in order to compare the traversal (or delay) time results relative to a beat-envelope signal with those as deduced from the distribution function of the randomized time and its analytical continuation in imaginary time. Subsequently, in tight analogy with a step signal in an electronic circuit (zero-dimensional system), we have searched for a simulation of traversal processes in real time, even for classically forbidden (tunneling) processes. First we have considered a finite series expansion of harmonic functions of the signal in the neighborhood of its rise, and applied the above mentioned procedure to each harmonic, implying analytical continuation in imaginary time and an arbitrary truncation in the range of the signal. Then, in order to avoid these shortcomings, we have considered a waveletlike description of the signal.
Both the physical picture of the dynamics of atoms and molecules in intense infrared fields and its theoretical description use the concept of electron trajectories. Here, we address a key question which arises in this context: Are distinctly quantum features of these trajectories, such as the complex-valued coordinates, physically relevant in the classically allowed region of phase space, and what is their origin? First, we argue that solutions of classical equations of motion can account for quantum effects. To this end, we construct an exact solution to the classical Hamilton–Jacobi equation which accounts for dynamics of the wave packet, and show that this solution is physically correct in the limit ħ→0. Second, we show that imaginary components of classical trajectories are directly linked to the finite size of the initial wave packet in momentum space. This way, if the electronic wave packet produced by optical tunnelling in strong infrared fields is localised both in coordinate and momentum, its motion after tunnelling ipso facto cannot be described with purely classical trajectories – in contrast to popular models in the literature.
Laser-induced electron tunneling ionization from atoms and molecules plays as the trigger for a broad class of interesting strong-field phenomena in attosecond community. Understanding the time of electron tunneling ionization is vital to achieving the ultimate accuracy in attosecond metrology. We propose a novel attosecond photoelectron interferometer, which is based on the interference of the direct and near-forward rescattering electron wave packets, to determine the time information characterizing the tunneling process. Adding a weak perturbation in orthogonal to the strong fundamental field, the phases of the direct and the near-forward rescattering electron wave packets are modified, leading to the shift of the interferogram in the photoelectron momentum distributions. By analyzing the response of the interferogram to the perturbation, the real part of the ionization time, which denotes the instant when the electron exits the potential barrier, and the associated rescattering time are precisely retrieved. Moreover, the imaginary part of the ionization time, which has been interpreted as a quantity related to electron motion under the potential barrier, is also unambiguously determined.