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arXiv:1903.06897

Wavelength Dependent Tunneling Delay Time

Xiaolei Hao, Zheng Shu, Weidong Li, Jing Chen

Published Mar 16, 2019• 0 citations
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Abstract

A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time $τ_{BL}$ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein $τ_{BL}$ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.

DOI
10.48550/arxiv.1903.06897

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A clear consensus on how long it takes a particle to tunnel through a potential barrier has never been so urgently required, since the electron dynamics in strong-field ionization can be resolved on attosecond time-scale in experiment and the exact nature of the tunneling process is the key to trigger subsequent attosecond techniques. Here a general picture of tunneling time is suggested by introducing a quantum travel time, which is defined as the ratio of the travel distance to the expected value of the velocity operator under the barrier. Specially, if applied to rectangular barrier tunneling, it can retrieve the Büttiker-Landauer time τBLτ_{BL}τBL​ in the case of an opaque barrier, and has a clear physical meaning in the case of a very thin barrier wherein τBLτ_{BL}τBL​ can not be well defined. In the case of strong-field tunneling process, with the help of the newly defined time, the tunneling delay time measured by attoclock experiment can be interpreted as a travel time spent by the electron to tunnel from a point under the barrier to the tunnel exit. In addition, a peculiar oscillation structure in the wavelength dependence of tunneling delay time in deep tunneling regime is observed, which is beyond the scope of adiabatic tunneling picture. This oscillation structure can be attributed to the interference between the ground state tunneling channel and the excited states tunneling channels.
Abstract Quantum tunnelling is a fundamental phenomenon in quantum physics, yet the question of how long it takes for a particle to tunnel through a barrier remains a topic of ongoing debate. Recent advances in attoclock experimental techniques in strong-field physics have renewed interest in this topic. While various definitions of tunneling time have been proposed, a consensus has yet to be reached. In this study, we adopt a definition of tunnelling time based on the ratio of the barrier width to the expected value of the velocity operator within the barrier. This approach reduces to the Büttiker-Landauer time (τBL\tau_{BL}τBL​) in the case of an opaque barrier and offers a clear physical interpretation for very thin barriers, where τBL\tau_{BL}τBL​ may not be well-defined. Additionally, by utilizing a virtual detector at the tunnel exit, we calculate the time delay between the peak of the electric field and the moment of ionization for long wavelengths ($\geq$6 μ\muμm). Our findings indicate that the time delay stabilizes around 190 as, remaining independent of the laser wavelength. Furthermore, through our quantum tunnelling time definition, this time delay can be interpreted as the time required for the electron to tunnel from a point within the barrier to the exit.
Abstract With the development of attosecond science, tunneling time can now be measured experimentally with the attoclock technique. However, there are many different theoretical definitions of tunneling time and no consensus has been achieved. Here, we bridge the relationship between different definitions of tunneling time based on a quantum travel time in one-dimensional rectangular barrier tunneling problem. We find that the real quantum travel time t Re is equal to the Bohmian time t Bohmian, which is related to the resonance lifetime of a bound state. The total quantum travel time τ t can perfectly retrieve the transversal time t x and the Büttiker–Landauer time τ BL in two opposite limits, regardless of the particle energy.
Tunnelling lies at the heart of quantum mechanics and is a fundamental process in attosecond science, molecular biology, and quantum devices. Whether tunnelling takes time and how a microscopic particle transits through a barrier have been debated since the early days of quantum mechanics. The time required for an electron to tunnel through an atomic potential barrier has been measured with attosecond angular streaking (attoclock), and a recent work on the hydrogen atom claimed that electron tunnelling is instantaneous. However, the time required for Rb atoms to tunnel through an optical potential barrier has been measured to be on the order of milliseconds with a recent Larmor clock measurement. The essence of electron and atom tunnelling is identical, but the reason for the contradictory conclusions remains unknown. Here, we demonstrate that the sub-barrier potential interaction is the root of the nonzero tunnelling delay time. We reveal that the wave-particle duality of a tunnelling electron must be fully taken into account when decoding the tunnelling delay time from the attoclock measurements. Based on energy-resolved attoclock measurements, we show that the tunnelling delay time of an electron ranges from 24 to 58 attoseconds, and the counterintuitive result that an electron with a lower energy may spend less time in the barrier is consistent with the velocity-dependent tunnelling time of atoms in Larmor clock measurements. Our results unify the tunnelling time of microscopic particles by highlighting the classically forbidden sub-barrier potential interactions of matter waves.
Tunneling delays represent a hotly debated topic, with many conflicting definitions and little consensus on when and if such definitions accurately describe the physical observables. Here, we relate these different definitions to distinct experimental observables in strong field ionization, finding that two definitions, Larmor time and Bohmian time, are compatible with the attoclock observable and the resonance lifetime of a bound state, respectively. Both of these definitions are closely connected to the theory of weak measurement, with Larmor time being the weak measurement value of tunneling time and Bohmian trajectory corresponding to the average particle trajectory, which has been recently reconstructed using weak measurement in a two-slit experiment [S. Kocsis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. M. Steinberg, Science 332, 1170 (2011)]. We demonstrate a big discrepancy in strong field ionization between the Bohmian and weak measurement values of tunneling time, and we suggest this arises because the tunneling time is calculated for a small probability postselected ensemble of electrons. Our results have important implications for the interpretation of experiments in attosecond science, suggesting that tunneling is unlikely to be an instantaneous process.
The conundrum parameter-operator of time in quantum mechanics, as well as the time-energy uncertainty relation and the tunneling delay time, have recently been addressed in attosecond optical ionization experiments. Dirac's formulation of the electron's relativistic quantum mechanics (RQM) allows the introduction of a dynamical self-adjoint time operator, representing an internal time of the system. Its relation to the parametric (laboratory) time is derived. Its relevance to the tunneling measurements in these experiments is exhibited, namely, as the time it takes the wave packet to cross the exit point of the barrier, thus justifying considering this time operator an additional observable in RQM.
The measurement of the tunneling time in attosecond experiments, termed attoclock, triggered a hot debate about the tunneling time, the role of time in quantum mechanics, where the interaction with the laser pulse involves two regimes of a different character, the multiphoton and the tunneling (field-) ionization. In the adiabatic field calibration, one of us (O. K.) developed in earlier works a real tunneling time model and showed that the model fits well to the experimental data of Landsmann et al. (Optica {\bf 1}, 343 2014). In the present work, it is shown that the model explains the experimental result in the nonadiabatic field calibration, where one reaches a good agreement with the experimental data of Hofmann et al. (J. of Mod. Opt. {\bf 66}, 1052, 2019). Furthermore, we confirm the result with the numerical integration of the time-dependent Schrödinger equation. The model is appealing because it offers a clear picture of the multiphoton and tunneling field-ionization regimes. In the nonadiabatic case (the nonadiabatic field calibration), the ionization is mainly driven by multiphoton absorption. Surprisingly, at a field strength F≤FaF \le F_aF≤Fa​ (FaF_aFa​ is the atomic field strength) the model always predicts a time delay with respect to the quantum limit τaτ_aτa​ at F=FaF=F_aF=Fa​. For an adiabatic tunneling the saturation at the limit (F=FaF=F_aF=Fa​) explains the well-known Hartman effect or Hartman paradox.