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arXiv:2001.06071

Quantum Travel Time and Tunnel Ionization Times of Atoms

D. Demir, Serkan Paçal

Published Jan 16, 2020• 0 citations
arXivHTMLPDF

Abstract

Time it takes to travel from one position to another, devoid of any quantum mechanical description, has been modeled variously, especially for quantum tunneling. The model time, if universally valid, must be subluminal, must hold everywhere (inside and outside the tunneling region), must comprise interference effects, and must have a sensible classical limit. Here we show that the quantum travel time, hypothesized to emerge with the state vector, is a function of the probability density and probability current such that all the criteria above are fulfilled. We compute it inside and outside a rectangular potential barrier and find physically sensible results. Moreover, we contrast it with recent ionization time measurements of the $\rm He$ as well as the $\rm Ar$ and $\rm Kr$ atoms, and find good agreement with data. The quantum travel time holds good for stationary systems, and can have applications in numerous tunneling-driven phenomena.

DOI
10.48550/arxiv.2001.06071

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Time it takes to travel from one position to another, devoid of any quantum mechanical description, has been modeled variously, especially for quantum tunneling. The model time, if universally valid, must be subluminal, must hold everywhere (inside and outside the tunneling region), must comprise interference effects, and must have a sensible classical limit. Here we show that the quantum travel time, hypothesized to emerge with the state vector, is a function of the probability density and probability current such that all the criteria above are fulfilled. We compute it inside and outside a rectangular potential barrier and find physically sensible results. Moreover, we contrast it with recent ionization time measurements of the He\rm HeHe as well as the Ar\rm ArAr and Kr\rm KrKr atoms, and find good agreement with data. The quantum travel time holds good for stationary systems, and can have applications in numerous tunneling-driven phenomena.
Quantum tunneling governs numerous phenomena in biology, chemistry, physics and technology. Tunneling time, formulated in various different forms due to the absence of a time operator in quantum theory, has been measured recently in experiments based on the attoclock in ultrafast laser ionization of Helium atoms [A.~Landsman {\it et al.}, Optica {\bf 1}, 343 (2014)]. The experiment performs a refined measurement with which no tunneling time formula in the literature exhibits adequate congruence. Here we show that, entropic considerations lead to a real tunneling time which shows remarkable agreement with the experimental data and stays always subluminal. Indeed, with phase space volume setting the number of microstates for a single evanescing particle in a state of definite momentum consistent with energy conservation, one is naturally led to a statistical description for quantum tunneling in which thermal energy sets the tunneling time. This entropic tunneling time is rather general and might also be extended to photon and phonon tunneling phenomena.
Tunneling and tunneling time are hot debated and very interesting issues because of their fundamental role in the quantum mechanics. The measurement of the tunneling time in today's attosecond and strong field (low-frequency) experiments, despite its controversial discussion offers a fruitful opportunity to understand the time measurement and the role of time in quantum mechanics. In previous work Kullie (2015 Phys. Rev. A 92 , 052118), we suggested a model and derived a simple relation to calculate the tunneling time, which showed a good agreement with the experimental result for He-atom. In the present work we analyze and discuss our model considering the experimental result for H-atom, which is obtained recently by Satya Sainadh et al (2017 arXiv: 1707.05445 ). In the tunneling region, we find that our model shows a good agreement with their experimental result (similar to the He-atom in our previous work), and with the accompanied time-dependent Schrödiger equation simulations. However, Sainadh et al use a different picture of the tunneling, in which tunneling time becomes an imaginary quantity. Whereas our model represents a real tunneling time picture, precisely a delay time with respect to the ionization time at the atomic field strength. Moreover, even though the above-threshold-ionization is beyond the tunneling regime, we still see that the actual ionization time is related to our model. However, crucial points arise and keep some questions open especially on the experimental side.
How much time does a tunneling wave packet spent in traversing a barrier? Quantum mechanical calculations result in zero time inside a barrier . In the nineties analogous tunneling experiments with microwaves were carried out. The results agreed with quantum mechanical calculations. Electron tunneling time is hard to measure being extremely short and parasitic effects due to the electric charge of electrons may be dominant. However, quite recently the atomic ionization tunneling time has been measured. Experimental data of photonic, phononic, and electronic tunneling time is available now and will be presented. It appears that the tunneling time is a universal property independent of the field in question.
Quantum tunnelling, a hallmark phenomenon of quantum mechanics, allows particles to pass through the classically forbidden region. It underpins fundamental processes ranging from nuclear fusion and photosynthesis to the operation of superconducting qubits. Yet the underlying dynamics of particle motion during tunnelling remain subtle and are still the subject of active debate. Here, by analyzing the temporal evolution of the tunnelling process, we show that the particle velocity inside the barrier continuously relaxes from a large initial value toward a smaller one, and may even approach zero in the evanescent regime. Meanwhile, the probability density within the barrier gradually builds up before reaching its stationary profile, in contrast to existing inherently. In addition, starting from the steady-state equations, we derive an explicit relation between the particle velocity and the barrier width, and show that the velocity in evanescent states approaches zero when the barrier is sufficiently wide. These findings resolve the apparent paradox of a vanishing steady-state velocity coexisting with a finite particle density. We point out that defining an effective speed from the probability density, rather than from the probability current, can lead to spuriously nonzero "stationary speed," as appears to be the case in Ref. [Nature 643, 67 (2025)]. Our work establishes a clear dynamical picture for the formation of tunnelling flow and provides a theoretical foundation for testing time-resolved tunnelling phenomena.
How much time does a wave packet spent in tunneling a barrier? Quantum mechanical calculations result in zero time inside a barrier. In the nineties analogous tunneling experiments with microwaves were carried out confirming quantum mechanics. Electron tunneling time is hard to measure being extremely short. However, quite recently the atomic ionization tunneling time has been measured. Experimental data of photonic, phononic, and electronic tunneling time is available now. It appears that the tunneling time is a universal property in the investigated time range of twelve orders of magnitude.
The quantum tunneling time remains the subject of heated debate, and one of its most curious features is faster-than-light or superluminal tunneling. Our tunnel-ionization model of the time-delay, presented in previous work, shows good agreement with the attoclock measurement in the adiabatic and nonadiabatic field calibrations, which also enables the determination of the barrier time-delay. In the present work, we show that the tunnel-ionization for H-like atoms with large nuclear charge can be superluminal (quantum superluminality), which in principle can be investigated experimentally using the attoclock scheme. We discuss the quantum superluminality in detail for the different regimes of the tunnel-ionization. Our result shows that quantum tunneling faster-than-light is indeed possible, albeit only under somewhat extreme conditions.