How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.
How much time does it take for a particle to tunnel through a barrier? This question continues to baffle up to this very day, one of the reasons being the non-existence of an energy–time commutator analogous to the position–momentum commutation relation. Instead of worrying about the definition of an operator one may consider time as a parameter in the time-dependent Schrödinger equation and study the time evolution of a wavepacket appropriately scattered on a potential barrier. Using this approach, one may formally define a tunnelling flight time that may be measured in a time-of-flight experiment. The resulting tunnelling flight time either vanishes or is very small. The implications of this observation with respect to recent hydrogen and helium atom attosecond photoionization experiments are discussed. The vanishing or small flight time does not contradict the finite time measured in Larmor clock experiments, where the tunnelling particle affects an external degree of freedom whose dynamics induced by the tunnelling may be interpreted in terms of a time scale. Flight times are also relevant to quantum reflection where coherences due to resonance scattering are not well accounted for by Wigner's phase time delay.
Attosecond ionization experiments have not resolved the question "What is the tunneling time?". Different definitions of tunneling time lead to different results. Second, a zero tunneling time for a material particle suggests that the nonrelativistic theory includes speeds greater than the speed of light. Chemical reactions, occurring via tunneling, should then not be considered in terms of a nonrelativistic quantum theory calling into question quantum dynamics computations on tunneling reactions. To answer these questions, we define a new experimentally measurable paradigm, the tunneling flight time, and show that it vanishes for scattering through an Eckart or a square barrier, irrespective of barrier length or height, generalizing the Hartman effect. We explain why this result does not lead to experimental measurement of speeds greater than the speed of light. We show that this tunneling is an incoherent process by comparing a classical Wigner theory with exact quantum mechanical computations.
Abstract Different approaches for considering barrier crossing times are analyzed, with special emphasis on recent experiments which attempt to measure what is commonly referred to as the Larmor tunneling time. We show that that these experiments cannot reveal the Larmor time, due to the finite energy width of the incident particles. The Larmor time, which measures changes in spin polarization, is classified together with other measurements such as the Buttiker–Landauer oscillating barrier time as indirect measurements of interaction times of scattered particles. In contrast, we present a direct quantum mechanical measure of a barrier crossing time taken to be the difference between the mean flight time for a particle transmitted through a potential barrier incident on a screen and the time it would take to reach the same screen without the barrier. These metrics are asymptotic, in the sense that they infer a time from a measurement after the scattering event is over, whereas other measures like the dwell time are local. Some time measures are well-defined only for incident states which are monochromatic in energy, others are well-defined also for incident wavepackets whose incident energy width is finite. In this paper we compare the different approaches to conclude that only the flight time can be used to answer the provocative (but ultimately ill-posed) question: how much time does it take to tunnel through a barrier?
The time it takes a particle to tunnel through the asymmetric Eckart barrier potential is investigated using Gaussian wavepackets, where the barrier serves as a model for the potential along a chemical reaction coordinate. We have previously shown that the, in principle experimentally measurable, tunneling flight time, which determines the time taken by the transmitted particle to traverse the barrier, vanishes for symmetric potentials like the Eckart and square barrier [ Petersen , J. ; Pollak , E. J. Phys. Chem. Lett. 2017 , 9 , 4017 ]. Here we show that the same result is obtained for the asymmetric Eckart barrier potential, and therefore, the zero tunneling flight time seems to be a general result for one-dimensional time-independent potentials. The wavepacket dynamics is simulated using both an exact quantum mechanical method and a classical Wigner prescription. The excellent agreement between the two methods shows that quantum coherences are not important in pure one-dimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.
Using the time parameter in the time-dependent Schr\"odinger equation, we study the time of flight for a particle tunneling through a square barrier potential. Comparing the mean and variance of the energy and the flight time for transmitted and reflected particles, using both density and flux distributions, we find that, when accounting for momentum filtering, the suitably normalized transmitted and reflected distributions are identical in both the density and flux cases. In contrast to previous studies, we demonstrate that these results do not imply a vanishing tunneling time, but rather that the time it takes to tunnel through a square barrier is precisely given by the reflected phase time. For wide barriers, this becomes independent of the barrier width, as predicted independently by MacColl and Hartman. We show that these conclusions can be reached using a variety of arguments, including purely quantum mechanical ones. Analysis of the shapes of the distributions under consideration reveals that wave-packet reshaping is not an explanation for the MacColl-Hartman effect. The results presented here have direct implications for understanding recent experimental results in the study of the barrier crossing of rubidium atoms. The finite width of an incident wave packet significantly ``masks'' the tunneling time, and induces substantial asymmetry between the flight times of transmitted and reflected atoms.
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.
The question of how long a tunneling particle spends in the forbidden region of a barrier has been a perplexing puzzle in foundational quantum mechanics for many decades. The Larmor time, one definition of tunneling time, uses an auxiliary degree of freedom of the tunneling particles to clock the time spent inside the barrier. Recently, we made our first measurement of the Larmor time for Bose-condensed 87Rb atoms tunneling through an optical barrier.1 Here, we report on follow up measurements with improved precision in the measured times and also study the time for the reflected atomic cloud. We observe significant discrepancies between our results and a simple theory based on weak measurement. We discuss our findings, hypothesize explanations for our results, and suggest future studies.