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arXiv:2305.09260

Partial and full tunneling processes across potential barriers

Philip Caesar Flores, Dean Alvin L. Pablico, Eric A. Galapon

Published May 16, 2023• 0 citations
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Abstract

We introduce the concept of partial and full tunneling processes to explain the seemingly contradictory non-zero and vanishing tunneling times often reported in the literature. Our analysis starts by considering the traversal time of a quantum particle through a potential barrier, including both above and below-barrier traversals, using the theory of time-of-arrival operators. We then show that there are three traversal processes corresponding to non-tunneling, full-tunneling, and partial tunneling. The distinction between the three depends on the support of the incident wavepackets energy distribution in relation to the shape of the barrier. Non-tunneling happens when the energy distribution of the quantum particle lies above the maximum of the potential barrier. Otherwise, full-tunneling process occurs when the energy distribution of the particle is below the minimum of the potential barrier. For this process, the obtained traversal time is interpreted as the tunneling time. Finally, the partial-tunneling process occurs when the energy distribution lies between the minimum and maximum of the potential barrier. This signifies that the quantum particle tunneled only through some portions of the potential barrier. We argue that the duration for a partial-tunneling process should not be interpreted as the tunneling time but instead as a partial traversal time to differentiate it from the full-tunneling process. We then show that a full-tunneling process is always instantaneous, while a partial-tunneling process takes a non-zero amount of time. We are then led to the hypothesis that experimentally measured non-zero and vanishing tunneling times correspond to partial and full-tunneling processes, respectively.

DOI
10.48550/arxiv.2305.09260

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We introduce the concept of partial and full tunneling processes to explain the seemingly contradictory non-zero and vanishing tunneling times often reported in the literature. Our analysis starts by considering the traversal time of a quantum particle through a potential barrier, including both above and below-barrier traversals, using the theory of time-of-arrival operators. We then show that there are three traversal processes corresponding to non-tunneling, full-tunneling, and partial tunneling. The distinction between the three depends on the support of the incident wavepackets energy distribution in relation to the shape of the barrier. Non-tunneling happens when the energy distribution of the quantum particle lies above the maximum of the potential barrier. Otherwise, full-tunneling process occurs when the energy distribution of the particle is below the minimum of the potential barrier. For this process, the obtained traversal time is interpreted as the tunneling time. Finally, the partial-tunneling process occurs when the energy distribution lies between the minimum and maximum of the potential barrier. This signifies that the quantum particle tunneled only through some portions of the potential barrier. We argue that the duration for a partial-tunneling process should not be interpreted as the tunneling time but instead as a partial traversal time to differentiate it from the full-tunneling process. We then show that a full-tunneling process is always instantaneous, while a partial-tunneling process takes a non-zero amount of time. We are then led to the hypothesis that experimentally measured non-zero and vanishing tunneling times correspond to partial and full-tunneling processes, respectively.
Abstract We introduce the concept of partial-tunneling and full-tunneling processes to explain the seemingly contradictory non-zero and vanishing tunneling times often reported in the literature. Our analysis starts by considering the traversal time of a quantum particle through a potential barrier, including both above- and below-barrier traversals, using the theory of time-of-arrival operators. We then show that there are three traversal processes corresponding to non-tunneling, full tunneling, and partial tunneling. The distinction among the three depends on the support of the incident wave packet's energy distribution in relation to the shape of the barrier. Non-tunneling happens when the energy distribution of the quantum particle lies above the maximum of the potential barrier. Otherwise, full-tunneling process occurs when the energy distribution of the particle is below the minimum of the potential barrier. For this process, the obtained traversal time is interpreted as the tunneling time. Finally, the partial-tunneling process occurs when the energy distribution lies between the minimum and maximum of the potential barrier. This signifies that the quantum particle tunneled only through some portions of the potential barrier. We argue that the duration for a partial-tunneling process should not be interpreted as the tunneling time but instead as a partial traversal time to differentiate it from the full-tunneling process. We then show that a full-tunneling process is always instantaneous, while a partial-tunneling process takes a non-zero amount of time. We are then led to the hypothesis that experimentally measured non-zero and vanishing tunneling times correspond to partial- and full-tunneling processes, respectively.
How long does a quantum particle take to traverse a classically forbidden energy barrier? In other words, what is the correct expression for quantum tunneling time? This seemingly simple question has inspired widespread debate in the physics literature. By drawing an analogy with the double-slit experiment, I argue that we should not even expect the standard interpretation of quantum mechanics to provide an expression for quantum tunneling time. I explain how this conclusion connects to time’s special status in quantum mechanics, the meaningfulness of classically inspired concepts in different interpretations of quantum mechanics, and the prospect of constructing experimental tests to distinguish between different interpretations.
Abstract How long does a quantum particle take to traverse a classically forbidden energy barrier? In other words, what is the correct expression for quantum tunnelling time? This seemingly simple question has inspired widespread debate in the physics literature. I argue that we should not expect the orthodox interpretation of quantum mechanics to provide a unique correct expression for quantum tunnelling time, because to do so it would have to provide a unique correct answer to a question whose assumptions are in tension with its core interpretational commitments. I explain how this conclusion connects to time’s special status in quantum mechanics, the meaningfulness of classically inspired concepts in different interpretations of quantum mechanics, the prospect of constructing experimental tests to distinguish between different interpretations, and the status of weak measurement in resolving questions about the histories of subensembles.
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.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
The question of how long a particle takes to tunnel through a potential barrier has remained unresolved for decades. Attoclock experiments report traversal times consistent with zero, while Larmor clock measurements yield small but nonzero durations---a discrepancy recently traced to the two methods measuring fundamentally different quantities. We propose a resolution based on a discrete field-quantum network model whose continuum limit is the nonlinear Klein--Gordon equation. In this framework, a particle is not a point object localized on one side of a barrier: it is an extended structure whose electromagnetic and gravitational stress fields---carrying a significant fraction of its total energy---penetrate the barrier and extend to the far side. When a detector on the far side couples resonantly to this pre-existing stress tail, the particle's global structural integrity fails instantaneously, delivering its full energy to the detection point without any transit through the barrier. The ``tunneling time'' is near-zero not because the particle moves superluminally, but because it was already present on both sides before detection. This picture naturally explains the attoclock's null result (no transit occurs), the Larmor clock's nonzero reading (spin precession accumulates in the extended field within the barrier), and the established exponential decay of tunneling probability with barrier width (the stress-field amplitude decays exponentially in classically forbidden regions). No new parameters or modifications to quantum mechanics are introduced.