Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Time-dependent quantum tunnelling via crossover processes

G J Papadopoulos

Published Mar 21, 1990• 19 citations
Publisher

Abstract

It is shown that the propagator K(xt mod x'0) for a particle in a potential field is derivable from the classical path starting from x' and reaching x at time t. Considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side. Tunnelling is then discussed in terms of the evolving wavefunction. It is argued that while the particle's expectation energy is taken to be lower than the barrier's height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process. The parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given. A computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles' injection rate.

DOI
10.1088/0305-4470/23/6/016

Related papers

Instantaneous velocity during quantum tunnelling

Xiao-Wen Shang, Jian-Peng Dou, Feng Lu +3 more

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.

quant-pharXiv:2512.16385•Dec 19, 2025•0 citations

Time dependence and line shape of spontaneous quantum tunneling

B. U. Felderhof

The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.

Journal of Physics A Mathematical and Theoretical•Oct 7, 2008•2 citations

Chapter 12. The Tunnelling Flight Time

Eli Pollak

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.

RSC theoretical and computational chemistry series•Jan 1, 2020•1 citation

A one-dimensional barrier and time-dependent tunnelling

K W H Stevens

There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.

Journal of Physics C Solid State Physics•Jul 10, 1983•83 citations

Time-dependent Interactions in Tunnelling Dynamics

Luca Nanni

In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrödinger equation of the model is exactly solved. The probability density inside the barrier is calculated from the obtained wave function, proving that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their interference. Furthermore, the interference term is time-dependent and contribute to the scattering process duration. The tunnelling time is calculated as the time needed to get the probability density inside the barrier to zero. This is the minimum duration of the measurement process before detecting the particle beyond the barrier. Based on this, a new method of estimating the tunnelling time by energy experimental measuring is proposed.

arXiv:2203.04882•Revista Mexicana de Física•Feb 9, 2022•1 citation

Semiclassical tunnelling of wavepackets with real trajectories

Ludovic D. C. Jaubert, Marcus A. M. de Aguiar

Semiclassical approximations for tunneling processes usually involve complex trajectories or complex times. In this paper we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunneling and interferences very accurately in the limit of small Plank's constant. We use these results to estimate the tunneling time of the wavepacket and find that, for high energies, the barrier slows down the wavepacket but that it speeds it up at energies comparable to the barrier height.

arXiv:quant-ph/0609016•Physica Scripta•Feb 8, 2007•4 citations
It is shown that the propagator K(xt mod x'0) for a particle in a potential field is derivable from the classical path starting from x' and reaching x at time t. Considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side. Tunnelling is then discussed in terms of the evolving wavefunction. It is argued that while the particle's expectation energy is taken to be lower than the barrier's height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process. The parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given. A computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles' injection rate.
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.
The time dependence of spontaneous quantum tunneling of a particle from a potential well through a barrier in one or three dimensions is studied on the basis of the solution of the time-dependent Schrödinger equation as an initial-value problem by the method of Laplace transform. The wavefunction in the outer space is expressed in terms of special functions known from the theory of classical diffusion. A concise expression is derived for the probability current into the outer space. At long times the current oscillates about zero before decaying with a t −4 long-time tail. A comparison is made with the solution obtained by an expansion of the initial wavefunction in terms of the complete set of scattering states and bound states.
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.
There has recently been a growth in interest in electron tunnelling through barriers, and in particular in the question of whether it is possible to talk of a tunnelling time and, if so, what is it? A brief account of a study of the signal velocity of wavepackets travelling into a rectangular barrier was given recently, and the present paper is an extended version of this theory. In particular it is shown that the signal velocity of a free electron is the same as its group velocity, which in turn is the same as that which a classical particle particle of the same energy would have. It is also shown that when the wavepacket enters a classically forbidden region the signal velocity continues to have a meaning, and indeed a pulse-like wavefunction travels very much like a light pulse in a dispersive and attenuating medium. The expression for the signal velocity is symmetrical with respect to the barrier height, so that pulses well below the barrier travel faster, though with greater attenuation, than pulses near the top of the barrier. This result agrees with conclusions reached by some other workers. It is also shown that an electron responds to an accelerating field in the barrier as would be expected for a classical particle. It is argued that the not uncommon statement that the non nor eigenstates of a free particle describe a beam of particles is incorrect for electrons, for such a solution can at most describe two electrons. The theory indicates that by using orthogonal wavepackets it is possible to construct many-electron wavefunctions which do describe beams of electrons, and then free electrons travel with the group (and signal) velocity.
In this paper, the tunnelling of a particle through a potential barrier is investigated in the presence of a time-dependent perturbation. The latter is attributed to the process of the energy measurement of the scattered particle. The time-dependent Schrödinger equation of the model is exactly solved. The probability density inside the barrier is calculated from the obtained wave function, proving that the tunnelling dynamics is determined not only by the transmitted and reflected waves but also by their interference. Furthermore, the interference term is time-dependent and contribute to the scattering process duration. The tunnelling time is calculated as the time needed to get the probability density inside the barrier to zero. This is the minimum duration of the measurement process before detecting the particle beyond the barrier. Based on this, a new method of estimating the tunnelling time by energy experimental measuring is proposed.
Semiclassical approximations for tunneling processes usually involve complex trajectories or complex times. In this paper we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunneling and interferences very accurately in the limit of small Plank's constant. We use these results to estimate the tunneling time of the wavepacket and find that, for high energies, the barrier slows down the wavepacket but that it speeds it up at energies comparable to the barrier height.