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arXiv:2504.16575 · Physical Review A

Impact of a tunneling particle on the quantum state of a barrier: Two-particle wave-packet model

Roman Michelko, Peter Bokes

Published Jul 2, 2025• 1 citation
arXivHTMLPDF

Abstract

We investigate the scattering of two distinguishable particles with unequal masses and a mutual short-range interaction with the aim of quantifying the impact of a tunneling ``projectile'' particle on the quantum mechanical state of the ``barrier'' particle. We find that the states of the barrier particle after the tunneling or reflection of the projectile are displaced by a finite distance that is given by the derivative of the phase of the transmission or reflection amplitudes multiplied by factors dependent on particles' masses, respectively. We demonstrate these results on a numerical example with a resonant interaction between a projectile and barrier. Our work demonstrates physical implication of the concept of phase time delay in the form of finite displacements of particles that are, in principle, experimentally measurable.

DOI
10.1103/g7yp-nmvy

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We investigate the scattering of two distinguishable particles with unequal masses and a mutual short-range interaction with the aim of quantifying the impact of a tunneling projectile'' particle on the quantum mechanical state of the barrier'' particle. We find that the states of the barrier particle after the tunneling or reflection of the projectile are displaced by a finite distance that is given by the derivative of the phase of the transmission or reflection amplitudes multiplied by factors dependent on particles' masses, respectively. We demonstrate these results on a numerical example with a resonant interaction between a projectile and barrier. Our work demonstrates physical implication of the concept of phase time delay in the form of finite displacements of particles that are, in principle, experimentally measurable.
A new variant of the packet analysis to solve the tunneling time problem for the so-called completed elastic scattering (a potential barrier of an arbitrary form is located in the finite spatial interval) is presented. In particular, we propose an explanation of the "paradoxes" of wave packet tunneling, and elaborate a rule enabling one to extract the tunneling times for a particle from the analysis of tunneling the corresponding wave packet. It is shown that the quantum ensembles of transmitted and reflected particles are statistically independent not only after the scattering but also before this event. We find the corresponding counterparts for the transmitted and reflected packets, which describe separately both scattering channels at early times, and define the expectation values of delay times for each channel. Besides, to describe jointly all particles at the stage when they cannot be divided into two independent subensembles, we find the scattering time taking into account broadening the packets. We derive the condition to be satisfied for the completed scattering. The times above are defined in terms of the well-known quantum-mechanical averages for the position and wave-number operators.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
We develop a new variant of the wave-packet analysis and solve the tunneling time problem for one particle. Our approach suggests an individual asymptotic description of the quantum subensembles of transmitted and reflected particles both at the final and initial stage of tunneling. We find the initial states of both subensembles, which are non-orthogonal. The latter reflects ultimately the fact that at the initial stage of tunneling it is impossible to predict whether a particle will be transmitted through or reflected off the barrier. At the same time, in this case, one can say about the to-be-transmitted and to-be-reflected subensembles of particles. We show that before the interaction of the incident packet with the barrier both the number of particles and the expectation value of the particle's momentum are constant for each subensemble. Besides, these asymptotic quantities coincide with those of the corresponding scattered subensemble. On the basis of this formalism we define individual delay times for both scattering channels. Taking into account the spreading of wave packets, we define the low bound of the scattering time to describe the whole quantum ensemble of particles. All three characteristic times are derived in terms of the expectation values of the position and momentum operators. The condition which must be fulfilled for completed scattering events is derived. We propose also the scheme of a {\it gedanken} experiment that allows one to verify our approach.
In order to better understand the trapping and response time of resonant electron wave packets in a resonant tunneling structure, we examined the time-dependent transport of a spherically symmetric Gaussian wave packet through a model double-barrier potential. Within the effective mass picture, the wave packet is projected onto the complete eigenstates of the model potential. The wave packet is initially localized to the left of the barriers and allowed to propagate through the structure. At wave packet energies corresponding to the resonant energies, the probability transmitted to the right of the barriers exhibits a steplike behavior as a function of time. A computation is made of the quantum mechanical delay time, to~tclass, where to is the packet transit time and tclass is the transit time for a classical particle. An increase in this delay time is observed when the packet energy corresponds to the energy of the resonant levels, and a comparison is made to the decay time of a particle in the first resonant state of the well. The delay time increases with increasing barrier area and decreasing packet energy width.
In this paper, we discuss the tunneling time of a quantum particle through a rectangular barrier. The reflection and transmission times associated with the wave packets representing the particle are discussed. By using an initial Gaussian momentum distribution, we carry out a comparative analysis of the stationary phases of the incident, reflected, and transmitted wave packets leading to the reflection and transmission times $\ensuremath{\Delta}{t}_{R}$ and $\ensuremath{\Delta}{t}_{T},$ respectively. In the present treatment of this old and very known problem we take into account the deformations of the reflected and transmitted momentum distributions. These deformations produce a dependence of the reflection and transmission times on the location of the initial wave packet. In a parallel calculation, by numerically monitoring the time evolution of the system, we characterize a reflection and a transmission time. Such times agree with the ones obtained via the stationary phase method.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.