[Paper Review] The method of multiple internal reflections in a description of tunneling evolution of nonrelativistic particles and photons

Sergei P. Maydanyuk, V. S. Olkhovsky+1 more|ArXiv.org|Jul 29, 2004

Cold Atom Physics and Bose-Einstein Condensates12 references13 citations

TL;DR

This paper introduces a non-stationary method based on multiple internal reflections to describe tunneling dynamics of nonrelativistic particles and photons through one-dimensional and spherically symmetric barriers. By modeling wave packets and tracking fluxes through repeated reflections, it enables time-resolved analysis of tunneling, yielding explicit expressions for transmitted and reflected wave packet amplitudes, tunneling and reflection times, and a decomposed S-matrix. The method confirms Hartman’s and Fletcher’s effects and provides a causal, physically consistent framework for superluminal tunneling phenomena.

ABSTRACT

A non-stationary method for tunneling description of non-relativistic particles and photons through a barrier on the basis of consideration of the multiple internal reflections of vawe packets in relation of barrier boundaries is presented. The method is described in details and proved in the case of the one-dimentional tunneling of the particle through the rectangular barrier. For problems of the tunneling of the particle through the spherically symmetric barrier and of the photon through the one-dimensional barrier the amplitudes of transmitted and reflected wave packets in relation to the barrier, times of the tunneling and the reflection are found using of the method. Hartman's and Fletcher's effect is analysed.

Motivation & Objective

• To develop a non-stationary, time-resolved approach to tunneling that overcomes limitations of stationary wave function methods in sub-barrier regions.
• To describe tunneling evolution using wave packets and multiple internal reflections, enabling physical interpretation of fluxes and time parameters.
• To extend the method to spherically symmetric barriers and demonstrate its applicability to photon tunneling via mathematical analogy.
• To analyze Hartman’s and Fletcher’s effects in a physically consistent, non-stationary framework.
• To provide a decomposition of the S-matrix into transmitted and reflected wave packet components, validating physical consistency.

Proposed method

• The method models tunneling via wave packets undergoing repeated internal reflections at barrier boundaries, tracking forward and backward fluxes in the sub-barrier region.
• It uses time-dependent wave packet propagation to compute amplitudes of transmitted and reflected components, avoiding unphysical stationary state assumptions.
• For one-dimensional rectangular barriers, the method derives explicit expressions for wave packet amplitudes and time delays using iterative reflection series.
• For spherically symmetric barriers, the method computes radial wave packet amplitudes and total tunneling/reflection times via angular momentum decomposition.
• A mathematical transformation maps the particle tunneling problem to the photon tunneling problem, preserving wave dynamics and time parameters.
• The S-matrix is expressed as a sum of two components: one for the transmitted wave packet and one for the reflected, ensuring physical consistency with standard stationary methods.

Experimental results

Research questions

• RQ1 How can tunneling dynamics be described in real time using wave packets and multiple internal reflections?
• RQ2 Can the method yield physically meaningful tunneling and reflection times for both particles and photons?
• RQ3 Does the method reproduce Hartman’s and Fletcher’s effects in a causal and consistent manner?
• RQ4 How can the S-matrix be decomposed into physically distinct transmitted and reflected wave packet components?
• RQ5 Can the method be extended to spherically symmetric barriers and applied to photon tunneling?

Key findings

• The method successfully computes time-resolved tunneling dynamics for nonrelativistic particles and photons through one-dimensional and spherically symmetric barriers.
• Tunneling and reflection times are derived for the first time using this non-stationary, multiple-reflection approach.
• The S-matrix is decomposed into two physically meaningful components corresponding to transmitted and reflected wave packets, with convergence to standard stationary results.
• Hartman’s and Fletcher’s effects are confirmed: superluminal group velocities are observed in wide, high barriers, but the wavefront velocity remains ≤ c, preserving causality.
• The method reveals that tunneling is a non-local process: the wave packet instantaneously senses both barrier boundaries upon entry.
• A mathematical transformation allows direct mapping of particle tunneling results to photon tunneling, demonstrating physical and mathematical analogy between the two systems.