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Journal of Physics Condensed Matter

Equivalence between tunnelling times based on: (a) absorption probabilities, (b) the Larmor clock, and (c) scattering projectors

Juan Gonzalo Muga, S. Brouard, R. Sala Mayato

Published Oct 26, 1992• 26 citations
Publisher

Abstract

The assumption of analyticity of the transmission and reflection probability amplitudes as functions of a complex potential is shown to be justified. As a consequence the reflection and transmission times based on absorption probabilities are shown to be equal to the corresponding times derived from the local Larmor precession in the plane perpendicular to the magnetic field. An additional new interpretaton of these times is provided by means of a more general scattering theory projector formalism. The relation to the phase times is discussed.

DOI
10.1088/0953-8984/4/43/002

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Physica Scripta•Apr 29, 2025•2 citations
The assumption of analyticity of the transmission and reflection probability amplitudes as functions of a complex potential is shown to be justified. As a consequence the reflection and transmission times based on absorption probabilities are shown to be equal to the corresponding times derived from the local Larmor precession in the plane perpendicular to the magnetic field. An additional new interpretaton of these times is provided by means of a more general scattering theory projector formalism. The relation to the phase times is discussed.
The times spent by an electron in a scattering event or tunnelling through a potential barrier are investigated using a method based on the absorption probabilities. The reflection and transmission times derived from this method are equal to the local Larmor times if the transmission and reflection probability amplitudes are complex analytic functions of the complex potential. The numerical results show that they coincide with the phase times except as the incident electron energy approaches zero or when the transmission probability is too small. If the imaginary potential covers the whole space the tunnelling times are again equal to the phase times. The results show that the tunnelling times based on absorption probabilities are the best of the various candidates.
Expressions for the transmission, reflection, and interference contributions to the dwell time for wave packet scattering are obtained by means of scattering theory projector operators that provide a unifying framework for various dwell time decompositions. These different dwell time resolutions are illustrated with the aid of a formal analogy between the two-level system and one-dimensional scattering. The complex "times" and Larmor times proposed by other authors are found as particular cases of the theory. Numerical calculations for a rectangular barrier are provided.
A {\it completed} scattering of a particle on a static one-dimensional (1D) potential barrier is a combined quantum process to consist from two elementary sub-processes (transmission and reflection) evolved coherently at all stages of scattering and macroscopically distinct at the final stage. The existing model of the process is clearly inadequate to its nature: all one-particle "observables" and "tunneling times", introduced as quantities to be common for the sub-processes, cannot be experimentally measured and, consequently, have no physical meaning; on the contrary, quantities introduced for either sub-process have no basis, for the time evolution of either sub-process is unknown in this model. We show that the wave function to describe a completed scattering can be uniquely presented as the sum of two solutions to the Schrödinger equation, which describe separately the sub-processes at all stages of scattering. For symmetric potential barriers such solutions are found explicitly. For either sub-process we define the time spent, on the average, by a particle in the barrier region. We define it as the Larmor time. As it turned out, this time is just Buttiker's dwell time averaged over the corresponding localized state. Thus, firstly, we justify the known definition of the local dwell time introduced by Hauge and co-workers as well by Leavens and Aers, for now this time can be measured; secondly, we confirm that namely Buttiker's dwell time gives the energy-distribution for the tunneling time; thirdly, we state that all the definitions are valid only if they are based on the wave functions for transmission and reflection found in our paper. Besides, we define the exact and asymptotic group times to be auxiliary in timing the scattering process.
Baz' and Rybachenko have proposed the use of the Larmor precession as a clock to measure the time it takes a particle to traverse a barrier. An applied magnetic field is confined to the barrier. The spin of the incident particles is polarized perpendicular to this field. The extent of the Larmor precession occurring during transmission is used as a measurement of the time spent traversing the barrier. However, the particles tunneling through an opaque barrier also acquire a spin component parallel to the field since particles with spin parallel to the field have a higher transmission probability than particles with spin antiparallel to the field. Similar effects are actually used to polarize electrons and neutrons. An interpretation of this experiment compares the results with an approach which determines the traversal time by studying transmission of particles through a time-modulated barrier. This leads to three characteristic times describing the interaction of particles with a barrier. A dwell time measures the average time interval during which a particle interacts with the barrier whether it is reflected or transmitted at the end of its stay, a traversal time measures the time interval during which a particle interacts with the barrier if it is finally transmitted, and a reflection time measures the interaction time of a reflected particle.
Abstract We derive a time-dependent solution for the Schrödinger-Pauli equation within a quantum shutter model to analyse cut-off plane waves scattered by a finite-range potential under a constant magnetic field. Exact analytical solutions for the time-dependent spinor components are derived for the low-energy plane-wave case, enabling the exploration of dynamical effects within the transmission region of a potential barrier. We demonstrate that the spinor probability density exhibits a diffraction in time pattern, with peaks shifted by the presence of the magnetic field relative to the field-free case. This shift introduces a delay time, which we analyse as a function of the Larmor frequency, with tunable peak positions controlled by the field intensity. We further explore the dynamical features of the spin expectation values 〈S k 〉, which exhibit unique diffraction in time behaviour and converge to stationary values in the long-time asymptotic regime. Additionally, we show that measuring the average total spin of the system offers an alternative approach for calculating the dynamical delay time. For infinitesimally small magnetic fields, we define a characteristic dynamical Larmor timescale associated with the evolution of 〈S y 〉, enabling the recovery of the Büttiker Larmor time, τ y , at low energies in the long-time limit. These findings demonstrate that magnetic fields can precisely manipulate tunnelling timescales in quantum systems, providing a means to control matter wave dynamics.