S. Brouard

Optimization of absorbing potentials

Abstract Different complex potentials are optimized and their absorption widths are compared. The best potential found is obtained from a simple systematic inversion technique. It is shown that discontinuities do not necessarily cause reflection, and that a real part in addition to the imaginary absorbing potential improves absorption.

Transmission and reflection tunneling times

Abstract Different separations of the dwell time into real reflection and transmission partial times for tunneling involving wave packet scattering are proposed. The formalism is valid in both quantum mechanics and classical statistical mechanics. Numerical examples are provided.

Wigner trajectories and Liouville’s theorem

It is found that, in general, Wigner trajectories satisfy Liouville’s theorem only locally, i.e., for restricted phase space and time domains. This fact limits their possible applications. Examples are provided to visualize the process of creation and destruction of Wigner trajectories. It is argued that Weyl transforms of Heisenberg operators are, however, viable alternatives to Wigner trajectories, even though they do not satisfy Liouville’s theorem either.

Perfect absorbers for stationary and wavepacket scattering

Complex potentials that absorb the incoming wave in a finite distance without reflection or transmission are found by a simple inversion technique, both for stationary and wavepacket scattering in one dimension.

Barrier traversal times using a phenomenological track formation model

Abstract A phenomenological model for a measurement of “barrier traversal times” for particles is proposed. Two idealized detectors for passage and arrival provide entrance and exit times for the barrier traversal. The averaged traversal time is computed over the ensemble of particles detected twice, before and after the barrier. The “Hartman effect” can still be found when passage detectors that conserve the momentum distribution of the incident packet are used.

Transmission, Reflection, and Interference Contributions to the Tunnelling Dwell Time

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.

Wigner function for the square barrier

Abstract The Wigner function for the scattering of an incident plane wave with definite momentum impinging on a square potential barrier is obtained explicitly. The associated Wigner trajectories are examined and compared with Wigner trajectories corresponding to other model potentials and boundary conditions. Conditions of applicability and non-classical features of Wigner trajectories are discussed.

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

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.

Explicit solution for a Gaussian wave packet impinging on a square barrier

The collision of a quantum Gaussian wave packet with a square barrier is solved explicitly in terms of known functions. The obtained formula is suitable for performing fast calculations or asymptotic analysis. It also provides physical insight since the description of different regimes and collision phenomena typically requires only some of the terms.

The influence functional: application to tunnelling

The influence functional is introduced as a kernel in an integral equation that gives the probability density at time t and position q in terms of the initial probability density. This functional is applied to tunnelling through a square barrier to determine the influence, at different times, of various regions of the incident packet on the transmitted peak.