From the delay time in Quantum Mechanics to the Goos-H\"anchen shift in Optics

TL;DR

This paper explores the connection between quantum mechanical delay times and optical phenomena like the Goos-H"anchen shift, providing a detailed analysis that enables optical experiments to simulate quantum delay measurements.

Contribution

It establishes a detailed theoretical link between quantum delay times and optical lateral displacements, enabling quantum phenomena to be studied through optical experiments.

Findings

Derived a closed-form expression for delay time at critical incidence energy.

Demonstrated excellent agreement between analytical and numerical delay time calculations.

Established a definitive connection between quantum wave packet reflection and optical beam reflection.

Abstract

Delay times in quantum mechanics always represented an intriguing challenge for physicists. Due to the fact that quantum mechanical experiments are, often, hard to be implemented, the possibility to connect delay times with laser lateral displacements gives us the opportunity to prepare, in optical laboratories, experiments which are equivalent to the quantum mechanical ones in detecting delay times. In this article, we will show in detail not only the relationship between delay times and Goos-Haenchen shifts, but also the close connection between the impulse change in quantum mechanics and angular deviations in optics. Lateral shifts are caused by the phase of Fresnel coefficients whereas angular deviations by the breaking of symmetry of the wave number distributions. The classical formula for the delay time is based on the use of the stationary phase method and contains a divergence for incidence at a critical potential energy. For Gaussian beams, the mean value calculation removes such a divergence. The closed expression for the delay time for incidence at critical energy show an excellent agreement with the numerical calculation. The three-dimensional analysis of delay times allow to find the final and definitive connection between wave packets reflected by a potential in quantum mechanics and optical beams reflected by a dielectric/air interface.