Dynamic and Geometric Shifts in Wave Scattering

TL;DR

This paper extends the geometric-dynamic phase decomposition from wave evolution to scattering problems, revealing how expectation-value shifts in wave observables can be understood through gauge-invariant geometric and dynamic contributions.

Contribution

It introduces a unified framework using the generalized Wigner-Smith operator to decompose observable shifts in wave scattering into geometric and dynamic parts, applicable across various physical systems.

Findings

Decomposition of shifts into gauge-invariant geometric and dynamic components.

Application to frequency, momentum, and time delays in diverse wave scattering scenarios.

Illustration of the theory with examples like polarized-light transmission and metasurface interactions.

Abstract

Since Berry's pioneering 1984 work, the separation of geometric and dynamic contributions in the {\it phase} of an evolving wave has become fundamental in physics, underpinning diverse phenomena in quantum mechanics, optics, and condensed matter. Here we extend this geometric-dynamic decomposition from the wave-evolution phase to a distinct class of wave scattering problems, where observables (such as frequency, momentum, or position) experience shifts in their expectation values between the input and output wave states. We describe this class of problems using a unitary scattering matrix and the associated generalized Wigner-Smith operator (GWSO), which involves gradients of the scattering matrix with respect to conjugate variables (time, position, or momentum, respectively). We show that both the GWSO and the resulting expectation-values shifts admit gauge-invariant decompositions into dynamic and geometric parts, related respectively to gradients of the eigenvalues and eigenvectors of the scattering matrix. We illustrate this general theory through a series of examples, including frequency shifts in polarized-light transmission through a time-varying waveplate (linked to the Pancharatnam-Berry phase), momentum shifts at spatially varying metasurfaces, optical forces, beam shifts upon reflection at a dielectric interface, and Wigner time delays in 1D scattering. This unifying framework illuminates the interplay between geometry and dynamics in wave scattering and can be applied to a broad range of physical systems.