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arXiv:quant-ph/9505024 · Physical Review A

Geometry of one-dimensional wave propagation

Masao Kitano

Published Jun 1, 1995• 3 citations
arXivHTMLPDF

Abstract

We investigate the geometrical features of one-dimensional wave propagation, whose dynamics is described by the (2+1)-dimensional Lorentz group. We find many interesting geometrical ingredients such as spinorlike behavior of wave amplitudes, gauge transformations, Bloch-type equations, and Lorentz-group Berry phases. We also propose an optical experiment to verify these effects.

DOI
10.1103/physreva.51.4427

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We investigate the geometrical features of one-dimensional wave propagation, whose dynamics is described by the (2+1)-dimensional Lorentz group. We find many interesting geometrical ingredients such as spinorlike behavior of wave amplitudes, gauge transformations, Bloch-type equations, and Lorentz-group Berry phases. We also propose an optical experiment to verify these effects.
Since Pancharatnam's 1956 discovery of optical geometric phase and Berry's 1984 discovery of geometric phase in quantum systems, researchers analyzing geometric phase have focused almost exclusively on algebraic approaches using the Jones calculus, or on spherical trigonometry approaches using the Poincaré sphere. The abstracted mathematics of the former and the abstracted geometry of the latter obscure the physical mechanism that generates geometric phase. We show that optical geometric phase derives entirely from the superposition of waves and the resulting shift in the location of the wave maximum. This wave-based model provides a way to visualize how geometric phase arises from relationships between waves, and from the transformations induced by optical elements. We also derive the relationship between the geometric phase of a wave by itself and the phase exhibited by an interferogram, and provide the conditions under which the two match one another.
The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing fields are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric fields, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcar\'e sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action flux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.
Why Is Group Theory Useful in Quantum Mechanics? Irreducible Representations of SU(2) and SO(3), Rotation Matrices The Symmetric Groups The Lie Algebra of SO(4) and the Hydrogen Atom Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole The Berry Phase in Molecular Dynamics The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions The Connection Is the Gauge Field and the Curvature Is the Force: Some Topology and Differential Geometry Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect Chern-Simons Theory: The Fractional Quantum Hall Effect and other chapters.
During the last few years, considerable interest has been focused on the phase that waves accumulate when the equations governing the waves vary slowly. The recent flurry of activity was set off by a paper by Michael Berry, where it was found that the adiabatic evolution of energy eigenfunctions in quantum mechanics contains a phase of geometric origin (now known as 'Berry's phase') in addition to the usual dynamical phase derived from Schrödinger's equation. This observation, though basically elementary, seems to be quite profound. Phases with similar mathematical origins have been identified