A Descriptive Approach to the Geometric Phase

Luis A. Garza-Soto, Nathan A. Hagen, Dorilian Lopez-Mago +1 more

We present a summarized and simplified introduction to our wave description of geometric phase. We start by discussing the addition of cosine waves of different amplitudes and a phase between them, from which immediately arises the geometric phase in 1D. We then expand on the analysis to the 2D case using the orthogonal components of a polarized light wave. We then show a graphic visualization that facilitates the analysis of geometric phase and use it to quantify the geometric phase obtained by passing different states of polarization through a quarter wave plate.

Armin Uhlmann

Hayato Goto, Kouichi Ichimura

A geometrical description of open quantum systems is presented. This is based on a connection on a vector bundle over a Grassmann manifold, as well as the Anandan geometric phase for pure states [J. Anandan, Phys. Lett. A 133, 171 (1988)]. The geometric phase proposed here also naturally includes the geometric phase in adiabatic open quantum systems proposed by Sarandy and Lidar [Phys. Rev. A 73, 062101 (2006)]. The present geometrical description can be applied to all the quantum systems described by time-local master equations for density operators.

Jerrold Eldon Marsden

Péter Lévay

Miguel Angel Alonso, Mark R. Dennis

We give a simple description of the Pancharatnam-Berry geometric phase and some of its applications in optics. Geometric phases are a universal phenomenon, but we focus here on the case of geometric phases caused by changes in polarization. The geometric origin of this phase is explained by analogy with the motion of an imaginary creature living on a small planet, inspired by Saint-Exupéry’s “The Little Prince”.