Berry and Pancharatnam topological phases of atomic and optical systems

Differences between the geometric phase and propagation phase: clarifying the boundedness problem

We show white light interferometer experiments that clearly demonstrate the basic differences between geometric and propagation phases. These experimental results also suggest a way to answer the "boundedness problem" in geometric phase-whether geometric phase is unbounded (i.e., can take on any values without limit) or bounded (i.e., limited to values between -π and +π). We show why the answer to this question is not as easy as it seems, from both a theoretical and an experimental perspective, and explain how the answer depends on one's choice of phase convention. We also hope that the videos provided will be pedagogically useful for explaining geometric phase.

How "Berry Phase" Analysis of Non-Adiabatic Non-Hermitian Systems Reflects Their Geometry

There is currently great interest in systems represented by non-Hermitian Hamiltonians, including a wide variety of real systems that may be dissipative and whose behaviour can be represented by a "phase" parameter that characterises the way "exceptional points" (singularities of various sorts) determine the system. These systems are briefly reviewed here with an emphasis on their geometrical thermodynamics properties.

Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

Geometric phase: two triangles on the Poincaré sphere

The method of obtaining the dynamic and geometric part of the phase introduced by a birefringent medium in two kinds of interferometric experiments is presented. The mathematical formulas for the phases obtained using Jones formalism are visualized with the specific triangles on the Poincaré sphere. Generally, these triangles are similar to those used in the Pancharatnam's original construction developed by Courtial to calculate the Pancharatnam geometric phase for the light passing through a single birefringent plate. In these graphical constructions following Courtial's idea, we used the points representing both of the birefringent medium's eigenvectors. This allowed the most intuitive explanation of the mechanism of dividing the whole phase shift introduced by the birefringent plate into two different parts: dynamical and geometrical. The considered constructions were used as a description of two simple experiments with a birefringent medium in a Mach-Zehnder interferometer and a polariscopic setup. The experimental verifications of our theoretical predictions should convince the reader of the correctness of the assumed model.

Self-induced quasistationary magnetic fields

The interaction of electromagnetic radiation with temporally dispersive magnetic solids of small dimensions may show very special resonant behaviors. The internal fields of such samples are characterized by magnetostatic-potential scalar wave functions. The oscillating modes have the energy orthogonality properties and unusual pseudoelectric (gauge) fields. Because of a phase factor, that makes the states single valued, a persistent magnetic current exists. This leads to appearance of an eigenelectric moment of a small disk sample. One of the intriguing features of the mode fields is dynamical symmetry breaking.