# Geometric Phase

In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase (named after S. Pancharatnam and Sir Michael Berry), Pancharatnam phase or most commonly Berry phase, is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was first discovered in 1956, and rediscovered in 1984. It can be seen in the Aharonov–Bohm effect and in the conical intersection of potential energy surfaces. In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it occurs whenever there are at least two parameters affecting a wave, in the vicinity of some sort of singularity or some sort of hole in the topology.

Waves are characterized by amplitude and phase, and both may vary as a function of those parameters. The Berry phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. Intuitively one expects that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a cyclic loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the Berry phase, and its occurrence typically indicates that the system's parameter dependence is singular (undefined) for some combination of parameters.

To measure the Berry phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle.

### Other articles related to "phase, geometric phase, phases":

Binoculars - Optical Coatings - Phase Correction Coatings
... The angle between the two polarization vectors is called the phase shift, or the geometric phase, or the Berry phase ... This interference between the two paths with different geometric phase results in a varying intensity distribution in the image reducing apparent contrast and ... depositing a special dielectric coating known as a phase-correction coating or P-coating on the roof surfaces of the roof prism ...
Examples of Geometric Phases - Exposing Berry/Pancharatnam Phases in Molecular Adiabatic Potential Surface Intersections
... There are several ways to compute the Berry phase in molecules within the Born Oppenheimer framework ... is considered) this matrix is diagonal with the diagonal elements equal to where are the Berry phases associated with the loop for the adiabatic electronic state ... symmetrical electronic Hamiltonians the Berry phase reflects the number of conical intersections encircled by the loop ...

### Famous quotes containing the words phase and/or geometric:

It no longer makes sense to speak of “feeding problems” or “sleep problems” or “negative behavior” is if they were distinct categories, but to speak of “problems of development” and to search for the meaning of feeding and sleep disturbances or behavior disorders in the developmental phase which has produced them.
Selma H. Fraiberg (20th century)

New York ... is a city of geometric heights, a petrified desert of grids and lattices, an inferno of greenish abstraction under a flat sky, a real Metropolis from which man is absent by his very accumulation.
Roland Barthes (1915–1980)