# geometric phase

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## Geometric phase

A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space. Under various aspects, this concept occurs in geometry, astronomy, classical mechanics, and quantum theory. In geometry it is known as holonomy. In quantum theory it is known as Berry's phase, after M. Berry, who isolated the concept (which was already known in special cases) and explained its wide-ranging signi-ficance.

A system is envisioned whose possible states can be visualized as points in a suitable abstract space. At the same time, the system has some position or orientation in another space. A history of internal states can be represented by a curve in the first space; and the effect of this history on the disposition of the system, by a curve in the second space. The mapping between these two curves is described by the geometric phase. Especially interesting is the case when a closed curve (cycle) in the first space maps onto an open curve in the second, for then there is no net change in internal state, yet the disposition of the system with respect to the outside world is altered.

The power of the geometric phase ideas is that they make it possible, in complex dynamical problems, to find some simple universal regularities without having to solve the complete equations. Significant uses of these ideas include demonstrations of the fractional electric charge and quantum statistics of the quasiparticles in the quantum Hall effect, and of the occurrence of anomalies in quantum field theory. See Anyons, Hall effect, Quantum field theory

## geometric phase

[‚jē·ə‚me·trik ′fāz]
(physics)
A unifying mathematical concept that describes the relation between the history of internal states of a system and the system's resulting orientation in space.
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References in periodicals archive ?
In 2010, using NSF funding, Dr Michael Escuti (a photonics and electro-optic materials expert) created a direct-write laser scanner (DWLS) that enabled near perfect geometric phase holograms to be produced.
To make geometric phase holograms, the DWLS uses an ultraviolet laser to 'print' on a super-thin film--only about 50 nanometers thick.
One reason why the DWLS is unique is that it produces geometric phase holograms that are smoothly varying and can create complex patterns seamlessly--there are no dividing lines or pixels (unless they are actually desired).
Let us formally establish the idea of a cyclic evolution to build the geometric phase and the non-Abelian Berry phase (non-adiabatic state evolution):
Hence, we end up with a total phase [phi] which is built of a dynamic phase [delta] that depends on the Hamiltonian H(t), and a geometric phase [gamma] that depends only on the path C, and is independent of the rate at which |[PSI](t)> completes C, the Hamiltonian, or the choice of reference {|[[PSI].
of Chicago) presents seven papers on: the influence of the geometric phase on reaction dynamics, optimal control theory for manipulating molecular processes, comprehension and control of nonadiabatic chemical dynamics and manifestation of molecular functions, exploration of multiple reaction paths to a single product channel, photoelectron circular dichroism in chiral molecules, spectroscopy of the potential energy surfaces for C-H and C-O bond activation by transition metal and metal oxide cation, and stabilization of different conformers of weakly bound complexes to access varying excited-state intermolecular dynamics.
Geometric phase induced false electric dipole moment signals for particles in traps, Phys.
The geometric phase of programming can be conducted at the CMM, at a CAD station, or at a desk with an off-line computer.
Abbas are physicists at the Solid State Physics Division of Bhabha Atomic Research Centre in Mumbai (India), working on neutron interferometry and geometric phase.
We present a split-beam neutron interferometric experiment to test the non-cyclic geometric phase tied to the spatial evolution of the system: the subjacent two-dimensional Hilbert space is spanned by the two possible paths in the interferometer and the evolution of the state is controlled by phase shifters and absorbers.
In all theses contexts the geometric phase is dependent only on the geometry of the subjacent Hilbert space, but not on the particular dynamics of the system under consideration.
Multiple Lie-Poisson structures reduction and geometric phases for the Maxwell-Bloch travelling wave equation.

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