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]
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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The latter is termed Berry phase or more general geometric phase in contrast to the former dynamical phase [[phi].
The geometric phase is associated with an evolution of a system governed by an Hamiltonian, e.
Let us briefly review the basic concepts of geometric phases: A geometric phase is a quantity which is deeply connected to the curvature of some underlying (state- or parameter-) space.
The geometric phase is a property of Ray space, where [~.
In both cases one can assign a geometric phase to the particular evolution of the initial state.
For testing the spatial geometric phase we use a double-loop interferometer (Fig.
To sum up, for a general evolution of a quantum state the state obtains a dynamical phase dependent on the energy and time as well as a geometric phase only dependent on the subjacent geometry of state space.
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.
Geometric phase induced false electric dipole moment signals for particles in traps, Phys.
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.
Multiple Lie-Poisson structures reduction and geometric phases for the Maxwell-Bloch travelling wave equation.