Geometric phase
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In quantum mechanics, the Geometric phase, or the Pancharatnam-Berry phase (named after S. Pancharatnam and Sir Michael Berry), also known as the Pancharatnam phase or Berry phase, is a phase acquired by quantum states when subjected to adiabatic processes, resulting from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was first discovered in 1956[1], and rediscovered in 1984[2]. It appears in particular in the theory of the Aharonov-Bohm effect and of the conical intersection of potential energy surfaces. In the case of the Aharonov-Bohm effect, the adiabatic parameter is the magnetic field inside the solenoid. 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. Generally speaking, it occurs whenever one can externally control at least two parameters affecting a wave.
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 e.g. 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 excursion is 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.
[edit] See also
- For the connection to mathematics, see curvature tensor,
- The Aharonov-Bohm effect,
- Conical intersections of potential energy surfaces.
[edit] Notes
[edit] References
- Jeeva Anandan, Joy Christian and Kazimir Wanelik (1997). "Resource Letter GPP-1: Geometric Phases in Physics". Am. J. Phys. 65: 180.
- Richard Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91), (2002) American Mathematical Society, ISBN 0-8218-1391-9. (See chapter 13 for a mathematical treatment)
- Connections to other physical phenomena (such as the Jahn-Teller effect) are discussed here: [1]
- Paper by Prof. Galvez at Colgate University, describing Geometric Phase in Optics: [2]
- Surya Ganguli, Fibre Bundles and Gauge Theories in Classical Physics: A Unified Description of Falling Cats, Magnetic Monopoles and Berry's Phase [3]
- Robert Batterman, Falling Cats, Parallel Parking, and Polarized Light [4]
