Geometric phase

In mechanics (including classical mechanics as well as 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, more commonly, Berry phase, is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, resulting 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 inside the solenoid, and cyclic means that the difference involved in measuring the effect by interference corresponds to a closed loop, in the usual way (see below). 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 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 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.

Examples of geometric phases

The Foucault Pendulum

One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by von Bergmann and von Bergmann:

How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2π. Finally, we can approximate any loop by a sequence of geodesic segments, so the most general result (on or off the surface of the sphere) is that the net precession is equal to the enclosed solid angle.

In summary, there are no inertial forces that could make the pendulum precess. Thus the orientation of the pendulum undergoes parallel transport along the path of fixed latitude. By the Gauss-Bonnet theorem the phase shift is given by the enclosed solid angle.

Polarized light in an optical fiber

Imagine linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions like a waveguide and the momentum of the light is at all times tangent to the fiber. The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map. There are no forces that could make the polarization turn, just the constraint to remain tangent to the sphere. Thus the polarization undergoes parallel transport and the phase shift is given by the enclosed solid angle (times the spin, which in case of light is 1).

Stochastic pump effect

A stochastic pump is a classical stochastic system that responds with nonzero, on average, currents to periodic changes of parameters. The stochastic pump effect can be interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents.

See also

• For the connection to mathematics, see curvature tensor,
• The Aharonov-Bohm effect,
• Conical intersections of potential energy surfaces.
• Chern class
• Wilson loop
• Winding number

References

• Jeeva Anandan, Joy Christian and Kazimir Wanelik (1997). "Resource Letter GPP-1: Geometric Phases in Physics". Am. J. Phys. 65: 180. doi:10.1119/1.18570.

• V. Cantoni and L. Mistrangioli (1992) "Three-Point Phase, Symplectic Measure and Berry Phase", International Journal of Theoretical Physics vol. 31 p. 937.

• 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]

• Frank Wilczek and Alfred Shapere, "Geometric Phases in Physics", World Scientific, 1989

• Jerrold E. Marsden, Richard Montgomery, Tudor S. Ratiu (1990). Reduction, Symmetry, and Phases in Mechanics. AMS Bookstore. p. 69. ISBN 0821824988.{{cite book}}: CS1 maint: multiple names: authors list (link)

• C. Pisani (1994). Quantum-mechanical Ab-initio Calculation of the Properties of Crystalline Materials (Proceedings of the IV School of Computational Chemistry of the Italian Chemical Society ed.). Springer. p. 282. ISBN 3540616454.

• L. Mangiarotti, Gennadiĭ Aleksandrovich Sardanashvili (1998). Gauge Mechanics. World Scientific. p. 281. ISBN 9810236034.

• Karin M Rabe, Jean-Marc Triscone, Charles H Ahn (2007). Physics of Ferroelectrics a Modern Perspective. Springer. p. 43. ISBN 3540345906.{{cite book}}: CS1 maint: multiple names: authors list (link)