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Birkhäuser Boston eBooks

Geometric Phases in Action

Dariusz Chruściński, Andrzej Jamiołkowski

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Both Berry’s phase and Hannay’s angles could have been discovered long before they were. In 1938 Russian physicist Sergei M. Rytov investigated the rotation of the polarization vector of light travelling along the coiled ray. Actually, as was shown by V.V. Vladimirskii in 1941, Rytov’s observation finds an elegant interpretation in terms of geometric properties of a coiled ray. It turns out that rotation of polarization may be interpreted as a simple manifestation of the geometric phase. Actually, the similar conclusion was made by Bortolotti in 1926, however, both Bortolotti and Rytov-Vladimirskii papers were completely unknown to optical community. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-0-8176-8176-0_6

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Both Berry’s phase and Hannay’s angles could have been discovered long before they were. In 1938 Russian physicist Sergei M. Rytov investigated the rotation of the polarization vector of light travelling along the coiled ray. Actually, as was shown by V.V. Vladimirskii in 1941, Rytov’s observation finds an elegant interpretation in terms of geometric properties of a coiled ray. It turns out that rotation of polarization may be interpreted as a simple manifestation of the geometric phase. Actually, the similar conclusion was made by Bortolotti in 1926, however, both Bortolotti and Rytov-Vladimirskii papers were completely unknown to optical community. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The wave superposition model of the geometric phase shows how the addition of waves creates a shift in the resulting wave position. While previous work focused on a basis of linearly polarized light waves and the Pancharatnam-Berry phase, we show how the spin-redirection phase (Rytov-Vladimirsky-Berry phase) can also be derived from the same approach of wave superposition, using rotating vectors to represent the superposing oscillations. The result is the first derivation of the spin-redirection phase using wave superposition. We illustrate this approach with two classic examples of the geometric phase of rotations in space: a system of three fold mirrors and the helically coiled fiber.
A mathematical treatment is presented of the known Berry, Wilczek-Zee, Aharonov-Anandan, and Pancharatnam topological phases, and simple illustrative examples of their quantum mechanics are presented. The continuity and connection is traced among the various phases, while filling the gap involved with the forgotten works of S. M. Rytov and V. V. Vladimirskiĭ in polarization optics. A set of current experiments in polarization optics where the topological phases are measured is discussed in detail. Additional information on recently obtained results involving manifestations of geometrical phases in quantum mechanics and other fields of physics is contained in the Appendix and in the studies cited there.
We report on two quantum polarization experiments. The first one shows that three geometrical rotations can in principle be distinguished with certainty using two photons. The second experiment shows quantum polarization properties of classically unpolarized light. We outline a new treatment of polarization in quantum optics. Our formalism differs from the usual quantum description, which uses the Stokes operators. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.