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 AspectsIn 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

InTech eBooksFeb 24, 201216 citations

Berry’s phase for a noncyclic rotation of light in a helically wound optical fiber

Atsuo Morinaga, Akinori Monma, Kazuhito Honda +1 more

We investigated Berry's phase for noncyclic evolution using the rotation of a polarization azimuth of linearly polarized light in a partially wound optical fiber over the surface of a cylinder. Using a rotation gauge around the rotation axis, the observed rotation of the polarization azimuth corresponds to the area of a spherical rectangle over the surface enclosed by the contour $C$ of actual evolution, a large circle on the equator, and a longitudinal line connecting them, whereas the rotation of the polar gauge encloses a spherical triangle connecting the zenith of the sphere. The observed values were converted to Berry's phase by transformation from the rotation gauge to the geodesic gauge. Consequently, we could confirm that Berry's phase for a noncyclic evolution is given by the geodesic rule proposed by Samuel and Bhandari.We investigated Berry's phase for noncyclic evolution using the rotation of a polarization azimuth of linearly polarized light in a partially wound optical fiber over the surface of a cylinder. Using a rotation gauge around the rotation axis, the observed rotation of the polarization azimuth corresponds to the area of a spherical rectangle over the surface enclosed by the contour $C$ of actual evolution, a large circle on the equator, and a longitudinal line connecting them, whereas the rotation of the polar gauge encloses a spherical triangle connecting the zenith of the sphere. The observed values were converted to Berry's phase by transformation from the rotation gauge to the geodesic gauge. Consequently, we could confirm that Berry's phase for a noncyclic evolution is given by the geodesic rule proposed by Samuel and Bhandari.

Physical Review ANov 16, 200714 citations

Berry phase from symmetry groups

Robert Olkiewicz

Reports on Mathematical PhysicsDec 1, 19934 citations

An Optical Measurement of Berry's Phase

Arthur L. Robinson

ScienceOct 24, 198615 citations

C

of actual evolution, a large circle on the equator, and a longitudinal line connecting them, whereas the rotation of the polar gauge encloses a spherical triangle connecting the zenith of the sphere. The observed values were converted to Berry's phase by transformation from the rotation gauge to the geodesic gauge. Consequently, we could confirm that Berry's phase for a noncyclic evolution is given by the geodesic rule proposed by Samuel and Bhandari.