Theoretical and experimental studies of Berry and Pancharatnam phases are reviewed. Basic elements of differential geometry necessary for understanding the topological nature of these phases are presented. The basic theory analysed by Berry in relation to magnetic monopoles is presented. The theory is generalized to nonadiabatic processes and to noncyclic Pancharatnam phases. Different systems are discussed including polarization optics, n -level atomic systems, neutron interferometry and molecular topological phases.
Theoretical and experimental studies of Berry and Pancharatnam phases are reviewed. Basic elements of differential geometry necessary for understanding the topological nature of these phases are presented. The basic theory analysed by Berry in relation to magnetic monopoles is presented. The theory is generalized to nonadiabatic processes and to noncyclic Pancharatnam phases. Different systems are discussed including polarization optics, n -level atomic systems, neutron interferometry and molecular topological phases.
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 study two types of topological phases, i.e., Aharonov-Bohm phases, by means of optical interferometry. The first arises from a cycle of redirections of the photon spin and is closely related to the Aharonov-Anandan phase. The second arises from a cycle of polarization changes on the Poincare sphere and was discovered by Pancharatnam. Both of these phases, and their combined effects, have been investigated by a nonplanar Mach-Zehnder interferometer. The signs of the fringe shifts arising from these topological effects have been determined and agree with theory.
Quantum eigenstates undergoing cyclic changes acquire a phase factor of geometric origin. This phase, known as the Berry phase, or the geometric phase, has found applications in a wide range of disciplines throughout physics, including atomic and molecular physics, condensed matter physics, optics, and classical dynamics. In this article, the basic theory of the geometric phase is presented along with a number of representative applications. The article begins with an account of the geometric phase for cyclic adiabatic evolutions. An elementary derivation is given along with a worked example for two-state systems. The implications of time-reversal are explained, as is the fundamental connection between the geometric phase and energy level degeneracies. We also discuss methods of experimental observation. A brief account is given of geometric magnetism; this is a Lorenz-like force of geometric origin which appears in the dynamics of slow systems coupled to fast ones. A number of theoretical developments of the geometric phase are presented. These include an informal discussion of fibre bundles, and generalizations of the geometric phase to degenerate eigenstates (the nonabelian case) and to nonadiabatic evolution. There follows an account of applications. Manifestations in classical physics include the Hannay angle and kinematic geometric phases. Applications in optics concern polarization dynamics, including the theory and observation of Pancharatnam's phase. Applications in molecular physics include the molecular Aharonov-Bohm effect and nuclear magnetic resonance studies. In condensed matter physics, we discuss the role of the geometric phase in the theory of the quantum Hall effect.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
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.
A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate collinear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic ‘curves' in Hilbert space. Explicit expressions for Pancharatnam, dynamic and geometric phases are derived for the transformations produced by linear phase converters. The problem of gauge invariance is treated in this article.