Journal of Physics A Mathematical and General

Analysis of Berry's phase by the evolution operator method

C M Cheng, P. C. W. Fung

Published Sep 7, 1989• 20 citations

Abstract

Berry (1984) discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase. Aharonov and Anandan (1987), on the other hand, showed that an eigenstate transporting round a closed circuit in the projective Hilbert space suffices to generate the topological phase. The authors employ the evolution operator method to study the propagation of an eigenstate. They show that Berry's phase can be represented as a closed path integral in the Hilbert space and is independent of the choice for the base ( mod n(t))). The treatments of Berry, and Aharonov and Anandan, are shown to correspond to two different choices of the base. Therefore their two approaches are unified; they have acquired a more general viewpoint on the origin of Berry's phase.

DOI: 10.1088/0305-4470/22/17/015

Related papers

Non-adiabatic Berry phase for periodic Hamiltonians

D J Moore, G E Stedman

Berry's (1984) formulation of the topological phase for periodic Hamiltonians is extended to the case of non-adiabatic evolution by proper choice of initial states. This restores a full parallel with the non-adiabatic formalism of Aharonov and Anandan (1987) for periodic orbits in a projective Hilbert space. The Berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by Aharonov and Anandan and by Page (1987).

Journal of Physics A Mathematical and GeneralJun 7, 199056 citations

Geometrical description of Berry's phase

Don Nelson Page

Berry, Simon, and Aharonov and Anandan have discovered, interpreted, and generalized a geometrical phase factor that occurs for a quantum state evolving around a closed path in the projective Hilbert space of rays. Here this phase is directly given in terms of the holonomies of several natural geometrical structures on the projective space.

Physical Review AOct 1, 1987177 citations

Study of the Aharonov-Anandan Phase by NMR Interferometry

Suter, Dieter, Karl Todd Mueller, Alexander Pines

Aharonov and Anandan have recently reformulated and generalized Berry’s phase by showing that a quantum system which evolves through a circuit C in projective Hilbert space acquires a geometrical phase β(C) related to the topology of the space and the geometry of the circuit. We present NMR interferometry experiments in a three-level system which demonstrate the Aharonov-Anandan phase and its topological invariance for different circuits.

Mar 1, 1988135 citations

Some geometrical considerations of Berry’s phase

J. Anandan, Leo Stodolsky

A group-theoretic treatment is given of the new phase found by Berry in the adiabatic evolution of a quantum-mechanical system in a finite-dimensional Hilbert space. It is shown how the Berry phases for the various eigenstates of the Hamiltonian are obtained from a set of angles associated with a group element. For the special case of a two-level system there is just one such angle which corresponds to the holonomy transformation associated with parallel transport around a closed curve on a sphere.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fieldsApr 15, 1987111 citations

Berry phase in nonlinear systems

Jian Liu, Libin Fu

The Berry phase acquired by an eigenstate that experienced a nonlinear adiabatic evolution is investigated thoroughly. The circuit integral of the Berry connection of the instantaneous eigenstate cannot account for the adiabatic geometric phase, while the Bogoliubov excitations around the eigenstates are found to be accumulated during the nonlinear adiabatic evolution and contribute a finite phase of geometric nature. A two-mode model is used to illustrate our theory. Our theory is applicable to Bose-Einstein condensate, nonlinear light propagation, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics.

Physical Review AMay 17, 201048 citations

The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

Francisco De

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

InTech eBooksFeb 24, 201216 citations