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Physical Review A

Berry’s phase for atomic levels

Pervez Amirali Hoodbhoy

Published Oct 1, 1988• 4 citations
Publisher

Abstract

The ground-state hyperfine structure of atomic states is shown to exhibit interesting new aspects of the topological phase discovered by M. V. Berry [Proc. R. Soc. London, Ser. A 392, 45 (1984)]. In particular, although a nontrivial topological phase exists, monopoles are absent. A possible experiment to probe these aspects is discussed.
DOI
10.1103/physreva.38.3766

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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.
The contract was for performing experiments on a recently discovered topological phase factor known as 'Berry's phase.' We proposed to examine this phase at both the quantum and classical levels, in its various manifestations. The goals of this contract were successfully accomplished; in particular, we succeeded in demonstrating this phase at the single photon level, thereby demonstrating that this phase did not originate at the classical level. Possible applications of this phase were explored. A two-photon light source, in the form a parametric down-converter of uv laser photons into highly correlated red photons was successfully constructed, and coincidence counting techniques were successfully applied to the detection of the conjugate signal and idler red photons. This formed the experimental basis for the examination of not only Berry's phase, but of the energy-time uncertainty relations, the Franson experiment, the measurement of the tunneling time of the photon, the quantum eraser, and other optical phenomena closely related to the Einstein-Podolsky- Rosen 'paradox.'
Two types of phases are discussed in this article: (1) The topological phase as introduced by Berry [Proc. R. Soc. London, Ser. A 392, 45(1984)] and Aharonov and Anandan [Phys. Rev. Lett. 58, 1593 (1987)] and (2) the Longuet–Higgins phase [Proc. R. Soc. London, Ser. A 344, 147 (1975)]. The two types of phases have a common origin, namely the multivaluedness of the electronic adiabatic basis, a phenomenon associated with the existence of a degeneracy in configuration space. It will be shown, by studying an electronic model Hamiltonian that arises from a two-state approximation to the Mathieu equation, that the two phases differ from each other substantially, coinciding only in the adiabatic limit upon completion of a cycle.
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
A theoretical analysis of Berry's phases, which is based on the Cartan subalgebra, is given for three-level atomic systems. The analysis is developed under the adiabatic approximation and is related to topological fibre bundle theories. The topological Berry's phases derived in the present study, are related to phase shifts and can be applied, also, for partial cycles. Some possible applications of the present theory are discussed, for three-level and two-level atomic systems, interacting with nearly resonant monochromatic fields, under the semiclassical and rotating-wave approximations.