Berry phases, geometric actions and applications

Mikhail Orionovich Katanaev

Ali Mostafazadeh

An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.

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

David H. Vanderbilt

Arno Böhm, Luis J. Boya, Brian K. Kendrick

Mustapha Maamache, Jean-Pierre Provost, Gérard Vallée

Through the use of coherent states describing the action-angle variables the authors establish a strict parallelism between Berry's phase and Hannay's angle. In particular they define a geometrical transport for classical tori which leads to Hannay's angle. As in the case of Berry's phase, the transport can be associated with the minimization of a distance.