A study is presented of Berry’s observation that when a quantum-mechanical system is transported on a closed adiabatic journey, a topological phase arises in addition to the usual dynamical phase expected from the adiabatic theorem. Consequences are explored in the case of a simple magnetic moment–magnetic field interaction and are shown to lead, among other things, to Dirac’s famous relation between electric and magnetic charges.
A study is presented of Berry’s observation that when a quantum-mechanical system is transported on a closed adiabatic journey, a topological phase arises in addition to the usual dynamical phase expected from the adiabatic theorem. Consequences are explored in the case of a simple magnetic moment–magnetic field interaction and are shown to lead, among other things, to Dirac’s famous relation between electric and magnetic charges.
Abstract Topological and geometrical effects appear in the solution of the Schrödinger equation due to special geometrical forms of the interaction, or due to interactions which do not result from conservative forces acting on the system (see Section 1.2). A conscientious reader may point out that all these effects could be entirely explained using the Schrödinger equation only. But the geometrical phase formalism is useful because it sometimes plays a role in fundamental quantum phenomena and it shows that additional phase factors exist which are easily overlooked, making the interpretation of measured effects more mysterious than necessary. Thus, our understand-ing of quantum mechanics has deepened since the existence of geometrical phases was explicitly revealed (Shapere and Wilczek 1989). This happened due to the seminal works of Pancharatnam (1956), Berry (1984), Anandan (1992), and Bhandari (1997). It also shows that a wavefunction often carries much more information than is usually extracted in a standard experiment. A typical example is the assumption of adiabaticity, where the magnetic moment of the neutron produces (inside a slowly varying magnetic field) a dynamical phase shift proportional to the path integral ∫ B ds, but when the direction of the field varies slowly an additional, i.e. geometrical, phase also appears. The back reaction of a quantum measurement can also be described as a kind of an induced topological phase (Aharonov et al. 1998).
The quantum adiabatic theorem incorporating the Berry phase phenomenon can be characterized as a factorization of the time evolution operator into a path-dependent geometric factor, a usual dynamical factor and a non-adiabatic factor that approaches the identity in the adiabatic limit. We study a case where all these three factors can be constructed explicitly and where the instantaneous Hamiltonian has infinitely degenerate energy eigenstates associated with magnetic translation symmetry. Significance of the non-adiabatic factor in terms of transition probabilities is discussed.
Berry's work on quantum systems described by adiabatically varying Hamiltonians is introduced. Some simple systems exhibiting the "Berry phase" phenomenon are discussed, including: a spin-½ particle in a slowly varying magnetic field, linearly polarised light propagating down a helically wound optical fibre, and dynamic Jahn-Teller E ⊗ systems. The way in which the Berry phase concept illuminates the origin of anomalies in chiral gauge field theories, and the fermionic behaviour of solitons in certain scalar field theories (skyrmions), is also indicated.
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
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
An introduction to topology in electrodynamics and the geometric phase con- cept is given. The form-invariance of the field-coupled Dirac equation is demon- strated, and it is shown, starting from the free Dirac equation, that demand of local phase invariance of the wave function dictates the presence of an elec- tromagnetic field. Afterwards the Dirac phase factor, Φ( C ), for a fixed-time loop is defined and discussed. It appears from The Principle of Local Phase Invariance that only the transverse part of the vector potential contributes to Φ( C ). The conditions ∇ × A T = 0 and ∇ · A T = 0 [A T being the transverse vector potential] in a certain finite spatial domain Ω do not imply that A T = 0 necessarily in Ω, as a certain integration involving A T over the domain outside Ω shows. The vector potential of a magnetic string with curvature and torsion is derived, and from this potential the result for a rectilinear magnetic string is obtained, In turn the Aharonov-Bohm effect is derived for a magnetic flux tube. Starting from the evolution operator in quantum physics an adiabatic theorem is established, and Berry's adiabatic geometric phase is introduced and discussed. An extended discussion shows that a geometric phase, called the Aharonov-Anandan phase, not restricted by the adiabatic principle can be introduced.