Geometrical Phases in Quantum Mechanics
Abstract
In quantum mechanics, the pathdependent geometrical phase associated with a physical system, over and above the familiar dynamical phase, was initially discovered in the context of adiabatically changing environments. Subsequently, Aharonov and Anandan liberated this phase from the original formulation of Berry, which used Hamiltonians, dependent on curves in a classical parameter space, to represent the cyclic variations of the environments. Their purely quantum mechanical treatment, independent of Hamiltonians, instead used the nontrivial topological structure of the projective space of onedimensional subspaces of an appropriate Hilbert space. The geometrical phase, in their treatment, results from a parallel transport of the timedependent pure quantum states along a curve in this space, which is endowed with an abelian connection. Unlike Berry, they were able to achieve this without resort to an adiabatic approximation or to a timeindependent eigenvalue equation. Prima facie, these two approaches are conceptually quite different. After a review of both approaches, an exposition bridging this apparent conceptual gap is given; by rigorously analyzing a model composite system, it is shown that, in an appropriate correspondence limit, the Berry phase can be recovered as a special case from the AharonovAnandan phase. Moreover, the model composite system is used to show that Berry's correction to the traditional BornOppenheimer energy spectra indeed brings the spectra closer to the exact results. Then, an experimental arrangement to measure geometrical phases associated with cyclic and noncyclic variations of quantum states of an entangled composite system is proposed, utilizing the fundamental ideas of the recently opened field of twoparticle interferometry. This arrangement not only resolves the controversy regarding the true nature of the phases associated with photon states, but also unequivocally predicts experimentally accessible geometrical phases in a truly quantum regime, and allows, for the first time, the measurements of such phases associated with arbitrary noncyclic evolutions of entangled linearmomentum photon states. This nonclassical manifestation of the geometrical phases is due to the entangled character of linearmomentum photonstates of two correlated photons produced by parametric downconversion in nonlinear crystals. Finally, the nonlocal aspect of the geometrical phase is contrasted with the fundamental nonlocality of quantum mechanics due to the entangled character of quantum states.
 Publication:

Ph.D. Thesis
 Pub Date:
 1991
 Bibcode:
 1991PhDT.......101C
 Keywords:

 BERRY'S PHASE;
 AHARONOV ANANDAN PHASE;
 GAUGE STRUCTURE;
 Physics: General; Physics: Elementary Particles and High Energy; Physics: Condensed Matter