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Journal of Physics A Mathematical and General

Quantum mechanics as a geometric phase: phase-space interferometers

Alfredo Luis

Published Sep 10, 2001• 30 citations
Publisher

Abstract

It is shown that basic quantum commutation relations are equivalent to a geometric phase. We propose two interferometric arrangements that are able to measure this quantum geometric phase via interference in phase space.
DOI
10.1088/0305-4470/34/37/317

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The wave function of a quantum system may acquire an additional phase factor (the geometric phase), when its parameters undergo a cyclic change. An optical manifestation of the geometric phase is the change in the phase of a light wave produced when its state of polarization is taken around a circuit on the Poincare sphere. This phase change can be verified by interferometric measurements. We describe an interferometer in which the two beams traverse the same path in opposite directions, so that their optical paths are always equal, and the phase difference between them is varied by operating on the geometric phase. Measurements made with this interferometer, at a light level at which the probability of more than one photon being present simultaneously in the interferometer was negligibly small, confirmed that the interference effects observed due to the geometric phase persist down to the single photon level.
Correlation relations for the spin measurements on a pair of entangled particles scattered by the two separate arms of interferometers in hybrid setups of different types are investigated. Concurrence, entanglement of formation, quantum fidelity, Bures distance are used to clarify how the geometric phase affects the initial bipartite state. This affect causes a quantum interference due to the movement of charged particles in regions where electromagnetic fields are not present. We shown that in some cases the geometric phase information is carried over to the final bipartite entangled state.
An experimental arrangement is proposed which combines the idea of two-photon interferometry with that of geometric phases. A pair of photons generated by parametric down-conversion is fed into a pair of Mach-Zehnder interferometers, in each of which there is phase shift produced by the helically wound optical fibre in the manner of Tomita and Chiao(1986). If the winding angle is a multiple of 2 pi there is a closed path in parameter space, in accordance with the pioneering analysis of Berry(1984). More general windings are also considered in which the path in parameter space is not closed, and the geometric phase for such open paths is calculated in accordance with the gauge-invariant prescription of Aitchison and Wanelik. If the proposed experiment is realized, it will provide a test for the noncyclic geometric phase for photons and for the set of prescriptions equivalent to that of Aitchison and Wanelik(1992). Furthermore, the geometric phase in this arrangement will be unequivocally quantum mechanical because of the non-classical character of two-photon interferometry.