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Journal of Physics A Mathematical and General

Invariance of the geometrical phase under time-dependent unitary transformations

Brian K. Kendrick

Published Feb 21, 1992• 6 citations
Publisher

Abstract

The effects of time-dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting. A detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle: Hilbert space to space of states changes. The measured relative phase, geometrical phase and dynamical phase are all invariant under these transformations.

DOI
10.1088/0305-4470/25/4/025

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The effects of time-dependent unitary transformations on the geometrical phase are investigated in the general non adiabatic setting. A detailed study considering both paths of an interference experiment shows that even though the geometry of the fibre bundle: Hilbert space to space of states changes. The measured relative phase, geometrical phase and dynamical phase are all invariant under these transformations.
Properties of geometric phase under Galilean transformation are investigated within a general framework. We found that the geometric phase in the system commonly met with in the laboratory is Galilean invariant. However, this is not always true for general non-relativistic quantum systems. The physical origins of the frame dependence of geometric phase is discussed in detail. We emphasize that the change of geometric phase caused by Galilean boosts does not destroy the results of measurement for two-state interference. It is consistent with the general constraint of Galilean invariance on a non-relativistic quantum system.
The geometric phase for a pure quantal state undergoing an arbitrary evolution is a "memory" of the geometry of the path in the projective Hilbert space of the system. We find that Uhlmann's geometric phase for a mixed quantal state undergoing unitary evolution depends not only on the geometry of the path of the system alone but also on a constrained bilocal unitary evolution of the purified entangled state. We analyze this in general, illustrate it for the qubit case, and propose an experiment to test this effect. We also show that the mixed state geometric phase proposed recently in the context of interferometry requires unilocal transformations and is therefore essentially a property of the system alone.
The interdependence of total, dynamic and geometric (Pancharatnam) phases of the state vector of a quantum system that are determined in different representations is considered. One finds that the phases calculated using two different bases of the Hilbert space, which are related by a time-dependent unitary transformation, are not invariant under this transformation; rather they are connected by non-trivial and well-defined relations. We determine the general relationships between the phases in different representations and then apply and verify these transformations in the case of the physically interesting model of the three-level Γ-type system specified by a pair of dipole-coupled excited states.
Neither geometric phases nor differences in geometric phases are generally invariant under time-dependent unitary transformations (unlike differences in total phases), in particular under local gauge transformations and Galilei transformations. (This was pointed out originally by Aharonov and Anandan, and in the case of Galilei transformations has recently been shown explicitly by Sjoeqvist, Brown and Carlsen.) In this paper, I introduce a phase, related to the standard geometric phase, for which phase differences are both gauge- and Galilei-invariant, and, indeed, invariant under transformations to linearly accelerated coordinate systems. I discuss in what sense this phase can also be viewed as geometric, what its relation is to earlier proposals for making geometric phases invariant under gauge or Galilei transformations, and what is its classical analogue. I finally apply this invariant phase to Berry's derivation of the Aharonov-Bohm effect.