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Physica Scripta

Change of the Pancharatnam phase under unitary transformation and application to a model three-level Γ-type system specified by a pair of dipole-coupled excited states

Istok P. Mendaš, Duska B Popovic

Published Dec 20, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0031-8949/85/01/015008

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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.
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
A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate collinear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic ‘curves' in Hilbert space. Explicit expressions for Pancharatnam, dynamic and geometric phases are derived for the transformations produced by linear phase converters. The problem of gauge invariance is treated in this article.
A theoretical analysis of Pancharatnam and Berry phases is made for biphoton three-level systems, which are produced via frequency degenerate collinear spontaneous parametric down conversion (SPDC). The general theory of Pancharatnam phases is discussed with a special emphasis on geodesic ‘curves' in Hilbert space. Explicit expressions for Pancharatnam, dynamic and geometric phases are derived for the transformations produced by linear phase converters. The problem of gauge invariance is treated in this article.
Geometric Phase in Quantum Mechanics is generally formulated entirely in terms of geometric structure of the Complex Hilbert Space. We will exploit this fact in case of mixed states for three level open systems undergoing depolarization using the eight di- mensional Poincare sphere and non unit vector rays in H3 within the limit of pure state approach may be found to be in agreement with the Pancharatnam Phase, Berry Phase and Aharonov-Anandan Phase. We will consider the Depolarization of the Three di- mensional fields by introducing the stokes parameters and by redefining the definition of Degree of Polarization using the SU(3) algebra to describe the quantum light depolariza- tion of Non-paraxial beams by minimal reservoir coupling models modeled by Lindblad Master equations and connect the solution of that with the geometric phase for systems undergoing depolarization.