531 research outputs found
The Schwinger SU(3) Construction - II: Relations between Heisenberg-Weyl and SU(3) Coherent States
The Schwinger oscillator operator representation of SU(3), studied in a
previous paper from the representation theory point of view, is analysed to
discuss the intimate relationships between standard oscillator coherent state
systems and systems of SU(3) coherent states. Both SU(3) standard coherent
states, based on choice of highest weight vector as fiducial vector, and
certain other specific systems of generalised coherent states, are found to be
relevant. A complete analysis is presented, covering all the oscillator
coherent states without exception, and amounting to SU(3) harmonic analysis of
these states.Comment: Latex, 51 page
Parametrizing the mixing matrix : A unified approach
A unified approach to parametrization of the mixing matrix for
generations is developed. This approach not only has a clear geometrical
underpinning but also has the advantage of being economical and recursive and
leads in a natural way to the known phenomenologically useful parametrizations
of the mixing matrix.Comment: 8 pages, LaTe
Entanglement and Complete Positivity: Relevance and Manifestations in Classical Scalar Wave Optics
Entanglement of states and Complete Positivity of maps are concepts that have
achieved physical importance with the recent growth of quantum information
science. They are however mathematically relevant whenever tensor products of
complex linear (Hilbert) spaces are involved. We present such situations in
classical scalar paraxial wave optics where these concepts play a role:
propagation characteristics of coherent and partially coherent Gaussian beams;
and the definition and separability of the family of Twisted Gaussian Schell
Model (TGSM) beams. In the former, the evolution of the width of a projected
one-dimensional beam is shown to be a signature of entanglement in a
two-dimensional amplitude. In the latter, the partial transpose operation is
seen to explain key properties of TGSM beams.Comment: 7 pages Revtex 4-
A classical optical approach to the `non-local Pancharatnam-like phases' in Hanbury-Brown-Twiss correlations
We examine a recent proposal to show the presence of nonlocal Pancharatnam
type geometric phases in a quantum mechanical treatment of intensity
interferometry measurements upon inclusion of polarizing elements in the setup.
It is shown that a completely classical statistical treatment of such effects
is adequate for practical purposes. Further we show that the phase angles that
appear in the correlations, while at first sight appearing to resemble
Pancharatnam phases in their mathematical structure, cannot actually be
interpreted in that manner. We also describe a simpler Mach-Zehnder type setup
where similar effects can be observed without use of the paraxial
approximation.Comment: Minor corrections, published versio
Wigner distributions for finite state systems without redundant phase point operators
We set up Wigner distributions for state quantum systems following a
Dirac inspired approach. In contrast to much of the work on this case,
requiring a phase space, particularly when is even, our
approach is uniformly based on an phase space grid and thereby
avoids the necessity of having to invoke a `quadrupled' phase space and hence
the attendant redundance. Both odd and even cases are analysed in detail
and it is found that there are striking differences between the two. While the
odd case permits full implementation of the marginals property, the even
case does so only in a restricted sense. This has the consequence that in the
even case one is led to several equally good definitions of the Wigner
distributions as opposed to the odd case where the choice turns out to be
unique.Comment: Latex, 14 page
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
Geometric phase for mixed states: a differential geometric approach
A new definition and interpretation of geometric phase for mixed state cyclic
unitary evolution in quantum mechanics are presented. The pure state case is
formulated in a framework involving three selected Principal Fibre Bundles, and
the well known Kostant-Kirillov-Souriau symplectic structure on (co) adjoint
orbits associated with Lie groups. It is shown that this framework generalises
in a natural and simple manner to the mixed state case. For simplicity, only
the case of rank two mixed state density matrices is considered in detail. The
extensions of the ideas of Null Phase Curves and Pancharatnam lifts from pure
to mixed states are also presented.Comment: 22 pages, revtex
Null Phase Curves and Manifolds in Geometric Phase Theory
Bargmann invariants and null phase curves are known to be important
ingredients in understanding the essential nature of the geometric phase in
quantum mechanics. Null phase manifolds in quantum-mechanical ray spaces are
submanifolds made up entirely of null phase curves, and so are equally
important for geometric phase considerations. It is shown that the complete
characterization of null phase manifolds involves both the Riemannian metric
structure and the symplectic structure of ray space in equal measure, which
thus brings together these two aspects in a natural manner.Comment: 10 pages, 1 figur
The Sampling Theorem and Coherent State Systems in Quantum Mechanics
The well known Poisson Summation Formula is analysed from the perspective of
the coherent state systems associated with the Heisenberg--Weyl group. In
particular, it is shown that the Poisson summation formula may be viewed
abstractly as a relation between two sets of bases (Zak bases) arising as
simultaneous eigenvectors of two commuting unitary operators in which geometric
phase plays a key role. The Zak bases are shown to be interpretable as
generalised coherent state systems of the Heisenberg--Weyl group and this, in
turn, prompts analysis of the sampling theorem (an important and useful
consequence of the Poisson Summation Formula) and its extension from a coherent
state point of view leading to interesting results on properties of von Neumann
and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late
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