177 research outputs found
On Geometric Phase from Pure Projections
The geometric phase is usually treated as a quantity modulo 2\pi, a
convention carried over from early work on the subject. The results of a series
of optical interference experiments involving polarization of light, done by
the present author (reviewed in R.Bhandari, Phys. Rep. 281 (1997) p.1) question
the usefulness of such a definition of the geometric phase in that it throws
away useful and measurable information about the system, for example strengths
of singularities giving rise to the geometric phase. Such singularities have
been directly demonstrated by phase-shift measurement in interference
experiments. In this paper, two recent polarization experiments (Hariharan
et.al., J.Mod.Opt. 44 (1997)p.707 and Berry and Klein, J.Mod.Opt. 43
(1996)p.165) are analysed and compared with previous experiments and
potentially detectible singularities in these experiments pointed out.Comment: Latex, 15 pages, 6 figures; ([email protected]
Non-Abelian Braiding of Lattice Bosons
We report on a numerical experiment in which we use time-dependent potentials
to braid non-abelian quasiparticles. We consider lattice bosons in a uniform
magnetic field within the fractional quantum Hall regime, where , the
ratio of particles to flux quanta, is near 1/2, 1 or 3/2. We introduce
time-dependent potentials which move quasiparticle excitations around one
another, explicitly simulating a braiding operation which could implement part
of a gate in a quantum computation. We find that different braids do not
commute for near and , with Berry matrices respectively
consistent with Ising and Fibonacci anyons. Near , the braids commute.Comment: 5 pages, 1 figur
On singularities of the mixed state phase
A recent proposal of Sjoqvist et.al. to extend Pancharatnam's criterion for
phase difference between two different pure states to the case of mixed states
in quantum mechanics is analyzed and the existence of phase singularities in
the parameter space of an interference experiment with particles in mixed
states pointed out. In the vicinity of such singular points the phase changes
sharply and precisely at these points it becomes undefined. A closed circuit in
the parameter space around such points results in a measurable phase shift
equal to 2n\pi, where n is an integer. Such effects have earlier been observed
in interference experiments with pure polarization states of light, a system
isomorphic to the spin-1/2 system in quantum mechanics. Implications of phase
singularities for the interpretation of experiments with partially polarized
and unpolarized neutrons are discussed. New kinds of topological phases
involving variables representing decoherence (depolarization) of pure states
are predicted and experiments to verify them suggested.Comment: 4 pages Latex, 1 postscript figure; submitted to Physical Review
Letters 12 Dec 2000; Revised on 13 August 200
Off-diagonal geometric phase for mixed states
We extend the off-diagonal geometric phase [Phys. Rev. Lett. {\bf 85}, 3067
(2000)] to mixed quantal states. The nodal structure of this phase in the qubit
(two-level) case is compared with that of the diagonal mixed state geometric
phase [Phys. Rev. Lett. {\bf 85}, 2845 (2000)]. Extension to higher dimensional
Hilbert spaces is delineated. A physical scenario for the off-diagonal mixed
state geometric phase in polarization-entangled two-photon interferometry is
proposed.Comment: small corrections; journal reference adde
Noncyclic geometric changes of quantum states
Non-Abelian quantum holonomies, i.e., unitary state changes solely induced by
geometric properties of a quantum system, have been much under focus in the
physics community as generalizations of the Abelian Berry phase. Apart from
being a general phenomenon displayed in various subfields of quantum physics,
the use of holonomies has lately been suggested as a robust technique to obtain
quantum gates; the building blocks of quantum computers. Non-Abelian holonomies
are usually associated with cyclic changes of quantum systems, but here we
consider a generalization to noncyclic evolutions. We argue that this open-path
holonomy can be used to construct quantum gates. We also show that a structure
of partially defined holonomies emerges from the open-path holonomy. This
structure has no counterpart in the Abelian setting. We illustrate the general
ideas using an example that may be accessible to tests in various physical
systems.Comment: Extended version, new title, journal reference adde
Interference of Quantum Channels
We show how interferometry can be used to characterise certain aspects of
general quantum processes, in particular, the coherence of completely positive
maps. We derive a measure of coherent fidelity, maximum interference visibility
and the closest unitary operator to a given physical process under this
measure.Comment: 4 pages, 5 figures, REVTeX 4, typographical corrections and added
acknowledgemen
Geometric phases and hidden local gauge symmetry
The analysis of geometric phases associated with level crossing is reduced to
the familiar diagonalization of the Hamiltonian in the second quantized
formulation. A hidden local gauge symmetry, which is associated with the
arbitrariness of the phase choice of a complete orthonormal basis set, becomes
explicit in this formulation (in particular, in the adiabatic approximation)
and specifies physical observables. The choice of a basis set which specifies
the coordinate in the functional space is arbitrary in the second quantization,
and a sub-class of coordinate transformations, which keeps the form of the
action invariant, is recognized as the gauge symmetry. We discuss the
implications of this hidden local gauge symmetry in detail by analyzing
geometric phases for cyclic and noncyclic evolutions. It is shown that the
hidden local symmetry provides a basic concept alternative to the notion of
holonomy to analyze geometric phases and that the analysis based on the hidden
local gauge symmetry leads to results consistent with the general prescription
of Pancharatnam. We however note an important difference between the geometric
phases for cyclic and noncyclic evolutions. We also explain a basic difference
between our hidden local gauge symmetry and a gauge symmetry (or equivalence
class) used by Aharonov and Anandan in their definition of generalized
geometric phases.Comment: 25 pages, 1 figure. Some typos have been corrected. To be published
in Phys. Rev.
Phases of quantum states in completely positive non-unitary evolution
We define an operational notion of phases in interferometry for a quantum
system undergoing a completely positive non-unitary evolution. This definition
is based on the concepts of quantum measurement theory. The suitable
generalization of the Pancharatnan connection allows us to determine the
dynamical and geometrical parts of the total phase between two states linked by
a completely positive map. These results reduce to the knonw expressions of
total, dynamical and geometrical phases for pure and mixed states evolving
unitarily.Comment: 2 figure
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
Topological phase for spin-orbit transformations on a laser beam
We investigate the topological phase associated with the double connectedness
of the SO(3) representation in terms of maximally entangled states. An
experimental demonstration is provided in the context of polarization and
spatial mode transformations of a laser beam carrying orbital angular momentum.
The topological phase is evidenced through interferometric measurements and a
quantitative relationship between the concurrence and the fringes visibility is
derived. Both the quantum and the classical regimes were investigated.Comment: 4 pages, 4 figure
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