Miller, Warner A.

As a subset of quantum optics, single photons are one of the competing physical resources for quantum information processing. When used as carriers of quantum information, they have no equal. For the processing of quantum information, single photons have proven difficult to scale beyond the order of ⇠ 10 photons. The lack of single-photon-level interaction has led to creative approaches which rely on postselection to filter the possible measured outcomes to those which appear as though
interaction occurred. This approach of post-selection leans heavily on the ability to
not only generate but also detect scores of single photons simultaneously and with
near perfect efficiency. Our work relaxes the emphasis which has been placed on single
photons for quantum information processing to that of states with, in principle, an
arbitrary number of photons. Central moment expectations on two-mode squeezed
states are shown to exhibit post-selection behavior which reflects the single-photon
counterpart. These measures are proven to be robust to loss and return entangled
state statistics on average. With naive estimation of the central moment, states
with ~ 20 modes are within reach with current technology, closing the gap between quantum states which can and cannot be classically simulated.

The angular momentum of light originates from two sources: one is the spin
angular momentum (SAM) of individual photons, which is related to the polarization
of light and the other is the orbital angular momentum (OAM) associated with helical
wavefront of the light if it is helically phased (complex phase front). A beam of light
that is composed of photons possessing both OAM and SAM states can be used in
different areas of study such as rotating microscopic particles, interacting with nonlinear
materials, investigating atom-light interactions, communication and medical
imaging technologies, quantum information, quantum entanglement and etc. In this
dissertation we study coherent beams that convey photons in superposition states
of polarization and complex phase front. Our study includes two fields: (I) classical
wave-like behavior with visible light in the field of singular optics. (II) quantum
particle-like behavior of photons of light in the field of quantum-entangled optics.
The approach is to investigate the state of such photons both mathematically and
experimentally in classical-singular and quantum-entangled fields. We discuss seven projects based on this research. In one project we present
a new method to encode OAM modes into perpendicular polarization components
and making superpositions of polarization and spatial modes mapped by Poincare
sphere. In another project using spatial light modulators (SLM) we realized highorder
disclination patterns in the polarization map of the cross section of the beam.
We also realize new forms of polarization disclination patterns (line patterns where
rotational invariance is violated) known as monstars that were not previously seen.
We proposed a new definition for characterizing these patterns since they can have
zero or negative singularity index. In another project, instead of SLM we used q-plates
to generate new forms of monstars. We proposed a robust and easy method for
determining the topological charge of a complex phase front beam by inspecting the
interference pattern the beam reflected from a wedged optical flat. In another project
we encoded OAM modes onto orthogonal polarization components of a photon from
an entangled pair and investigated the quantum entanglement. We also prepared
a polarization entangled state and calculated some measures of entanglement. We
summarize the projects and discuss the future prospects.

The new formalism for quantization of gauge systems based on the concept of the
dynamical Hamiltonian recently introduced as a basis for the canonical theory of
quantum gravity was considered in the context of general gauge theories. This and
other Hamiltonian methods, that include Dirac's theory of extended Hamiltonian
and the Hamiltonian reduction formalism were critically examined. It was established
that the classical theories of constrained gauge systems formulated within the
framework of either of the approaches are equivalent. The central to the proof of
equivalence was the fact that the gauge symmetries resuIt in the constraints of the
first class in Dirac's terminology that Iead to redundancy of equations of motion
for some of the canonica variables. Nevertheless, analysis of the quantum theories
showed that in general, the quantum theory of the dynamical Hamiltonian is inequivalent
to those of the extended Hamiltonian and the Hamiltonian reduction. The
new method of quantization was applied to a number of gauge systems, including
the theory of relativistic particle, the Bianchi type IX cosmological model and spinor electrodynamics along side with the traditional methods of quantization. In all of the
cases considered the quantum theory of the dynamical Hamiltonian was found to be
well-defined and to possess the appropriate classical limit. In particular, the quantization
procedure for the Bianchi type IX cosmological spacetime did not run into
any of the known problems with quantizing the theory of General Relativity. On the
other hand, in the case of the quantum electrodynamics the dynamical Hamiltonian
approach led to the quantum theory with the modified self-interaction in the matter
sector. The possible consequence of this for the quantization of the full theory of
General Relativity including the matter fields are discussed.

One of the most fundamental problems in classical general relativity is the
measure of e↵ective mass of a pure gravitational field. The principle of equivalence
prohibits a purely local measure of this mass. This thesis critically examines the most
recent quasi-local measure by Wang and Yau for a maximally rotating black hole
spacetime. In particular, it examines a family of spacelike 2-surfaces with constant
radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which, the
Wang and Yau quasi-local energy has yet to be explored. In this region, the results of
this thesis indicate that the Wang and Yau quasi-local energy yields complex values
and is essentially equivalent to the previously defined Brown and York quasi-local
energy. However, an application of their quasi-local mass is suggested in a dynamical
setting, which can potentially give new and meaningful measures. In supporting this
thesis, the development of a novel adiabatic isometric mapping algorithm is included.
Its purpose is to provide the isometric embedding of convex 2-surfaces with spherical
topology into Euclidean 3-space necessary for completing the calculation of quasilocal
energy in numerical relativity codes. The innovation of this algorithm is the
guided adiabatic pull- back routine. This uses Ricci flow and Newtons method to give isometric embeddings of piecewise simplicial 2-manifolds, which allows the algorithm
to provide accuracy of the edge lengths up to a user set tolerance.

Images of highly idealized Bardeen-Petterson accretion disks around Kerr black holes and their generated energy spectra line profiles are created via computer simulations in this work. The line profiles are examined in the relation to the original disk parameters to demonstrate that Bardeen-Petterson disks can be the source of the spectra we observe, even of actual systems. The challenges in computer programming the simulations and methods to overcome the challenges are discussed in the paper. Also discussed are the future work, improvements and intentions for the simulations. Finally, the discussion turns to the usage of indicators in these line profiles to predict parameters of the original disk systems.