Towards the Formalization of Fractional Calculus in Higher-Order Logic

Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to analyze a wide class of physical systems in various fields of science and engineering. In this paper, we describe an ongoing project which aims at formalizing the basic theories of fractional calculus in the HOL Light theorem prover. Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones.Comment: 9 page

Quantum holonomies in photonic waveguide systems

The thesis at hand deals with the emergence of quantum holonomies in systems of coupled waveguides. Several proposals for their realisation in arrays of laser-written fused-silica waveguides are presented, including experimental results. I develop an operator-theoretic framework for the photon-number independent description of these optical networks. Finally, quantum holonomies will be embedded into schemes for measurement-based quantum computation, with the aim of approximating Jones polynomials.Die vorliegende Arbeit untersucht die Konzipierung von Quantenholonomien in Systemen gekoppelter Wellenleiter. Eine Vielzahl möglicher Realisierungen mittels lasergeschriebener Wellenleiter in Quarzglas wird präsentiert und zugehörige experimentelle Ergebnisse erläutert. Die Entwicklung einer operatortheoretischen Darstellung für die photonenzahlunabhängige Beschreibung dieser optischen Netzwerke wird vorgenommen. Abschließend werden Quantenholonomien für die messinduzierte Quantenberechnung von Jones-Polynomen verwendet

On-chip fluorescence detection using photonic bandgap guiding optofluidic hollow-core light cage

The on-chip detection of fluorescent light is essential for many bioanalytical and life-science related applications. Here, the optofluidic light cage consisting of a sparse array of micrometer encircling a hollow core represents an innovative concept, particularly for on-chip waveguide-based spectroscopy. In the present work, we demonstrate the potential of the optofluidic light cage concept in the context of integrated on-chip fluorescence spectroscopy. Specifically, we show that fluorescent light from a dye-doped aqueous solution generated in the core of a nanoprinted dual-ring light cage can be efficiently captured and guided to the waveguide ports. Notably, the fluorescence collection occurs predominantly in the fundamental mode, a property that distinguishes it from evanescent field-based waveguide detection schemes that favor collection in higher-order modes. Through exploiting the flexibility of waveguide design and 3D nanoprinting, both excitation and emission have been localized in the high transmission domains of the fundamental core mode. Fast diffusion, detection limits comparable to bulk measurements, and the potential of this approach in terms of device integration were demonstrated. Together with previous results on absorption spectroscopy, the achievements presented here suggest that the optofluidic light cage concept defines a novel photonic platform for integrated on-chip spectroscopic devices and real-time sensors compatible with both the fiber circuitry and microfluidics. Applications in areas such as bioanalytics and environmental sciences are conceivable, while more sophisticated applications such as nanoparticle tracking analysis and integrated Raman spectroscopy could be envisioned

Photon correlations in a two-site non-linear cavity system under coherent drive and dissipation

We calculate the normalized second-order correlation function for a system of two tunnel-coupled photonic resonators, each one exhibiting a single-photon nonlinearity of the Kerr type. We employ a full quantum formulation: the master equation for the model, which takes into account both a coherent continuous drive and radiative as well as non-radiative dissipation channels, is solved analytically in steady state through a perturbative approach, and the results are compared to exact numerical simulations. The degree of second-order coherence displays values between 0 and 1, and divides the diagram identified by the two energy scales of the system - the tunneling and the nonlinear Kerr interaction - into two distinct regions separated by a crossover. When the tunneling term dominates over the nonlinear one, the system state is delocalized over both cavities and the emitted light is coherent. In the opposite limit, photon blockade sets in and the system shows an insulator-like state with photons locked on each cavity, identified by antibunching of emitted light.Comment: 9 pages, 4 figures, to appear in Phys. Rev.

Describing many-body bosonic waveguide scattering with the truncated Wigner method

We consider quasi-stationary scattering of interacting bosonic matter waves in one-dimensional waveguides, as they arise in guided atom lasers. We show how the truncated Wigner (tW) method, which corresponds to the semiclassical description of the bosonic many-body system on the level of the diagonal approximation, can be utilized in order to describe such many-body bosonic scattering processes. Special emphasis is put on the discretization of space at the exact quantum level, in order to properly implement the semiclassical approximation and the tW method, as well as on the discussion of the results to be obtained in the continuous limit.Comment: 9 pages, 3 figure

Waveguiding in electrooptic langmuir-blodgett films

This thesis describes work on the waveguiding and second-order nonlinear optics of Langmuir-Blodgett (LB) films. A number of new monomeric materials are assessed for their LB deposition and their second-order nonlinear properties. Both the Pockels effect and second harmonic generation are studied. One new chromophore, diphenyl butadiyne is shown to possess exceptional nonlinear effects in monolayer form with x((^2))(-w; w, 0) = 2.7 x 10(^-10) mV(^-1) A novel aminonitrostilbene carboxylic acid with x((^2))(-w;w,0) = 1.43 x 10(^-10) m V(^-1) is also described. Polymeric LB materials have been found to possess greater stability than monomers but the deposition properties are often poor. Two oligomers are investigated, these both form high-quality LB layers and one gave X((^2))(-w;w,0) = (3.4-1-1.0j) x 10(^-11) m V(^-1). Two alternate-layer systems are considered. The first, of an amidonitrostilbene and a novel functionalised diarylalkyne deposits to 150 bilayers, but the nonlinear coefficients are shown to diminish after fewer than ten layers. X-ray diffraction, waveguiding and Fourier transform infra-red measurements are also performed. The second system alternates the two oligomers; only films of fewer than seven layer are studied due to the low value of Pockels effect and second-harmonic generation detected. A novel method of waveguide fabrication combining solution dipped polymer layers with LB films is introduced. This technique allows the rapid fabrication of waveguides with nonlinear properties. Monomode waveguides are produced and their electrooptic properties measured. These are the first such measurements on an LB film waveguide. It is shown theoretically that such waveguides could be used to produce modulators of similar figures of merit to those made from poled-polymers but that an order of magnitude improvement in X((^2))(-w; w, 0) is needed to match lithium niobate systems