Sampling Colourings of the Triangular Lattice

We show that the Glauber dynamics on proper 9-colourings of the triangular lattice is rapidly mixing, which allows for efficient sampling. Consequently, there is a fully polynomial randomised approximation scheme (FPRAS) for counting proper 9-colourings of the triangular lattice. Proper colourings correspond to configurations in the zero-temperature anti-ferromagnetic Potts model. We show that the spin system consisting of proper 9-colourings of the triangular lattice has strong spatial mixing. This implies that there is a unique infinite-volume Gibbs distribution, which is an important property studied in statistical physics. Our results build on previous work by Goldberg, Martin and Paterson, who showed similar results for 10 colours on the triangular lattice. Their work was preceded by Salas and Sokal's 11-colour result. Both proofs rely on computational assistance, and so does our 9-colour proof. We have used a randomised heuristic to guide us towards rigourous results.Comment: 42 pages. Added appendix that describes implementation. Added ancillary file

Three lectures on random proper colorings of Zd\mathbb{Z}^d

A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on Random Walks, Random Graphs and Random Media, September 2019, Munich and at the school Lectures on Probability and Stochastic Processes XIV, December 2019, Delh

Fabricate 2020

Fabricate 2020 is the fourth title in the FABRICATE series on the theme of digital fabrication and published in conjunction with a triennial conference (London, April 2020). The book features cutting-edge built projects and work-in-progress from both academia and practice. It brings together pioneers in design and making from across the fields of architecture, construction, engineering, manufacturing, materials technology and computation. Fabricate 2020 includes 32 illustrated articles punctuated by four conversations between world-leading experts from design to engineering, discussing themes such as drawing-to-production, behavioural composites, robotic assembly, and digital craft

Cold-Atom Loading of Hollow-Core Photonic Crystal Fibre for Quantum Technologies

Ultra-strong light-atom interaction is a key resource for numerous applications in quantum-information processing, nonlinear optics, and quantum sensing. Maximising the strength of the interaction requires optimising the combination of light-atom coherent interaction time, spatial overlap between the optical mode and the atomic cross section, and the number of participating atoms. An exciting approach to achieving these goals is to use a collection of laser-cooled atoms inside a hollow-core photonic crystal fibre. Here the tight transverse confinement provided by fibre guarantees overlap between the atomic sample and guided optical modes over an arbitrarily long distance. Laser cooling improves the effective atom number of the sample by increasing the fraction that participate in the interaction and significantly improves the coherent interaction time by reducing the spatial decoherence rate of the ensemble. This project focuses around the development of an apparatus that realises the lasercooling, trapping, and loading of atoms into a kagome-lattice hollow-core fibre. In this thesis we describe the development of the elements required to realise this task, including the vacuum system, laser sources, computer oversight, and theoretical models employed. The resulting platform is capable of achieving the ultra-high optical depths required for exciting quantum-optics applications such as long-lived coherent optical pulse storage. We have demonstrated high-efficiency transport of cold rubidium atoms from a magneto-optical trap into a hollow-core fibre, measuring a peak optical depth of 600 with only 3£106 atoms. These experiments were guided by a Monte-Carlo simulation that has been shown to have excellent agreement with the physical system. The results show that this platform is in an excellent position to investigate coherent optical phenomena at the few-photon level. Along the way we investigated the application of light-shift engineering to both measure and compensate for the perturbative effects the strong light fields present in the experiment have on atomic states. We extend the ‘magic-wavelength’ technique used in the atomic lattice clock community to nullify the lineshape broadening of the target ensemble by introducing an additional light field. This allows the technique to be implemented in a broad range of atomic species and transitions, where the original technique was only accessible for limited species with specific energy-level structures. We also take advantage of light-shift engineering to extract a detailed model of the spatial distribution of an optically-trapped ensemble through a simple spectroscopic technique. We use this model to infer the temperature, coherence time, and number of atoms in the trap in addition to the depth of the trap itself. Experimentally we demonstrate this on our cold-atom-filled fibre platform, showing that this information can be extracted from a system with limited optical access and where conventional techniques cannot be applied. The apparatus and experimental techniques we have developed place this project in an excellent position to perform cutting-edge research in the fields of quantum information processing and nonlinear optics.Thesis (Ph.D.) -- University of Adelaide, School of Physical Sciences, 202

Biosignature storage in sulfate minerals- synthetic and natural investigations of the jarosite group minerals

The discovery of jarosite on Mars in 2004 generated increased interest in the properties of the mineral related to the search for life on other planets. Several studies indicate that the formation of jarosite can be linked to biological activity on Earth and biomolecules such as amino acids have been found associated with terrestrial jarosite samples. A series of natural and synthetic investigations using different jarosite end-members has been conducted and is presented in this dissertation to investigate the possibility that jarosite can store biosignatures. Natural samples were analyzed by x-ray diffraction, elemental carbon analysis and laser-desorption Fourier transform mass spectrometry (LD-FTMS) and were found to contain the amino acid glycine. Synthetic experiments were conducted in which the different end-members were synthesized in the presence of glycine as well as the amino acid alanine and the amino acid breakdown product methylamine. These samples were analyzed by x-ray diffraction, neutron diffraction, LD-FTMS and thermogravimetric analysis (TGA) techniques. Results of these experiments show that the detection of the biosignature and the effect that biomolecule has on the jarosite minerals is dependent on the end-member and indicate that the jarosite minerals are an excellent target for detecting potential signs of past life on other planets

Colloidal Self-Assembly of Novel Materials Displaying Structural Colour

Self-assembly of colloids is a powerful technique for the synthesis of novel materials. While top-down manufacturing methods can also produce structures patterned on the mesoscale, they are frequently limited in resolution, slow to employ or restricted to two-dimensional systems; in contrast, self-assembly theoretically offers full control over the three-dimensional microstructure of a material and can be tuned by varying initial conditions and external forces. Self-assembly is particularly attractive because of its prevalence in the natural world; most naturally occurring nanostructures with interesting properties are produced through self-assembly processes that can then be mimicked in synthetic systems. One such class of materials exhibits structural colour, where light is differentially scattered or reflected based not on absorption but on the physical arrangement of the material on the nanoscale. Structurally produced colours tend to be brighter and more vivid than pigment-based ones and do not fade over time. As such they have numerous potential applications not only as a source of colouration but also in next-generation non-backlit displays and optoelectronic systems. This thesis discusses the synthesis, functionalisation and self-assembly of colloids into novel functional materials, particularly those that mimic naturally occurring structural colour based on disordered and ordered systems. The functionalisation of colloids with DNA introduces a specific, tuneable and thermally reversible attractive potential between particles, making it an excellent system to explore self-assembly. DNA-coated polystyrene particles were used to investigate gel formation through a spinodal decomposition mechanism and how the resulting gel structure reproducibly depends upon suspension properties. Subsequently, the tendency of such gels to produce structural colour was investigated using both polystyrene and silica DNA-coated colloidal systems, in comparison with similar natural materials. Finally, soft solution-phase photonic crystals were assembled using highly charged colloids of a low refractive index. This system was found to exhibit isotropic structural colour tuneable throughout the visible range, coupled with high transparency.This work was supported by the EPSRC Cambridge NanoDTC, EP/L015978/