The laws of quantum mechanics are the most fundamental laws of physics that we know of. They have been stringently tested in a variety of situations. Even so, there are still basic unanswered questions concerning our understanding and interpretation of some of the results. Despite this, it is very important to make practical use of what we already know about quantum mechanics. In this sense, quantum physics is both a fundamental science and new engineering. It seems a certainty that in the forthcoming century we will progress in our understanding and technical mastery of quantum effects as quickly as we have done with electricity in the last. Our research proposal is based on one of the most exciting recent results. In 1999 Japanese researchers, building on other work, showed that it is possible to make an electrical circuit that obeys the laws of quantum physics. Normally objects that obey quantum mechanics are 'natural' single particles such as electrons and photons, never before have we had the opportunity to study or exploit an artificial quantum circuit. Presently such circuits are made from Aluminium, they operate at very low temperatures, below 100mK where the Aluminium is superconducting and at very high frequencies, typically 10 GHz. It is now possible to observe the discrete (quantised) changes in energy, to manipulate the circuit at will into its different quantum states, and to perform all the basic atomic physics experiments on these man made electrical circuits. Five research groups in the world have so far been able to reproduce and improve on the early results using different designs of circuits and with varying degrees of success. However, it is now clear that none of these five experiments operate perfectly. It has proven difficult to measure reliably the quantum state of the circuit for reasons that are not yet fully understood, this is known as the readout problem. In addition the circuits are not completely stable in the sense that microscopic changes in the environment around them interfere with their operation, an effect known as environmental decoherence. Our research is dedicated to solving these problems. We plan to take the best available readout technology, a quantised photon cavity resonator developed at Yale University in the USA and use it on the best available quantum circuit, the quantronium circuit developed at the CEA-Saclay, France. The fastest way to establish a serious independent research effort in the UK is to collaborate with one of the best current research groups. With this in mind, the proposer of this research has spent the past year working with the CEA-Saclay group. Now we will initiate a new research effort at Royal Holloway, University of London, already well known for its contributions to quantum computing. The collaboration with the CEA-Saclay will continue and there will be distinct but complementary research programmes.The research programme is dedicated to understanding and eliminating the problems referred to above and to building better circuits. Quantum circuits offer a very promising route to building a quantum computer and superconducting qubits are presently the best available solid state qubits. We wish to produce a device that couples two qubits together, this is the necessary next step in the production of a quantum computer. Such a device would also allow us to make systematic studies of quantum entanglement, perhaps the least well understood area of quantum mechanics. We also plan to explore how it is that quantum mechanics makes the transition to classical mechanics. It is thought that this proceeds through the process of environmental decoherence, which is precisely the effect to which a quantum circuit is most vulnerable, hence presenting a unique opportunity to study this problem in a very direct way.

Rotating fluid flow is ubiquitous in many naturally occurring and engineering systems and plays a crucial role. For instance, the turbulence of geophysical vortices in the oceans is responsible for the mixing of fluid momentum and scalars such as salinity or planktons. Rotating flow is also important in industrial processes to produce homogenised products by efficient turbulent mixing (e.g. glass or polymer manufacturing processes). Rotation profiles of fluid flow are often differential, i.e. the angular speed varies with radius from the rotation axis. Such differentially-rotating flow can become centrifugally unstable when an imbalance exists between the pressure gradient and centrifugal force, a situation arising when the angular momentum decreases with the radius. This centrifugal instability is very destructive and thus an important source of turbulence. Most of the studies on centrifugal instability have considered linear analyses in which perturbations that drive the instability are assumed to be small enough to neglect nonlinear terms in the governing equations. On the other hand, nonlinear development processes of the instability, such as saturation or laminar-turbulent transition, have not been thoroughly investigated. In particular, the nonlinear centrifugal instability is not fully understood under the combined effects of thermal diffusion and stratification. Fluid flow with heat transfer is a very common configuration in various natural and engineering systems, thus revealing the role of such thermal effects on turbulence can significantly contribute to our knowledge of multi-physical flow systems in physical sciences and engineering. This situation motivates the current research programme with two main objectives: (i) Investigate nonlinear development processes of the centrifugal instability under the effects of thermal diffusion and stratification, and; (ii) Develop a new turbulence model to apply to multi-physics simulations. In the first part of the programme, we will examine linear and nonlinear centrifugal instability of a famous rotating shear flow called Taylor-Couette (TC) flow, the flow between two concentric cylinders that rotate independently. We will first analyse linear centrifugal instability of the TC flow in thermally diffusive and stratified fluids using the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) method. The linear analysis will reveal how the thermal effects affect the initial growth of small-amplitude perturbations and the WKBJ method will allow us to derive explicit mathematical expressions of the instability growth. Nonlinear instability will then be investigated by both direct numerical simulations and a semi-linear model. Such nonlinear analyses can demonstrate how nonlinear interactions between perturbations and base flow lead to the saturation or laminar-turbulent transition processes. The second part of the programme will focus on developing a new turbulence model. Results from linear and nonlinear stability analyses will be used to construct a turbulent viscosity to apply to multi-physics simulations. More specifically, we will apply the new model to the state-of-the-art code for stellar physics simulations of the evolution of rotating stars. The updated code will simulate the stellar evolution and produce results such as radial distributions of mass, angular momentum or chemicals in the stellar interior. The outcomes will be compared with those from other stellar evolution simulations and observations. By achieving the main objectives of the proposed research, we will advance our understanding of instability-induced turbulence and its role in the multi-physics processes of the evolution of star, as just one example. Such turbulence modelling will also be beneficial for researchers in other fields of physical sciences and engineering.

Providing a long-term solution to the world energy problem and climate change is one of the most scientifically challenging endeavours that faces humanity on the global scale. Fusion energy is a particularly attractive solution and is poised to become a viable energy source in the coming decades by providing carbon-free, steady-state, high energy density in the absence of radioactive waste. However, confinement of the hot plasma is remarkably complicated to achieve in fusion conditions. Magnetic confinement fusion is championed by the tokamak concept, which uses strong magnetic fields in a donut-shaped device to produce the confinement. Despite the confining magnetic fields, experiments and theory have provided strong evidence that turbulent processes in the plasma produce a constant leakage of heat and particles out of the hot confining core, which impedes efficient generation of the fusion processes. This motivates understanding the plasma turbulent processes leading to heat and particle transport from a fundamental perspective, as well as its implications for real life tokamak experiments. Recent numerical and theoretical studies have shown that fluctuations in the electromagnetic field can lead to enhanced transport losses in the spherical tokamak core at sufficiently high values of beta (ratio of plasma to magnetic pressure), leading to a paradigm shift between the traditional electrostatic description of the 'so-called' ion-scale turbulence (characteristic of the outer-core conventional tokamak) to a fully electromagnetic description and new transport processes. Electromagnetic, meso-scale instabilities (Alfvén eigenmodes) can also be driven by the presence of supra-thermal particles, and are capable of stabilising electrostatic ion-scale turbulence fluctuations. The stabilisation of ion-scale turbulence in the ST and the inner tokamak core leads to unexplored confinement regimes that are likely dominated by the electron heat losses. These are traditionally subdominant to ion heat losses in conventional core tokamak plasma scenarios. These regimes are critically relevant for future fusion reactors which are expected to have dominant electron heating and transport. The novelty of this research is to study these unexplored confinement regimes by combining experimental measurements of the electron fluctuations that are believed to be responsible for the electron heat transport (extremely scarce to date), direct numerical turbulence simulation (gyrokinetic simulation), synthetic diagnostics for the quantitative interpretation of the experimental measurements, and analytical theory to yield a fundamental understanding of the experimental and numerical findings. Programmatically, the study of electromagnetic fluctuations and Alfvén eigenmodes driven by fast particles is important as they bridge the gap between current machine operation and future fusion reactors. The upcoming JET DT campaign (CCFE, UK), MAST-U (CCFE, UK) and NSTX-U experiments (Princeton, USA) are the missing link between present-day machines and fusion burning plasma experiments such as ITER, STEP (UK) and SPARC (US). Scientifically, this research will lead to ground-breaking discoveries such as new interaction mechanisms between supra-thermal particles and turbulence or the discovery of enhanced confinement regimes expected of future fusion reactors. This will have direct influences affecting the projections and design of the future STEP and ITER burning-plasma experiments, and will enable the UK to gain full in-house energy independence from magnetic fusion in the coming decades.

This proposal addresses the problem of studying the heterogeneity of a variable of interest in spatially-distributed structures. Quite often the spatial domain under study can be coarsened and represented as a graph, and the variable of interest can take a discrete number of different values, indicated by labels or colours. Hence, the quantification of the spatial heterogeneity of a variable can be mapped onto the study of the distribution of labels in a graph with coloured nodes. A typical example is the problem of identifying the social, economic, or ethnic segregation of an urban area. We say that the population of a city is segregated with respect to ethnicity or income level if people prefer to live in areas where other people of the same ethnicity or income level already live, avoiding areas that are inhabited by people of other ethnic groups or income levels. In this case, the system consists of a discrete set of regions (neighbourhoods, boroughs, etc), and can thus be represented by a planar graph of adjacency among regions, where each region (node) is assigned a label or colour according to the income level or most abundant ethnicity of its inhabitants. To determine the level of segregation of a city, one needs to quantify the heterogeneity of the distribution of those labels, under the assumption that higher heterogeneity usually corresponds to higher segregation. In general, the presence of high levels of segregation is the cause of economic and social tension. Consequently, assessing the level of segregation of a urban area is fundamental for policy makers, with potential repercussions on millions of lives. Despite being a long-standing question in urbanism and economics, there is currently no universal agreement about how segregation should be quantified, or on how to compare the levels of segregation measured in different areas. A very different but relevant example is that of tumor growth. The cells of a tumor normally belong to several strains or sub-clones, due to the occurrence of genetic mutations, and some of those sub-clones often develop resistance to chemotherapy, accelerating the growth of the tumor. Recent research suggested that the number of distinct sub-clones of a tumor and the irregularity of their spatial distribution correlate with the speed at which the tumor grows. Also this system can be represented as a spatial graph, where each region of the tumor (node) is associated to a label or colour corresponding to the most abundant cancer sub-clone in that region. And also in this case, the quantification of the spatial heterogeneity of the distribution of labels on a graph can have important repercussions on the life of thousands of people. This project proposes a novel way of quantifying heterogeneity in spatial systems, based on the trajectories of random walks over the associated adjacency graphs with coloured nodes. A random walk is the simplest way to explore a graph -if you are on a node, you jump to one if its neighbours chosen at random- yet its trajectories retain a lot of information about the correlations among node properties. The main aim of this project is to study the statistical properties of trajectories of random walks on graphs with coloured nodes, focusing in particular on inter-class first passage times (the number of jumps needed to a random walk to get from a node of a certain colour to a node of another colour) and cover times (the number of jumps needed to visit at least one node of each colour). The Principal Investigator will provide analytical expressions for inter-class passage and cover times in different real and synthetic spatial graphs with coloured nodes, will use those expressions to quantify the heterogeneity of a given colour distribution, and, thanks to the collaboration with several research groups in the UK and abroad, will apply those measures to urban segregation, plant biology, and cancer research.

During this Fellowship, I intend to develop a multi-scale approach that, revealing the structure-relaxation dynamics correlation over a wide time- and length-scale, will direct the design of smart membranes with customised functionalities (while providing a fundamental understanding of commercially available materials). My methodology addressing the microstructure/processing/performance triangle aims to facilitate the transition from theoretical properties to practical applications in materials designed for energy conversion applications and separation science (while it can be extended to biomedical applications). I intend to develop my research at UCL Chemistry, which provides the ideal scientific framework enabling close collaborations with Physical Sciences and Engineering Departments.