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Title  Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wavechaos theory  

Group(s)  Industrial and Applied Mathematics, Mathematical Physics  
Proposer(s)  Dr Stephen Creagh, Prof Gregor Tanner  
Description  The focus of this project is the development of a mathematical framework to understand the propagation of electromagnetic fields within complicated environments – a challenging task especially in the high frequency limit. Modern technology is typically stuffed with electronic componentry. Devices ranging from a mobile phone to a pc to an Airbus A380 will have many internal electronic components operating at high frequencies and therefore radiating electromagnetic waves. If the waves radiated from one component are strong enough, they can interfere with the functioning of another component somewhere else in the unit. The field of Electromagnetic Compatibility (EMC) aims to mitigate these effects by better understanding the emitted radiation. The outcome of the research will help to design electronic devices, which are protected from interference from other EM sources within buildings, pc enclosures or even planes. The innovative idea in the proposed approach rests on combining EMfield propagation with ideas of chaos theory and nonlinear dynamics. In particular, the representation of waves emitted from a complex source is described in terms of their raydynamics in phase space using the socalled Wigner distribution function (WDF) formalism. It allows us to replace the wave propagation problem with one of propagating classical densities within phase space.


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Title  Wave propagation in complex builtup structures – tackling quasiperiodicity and inhomogeneity  

Group(s)  Industrial and Applied Mathematics, Mathematical Physics  
Proposer(s)  Prof Gregor Tanner, Dr Stephen Creagh  
Description  Computing the dynamic response of modern aerospace, automotive and civil structures can be a computationally challenging task. Characterising the structural dynamics in terms of waves in a uniform or periodic medium is often an important first step in understanding the principal propagating wave modes. Real mechanical structures are rarely fully periodic or homogeneous – variations in shape or thickness, boundaries and intersections as well as curvature destroy the perfect symmetry. The aim of the project is to extend periodic structure theory to wave propagation in quasiperiodic and inhomogeneous media such as stiffened structures. The modelling of waves can then be recast in terms of Bloch theory, which will be modified by using appropriate energy or flux conservation assumptions. The information about the propagating modes will then be implemented into modern highfrequency wave methods – such as the socalled Dynamical Energy Analysis developed in Nottingham  making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.


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Title  Operational tasks in quantum information science and beyond  

Group(s)  Mathematical Physics  
Proposer(s) 
Prof Gerardo Adesso Notice: Undefined index: pmzll in /maths/www/html/postgraduate/projects/index.php on line 560 Notice: Undefined index: pmzll in /maths/www/html/postgraduate/projects/index.php on line 560 , 

Description  Over the last decades, quantum information has taught us that many operational tasks are much more efficiently performed exploiting quantum mechanics than it would be possible in a purely classical world. If one wants to assess the performance of a real device, a comparison with its ‘intrinsic’ limitations as deriving from theoretical considerations has to be made. Determining those bounds is one of our goals as information scientists. This raises the problem of choosing what theoretical framework to use for the above considerations. There are two main alternatives: (i) one could choose to ‘trust’ quantum mechanics; or (ii) one could instead look at limitations arising only from ‘deeper’ requirements, such as the nosignaling constraint. Limitations of this second kind do not determine uniquely the mathematical features of the theory, leaving room for several different alternatives. Thus, in order to pursue the program (ii), one needs to handle a wider class of theories (generically called ‘general probabilistic theories’ or GPTs) that encompasses classical probability theory and quantum mechanics as special cases. For an introduction to the GPT formalism, see for instance [1] or [2, §II]. The main goal of this PhD project is to look at various operationally relevant tasks and analyse their ultimate performances in the sense of (ii), following the blueprint of [2] at a higher degree of generality. Despite the ‘postquantum’ nature of the project, an analysis of the quantum case (i) has to be carried out whenever it is not already covered by the existing literature. A distinctive feature of this project is that it can be adapted to the personal taste and preferences of the candidate. In fact, many possible directions can be taken, one for each task one chooses to focus on. For the sake of concreteness, let us give three explicit examples: a) Shared quantum entanglement can enhance the classical communication capability of a noiseless channel, a property known as quantum superdense coding (see [3, §6.2.3] or [4, §2.3]. Is there an analogous protocol for other GPTs? And if the answer is affirmative, what is the ultimate limit on the information gain that one can achieve by sending a local share of dimension d of a suitable entangled state? b) Grover’s search algorithm [4, §6.1] allows us to exploit quantum mechanics to conduct a search in an unstructured database of N objects using only √N oracle calls. This is known to be optimal in the quantum case [4, §6.6]. Can one do better in other GPTs? How much better? c) Quantum data locking [5] rests on the existence of bipartite quantum states whose locally accessible correlations can be greatly boosted by means of a small amount of classical communication between the parties. Are there GPTs that perform better than quantum mechanics at data locking? 

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Title  Quantum Thermal Engineering  

Group(s)  Mathematical Physics  
Proposer(s) 
Prof Gerardo Adesso Notice: Undefined index: pmzlac in /maths/www/html/postgraduate/projects/index.php on line 560 Notice: Undefined index: pmzlac in /maths/www/html/postgraduate/projects/index.php on line 560 , 

Description  In essence, quantum thermodynamics aims at unveling the connections between quantum physics and thermodynamics. Even if statistical mechanics teaches us that thermodynamics is only an effective theory emerging in the limit of large particle numbers, an individual quantum system can, in fact, convert heat to work or produce refrigeration in pretty much the same way as a car engine or a household refrigerator. Understanding the extent of this analogy would help us answer fundamental questions such as: Which ingredients are necessary for thermodynamic behaviour on an individual quantum system?, as well as more pragmatic questions such as: Can quantum effects help to bend the laws of quantum thermodynamics? Even if they cannot, can quantum thermodynamic devices assist in the deployment of quantum technologies? Quantum thermodynamics was born side by side with the theory of open quantum systems, back in the late 1970s, and developped slowly for over two decades. Only very recently, has the field started to attract an everincreasing interest from a wide variety of communnities, encompassing condensed matter physics, quantum information theory, statistical mechanics, or quantum manybody physics. Most of the existing restuls in the field are limited to the canonical scenario of systems weakly coupled to equilibrium environments. In this research programme, we shall use instead cuttingedge tools for the treatment of open quantum systems srontgly coupled to their environments—i.e. exact methods, such as quantum Langevin equations, as well as perturbative techniques such as the polaron transformation, the reaction coordinate mapping, collisional models, dissipation into finite baths, etc.—. The goal of this programme is to generalise the existing results in the literature about the performance limitations of quantum thermodynamic cycles to the strong coupling regime. In turn, understanding the thermodynamics of strongly coupled open quantum systems would ultimately allow to formulate systembath coupling schemes tailormade so as to e.g. maximise the energy efficiency of a heat devise, or its power output, or both. The successful candidate should have a strong background in quantum mechanics and, ideally, also on thermodynamics, statistical mechanics, fluid mechanics, quantum optics, and/or molecular physics. Good analytical skills and high motivation are essential. No specific programming/numerical skills are required, although familiarity with e.g. Matlab, Mathematica or Python would be desirable. 

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Title  Coherent state path integrals in manybody physics  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Alexander Ossipov  
Description  Path integrals were originally introduced by Richard Feynman as an alternative formulation of a single particle quantum mechanics [1]. The advantage of this approach is that it deals with the classical action calculated on all possible quantum trajectories instead of noncommutative operators.
The coherent state path integrals generalise this idea and can be applied to more complicated Hamiltonians such as manybody interacting systems [2]. Despite a wide range of successful applications of the coherent state path integrals in many areas of modern theoretical physics, usually they are calculated only perturbatively.
The aim of this project is to develop a new nonperturbative approach for evaluation of the coherent state path integrals applied to manybody systems such as the BoseHubbard model. 

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Title  Quantum Resource Theories  

Group(s)  Mathematical Physics  
Proposer(s)  Prof Gerardo Adesso  
Description  The emergence of quantum information theory in the last three decades has led to a crucial reassessment of quantum effects such as superposition and entanglement: from poorly understood and even paradoxical concepts, these are now regarded as fundamental ingredients to achieve tasks otherwise impossible within the realm of classical physics, thus enabling a wealth of innovative technologies. At the core of this revolution lies the formalisation and characterisation of such phenomena as physical resources. Initiated with quantum entanglement, and successfully applied e.g. to purity, coherence [1], and informational nonequilibrium in thermodynamics, this applicationdriven viewpoint motivates the formulation of resource theories, that is, quantitative theories capturing the resource character of physical traits in a mathematically rigorous fashion. In very general terms, resource theories can be thought of as a radically new way of doing science, that takes the perspective of restricted agents who want to optimize given tasks of practical relevance. This project aims to advance the current frontiers of knowledge on resource theories for quantum phenomena and physics more broadly. In particular, possible directions include: i) Develop the mathematical foundations of resource theories, including an operational formalism for multiple, possibly competing resources and their interconnections, and a general framework to study the relations between different subjective agents; ii) Further apply the concept of resource theories to physical quantities such as nonclassicality, coherence, uncertainty, etc., providing rigorous methods to validate existing experimental approaches to quantum optics and information, and delivering new tools to quantify the tradeoff of different resources in operational tasks (e.g. distillation, activation, etc.); iii) Apply results from resource theories to draw conclusions on fundamental limitations of physical processes, e.g. to characterise the structure of thermodynamics from macro to nano and quantum scale, and the ultimate performance of quantum clocks and sensors in realistic conditions.
More details on resource theories are provided in the following. A resource theory is a framework to study the possible actions of agents given certain constraints, which may stem e.g. from fundamental physical laws such as energy conservation, or technical limitations in experimental settings. The agents’ constraints are specified by a set of operations which are considered free to implement. Any other operation comes at a cost, requiring the use of a resource. Similarly, the states of the system which can be prepared using only free operations are named free states, while any other is a resource state. Once this structure is in place, one can address a number of key questions, such as: Under which conditions can a state be converted into another by using only free operations? What makes a good quantifier of the resource content of a state? Which are the most resourceful states for practical applications given the physical constraints in place? The conventional paradigm of entanglement in quantum information theory is based on two or more agents in distant laboratories, so that local operations and classical communication (LOCC) are assumed to be free. Quantum entanglement is then the resource which allows agents to perform operations beyond LOCC, enabling tasks such as teleportation and dense coding. The theory of entanglement manipulation, quantification, and operational interpretation is precisely the resource theory defined by LOCC as the free operations. It is a remarkable fact that the developments in this resource theory have been pivotal for the quantum technology revolution we are witnessing in this century, leading e.g. to commercial quantum communication devices. Similarly, successful applications of resource theories to informational nonequilibrium, asymmetry, and coherence (among all) have led in the last decade to fundamental insights, such as revisions to the second law of thermodynamics for smallscale systems and an extension of Noether’s theorem, and hold a tremendous – and still largely unexplored – potential impact on further technological advances.
This project is motivated by questions such as:  Can we describe all useful signatures of nonclassicality from first principles, compiling a family tree of quantum resources and their functional interrelations?  Are there universal laws underpinning the conversion between different resources and entailing an operational framework to quantify them, irrespective of specific nature?  How best can quantum devices operate under competing constraints, such as energy conservation, locality, and the inability to create superposition in a reference basis?  What does thermodynamics allow or deny for physical machines at the quantum scale, once coherence effects and the subjectivity of agents are incorporated into the theory?  Can we derive practical bounds on the performance of quantum technologies, i.e. metrological precision and secure key rate, tailored to experimental costs and imperfections?


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Title  Models of Quantum Geometry  

Group(s)  Mathematical Physics  
Proposer(s)  Prof John Barrett  
Description  Noncommutative geometry is a generalisation of differential geometry where the "functions" on the space are not required to commute when multiplied together. This study is based on the approach to noncommutative geometry pioneered by Alain Connes. It has a number of applications, the most spectacular being the discovery that the fields in the standard model of particle physics have the structure of a noncommutative geometry. This noncommutativity relates to the "internal space" i.e. a geometric structure at every point of spacetime, and reveals itself in the nonabelian gauge groups, the Higgs and their couplings to fermion fields. The new idea is to use the noncommutative geometry also for spacetime itself, which one hopes will eventually give a coherent explanation of the structure of spacetime at the Planck scale. There are a number of projects investigating aspects of these quantum geometry models and related mathematics. It also uses techniques from topology, algebra, category theory and geometry, as well as numerical computations. The motivation is to study models that include gravity, working towards solving the problem of quantum gravity, and to study implications for particle physics. For the latest information on this research, please see my homepage https://johnwbarrett.wordpress.com/ 

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Title  Hydrodynamic simulations of rotating black holes  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Silke Weinfurtner  
Description  We are currently carrying out an experiment to study the effects occurring around effective horizons in an analogue gravity system. In particular, the scientific goals are to explore superradiant scattering and the black hole evaporation process. To address this issue experimentally, we utilize the analogy between waves on the surface of a stationary draining fluid/superfluid flows and the behavior of classical and quantum field excitations in the vicinity of rotating black. This project will be based at the University of Nottingham at the School of Mathematical Sciences. The two external collaborators are Prof. Josef Niemela (ICTP, Trieste in Italy) and Prof. Stefano Liberati (SISSA, Trieste in Italy). The external consultant for the experiment is Prof. Bill Unruh, who will be a regular visitor. The PhD student will be involved in all aspects of the experiments theoretical as well experimental. We require an enthusiastic graduate with a 1st class degree in Mathematics/Physics/Engineering (in exceptional circumstances a 2(i) class degree can be considered), preferably of the MMath/MSc level. Candidates would need to be keen to work in an interdisciplinary environment and interested in learning about quantum field theory in curved spacetimes, fluid dynamics, analogue gravity, and experimental techniques such as flow visualisation (i.g. Particle Imaging or Laser Doppler Velocimetry) and surface measurements (i.g. profilometry methods). 

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Title  Acceleration, black holes and thermality in quantum field theory  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Jorma Louko  
Description  Hawking's 1974 prediction of black hole radiation continues to inspire the search for novel quantum phenomena associated with global properties of spacetime and with motion of observers in spacetime, as well as the search for laboratory systems that exhibit similar phenomena. At a fundamental level, a study of these phenomena provides guidance for developing theories of the quantum mechanical structure of spacetime, including the puzzle of the microphysical origin of black hole entropy. At a more practical level, a theoretical control of the phenomena may have applications in quantum information processing in situations where gravity and relative motion are significant, such as quantum communication via satellites. Specific areas for a PhD project may include: Model particle detectors as a tool for probing nonstationary quantum phenomena in spacetime, such as the onset of Hawking radiation during gravitational collapse. See arXiv:1406.2574, arXiv:1605.01316 and arXiv:1610.08455, and the references therein. Black hole structure behind the horizons as revealed by quantum field observations outside the horizons. See arXiv:1001.0124 and references therein. Quantum fields in accelerated cavities. See arXiv:1210.6772 and arXiv:1411.2948 and references therein. Theory supporting analogue spacetime laboratory experiments for detecting Hawking radiation and related effectsl. See arXiv:1807.04584 and references therein. 

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Title  Quantum learning for large dimensional quantum systems  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Madalin Guta  
Description  Statistical inference and learning play an increasing role in Quantum Engineering and Quantum Metrology. The efficient statistical reconstruction of quantum states is a crucial enabling tool for current quantum engineering experiments in which multiple qubits can be prepared in exotic states. However, standard estimation methods such as maximum likelihood become practically unfeasible for systems of merely 10 qubits, due to the exponential growth in size of the Hilbert space. The aim of this project is to develop mathematical theory and investigate new methods for learning quantum states of large dimensional quantum systems. This stems from ongoing collaborations with Theo Kypraios and Ian Dryden (Statistics group, Nottingham), Cristina Butucea (Univesite Paris Est), Michael Nussbaum (Cornell), Jonas Kahn (Toulouse) and Richard Kueng (Caltech). In [1,2] we proposed and analysed faster estimation methods with close to optimal accuracy. The first goal is to better understand the behaviour of the estimators with respect to different measurement scenarios. Next, we would like to equip them with reliable confidence regions (error bars) which are crucial for experimental applications. Going beyond "full state tomography" new methods are needed which are able to "learn" the structure of the quantum state by making use of prior information encoded in physically relevant low dimensioanal models. Possible directions to be explored include models based on matrix product states, neural networks, quantum time series, compressed sensing [3] and the study of the asymptotical structure of the statistical models [4]. The project will involve both theoretical and computational work at the overlap between quantum information theory and modern statistical inference. 

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Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Feedback control of quantum dynamical systems and applications in metrology  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Madalin Guta  
Description 
The ability to manipulate, control and measure quantum systems is a central issue in Quantum Technology applications such as quantum computation, cryptography, and high precision metrology [1]. Most realistic systems interact with an environment and it is important to understand how this affects the performance of quantum protocols and how it can be used to improve it. The inputoutput theory of quantum open systems [2] offers a clear conceptual understanding of quantum dynamical systems and continuoustime measurements, and has been used extensively at interpreting experimental data in quantum optics. Mathematically, we deal with an extension of the classical filtering theory used in control engineering at estimating an unobservable signal of interest from some available noisy data [3]. This projects aims at investigating the identification and control of quantum dynamical systems in the framework of the inputoutput formalism. As an example, consider a quantum system (atom) interacting with an incoming "quantum noise" (electromagnetic field); the output fields (emitted photons) emerging from the interaction can be measured, in order to learn about the system's dynamical parameters (e.g. its hamiltonian). The goal is to find optimal system identification strategies which may involve input state preparation, output measurement design, and quantum feedback control. An interesting related question is to understand the informationdisturbance tradeoff which in the context of quantum dynamical systems becomes identificationcontrol tradeoff. The first steps in this direction were made in [4] which introduce the concept of asymptotic quantum Fisher information for "nonlinear" quantum Markov processes, and [5] which investigates system identification for linear quantum systems, using transfer functions techniques from control theory. A furhter goal is to develop genearal Central Limit theory for quantum output processes as a probablistic underpinning of the asymptotic estimation theory. Another direction is the recently found connection between dynamical phase transitions in manybody open systems and high precision metrology for dynamical parameters (see arXiv:1411.3914).


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Other information  Click here to find more information on this topic and some illustrations of different types of estimators. For more about my reasearch interests you can visit my homepage. 
Title  Using artificial intelligence to design quantum optics experiments  

Group(s)  Mathematical Physics  
Proposer(s) 
Prof Gerardo Adesso Notice: Undefined index: pmzpk in /maths/www/html/postgraduate/projects/index.php on line 560 Notice: Undefined index: pmzpk in /maths/www/html/postgraduate/projects/index.php on line 560 , 

Description  In order for technologies to benefit from the inherent power of quantum mechanics, quantum states with specific properties must be engineered for particular applications. This project explores the use of an evolutionary algorithm – a subset of artificial intelligence which mimics natural selection – that has been previously developed to design quantum states for performing quantumenhanced measurements [1]. PhD students will have the opportunity to extend this work in one or more of the following veins: i) Extend this technique to design quantum states and quantum experiments for a wide range of applications, including quantum computing, quantum cryptography, high precision measurements, and tests of fundamental physics. This strand will involve developing an understanding of the structure of quantum states – and the state space they live in – in order to design new quantum states with novel properties. ii) Incorporate realistic experimental imperfections, such as photon loss and imperfect detectors, into the evolutionary algorithm. This will enable the algorithm to find quantum states that can then be made in the laboratory, including in our collaborators’ labs in Bristol, Oxford, Stuttgart (Germany), and MIT (Boston, USA). This strand of the project will involve working closely with experimentalists to enhance and optimise their current experiments, and to design new experiments that they will subsequently perform. iii) As the algorithms used in i) and ii) become more complicated, more advanced artificial intelligence methods will need to be utilised and developed in order to make the simulations tractable. The use of machine learning (in particular reinforcement learning) and genetic algorithms will be explored, and the student will have the opportunity to collaborate with artificial intelligence researchers. 

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Title  Developing new relativistic quantum technologies  

Group(s)  Mathematical Physics  
Proposer(s)  Prof Ivette Fuentes  
Description  Relativistic quantum information is an emerging field which studies how to process information using quantum systems taking into account the relativistic nature of spacetime. The main aim of this PhD project is to find ways to exploit relativity to improve quantum information tasks such as teleportation and to develop new relativistic quantum technologies. Moving cavities and UnruhDewitt type detectors promise to be suitable systems for quantum information processing [1,2]. Interestingly, motion and gravity have observable effects on the quantum properties of these systems [2,3]. In this project we will find ways to implement quantum information protocols using localized systems such as cavities and detectors. We will focus on understanding how the protocols are affected by taking into account the nontrivial structure of spacetime. We will look for new protocols which exploit not only quantum but also relativistic resources for example, the nonlocal quantum correlations present in relativistic quantum fields. 

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Title  Homotopical algebra and quantum gauge theories  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Alexander Schenkel  
Description  A problem which frequently arises in mathematics is that one would like to treat certain classes of maps as if they were isomorphisms, even though they are not in the strict sense. Examples are homotopy equivalences between topological spaces  remember the famous doughnut and coffee mug  or quasiisomorphisms between chain complexes of modules. Homotopical algebra was introduced by Quillen in the late 1960s as an abstract framework to address these and related problems. Since then it has found many important applications in algebra, topology, geometry and also in mathematical physics. In quantum field theory, homotopical algebra turns out to be essential as soon as one deals with models involving gauge symmetries. Recent results showed that quantum gauge theories do not satisfy the standard axioms of algebraic quantum field theory (AQFT), hence they are not quantum field theories in this strict sense. To solve these problems, we initiated the development of a novel and promising approach called “homotopical algebraic quantum field theory”, which combines the basic concepts of AQFT with homotopical algebra and which is expected to be a suitable mathematical framework for quantum gauge theories. Specific problems for a PhD project include modelindependent developments in hoAQFT or explicit contructions of examples of quantum gauge theories within this framework. 

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Title  Quantum optomechanics: radiation pressure at the single photon level  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Tommaso Tufarelli  
Description  Optomechanics investigates the interaction of quantised light (photons) with microscopic vibrating objects such as mirrors, dielectric membranes or levitated nanoparticles [1]. Such interaction takes place via radiation pressure, a phenomenon initially predicted by Johannes Kepler in 1619 in the context of astronomy, and nowadays observed even at the singlephoton level. Among other applications, optomechanics embodies a promising experimental platform to probe quantum effects in massive objects, and hence investigate the classical/quantum boundary. The student will initially review a widely used effective Hamiltonian for cavity optomechanics (the "linear model") [2], which is analytically solvable and predicts the generation of nonclassical states of light as well as lightmatter entanglement. Subsequently, he/she will delve into the more rigorous canonical quantization of an optomechanical system [3], and assess the crucial limitations of the more basic model. Unfortunately, the more rigorous "mircoscopic" Hamiltonian is currently intractable both analytically and numerically, so that novel approximation techniques will need to be developed to improve the basic model while retaining computability. An example of a preliminary study in this direction can be found in Ref. [4]. The broad objective of the project will be to develop new optomechanical models that strike an optimal balance between reliability and tractability. These will then be used to verify and refine a number of theoretical predictions that have been made in the literature, tipically based on the linear model alone. Such predictions pertain a variety of applications of optomechanical systems, ranging from quantum information science to gravitational wave detection and Planckscale physics [5]. At the same time, an improved theoretical description will give us an opportunity to explore new physical effects and applications of these systems. A futher ambitious goal of the project will be to develop a rigorous open quantum system model for optomechanics, improving the phenomenological approaches that are currently used in the literature. Depending on the inclinations of the student, more emphasis can be put on either analytical or numerical work (e.g. via Matlab, Python or Mathematica). 

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Title  Manybody localization in quantum spin chains and Anderson localization  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Alexander Ossipov  
Description  Properties of wave functions in manybody systems is very active topic of research in modern condensed matter theory. Quantum spin chains are very useful models for studying quantum manybody physics. They are known to exhibit complex physical behaviour such as quantum phase transitions. Recently, they have been studied intensively in the context of manybody localization. 

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Title  Entanglement of noninteracting fermions at criticality  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Alexander Ossipov  
Description  Entanglement of the ground state of manyparticle systems has recently attracted a lot of attention. For noninteracting fermions, the ground state entanglement can be calculated from the eigenvalues of the correlation matrix of the single particle wavefunctions. For this reason, the nature of the single particle wavefunctions is crucially important for understanding of the entanglement properties of a manybody system. 

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Title  Gravity and differential forms  

Group(s)  Mathematical Physics  
Proposer(s)  Prof Kirill Krasnov  
Description  It turns out that solutions of 4D gravity can be lifted to solutions of a certain theory of differential 3forms in 7 dimensions. This suggests a new perspective on 4D gravity and its problems, such as nonrenormalisability. The aim of this project is to further study the geometry in 7 dimensions, in its relation to 4D geometry. Another aim is to understand the quantum properties of diffeomorphism invariant theories of differential forms in dimensions 7 and 6, and in particular understand how such theories renormalise at one and possibly two loops. This project lies at the intersection of differential geometry and quantum field theory. 

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Title  Gravity at all scales  

Group(s)  Mathematical Physics  
Proposer(s)  Prof Thomas Sotiriou  
Description  Various projects are available on the interplay between any of the following areas: quantum gravity, alternative theories of gravity, strong gravity and black holes. 

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Title  Quantum control of nonlinear interactions  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Tommaso Tufarelli  
Description  Photons do not directly interact with each other, but effective interactions between them can be obtained by exploiting the mediation of matter. Perhaps the most common systems in which these effective interactions are achieved are nonlinear crystals. The sophistication of modern experiments, however, allows us to consider the mediation of quantum systems such as single atoms and optomechanical devices, where light is coupled to the vibrational degrees of freedom of a mesoscopic mirror via radiation pressure.


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Other information  This project will involve close collaboration with Dr. Florian Mintert of the Controlled Quantum Dynamics theory group, Imperial College London. 
Title  Scattering approach to topological insulators and superconductors  

Group(s)  Mathematical Physics  
Proposer(s)  Dr Alexander Ossipov  
Description  Topological insulators and superconductors are one of the hottest topics in the modern condensed matter theory. They represent a new phase of matter characterised by a rather unusual quantummechanical behaviour of electrons. Topological insulators are insulating materials, but they conduct electricity via special surface states. The surface states are topologically protected due to quantum symmetries and cannot be destroyed by disorder. Nevertheless the role of disorder is essential for understanding the properties of topological insulators.
A novel approach to disordered quantum system based on the scattering formalism has been proposed recently [1]. The aim of the project is to generalise and apply this approach to topological insulators and superconductors. 

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Title  Topological Resonances on Graphs  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  If a light wave in a resonator between two almost perfect mirrors shows resonance if the wavelength is commensurate with the distance between the two mirrors. If this condition is satisfied it will decay much slower than at other wavelengths which are not commensurate. This is one of the simplest mechanisms for a resonace in a wave system. There are other weill known mechanisms that rely on complexity and disorder. It has recently been observed that a netork of wire may have a further mechanism that leads to resonances. This mechanism relies on cycles in the network and leads to various signatures which cannot be explained using other wellknown mechanisms for resonances. In this project these signatures will be analysed in detail. 

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Title  Quantum Chaos in Combinatorial Graphs  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  Graphs consist of V vertices connected by B bonds (or edges). They are used in many branches of science as simple models for complex structures. In mathematics and physics one is strongly interested in the eigenvalues of the V x V connectivity matrix C of a graph. The matrix element C_ij of the latter is defined to be the number of bonds that connect the i'th vertex to the j'th vertex. In this PhD project the statistical properties of the connectivity spectra in (generally large) graph structures will be analysed using methods known from quantum chaos. These methods have only recently been extended to combinatorial graphs (Smilansky, 2007) and allow to represent the density of states and similar spectral functions of a graph as a sum over periodic orbits. The same methods have been applied successfully to metric graphs and quantum systems in the semiclassical regime for more than two decades. 

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Title  Quantum Searching in Random Networks  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Prof Gregor Tanner, Dr Sven Gnutzmann  
Description  The project deals with properties of quantum networks, that is, of networks on which unitary (wave) evolution takes place along edges with scattering at the vertices. Such systems have been studied in the context of quantum information as well as in quantum chaos. It has been noted that a quadratic speed up of quantum random walks on these networks over Recently, it has been shown, that quantum searching can also been undertaken on random graphs, that is, graphs for which connections between edges are given only wth a certain probability  so called ErdösRényi graphs. We will explore this new setup for quantum searching and develop statisticsal models for the arrival times and success probabilities as well extend the model to realistic graph setups. 

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Title  Supersymmetric field theories on quantum graphs and their application to quantum chaos  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  Quantum graphs are a paradigm model for quantum chaos. They consist of a system of wires along which waves can propagate. Many properties of the excitation spectrum and the spatial distribution of standing waves can be mapped exactly onto a supersymmetric field theory on the network. In a meanfield approximation one may derive various universal properties for large quantum graphs. In this project we will focus on deviations from universal behaviour for finite quantum graphs with the fieldtheoretic approach. 

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Title  Pseudoorbit expansions in quantum graphs and their application  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  Quantum graphs are a paradigm model to understand and analyse the effect of complexity on wave propagation and excitations in a network of wires. They have also been used as a paradigm model to understand topics in quantum and wave chaos where the complexity has a different origin while the mathematical framework is to a large extent analogous. Many properties of the waves that propagate through such a network can be described in terms of trajectories of a point particle that propagates through the network. The ideas is to write a property of interest as a sum over amplitudes (complex numbers) connected to all possible trajectories of the point particle. These sums remain challenging objects for explicit evaluations. Recently a numer of advanced methods for their summation have been introduced. The latter are built on socalled pseudoorbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and randommatrix theory. 

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Title  The tenfold way of symmetries in quantum mechanics. An approach using coupled spin operators.  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  About 50 years ago Wigner and Dyson proposed a threefold symmetry classification for quantum mechanical systems  these symmetry classes consisted of timereversal invariant systems with integer spin which can be described by real symmetric matrices, timereversal invariant systems with halfinteger spin which can be described by real quaternion matrices, and systems without any timereversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of randommatrix theory: the Gaussian orthogonal ensemble GOE, the Gaussian symplectic ensemble GSE, and the Gaussian unitary ensemble GUE. In the 1990's this classification was extended by adding charge conjugation symmetries  symmetries which relate the positive and negative part of a spectrum and which are described by anticommutators. 

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Title  Nonlinear waves in waveguide networks  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  Many wave guides (such as optical fibres) show a Kerrtype effect that leads to nonlinear wave propagation. If th wave guides are coupled at junctions then there is an additional element of complexity due to the nontrivial connectivity of wave guides. In this project the impact of the structure and topology of the network on wave propagation will be studied starting from simple geometries such as a Yjunctions (three waveguides coupled at one junction), a star (many waveguides at one junction), or a lasso (a waveguide that forms a loop and is connected at one point to a second waveguide). 

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Title  The statistics of nodal sets in wavefunctions  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  If a membrane vibrates at one of its resonance frequencies there are certain parts of the membrane that remain still. These are called nodal points and the collection of nodal points forms the nodal set. Building on earlier work this project will look at the statistical properties of the nodal set  e.g. for 3dimensional waves the nodal set consists of a coillection of surfaces and one may ask questions about how the area of the nodal set is distributed for an ensemble of membranes or for an ensemble of different resonances of the same membrane. This project will involve a strong numerical component as wavefunctions of irregular membranes need to be found and analysed on the computer. Effective algorithms to find the area of the nodal set, or the number of domain in which the sign does not change (nodal domains) will need to be developed andimplemented. 

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Title  Coherent states, nonhermitian Quantum Mechanics and PTsymmetry  

Group(s)  Mathematical Physics, Industrial and Applied Mathematics  
Proposer(s)  Dr Sven Gnutzmann  
Description  Heisenberg's uncertainty principle states that momentum and position cannot be sharp at the same time because there is a lower bound for the product of the uncertaincies. Coherent states can be defined as the states that minimize the uncertainty  in this sense they are as close as quantum mechanics allows to describe a classical point particle. When a quantum system starts in a coherent states it's expectation values follow the classical equations of motion while the shape of the wave function often changes only very slowly. Coherent states are an important tool to understand the corresp[ondence between quantum and classical dynamics. In this project this correspondence will be analysed for a generalized quantum dynamics where the Hamilton operator is not required to be Hermitian. Such dynamics can arise in practice as an effective description for an open quantum system with eitehr decay or gain. Accordingly the energy eigenvalues may have an imaginary part that describes the loss or gain. Recently there have also be suggestions that nonhermitian Hamilton operators could play a fundamental role in quantum mechanics if the Hamilton operator remains symmetric with respect to a combined operatyion of parity P and time reversal T. Such PTsymmetric dynamics have a balance between gain and loss which can lead to real energy eigenvalues. Classical to quantum correspondence for such systems remains an open research topic and this project will aim at getting a clear understanding of the underlying classical dynamics using coherent states as the main tool. 

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