You may contact a Proposer directly about a specific project or contact the Postgraduate Admissions Secretary with general enquiries.

Title Classical and quantum Chaos in 3-body Coulomb problems
Group(s) Industrial and Applied Mathematics, Mathematical Physics
Proposer(s) Prof Gregor Tanner
Description

The realisation that the dynamics of 2 particles interacting via central forces is fundamentally different from the dynamics of three particles can be seen as the birth of modern dynamical system theory. The motion of two particles (for example the earth-moon problem neglecting the sun and other planets) is regular and thus easy to predict. This is not the case for three or more particles (especially if the forces between all these particles are of comparable size) and the resulting dynamics is in general chaotic, a fact first spelt out be Poincaré at the end of the 19th century. An important source for chaos in the three-body problem is the possibility of triple collisions, that is, events where all three particles collide simultaneously. Triple collisions form essential singularities in the equation of motions, that is, trajectories can not be smoothly continued through triple collision events. This is related to the fact, that the dynamics at the triple collision point itself takes place on a collision manifold of non-trivial topology.

During the project, the student will be introduced to scaling techniques which allow to study the dynamics at the triple collision point. We will in particular consider three-body Coulomb problems, such as two-electron atoms, and study the influence of the triple-collision on the total dynamics of the problem. As a long term goal, we will try to uncover the origin of approximate invariants of the dynamics whose existence is predicted by experimental and numerical quantum spectra of two-electron atoms such as the helium atom.

Relevant Publications
  • The semiclassical helium atom, G. Tanner and K Richter, www.scholarpedia.org/article/Semiclassical_theory_of_helium_atom
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Title Electromagnetic compatibility in complex environments: predicting the propagation of electromagnetic waves using wave-chaos 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 EM-field 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 ray-dynamics in phase space using the so-called Wigner distribution function (WDF) formalism.  It allows us to replace the wave propagation problem with one of propagating classical densities within phase space. 

 

Relevant Publications
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Title Wave propagation in complex built-up structures – tackling quasi-periodicity 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 quasi-periodic 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 high-frequency wave methods – such as the so-called Dynamical Energy Analysis developed in Nottingham - making it possible to compute the vibrational response of structures with arbitrary complexity at large frequencies.



 

Relevant Publications
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Title Critical random matrix ensembles
Group(s) Mathematical Physics
Proposer(s) Dr Alexander Ossipov
Description

In Random Matrix Theory (RMT) one deals with matrices whose entries are given by random variables. RMT has a great number of applications in physics, mathematics, engineering, finance etc. In this project, a particular class of random matrix ensembles --- critical random matrix models will be studied. These models describe statistical properties of disordered systems at a point of the quantum phase transition. Using RMT one can compute various critical exponents, correlation functions and other physically important quantities.


In the language of RMT, the criticality implies very special properties of eigenvalues and eigenvectors of random matrices. One can show, for example, that eigenvectors in critical models are fractal. For certain models fractality of eigenvectors can be investigated with the help of rather simple random matrices, such as two by two matrices in the simplest case. In other cases, much more sophisticated methods, such as supersymmetry, should be employed.  Some steps in this direction have been taken recently, but very few general results are available at the moment. The aim of the project is to close this gap.

Relevant Publications
  • Y. V. Fyodorov, A. Ossipov, and A. Rodriguez, Anderson localization transition and eigenfunction multifractality in ensemble of ultrametric random matrices, J. Stat. Mech., L12001 (2009).
  • V.E. Kravtsov, A. Ossipov, O.M. Yevtushenko, E. Cuevas, Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?, Phys. Rev. B 82, 161102(R) (2010).
  • A. Ossipov, I. Rushkin, and E. Cuevas, Level number variance and spectral compressibility in a critical two-dimensional random matrix model, Phys. Rev. E 85, 021127 (2012).
  • A. Ossipov, Virial expansion of the non-linear sigma model in the strong coupling limit, J. Phys. A: Math. Theor. 45, 335002 (2012).
  • A. Ossipov, Anderson localization on a simplex, http://arxiv.org/abs/1211.2643
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Title Models of Quantum Geometry
Group(s) Mathematical Physics
Proposer(s) Prof John Barrett
Description

Non-commutative 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 non-commutative 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 non-commutative geometry. This non-commutativity relates to the "internal space" i.e. a geometric structure at every point of space-time, and reveals itself in the non-abelian gauge groups, the Higgs and their couplings to fermion fields.

The new idea is to use the non-commutative geometry also for space-time itself, which one hopes will eventually give a coherent explanation of the structure of space-time 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/

Relevant Publications
  • Jean Petitot: Noncommutative geometry and physics. https://arxiv.org/abs/1505.00132
  • John W. Barrett, Lisa Glaser: Monte Carlo simulations of random non-commutative geometries. https://arxiv.org/abs/1510.01377
  • John W. Barrett: Matrix geometries and fuzzy spaces as finite spectral triples. https://arxiv.org/abs/1502.05383
  • Walter D. van Suijlekom: Noncommutative Geometry and Particle Physics. http://www.springer.com/gb/book/9789401791618
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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). 

Relevant Publications
  • Carlos Barceló and Stefano Liberati and Matt Visser, "Analogue Gravity", Living Rev. Relativity 14, (2011), 3. URL: http://www.livingreviews.org/lrr-2011-3
  • W. G. Unruh, “Experimental Black-Hole Evaporation?” Phys. Rev. Lett. 46, 1351 – Published 25 May 1981.
  • Silke Weinfurtner, Edmund W. Tedford, Matthew C. J. Penrice, William G. Unruh, and Gregory A. Lawrence, “Measurement of Stimulated Hawking Emission in an Analogue System”, Phys. Rev. Lett. 106, 021302 – Published 10 January 2011
  • Mauricio Richartz, Silke Weinfurtner, A. J. Penner, W. G. Unruh, “Generalised superradiant scattering”, Phys. Rev. D80:124016,2009
  • Mauricio Richartz, Angus Prain, Stefano Liberati, Silke Weinfurtner, "Rotating black holes in a draining bathtub: superradiant scattering of gravity waves ", arXiv:1411.1662 [gr-qc]
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Title Gravity as a theory of connections
Group(s) Mathematical Physics
Proposer(s) Prof Kirill Krasnov
Description

General Relativity is normally described as a dynamical theory of spacetime metrics. However, GR is a rather complicated theory - think about the rather non-trivial exercise of deriving Schwarzschild solution, with its computation of Christoffel symbols, then the curvature tensor, then Ricci tensor. At the same time, it has been appreciated for a long time that one can simplify the GR formalism by using differential forms. Indeed, the exercise leading to Schwarzschild solution does become simpler if one uses tetrads and the spin connection instead of the metric and the affine connection. Also in 3 spacetime dimensions General Relativity is best thought of as a theory of flat Poincare connections, with the action describing the dynamics being that of Chern-Simons theory. A point of view on 3D gravity as a theory of connections has been extremely successful both classically (in describing the space of all possible solutions of 3D GR) and quantum mechanically (in quantising the space of solutions and obtaining an explicit description of the arising Hilbert space). 

This PhD project will concern itself with developing a similar language for 4D GR. Thus, it turns out to be possible to describe 4D GR as a dynamical theory of connections rather than metrics. The metric appears as a derived notion, and is constructed in a certain way from the curvature of the connection. There are many possible projects within this general area of development. One can either explore how some concrete solutions of GR are obtained in this way, or study the quantum mechanics of gravity (i.e. perturbative quantum gravity) in this language. The language of connections is simpler in many aspects than the usual metric formalism for GR, and the hope is that this simplicity will lead to qualitatively new understanding of what gravity really is.

Relevant Publications
  • http://arXiv.org/abs/arXiv:1202.6183
  • http://arxiv.org/abs/1312.2831
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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 could 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 and 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.

Relevant Publications
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Title Scattering in disordered systems with absorption: beyond the universality
Group(s) Mathematical Physics
Proposer(s) Dr Alexander Ossipov
Description

The study of wave scattering in quantum systems with disorder or underlying classical chaotic dynamics is essential for an understanding of many different physical systems. These include, for example, light propagation in random media, transport of electrons in quantum dots, transmission of microwaves in waveguides and cavities, and many others.

An important feature of any real experiment on scattering is the presence of absorption. As the result, not all the incoming flux is either reflected or transmitted through system, but part of it is irreversibly lost in the environment.

In recent years, considerable progress has been made in the study of scattering in disordered or chaotic quantum systems in the presence of absorption, see e.g Fyodorov, Savin & Sommers, (2005). However almost all results known so far are restricted by the so called "universal limit" described by the conventional Random Matrix Theory. The idea of the suggested project is to go beyond the "universal limit" and to investigate properties of the scattering matrix in lossy systems for the case of a quasi-one-dimensional disordered waveguide. This model describes for example electron dynamics in a thick disordered wire or propagation of light or microwave radiation in a slab geometry. There are two recent advances making an analytical treatment of this problem feasible. The first one is a discovery of a kind of fluctuation dissipation relation between the properties of an open system in the presence of absorption and a certain correlation function of its closed counterpart. This can be exploited, for example, to relate statistics of scattering characteristics to eigenfunction fluctuations in closed systems (Ossipov & Fyodorov, 2005). The second one is a new analytical insight into properties of quasi-one-dimensional disordered conductors, see Skvortsov & Ostrovsky, (2006).

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Title Quantum learning for large dimensional quantum systems
Group(s) Mathematical Physics
Proposer(s) Dr Madalin Guta
Description

This project stems from the ongoing collaboration with Theo Kypraios and Ian Dryden (Statistics group, Nottingham), Cristina Butucea (Univesite Paris Est) and Thomas Monz and Philipp Schindler (Rainer Blatt trapped ions experimental group, University of Innsbruck). The aim is to explore and investigate new methods for learning quantum states of large dimensional quantum systems. The efficient statistical reconstruction of such states is a crucial enabling tool for current quantum engineering experiments in which multiple qubits can be controlled and prepared in exotic entangled states. However, standard estimation methods such as maximum likelihood become practically unfeasible for systems of merely 10 qubits, due to the exponential growth of the Hilbert space with the number of qubits. Therefore 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.

In [1] we investigated the use of model selection methods for state estimation, in particular the Akaike information criterion and  the Bayesian information criterion.  The general principle is to find the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity, the latter being given by the rank of the density matrix. Another rank selection technique was considered in [2] and its performance under compressed sensing [3,4] measurements is currently analysed from a statistical viewpoint.

The goal of the project is to compare the efficiency of the different methods, and explore new, possibly hybrid estimators which are both accurate and computationally efficient. Possible directions to be explored include models bases on matrix product states, optimal design of experiments, quantum pattern recognition, asymptotical structure of the statistical models. The project will involve both theoretical and computational work at the overlap between quantum information theory and modern statistical inference.

Relevant Publications
  • [1] M. Guta, T. Kypraios and I. Dryden, Rank based model selection for multiple ions quantum tomography New Journal of Physics, 14, 105002 (2012), Arxiv: 1206.4032
  • [2] C. Butucea, M. Guta, T. Kypraios, Spectral thresholding quantum tomography for low rank states, New Journal of Physics, 17 113050 (2015), ArXiv:1504.08295
  • [3] D. Gross, Y. K. Liu, S. Flammia, S. Becker and J. Eisert, Physical Review Letters 105 150401 (2010) Arxiv:0909.3304
  • [4] S. Flammia, D. Gross, Y.K. Liu and J. Eisert Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators New Journal of Physics 14, 095022 (2012) ArXiv:1205.2300
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 input-output theory of quantum open systems [2] offers a clear conceptual understanding of quantum dynamical systems and continuous-time 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 input-output 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 information-disturbance trade-off which in the context of quantum dynamical systems becomes identification-control trade-off.

The first steps in this direction were made in [4] which introduce the concept of asymptotic quantum Fisher information for "non-linear" 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 many-body open systems and high precision metrology for dynamical parameters (see arXiv:1411.3914).

 

 

Relevant Publications
  • [1] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Universtiy Press (2000)
  • [2] C. Gardiner, P. Zoller, Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics, Springer (2004)
  • [3] K. Zhou, J.C. Doyle and K. Glover, Robust and Optimal Control, Prentice Hall, (1995)
  • [4] M. Guta, Quantum information Fisher information and asymptotic normality in system identification for quantum Markov chains, Physical Review A, 83, 062324 (2011) Arxiv:1007.0434; M. Guta, J. Kiukas, Equivalence classes and local asymptotic normality in system identification for quantum Markov chains, Commun. Math. Phys. 335, 1397–1428 (2014)
  • [5] M. Guta and N. Yamamoto, System identification for linear passive quantum systems Short version of the archive paper appeared in Proc. 52 IEEE CDC Conference Florence 2013 arXiv:1303.3771v2
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 Quantum correlations in many-body systems
Group(s) Mathematical Physics
Proposer(s) Prof Gerardo Adesso
Description

The behaviour of physical systems at the microscopic scale obeys the laws of quantum mechanics. Quantum systems can share a form of quantum correlations known as entanglement, which is nowadays acknowledged as a resource for enhanced information processing. However, there are more general types of quantum correlations, beyond entanglement, that can be present in separable quantum states.

This project deals with the characterisation of the nonclassicality of correlations in multipartite quantum systems. Interesting aspects of this project are the elucidation of the relationship between these more general forms of quantum correlations, as quantified e.g. by the "quantum discord", and entanglement in mixed multipartite quantum states. Another theme will be the identification of experimentally friendly schemes to engineer quantum correlations, and detect them in practical demonstrations, as well as rigorously assessing the usefulness of quantum correlations beyond entanglement as resources for next-generation quantum information protocols.

Relevant Publications
  • C. Rodo', G. Adesso, A. Sanpera; Operational Quantification of Continuous-Variable Correlations; Phys. Rev. Lett. 100, 110505 (2008)
  • G. Adesso, A. Datta; Quantum versus classical correlations in Gaussian states; Phys. Rev. Lett. 105, 030501 (2010)
  • D. Girolami, M. Paternostro, G. Adesso; Non-classicality indicators and extremal quantum correlations in two-qubit states; arXiv:1008.4136
Other information

http://www.maths.nottingham.ac.uk/personal/ga

Title Quantum aspects of frustration in spin lattices
Group(s) Mathematical Physics
Proposer(s) Prof Gerardo Adesso
Description

Recently, a number of tools developed in the framework of quantum information theory have proven useful to tackle founding open questions in condensed matter physics, such as the characterization of quantum phase transitions and the scaling of correlations at critical points.  Our contribution to the field dealt  with a method, based on quantum informational concepts, to identify analytically factorized (unentangled) ground states in many-body spin models, which constitute an exact solution to generally non-exactly solvable models for specific values of the Hamiltonian parameters. In presence of frustration, ground state factorization is suppressed. Therefore the factorizability provides a qualitative handle on the degree of quantum frustration.

This project will build on these premises and will seek for genuine signatures of quantum versus classical frustration in spin systems, a topic of great relevance for condensed matter. Frustrated quantum models may play a key role for high-temperature superconductivity and for certain biological processes. The relationship between frustration, disorder and entanglement is yet largely unexplored.

Relevant Publications
  • S. M. Giampaolo, G. Adesso, F. Illuminati; Theory of Ground State Factorization in Quantum Cooperative Systems; Phys. Rev. Lett. 100, 197201 (2008)
  • S. M. Giampaolo, G. Adesso, F. Illuminati; Separability and ground-state factorization in quantum spin systems; Phys. Rev. B 79, 224434 (2009)
  • S. M. Giampaolo, G. Adesso, F. Illuminati; Probing Quantum Frustrated Systems via Factorization of the Ground State; Phys. Rev. Lett. 104, 207202 (2010)
Other information

http://www.maths.nottingham.ac.uk/personal/ga

Title Quantum information with non-Gaussian states
Group(s) Mathematical Physics
Proposer(s) Prof Gerardo Adesso
Description

Quantum information with continuous variable systems is a burgeoning area of research which has recorded astonishing theoretical and experimental successes, mainly thanks to the manipulation and exploitation of Gaussian states of light and matter. However, quite recently a number of tasks have been individuated which can not be perfectly implemented by using Gaussian states and operations only, and another set of processes is being explored where some non-Gaussianity has been recognised as an advantageous ingredient to sharply improve performances of quantum communication.

In this project the student will investigate the limitations of the Gaussian scenario in different contexts such as quantum communication, computation and estimation and, more generally, quantum technology. This is paralleled by recent progresses in the experimental generation of non-Gaussian states, which further motivate their application in quantum information science. Special emphasis will be put on devising efficient methods to quantify the entanglement in selected classes of non-Gaussian states, using techniques whose complexity is not exceedingly large compared to the usual tools (quadrature measurements, homodyne detection) which are effective for Gaussian states.

Relevant Publications
  • G. Adesso, F. Illuminati; Entanglement in continuous-variable systems: recent advances and current perspectives; J. Phys. A 40, 7821 (2007)
  • G. Adesso; Experimentally friendly bounds on non-Gaussian entanglement from second moments; Phys. Rev. A 79, 022315 (2009)
  • G. Adesso, F. Dell'Anno, S. De Siena, F. Illuminati, L. A. M. Souza; Optimal estimation of losses at the ultimate quantum limit with non-Gaussian states; Phys. Rev. A 79, 040305(R) (2009)
Other information

http://www.maths.nottingham.ac.uk/personal/ga

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 Unruh-Dewitt 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 non-trivial structure of spacetime. We will look for new protocols which exploit not only quantum but also relativistic resources for example, the non-local quantum correlations present in relativistic quantum fields.

Relevant Publications
  • T. Downes, I. Fuentes, & T. C. Ralph Entangling moving cavities in non-inertial frames Physics Review Letters, 106, 210502 (2011)
  • D. E. Bruschi, I. Fuentes, & J. Louko Voyage to Alpha Centauri: Entanglement degradation of cavity modes due to motion accepted in Physical Review D Rapid Communications
  • N. Friis, D. E. Bruschi, J. Louko & I. Fuentes Motion generates entanglement Submitted to Physical Review D
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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 quasi-isomorphisms 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, 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 algebraic quantum field theory with homotopical algebra and which is expected to be a suitable mathematical framework for quantum gauge theories.

Specific areas for a PhD project could include:

1.) Examples of homotopical algebraic quantum field theory.

This project is about investigating the symplectic geometry of solution "spaces" of gauge theories, which are generalised spaces called stacks, and developing new techniques for their quantisation. An important part will be to analyse local-to-global properties (called descent) of the resulting quantum gauge theories.

2.) Operadic structure of homotopical algebraic quantum field theory.

The algebraic operations in homotopical algebraic quantum field theory are expected to be captured in an abstract structure called a coloured operad. This project is about constructing this coloured operad and using it to obtain model-independent results in homotopical algebraic quantum field theory.

Relevant Publications
  • M. Benini, A. Schenkel and R. J. Szabo, Homotopy colimits and global observables in Abelian gauge theory, Lett. Math. Phys. 105, no. 9, 1193 (2015) [arXiv:1503.08839 [math-ph]].
  • M. Benini and A. Schenkel, Quantum field theories on categories fibered in groupoids, arXiv:1610.06071 [math-ph].
  • W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, in: Handbook of Algebraic Topology, North-Holland, Amsterdam (1995).
  • S. Hollander, A homotopy theory for stacks, Israel J. Math. 163, 93 (2008) [arXiv:math.AT/0110247].
  • D. Yau, Colored Operads, AMS (2016).
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Title Many-body localization in quantum spin chains and Anderson localization
Group(s) Mathematical Physics
Proposer(s) Dr Alexander Ossipov
Description

Properties of wave functions in many-body systems is very active topic of research in modern condensed matter theory. Quantum spin chains are very useful models for studying quantum many-body physics. They are known to exhibit complex physical behaviour such as quantum phase transitions. Recently, they have been studied intensively in the context  of many-body localization.
 
The key idea of this project is to explore the similarity between Hamiltonians of the spin chain models and the Anderson models on d-dimensional hypercube. In such models, single particle wave functions can be localized in space due to the famous phenomenon of the Anderson localization.

Understanding of the relation between many-body localization and Anderson localization, quantum phase transitions in spin chains and the Anderson metal-insulator transition will be the main topic of this project.

Relevant Publications
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Title Entanglement of non-interacting fermions at criticality
Group(s) Mathematical Physics
Proposer(s) Dr Alexander Ossipov
Description

Entanglement of the ground state of many-particle systems has recently attracted a lot of attention. For non-interacting 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 many-body system.

The ground state entanglement is well understood now for free fermions, whose wavefuctions are simple plane waves. However, there are almost no analytical results available in the case where wavefuctions are non-trivial.

This project will explore the ground state entanglement at the quantum critical point of the metal-insulator transition, where single particle wavefunctions are known to have self-similar fluctuations, characterised by non-trivial fractal dimensions.

Relevant Publications
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Title Gravity at all scales
Group(s) Mathematical Physics
Proposer(s) Dr 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.

The description of phenomena for which gravity is important and also are in the realm of quantum physics requires a quantum gravity theory. Developing candidate theories to the extent that they can be confronted with observations is a very challenging task. In their classical limit these theories are mostly expected to deviate from General Relativity. In this sense, classical alternative gravity theories can be the interface between quantum gravity theory and classical phenomenology.

The gravitational interaction is much less explored in regimes where gravity is strong, such as in the vicinity of black holes or veer compact stars. These system can be thought of as natural laboratories for gravity.

The overall scope is to follow a synthetic approach which will combine results about the behavior of gravity at all different scales - from the quantum to astrophysical and cosmological system - in order to provide new insights.

See also http://thomassotiriou.wix.com/challenginggr for further info

Relevant Publications
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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 well-known mechanisms for resonances. In this project these signatures will be analysed in detail.

Relevant Publications
  • GNUTZMANN, S., SMILANSKY, U. and DEREVYANKO, S., 2011. Stationary scattering from a nonlinear network Physical Review A. 83, 033831
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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.

Relevant Publications
  • Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (I) J. Phys. A: Math. Theor. 42 (2009) 415101
  • Idan Oren, Amit Godel and Uzy Smilansky Trace formulae and spectral statistic for discrete Laplacians on regular graphs (II) J. Phys. A: Math. Theor. 43 (2010) 225205
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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 mean-field 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 field-theoretic approach.

Relevant Publications
  • GNUTZMANN, S, KEATING, J.P. and PIOTET, F., 2010. Eigenfunction statistics on quantum graphs Annals of Physics. 325(12), 2595-2640
  • GNUTZMANN, S., KEATING, J.P. and PIOTET, F., 2008. Quantum Ergodicity on Graphs PHYSICAL REVIEW LETTERS. VOL 101(NUMB 26), 264102
  • GNUTZMANN, S. and SMILANSKY, U., 2006. Quantum graphs: applications to quantum chaos and universal spectral statistics Advances in Physics. 55(5-6), 527-625
  • GNUTZMANN, S. and ALTLAND, A., 2005. Spectral correlations of individual quantum graphs. Physical Review E. 72, 056215
  • GNUTZMANN, S. and ALTLAND, A., 2004. Universal spectral statistics in quantum graphs. Physical Review Letters. 93(19), 194101
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Title Pseudo-orbit 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 so-called pseudo-orbits. In this project these methods will be develloped further and applied to questions related to quantum chaos and random-matrix theory.

Relevant Publications
  • Daniel Waltner, Sven Gnutzmann, Gregor Tanner, Klaus Richter, A sub-determinant approach for pseudo-orbit expansions of spectral determinants, arXiv:1209.3131 [nlin.CD]
  • Ram Band, Jonathan M. Harrison, Christopher H. Joyner, Finite pseudo orbit expansions for spectral quantities of quantum graphs, arXiv:1205.4214 [math-ph]
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Title The ten-fold 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 three-fold symmetry classification for quantum mechanical systems -- these symmetry classes consisted of time-reversal invariant systems with integer spin which can be described by real symmetric matrices, time-reversal invariant systems with half-integer spin which can be described by real quaternion matrices, and systems without any time-reversal symmetry which are described by complex hermitian matrices. These three symmetry classes had their immediate application in the three classical Gaussian ensembles of random-matrix 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 anti-commutators.
The classification was completed by Altland and Zirnbauer who have shown that there are essentially only seven further symmetry classes on top of the Wigner-Dyson classes leading to what is now known as the 'ten-fold way'. All symmetry classes have applications in physics. The new symmetry classes are realised by various cases of the Dirac equation and the Bogoliubov-de Gennes equation. For a long time people have thought of these symmetries only in the context of many-body physics or quantum field theory. However there are simple quantum mechanical realisations of all ten symmetry classes which in terms of two coupled spins where the classification follows from properties of the coupling parameters and of the irreducible SU(2) representations on which the spin operators act. This project will explore these simple representations in the quantum mechanical and semiclassical context. One goal will be to understand the implications of the quantum mechanical symmetries for the corresponding classical dynamics which appears in the semiclassical limit of large spins.

Relevant Publications
  • M.R. Zirnbauer, Riemannian symmetric superspaces and their origin in random matrix theory, J. Math. Phys. 37, 4986 (1996)
  • A. Altland, M.R. Zirnbauer, Non-standard symmetry classes in mesoscopic normal-/superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997)
  • S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. I. Construction and numerical results, Physical Review E 69, 056219 (2004)
  • S. Gnutzmann and B. Seif, Universal spectral statistics in Wigner-Dyson, chiral, and Andreev star graphs. II. Semiclassical approach. Physical Review E 69, 056220 (2004)
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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 Kerr-type 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 non-trivial 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 Y-junctions (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).

Relevant Publications
  • Sven Gnutzmann, Uzy Smilansky, and Stanislav Derevyanko, Stationary scattering from a nonlinear network, Phys. Rev. A 83, 033831 (2011)
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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 3-dimensional 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.

Relevant Publications
  • Galya Blum, Sven Gnutzmann, Uzy Smilansky, Nodal domains statistics- a criterion for quantum chaos, Phys. Rev. Lett. 88, 114101 (2002)
  • Alejandro G. Monastra, Uzy Smilansky and Sven Gnutzmann, Avoided intersections of nodal lines, J. Phys. A. 36, 1845-1853 (2003)
  • G. Foltin, S. Gnutzmann and U. Smilansky, The morphology of nodal lines- random waves vs percolation, J. Phys. A 37, 11363 (2004)
  • Yehonatan Elon, Sven Gnutzmann, Christian Joas and Uzy Smilansky, Geometric characterization of nodal domains: the area-to-perimeter ratio, J. Phys. A 40, 2689 (2007)
  • S. Gnutzmann, P. D. Karageorge and U Smilansky, Can one count the shape of a drum?, Phys. Rev. Lett. 97, 090201 (2006)
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Title Coherent states, nonhermitian Quantum Mechanics and PT-symmetry
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 non-hermitian 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 PT-symmetric 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.

Relevant Publications
  • S.Gnutzmann, M Kus, Coherent states and the classical limit on irreducible SU3 representations, J. Phys. A 31, 9871 (1998)
  • E.-M. Graefe, M. Höning, H.J. Korsch, Classical limit of non-Hermitian quantum dynamic - a generalized canonical structure, J. Phys A 43, 075306 (2010)
  • E.-M. Graefe, R. Schubert, Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians, J. Phys. A 45, 244033 (2012)
  • C.M. Bender, M. DeKieviet, S.P. Klevansky, PT Quantum Mechanics, Philosophical Transactions of the Royal Society A 371, 20120523 (2013)
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