Department of Applied Mathematics and Theoretical Physics |

## Natalia G Berloff## Professor of Applied Mathematics## Fellow of Jesus College
Most resent research focuses on the new physical platform that acts as an analog simulator of hard optimization problems: Several platforms are currently being explored for simulating physical systems whose complexity increases faster than polynomially with the number of particles or degrees of freedom in the system. Many of these computationally intractable problems can be mapped into classical spin models such as the Ising and the XY models and be simulated by a suitable physical system. Polariton condensates can be imprinted into any two-dimensional graph by spatial modulation of the pumping source. As we have recently shown in such a graph, polariton condensation occurs at the state with the phase-configuration that carries the highest polariton occupation and as a result he system of an arbitrary polariton graph condenses into the global minimum of the XY Hamiltonian which is one of the universal spin models that any other model can be mapped into. This opens a road to mapping hard optimization problems onto a polariton graph and read out the optimal result. One direction of my research is to relate and improve different mathematical models of superfluid turbulence by developing hierarchies of new stochastic models of vortex motion and turbulence. Mathematical models of superfluidity and superfluid turbulence have used four approaches: (1) the phenomenological Landau two-fluid model, (2) the phenomenological Hall-Vinen-Bekherevich-Khalatnikov (HVBK) model, (3) classical inviscid model of vortex motion with ad hoc reconnections, and (4) the Gross-Pitaevskii (GP) semi-classical model. The first of these was designed to describe the superfluid motion at all temperatures at which superfluidity exists. It requires modification when the vorticity is present. The HVBK model is intended for situations in which superfluid lines are dense. The GP model is applicable at very low temperatures where normal fluid is absent; it has proved its worth especially recently in describing dilute condensates. (Image at right adapted from N.G.Berloff and B.V. Svistunov, Phys. Rev. A, 66 , 013603, 2002.) The aim of my research on superfluid turbulence is to model and investigate the fundamental processes of
superfluid turbulence in Bose-Einstein
condensates (BEC). The central idea is to develop a hierarchy of new models of
vortex motion in which the action of sound waves on the large-scale superfluid
motion is parametrised in terms of simple random dynamic forces representing sound
acting on nonlinear `reduced' dynamic equations with relatively few
degrees of freedom representing the vortex motion. Spatially
inhomogeneous parameters of the random forcing will be estimated from
the statistics diagnosed from the Gross-Pitaevskii (GP) equation. There are
three major tasks to achieve this goal: (A) To use GP theory to study the reconnection
process in order to evaluate quantitatively the
associated radiation of sound and Kelvin waves and to define the
reconnection rules for the vortex dynamics; (B) To develop a new method
of decomposing the superfluid turbulence into the time-dependent
large-scale vortex motion and sound components which makes transparent
the sound/sound and vortex/sound nonlinear interactions and allows us
to understand the fundamental dynamics involved; (C) To repeat (A) and
(B) for nonlocal and dissipative GP models that have different
acoustic properties.
Finally, there is a hope that the random-forcing parameters can be
expressed in terms of the large-scale flow characteristics -- this
would be a new turbulent closure. (Image at left: cover page of PRL featuring N. G. Berloff and C. F. Barenghi, Phys. Rev. Lett.
A separate line of investigation focuses on exciton-polariton condensates. Microcavity exciton-polaritons are quasi-particles that result from the hybridisation of excitons and photons confined inside semiconductor microcavities. At low enough densities, they behave as bosons according to Bose-Einstein statistics, and so one may investigate Bose-Einstein condensation of these quasi-particles. Because of the imperfect confinement of the photon component, exciton-polaritons have a finite lifetime, and have to be continuously re-populated. Therefore, exciton-polariton condensates lie somewhere between equilibrium Bose-Einstein condensates and lasers. Similar to other laser systems nonequlibrium nature of the polariton condensates gives rise to different kinds of interesting pattern formation. Specific to polariton condensates, the mechanism responsible for pattern formation is an intricate interplay between nonlinear interactions, forcing, dissipation, dispersion and intrincic disorder in the material. The key to understanding the universal behaviour of pattern-forming systems lies in a common description obtained when particular microscopic models reduced to an order parameter equation. This project concerns with development of such models in close interaction with experimental groups such as Nanophotonics group of Professor Jeremy Baumberg (Cavendish). (Image at right adapted from G. Tosi et al Nature Physics, 8, 190-194 2012 ) Exciton-polaritons are not the only quasi-particles in solids in which condensation has been sought or realised and I am interested in other solid-state systems that exhibit similar coherent motion, such as magnon condensates. Magnons, which are elementary excitations -- quantised spin waves -- of a magnetic system have been observed to condense in ferromagnets and in superfluid 3He-B and in compressed aerogel of superfluid 3He-A. Similar to the phases of non-condensed Bose gas, atomic spins of normal magnetic materials are in a disordered paramagnetic state. In the ordered state the spins develop the common global frequency and phase of precession. The magnon condensation of room temperature yttrium-iron garnet (YIG) films magnetised by in-plane fields is driven by microwave radiation. A microwave photon excites two primary magnons that relax forming a magnon gas with the Bose distribution. The chemical potential increases with pumping power. When the microwave power exceeds a threshold value, the magnon population condenses at finite minima of the dispersion spectrum formed by the combined effects of the exchange and magnetic dipolar interactions. I am interested in modelling these condensates and understanding the structure of the ground state and excitations such as sound waves. On this project we work in close collaboration with professor Demokritov group in Munster, Germany. |