Vortex-impurity interactions in two-dimensional quantum fluids

Abstract

This thesis investigates the behaviour of a quantum impurity coupled to a Bose–Einstein condensate. The condensate dynamics are described by the Gross–Pitaevskii equation, while the impurity is modelled by a coupled Schr¨odinger equation, forming a Gross–Clark- system. The work focuses on impurity localisation, vortex structure, and the dynamical consequences of impurity-condensate coupling in effectively two-dimensional geometries. The thesis begins by establishing the Gross–Pitaevskii model and its dimensionless formulation, together with the structure and stability of quantised vortices. Both numer- ical and approximate analytical descriptions of vortex profiles are developed and used throughout the thesis. The localisation of impurities is then examined in a range of static settings, including idealised trapping potentials and vortex cores, revealing how condensate density depletion can generate effective confinement for the impurity. Self-consistent impurity-vortex bound states are obtained by solving coupled station- ary equations. The spectrum of localised impurity states is analysed, and the dependence of localisation on interaction parameters and angular momentum is characterised. These static results provide the foundation for subsequent dynamical studies. Real-time simulations are used to investigate the stability of impurity-bound config- urations. For singly quantised vortices, impurity ground states are shown to be dynam- ically robust, while excited impurity states relax under time evolution despite remain- ing localised. A quantitative overlap measure is introduced to characterise the dynam- ical stability of impurity states. The thesis then examines multiply quantised vortices, demonstrating that impurity coupling can modify vortex dynamics and, in certain cases, suppress intrinsic instabilities. Overall, this work provides a systematic analysis of impurity localisation and impu- rity–vortex interactions in Bose-Einstein condensates, combining theoretical modelling, numerical methods, and dynamical simulations to clarify the role of impurities in coupled quantum fluids.