Abstract Understanding quantum dynamics in a two-dimensional Bose–Einstein condensate relies on understanding how vortices interact with each other microscopically and with local imperfections of the potential which confines the condensate. Within a system consisting of many vortices, the trajectory of a vortex–antivortex pair is often scattered by a third vortex, an effect previously characterised. However, the natural question remains as to how much of this effect is due to the velocity induced by this third vortex and how much is due to the density inhomogeneity which it introduces. In this work, we describe the various qualitative scenarios which occur when a vortex–antivortex pair interacts with a smooth density impurity whose profile is identical to that of a vortex but lacks the circulation around it.
Abstract Understanding quantum dynamics in a two-dimensional Bose–Einstein condensate relies on understanding how vortices interact with each other microscopically and with local imperfections of the potential which confines the condensate. Within a system consisting of many vortices, the trajectory of a vortex–antivortex pair is often scattered by a third vortex, an effect previously characterised. However, the natural question remains as to how much of this effect is due to the velocity induced by this third vortex and how much is due to the density inhomogeneity which it introduces. In this work, we describe the various qualitative scenarios which occur when a vortex–antivortex pair interacts with a smooth density impurity whose profile is identical to that of a vortex but lacks the circulation around it.
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 Schrödinger 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 numerical 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 stationary 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 configurations. For singly quantised vortices, impurity ground states are shown to be dynamically robust, while excited impurity states relax under time evolution despite remaining localised. A quantitative overlap measure is introduced to characterise the dynamical 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 impurity–vortex interactions in Bose-Einstein condensates, combining theoretical modelling, numerical methods, and dynamical simulations to clarify the role of impurities in coupled quantum fluids.
The dynamics of interacting quantum vortices in a quasi-two-dimensional spatially inhomogeneous Bose–Einstein condensate, whose equilibrium density vanishes at two points of the plane with a possible presence of an immobile vortex with a few circulation quanta at each point, has been considered in a hydrodynamic approximation. A special class of density profiles has been chosen, so that it proves possible to calculate analytically the velocity field produced by point vortices. The equations of motion have been given in a noncanonical Hamiltonian form. The theory has been generalized to the case where the condensate forms a curved quasi-two-dimensional shell in the three-dimensional space.
Abstract We numerically investigate the impact of various impurities on rotating Bose–Einstein condensates confined within two-dimensional harmonic and Gaussian distributed square lattice potentials. Without impurities, the rotating condensates display an organized square lattice pattern of vortices due to the influence of Gaussian distributed square lattice potential. The introduction of impurity potentials disrupts this lattice structure, inducing a phase transition from an ordered state to a disordered state. Our analysis encompasses both static and dynamic types of impurities. The static impurities are implemented using a randomly varying potential with a spatially random amplitude. The transformation of the vortex lattice structure, in this case, relies on the strength and lattice constant of the impurity potential. For dynamical impurities, we employ a Gaussian obstacle that orbits around the condensate at a specific distance from its center. In this scenario, the vortex lattice melting occurs beyond a certain threshold radius and frequency of oscillation of the rotating obstacle. We characterize the melting of the vortex lattice due to impurities using various quantities, such as the structure factor and angular momentum. Notably, in the vortex-melted state, the angular momentum follows a power-law dependence with an exponent of approximately 1.73, regardless of the type of impurity. Finally, we demonstrate the signature of the presence of a turbulent state within the vortex-melted state generated by both static and dynamical impurities.
We study the interaction and dynamics of two half-quantized vortices in two-component Bose-Einstein condensates. Using the Pad\'e approximation for the vortex core profile, we calculate the intervortex potential, whose asymptotic form for a large distance has been derived by Eto et al. [Phys. Rev. A 83, 063603 (2011)]. Through numerical simulations of the two-dimensional Gross-Pitaevskii equations, we reveal different kinds of dynamical trajectories of the vortices depending on the combinations of signs of circulations and the intercomponent density coupling. Under the adiabatic limit, we derive the equations of motion for the vortex coordinates, in which the motion is caused by the balance between Magnus force and the intervortex forces. The initial velocity of the vortex motion can be explained quantitatively by this point vortex approximation, but understanding the long-time behavior of the dynamics needs more consideration beyond our model.
We investigate the impact of various impurities on rotating Bose-Einstein condensates confined within two-dimensional harmonic and optical lattice potentials. Without impurities, the rotating condensates display an organized square lattice pattern of vortices due to the influence of a square optical lattice. The introduction of impurity potentials disrupts this lattice structure, inducing a phase transition from an ordered state to a disordered state. Our analysis encompasses both static and dynamic types of impurities. The static impurities are implemented using a randomly varying potential with a spatially random amplitude. The transformation of the vortex lattice structure, in this case, relies on the strength and lattice constant of the impurity potential. For dynamical impurities, we employ a Gaussian obstacle that orbits around the condensate at a specific distance from its center. In this scenario, the vortex lattice melting occurs beyond a certain threshold radius and frequency of oscillation of the rotating obstacle. We characterize the melting of the vortex lattice due to impurities using various quantities, such as the structure factor and angular momentum. Notably, in the vortex-melted state, the angular momentum follows a power-law dependence with an exponent of approximately 1.73, regardless of the type of impurity. Finally, we demonstrate the signature of the presence of a turbulent state within the vortex-melted state generated by both static and dynamical impurities.
We study the process of scattering of two-dimensional dark solitons, and their vortex-antivortex pairs as a specific case, by a single quantum vortex in a Bose-Einstein condensate with repulsive interaction between atoms. An asymptotic theory describing the dynamics of such solitonlike structures in a smoothly inhomogeneous flow of ultracold Bose gas is developed. An analytical expression for the angle of scattering of two-dimensional dark solitons (including vortex pairs) by a single-phase singularity is obtained in the limit of large impact parameters. All theoretical concepts are confirmed by numerical calculations performed directly within the Gross-Pitaevskii equation. It is shown that for small impact parameters, the considered solitonlike structures interact inelastically with the core of a single quantum vortex, scattering over large angles and radiating sound waves.