Gross-Pitaevskii vortex motion with critically scaled inhomogeneities

We study the dynamics of vortices in an inhomogeneous Gross--Pitaevskii equation iδtu = Δu + 1/ε²(ρ² ε(ᵡ) - |u|²). For a unique scaling regime |ρε(ᵡ) - 1 = O(logε¯¹), it is shown that vortices can interact both with the background perturbation and with each other. Results for associated parabolic and elliptic problems are discussed

Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation

Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.Comment: 39 pages, 2 figure

Aspects of Quantum Fluctuations under Time-dependent External Influences

The vacuum of quantum field theory is not empty space but filled with quantum vacuum fluctuations, which give rise to many intriguing effects. The first part of this Thesis addresses cosmic inflation, where the quantum fluctuations of the inflaton field freeze and get amplified in the expanding universe. Afterwards, we turn our attention towards Bose-Einstein condensates, a laboratory system. Since most of our calculations are performed using a mean-field expansion, we will study the accuracy of a finite-range interaction potential onto such an expansion. Exploiting the universality of quantum fluctuations, several aspects of cosmic inflation will be identified in ballistically expanding Bose-Einstein condensates. The effective action technique for calculating the quantum backreaction will be scrutinized. Finally, we consider dynamic quantum phase transitions in the last part of this Thesis. To this end two specific scenarios will be investigated: firstly, the structure formation during the superfluid to Mott-insulator transition in the Bose-Hubbard model; and secondly, the formation of spin domains as a two-dimensional spin-one Bose gas is quenched from the (polar) paramagnetic to the (planar) ferromagnetic phase. During this quench, the symmetry of the ground state is spontaneously broken and vortices (topological defects) form

Well-posedness for mean-field evolutions arising in superconductivity

We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in a joint work with Serfaty in certain regimes as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.Comment: 53 pages; revised version, including an appendix jointly written with Julian Fischer about global existence for the degenerate parabolic cas

Quantum properties of Bose-Einstein condensates coupled to semiconductor heterojunctions

In this thesis, we present a theoretical study of the effects that a current-carrying Two-Dimensional Electron Gas (2DEG) produce on a neighbouring magnetically trapped Bose-Einstein Condensate of alkali atoms (BEC). We suggest that technology used for magnetic micro-controlling of cold gases could be improved by replacing or combining the metallic wires used in such structures with 2DEG-based conductors or quantum electronic devices. All calculations presented in this thesis consider parameters attainable with present technology, suggesting that experimental realization of the scenarios proposed here is already feasible. In Chapter one we present the general context in which this thesis is developed. It includes a definition of the Bose-Einstein condensate state, a description of the principles of magnetic trapping and a brief review of the developments in the area of micromanipulation of atomic BECs. Chapter two is devoted to describe the characteristics of the heterojunction considered in the thesis, and a simple model used to evaluate the electron flow in the 2DEG it contains. Chapter three shows in detail the properties of magnetic trapping configurations considered afterwards. We study two simple applications that can be developed by bringing a BEC near to a current carrying 2DEG. Firstly, in Chapter four, we demonstrate the feasibility of creating magnetic trapping potentials using a current-carrying 2DEG and an external magnetic field. We identify the advantages of such a 2DEG-based trap over traditional metal-based traps and the conditions needed for operability. Recently developed techniques of magnetic field microscopy with BECs motivate our second considered application, namely, using a Bose-Einstein Condensate to probe the electron transport in 2DEGs and structural characteristics of the heterojunction. In Chapter five, we demonstrate how the quantization of conductance occurring in Quantum Point Contacts (QPC) fabricated from the heterojunction in the 2DEG, can be detected through a localized depletion of the BEC caused by a small inhomogeneity of the magnetic field that originates from a current through the QPC. In addition, we show that the electron density fluctuations in the 2DEG can be measured by detecting the corresponding inhomogeneous magnetic field produced when current flows, via modulation of the BEC’s density. We establish the conditions under which a sensitive response of the BEC to the magnetic field is possible. We also derive a general relation between the modulation of the magnetic field affecting the BEC and the distribution of ionized donors in the heterojunction. Creating semiconductor-cold-atom hybrid systems, where electrons in the semiconductor and atoms in the BEC are coupled to each other, requires a full understanding of the properties of both systems. Since typical micro-traps have an elongated geometry, in Chapter six we present a study of phase correlations of BECs in such geometries. To do this, we use a recently proposed effective one-dimensional equation that takes into account the 3D character of the BEC. Finally, in Chapter seven, we conclude and identify directions for future work emerging from this thesis

Universality of Bose-Einstein Condensation and Quenched Formation Dynamics

The emergence of macroscopic coherence in a many-body quantum system is a ubiquitous phenomenon across different physical systems and scales. This Chapter reviews key concepts characterizing such systems (correlation functions, condensation, quasi-condensation) and applies them to the study of emerging non-equilibrium features in the dynamical path towards such a highly-coherent state: particular emphasis is placed on emerging universal features in the dynamics of conservative and open quantum systems, their equilibrium or non-equilibrium nature, and the extent that these can be observed in current experiments with quantum gases. Characteristic examples include symmetry-breaking in the Kibble-Zurek mechanism, coarsening and phase-ordering kinetics, and universal spatiotemporal scalings around non-thermal fixed points and in the context of the Kardar- Parisi-Zhang equation; the Chapter concludes with a brief review of the potential relevance of some of these concepts in modelling the large-scale distribution of dark matter in the universe.Comment: Invited contribution to the Encyclopedia of Condensed Matter Physics (Elsevier, 2nd Edition

An inverse problem from condensed matter physics

We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.Peer reviewe

Quantum properties of Bose-Einstein condensates coupled to semiconductor heterojunctions

In this thesis, we present a theoretical study of the effects that a current-carrying Two-Dimensional Electron Gas (2DEG) produce on a neighbouring magnetically trapped Bose-Einstein Condensate of alkali atoms (BEC). We suggest that technology used for magnetic micro-controlling of cold gases could be improved by replacing or combining the metallic wires used in such structures with 2DEG-based conductors or quantum electronic devices. All calculations presented in this thesis consider parameters attainable with present technology, suggesting that experimental realization of the scenarios proposed here is already feasible. In Chapter one we present the general context in which this thesis is developed. It includes a definition of the Bose-Einstein condensate state, a description of the principles of magnetic trapping and a brief review of the developments in the area of micromanipulation of atomic BECs. Chapter two is devoted to describe the characteristics of the heterojunction considered in the thesis, and a simple model used to evaluate the electron flow in the 2DEG it contains. Chapter three shows in detail the properties of magnetic trapping configurations considered afterwards. We study two simple applications that can be developed by bringing a BEC near to a current carrying 2DEG. Firstly, in Chapter four, we demonstrate the feasibility of creating magnetic trapping potentials using a current-carrying 2DEG and an external magnetic field. We identify the advantages of such a 2DEG-based trap over traditional metal-based traps and the conditions needed for operability. Recently developed techniques of magnetic field microscopy with BECs motivate our second considered application, namely, using a Bose-Einstein Condensate to probe the electron transport in 2DEGs and structural characteristics of the heterojunction. In Chapter five, we demonstrate how the quantization of conductance occurring in Quantum Point Contacts (QPC) fabricated from the heterojunction in the 2DEG, can be detected through a localized depletion of the BEC caused by a small inhomogeneity of the magnetic field that originates from a current through the QPC. In addition, we show that the electron density fluctuations in the 2DEG can be measured by detecting the corresponding inhomogeneous magnetic field produced when current flows, via modulation of the BEC’s density. We establish the conditions under which a sensitive response of the BEC to the magnetic field is possible. We also derive a general relation between the modulation of the magnetic field affecting the BEC and the distribution of ionized donors in the heterojunction. Creating semiconductor-cold-atom hybrid systems, where electrons in the semiconductor and atoms in the BEC are coupled to each other, requires a full understanding of the properties of both systems. Since typical micro-traps have an elongated geometry, in Chapter six we present a study of phase correlations of BECs in such geometries. To do this, we use a recently proposed effective one-dimensional equation that takes into account the 3D character of the BEC. Finally, in Chapter seven, we conclude and identify directions for future work emerging from this thesis