The subject of the present thesis is the study of the interaction between a Bose-Einstein condensate (BEC) and a superconductor.The BEC performs center-of-mass oscillations perpendicular to the superconducting surface.Each atom in the BEC carries a magnetic dipole moment, which induces eddy currents in the surface.These eddy currents are a source of a magnetic field which modifies the potential in which the BEC is trapped.This leads to a change of the dynamical behavior of the BEC.In particular the BEC center-of-mass oscillation frequency is shifted compared to the case without the surface interaction.Additionally, the potential generated by the superconductor excites shape oscillations of the BEC.To investigate these effects, the Gross-Pitaevskii equation for a dipolar BEC close to a superconducting surface is solved numerically.Also analytical approximations are presented and compared to the numerically obtained results.It is shown that the interaction with the superconducting surface generates a frequency shift, that is large enough to be detected in an experiment.Furthermore, different methods to identify the eddy current effect are presented.It is also discussed how the effects on the BEC can be enhanced by choosing the proper dipole orientation and by adjusting the geometry of the setup.The BEC shape oscillations can be additionally enhanced by making use of a resonance phenomenon.4 Center-of-mass frequency shift 4.1 Frequency shift within TF approximation . . . . . . . . . .4.1.1Frequency shift based on a Thomas-Fermi ellipsoid 4.1.2Frequency shift based on the column density model 4.2 Numerical results for the frequency shift . . . . . . . . . . .4.3 Different polarizations of the BEC . . . . . . . . . . . . . .4.3.1 Polarization in the y-direction . . . . . . . . . . . .4.3.2Polarization in the x-direction (perpendicular to the surface) . . . . . . . . . . .
The subject of the present thesis is the study of the interaction between a Bose-Einstein condensate (BEC) and a superconductor.The BEC performs center-of-mass oscillations perpendicular to the superconducting surface.Each atom in the BEC carries a magnetic dipole moment, which induces eddy currents in the surface.These eddy currents are a source of a magnetic field which modifies the potential in which the BEC is trapped.This leads to a change of the dynamical behavior of the BEC.In particular the BEC center-of-mass oscillation frequency is shifted compared to the case without the surface interaction.Additionally, the potential generated by the superconductor excites shape oscillations of the BEC.To investigate these effects, the Gross-Pitaevskii equation for a dipolar BEC close to a superconducting surface is solved numerically.Also analytical approximations are presented and compared to the numerically obtained results.It is shown that the interaction with the superconducting surface generates a frequency shift, that is large enough to be detected in an experiment.Furthermore, different methods to identify the eddy current effect are presented.It is also discussed how the effects on the BEC can be enhanced by choosing the proper dipole orientation and by adjusting the geometry of the setup.The BEC shape oscillations can be additionally enhanced by making use of a resonance phenomenon.4 Center-of-mass frequency shift 4.1 Frequency shift within TF approximation . . . . . . . . . .4.1.1Frequency shift based on a Thomas-Fermi ellipsoid 4.1.2Frequency shift based on the column density model 4.2 Numerical results for the frequency shift . . . . . . . . . . .4.3 Different polarizations of the BEC . . . . . . . . . . . . . .4.3.1 Polarization in the y-direction . . . . . . . . . . . .4.3.2Polarization in the x-direction (perpendicular to the surface) . . . . . . . . . . .
We study the dynamics of a dipolar Bose-Einstein condensate, like, for example, a 52Cr or 164Dy condensate, interacting with a superconducting surface. The magnetic dipole moments of the atoms in the Bose-Einstein condensate induce eddy currents in the superconductor. The magnetic field generated by eddy currents modifies the trapping potential such that the center-of-mass oscillation frequency is shifted. We numerically solve the Gross-Pitaevskii equation for this system and compare the results with analytical approximations. We present an approximation that gives excellent agreement with the numerical results. The eddy currents give rise to anharmonic terms, which leads to the excitation of shape fluctuations of the condensate. We discuss how the strength of the excitation of such modes can be increased by exploiting resonances, and we examine the strength of the resonances as a function of the center-of-mass oscillation amplitude of the condensate. Finally, we study different orientations of the magnetic dipoles and discuss favorable conditions for the experimental observation of the eddy-current effect.
We study center-of-mass oscillations of a dipolar Bose–Einstein condensate in the vicinity of a superconducting surface. We show that the magnetic field of the magnetic dipoles induces eddy currents in the superconductor, which act back on the Bose–Einstein condensate. This leads to a shift of its oscillation frequency and to an anharmonic coupling of the Bose–Einstein condensate with the superconductor. The anharmonicity creates a coupling to one of the collective modes of the condensate that can be resonantly enhanced if the parameters of the condensate are chosen properly. This provides a new physical mechanism to couple a Bose–Einstein condensate and a superconductor, which becomes significant for 52 Cr, 168 Er or 164 Dy condensates in superconducting microtraps.
In this thesis we study the effects of the dipole-dipole interaction on the physics of ultracold quantum gases, both bosonic and fermionic, within the theoretical framework provided by the mean-field regime. This kind of interaction takes place in ultracold atomic gases (for instance 52Cr or 164Dy) due to their atomic magnetic dipole moment, and in ultracold molecular gases due to the magnetic or electric dipole moment. \n \nIn the case of quantum gases of bosonic atoms, or Bose-Einstein condensates, the dipole-dipole interaction can be studied within mean-field approximation using the Gross-Pitaevskii equation, which now contains a new non-linear term due to the dipole-dipole interaction. We investigate, on the one hand dipolar condensates confined in harmonic traps, and on the other dipolar condensates confined in toroidal traps. In the harmonic geometry, our focus is on the study of the ground state and the quantized vortex state, where the density profile is characterized as well as some properties leading to the process of vortex formation, such as the critical frequency and the energy barrier that has to be overcome to bring the vortex from the surface to the centre of the gas. We finish the study of dipolar condensates in harmonic traps by dynamically simulating the precession frequency of an off-center vortex in a non-rotating condensate. \n \nIn the toroidal geometry the dipolar effects are strongly magnified when the polarization axis of the dipoles is perpendicular to the trap symmetry axis. In this case, the anisotropic structure of the density can be understood as the response of the system to the double-well effective potential along the ring. We have studied the dynamics of this system when the initial number of atoms in the left and right wells is imbalanced, predicting Josephson and self-trapping oscillations depending on the initial condition. This has led us to name this new system as Self-induced Josephson Junction. We have studied in detail the self-trapping regime and we have seen that the particle flux inversion is closely related to the crossing of vortices across the Josephson junctions. This result opens the door to establishing a more direct connection between the phase-slip regime, widely addressed in superfluid helium, and the self-trapping regime of condensates. \n \nIn the case of quantum gases of fermionic dipolar particles, we have studied how the radial quadrupole mode allows one to distinguish between hydrodynamic and collisionless regimes. We have analytically calculated the frequency of this mode in the mean-field approximation, generalizing the results from the Thomas-Fermi approximation for trapped ideal Fermi gases. On the one hand, we observe that the frequency in the hydrodynamic regime is smaller than in non-dipolar Fermi gases, while in the collisionless regime the frequency is larger or smaller than that corresponding to the non-interacting system depending on the geometry of the harmonic trap. On the other hand, we predict that reducing the trap deformation (aspect ratio) an observable jump in the frequency of the radial quadrupole mode would take place, which would correspond to the transition between the collisionless and hydrodynamic regimes, for instance when the gas undergoes the transition to the superfluid state.
The center-of-mass dynamics of a vortex in the surface region of a Bose-Einstein condensate is investigated both analytically using a variational calculation and numerically by solving the time-dependent Gross-Pitaevskii equation. We find, in agreement with previous works, that away from the Thomas-Fermi surface, the vortex moves parallel to the surface of the condensate with a constant velocity. We obtain an expression for this velocity in terms of the distance of the vortex core from the Thomas-Fermi surface that fits accurately with the numerical results. We find also that, coupled to its motion parallel to the surface, the vortex oscillates along the direction normal to the surface around a minimum point of an effective potential.
We investigate the superfluid properties of a Bose–Einstein condensate (BEC) trapped in a one-dimensional periodic potential. We study, both analytically (in the tight binding limit) and numerically, the Bloch chemical potential, the Bloch energy and the Bogoliubov dispersion relation, and we introduce two different, density dependent, effective masses and group velocities. The Bogoliubov spectrum predicts the existence of sound waves, and the arising of energetic and dynamical instabilities at critical values of the BEC quasi-momentum which dramatically affect its coherence properties. We investigate the dependence of the dipole and Bloch oscillation frequencies in terms of an effective mass averaged over the density of the condensate. We illustrate our results with several animations obtained solving numerically the time-dependent Gross–Pitaevskii equation.
There are many efforts to understand better the dipolar long-range interaction, the shell potential confinement, and the vortices in Bose-Einstein condensates (BECs). Furthermore, combining these three areas suggests that it can give rise to new phenomena. In this thesis, we studied the dynamics of vortices in a shell-trapped dipolar BEC. The long-range dipolar interaction induces an inhomogeneous density distribution on the shell, affecting the vortex velocity field. We analyzed the dynamics of one and two pairs of vortices symmetrically located in the shell to separate the vortex-vortex interaction contribution from that of the density gradient due to the inhomogeneous background. In addition to the numerical studies, a Lagrangian formalism allowed us to derive semi-analytical expressions which are in agreement with the results of the Gross-Pitaevskii equation (GPE) simulations, giving us better insights into the influence of the core in the vortex dynamics outside the validity of the point-vortex model. In the second part of this thesis, we describe an actual experimental implementation of the shell BEC. Using the Stochastic Projected GPE simulations, we determine the configuration of a finite temperature vortex lattice produced in this potential over the action of gravity. More specifically, we studied the transition from a perfectly hexagonal (triangular) lattice to the hexatic and liquid phases by detecting the vortex positions to calculate the angular and pair correlation functions of this system for different values of temperatures and trap rotation frequencies. Our results for the vortex lattice melting are consistent with the hexatic to liquid transition in the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) scenario and can be extended to predict the melting temperature of other finite-size systems.