We present a rigorous proof of the appearance of quantized vortices in dilute trapped Bose gases with repulsive two-body interactions subject to rotation, which was obtained recently in joint work with Elliott Lieb. Starting from the many-body Schroedinger equation, we show that the ground state of such gases is, in a suitable limit, well described by the nonlinear Gross-Pitaevskii equation. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.
We present a rigorous proof of the appearance of quantized vortices in dilute trapped Bose gases with repulsive two-body interactions subject to rotation, which was obtained recently in joint work with Elliott Lieb. Starting from the many-body Schroedinger equation, we show that the ground state of such gases is, in a suitable limit, well described by the nonlinear Gross-Pitaevskii equation. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.
In this thesis, I explore the behavior of rotating ultra-cold Bose gases, by diagonalizing the Hamiltonian. This method has the advantage of being exact in the limit of weak interactions, and thus is a useful complement to the more commonly used Gross-Pitaevskii (GP) equation, which relies on a mean-field approximation. It is generally known that an ultra-cold Bose gas shows some remarkable properties under rotation, such as quantized vortices and persistent currents, and these are one main reason why we choose to study rotation. Abstract A repulsively interacting Bose-Einstein condensate (BEC) in a harmonic trapping potential will form singly quantized vortices when rotated. Papers I-V are examples of different kinds of generalizations to this basic system, each showing its particular non-trivial new features. Abstract In Paper I we make the trapping potential weakly anharmonic, and confirm the existence of multiply quantized vortices in both repulsively and attractively interacting condensates, which were previously predicted in Gross-Pitaevskii calculations. Abstract In Papers II and III, we generalize to a two-component system in a harmonic trap. The vortices that form under rotation will in general be coreless, i.e. one component will form a vortex and rotate around the other component. In Paper II we give some analytical expressions for the dispersion relation and occupation numbers at low angular momenta, and in Paper III we show how multiply quantized vortices can be stable in these systems. Abstract Finally we investigate bosons in potentials which can support persistent currents, i.e. rotating states that do not decay in a finite amount of time. These are possible if there is an energy barrier separating the rotating state from the non-rotating ground state. In Paper IV, we have a two-component condensate in a one-dimensional potential and we calculate among other things the minimum interaction strength where persistent currents can occur. In Paper V, we study an experimentally relevant two-dimensional annulus. Again we find that persistent currents are possible above a certain interaction strength, and we find this strength as a function of the width of the annulus, as well as how it depends on the relative population of the two species.
Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.
We study numerically the stability of axially symmetric vortex lines in trapped dilute gases subject to rotation. For this purpose, we solve numerically both the Gross-Pitaevskii and the Bogoliubov equations for a three-dimensional condensate in spherically and cylindrically symmetric traps, from small to very large nonlinearities. In the stationary case we find that the vortex states with m=1 and m=2 are energetically unstable. In the rotating trap it is found that this energetic instability may only be suppressed for the m=1 vortex line, and that the multicharged vortices are never a local minimum of the energy functional. This result implies that the absolute minimum of the energy is not an eigenstate of the Lz operator, when the angular speed is above a certain value.
We review the topic of quantized vortices in multicomponent Bose–Einstein condensates of dilute atomic gases, with an emphasis on the two-component condensates. First, we review the fundamental structure, stability and dynamics of a single vortex state in a slowly rotating two-component condensates. To understand recent experimental results, we use the coupled Gross–Pitaevskii equations and the generalized nonlinear sigma model. An axisymmetric vortex state, which was observed by the JILA group, can be regarded as a topologically trivial skyrmion in the pseudospin representation. The internal, coherent coupling between the two components breaks the axisymmetry of the vortex state, resulting in a stable vortex molecule (a meron pair). We also mention unconventional vortex states and monopole excitations in a spin-1 Bose–Einstein condensate. Next, we discuss a rich variety of vortex states realized in rapidly rotating two-component Bose–Einstein condensates. We introduce a phase diagram with axes of rotation frequency and the intercomponent coupling strength. This phase diagram reveals unconventional vortex states such as a square lattice, a double-core lattice, vortex stripes and vortex sheets, all of which are in an experimentally accessible parameter regime. The coherent coupling leads to an effective attractive interaction between two components, providing not only a promising candidate to tune the intercomponent interaction to study the rich vortex phases but also a new regime to explore vortex states consisting of vortex molecules characterized by anisotropic vorticity. A recent experiment by the JILA group vindicated the formation of a square vortex lattice in this system.
The ground state of a gas of Bosons confined in an external trap potential and interacting via repulsive two-body forces has recently been shown to exhibit complete Bose-Einstein condensation in the dilute limit, yielding for the first time a rigorous proof of this phenomenon in a physically realistic setting. We give here an account of this work about the Gross-Pitaevskii limit where the particle number N goes to infinity with Na fixed, where a is the scattering length of the interparticle interaction, measured in units of the trap size.
This thesis contains theoretical studies of weakly interacting, trapped Bose-Einstein condensed gases. The main emphasis is on the rotational properties of these systems. A typical Bose-Einstein condensate reacts to external rotation by forming singly quantized vortices. By the word ”typical” we refer to a condensate with repulsive interparticle interactions and with a frozen spin degree of freedom. However, there are some atomic species for which the low-energy interactions are attractive. Moreover, the tuning of different condensate properties is relatively easy. For example, there are ways to experimentally change the effective interaction from repulsive to attractive. For an attractively interacting condensate the vortex is not the only possible rotational state but the sample may also experience rotation through its center of mass. In the first part of the thesis, we analyze some properties of a rotating attractive Bose gas at the zero temperature limit. First, we note that in order to maintain energetical stability, the gas has to be rotated in an external trap which is steeper than a harmonic trap. We study the system in an anharmonic quadratic + quartic potential. The rotating ground state phase-space is determined with analytical and numerical methods. If the interactions are strong enough, the rotating ground state appears to be a state with the rotating center of mass. On the other hand, at the limit of weak interactions the system experiences rotation by forming a multiply quantized vortex. We also determine the spectrum of elementary excitations for these quantum many-particle states. We find that the broken symmetry of the center of mass rotating state induces an additional zero-energy (Goldstone) mode and that the small oscillations of the ground state exhibit the decoupling between the center of mass and relative motions. If the spin of the atoms is allowed to change freely, a rich variety of topological excitations in condensates can be studied. External rotation is one of the tools for examining the excitations in spinor condensates. In a spinless scalar condensate the appearance of a quantized vortex is manifested by the vortex line in which the condensate density vanishes. In the plane perpendicular to the vortex line, one speaks of the vortex core for the density minimum. We study the properties of slowly rotated F = 1 spinor condensates and show that for ferromagnetic interactions the ground state is a coreless vortex. While the individual spin components may carry ordinary vortices the overall density of the system is coreless. We also study a method to create a point-like monopole structure in a ferromagnetic condensate by suitably imprinting vortices and solitons for different spin components of the condensate.