A mathematical lattice, called the von Neumann lattice, is a subset of coherent states and exists periodically in the phase space. It is unlike solids or Abrikosov lattices that are observable in physical systems. Abrikosov lattices are vortices closely packed into a lattice with a flux quantum through a unit cell. Although Abrikosov lattices appear generally in various physical systems, vortex lattices with multiple-flux quantums through a unit cell are more stable than Abrikosov lattices in some physical regimes of the systems with non-local interactions between particles. No theory is able to describe these vortex lattices today. Here, we develop a theory for these vortex lattices by extending von Neumann lattices to the coordinate space with a unit cell of area that is proportional to flux quantums through a unit cell. The von Neumann lattices not only show the same physical properties as the Abrikosov lattice, but also describe vortex lattices with multiple-flux quantums through a unit cell. From numerical simulations of a rapidly rotating dipole-blockaded gas, we confirm that vortex lattices showed in our simulations are the representation of von Neumann lattices in the coordinate space. We anticipate our theory to be a starting point for developing more sophisticated vortex-lattice models. For example, the effect of Landau-level mixing on vortex lattice structures, vortices formed inside superfluid droplets and structural phase transitions of vortex matter in two-component Bose-Einstein condensates will be relevant for such developments.
A mathematical lattice, called the von Neumann lattice, is a subset of coherent states and exists periodically in the phase space. It is unlike solids or Abrikosov lattices that are observable in physical systems. Abrikosov lattices are vortices closely packed into a lattice with a flux quantum through a unit cell. Although Abrikosov lattices appear generally in various physical systems, vortex lattices with multiple-flux quantums through a unit cell are more stable than Abrikosov lattices in some physical regimes of the systems with non-local interactions between particles. No theory is able to describe these vortex lattices today. Here, we develop a theory for these vortex lattices by extending von Neumann lattices to the coordinate space with a unit cell of area that is proportional to flux quantums through a unit cell. The von Neumann lattices not only show the same physical properties as the Abrikosov lattice, but also describe vortex lattices with multiple-flux quantums through a unit cell. From numerical simulations of a rapidly rotating dipole-blockaded gas, we confirm that vortex lattices showed in our simulations are the representation of von Neumann lattices in the coordinate space. We anticipate our theory to be a starting point for developing more sophisticated vortex-lattice models. For example, the effect of Landau-level mixing on vortex lattice structures, vortices formed inside superfluid droplets and structural phase transitions of vortex matter in two-component Bose-Einstein condensates will be relevant for such developments.
This paper reports a novel type of vortex lattice, referred to as a bubble crystal, which was discovered in rapidly rotating Bose gases with long-range interactions. Bubble crystals differ from vortex lattices which possess a single quantum flux per unit cell, while atoms in bubble crystals are clustered periodically and surrounded by vortices. No existing model is able to describe the vortex structure of bubble crystals; however, we identified a mathematical lattice, which is a subset of coherent states and exists periodically in the physical space. This lattice is called a von Neumann lattice, and when it possesses a single vortex per unit cell, it presents the same geometrical structure as an Abrikosov lattice. In this report, we extend the von Neumann lattice to one with an integral number of flux quanta per unit cell and demonstrate that von Neumann lattices well reproduce the translational properties of bubble crystals. Numerical simulations confirm that, as a generalized vortex, a von Neumann lattice can be physically realized using vortex lattices in rapidly rotating Bose gases with dipole interatomic interactions.
Atomic clouds in a rotating optical lattice are at the intellectual intersection of several major paradigms of condensed matter physics. An optical lattice simulates the periodic potential ubiquitous in solid state physics, while rotation probes the superfluid character of these cold atomic gases by driving the formation of quantized vortices. Here we explore the theory of vortices in an optical lattice. We first provide a detailed introduction section aimed at providing the reader with the information necessary to understand and appreciate the research presented in later chapters. Next we study an infinite square lattice configuration of vortices in a rotating optical lattice near the superfluid-Mott-insulator transition. We find a series of abrupt structural phase transitions where vortices are pinned with their cores only on plaquettes or only on sites. We discuss connections between these vortex structures and the Hofstadter-butterfly spectrum of free particles on a rotating lattice. We then investigate vortex configurations within a harmonically trapped Bose-Einstein condensate in a rotating optical lattice. We find that proximity to the Mott insulating state dramatically affects the vortex structures. To illustrate we give examples in which the vortices: (i) all sit at a fixed distance from the center of the trap, forming a ring, or (ii) coalesce at the center of the trap, forming a giant vortex. We model time-of-flight expansion to demonstrate the experimental observability of our predictions. Finally for a trapped gas far from the Mott regime, the competition between vortex-vortex interactions and pinning to the optical lattice results in a complicated energy landscape, which leads to hysteretic evolution. The qualitative structure of the vortex configurations depends on the commensurability between the vortex density and the site density - with regular lattices when these are commensurate, and concentric rings when they are not. Again we model the imaging of these structures by calculating time-of-flight column densities. As in the absence of the optical lattice, the vortices are much more easily observed in a time-of-flight image than in-situ.
In 1995, atomic physics took a quantum leap with the experimental discovery of almost ideal Bose-Einstein condensation in dilute gases of alkali atoms. The gaseous condensates offer a rare chance to study an interacting many-particle system accurately from first-principles theories, and they can be used to simulate many seminal models that have proven elusive in their original context of, e.g., solid-state or high-energy physics. An important topic in the research has been the quantized vortex, which was experimentally realized in 1999 and whose existence is intimately related to the concepts of quantum phase coherence and superfluidity.\nIn this dissertation, unconventional vortex structures are investigated in the dilute condensates at ultralow temperatures. A combination of analytical and numerical methods is used to examine the structure, stability, and dynamical behavior of multiquantum vortices and vortex-antivortex pairs in spin-polarized condensates, unusual vortex lattices in two-species condensates, and spin textures in condensates with dipolar interactions. In studying the properties of the multiquantum vortices, particular emphasis is placed on exploring the practical limits of producing them through adiabatic pumping of vorticity. The majority of the research is conducted by solving the mean-field Gross-Pitaevskii and Bogoliubov equations.\nVarious original, experimentally verifiable results concerning the exotic vortex structures are presented. Novel splitting patterns of vortices with large quantum numbers are introduced, and it is shown that such vortices can be feasibly stabilized by piercing them with a focused laser beam. A recent experiment on vortex-antivortex pairs is simulated, and an excellent quantitative agreement with the experimental data is obtained. Unconventional ground-state vortex lattices, such as ones having a square geometry or consisting of two-quantum vortices, are shown to exist in rotating two-species condensates. Dipolar interactions are found to support helical spin-vortex states, and the energies of spin-wave excitations are observed to increase rapidly with the dipolar coupling strength.\nThis dissertation contributes to the understanding of superfluid phenomena in Bose-Einstein condensates and has significant implications for the prospects of detecting novel vortex structures in current experiments.
The experimental realization of quantum-degenerate Bose gases made of atoms with sizeable magnetic dipole moments has created a new type of fluid, known as a quantum ferrofluid, which combines the extraordinary properties of superfluidity and ferrofluidity. A hallmark of superfluids is that they are constrained to rotate through vortices with quantized circulation. In quantum ferrofluids the long-range dipolar interactions add new ingredients by inducing magnetostriction and instabilities, and also affect the structural properties of vortices and vortex lattices. Here we give a review of the theory of vortices in dipolar Bose-Einstein condensates, exploring the interplay of magnetism with vorticity and contrasting this with the established behaviour in non-dipolar condensates. We cover single vortex solutions, including structure, energy and stability, vortex pairs, including interactions and dynamics, and also vortex lattices. Our discussion is founded on the mean-field theory provided by the dipolar Gross-Pitaevskii equation, ranging from analytic treatments based on the Thomas-Fermi (hydrodynamic) and variational approaches to full numerical simulations. Routes for generating vortices in dipolar condensates are discussed, with particular attention paid to rotating condensates, where surface instabilities drive the nucleation of vortices, and lead to the emergence of rich and varied vortex lattice structures. We also present an outlook, including potential extensions to degenerate Fermi gases, quantum Hall physics, toroidal systems and the Berezinskii-Kosterlitz-Thouless transition.
Vortex lattices have become one of the most interesting areas of research in the area of cold atomic physics in recent years. When inducing a large amount of angular momentum into the ground state of a superfluid Bose-Einstein condensate, the sample responds by forming a so-called Abrikosov lattice. Within this lattice, vortices with winding number one are arranged in a triangular pattern that extends over the whole size of the sample. While the Abrikosov lattice is the exact solution for homogeneous condensates, it is also a very good solution for systems in isotropic harmonic traps. Due to the inhomogeneous potential however, the vortex lattice as a whole rotates within the trap.
In this article, we review developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single-component atomic Bose gas, which (at least at rest) forms a Bose–Einstein condensate. Rotation leads to the formation of quantized vortices which order into a vortex array, in close analogy with the behaviour of superfluid helium. Under conditions of rapid rotation, when the vortex density becomes large, atomic Bose gases offer the possibility to explore the physics of quantized vortices in novel parameter regimes. First, there is an interesting regime in which the vortices become sufficiently dense that their cores, as set by the healing length, start to overlap. In this regime, the theoretical description simplifies, allowing a reduction to single-particle states in the lowest Landau level. Second, one can envisage entering a regime of very high vortex density, when the number of vortices becomes comparable to the number of particles in the gas. In this regime, theory predicts the appearance of a series of strongly correlated phases, which can be viewed as bosonic versions of fractional quantum Hall states. In this article, we describe the equilibrium properties of rapidly rotating atomic Bose gases in both the mean-field and the strongly correlated regimes, and related theoretical developments for Bose gases in lattices, for multi-component Bose gases and for atomic Fermi gases. The current experimental situation and outlook for the future are discussed in light of these theoretical developments.