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arXiv:0710.1090 · Physical Review A

Vortex lattices of bosons in deep rotating optical lattices

Daniel S. Goldbaum, Erich J. Mueller

Published Mar 27, 2008• 38 citations
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Abstract

We study vortex-lattice phases for a Bose gas trapped 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.

DOI
10.1103/physreva.77.033629

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We study vortex-lattice phases for a Bose gas trapped 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.
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.
Rotational states for trapped bosons in an optical lattice are studied in the framework of the Hubbard model. Critical frequencies are calculated and the main parameter regimes are identified. Transitions are observed from edge superfluids to vortex lattices with Mott insulating cores, and subsequently to lattices of interstitial vortices. The former transition coincides with the Mott transition. Changes in symmetry of the vortex lattices are observed as a function of lattice depth. Predictions for experimental signatures are presented.
Quantized vortices stunningly illustrate the coherent nature of a superfluid Bose condensate of alkali atoms. Introducing an optical lattice depletes this coherence. Consequently, novel vortex physics may emerge in an experiment on a harmonically trapped gas in the presence of a rotating optical lattice. The most dramatic effects would occur in proximity to the Mott state, an interaction dominated insulator with a fixed integer number of particles per site. We model such a rotating gas, showing that the lattice-induced spatial profile of the superfluid density drives a gross rearrangement of vortices. For example, instead of the uniform vortex lattices commonly seen in experiments, we find parameters for which the vortices all sit at a fixed distance from the center of the trap, forming a ring. Similarly, they can coalesce at the center, forming a giant vortex. We find that the properties of this system are hysteretic, even far from the Mott state. We explain this hysteresis in terms of vortex pinning, commensurability between vortex density and pinning site density, and energy barriers against changing the number of vortices. Finally, we model time-of-flight expansion, demonstrating the experimental observability of our predictions.
The single-vortex problem in a strongly correlated bosonic system is investigated self-consistently within the mean-field theory of the Bose-Hubbard model. Near the superfluid--Mott-insulator transition, the vortex core has a tendency toward the Mott-insulating phase, with the core particle density approaching the nearest commensurate value. If the nearest-neighbor repulsion exists, the charge-density wave order may develop locally in the core. The evolution of the vortex configuration from the strong- to weak-coupling regions is studied. This phenomenon can be observed in systems of rotating ultracold atoms in optical lattices and Josephson-junction arrays.