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arXiv:1106.5757

Rotating Bose-Einstein condensate in an square optical lattice: vortex configuration for ground state in Josephson junction arrays regime

Yousef Azizi, Alireza Valizadeh

Published Jun 28, 2011• 0 citations
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Abstract

We consider a rotating Bose-Einstein condensate in a square optical lattice in the regime in which the Hamiltonian of the system can be mapped onto a Josephson junction array. In an approximate scheme where the couplings are assumed uniform, the ground state energy is formulated in terms of the vortex configuration. The results are compared with experimental results and also previously reported results for frustrated XY model. We also show that vortex configuration is robust with respect to change of couplings and therefore the results remain valid when we consider more realistic model with non-uniform couplings.

DOI
10.48550/arxiv.1106.5757

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We consider a rotating Bose-Einstein condensate in a square optical lattice in the regime in which the Hamiltonian of the system can be mapped onto a Josephson junction array. In an approximate scheme where the couplings are assumed uniform, the ground state energy is formulated in terms of the vortex configuration. The results are compared with experimental results and also previously reported results for frustrated XY model. We also show that vortex configuration is robust with respect to change of couplings and therefore the results remain valid when we consider more realistic model with non-uniform couplings.
We investigate the ground state structure of two-dimensional Josephson junction arrays consisting of Bose-Einstein condensates trapped by both a rotating harmonic potential and a corotating deep optical lattice with square symmetry. Monte Carlo simulations of the uniformly frustrated XY model in a wide range of the frustration parameter f reveal that the stable configuration of vortices for a simple fractional number f is a staircase structure, consisting of diagonal chains of vortices with periodic lattice of unit cells.
Under suitable conditions, the Hamiltonian of a gas of cold bosonic atoms inside a rotating optical lattice can be mapped onto that of a frustrated Bose-Hubbard model. Here, the role of magnetic frustration is played by the angular frequency of rotation of the lattice. The system can be fully frustrated at experimentally attainable values of the frequency of rotation, and a superlattice of vortices appears in the ground state. This has several implications on observable quantities such as the superfluid density and the momentum distribution function. In the limit of large on-site occupation and in dimensionality D ≥ 2, the system can be mapped into an array of Josephson junctions in a magnetic field.
We derive a uniformly frustrated XYXYXY model that describes two-dimensional Josephson-junction arrays consisting of rotating Bose-Einstein condensates trapped by both a harmonic trap and a corotating deep optical lattice. The harmonic trap makes the coupling constant of the model have a nonuniform parabolic dependence. We study the ground state through Monte Carlo simulations in a wide range of the frustration parameter fff, revealing a rich variety of vortex patterns.
We consider the ground state of vortices in a rotating Bose-Einstein condensate that is loaded in a corotating two-dimensional optical lattice. Due to the competition between vortex interactions and their potential energy, the vortices arrange themselves in various patterns, depending on the strength of the optical potential and the vortex density. We outline a method to determine the phase diagram for arbitrary vortex filling factor. Using this method, we discuss several filling factors explicitly. For increasing strength of the optical lattice, the system exhibits a transition from the unpinned hexagonal lattice to a lattice structure where all the vortices are pinned by the optical lattice. The geometry of this fully pinned vortex lattice depends on the filling factor and is either square or triangular. For some filling factors there is an intermediate half-pinned phase where only half of the vortices is pinned. We also consider the case of a two-component Bose-Einstein condensate, where the possible coexistence of the above-mentioned phases further enriches the phase diagram. In addition, we calculate the dispersion of the low-lying collective modes of the vortex lattice and find that, depending on the structure of the ground state, they can be gapped or gapless. Moreover, in the half-pinned and fully pinned phases, the collective mode dispersion is anisotropic. Possible experiments to probe the collective mode spectrum, and in particular the gap, are suggested.
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.
We report the observation of vortex nucleation in a rotating optical lattice. A 87Rb Bose-Einstein condensate was loaded into a static two-dimensional lattice and the rotation frequency of the lattice was then increased from zero. We studied how vortex nucleation depended on optical lattice depth and rotation frequency. For deep lattices above the chemical potential of the condensate we observed a linear dependence of the number of vortices created with the rotation frequency, even below the thermodynamic critical frequency required for vortex nucleation. At these lattice depths the system formed an array of Josephson-coupled condensates. The effective magnetic field produced by rotation introduced characteristic relative phases between neighboring condensates, such that vortices were observed upon ramping down the lattice depth and recombining the condensates.