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arXiv:cond-mat/0401173 · Physical Review A

Double-layer Bose-Einstein condensates with a large number of vortices

Hui Zhai, Qi Zhou, Rong Lü, Lee Chang

Published Jun 15, 2004• 13 citations
arXivHTMLPDF

Abstract

In this paper we systematically study the double-layer vortex lattice model, which is proposed to illustrate the interplay between the physics of a fast rotating Bose-Einstein condensate and the macroscopic quantum tunneling. The phase diagram of the system is obtained. We find that under certain conditions the system will exhibit a phase transition which is a consequence of the competition between interlayer coherent hopping and the interlayer density-density interaction. In one phase the vortices in one layer coincide with those in the other layer. In another phase two sets of vortex lattices are staggered, and as a result the quantum tunneling between two layers is suppressed. To obtain the phase diagram we use the quantum Hall mean field and Thomas-Fermi mean field theories. Two different criteria for the transition taking place are obtained, which reveals some fundamental differences between these two mean-field states. The sliding mode excitation is also discussed.

DOI
10.1103/physreva.69.063609

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In this paper we systematically study the double-layer vortex lattice model, which is proposed to illustrate the interplay between the physics of a fast rotating Bose-Einstein condensate and the macroscopic quantum tunneling. The phase diagram of the system is obtained. We find that under certain conditions the system will exhibit a phase transition which is a consequence of the competition between interlayer coherent hopping and the interlayer density-density interaction. In one phase the vortices in one layer coincide with those in the other layer. In another phase two sets of vortex lattices are staggered, and as a result the quantum tunneling between two layers is suppressed. To obtain the phase diagram we use the quantum Hall mean field and Thomas-Fermi mean field theories. Two different criteria for the transition taking place are obtained, which reveals some fundamental differences between these two mean-field states. The sliding mode excitation is also discussed.
We investigate vortex states and quantum fluctuations of bilayer atomic gases in rapid rotation. Among mean-field solutions of the Gross-Pitaevskii equation, we consider two types of vortex configurations: the vortex core positions of the layers are coincident or staggered. It is found that the coincident type is energetically preferred in the practical parameter regime. We also calculate the dispersion relations of collective modes and the filling factor for bosons outside the condensates. In the double layer system, quantum depletion is found to be suppressed due to interlayer tunneling. This means that Bose-Einstein condensation is stabilized compared to the single layer case.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.
We investigate the vortex-lattice structure for single- and two-component Bose-Einstein condensates in the presence of an optical lattice, which acts as a pinning potential for the vortices. The problem is considered in the mean-field quantum Hall regime, which is reached when the rotation frequency $\ensuremath{\Omega}$ of the condensate in a radially symmetric trap approaches the (radial) trapping frequency $\ensuremath{\omega}$ and the interactions between the atoms are weak. We determine the vortex-lattice phase diagram as a function of optical-lattice strength and geometry. In the limit of strong pinning the vortices are always pinned at the maxima of the optical-lattice potential, similar to the slow-rotation case. At intermediate pinning strength, however, due to the competition between interactions and pinning energy, a structure arises for the two-component case where the vortices are pinned on lines of minimal potential.
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.
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.
We study the ground states of rotating atomic Bose-Einstein condensates with dipolar interactions. We present the results of numerical studies on a periodic geometry which show vortex lattice ground states of various symmetries: triangular and square vortex lattices, ``stripe crystal,'' and ``bubble crystal.'' We present the phase diagram (for systems with a large number of vortices) as a function of the ratio of dipolar to contact interactions and of the chemical potential. We discuss the experimental requirements for observing transitions between vortex lattice ground states via dipolar interactions. We finally investigate the stability of mean-field supersolid phases of a quasi-two-dimensional nonrotating Bose gas with dipolar interactions.