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Chinese Physics Letters

Topological Structure of Vortices in Multicomponent Bose—Einstein Condensates

Xu Tao

Published Sep 1, 2009• 0 citations
Publisher

Abstract

The structure of the vortex in a two-component Bose—Einstein condensate is studied by the method of Dirac δ function. The vortex can be characterized by the Brouwer degree and Hopf index, i.e., β 1 η 1 , β 2 η 2 . The circulation of the vortex can be a fraction, which is different from the usual result for a one-component condensate. The kinetic helicity of vortices is calculated.

DOI
10.1088/0256-307x/26/9/096701

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The structure of the vortex in a two-component Bose—Einstein condensate is studied by the method of Dirac δ function. The vortex can be characterized by the Brouwer degree and Hopf index, i.e., β 1 η 1 , β 2 η 2 . The circulation of the vortex can be a fraction, which is different from the usual result for a one-component condensate. The kinetic helicity of vortices is calculated.
Abstract In contrast to one-component Bose-Einstein condensate case, the vortices in two-component condensate can have various complicated structures. The vortices in a space-homogeneous Bose-Einstein condensate have been studied in this paper. It is shown that the vortex structure is described by three dimensionless parameters. This is totally different from the usual one-component condensate case,where an isolated vortex is described by a parameterless dimensionless equation. The two-component vortex structure strongly depends on the sign of “interaction” constant of the components. A few types of vortices with different qualitative structure are explored. We show that the “super-density vortices” can exist, when the “interaction” constant is positive. The “super-density vortices” have the near-axis density greater than the equilibrium density of a homogeneous space Bose-Einstein condensate. We also show that the vortices with opposite direction of the condensate component rotation near the axis and far off the axis can exist.
The vortex structure in a space-homogeneous Bose-Einstein condensate of two particle species with different masses has been studied. It is shown that the vortex structure in the two-component condensate is described by three dimensionless parameters, which define the system. That is different from the usual one-component condensate, where the vortex is described by a parameter-less dimensionless equation with a fixed structure. A few types of vortices with different qualitative structure are explored. In particular, it is shown, that the vortex structure strongly depends on the sign of “interaction” constant of the components. There can exist the super-density vortices, when the “interaction” constant is positive. These vortices have the near-axis density greater than the equilibrium density of a homogeneous space Bose-Einstein condensate. We show that there can exist the vortices with opposite direction of the condensate rotation near the axis and far off the axis.
The vortex-vortex interaction in two-component Bose-Einstein condensates is shown to present characteristic effects not possible in single-component condensates. In particular, vortices in different components undergo separate, but concentric, orbits for both the same and opposite circulation. In addition, contrary to the well-known behavior in single-component condensates, vortices with the same circulation created in the less dense component merge in the presence of dissipation, allowing for the creation of metastable vortices with more than one quantum of circulation.
We explore vortices in non-equilibrium Dirac Bose-Einstein condensates (Dirac BEC) described by a stationary Dirac Gross-Pitaevskii equations (GPE). We find that the multi-component structure of Dirac equation enables the difference in phase winding of two condensates with respective phase winding number differing by one, ℓa−ℓb=±1\ell_a - \ell_b = \pm 1ℓa​−ℓb​=±1. We observe three classes of vortex states distinguished by their far-field behavior: A ring soliton on either of the two components in combination with a vortex on the other component, and, in the case of strong inter-component interactions, a vortex profile on both components. The latter are multiple core vortices due to the phase winding difference between the components. We also address the role of a Haldane gap on these vortices, which has a similar effect than inter-component by making the occupation on either sublattice more costly. We employ a numerical shooting method to reliably identify vortex solutions and use it to scan large parts of the phase space. We then use a classification algorithm on the integrated wavefunctions to establish a phase diagram of the different topological sectors.
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 demonstrate the existence of the helical vortex solution in two-component Bose-Einstein condensates which can be identified as a twisted vorticity flux. Based on this we argue that the recently proposed knot in two-component Bose-Einstein condensates can be interpreted as a vorticity knot, a vortex ring made of the helical vortex. This picture shows that the knot is made of two quantized vorticity fluxes linked together, whose topology π3(S2)π_3(S^2)π3​(S2) is fixed by the linking number of two vorticity fluxes. Due to the helical structure the knot has both topological and dynamical stability. We estimate the energy of the lightest knot to be about 3×10−3eV3\times 10^{-3} eV3×10−3eV.