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Journal of Mathematical Physics

Incompressible and compressible limits of two-component Gross–Pitaevskii equations with rotating fields and trap potentials

Chiun‐Chang Lee, Tai‐Chia Lin

Published Apr 1, 2008• 10 citations
Publisher

Abstract

The relaxation of turbulence is crucial for the formation of vortex lattices and vortex sheets separated by an interface in rapidly rotating two-component Bose–Einstein condensates (BECs) depicted by two-component systems of Gross–Pitaevskii equations (GPEs). To see the relaxation, we study the incompressible limit of two-component systems of GPEs with initial data having a large number of vortices and an interface. The incompressible limit may provide the relaxation of turbulence such that the turbulent flow can be self-organized into vortex crystals. Recently, a rich variety of dynamical phenomena ranging from shock-wave formation to anisotropic sound propagation has been observed in rotating BECs. To see the dynamical phenomena of rotating two-component BECs, we study the compressible limit of two-component systems of GPEs. The compressible limit may describe geostrophic flow and provide rich dynamical phenomena in rotating two-component BECs.

DOI
10.1063/1.2912716

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The relaxation of turbulence is crucial for the formation of vortex lattices and vortex sheets separated by an interface in rapidly rotating two-component Bose–Einstein condensates (BECs) depicted by two-component systems of Gross–Pitaevskii equations (GPEs). To see the relaxation, we study the incompressible limit of two-component systems of GPEs with initial data having a large number of vortices and an interface. The incompressible limit may provide the relaxation of turbulence such that the turbulent flow can be self-organized into vortex crystals. Recently, a rich variety of dynamical phenomena ranging from shock-wave formation to anisotropic sound propagation has been observed in rotating BECs. To see the dynamical phenomena of rotating two-component BECs, we study the compressible limit of two-component systems of GPEs. The compressible limit may describe geostrophic flow and provide rich dynamical phenomena in rotating two-component BECs.
Counter-rotating vortices in miscible two-component Bose-Einstein condensates, in which superflows counter-rotate between the two components around the overlapped vortex cores, are studied theoretically in a pancake-shaped potential. In a linear stability analysis with the Bogoliubov--de Gennes model, we show that counter-rotating vortices are dynamically unstable against splitting into multiple vortices. The instability shows characteristic behaviors as a result of counter-superflow instability, which causes relaxation of relative flows between the two components in binary condensates. The characteristic behaviors are completely different from those of multiquantum vortices in single-component Bose-Einstein condensates; the number of vortices generated by the instability can become larger than the initial winding number of the counter-rotating vortex. We also investigate the nonlinear dynamics of the instability by numerically solving the Gross-Pitaevskii equations. The nonlinear dynamics drastically changes when the winding number of counter-rotating vortices becomes larger, which lead to nucleation of vortex pairs outside of the vortex core. The instability eventually develops into turbulence after the relaxation of the relative rotation between the two components.
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.
This report addresses our recent study on pattern formation in two-component Bose-Einstein condensates(BECs), focusing on the dynamics of multiple-domain formation and the structure of vortex states in rotating condensates. Cross-phase modulation instability gives rise to formation of multiple domains that alternate two components, which are consistent with the experimental observation by the MIT group. Rotating two-component BECs exhibit a rich variety of vortex states such as square lattices and vortex sheets, which depend on the strength of intercomponent interaction and rotation frequency.
We present a study on the development of rotating turbulence in Bose-Einstein condensates with a dissipative Gross-Pitaevskii model. Turbulence is generated by driving the lattice of quantized vortices in a harmonic potential with a random forcing potential. As the turbulence progressed, the initial alignment of vortices underwent slight disruptions, thereby increasing the high-wavenumber components of the kinetic energy. In the turbulent state, the distribution of incompressible kinetic energy exhibits milder anisotropy than that in the initial lattice state and demonstrates a scaling behavior of kz−2.5k_z^{-2.5}kz−2.5​ in the direction parallel to the rotation axis. In contrast, the compressible kinetic energy exhibits an isotropic scaling behavior at high wavenumbers.
We consider vortex dynamics in the context of Bose–Einstein condensates (BECs) with a rotating trap, with or without anisotropy. Starting with the Gross–Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.