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arXiv:0912.1276 · New Journal of Physics

Rossby waves in rapidly rotating Bose–Einstein condensates

Hugo Terças, João P. A. Martins, José Tito Mendonça

Published Sep 1, 2010• 5 citations
arXivHTMLPDF

Abstract

We predict and describe a new collective mode in rotating Bose–Einstein condensates, which is very similar to Rossby waves in geophysics. In the regime of fast rotation, the Coriolis force dominates the dynamics and acts as a restoring force for acoustic-drift waves along the condensate. We derive a nonlinear equation that includes the effects of both zero-point pressure and inhomogeneity of the gas. It is shown that such waves have negative phase speed, propagating in the opposite sense of the rotation. We discuss different equilibrium configurations and compare them to those resulting from the Thomas–Fermi approximation.

DOI
10.1088/1367-2630/12/9/093001

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We predict and describe a new collective mode in rotating Bose–Einstein condensates, which is very similar to Rossby waves in geophysics. In the regime of fast rotation, the Coriolis force dominates the dynamics and acts as a restoring force for acoustic-drift waves along the condensate. We derive a nonlinear equation that includes the effects of both zero-point pressure and inhomogeneity of the gas. It is shown that such waves have negative phase speed, propagating in the opposite sense of the rotation. We discuss different equilibrium configurations and compare them to those resulting from the Thomas–Fermi approximation.
As the Earth rotates, the Coriolis force causes various oceanic and atmospheric waves to be trapped along the equator, including Kelvin, Yanai, Rossby, and Poincar\'e modes. It has been demonstrated that the mathematical origin of these waves is related to the nontrivial topology of the underlying hydrodynamic equations. Inspired by recent observations of Bose-Einstein condensation (BEC) in bubble-shaped traps in microgravity ultracold quantum gas experiments, we demonstrate that equatorial modes are supported by a rapidly rotating condensate in a spherical geometry. Using a zero-temperature coarse-grained hydrodynamic framework, we reformulate the coupled oscillations of the superfluid and the Abrikosov vortex lattice resulting from rotation as a Schr\"odinger-like eigenvalue problem. The resulting non-Hermitian Hamiltonian is topologically nontrivial. We also solve the hydrodynamic equations for a spherical geometry and find that the rotating superfluid hosts Kelvin, Yanai, and Poincar\'e equatorial modes, but not the Rossby mode. Our predictions can be tested with state-of-the-art bubble-shaped trapped BEC experiments.
A Rossby-Haurwitz (RH) wave is an excitation mode of a fluid on a rotating spherical surface, which propagates westward in the rotating frame of reference. Motivated by the recent realization of the shell-shaped Bose-Einstein condensate in microgravity, we investigate the RH wave in a superfluid rotating on a spherical-surface geometry. We employ the point-vortex model and the three-dimensional Gross-Pitaevskii equation, and numerically demonstrate that RH waves can be observed in the system of a superfluid with quantized vortices.
The present paper suggests the continuum theory of Tkachenko modes in a rotating two-dimensional Bose-Einstein condensate taking into account density inhomogeneity and compressibility of the condensate. The theory is based on the solution of coupled hydrodynamic equations for vortex and liquid motion with proper boundary conditions, which were derived for the condensate described by the Thomas-Fermi approximation. Compressibility becomes essential at rapid rotation with angular velocity close to the trap frequency. The theory is in reasonable agreement with experimental observation of Tkachenko modes.
Abstract A general approach to equilibrium and oscillations in a turbulent Bose–Einstein condensate, valid in the mean-field approximation, is proposed. Two different wave equations, describing the quasi-static equilibrium and the elementary excitations of the medium are derived. In particular, it is shown that a modified Thomas–Fermi equilibrium can be defined in the presence of an arbitrary spectrum of oscillations. The oscillating modes are also determined using Bogoliubov–de Gennes (BdG) equations, valid in different geometrical configurations, such as infinite, cylindrical and toroidal. The case of a non-homogenous density is considered, which includes the modified equilibrium profiles. A general form of dispersion relation for the elementary excitations is derived. Twisted phonons, carrying a finite amount of angular momentum, are shown to be a particular class of BdG modes. They were previously derived from fluid equations. Nonlinear analysis also shows that envelope dark solitons can exist, and correspond to a rarefaction of twisted phonons.
After delineating the physical regimes that vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate $\ensuremath{\Omega}$ increases, we derive the normal modes of the vortex lattice in two dimensions at zero temperature. Taking into account effects of the finite compressibility, we find an inertial mode of frequency $\ensuremath{\geqslant}2\ensuremath{\Omega}$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\ensuremath{\Omega}$ is small compared with the lowest compressional mode frequency, to quadratic in the wave vector in the opposite limit. We calculate the correlation functions of vortex displacements and phase, density and superfluid velocities, and find that the zero-point excitations of the soft quadratic Tkachenko modes lead in a large system to a loss of long-range phase correlations, growing logarithmically with distance, and hence lead to a fragmented state at zero temperature. The vortex positional ordering is preserved at zero temperature, but the thermally excited Tkachenko modes cause the relative positional fluctuations to grow logarithmically with separation at finite temperature. The superfluid density, defined in terms of the transverse velocity autocorrelation function, vanishes at all temperatures. Finally we construct the long-wavelength single-particle Green's function in the rotating system and calculate the condensate depletion as a function of temperature.
We study a rapidly rotating Bose–Einstein condensate confined to a finite trap in the framework of two-dimensional Gross–Pitaevskii theory in the strong coupling (Thomas–Fermi) limit. Denoting the coupling parameter by 1/ε 2 and the rotational velocity by Ω, we evaluate exactly the next to the leading-order contribution to the ground-state energy in the parameter regime |log ε| ≪ Ω ≪ 1/(ε 2 |log ε|) with ε → 0. While the TF energy includes only the contribution of the centrifugal forces the next order corresponds to a lattice of vortices whose density is proportional to the rotational velocity.