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arXiv:cond-mat/0306549 · Physical Review A

Excitation spectrum of vortex lattices in rotating Bose-Einstein condensates

Seok Jin Choi, Leslie O. Baksmaty, Sungjong Woo, N. P. Bigelow

Published Sep 29, 2003• 19 citations
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Abstract

Using the coarse-grain averaged hydrodynamic approach, we calculate the excitation spectrum of vortex lattices sustained in rotating Bose-Einstein condensates. The spectrum gives the frequencies of the common-mode longitudinal waves in the hydrodynamic regime, including those of the higher-order compressional modes. Reasonable agreement with the measurements taken in a recent experiment is found, suggesting that one of the longitudinal modes reported in the experiment is likely to be the $n=2,m=0$ mode.

DOI
10.1103/physreva.68.031605

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Using the coarse-grain averaged hydrodynamic approach, we calculate the excitation spectrum of vortex lattices sustained in rotating Bose-Einstein condensates. The spectrum gives the frequencies of the common-mode longitudinal waves in the hydrodynamic regime, including those of the higher-order compressional modes. Reasonable agreement with the measurements taken in a recent experiment is found, suggesting that one of the longitudinal modes reported in the experiment is likely to be the n=2,m=0n=2,m=0n=2,m=0 mode.
The complete low-energy collective-excitation spectrum of vortex lattices is discussed for rotating Bose-Einstein condensates by solving the Bogoliubov-de Gennes equation, yielding, e.g., the Tkachenko mode recently observed at JILA. The totally symmetric subset of these modes includes the transverse shear, common longitudinal, and differential longitudinal modes. We also solve the time-dependent Gross-Pitaevskii equation to simulate the actual JILA experiment, obtaining the Tkachenko mode and identifying a pair of breathing modes. Combining both approaches allows one to unambiguously identify every observed mode.
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.
Using the coarse-grain averaged hydrodynamic approach, we calculate all low energy transverse excitation spectrum of a rotating Fermi superfluid containing vortex lattices for all regimes along the Bose-Einstein condensate--Bardeen-Cooper-Schrieffer (BEC-BCS) crossover. In the fast rotating regime, the molecular BEC enters into the lowest Landau level, but the superfluid in the unitarity and the BCS regimes occupies many low-lying Landau levels. The difference between the breathing mode frequencies at the BEC and unitarity limit shrinks to zero as the rotation speed approaches the radial trap frequency, in contrast to the finite difference in the nonrotating systems.
In a rotating two-component Bose-Einstein condensate (BEC), the traditional triangular vortex lattice can be replaced by a rectangular vortex lattice or even a structure characterized in terms of vortex sheets, depending on the interspecies interactions. We study the dynamics of this system by analyzing the Bogoliubov excitation spectrum. Excitations familiar to BEC vortex systems are found such as Tkachenko modes, hydrodynamic modes, and surface waves, however, the complex two-component morphology also gives rise to phenomena including shear flow between vortex sheets.