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arXiv:cond-mat/0201115 · Physical Review Letters

Quadrupole Collective Modes in Trapped Finite-Temperature Bose-Einstein Condensates

B. Jackson, E. Zaremba

Published Apr 19, 2002• 75 citations
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Abstract

Finite-temperature simulations are used to study quadrupole excitations of a trapped Bose-Einstein condensate. We focus specifically on the m = 0 mode, where a long-standing theoretical problem is to account for an anomalous variation of the mode frequency with temperature. We explain this behavior in terms of the excitation of two separate modes, corresponding to the coupled motion of the condensate and the thermal cloud. The relative amplitude of the modes depends sensitively on the temperature and on the frequency of the harmonic drive used to excite them. Good agreement with experiment is found for appropriate drive frequencies.

DOI
10.1103/physrevlett.88.180402

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Finite-temperature simulations are used to study quadrupole excitations of a trapped Bose-Einstein condensate. We focus specifically on the m = 0 mode, where a long-standing theoretical problem is to account for an anomalous variation of the mode frequency with temperature. We explain this behavior in terms of the excitation of two separate modes, corresponding to the coupled motion of the condensate and the thermal cloud. The relative amplitude of the modes depends sensitively on the temperature and on the frequency of the harmonic drive used to excite them. Good agreement with experiment is found for appropriate drive frequencies.
We derive a simple formula for the finite-temperature shifts of the collective excitations of a Bose-Einstein condensate. To test the validity of our treatment we apply it to explain the anomalous behavior of the ``m=2'' excitation frequency in the recent JILA experiments (D. S. Jin et al, PRL, 78 (1997) 764), where this frequency does not approach its ``noninteracting'' value at ω=2ωρω= 2 ω_ρω=2ωρ​ but, instead, decreases with temperature. It turns out that the effect is mainly governed by a resonance between the condensate excitation frequency and one of the quantum eigenfrequencies of the thermal cloud. Comparison of our predictions with the JILA experimental data shows a good agreement.
We present a finite temperature field theory for collective excitations of trapped Bose condensates which includes the dynamics of the thermal cloud. In spherical traps we show that excitations couple strongly to a small number of modes, giving resonance structure in their frequency spectra. Where possible, we derive energy shifts and lifetimes of excitations. For the l = 0 mode we show that the simple picture of a decay rate fails, which should be observable in suitable experiments. It also suggests a possible explanation for the anomalous behavior of the m = 0 mode observed in anisotropic traps.
The dynamics of Bose-Einstein condensate (BEC) is studied at nonzero temperatures using our variational time-dependent-Hartree-Fock-Bogoliubov formalism. We have shown that this approach is an efficient tool to study the expansion and collective excitations of the condensate, the thermal cloud, and the anomalous correlation function at nonzero temperatures. We have found that the condensate and the anomalous density have the same breathing oscillations. We have investigated, on the other hand, the behavior of a single quantized vortex in a harmonically trapped BEC at nonzero temperatures. Generalized expressions for vortex excitations, vortex core size, and Kelvin modes have been derived. An important and somehow surprising result is that the numerical solution of our equations predicts that the vortex core is partially filled by the thermal atoms at nonzero temperatures. We have shown that the effect of thermal fluctuations is important and it may lead to enhancing the size of the vortex core. The behavior of the singly anomalous vortex has also been studied at nonzero temperatures.
Recent experimental observations of collective excitations of Bose condensed atomic vapours have stimulated interest in the microscopic description of the dynamics of a Bose-Einstein condensate confined in an external potential. We present a finite temperature field theory for collective excitations of trapped Bose-Einstein condensates and use a finite-temperature linear response formalism, which goes beyond the simple mean-field approximation of the Gross-Pitaevskii equation. The effect of the non-condensed thermal atoms we include using perturbation theory in a quasiparticle basis. This presents a simple scheme to understand the interaction between condensate and non-condensed atoms and enables us to include the effect the condensate has on collision dynamics. At first we limit our treatment to the case of a spatially homogeneous Bose gas. We include the effect of pair and triplet anomalous averages and thus obtain a gapless theory for the excitations of a weakly interacting system, which we can link to well known results for Landau and Beliaev damping rates. A gapless theory for trapped systems with a static thermal component follows straightforwardly. We then investigate finite temperature excitations of a condensate in a spherically symmetric harmonic trap. We avoid approximations to the density of states and thus emphasise finite size aspects of the problem. We show that excitations couple strongly to a restricted number of modes, giving rise to resonance structure in their frequency spectra. Where possible we derive energy shifts and lifetimes of excitations. For one particular mode, the breathing mode, the effects of the discreteness of the system are sufficiently pronounced that the simple picture of an energy shift and width fails. Experiments in spherical traps have recently become feasible and should be able to test our detailed quantitative predictions.
We have measured the effect of dipole-dipole interactions on the frequency of a collective mode of a Bose-Einstein condensate. At relatively large numbers of atoms, the experimental measurements are in good agreement with zero temperature theoretical predictions based on the Thomas-Fermi approach. Experimental results obtained for the dipolar shift of a collective mode show a larger dependency to both the trap geometry and the atom number than the ones obtained when measuring the modification of the condensate aspect ratio due to dipolar forces. These findings are in good agreement with simulations based on a Gaussian ansatz.
We optimize the classical field approximation of the version described by M. Brewczyk, M. Gajda, and K. Rz\k{a}\ifmmode \dot{z}\else \.{z}\fi{}ewski [J. Phys. B 40, R1 (2007)] for the oscillations of a Bose gas trapped in a harmonic potential at nonzero temperatures, as experimentally investigated by Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. Similar to experiment, the system response to external perturbations strongly depends on the initial temperature and the symmetry of perturbation. While for lower temperatures the thermal cloud follows the condensed part, for higher temperatures the thermal atoms oscillate rather with their natural frequency, whereas the condensate exhibits a frequency shift toward the thermal cloud frequency (m=0m=0m=0 mode) or in the opposite direction (m=2m=2m=2 mode). In the latter case, for temperatures approaching critical, we find that the condensate begins to oscillate with the frequency of the thermal atoms, as in the m=0m=0m=0 mode. A broad range of frequencies of the perturbing potential is considered.