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arXiv:cond-mat/0109149 · Physical Review A

Finite-temperature theory of the scissors mode in a Bose gas using the moment method

Tetsuro Nikuni

Published Feb 11, 2002• 15 citations
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Abstract

We use a generalized Gross-Pitaevskii equation for the condensate and a semiclassical kinetic equation for the noncondensate atoms to discuss the scissors mode in a trapped Bose-condensed gas at finite temperatures. Both equations include the effect of collisions between the condensate and noncondensate atoms. We solve the coupled moment equations describing oscillations of the quadrupole moments of the condensate and noncondensate components to find the collective-mode frequencies and collisional damping rates as a function of temperature. Our calculations extend those of Gu\'ery-Odelin and Stringari at $T=0$ and in the normal phase. They complement the numerical results of Jackson and Zaremba, although Landau damping is left out of our approach. Our results are also used to calculate the quadrupole response function, which is related to the moment of inertia. It is shown explicitly that the moment of inertia of a trapped Bose gas at finite temperatures involves a sum of an irrotational component from the condensate and a rotational component from the thermal-cloud atoms.

DOI
10.1103/physreva.65.033611

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We use a generalized Gross-Pitaevskii equation for the condensate and a semiclassical kinetic equation for the noncondensate atoms to discuss the scissors mode in a trapped Bose-condensed gas at finite temperatures. Both equations include the effect of collisions between the condensate and noncondensate atoms. We solve the coupled moment equations describing oscillations of the quadrupole moments of the condensate and noncondensate components to find the collective-mode frequencies and collisional damping rates as a function of temperature. Our calculations extend those of Gu'ery-Odelin and Stringari at T=0T=0T=0 and in the normal phase. They complement the numerical results of Jackson and Zaremba, although Landau damping is left out of our approach. Our results are also used to calculate the quadrupole response function, which is related to the moment of inertia. It is shown explicitly that the moment of inertia of a trapped Bose gas at finite temperatures involves a sum of an irrotational component from the condensate and a rotational component from the thermal-cloud atoms.
The dynamics of a trapped Bose-condensed gas at finite temperatures is described by a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud, solved using N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. We use this scheme to investigate scissors modes in anisotropic traps as a function of temperature. Frequency shifts and damping rates of the condensate mode are extracted, and are found to be in good agreement with recent experiments.
We utilize a two-gas model to simulate collective oscillations of a Bose-Einstein condensate at finite temperatures. The condensate is described using a generalized Gross-Pitaevskii equation, which is coupled to a thermal cloud modeled by a Monte Carlo algorithm. This allows us to include the collective dynamics of both the condensed and noncondensed components self-consistently. We simulate quadrupolar excitations, and measure the damping rate and frequency as a function of temperature. We also observe revivals in condensate oscillations at high temperatures, and in the thermal cloud at low temperature. Extensions of the model to include nonequilibrium effects and describe more complex phenomena are discussed.
In this paper, we develop a numerical procedure for investigating the dynamics of trapped Bose gases based on the ZGN theory. The dynamical equations used consist of a generalized Gross-Pitaevskii equation for the condensate order parameter and a semiclassical kinetic equation for the thermal cloud. The former is solved using a fast Fourier transform split-operator technique while the Boltzmann equation is treated by means of N-body simulations. The two components are coupled by mean fields as well as collisional processes that transfer atoms between the two. This scheme has been applied to a model equilibration problem, dipole oscillations in isotropic traps and scissors modes in anisotropic traps. In the case of the latter, the frequencies and damping rates of the condensate mode have been extracted from the simulations for a wide range of temperatures. Good agreement with recent experiments has been found.
We consider a model of a dilute Bose-Einstein condensed gas at finite temperatures, where the condensate coexists in a trap with a cloud of thermal excitations. Within the Zaremba, Nikuni, and Griffin formalism, the dynamics of the condensate is described by a generalized Gross-Pitaevskii equation, while the thermal cloud is represented by a semiclassical kinetic equation. Our numerical approach simulates the kinetic equation using a cloud of representative test particles, while collisions are treated by means of a Monte Carlo sampling technique. A full three-dimensional split-operator fast Fourier transform method is used to evolve the condensate wave function. We give details regarding the numerical methods used and discuss simulations carried out to test the accuracy of the numerics. We use this scheme to simulate the monopole mode in a spherical trap. The dynamical coupling between the condensate and thermal cloud is responsible for frequency shifts and damping of the condensate collective mode. We compare our results to previous theoretical approaches, not only to confirm the reliability of our numerical scheme, but also to check the validity of approximations which have been used in the past.
We have studied the properties of the scissors mode of a trapped Bose-Einstein condensate of 87Rb atoms at finite temperature. We measured a significant shift in the frequency of the mode below the hydrodynamic limit and a strong dependence of the damping rate as the temperature increased. We compared our damping rate results to recent theoretical calculations for other observed collective modes, finding a fair agreement. From the frequency measurements we deduce the moment of inertia of the gas and show that it is quenched below the transition point, because of the superfluid nature of the condensed gas.
We develop a formalism based on the projected Gross-Pitaevskii equation to simulate the finite-temperature collective mode experiments of Jin et al. [Phys. Rev. Lett. 78, 764 (1997)]. We examine the m=0m=0m=0 and m=2m=2m=2 quadrupolar modes on the temperature range 0.51Tc−−0.83Tc0.51{T}_{c}--0.83{T}_{c}0.51Tc​−−0.83Tc​ and calculate the frequencies of, and phase between, the condensate and noncondensate modes, and the condensate mode damping rate. This study is the first quantitative comparison of the projected Gross-Pitaevskii equation to experimental results in a dynamical regime.