Density excitations of a harmonically trapped ideal gas

📝 Abstract

The dynamic structure factor of a harmonically trapped Bose gas has been calculated well above the Bose-Einstein condensation temperature by treating the gas cloud as a canonical ensemble of noninteracting classical particles. The static structure factor is found to vanish as wavenumber squared in the long-wavelength limit. We also incorporate a relaxation mechanism phenomenologically by including a stochastic friction force to study the dynamic structure factor. A significant temperature dependence of the density-fluctuation spectra is found. The Debye-Waller factor has been calculated for the trapped thermal cloud as function of wavenumber and of particle number. A substantial difference is found between clouds of small and large particle number.

💡 Analysis

The dynamic structure factor of a harmonically trapped Bose gas has been calculated well above the Bose-Einstein condensation temperature by treating the gas cloud as a canonical ensemble of noninteracting classical particles. The static structure factor is found to vanish as wavenumber squared in the long-wavelength limit. We also incorporate a relaxation mechanism phenomenologically by including a stochastic friction force to study the dynamic structure factor. A significant temperature dependence of the density-fluctuation spectra is found. The Debye-Waller factor has been calculated for the trapped thermal cloud as function of wavenumber and of particle number. A substantial difference is found between clouds of small and large particle number.

📄 Content

arXiv:0912.2849v1 [cond-mat.quant-gas] 15 Dec 2009 Density excitations of a harmonically trapped ideal gas Jai Carol Cruz[1], C. N. Kumar, K. N. Pathak Department of Physics, Panjab University, Chandigarh, India and J. Bosse Institute of Theoretical Physics, Freie Universit¨at Berlin, Berlin, Germany ( 6 th March, 2009 ) Abstract The dynamic structure factor S(q, ω) of a harmonically trapped Bose gas has been calculated well above the Bose–Einstein con- densation temperature by treating the gas cloud as a canonical ensemble of noninteracting classical particles. The static struc- ture factor is found to vanish ∝q2 in the long–wavelength limit. We also incorporate a relaxation mechanism phenomenologically by including a stochastic friction force to study S(q, ω). A signif- icant temperature dependence of the density–fluctuation spectra is found. The Debye–Waller factor has been calculated for the trapped thermal cloud as function of q and the number N of atoms. A substantial difference is found for small– and large–N clouds. Keywords: Trapped classical gas, dynamical structure factor, Debye–Waller factor PACS: 51.90+r, 05.20.Jj, 61.20.Lc 1 1 Introduction There has been much interest in the study of trapped atomic gases during more than a decade.[2, 3] Experimentalists have observed not only density profiles of these systems but they have successfully detected particle–number and –density fluctuations in ultra–cold atomic gases.[4] Progress has been made in understanding spatial and temporal correlations of density fluctua- tions in a variety of inhomogeneous cold–atom systems [5, 6, 7, 8] for T=0 as well as below the Bose–Einstein condensation (BEC) temperature Tc. In addition there have been extensive studies of dynamical density fluctuations for homogeneous systems. [9, 10] The purpose of this work is to calculate the diagonal elements of the ma- trix of time–dependent correlation functions of density fluctuations for a harmonically trapped classical ideal gas, which determine the coherent scat- tering properties of the system. Motivation for calculating the dynamical structure factor is its expected strong temperature dependence which could be of use in estimating the cloud temperature experimentally. Moreover, the exactly solvable model not only provides the limiting case of a system in the absence of interactions but also furnishes a useful result for trapped dilute gases well above the BEC temperature. Justification for a classical treatment is based on the fact that for an atom trap of typical trap fre- quency Ω=120×2π s−1 containing N =2×104 rubidium atoms which implies a critical temperature Tc=¯hΩN 1/3/[ζ(3)1/3kB] ≈147 nK of Bose–Einstein condensation [2, Chap. 2.2], the ratio ¯hΩover kBT is very small in the temperature range well above Tc, where many experiments are performed, because ¯hΩ/(kBTc) ≈0.04. Moreover, at these temperatures the reduced density ρ=nλ3 ≈0.01 is small implying λ ≈0.008 L, i.e. the deBroglie wave- length λ is still very small compared to the linear extension L of the atom cloud. In addition to the restoring force acting on each particle due to the trap potential, we allow for a frictional force when studying the dynamical be- havior of the trapped gas. Simultaneously, a time–dependent random force is assumed to act on each oscillator. This stochastic force will keep particles moving, thus preventing them from being slowed down by friction and com- ing to a rest in the trap–potential minimum. Thermal averages are therefore understood to include additional averaging over stochastic variables. In this 2 way, the assumption of time–translational invariance will remain justified throughout in spite of frictional forces. In Sec. 2, we present the basic definitions and the correlation functions to be studied. In Sec. 3, the one–and two–particle densites and the static structure factor are obtained. In Sec. 4, the intermediate scattering function has been calculated in the presence of damping including a stochastic force with Gaussian noise. Resulting low–friction density–relaxation spectra are discussed as function of temperature. In Sec. 5, density excitations for the limiting case of zero friction and the Debye–Waller factor (DWF) are discussed. Finally, concluding remarks are given in Sec. 6. 2 Theoretical Considerations In the description of dynamical properties of a many–particle system, the correlation function of density fluctuations δN(r, t), F(r, r′, t) = 1 N ⟨δN(r, t) δN(r′, 0) ⟩, (1) and its spatial Fourier transform Fq q′(t) = Z d3r Z d3r′ e−iq·r+iq′·r′F(r, r′, t) = 1 N ⟨δNq(t) δNq′(0)∗⟩ (2) play an important role. Here ⟨. . . ⟩denotes a thermal equilibrium average, and the dynamical variable N(r, t)= PN j=1 δ(r −rj(t)) describes the particle density at space point r and time t, which, when integrated over all space, sums up to the total number of particles, N = R d3r N(r, t). Correspondingly, the density variable in Fourier space is given by Nq(t) = Z d3r e−iq·r N(r, t) = N X j

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