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Physical Review A

Bose-Einstein condensates in 85Rb gases at higher densities

Asaad R. Sakhel, Jonathan L. DuBois, H. R. Glyde

Published Dec 31, 2002• 11 citations
Publisher

Abstract

The Bose-Einstein condensation in trapped gases of ${}^{85}\mathrm{Rb}$ at densities beyond the dilute regime can now be realized taking advantage of Feshbach resonances. The gas density may be characterized by the parameter ${\mathrm{na}}^{3},$ where $n=N/V$ is the number density and a is the scattering length which is tuned using Feshbach resonance. We explore the properties of interacting trapped bosons in the density range $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}<~{\mathrm{na}}^{3}<~5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ using Monte Carlo methods to compare with experiments. We find that there is a significant depletion of the condensate at $T=0\mathrm{K},$ for example, $25%$ at ${\mathrm{na}}^{3}{=10}^{\ensuremath{-}2}.$ The condensate is not concentrated at the center of the trap but is spread out over four or five trap lengths. The condensate density distributions and the total density distributions within the trap are very similar and the condensate fraction is $100%$ at the edges of the trap.

DOI
10.1103/physreva.66.063610

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The Bose-Einstein condensation in trapped gases of 85Rb{}^{85}\mathrm{Rb}85Rb at densities beyond the dilute regime can now be realized taking advantage of Feshbach resonances. The gas density may be characterized by the parameter na3,{\mathrm{na}}^{3},na3, where n=N/Vn=N/Vn=N/V is the number density and a is the scattering length which is tuned using Feshbach resonance. We explore the properties of interacting trapped bosons in the density range 5\ifmmode×\else\texttimes\fi10\ensuremath−3< na3< 5\ifmmode×\else\texttimes\fi10\ensuremath−25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}<~{\mathrm{na}}^{3}<~5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}5\ifmmode×\else\texttimes\fi10\ensuremath−3< na3< 5\ifmmode×\else\texttimes\fi10\ensuremath−2 using Monte Carlo methods to compare with experiments. We find that there is a significant depletion of the condensate at T=0K,T=0\mathrm{K},T=0K, for example, 2525%25 at na3=10\ensuremath−2.{\mathrm{na}}^{3}{=10}^{\ensuremath{-}2}.na3=10\ensuremath−2. The condensate is not concentrated at the center of the trap but is spread out over four or five trap lengths. The condensate density distributions and the total density distributions within the trap are very similar and the condensate fraction is 100100%100 at the edges of the trap.
Bose-Einstein condensation has been achieved in a magnetically trapped sample of 85Rb atoms. Stable condensates of up to 104 atoms have been created by using a magnetic-field-induced Feshbach resonance to reverse the sign of the zero-field scattering length. These condensates provide unique opportunities for the study of BEC physics. The variation of the scattering length near the resonance has been used to magnetically tune the condensate self-interaction energy over a very wide range. This range extended from very strong repulsive self-interactions to large attractive ones. The effect of moving the condensate through the Feshbach resonance has been studied and compared with theory. Long-lived metastable condensates with attractive interactions have been produced near the zero of the Feshbach resonance. The transition from repulsive to attractive interactions can lead to a “collapse” of the condensate in which the cloud shrinks below our resolution limit, loses a significant number of atoms due to inelastic losses, and emits a burst of high-energy atoms.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
We present semianalytical many-body results for energies and excitation frequencies for an inhomogeneous Bose-Einstein condensate over a wide range of atom numbers NNN for both small sss-wave scattering lengths, typical of most laboratory experiments, and large scattering lengths, achieved by tuning through a Feshbach resonance. Our dimensional perturbation treatment includes two-body correlations at all orders and yields analytical results through first order by taking advantage of the high degree of symmetry of the condensate at the zeroth-order limit. Because NNN remains a parameter in our analytical results, the challenge of calculating energies and excitation frequencies does not rise with the number of condensate atoms. In this proof-of-concept paper the atoms are confined in a spherical trap and are treated as hard spheres. Our many-body calculations compare well to Gross-Pitaevskii results in the weakly interacting regime and depart from the mean-field approximation as the density approaches the strongly interacting regime. The excitation frequencies provide a particularly sensitive test of beyond-mean-field corrections. For example, for N=2000N=2000N=2000 atoms and an experimentally realized large scattering length of $a=0.433{a}_{ho}\phantom{\rule{0.3em}{0ex}}({a}_{ho}=\sqrt{\ensuremath{\hbar}∕m{\ensuremath{\omega}}_{ho}})$ we predict a 7575%75 shift from the mean-field breathing mode frequency.
Bose-Einstein condensation in trapped gases of 85 Rb atoms can now be obtained in the large gas-parameter region in presence of Feshbach resonance and the gas density which is characterized by peak gas parameter x pk can attain the value ~ 10 -2 . For such large gas-parameter, the first correction term to the Bogoliubov energy functional is considered to give modified Gross-Pitaevskii (MGP) equation. In this work analytical expressions for the chemical potential, the density of particles and the excitation frequency of the monopole mode have been derived in Thomas-Fermi(TF) approximated approach to the MGP equation (modified TF approach) following method of higher order iteration for Bose-Einstein condensate (BEC) of atoms in spherically symmetric trap. We have also solved the MGP equation by using Runge-Kutta method for integration of nonlinear differential equation considering spherically symmetric trap. The analytical results for chemical potential and density of particles for BEC of 10 3 and 10 4 85 Rb atoms in a spherically symmetric trap are in very good agreement with the numerical results from MGP equation at low as well as at large scattering length i.e, at large gas-parameter region. The frequency of monopole mode calculated using the density of particles in the ground state (found out in higher order iterative method) are also in very good agreement with the existing numerical results. The difference between the analytical and numerical results decreases as the value of scattering length i.e, gas-parameter is increased.
Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al. , 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al. , 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, T BEC ), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ( r , t ). As shown by research since 1995, this pure condensate domain is very rich in physics. The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.
Bose-Einstein condensates of 10485Rb{10}^{4}{}^{85}\mathrm{Rb}10485Rb atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter xpk{x}_{\mathrm{pk}}xpk​ of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at z=0z=0z=0 may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.