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.
We investigate the low-energy excitations of the spherically and axially trapped atomic Bose-Einstein condensate coupled to a molecular Bose gas by coherent Raman transitions. We apply the sum-rule approach of many-body response theory to derive the low-lying collective excitation frequencies of the hybrid atom-molecular system. The atomic and molecular ground-state densities obtained in Gross-Pitaevskii and modified Gross-Pitaevskii (including the higher order Lee-Huang-Yang term in interatomic interaction) approaches are used to find out the individual energy components and hence the excitation frequencies. We obtain different excitation energies for different angular momenta and study their characteristic dependence on the effective Raman detuning, the scattering length for atom-atom interaction, and the intensities of the coupling lasers. We show that the inclusion of the higher-order nonlinear interatomic interaction in modified Gross-Pitaevskii approach introduces significant corrections to the ground-state properties and the excitation frequencies both for axially and spherically trapped coupled
87Rb condensate system with the increase in the
s-wave scattering length (for peak gas-parameter $\ensuremath{\geqslant}{10}^{\ensuremath{-}3}$). It has been shown that the excitation frequencies decrease with the increase in the effective Raman detuning as well as the
s-wave scattering length, whereas excitation frequencies increase with the increase in the atom-molecular coupling strength. The frequencies in modified Gross-Pitaevskii approximation exhibit an upward trend after a certain value of scattering length and also largely deviate from the Gross-Pitaevskii results with the increase in
s-wave scattering length. The strong dependence of excitation frequencies on the laser intensities used for Raman transitions manifests the role of atom-molecular coupling strength on the control of collective excitations. The collective excitation frequencies for the hybrid atom-molecular BEC differ significantly from the excitation frequencies of a pure atomic BEC system when the atom-to-molecule conversion efficiency increases due to the decrease in the effective Raman detuning and increase in the atom-molecule coupling strength.
Abstract We theoretically investigate measurement-based feedback control over the motional degrees of freedom of an oblate quasi-2D atomic Bose-Einstein condensate (BEC) subject to continuous density monitoring. We develop a linear-quadratic-Gaussian (LQG) model that describes the multi-mode dynamics of the condensate's collective excitations under continuous measurement and control. Crucially, the multi-mode cold-damping feedback control we consider uses a realistic state-estimation scheme that does not rely upon a particular model of the atomic dynamics. We present analytical results showing that collective excitations can be cooled to below single-phonon average occupation (ground-state cooling) across a broad parameter regime, and identify the conditions under which the lowest steady-state phonon occupation is asymptotically achieved. Further, we develop multi-objective optimization methods that explore the trade-off between cooling speed and the final energy of the cloud, and provide numerical simulations demonstrating the ground-state cooling of the lowest ten motional modes above the condensate ground state. Our investigation provides concrete guidance on the feedback control design and parameters needed to experimentally realize a feedback-cooled BEC.