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Physical Review A

Photoexcitation of Na Bose-Einstein condensates

Michael Stuart Pindzola, Sh. A. Abdel-Naby, F. Robicheaux

Published Nov 5, 2012• 0 citations
Publisher

Abstract

The photoexcitation of Na Bose-Einstein condensates is studied by direct solution of the Gross-Pitaevskii equation in both cylindrical and Cartesian coordinates. As the frequency of a Bragg scattering potential is varied the first dipole mode of the ground state is brought into resonance. The dipole mode frequency is verified by diagonalization of the Bogoliubov--de Gennes matrix in cylindrical coordinates. Dipole mode frequencies are found for Na condensates bound in both isotropic and nonisotropic magnetic traps.

DOI
10.1103/physreva.86.055601

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The photoexcitation of Na Bose-Einstein condensates is studied by direct solution of the Gross-Pitaevskii equation in both cylindrical and Cartesian coordinates. As the frequency of a Bragg scattering potential is varied the first dipole mode of the ground state is brought into resonance. The dipole mode frequency is verified by diagonalization of the Bogoliubov--de Gennes matrix in cylindrical coordinates. Dipole mode frequencies are found for Na condensates bound in both isotropic and nonisotropic magnetic traps.
Collective modes of the ground and excited states of Bose?Einstein condensates are calculated by diagonalization of the Bogoliubov?de Gennes matrix in cylindrical coordinates. For an isotropic magnetic trap, the first breathing mode frequencies of the ground and dark soliton excited states are found to be in excellent agreement with a direct solution of the Gross?Pitaevskii equation with a time-dependent driving potential. In addition, the matrix diagonalization yields densities and frequencies for second breathing modes and higher order multipole modes. For an axially symmetric anisotropic trap, a considerable number of collective modes of the ground and dark soliton excited states are calculated. Prolate traps are found to have many more low-frequency multipolar modes than found in oblate traps for both ground and dark soliton excited states.
Collective modes of the ground and excited states of Bose–Einstein condensates are studied by numerical solution of the time-dependent Gross–Pitaevskii equation in three spatial dimensions. Scissors modes are studied by quadrupole excitation of the ground and excited states of 23 Na condensates in a fully anisotropic magnetic trap. The frequencies of the collective modes for the ground state condensates are found to be in good agreement with previous linear response theory. New results are presented for the frequencies of the collective modes for excited state condensates which exhibit solitary density depressions. A key to the proper characterization of the wide variety of dark-soliton states in atomic condensates is the experimental observation of collective modes.
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.
Dark soliton states of Bose-Einstein condensates in harmonic traps are studied both analytically and computationally by the direct solution of the Gross-Pitaevskii equation in three dimensions. The ground and self-consistent excited states are found numerically by relaxation in imaginary time. The energy of a stationary soliton in a harmonic trap is shown to be independent of density and geometry for large numbers of atoms. Large-amplitude field modulation at a frequency resonant with the energy of a dark soliton is found to give rise to a state with multiple vortices. The Bogoliubov excitation spectrum of the soliton state contains complex frequencies, which disappear for sufficiently small numbers of atoms or large transverse confinement. The relationship between these complex modes and the snake instability is investigated numerically by propagation in real time.
Bogoliubov–de Gennes equations (BdGEs) for collective excitations from a trapped Bose–Einstein condensate described by a spatially smooth ground-state wavefunction can be treated analytically. A new class of closed solutions for the BdGEs is obtained for the one-dimensional (1D) and 3D spherically harmonic traps. The solutions of zero-energy mode of the BdGEs are also provided. The eigenfunctions of the excitations consist of zero-energy mode, zero-quantum-number mode and entire excitation modes when the approximate ground state is a background Bose gas sea.
The calculation of properties of Bose-Einstein condensates with dipolar interactions has proven a computationally intensive problem due to the long range nature of the interactions, limiting the scope of applications. In particular, the lowest lying Bogoliubov excitations in three-dimensional harmonic trap with cylindrical symmetry were so far computed in an indirect way, by Fourier analysis of time-dependent perturbations, or by approximate variational methods. We have developed a very fast and accurate numerical algorithm based on the Hankel transform for calculating properties of dipolar Bose-Einstein condensates in cylindrically symmetric traps. As an application, we are able to compute many excitation modes by directly solving the Bogoliubov-De Gennes equations. We explore the behavior of the excited modes in different trap geometries. We use these results to calculate the quantum depletion of the condensate by a combination of a computation of the exact modes and the use of a local density approximation.