Based on the energy functional and variational method, we present a new method to investigate the ground state properties for a weakly interacting Bose-condensed gas in an anisotropic harmonic trap at zero temperature. With this method we are able to find the analytic expression of the ground-state wavefunction and to explore the relevant quantities, such as energy, chemical potential, and the aspect ratio of the velocity distribution. These results agree well with previous ground state numerical solutions of the Gross–Pitaevskii equation given by Dalfovo et al. [Phys. Rev. A 53 (1996) 2477] This new method is simple compared to other methods used to solve numerically the Gross–Pitaevskii equation, and one can obtain analytic and reliable results.
Based on the Gross-Pitaevskii(G-P) energy functional and variational method, we present a new method to find the ground state wave function for Bose-condensed gas in a harmonic trap at zero temperature. With this method we are able to find the analytic expression of ground state wave function and explore the relevant quantities, such as energy, chemical potential and so on. These results agree well with previous ground state numerical solutions of the G-P equation given by Dalfovo et al. Under the large number of particles N limit for repulsive interaction, our results and Thomas-Fermi approximation model approach to reaching unanimity. This new method is simple compared to other methods used to solve numerically the G-P equation, and one can obtained the analytic and reliable results.
We solve the Gross-Pitaevskii equations for a dilute atomic gas in a magnetic trap, modeled by an anisotropic harmonic potential. We evaluate the wave function and the energy of the Bose-Einstein condensate as a function of the particle number, both for positive and negative scattering length. The results for the transverse and the vertical size of the cloud of atoms, as well as for the kinetic and potential energy per particle, are compared with the predictions of approximated models. We also compare the aspect ratio of the velocity distribution with experimental estimates available for
87Rb. Vortex states are considered and the critical angular velocity for production of vortices is calculated. We show that the presence of vortices significantly increases the stability of the condensate in the case of attractive interactions. \textcopyright{} 1996 The American Physical Society.
For Bose–Einstein condensation of neutral atoms in anisotropic traps at zero temperature, we present simple analytical methods for computing the properties of ground state and single vortex of Bose–Einstein condensates, and compare those results to extensive numerical simulations. The critical angular velocity for production of vortices is calculated for both positive and negative scattering lengths a, and find an analytical expression for the large- N limit of the vortex critical angular velocity for a >0, and the critical number for condensate population approaches the point of collapse for a <0, by using approximate variational method.
We consider a gas of
N(< 15) Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\ensuremath{\Omega}$ about the z axis. Exact diagonalization of the $n\ifmmode\times\else\texttimes\fi{}n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum
Lz, with $n\ensuremath{\sim}{10}^{5}$ (e.g., for
Lz=N=15,n=240782) was carried out using Davidson's algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\ensuremath{\rho}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}})={\ensuremath{\sum}}_{\ensuremath{\mu}}{\ensuremath{\lambda}}_{\ensuremath{\mu}}{\ensuremath{\chi}}_{\ensuremath{\mu}}^{*}(\mathbf{r}){\ensuremath{\chi}}_{\ensuremath{\mu}}({\mathbf{r}}^{\ensuremath{'}})$ were determined. With the usual identification of $\ensuremath{\Omega}$ as the Lagrange multiplier associated with
Lz for a rotating system, the ${L}_{z}\ensuremath{-}\ensuremath{\Omega}$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{},$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}}, i=1,2,3,\dots{}.$ All the states in the regime
N>Lz>0 are metastable. For
Lz>N, the
Lz values for the stable ground states generally increased with increasing critical angular velocities ${\ensuremath{\Omega}}_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\ensuremath{\Omega}}_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the
Lz=N single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $(\ensuremath{\sim}4)$ orders of magnitude in the moderately to the weakly interacting regime.
We discuss various properties of the ground state of a Bose-condensed dilute gas confined by an external potential. We devote particular attention to the role played by the interaction in determining the kinetic energy of the system and the aspect ratio of the velocity distribution. The structure of the wave function near the classical turning point is discussed and the drawback of the Thomas-Fermi approximation is explicitly pointed out. We consider also states with quantized vorticity and calculate the critical angular velocity for the production of vortices. The presence of vortex states is found to increases the stability of the condensate in the case of attractive interactions.