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Chinese Physics Letters

Variational Calculations for the Spectrum of Finite Bosons in Harmonic Potentials

Ma Yong-Li, Zhongzhou Chen

Published Aug 29, 2003• 5 citations
Publisher

Abstract

Using both the Gaussian and Fetter's variational calculations for the N -body ground-state wavefunction of the trapped Bose-Einstein condensate, we give explicit analytic formulae for the spectrum of finite bosons in harmonic potentials based on the corrected sum rules and generalized virial identities. We compare the low-lying excitation spectra among the Gaussian and Fetter's variational calculations and the exact numerical results. The Gaussian approximation has the simplest reasonable results, valid for N → ∞ and high-lying excitations.

DOI
10.1088/0256-307x/20/9/307

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Using both the Gaussian and Fetter's variational calculations for the N -body ground-state wavefunction of the trapped Bose-Einstein condensate, we give explicit analytic formulae for the spectrum of finite bosons in harmonic potentials based on the corrected sum rules and generalized virial identities. We compare the low-lying excitation spectra among the Gaussian and Fetter's variational calculations and the exact numerical results. The Gaussian approximation has the simplest reasonable results, valid for N → ∞ and high-lying excitations.
An analytic expression is presented for the elementary excitation spectrum of the Bose-Einstein condensate of a trapped boson system in the weakly interacting, low-density limit. Explicit analytic formulas for the elementary excitation spectrum are obtained for harmonic-oscillator traps. These formulas provide information about the behavior of the elementary excitation levels as a function of the number of atoms and their interaction strength for a given trap geometry. They also provide a low-density benchmark for results of fully numerical calculations.
This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic-interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum.
We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian stability matrix in a variational approach that maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r1/r1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibit a Rydberg-like structure, which can be explained by means of a quantum-defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussian functions combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates and the eigenfrequencies of the excitation modes are compared.
We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.
The motivation in this paper is to simulate numerically some properties of an interacting Bose–Einstein condensate at zero temperature in an axial symmetry trapping potential with finite amplitude for modeling the practical experimental cases. By use of the basis expansion using three-dimensional harmonic oscillator eigenfunctions, we obtain the ground-state wavefunction and the collective excitation spectra of the system in both usual harmonic potential and different amplitudes of the finite potential. After comparing our results for the finite potential with the data derived from the harmonic potential, we conclude that the finite trap in the practical experiments decreases the entire excitation frequencies in the whole regimes. This decrease is consistent with our analytic prediction qualitatively and agrees well with the experimental data quantitatively.