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arXiv:cond-mat/9603126 · Physical Review Letters

Collective Excitations of a Trapped Bose-Condensed Gas

S. Stringari

Published Sep 16, 1996• 922 citations
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Abstract

We investigate the low energy excitations of a dilute atomic Bose gas confined in a harmonic trap of frequency ${\ensuremath{\omega}}_{0}$ and interacting with repulsive forces. The dispersion law $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}({2n}^{2}+2n\ensuremath{\ell}+3n+\ensuremath{\ell}{)}^{1/2}$ for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}(2n+\ensuremath{\ell})$ of the noninteracting harmonic oscillator model. Here $n$ is the number of radial nodes and $\ensuremath{\ell}$ is the orbital angular momentum. The effects of the kinetic energy pressure are estimated using a sum rule approach. Results are also presented for deformed traps and attractive forces.

DOI
10.1103/physrevlett.77.2360

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We investigate the low-energy excitations of a dilute atomic Bose gas confined in a anharmonic trap interacting with repulsive forces. The dispersion law of both surface and compression modes is derived and analyzed for large numbers of atoms in the trap, which show two branches of excitation and appear two critical values, where one of them indicates collective excitation which would be unstable dynamically, and the other one indicates the existing collective mode with lower frequency under anharmonic influence than that in harmonic trapping case. Our work reveals the key role played by the anharmonicity and interatomic forces which introduce a rich structure in the dynamic behavior of these new many-body systems.
We investigate the thermodynamics of a Bose gas interacting with repulsive forces and confined in a harmonic trap. We show that the relevant parameters of the system (temperature, number NNN of atoms, harmonic oscillator length, deformation of the trap, sss-wave scattering length) fix its thermodynamic behavior in the large NNN limit through two dimensionless scaling parameters. These are the reduced temperature t=T/Tc0t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T/{T}_{c}^{0}t=T/Tc0​ and the ratio $\ensuremath{\eta}$ between the T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 value of the chemical potential, evaluated in the Thomas-Fermi limit, and the critical temperature Tc0{T}_{c}^{0}Tc0​ of the noninteracting model. The scaling functions relative to the condensate fraction, energy, chemical potential, and moment of inertia are calculated within the Popov approximation.
The density of states of a Bose-condensed gas confined in a harmonic trap is investigated. The predictions of Bogoliubov theory are compared with those of Hartree-Fock theory and of the hydrodynamic model. We show that the Hartree-Fock scheme provides an excellent description of the excitation spectrum in a wide range of energy, revealing a major role played by single-particle excitations in these confined systems. The crossover from the hydrodynamic regime, holding at low energies, to the independent-particle regime is explicitly explored by studying the frequency of the surface mode as a function of their angular momentum. The applicability of the semiclassical approximation for the excited states is also discussed. We show that the semiclassical approach provides simple and accurate formulas for the density of states and the quantum depletion of the condensate.
In the Thomas–Fermi (TF) regime, S. Stringari [Phys. Rev. Lett., 77, 2360 (1996)] established the hydrodynamic normal modes of an interacting dilute Bose-condensed fluid confined in an isotropic harmonic trap. Here, we extend this treatment beyond the TF condensate using the work of B. Hu, G. Huang and Y. Ma [Phys. Rev. A, 69, 063608 (2004)] as a starting point. Both numerical results for low-lying eigenfrequencies, for various angular momentum quantum numbers ℓ and n = 0, and also samples of corresponding eigenfunctions, are presented. Finally, eigenfrequencies are also given for some collective excitations in an axially symmetrical trap.
The Bogoliubov approximation is used to study the ground state and low-lying excited states of a dilute gas of N atomic bosons held in an isotropic harmonic potential characterized by frequency \ensuremath{\omega} and oscillator length d0{\mathit{d}}_{0}d0​=\ensuremath{\surd}\ensuremath{\Elzxh}/m\ensuremath{\omega}. By assumption, the self-consistent condensate has a macroscopic occupation number N0{\mathit{N}}_{0}N0​\ensuremath{\gg}1, with N-N0{\mathit{N}}_{0}N0​\ensuremath{\ll}N0{\mathit{N}}_{0}N0​. A linearized hydrodynamic description yields operator forms of the particle-conservation law and Bernoulli's theorem, expressed in terms of the small density fluctuation operator \ensuremath{\rho}^\ensuremath{'} and velocity potential operator \ensuremath{\Phi}^\ensuremath{'}, along with the condensate density n0{\mathit{n}}_{0}n0​ and velocity v0{\mathbf{v}}_{0}v0​. For positive scattering length a and large stationary condensate (N0{\mathit{N}}_{0}N0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​/a and v0{\mathbf{v}}_{0}v0​=0), the spherical condensate has a well-defined radius R0{\mathit{R}}_{0}R0​\ensuremath{\gg}d0{\mathit{d}}_{0}d0​, and the low-lying excited states are irrotational compressional waves localized near the surface. Approximate variational energies E0l{\mathit{E}}_{0\mathit{l}}E0l​ of the lowest radial modes (n=0) for successive values of orbital angular momentum l form a rotational band given by E0l{\mathit{E}}_{0\mathit{l}}E0l​\ensuremath{\approxeq}E00{\mathit{E}}_{00}E00​+1/2${\mathrm{\ensuremath{\Elzxh}}}^{2}$l(l+1)/mR02{\mathrm{mR}}_{0}^{2}mR02​, with radial zero-point energy E00{\mathit{E}}_{00}E00​\ensuremath{\propto}\ensuremath{\Elzxh}\ensuremath{\omega}(R0{\mathit{R}}_{0}R0​/d0{\mathit{d}}_{0}d0​)2/3{)}^{2/3})2/3=(${\mathrm{\ensuremath{\Elzxh}}}^{2}$ m${\mathrm{\ensuremath{\omega}}}^{4}$R02{\mathit{R}}_{0}^{2}R02​)1/3{)}^{1/3})1/3. \textcopyright{} 1996 The American Physical Society.
We observe phononlike excitations of a Bose-Einstein condensate (BEC) in a dilute atomic gas. 87{}^{87}87Rb atoms are optically trapped and precooled, loaded into a magnetic trap, and then evaporatively cooled through the BEC phase transition to form a condensate. We excite the condensate by applying an inhomogeneous oscillatory force with adjustable frequency and symmetry. We have observed modes with different angular momenta and different energies and have studied how their characteristics depend on interaction energy. We find that the condensate excitations persist longer than their counterparts in uncondensed clouds.