Physical Review A

Noncondensate atoms in a trapped Bose gas

Juha Javanainen

Published Nov 1, 1996• 37 citations

Abstract

We formulate Bogoliubov theory for a weakly interacting Bose-Einstein condensate confined to a spherically symmetric harmonic-oscillator potential. The theory is solved numerically by diagonalizing a Hamiltonian that is second order in boson creation and annihilation operators with the aid of a symplectic transformation. At zero temperature, the number of noncondensate atoms is well approximated by a heuristic expression based on Bogoliubov theory for free atoms.

DOI: 10.1103/physreva.54.r3722

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Physical Review A

Noncondensate atoms in a trapped Bose gas

Juha Javanainen

Published Nov 1, 1996• 37 citations

Publisher

Abstract

DOI: 10.1103/physreva.54.r3722

Related papers

Weakly-Interacting Bosons in a Trap within Approximate Second Quantization Approach

Andrij A. Rovenchak

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Alexander L. Fetter, Daniel S. Rokhsar

The low-lying hydrodynamic normal modes of a dilute Bose-Einstein gas in an isotropic harmonic trap determine the corresponding Bogoliubov amplitudes. In the Thomas-Fermi limit, these modes have large low-temperature occupation numbers and they permit an explicit construction of the dynamic structure function $S(\mathbf{q},\ensuremath{\omega})$. The total noncondensate number ${N}^{\ensuremath{'}}(0)$ at zero temperature increases like ${R}^{6}$ , where $R$ is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.The low-lying hydrodynamic normal modes of a dilute Bose-Einstein gas in an isotropic harmonic trap determine the corresponding Bogoliubov amplitudes. In the Thomas-Fermi limit, these modes have large low-temperature occupation numbers and they permit an explicit construction of the dynamic structure function $S(\mathbf{q},\ensuremath{\omega})$. The total noncondensate number ${N}^{\ensuremath{'}}(0)$ at zero temperature increases like ${R}^{6}$, where $R$ is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.

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We present an analysis of Bose-Einstein condensation for a system of non-interacting spin-0 particles in a harmonic oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quantitatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic oscillator confining potential.We present an analysis of Bose-Einstein condensation for a system of non-interacting spin-0 particles in a harmonic oscillator confining potential trap. We discuss why a confined system of particles differs both qualitatively and quantitatively from an identical system which is not confined. One crucial difference is that a confined system is not characterized by a critical temperature in the same way as an unconfined system such as the free boson gas. We present the results of both a numerical and analytic analysis of the problem of Bose-Einstein condensation in a general anisotropic harmonic oscillator confining potential.

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Numerical approach to the ground and excited states of a Bose-Einstein condensed gas confined in a completely anisotropic trap

Barry I. Schneider, David L. Feder

The ground and excited states of a weakly interacting and dilute Bose-Einstein condensed gas, confined in a completely anisotropic harmonic oscillator potential, are determined at zero temperature within the Bogoliubov approximation. The numerical calculations employ a computationally efficient procedure based on a discrete variable representation (DVR) of the Hamiltonian. The DVR is efficient for problems where the interaction potential may be expressed as a local function of interparticle coordinates. In order to address condensates that are both very large ($\ensuremath{\sim}{10}^{6}$ atoms) and fully anisotropic, the ground state is found using a self-consistent field approach. Experience has demonstrated, however, that standard iterative techniques applied to the solution of the nonlinear partial differential equation for the condensate are nonconvergent. This limitation is overcome using the method of direct inversion in the iterated subspace (DIIS). In addition, the sparse structure of the DVR enables the efficient application of iterative techniques such as the Davidson and/or Lanczos methods, to extract the eigenvalues of physical interest. The results are compared with recent experimental data obtained for Bose-Einstein condensed alkali-metal vapors confined in magnetic traps.The ground and excited states of a weakly interacting and dilute Bose-Einstein condensed gas, confined in a completely anisotropic harmonic oscillator potential, are determined at zero temperature within the Bogoliubov approximation. The numerical calculations employ a computationally efficient procedure based on a discrete variable representation (DVR) of the Hamiltonian. The DVR is efficient for problems where the interaction potential may be expressed as a local function of interparticle coordinates. In order to address condensates that are both very large ($\ensuremath{\sim}{10}^{6}$ atoms) and fully anisotropic, the ground state is found using a self-consistent field approach. Experience has demonstrated, however, that standard iterative techniques applied to the solution of the nonlinear partial differential equation for the condensate are nonconvergent. This limitation is overcome using the method of direct inversion in the iterated subspace (DIIS). In addition, the sparse structure of the DVR enables the efficient application of iterative techniques such as the Davidson and/or Lanczos methods, to extract the eigenvalues of physical interest. The results are compared with recent experimental data obtained for Bose-Einstein condensed alkali-metal vapors confined in magnetic traps.

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Klaus Kirsten, David J. Toms

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arXiv:cond-mat/9604031Physical Review ANov 1, 1996134 citations

Bose condensates in a harmonic trap near the critical temperature

T. H. Bergeman, David L. Feder, Nandor L. Balazs +1 more

The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point ${T}_{c}$ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length ${a}_{\mathrm{sc}},$ increases the number of condensate atoms at each temperature, and raises the value of ${T}_{c}$ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near ${T}_{c}.$ Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local-density approximation. Consistency of the theory for temperatures through the Bose condensation point ${T}_{c}$ requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length ${a}_{\mathrm{sc}},$ increases the number of condensate atoms at each temperature, and raises the value of ${T}_{c}$ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near ${T}_{c}.$ Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.

arXiv:cond-mat/0001079Physical Review AMay 11, 200031 citations

{R}^{6}

, where

R

is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.

{T}_{c}

requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length

{a}_{\mathrm{sc}},

increases the number of condensate atoms at each temperature, and raises the value of

{T}_{c}

relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near

{T}_{c}.

Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.