Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9711140 · Physical Review A

Quasiparticle excitations and dynamical structure function of trapped Bose condensates in the WKB approximation

András Csordás, Robert Graham, P. Szépfalusy

Published Jun 1, 1998• 15 citations
arXivHTMLPDF

Abstract

The Bogoliubov equations of the quasiparticle excitations in a weakly interacting trapped Bose condensate are solved in the WKB approximation in an isotropic harmonic trap, determining the discrete quasiparticle energies and wave functions by torus (Bohr-Sommerfeld) quantization of the integrable classical quasiparticle dynamics. The results are used to calculate the position and strengths of the peaks in the dynamic structure function, which can be observed by off-resonance inelastic light scattering.

DOI
10.1103/physreva.57.4669

Related papers

Semiclassical wave functions and energy levels of Bose-condensed gases in spherically symmetric traps

András Csordás, Robert Graham, P. Szépfalusy

The WKB approximation for the Bogoliubov equations of the quasiparticle excitations in Bose gases with a condensate is worked out in the case of spherically symmetric trap potentials on the basis of the resulting quantization rule. The excitation spectrum is calculated numerically and also analytically in certain limiting cases.

arXiv:cond-mat/9705133•Physical Review A•Dec 1, 1997•17 citations

Off-resonance light scattering from Bose condensates in traps

András Csordás, Robert Graham, P. Szépfalusy

The density-density correlation function determining off-resonance light scattering in weakly interacting Bose-condensed gases in a trapping potential is calculated in the Bogoliubov approximation. The evaluation of the necessary matrix elements is carried out for positive scattering length in the WKB approximation using the Thomas-Fermi approximation for the condensate.

Physical Review A•Oct 1, 1996•18 citations

Classical quasiparticle dynamics in trapped Bose condensates

Martin Fliesser, András Csordás, Robert Graham +1 more

The dynamics of quasiparticles in repulsive Bose condensates in a harmonic trap is studied in the classical limit. In isotropic traps the classical motion is integrable and separable in spherical coordinates. In anisotropic traps the classical dynamics is found, in general, to be nonintegrable. For quasiparticle energies $E$ much smaller than the chemical potential $\ensuremath{\mu}$, besides the conserved quasiparticle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$. However, there coexists at the same energy a dynamics confined to the surface of the condensate, which is governed by a classical Hartree-Fock Hamiltonian. We find that also this dynamics becomes integrable for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$ because of the appearance of an adiabatic invariant. For $E/\ensuremath{\mu}$ of order 1 a large portion of the phase-space supports chaotic motion, both for the Bogoliubov Hamiltonian and its Hartree-Fock approximant. To exemplify this we exhibit Poincar\'e surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For $E/\ensuremath{\mu}\ensuremath{\gg}1$ the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasiintegrable because the remaining small chaotic components in phase space are tightly confined by tori.

arXiv:cond-mat/9707122•Physical Review A•Dec 1, 1997•23 citations

Excited states of a dilute Bose-Einstein condensate in a harmonic trap

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.

arXiv:cond-mat/9704234•Physical Review A•Feb 1, 1998•62 citations

Analytical calculations on Landau damping of collective modes in anisotropic Bose-Einstein condensates

Xiaodong Ma, Yong-li Ma, Guoxiang Huang

We develop an analytical approach for calculating the Landau damping of collective modes in the Bose-Einstein condensate trapped by an anisotropic harmonic potential. Based on a variational ground-state wave function obtained by solving the time-independent Gross-Pitaevskii equation beyond Thomas-Fermi approximation, we solve the Bogoliubov--de Gennes equations describing thermal excited quasiparticles and provide divergence-free analytical solutions for the Bogoliubov amplitudes and coupling matrix elements that characterize the interaction between the collective modes and the quasiparticles. With these analytical results we evaluate the Landau damping rates of several collective modes for various anisotropic parameters of the trapping potential in terms of the formulas derived from a time-dependent mean-field theory. In addition, we discuss the dependence of the damping rates on temperature, particle number, trapping frequency, and anisotropic parameter of the system, and compare our theoretical results with experimental and numerical data reported in literature.

Physical Review A•Jan 31, 2007•5 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.

arXiv:cond-mat/0001079•Physical Review A•May 11, 2000•31 citations
The Bogoliubov equations of the quasiparticle excitations in a weakly interacting trapped Bose condensate are solved in the WKB approximation in an isotropic harmonic trap, determining the discrete quasiparticle energies and wave functions by torus (Bohr-Sommerfeld) quantization of the integrable classical quasiparticle dynamics. The results are used to calculate the position and strengths of the peaks in the dynamic structure function, which can be observed by off-resonance inelastic light scattering.
The dynamics of quasiparticles in repulsive Bose condensates in a harmonic trap is studied in the classical limit. In isotropic traps the classical motion is integrable and separable in spherical coordinates. In anisotropic traps the classical dynamics is found, in general, to be nonintegrable. For quasiparticle energies EEE much smaller than the chemical potential $\ensuremath{\mu}$, besides the conserved quasiparticle energy, we identify two additional nearly conserved phase-space functions. These render the dynamics inside the condensate (collective dynamics) integrable asymptotically for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$. However, there coexists at the same energy a dynamics confined to the surface of the condensate, which is governed by a classical Hartree-Fock Hamiltonian. We find that also this dynamics becomes integrable for $E/\ensuremath{\mu}\ensuremath{\rightarrow}0$ because of the appearance of an adiabatic invariant. For $E/\ensuremath{\mu}$ of order 1 a large portion of the phase-space supports chaotic motion, both for the Bogoliubov Hamiltonian and its Hartree-Fock approximant. To exemplify this we exhibit Poincar\'e surface of sections for harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps. For $E/\ensuremath{\mu}\ensuremath{\gg}1$ the dynamics is again governed by the Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes quasiintegrable because the remaining small chaotic components in phase space are tightly confined by tori.
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 R6{R}^{6}R6, where RRR is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.
We develop an analytical approach for calculating the Landau damping of collective modes in the Bose-Einstein condensate trapped by an anisotropic harmonic potential. Based on a variational ground-state wave function obtained by solving the time-independent Gross-Pitaevskii equation beyond Thomas-Fermi approximation, we solve the Bogoliubov--de Gennes equations describing thermal excited quasiparticles and provide divergence-free analytical solutions for the Bogoliubov amplitudes and coupling matrix elements that characterize the interaction between the collective modes and the quasiparticles. With these analytical results we evaluate the Landau damping rates of several collective modes for various anisotropic parameters of the trapping potential in terms of the formulas derived from a time-dependent mean-field theory. In addition, we discuss the dependence of the damping rates on temperature, particle number, trapping frequency, and anisotropic parameter of the system, and compare our theoretical results with experimental and numerical data reported in literature.
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 Tc{T}_{c}Tc​ 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 asc,{a}_{\mathrm{sc}},asc​, increases the number of condensate atoms at each temperature, and raises the value of Tc{T}_{c}Tc​ relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near Tc.{T}_{c}.Tc​. Comparisons with the results of quantum Monte Carlo calculations and various local-density approximations are presented, and experimental consequences are discussed.