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

Physical Review A

Hartree-Fock theory for Bose-Einstein condensates and the inclusion of correlation effects

Brett D. Esry

Published Feb 1, 1997• 114 citations
Publisher

Abstract

I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.

DOI
10.1103/physreva.55.1147

Related papers

Configuration-interaction studies of Bose-Einstein condensates

Tommy Vänskä, Dage Sundholm, Markus Lindberg

A direct configuration-interaction (CI) approach for studies of Bose-Einstein condensates has been developed and implemented. The reference state is obtained by solving the Hartree-Fock equation in a nonorthogonal Gaussian basis. The CI problem is solved by diagonalizing the Hamiltonian constructed in a many-body basis generated using the orthogonal canonical orbital basis obtained in a Hartree-Fock calculation. For Hamiltonian matrices, which are too large for full diagonalization, a direct iterative diagonalization scheme is employed for obtaining the lowest-lying eigenstates. The computational methods were tested by applications to Bose-Einstein condensates using various isotropic and anisotropic trap geometries.

Physical Review A•Feb 26, 2007•3 citations

Hartree-Fock calculation for trapped, two-dimensional interacting Bose gases of finite size

Jae Gil Kim, Kab Seok Kang, Bong Soo Kim +1 more

The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.

Journal of Physics B Atomic Molecular and Optical Physics•Jun 22, 2000•7 citations

Comparison of gapless mean-field theories for trapped Bose-Einstein condensates

Nick P. Proukakis, S. A. Morgan, Seok Jin Choi +1 more

We present two gapless mean-field theories for trapped Bose-Einstein condensates, and compare them to the better known Hartree-Fock-Bogoliubov (HFB)-Popov approach. The proposed theories are based on suitable inclusion of the simplest anomalous average of the Bose field operator. This leads to an effective interaction between two atoms which is both temperature and density dependent, as opposed to the HFB-Popov approach, for which it is constant. These theories are compared to the HFB-Popov approach for a trapped one-dimensional condensate of 2000 atoms. The density profiles and excitation frequencies of the proposed theories differ from the corresponding HFB-Popov ones by at most a few percent.

Physical Review A•Sep 1, 1998•61 citations

Hartree–Fock mean-field theory for trapped dirty bosons

Tama Khellil, Axel Pelster

Abstract Here we work out in detail a non-perturbative approach to the dirty boson problem, which relies on the Hartree–Fock theory and the replica method. For a weakly interacting Bose gas within a trapped confinement and a delta-correlated disorder potential at finite temperature, we determine the underlying free energy. From it we determine via extremization self-consistency equations for the three components of the particle density, namely the condensate density, the thermal density, and the density of fragmented local Bose–Einstein condensates within the respective minima of the random potential landscape. Solving these self-consistency equations in one and three dimensions in two other publications has revealed how these three densities change for increasing disorder strength.

arXiv:1511.08882•Journal of Statistical Mechanics Theory and Experiment•Jun 1, 2016•16 citations

Cranked Hartree-Fock-Bogoliubov calculation for rotating Bose-Einstein condensates

Nobukuni Hamamoto, Makito Oi, Naoki Onishi

A rotating bosonic many-body system in a harmonic trap is studied with the three-dimensional cranked Hartree-Fock-Bogoliubov method at zero temperature, which has been applied to nuclear many-body systems at high spin. This method is a variational method extended from Hartree-Fock theory, which can treat the pairing correlations in a self-consistent manner. An advantage of this method is that a finite-range interaction between constituent particles can be used in the calculation, unlike the original Gross-Pitaevskii approach. To demonstrate the validity of our method, we present a calculation for a toy model---that is, a rotating system of ten bosonic particles interacting through the repulsive quadrupole-quadrupole interaction in a harmonic trap. It is found that the yrast states, the lowest-energy states for the given total angular momentum, do not correspond to the Bose-Einstein condensate, except for a few special cases. One such case is a vortex state, which appears when the total angular momentum $L$ is twice the particle number $N$ (i.e., $L=2N$).

arXiv:0705.0196•Physical Review A•Jun 14, 2007•1 citation

Dipoles and Disorder: Applications of Hartree-Fock-Bogoliubov Theory

Sam Cormack

We apply the Hartree-Fock-Bogoliubov-Popov (HFBP) method to two problems. The first is the phenomenon of Anderson localisation of a BEC in a two-dimensional bichromatic lattice. The second is the case of a harmonically trapped BEC with an-isotropic, long-range, dipolar interactions between particles. The HFBP method allows us to calculate the condensate properties and excitation spectrum of the sys-tem at finite temperature. For the bichromatic lattice, we examine the effects of in-teractions and finite temperature on the the localisation of the condensate. We also include a synthetic magnetic field and determine its effect. For the dipolar BEC, we examine the effect of the dipolar interactions on the condensate and thermal cloud as well as the excitation spectrum of the system. At finite temperature, we calculate the effects of both direct and exchange dipolar interactions of the thermal cloud on the condensate. The exchange interactions are numerically intensive to calculate and have been excluded from most previous studies. i

•Jan 1, 2012•0 citations
I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.
A direct configuration-interaction (CI) approach for studies of Bose-Einstein condensates has been developed and implemented. The reference state is obtained by solving the Hartree-Fock equation in a nonorthogonal Gaussian basis. The CI problem is solved by diagonalizing the Hamiltonian constructed in a many-body basis generated using the orthogonal canonical orbital basis obtained in a Hartree-Fock calculation. For Hamiltonian matrices, which are too large for full diagonalization, a direct iterative diagonalization scheme is employed for obtaining the lowest-lying eigenstates. The computational methods were tested by applications to Bose-Einstein condensates using various isotropic and anisotropic trap geometries.
The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.
We present two gapless mean-field theories for trapped Bose-Einstein condensates, and compare them to the better known Hartree-Fock-Bogoliubov (HFB)-Popov approach. The proposed theories are based on suitable inclusion of the simplest anomalous average of the Bose field operator. This leads to an effective interaction between two atoms which is both temperature and density dependent, as opposed to the HFB-Popov approach, for which it is constant. These theories are compared to the HFB-Popov approach for a trapped one-dimensional condensate of 2000 atoms. The density profiles and excitation frequencies of the proposed theories differ from the corresponding HFB-Popov ones by at most a few percent.
Abstract Here we work out in detail a non-perturbative approach to the dirty boson problem, which relies on the Hartree–Fock theory and the replica method. For a weakly interacting Bose gas within a trapped confinement and a delta-correlated disorder potential at finite temperature, we determine the underlying free energy. From it we determine via extremization self-consistency equations for the three components of the particle density, namely the condensate density, the thermal density, and the density of fragmented local Bose–Einstein condensates within the respective minima of the random potential landscape. Solving these self-consistency equations in one and three dimensions in two other publications has revealed how these three densities change for increasing disorder strength.
A rotating bosonic many-body system in a harmonic trap is studied with the three-dimensional cranked Hartree-Fock-Bogoliubov method at zero temperature, which has been applied to nuclear many-body systems at high spin. This method is a variational method extended from Hartree-Fock theory, which can treat the pairing correlations in a self-consistent manner. An advantage of this method is that a finite-range interaction between constituent particles can be used in the calculation, unlike the original Gross-Pitaevskii approach. To demonstrate the validity of our method, we present a calculation for a toy model---that is, a rotating system of ten bosonic particles interacting through the repulsive quadrupole-quadrupole interaction in a harmonic trap. It is found that the yrast states, the lowest-energy states for the given total angular momentum, do not correspond to the Bose-Einstein condensate, except for a few special cases. One such case is a vortex state, which appears when the total angular momentum LLL is twice the particle number NNN (i.e., L=2NL=2NL=2N).
We apply the Hartree-Fock-Bogoliubov-Popov (HFBP) method to two problems. The first is the phenomenon of Anderson localisation of a BEC in a two-dimensional bichromatic lattice. The second is the case of a harmonically trapped BEC with an-isotropic, long-range, dipolar interactions between particles. The HFBP method allows us to calculate the condensate properties and excitation spectrum of the sys-tem at finite temperature. For the bichromatic lattice, we examine the effects of in-teractions and finite temperature on the the localisation of the condensate. We also include a synthetic magnetic field and determine its effect. For the dipolar BEC, we examine the effect of the dipolar interactions on the condensate and thermal cloud as well as the excitation spectrum of the system. At finite temperature, we calculate the effects of both direct and exchange dipolar interactions of the thermal cloud on the condensate. The exchange interactions are numerically intensive to calculate and have been excluded from most previous studies. i