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

The Road Less Traveled: Resonant Bose-Einstein Condensates via a Hyperspherical Lowest-Order Constrained Variational Approach

Michelle Wynne Sze

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

In this work, we study the ground state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained variational (LOCV) method, we are able to semi-quantitatively attain Bose-Einstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle $E_0/N \\approx 2.5 N^{1/3} \\hbar \\omega$ and two-body contact $C_2/N \\approx 16 N^{1/6}\\sqrt{m\\omega/\\hbar}$ for a Bose gas on resonance, where $\\omega$ is the trap frequency. Further, by considering only two-body correlations, we note that a sudden quench from small to large scattering lengths leads to out-of-equilibrium resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between $0$ and $\\infty$, which serves to maximize the transfer probability of $N$ bosons in a harmonic trap with frequency $\\omega$ to the resonant state. We find that the intermediate scattering length should be $a\\approx3.16N^{-2/3}\\sqrt{\\hbar/(m\\omega)}$, and that it produces an optimum transition probability of $1.03N^{-1/6}$.

Related papers

Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

Michelle Wynne Sze, A. G. Sykes, D. Blume +1 more

We study the ground-state properties of a system of $N$ harmonically trapped bosons of mass $m$ interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle ${E}_{0}/N\ensuremath{\approx}2.5{N}^{1/3}\ensuremath{\hbar}\ensuremath{\omega}$ and two-body contact ${C}_{2}/N\ensuremath{\approx}16{N}^{1/6}\sqrt{m\ensuremath{\omega}/\ensuremath{\hbar}}$ for a Bose gas on resonance, where $\ensuremath{\omega}$ is the trap frequency.

arXiv:1802.02103•Physical Review A•Mar 16, 2018•7 citations

Two-step production of resonant Bose-Einstein condensates

Michelle Wynne Sze, John L. Bohn

Producing a substantial and stable resonant Bose-Einstein condensate (BEC) has proven to be a challenging experimental task due to heating and three-body losses that may occur even before the gas comes to thermal equilibrium. In this paper, by considering two-body correlations, we note that a sudden quench from small to large scattering lengths is not an efficient way to prepare a resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between 0 and $\ensuremath{\infty}$, which serves to maximize the transfer probability of $N$ bosons of mass $m$ in a harmonic trap with frequency $\ensuremath{\omega}$ to the resonant state. We find that the intermediate scattering length should be $a\ensuremath{\approx}3.16{N}^{\ensuremath{-}2/3}\sqrt{\ensuremath{\hbar}/(m\ensuremath{\omega})}$, and that it produces an optimum transition probability of $1.03{N}^{\ensuremath{-}1/6}$.

arXiv:1812.08699•Physical Review A•Mar 11, 2019•2 citations

Behavior of trapped ultracold dilute Bose gases at large scattering length near a Feshbach resonance

M. L. Lekala, Barnali Chakrabarti, Gaotsiwe Joel Rampho +3 more

We calculate the ground-state energy and the collective excitation frequency of trapped bosons at large scattering length interacting via the realistic two-body van der Waals potential. Our many-body method keeps two-body correlations produced by all interacting pairs. When the scattering length is small compared to the trap size and the number of bosons in the trap is of the order of a few thousands, the mean-field results are in good agreement with the many-body results. However for large particle numbers, even when the condensate is sufficiently dilute, the interatomic correlation comes into the picture. When the scattering length is quite large near the Feshbach resonance, the Bose gas becomes highly correlated. The many-body results are close to the Gross-Pitaevskii results for a small number of bosons, however, large deviations are noted in the large particle limit. We also calculate the lowest collective excitation and the interaction energy for large scattering lengths. The monopole excitation frequency exhibits a pronounced dependence on the scattering length. We also observe a universal behavior for the interaction energy at the limit of large scattering length.

Physical Review A•Feb 18, 2014•10 citations

Equation of state of cold atoms: A lowest-order-constrained variational study of systems with large non- s -wave scattering lengths

Ryan M. Kalas, D. Blume

Dilute Fermi systems with large $s$-wave scattering length ${a}_{s}$ exhibit universal properties if the interparticle spacing ${r}_{0}$ greatly exceeds the range of the underlying two-body interaction potential. In this regime, ${r}_{0}$ is the only relevant length scale and observables such as the energy per particle depend only on ${r}_{0}$ (or, equivalently, the energy ${E}_{\mathrm{FG}}$ of the free Fermi gas). This paper investigates Bose and Fermi systems with nonvanishing angular momentum $l$ using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of an $l$-dependent length scale ${\ensuremath{\xi}}_{l}$, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite $l$. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.

Physical Review A•Jul 18, 2007•3 citations

Potential harmonics expansion method for trapped interacting bosons: Inclusion of two-body correlation

Tapan Kumar Das, Barnali Chakrabarti

We study a system of $A$ identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as $A$ increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital $l$, azimuthal $m$, and the grand orbital $2K+l$ quantum numbers for any arbitrary $A$. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any $A$. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.

arXiv:quant-ph/0408061•Physical Review A•Dec 1, 2004•54 citations

Orbital variational adiabatic hyperspherical method applied to Bose-Einstein condensates

Hyunwoo Lee, Chris H. Greene

A variational basis set motivated by mean-field theory is utilized to describe the Bose-Einstein condensate within the adiabatic hyperspherical coordinate framework. The simplest single-orbital variant of this treatment reproduces many of the ground-state properties predicted by the Gross-Pitaevskii equation. But a multiorbital improvement to the basis set yields a better representation of particle correlations and of the critical number where the condensate collapses for a negative two-body scattering length. The method also produces systematic deviations from Bogoliubov theory for the fundamental monopole excitation frequency.

arXiv:2012.11784•Physical Review A•Feb 23, 2021•2 citations
In this work, we study the ground state properties of a system of NNN harmonically trapped bosons of mass mmm interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained variational (LOCV) method, we are able to semi-quantitatively attain Bose-Einstein condensate ground state energies even for gases with infinite scattering length. In the large particle number limit, our method provides analytical estimates for the energy per particle E0/Napprox2.5N1/3hbaromegaE_0/N \\approx 2.5 N^{1/3} \\hbar \\omegaE0​/Napprox2.5N1/3hbaromega and two-body contact C2/Napprox16N1/6sqrtmomega/hbarC_2/N \\approx 16 N^{1/6}\\sqrt{m\\omega/\\hbar}C2​/Napprox16N1/6sqrtmomega/hbar for a Bose gas on resonance, where omega\\omegaomega is the trap frequency. Further, by considering only two-body correlations, we note that a sudden quench from small to large scattering lengths leads to out-of-equilibrium resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between 000 and infty\\inftyinfty, which serves to maximize the transfer probability of NNN bosons in a harmonic trap with frequency omega\\omegaomega to the resonant state. We find that the intermediate scattering length should be aapprox3.16N−2/3sqrthbar/(momega)a\\approx3.16N^{-2/3}\\sqrt{\\hbar/(m\\omega)}aapprox3.16N−2/3sqrthbar/(momega), and that it produces an optimum transition probability of 1.03N−1/61.03N^{-1/6}1.03N−1/6.
We study the ground-state properties of a system of NNN harmonically trapped bosons of mass mmm interacting with two-body contact interactions, from small to large scattering lengths. This is accomplished in a hyperspherical coordinate system that is flexible enough to describe both the overall scale of the gas and two-body correlations. By adapting the lowest-order constrained-variational method, we are able to semiquantitatively attain Bose-Einstein condensate ground-state energies even for gases with infinite scattering length. In the large-particle-number limit, our method provides analytical estimates for the energy per particle ${E}_{0}/N\ensuremath{\approx}2.5{N}^{1/3}\ensuremath{\hbar}\ensuremath{\omega}$ and two-body contact ${C}_{2}/N\ensuremath{\approx}16{N}^{1/6}\sqrt{m\ensuremath{\omega}/\ensuremath{\hbar}}$ for a Bose gas on resonance, where $\ensuremath{\omega}$ is the trap frequency.
Producing a substantial and stable resonant Bose-Einstein condensate (BEC) has proven to be a challenging experimental task due to heating and three-body losses that may occur even before the gas comes to thermal equilibrium. In this paper, by considering two-body correlations, we note that a sudden quench from small to large scattering lengths is not an efficient way to prepare a resonant BEC. As an alternative, we propose a two-step scheme that involves an intermediate scattering length, between 0 and $\ensuremath{\infty}$, which serves to maximize the transfer probability of NNN bosons of mass mmm in a harmonic trap with frequency $\ensuremath{\omega}$ to the resonant state. We find that the intermediate scattering length should be $a\ensuremath{\approx}3.16{N}^{\ensuremath{-}2/3}\sqrt{\ensuremath{\hbar}/(m\ensuremath{\omega})}$, and that it produces an optimum transition probability of $1.03{N}^{\ensuremath{-}1/6}$.
We calculate the ground-state energy and the collective excitation frequency of trapped bosons at large scattering length interacting via the realistic two-body van der Waals potential. Our many-body method keeps two-body correlations produced by all interacting pairs. When the scattering length is small compared to the trap size and the number of bosons in the trap is of the order of a few thousands, the mean-field results are in good agreement with the many-body results. However for large particle numbers, even when the condensate is sufficiently dilute, the interatomic correlation comes into the picture. When the scattering length is quite large near the Feshbach resonance, the Bose gas becomes highly correlated. The many-body results are close to the Gross-Pitaevskii results for a small number of bosons, however, large deviations are noted in the large particle limit. We also calculate the lowest collective excitation and the interaction energy for large scattering lengths. The monopole excitation frequency exhibits a pronounced dependence on the scattering length. We also observe a universal behavior for the interaction energy at the limit of large scattering length.
Dilute Fermi systems with large sss-wave scattering length as{a}_{s}as​ exhibit universal properties if the interparticle spacing r0{r}_{0}r0​ greatly exceeds the range of the underlying two-body interaction potential. In this regime, r0{r}_{0}r0​ is the only relevant length scale and observables such as the energy per particle depend only on r0{r}_{0}r0​ (or, equivalently, the energy EFG{E}_{\mathrm{FG}}EFG​ of the free Fermi gas). This paper investigates Bose and Fermi systems with nonvanishing angular momentum lll using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of an lll-dependent length scale ${\ensuremath{\xi}}_{l}$, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite lll. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.
We study a system of AAA identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as AAA increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital lll, azimuthal mmm, and the grand orbital 2K+l2K+l2K+l quantum numbers for any arbitrary AAA. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any AAA. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.
A variational basis set motivated by mean-field theory is utilized to describe the Bose-Einstein condensate within the adiabatic hyperspherical coordinate framework. The simplest single-orbital variant of this treatment reproduces many of the ground-state properties predicted by the Gross-Pitaevskii equation. But a multiorbital improvement to the basis set yields a better representation of particle correlations and of the critical number where the condensate collapses for a negative two-body scattering length. The method also produces systematic deviations from Bogoliubov theory for the fundamental monopole excitation frequency.