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arXiv:1802.02103 · Physical Review A

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

Michelle Wynne Sze, A. G. Sykes, D. Blume, John L. Bohn

Published Mar 16, 2018• 7 citations
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Abstract

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.

DOI
10.1103/physreva.97.033608

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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 E0/N\ensuremath≈2.5N1/3\ensuremathℏ\ensuremathω{E}_{0}/N\ensuremath{\approx}2.5{N}^{1/3}\ensuremath{\hbar}\ensuremath{\omega}E0​/N\ensuremath≈2.5N1/3\ensuremathℏ\ensuremathω and two-body contact C2/N\ensuremath≈16N1/6m\ensuremathω/\ensuremathℏ{C}_{2}/N\ensuremath{\approx}16{N}^{1/6}\sqrt{m\ensuremath{\omega}/\ensuremath{\hbar}}C2​/N\ensuremath≈16N1/6m\ensuremathω/\ensuremathℏ​ for a Bose gas on resonance, where \ensuremathω\ensuremath{\omega}\ensuremathω is the trap frequency.
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 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.
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.
A two-body correlated basis set is used to develop a many-body theory which is valid for any number of bosons in the trap. The formalism incorporates the van der Waals interaction and two-body correlations in an exact way. The theory has successfully been applied to Bose-Einstein condensates---dilute weakly interacting and also dilute but having a large scattering length. Even in the extreme dilute condition, we observe the breakdown of the shape-independent approximation and the interatomic correlation plays an important role in the large particle-number limit. This correlated many-body calculation can handle, within the two-body correlation approximation, the entire range of atom number of experimentally achieved condensates. Next we successfully push the basis function for large scattering lengths where the mean-field results are manifestly bad. The sharp increase in correlation energy clearly shows the beyond-mean-field effect. We also calculate one-particle densities for various scattering lengths and particle numbers. Our many-body calculation exhibits the finite-size effect in the one-body density.
We present a detailed description of the equivalent linear two-body method for the many-body problem, which is based on an approximate reduction of the many-body Schrödinger equation by the use of a variational principle. The method has been applied to the one-dimensional N -body problem with pair-wise contact interactions (McGurie-Yang N -body problem) and to the dilute Bose-Einstein condensation of atoms in harmonic traps at zero temperature for both positive and negative scattering lengths. The ground state energy and wavefunction for a dilute Bose gas are obtained analytically for large value of N and it is shown that the method gives excellent results in the large- N limit.