arXiv:0712.3597 · Physical Review A

Dilute Bose gases interacting via power-law potentials

Ryan M. Kalas, D. Blume

Published Mar 10, 2008• 8 citations

Abstract

Neutral atoms interact through a van der Waals potential which asymptotically falls off as ${r}^{\ensuremath{-}6}$. In ultracold gases, this interaction can be described to a good approximation by the atom-atom scattering length. However, corrections arise that depend on the characteristic length of the van der Waals potential. We parametrize these corrections by analyzing the energies of two- and few-atom systems under external harmonic confinement, obtained by numerically and analytically solving the Schr\"odinger equation. We generalize our results to particles interacting through a longer-ranged potential which asymptotically falls off as ${r}^{\ensuremath{-}4}$.

DOI: 10.1103/physreva.77.032703

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arXiv:0712.3597 · Physical Review A

Dilute Bose gases interacting via power-law potentials

Ryan M. Kalas, D. Blume

Published Mar 10, 2008• 8 citations

arXiv HTML PDF

Abstract

Neutral atoms interact through a van der Waals potential which asymptotically falls off as

{r}^{\ensuremath{-}6}

. In ultracold gases, this interaction can be described to a good approximation by the atom-atom scattering length. However, corrections arise that depend on the characteristic length of the van der Waals potential. We parametrize these corrections by analyzing the energies of two- and few-atom systems under external harmonic confinement, obtained by numerically and analytically solving the Schr"odinger equation. We generalize our results to particles interacting through a longer-ranged potential which asymptotically falls off as

{r}^{\ensuremath{-}4}

DOI: 10.1103/physreva.77.032703

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Bose atoms in a trap: A variational Monte Carlo formulation for the universal behavior at the van der Waals length scale

Imran Khan, Bo Gao

We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for $N$ Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for $N=3$ and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement.We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for $N$ Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for $N=3$ and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement.

arXiv:cond-mat/0603056Physical Review AJun 23, 200611 citations

Neutral atoms interact through a van der Waals potential which asymptotically falls off as

{r}^{\ensuremath{-}6}

{r}^{\ensuremath{-}4}

{C}_{6}

parameters has been critically examined, starting from very few to 14 000 atoms to analyze and justify the idea of a shape-independent approximation. It is found that the condensate properties almost remain unchanged when the number of atoms are quite small $(\ensuremath{\sim}100)$, in good agreement with earlier results of Blume and Greene [Phys. Rev. A 63, 063601 (2001)], but we observe the appreciable effect of a long attractive tail when

N

is large, even in the low-density limit. The above reference considered only 20 atoms which is far from a real experimental situation. Our calculation gives a realistic scenario which justifies the use of a shape-dependent two-body interaction in many-body theories.

At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by

s

-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range

{r}_{0}

. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where

A

B

, and

C

depend on the dispersion coefficients

{C}_{n}

and the reduced diatom mass. We confront this formula to about 100 determinations of

{r}_{0}

and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient

{C}_{6}

. Universality and scaling extend much beyond naive dimensional analysis estimates.

We present a variational Monte Carlo formulation for the universal equations of state at the van der Waals length scale [B. Gao, J. Phys. B 37, L227 (2004)] for

N

Bose atoms in a trap. The theory illustrates both how such equations of state can be computed exactly, and the existence and the importance of long-range atom-atom correlation under strong confinement. Explicit numerical results are presented for

N=3

and 5, and used to provide a quantitative understanding of the shape-dependent confinement correction that is important for few atoms under strong confinement.