We show that universal properties of a many-atom quantum system exist at a much higher density, or a much shorter length scale, than those implied by traditional theories for a dilute Bose gas. In particular, a universal equation of state at length scale β6 = (mC6/2)1/4 (corresponding to the van der Waals interaction) is defined and investigated by combining the concept of effective potential, the constrained variational method and the analytic solution for the van der Waals potential. In one application, we show that a many-atom Bose system with negative scattering length may form a metastable liquid Bose–Einstein condensate (BEC) state with an equilibrium density that is controlled by a scaled scattering length.We show that universal properties of a many-atom quantum system exist at a much higher density, or a much shorter length scale, than those implied by traditional theories for a dilute Bose gas. In particular, a universal equation of state at length scale β6 = (mC6/2)1/4 (corresponding to the van der Waals interaction) is defined and investigated by combining the concept of effective potential, the constrained variational method and the analytic solution for the van der Waals potential. In one application, we show that a many-atom Bose system with negative scattering length may form a metastable liquid Bose–Einstein condensate (BEC) state with an equilibrium density that is controlled by a scaled scattering length.

APSMay 1, 20040 citations

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

Dilute Bose gases interacting via power-law potentials

Ryan M. Kalas, D. Blume

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

arXiv:0712.3597Physical Review AMar 10, 20088 citations

Low-energy universality and scaling of van der Waals forces

A. Calle Cordón, E. Ruiz Arriola

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.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.

arXiv:0912.1714Physical Review AApr 23, 201018 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.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 AJul 18, 20073 citations

Efimov–van der Waals universality for ultracold atoms with positive scattering lengths

P. M. A. Mestrom, Jia Wang, Chris H. Greene +1 more

The universality of the three-body parameter in ultracold atomic systems is studied with a single-channel model when the scattering length is positive, where the parameter is found to be constrained by a variational principle and can be strongly affected by $d$ -wave interactions.The universality of the three-body parameter in ultracold atomic systems is studied with a single-channel model when the scattering length is positive, where the parameter is found to be constrained by a variational principle and can be strongly affected by $d$-wave interactions.

arXiv:1609.02857Physical Review AMar 30, 201734 citations

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.

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.

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.

d

-wave interactions.