Three-body recombination of identical bosons with a large positive scattering length at nonzero temperature

For identical bosons with a large scattering length, the dependence of the three-body recombination rate on the collision energy is determined in the zero-range limit by universal functions of a single scaling variable. There are six scaling functions for angular momentum zero and one scaling function for each higher partial wave. We calculate these universal functions by solving the Skorniakov-Ter-Martirosian equation. The results for the three-body recombination as a function of the collision energy are in good agreement with previous results from solving the three-body Schrödinger equation for ${}^4\mathrm{He}$ atoms. The universal scaling functions can be used to calculate the three-body recombination rate at nonzero temperature. We obtain an excellent fit to the data from Grimm and co-workers for ${}^{133}\mathrm{Cs}$ atoms with a large positive scattering length.

Figure 1

(Color online) Universal scaling functions $f_J\left(x\right)=1-\exp [-4\mathrm{Im}{\delta }_{AD}^{\left(J\right)}\left(E\right)]$ for $J=1,\dots ,6$ as functions of the scaling variable $x$.

Figure 2

Universal scaling functions $s_{12}\left(x\right)$ (top panels), $s_{22}\left(x\right)$ (middle panels), and $s_{11}\left(x\right)$ (bottom panels) as functions of $x$. The modulus $\mid s_{ij}\left(x\right)\mid$ and the phase $\arg s_{ij}\left(x\right)$ of the function are shown in the left and the right panels, respectively.

Figure 3

(Color online) $J=0$ contribution to the three-body recombination rate into the shallow dimer for the case in which there are no deep dimers. The hyperangular average $K^{\left(0\right)}\left(E\right)$ is shown as a function of the collision energy $E$ for six values of ${\theta }_{*0}$: $\pi ∕2$ (upper solid line), $9\pi ∕10$ and $\pi ∕10$ (upper and lower dashed lines), $29\pi ∕30$ and $\pi ∕30$ (upper and lower dash-dotted lines), and 0 or $\pi$ (lower solid line).

Figure 4

(Color online) Contributions to the three-body recombination rate into the shallow dimer from angular momenta $J=1,\dots ,6$. The hyperangular averages $K^{\left(J\right)}\left(E\right)$ are shown as functions of the collision energy $E$.

Figure 5

(Color online) Three-body recombination rates $K^{\left(J\right)}\left(E\right)$ for ${}^4\mathrm{He}$ atoms (in units of ${\mathrm{cm}}^6∕\mathrm{s}$) as functions of the collision energy $E$ (in units of mK) for $J=0–6$. The solid lines are our zero-range results for $a=164.5a_0$ and $a_{*0}=143.1a_0$. The dots are the results obtained in Ref. 38 by solving the three-body Schrödinger equation for the HFD-B3-FCI1 potential.

Figure 6

(Color online) $J=0$ contribution to the three-body recombination rate into the shallow dimer for cases in which there are one or more deep dimers. The hyperangular average $K^{\left(0\right)}\left(E\right)$ is shown as a function of the collision energy $E$ for six values of ${\theta }_{*0}$: $\pi ∕2$ (upper solid line), $9\pi ∕10$ and $\pi ∕10$ (upper and lower dashed lines), $29\pi ∕30$ and $\pi ∕30$ (upper and lower dash-dotted lines), and 0 or $\pi$ (lower solid line). The left and right panels are for ${\eta }_{*}=0.1$ and 0.2, respectively.

Figure 7

(Color online) Three-body recombination rate into deep dimers. The maximum and minimum values of $K_{\text{deep}}\left(E\right)$ with respect to variations of ${\theta }_{*0}$ are shown as functions of the collision energy $E$ for three values of ${\eta }_{*}$: 0.01, 0.1, and 0.2.

Figure 8

(Color online) Three-body recombination length ${\rho }_3$ for ${}^{133}\mathrm{Cs}$ atoms as a function of $a$ for $T=200\mathrm{nK}$. The data points are from Ref. 16. The curves are the zero-range predictions for three values of ${\eta }_{*}$: 0 (solid line), 0.1 (dashed line), and 0.2 (dotted line).

Figure 9

(Color online) Three-body recombination length ${\rho }_3$ for ${}^{133}\mathrm{Cs}$ atoms as a function of $a$ for $T=200$ (left panel) and $40\mathrm{nK}$ (right panel). The data points for $T=200\mathrm{nK}$ are from Ref. 16. The curves are the zero-range predictions for three values of ${\eta }_{*}$: 0 (solid lines), 0.1 (dashed lines), and 0.2 (dotted lines).

Figure 10

(Color online) Three-body recombination length ${\rho }_3$ for ${}^{133}\mathrm{Cs}$ atoms as a function of $a$ for ${\eta }_{*}=0.2$. The curves are the zero-range predictions for three values of the temperature: $T=0$ (solid line), 40 (dashed line), and $200\mathrm{nK}$ (dotted line). The data points are for $T=200\mathrm{nK}$ from Ref. 16.