Limits on Universality in Ultracold Three-Boson Recombination

The recombination rate for three identical bosons has been calculated to test the limits of its universal behavior. It has been obtained for several different collision energies and scattering lengths $a$ up to ${10}^5\mathrm{a}\mathrm{.}\mathrm{u}\mathrm{.}$, giving rates that vary over 15 orders of magnitude. We find that universal behavior is limited to the threshold region characterized by $E\lesssim {\hslash }^2/2{\mu }_{12}{a}^2$, where $E$ is the total energy and ${\mu }_{12}$ is the two-body reduced mass. The analytically predicted infinite series of resonance peaks and interference minima is truncated to no more than three of each for typical experimental parameters.

Figure 1

Three-body recombination rate for (a) $a<0$ and (b) $a>0$. The horizontal lines represent the unitarity limit $u_N$ for each energy (reported as a temperature using $E=k_{B}T$). The vertical lines represent the boundary of the threshold regime $a_N$. The solid curve represents the analytical formula Eq. (3). The insets show the third resonance peak (a) and third interference minima (b) from $a=0$.

Figure 2

Thermally averaged recombination rate (a) near the second resonance peak and (b) near the second interference minimum. Circles and diamonds represent $K_3$ and $\langle K_3\rangle$, respectively, at $0.1\mu \mathrm{K}$ (empty symbols) and $1\mu \mathrm{K}$ (filled symbols).

Figure 3

Comparison of the $0^{+}$ and $2^{+}$ recombination rate for $a>0$ and temperatures of (a) 10 nK and (b) $1\mu \mathrm{K}$.