Three-body problem in heteronuclear mixtures with resonant interspecies interaction

We use the zero-range approximation to study a system of two identical bosons interacting resonantly with a third particle. The method is derived from effective field theory. It reduces the three-body problem to an integral equation which we then solve numerically. We also develop an alternative approach which gives analytic solutions of the integral equation in coordinate representation in the limit of vanishing total energy. The atom-dimer scattering length, the rates of atom-dimer relaxation, and the three-body recombination to shallow and to deep molecular states are calculated either analytically or numerically with a well-controlled accuracy for various energies as functions of the mass ratio, scattering length, and three-body parameter. We discuss in detail the relative positions of the recombination loss peaks, which in the universal limit depend only on the mass ratio. Our results have implications for ongoing and future experiments on Bose-Bose and Bose-Fermi atomic mixtures.

Figure 1

Integral equation for the atom-dimer scattering amplitude $\mathcal{A}$. Solid (dashed) lines denote atom species $2$ ($1$). Mixed double lines denote the full dimer propagator.

Figure 2

Dependence of the trimer energy on the inverse scattering length $1/a$ in arbitrary units (dashed line). The parameters $a_{-}$ and $a_{*}$ specify where the trimer state hits the three-atom and atom-dimer thresholds, respectively.

Figure 3

Solid line: $a_{*}^{(n)}/\left|a_{-}^{(n)}\right|$ vs $\delta$, where $n$ is the index of the Efimov state [see Eq. (1)]. Dashed line: $a_{*}^{(n+1)}/\left|a_{-}^{(n)}\right|=\exp (\pi /s_0)a_{*}^{(n)}/\left|a_{-}^{(n)}\right|$.

Figure 4

Diagrammatic representation of (a) the three-body recombination amplitude and (b) the elastic three-body scattering. Line patterns are the same as in Fig. 1.

Figure 5

The coefficient $C$ for the different three-body recombination rates in dependence of the mass ratio $\delta =m_1/m_2$. The inset shows $C$ for larger values of $\delta$.

Figure 6

The parameters $C_1(\delta )$ and $C_2(\delta )$ in the expression for the atom-dimer scattering length, Eq. (18).

Figure 7

The dimer relaxation rate constant $\beta$ in units of $\hslash a/m_1$ for ${\eta }_{*}=0.1$ and $\delta =0.471$ as function of $a/a_{*}$. The solid, short-dashed, long-dashed, and dot-dashed lines show $\beta$ for $E/B_d=-1$, $-$0.95, $-$0.5, and 0, respectively. The double-dot-dashed line indicates the trajectory of the resonance maximum as the energy is increased from $-B_d$ to zero.

Figure 8

${\alpha }_d$ vs $a$ for the ${}^{41}\mathrm{K}$-${}^{87}\mathrm{Rb}$-${}^{87}\mathrm{Rb}$ system assuming $a_{-}=-246$ $a_{\mathrm{Bohr}}$ and three values of ${\eta }_{*}$: ${\eta }_{*}=0.12$ (dashed line), ${\eta }_{*}=0.01$ (solid line), and ${\eta }_{*}=0.4$ (dash-dotted line). Data points indicated by diamonds are taken from Refs. [15, 42]; see text.