Effective-field-theory analysis of Efimov physics in heteronuclear mixtures of ultracold atomic gases

We use an effective-field-theory framework to analyze the Efimov effect in heteronuclear three-body systems consisting of two species of atoms with a large interspecies scattering length. In the leading-order description of this theory, various three-body observables in heteronuclear mixtures can be universally parametrized by one three-body parameter. We present the next-to-leading corrections, which include the effects of the finite interspecies effective range and the finite intraspecies scattering length, to various three-body observables. We show that only one additional three-body parameter is required to render the theory predictive at this order. By including the effective range and intraspecies scattering length corrections, we derive a set of universal relations that connect the different Efimov features near the interspecies Feshbach resonance. Furthermore, we show that these relations can be interpreted in terms of the running of the three-body counterterms that naturally emerge from proper renormalization. Finally, we make predictions for recombination observables of a number of atomic systems that are of experimental interest.

Figure 1

STM equation for the LO scattering amplitude. The dashed and the solid lines represent the propagators of atoms 1 and 2 respectively, and the thick gray line represents the dressed two-body propagator $i{\mathcal{D}}_{12}^{\left(0\right)}$.

Figure 2

Numerical prefactor $c$ appearing in the LO three-body force $H_0\left(\mathrm{\Lambda }\right)$ as a function of the mass ratio $\delta =m_1/m_2$.

Figure 3

The upper equation shows the NLO scattering amplitude $i{\mathcal{A}}_1$. The thick gray line with a black square represents the NLO dressed propagator $i{\mathcal{D}}_{12}^{\left(1\right)}$, the double line represents the bare propagator of the $d_{22}$ field, and the three-atom vertex with circle represents the NLO three-body force. The lower equation defines the coupled-channel amplitude, which, up to constant factors, is equal to $\mathcal{M}$ defined in Eq. (11).

Figure 4

Amplitude for the recombination of free atoms into a shallow two-atom bound state and a residual atom. The first term on the right-hand side is the LO amplitude and the rest are NLO corrections.

Figure 5

Functions $\mathrm{\Xi }\left(\mathrm{\Lambda }\right)\equiv \xi \left(\mathrm{\Lambda }\right){sin}^2\left(s_{0}ln\frac{\mathrm{\Lambda }}{{\mathrm{\Lambda }}_{*}}-arctan{s}_0\right)$ and $\overline{\mathrm{\Xi }}\equiv \overline{\xi }\left(\mathrm{\Lambda }\right){sin}^2\left(s_{0}ln\frac{\mathrm{\Lambda }}{{\mathrm{\Lambda }}_{*}}-arctan{s}_0\right)$ for the ${}^6\mathrm{Li}$-Cs-Cs system. The renormalization conditions were $E_1^{\left(0\right)}\left(0\right)=0$ and $\mathrm{\Delta }{\gamma }_0=0$. $Q_{*}$ was chosen to be equal to ${\kappa }_{*}$. Periodicity of these functions in $ln\mathrm{\Lambda }$ could be obtained by requiring both $d_R$ and ${\overline{d}}_R$ to have the value 0.981.