Universal energy-dependent pseudopotential for the two-body problem of confined ultracold atoms

Abstract

The two-body scattering amplitude and energy spectrum of confined ultracold atoms are of fundamental importance for both theoretical and experimental studies of ultracold-atom physics. For many systems, one can efficiently calculate these quantities via the zero-range Huang-Yang pseudopotential (HYP), in which the interatomic interaction is characterized by the scattering length $a$. Furthermore, when the scattering length is dependent on the kinetic energy ${\ensuremath{\varepsilon}}_{\mathrm{r}}$ of two-atom relative motion, i.e., $a=a({\ensuremath{\varepsilon}}_{\mathrm{r}})$, the results are applicable for a broad energy region. However, when the free Hamiltonian of atomic internal state (e.g., the Zeeman Hamiltonian) does not commute with the interatomic interaction, or the center-of-mass (c.m.) motion is coupled to the relative motion, the generalization of this technique is still lacking. In this work we solve this problem and construct a reasonable energy-dependent multichannel HYP, which is characterized by a ``scattering length operator'' ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$, for the above complicated cases. Here, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ is an operator for atomic internal states and c.m. motion, and depends on both the total two-atom energy and the external field as well as the trapping parameter. The effects from the internal-state or c.m.-relative motion coupling can be self-consistently taken into account by ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$. We further show a method based on the quantum defect theory, with which ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ can be analytically derived for systems with van der Waals interatomic interaction. To demonstrate our method, we calculate the spectrum of two ultracold fermionic alkaline-earth-like atoms [in electronic $^{1}S_{0}$ $(|g\ensuremath{\rangle})$ and $^{3}P_{0}$ $(|e\ensuremath{\rangle})$ states, respectively] confined in an optical lattice. By comparing our results with the recent experimental measurements for two $^{173}\mathrm{Yb}$ atoms and two $^{171}\mathrm{Yb}$ atoms, we calibrate the scattering lengths ${a}_{\ifmmode\pm\else\textpm\fi{}}$ with respect to antisymmetric and symmetric nuclear-spin states to be ${a}_{+}=2012(19){a}_{0}$ and ${a}_{\ensuremath{-}}=193(4){a}_{0}$ for $^{173}\mathrm{Yb}$, and ${a}_{+}=232(3){a}_{0}$ and ${a}_{\ensuremath{-}}=372(1){a}_{0}$ for $^{171}\mathrm{Yb}$.