Predicting scattering properties of ultracold atoms: Adiabatic accumulated phase method and mass scaling

Ultracold atoms are increasingly used for high-precision experiments that can be utilized to extract accurate scattering properties. This results in a stronger need to improve on the accuracy of interatomic potentials, and in particular the usually rather inaccurate inner-range potentials. A boundary condition for this short range can be conveniently given via the accumulated phase method. However, in this approach one should satisfy three conditions, two of which are in principle conflicting, and the validity of these approximations comes under stress when higher precision is required. We show that a better compromise between the two is possible by allowing for an adiabatic change in the hyperfine mixing of singlet and triplet states for interatomic distances smaller than the separation radius. Results we presented previously in a brief publication using this method show a high precision and extend the set of predicted quantities. The purpose of this paper is to describe its background. A mass-scaling approach to relate accumulated phase parameters in a combined analysis of isotopically related atom pairs is described in detail and its accuracy is estimated, taking into account both Born-Oppenheimer and Wentzel-Kramers-Brillouin breakdown. We demonstrate how numbers of singlet and triplet bound states follow from the mass scaling.

Figure 1

Main figure: Singlet $\left(S=0\right)$ and triplet $\left(S=1\right)$ potentials for a pair of rubidium atoms in the electronic ground state. Inset: $S=0\leftrightarrow S=1$ energy splitting of two ground-state rubidium atoms (equal to $2V^{\text{exch}}$) versus interatomic separation. The hyperfine energies for the isotopes ${}^{85}\text{R}\text{b}$ and ${}^{87}\text{R}\text{b}$ are indicated for comparison.

Figure 2

(Color online) Part A illustrates the behavior of the wave-function phase near $r_0=16a_0$ for three different energies. A comparison between the true accumulated phase (dots) and a first-order approximation (solid lines for triplet, dashed lines for singlet) is shown in part B. As a function of $E$ and $l\left(l+1\right)$ the graph shows the difference in accumulated phase $\varphi \left(E,l\right)$ at $r=r_0$ as compared to the $E=0$, $l=0$ situation: $\Delta \varphi \left(E,0\right)=\varphi \left(E,0\right)-\varphi \left(0,0\right)$ and $\Delta \varphi \left(0,l\right)=\varphi \left(0,l\right)-\varphi \left(0,0\right)$, respectively. The horizontal arrow indicates the typical $E$ and $l$ ranges for which we apply the first-order approximation. Typical rubidium potentials are used for this calculation. Note that for clarity the energy intervals for the wave functions in part A exceed the energies occurring in practice by far.

Figure 3

Subdivision of radial ranges to illustrate choices of $r_0$. Part A distinguishes three radial ranges. In the left interval $S$ is a good quantum number. In the right interval the individual atomic hyperfine labels $f_1,m_{f1},f_2,m_{f2}$ characterize the spin states. Conventionally, $r_0$ is chosen as far right as possible in the $V^{\text{hf}}⪡V^{\text{exch}}$ interval. Part B shows the radial intervals as they occur in the adiabatic accumulated phase method. The intermediate radial interval is subdivided in one in which the influence of $V^{\text{hf}-}$ is small and adiabatic and one in which it is not. The separation radius $r_0$ is chosen as far right as possible in the former interval.

Figure 4

Comparison of conventional accumulated phase method and alternative approach for ${}^{87}\text{R}\text{b}+{}^{87}\text{R}\text{b}$ scattering with initial spin state $|f_1,m_{f1},f_2,m_{f2}⟩=|1,-1,1,-1⟩$. Solid line: largest $C$ matrix element $C_{ij}$ in absolute value for “rigorous” coupled-channel calculation with $r_0$ in range $\left[11.75,16.0\right]a_0$. Dashed line: analogous result for $C_{ij}^{\text{ad}}$ from adiabatic accumulated phase method. Conventional method corresponds to $C_{ij}=0$

Figure 5

Predicted value of $C_8$ versus $r_0$. The dashed line connects points calculated with the traditional accumulated phase method, the solid curve corresponds similarly to the adiabatic accumulated phase method.