Determination of the $s$-Wave Scattering Length and the $C_6$ van der Waals Coefficient of ${}^{174}\mathrm{Yb}$ via Photoassociation Spectroscopy

We report photoassociation spectroscopy of ${}^{174}\mathrm{Yb}$ for the ${}^{1}S_0\mathrm{\text{−}}{}^{1}P_1$ transition at $1\mu \mathrm{K}$, where only the $s$-wave scattering state contributes to the spectra. The wave function of the $s$-wave scattering state is obtained from the photoassociation efficiency, and we determine that the $C_6$ potential coefficient is $2300\pm 250\mathrm{a}.\mathrm{u}.$ and the $s$-wave scattering length is $5.53\pm 0.11\mathrm{nm}$. Based on these parameters, we discuss the scattering properties of $s$- and $d$-wave states.

Figure 1

Example of the observed spectra. The number of atoms after the PA laser irradiation is plotted as a function of the laser frequency detuning from the ${}^{1}S_0\mathrm{\text{−}}{}^{1}P_1$ atomic resonance. The labels indicate the vibrational quantum numbers $v^{\prime }$.

Figure 2

Reconstructed squared wave function of the zero-energy $s$-wave scattering state. ○ indicate experimental results derived from Eq. (3). The uncertainty of the data is shown at the lower right corner. ● representing data for $v^{\prime }>165$ are not taken into account for the fitting. The solid line is the best-fitted theoretical wave function with $C_{6g}=2340\mathrm{a}.\mathrm{u}.$ and $a=5.53\mathrm{nm}$.

Figure 3

Contour lines of ${\chi }^2$ for the fitting of the wave function to the experimental data shown in Fig. 2. This value of ${\chi }^2$ is measured from the minimal point. The solid lines are the results for the ground-state potential of $V_g(R)=-C_{6g}/R^6$, and the dashed lines are for $V_g(R)=-C_{6g}/R^6-C_{8g}/R^8$ with $C_{8g}=5\times {10}^5\mathrm{a}.\mathrm{u}.$

Figure 4

(a) Wave functions and potentials of the $s$- and $d$-wave scattering states calculated from the best-fitted parameters $C_{6g}$ and $a$ without the $C_{8g}/R^8$ term. Both the wave functions are calculated for the energy of $50\mu \mathrm{K}$ and are normalized to have the same amplitude at $R\to \infty$. (b) Elastic-collision cross sections of the $s$- and $d$-wave scatterings as functions of the collision energy.