Two-color photoassociation spectroscopy of ytterbium atoms and the precise determinations of $s$-wave scattering lengths

By performing high-resolution two-color photoassociation spectroscopy, we have successfully determined the binding energies of several of the last bound states of the homonuclear dimers of six different isotopes of ytterbium. These spectroscopic data are in excellent agreement with theoretical calculations based on a simple model potential, which very precisely predicts the $s$-wave scattering lengths of all 28 pairs of the seven stable isotopes. The $s$-wave scattering lengths for collision of two atoms of the same isotopic species are 13.33(18) nm for ${}^{168}\text{Y}\text{b}$, 3.38(11) nm for ${}^{170}\text{Y}\text{b}$, $-0.15\left(19\right)\text{nm}$ for ${}^{171}\text{Y}\text{b}$, $-31.7\left(3.4\right)\text{nm}$ for ${}^{172}\text{Y}\text{b}$, 10.55(11) nm for ${}^{173}\text{Y}\text{b}$, 5.55(8) nm for ${}^{174}\text{Y}\text{b}$, and $-1.28\left(23\right)\text{nm}$ for ${}^{176}\text{Y}\text{b}$. The coefficient of the lead term of the long-range van der Waals potential of the ${\text{Yb}}_2$ molecule is $C_6=1932\left(30\right)$ atomic units $\left(E_h{a}_0^6\approx 9.573\times {10}^{-26}\text{J}{\text{nm}}^6\right)$.

Figure 1

(Color online) Schematic description of the two-color PA spectroscopy. The laser $L_1$ drives the one-color PA transition. The laser $L_2$ couples the bound state in the excited molecular potential to the one in the ground molecular potential. The detuning $\Delta$ of the PA laser with respect to the one-color PA resonance is set to several MHz for the Raman spectroscopy, while $\Delta$ is set to zero for the Autler-Townes spectroscopy.

Figure 2

(Color online) Schematic description of an experimental setup for PA lasers. The two lasers were prepared by splitting one laser, and two double-pass acousto-optic modulators (AOMs) were employed to tune them. Then the two lasers were aligned in the same path. The solid lines and broken lines indicate the laser beams before and after the double pass, respectively. A polarizing beam splitter (PBS) and a $\lambda /2$ plate after the optical fiber were employed to fix polarization of both lasers in the same direction.

Figure 3

(Color online) Two-color PA spectra at about $1\mu \text{K}$. The horizontal and vertical axes are the frequency difference between the two lasers and the number of remaining atoms, respectively. The vertical error bars represent the fluctuations of the number of atoms. The solid lines are fits of the Lorentz functions. The Autler-Townes spectroscopy is applied to the $v=1$, $J=0$,2 states of ${{}^{171}\text{Y}\text{b}}_2$ and the $v=1$, $J=0$ state of ${{}^{172}\text{Y}\text{b}}_2$, while the Raman spectroscopy is applied to the others.

Figure 4

(Color online) Resonance positions $f_1-f_2$ of the $v=1$, $J=0$ state of ${{}^{174}\text{Y}\text{b}}_2$ as a function of ${\delta }_{\text{LS}}$. The temperature shift is not compensated. The vertical error bars represent the uncertainty estimates for the temperature and the center frequency of the resonance. The horizontal error bars include the fluctuations of the laser intensities. The data are fitted with a linear function represented by a solid line.

Figure 5

(Color online) Resonance positions $f_1-f_2$ of the $v=1$, $J=0$ state of ${{}^{174}\text{Y}\text{b}}_2$ as a function of the temperature. The light shift is not compensated. The vertical error bars represent the uncertainty estimates for the center frequency of the resonance. The horizontal error bars include the heating effect due to the irradiation of the PA lasers. The data are fitted with a linear function represented by a solid line.

Figure 6

(Color online) (a) Calculated scattering lengths and (b) binding energies $E_b$ vs twice the reduced mass, $2\mu$, in the atomic mass unit, using the three-parameter potential energy model in Eq. (5) with the parameters given in the text. The vertical lines show the masses for the seven like-atom pairs. The solid and dashed lines in (b) show the $J=0$ and two eigenvalues, respectively. The measured values with vertical error bars are shown for the levels for which they have been measured. The measured scattering lengths of ${}^{170}\text{Y}\text{b}$ [20], ${}^{173}\text{Y}\text{b}$ [18], and ${}^{174}\text{Y}\text{b}$ [19] are also shown in (a). The horizontal dashed line in (a) shows the van der Waals length $\overline{a}$.

Figure 7

(Color online) Calculated cross section $\sigma \left(E\right)$ vs collision energy $E/k_B$ for two atoms of the same isotope. The label for each curve shows the isotopic mass number. Even mass number corresponds to identical bosons and odd mass number corresponds to two fermions with different spin projections. The dashed lines show the $s$-wave contribution to the total cross section. The solid lines show the contribution from $s$- and $d$-waves for the like boson cases, and the contribution from $s$-, $p$, and $d$-waves for the fermion case with different spin components. The $E\to 0$ cross sections are $8\pi a^2$ and $4\pi a^2$ for the respective boson and fermion cases.