Potassium ground-state scattering parameters and Born-Oppenheimer potentials from molecular spectroscopy

We present precision measurements with MHz uncertainty of the energy gap between asymptotic and well bound levels in the electronic ground state $X{{}^1\Sigma }_{\text{g}}^{+}$ of the ${{}^{39}\text{K}}_2$ molecule. The molecules are prepared in a highly collimated particle beam and are interrogated in a $\Lambda$-type excitation scheme of optical transitions to long range levels close to the asymptote of the ground state, using the electronically excited state ${A{}^1\Sigma }_{\text{u}}^{+}$ as the intermediate one. The transition frequencies are measured either by comparison with ${\text{I}}_2$ lines or by absolute measurements using a fs-frequency comb. The determined level energies were used together with Feshbach resonances from cold collisions of ${}^{39}\text{K}$ and ${}^{40}\text{K}$ reported from other authors to fit ground state potentials. Precise scattering lengths are determined and tests of the validity of the Born-Oppenheimer approximation for the description of cold collisions at this level of precision are performed.

Figure 1

(Color online) Simplified excitation scheme of ${\text{K}}_2$. Initially, the molecules are in the vibrational ground state of the $X{{}^1\Sigma }_g^{+}$ state. The laser ${\text{L}}_1$ is applied first and drives Franck-Condon pumping to populate $v=41$, $J$ in the ground state. Asymptotic levels are observed as dark resonances induced by laser ${\text{L}}_2$ (on resonance) and laser ${\text{L}}_3$ (tuned).

Figure 2

(Color online) Laser system used for frequency measurement and tuning of ${\text{L}}_3$. OD: optical diode; BS: beam splitter; PBS: polarizing beam splitter; PZT: piezoelectric actuator; $\lambda /4$: $\lambda /4$ retarder; and AOM: acousto-optical modulator.

Figure 3

(Color online) Spectra of dark resonances observed in two $\Lambda$ configurations. Left-hand side: coupling between $\left(v=41,J=7\right)-\left(v^{\prime }=143,J^{\prime }=8\right)-\left(v^{″}=80,J^{″}=7\right)$. Right-hand side: case $\left(v=41,J=7\right)-\left(v^{\prime }=159,J^{\prime }=6\right)-\left(v^{″}=83,J^{″}=5\right)$, two dips are visible, which belong to the small hyperfine splitting of asymptotic levels in the state $\mathit{X}$. The hyperfine structure stems from coupling to the $a{{}^3\Sigma }_u^{+}$ state and is so small in potassium that it is only resolved for $v^{″}=82$ and higher. The frequency scale is given with respect to iodine lines.

Figure 4

Difference of vibrational ladders for ${{}^{40}\text{K}}_2$ calculated with the potential from Table with adiabatic correction taken into account or setting it to zero. The actual functional form is far from being fixed by this study, only the magnitudes should be considered. For low $v$, the magnitude should be compared to the average uncertainty of $0.003{\text{cm}}^{-1}$ of the Fourier transform spectroscopy data. For $v$ close to the asymptote, data with smaller uncertainty are available in the present study ($0.0001{\text{cm}}^{-1}$ or 3 MHz) and from Feshbach resonances (corresponding uncertainty better than 1 MHz).