High-precision measurement and ab initio calculation of the $(6s^26p^2),^3!P_0 rightarrow , ^3!P_2$ electric quadrupole transition amplitude in $^{208}$Pb

We have completed a measurement of the $(6s^26p^2), ^3!P_0 rightarrow , ^3!P_2$ 939 nm electric quadrupole ($E2$) transition amplitude in atomic lead. Using a Faraday rotation spectroscopy technique and a sensitive polarimeter, we have measured this very weak $E2$ transition for the first time, and determined its amplitude to be $langle ^3!P_2 || Q || ^3!P_0 rangle$ = 8.91(9) a.u.. We also present an ab initio theoretical calculation of this matrix element, which agrees with experiment at the 0.5% level. We heat a quartz vapor cell containing $^{208}$Pb to between 800 and 940 $^{circ}$C, apply a $sim ! 10 , {rm G}$ longitudinal magnetic field, and use polarization modulation/lock-in detection to measure optical rotation amplitudes of order 1 mrad with noise near 1 $mu$rad. We compare the Faraday rotation amplitude of the $E2$ transition to that of the $^3!P_0 -, ^3!P_1$ 1279 nm magnetic dipole ($M1$) transition under identical sample conditions.