Motivated by the potential use of electromagnetically induced transparency (EIT) to measure atomic parity nonconservation (PNC), we have studied EIT and associated optical rotation for a three-level system in thallium vapor. EIT allows sub-Doppler resolution of the ${6P}_{1/2}\ensuremath{\rightarrow}{6P}_{3/2}$ mixed magnetic dipole, electric quadrupole transition at $1.28\ensuremath{\mu}\mathrm{m}$ when $0.535\ensuremath{-}\ensuremath{\mu}\mathrm{m}$ radiation acts on the ${6P}_{3/2}\ensuremath{\rightarrow}{7S}_{1/2}$ transition. Our measurements include rotation due to electromagnetically induced birefringence and Faraday rotation perturbed by EIT. We also identify a new method for determining the amplitude ratio
E2/M1 for the $1.28\ensuremath{-}\ensuremath{\mu}\mathrm{m}$ transition. The possible advantages of an EIT technique for measuring atomic PNC are discussed in the context of our results.
We propose the measurement of cavity-enhanced parity-nonconserving (PNC) optical rotation in several transitions of metastable Xe and Hg, including Xe ${(}^{2}{P}_{3/2}^{o})6s\text{ }^{2}[3/2{]}_{2}^{o}\ensuremath{\rightarrow}{(}^{2}{P}_{1/2}^{o})6s\text{ }^{2}[1/2{]}_{1}^{o}$ and Hg $6s6p\text{ }^{3}P_{2}^{o}\ensuremath{\rightarrow}6s6p\text{ }^{1}P_{1}^{o}$, with calculated amplitude ratios of ${E}_{1}^{\mathrm{PNC}}/M1=11\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ and $10\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$, respectively. We demonstrate the use of a high-finesse bow-tie cavity with counterpropagating beams and a longitudinal magnetic field, which allows the absolute measurement of chiral optical rotation, with a path length enhancement of about
104, necessary for PNC measurement from available column densities of ${10}^{14}\text{ }\text{ }{\mathrm{cm}}^{\ensuremath{-}2}$ for metastable Xe or Hg. Rapid PNC-signal reversal, allowing robust background subtraction, is achieved by shifting the cavity resonance to an opposite polarization mode or by inverting the magnetic field. The precise measurement of isotope and nuclear-spin dependent
E1PNC amplitudes provides a sensitive low-energy test of the standard model.