$mathcal{P}$,$mathcal{T}$-odd Faraday rotation on atoms and molecules in intra-cavity absorption spectroscopy as an alternative way to search for the $mathcal{P}$,$mathcal{T}$-odd effects in nature

Present limit on the electron electric dipole moment ($e$EDM) is based on the electron spin precession measurement. We propose an alternative approach - observation of the $mathcal{P}$,$mathcal{T}$-odd Faraday effect in an external electric field on atoms and molecules using cavity-enhanced polarimetric scheme in combination with molecular (atomic) beam crossing the cavity. Our calculations of the effective electric fields and theoretical simulation of the proposed experiment on Tl and Pb atoms, PbF, YbF, ThO, and YbOH show that the present limit on the $e$EDM can be improved by 6-7 orders of magnitude.