Linear Stark Shifts to Measure the Fr Weak Nuclear Charge with Small Atom Samples

We study the chirality of ground-state alkali atoms in E and B fields, dressed with a circularly-polarized laser beam close-detuned from an E-field-assisted forbidden transition, such as 7S-8S in Fr. We predict a parity violating energy shift of their sublevels, linear in E, and the weak nuclear charge Q_W$\mathrm{.}Adressingbeamof10kW\mathrm{/}cm$^2$at506nmproducesashiftof$\sim100 \mu$HzatE=100V\mathrm{/}cm,B$\gtrsim$30mG\mathrm{.}Itshouldbeobservablewith$\sim 10^4$ Fr atoms confined in an optical dipole trap. We discuss optimal conditions, parameter reversals and a calibration procedure to measure $Q_W$.