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$. A dressing beam of 10 kW/cm$^2$at 506 nm produces a shift of$\sim100 \mu$Hz at E=100 V/cm, B$\gtrsim$30 mG. It should be observable with$\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$.