Valley Polarization-Electric Dipole Interference and Nonlinear Chiral Selection Rules in Monolayer WSe$_2$

In monolayer transition metal dichalcogenides time-reversal symmetry, combined with space-inversion symmetry, defines the spin-valley degree of freedom. As such, engineering and control of time-reversal symmetry by optical or magnetic fields constitutes the foundation of valleytronics. Here, we propose a new approach for the detection of broken time-reversal symmetry and valley polarization in monolayer WSe$_2$ based on second harmonic generation. Our method can selectively and simultaneously generate and detect a valley polarization at the $\pm K$ valleys of transition metal dichalcogenides at room temperature. Furthermore, it allows to measure the interference between the real and imaginary parts of the intrinsic (electric dipole) and valley terms of the second order nonlinear susceptibility. This work demonstrates the potential and unique capabilities of nonlinear optics as a probe of broken time-reversal symmetry and as a tool for ultrafast and non-destructive valleytronic operations.