  We developed a comprehensive theoretical framework focusing on many-body
effects of exciton species in monolayer WS$_2$, including bright and dark
excitons, and intra- and inter-valley biexcitons, to investigate valley
dynamics in monolayer WS$_2$ subjected to a tilted magnetic field B. In
particular, the evolution of the exciton population densities and the many-body
particle scatterings are considered to calculate the valley polarization (V P )
as a function of the magnetic field. We found valley splittings for the dark
exciton and biexciton energy levels that are larger than those of bright
excitons, of -0.23 meV/T. For example, -0.46 meV/T for dark excitons and -0.69
meV/T for bright-dark intra-valley biexcitons. Furthermore, inter-valley
bright-dark excitons have an opposite valley energy splittings of +0.23 meV/T.
For the samples pumped by linearly polarized light, V P exhibits distinct
magnetic field dependence for different types of many-body particle states.
Among them, the V P of the intra-valley bright-dark biexcitons increases with
increasing magnetic field and reaches nearly 50% at B=65 T. The brightened dark
exciton, on the other hand, exhibits vanishing VP, indicating long valley
relaxation time. Remarkably, the inter-valley bright-dark biexciton shows
unconventional behavior with an inverted VP. The opposite VP for intra- and
inter-valley bright-dark biexcitons, coupled with their large valley splitting
and long valley lifetime may facilitate their coherent manipulation for quantum
computing.
