Harmonic to anharmonic tuning of moir\'e potential leading to unconventional Stark effect and giant dipolar repulsion in WS$_2$/WSe$_2$ heterobilayer

Excitonic states trapped in harmonic moir\'e wells of twisted heterobilayers is an intriguing testbed. However, the moir\'e potential is primarily governed by the twist angle, and its dynamic tuning remains a challenge. Here we demonstrate anharmonic tuning of moir\'e potential in a WS$_2$/WSe$_2$ heterobilayer through gate voltage and optical power. A gate voltage can result in a local in-plane perturbing field with odd parity around the high-symmetry points. This allows us to simultaneously observe the first (linear) and second (parabolic) order Stark shift for the ground state and first excited state, respectively, of the moir\'e trapped exciton - an effect opposite to conventional quantum-confined Stark shift. Depending on the degree of confinement, these excitons exhibit up to twenty-fold gate-tunability in the lifetime ($100$ to $5$ ns). Also, exciton localization dependent dipolar repulsion leads to an optical power-induced blueshift of $\sim$1 meV/$\mu$W - a five-fold enhancement over previous reports.