Here, W is a quasi-probability density encapsulating the proton's quantum state as a function of both position (q) and momentum (p). The first set of terms in Eq. 1 corresponds to the Schrödinger dynamics of a particle with effective mass µ. The subsequent two terms correspond to the dissipation and decoherence arising from the coupling to the quantum bath. Here, γ is the phenomenological friction constant that describes the strength of the coupling to the bath, k B is Boltzmann's constant and T represents the effective bath temperature.