Exact solution to a Bhatnagar-Gross-Krook-type equation for quantum lattice gases with dephasing noise

The mean-state dynamics of free fermions subject to random projective measurements of local occupation number operators is governed by a Lindblad equation with dephasing noise. In the continuum limit, the equation of motion for the correlation matrix is mapped to a kinetic equation for the Wigner function, which corresponds to a special case of the Bhatnagar-Gross-Krook (BGK) equation without energy conservation. Our main result is the solution to the kinetic equation, showing that the Wigner dynamics emerges from stochastic sampling of classical run-and-tumble processes. As an application, we recover the crossover between ballistic and diffusive transport regimes.