  Driven-dissipative protocols are proposed to control and create nontrivial
quantum many-body correlated states. Protocols conserving the number of
particles stand apart. As well-known, in quantum systems with the unitary
dynamics the particle number conservation and random scattering yield diffusive
behavior of two-particle excitations (diffusons and cooperons). Existence of
diffusive modes in the particle-number-conserving dissipative dynamics is not
well studied yet. We explicitly demonstrate the existence of diffusons in a
paradigmatic model of a two-band system, with dissipative dynamics aiming to
empty one fermion band and to populate the other one. The studied model is
generalization of the model introduced in F. Tonielli, J. C. Budich, A.
Altland, and S. Diehl, Phys. Rev. Lett. 124, 240404 (2020). We find how the
diffusion coefficient depends on details of a model and the rate of
dissipation. We discuss how the existence of diffusive modes complicates
engineering of macroscopic many-body correlated states.
