  Our understanding of neutrino flavor conversion in the innermost regions of
core-collapse supernovae and neutron star mergers is mostly limited to
spherically symmetric configurations that facilitate the numerical solution of
the quantum kinetic equations. In this paper, we simulate neutrino quantum
kinetics within a (2+1+1) dimensional setup: we model the flavor evolution
during neutrino decoupling from matter in two spatial dimensions, one neutrino
momentum variable, and time; taking into account non-forward neutral current
and charged current collisions of neutrinos with the matter background, as well
as neutrino advection. In order to mimic fluctuations in the neutrino emission
and matter background, and explore their effect on the flavor evolution, we
introduce perturbations in the collision term as well as in the vacuum term of
the Hamiltonian. Because of such perturbations, the initial symmetry of the
neutrino field across the simulation annulus is broken and flavor conversion is
qualitatively affected, with regions of larger flavor conversion alternating
across the simulation annulus. In addition, neutrino advection is responsible
for spreading flavor waves across neighboring spatial regions. Although based
on a simplified setup, our findings highlight the importance of modeling
neutrino quantum kinetics in multi-dimensions to assess the impact of neutrinos
on the physics of compact astrophysical sources and nucleosynthesis.
