  We present and analyze a cut finite element method for the weak imposition of
the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed
element on both triangular and quadrilateral meshes is considered. Our method
is based on the Nitsche formulation studied in [10.1515/jnma-2021-0042] and can
be considered as a first attempt at extension in the unfitted case. The key
feature is to add two ghost penalty operators to stabilize both the velocity
and pressure fields. We rigorously prove our stabilized formulation to be
well-posed and derive a priori error estimates for the velocity and pressure
fields. We show that an upper bound for the condition number of the stiffness
matrix holds as well. Numerical examples corroborating the theory are included.
