  Maxwell interface problems are of great importance in many electromagnetic
applications. Unfitted mesh methods are especially attractive in 3D computation
as they can circumvent generating complex 3D interface-fitted meshes. However,
many unfitted mesh methods rely on non-conforming approximation spaces, which
may cause a loss of accuracy for solving Maxwell equations, and the widely-used
penalty techniques in the literature may not help in recovering the optimal
convergence. In this article, we provide a remedy by developing
N\'ed\'elec-type immersed finite element spaces with a Petrov-Galerkin scheme
that is able to produce optimal-convergent solutions. To establish a systematic
framework, we analyze all the $H^1$, $\mathbf{H}(\text{curl})$ and
$\mathbf{H}(\text{div})$ IFE spaces and form a discrete de Rham complex. Based
on these fundamental results, we further develop a fast solver using a modified
Hiptmair-Xu preconditioner which works for both the GMRES and CG methods.
