  It was recently found that the electric local-field effect (LFE) can lead to
strong coupling of atomic Bose-Einstein condensates (BECs) to off-resonant
optical fields. We demonstrate that the magnetic LFE gives rise to a previously
unexplored mechanism for coupling a (pseudo) spinor BEC or fermion gas to
microwaves (MWs). We present a theory for the magnetic LFE, and find that it
gives rise to a short-range attractive interaction between two components of
the (pseudo) spinor, and a long-range interaction between them. The latter
interaction, resulting from deformation of the magnetic field, is locally
repulsive but globally attractive, in sharp contrast with its counterpart for
the optical LFE, produced by phase modulation of the electric field. Our
analytical results, confirmed by the numerical computations, show that the
long-range interaction gives rise to modulational instability of the spatially
uniform state, and creates stable ground states in the form of hybrid
matter-wave-microwave solitons (which seem like one-dimensional magnetic
monopoles), with a size much smaller than the MW wavelength, even in the
presence of arbitrarily strong contact inter-component repulsion. The setting
is somewhat similar to exciton-polaritonic condensates in semiconductor
microcavities. The release of matter waves from the soliton may be used for the
realization of an atom laser. The analysis also applies to molecular BECs with
rotational states coupled by the electric MW field.
