  Adding a dopant to an antiferromagnetic spin background disturbs the magnetic
order and leads to the formation of a quasiparticle coined the magnetic
polaron, which plays a central role in understanding strongly correlated
materials. Recently, remarkably detailed insights into the spatial properties
of such polarons have been obtained using atoms in optical lattices. Motivated
by this we develop a nonperturbative scheme for calculating the wave function
of the magnetic polaron in a bilayer antiferromagnet using the self-consistent
Born approximation. The scheme includes an infinite number of spin waves, which
is crucial for an accurate description of the most interesting regime of strong
correlations. Utilizing the developed wave function, we explore the spatial
structure of the polaron dressing cloud consisting of magnetically frustrated
spins surrounding the hole. Mimicking the nonmonotonic behavior of the
antiferromagnetic order, we find that the dressing cloud first decreases and
then increases in size with increasing interlayer hopping. The increase
reflects the decrease in the magnetic order as a quantum phase transition to a
disordered state is approached for large interlayer hopping. We, furthermore,
find that the symmetry of the ground state dressing cloud changes as the
interlayer coupling increases. Our results should be experimentally accessible
using quantum simulation with optical lattices.
