  We demonstrate a method to obtain homogeneous atom-cavity coupling by
selecting and keeping $^{87}$Rb atoms that are near maximally coupled to the
cavity's standing-wave mode. We select atoms by imposing an AC Stark shift on
the ground state hyperfine microwave transition frequency with light injected
into the cavity. We then induce a spin flip with microwaves that are resonant
for atoms that are near maximally coupled to the cavity mode of interest, after
which, we use radiation pressure forces to remove from the cavity all the atoms
in the initial spin state. Achieving greater homogeneity in the atom-cavity
coupling will potentially enhance entanglement generation, intracavity driving
of atomic transitions, cavity-optomechanics, and quantum simulations. This
approach can easily be extended to other atomic species with microwave or
optical transitions.
