  We study an atom-cavity system in which the cavity has several degenerate
transverse modes. Mode-resolved cavity transmission spectroscopy reveals
well-resolved atom-cavity resonances for several cavity modes, a signature of
collective strong coupling for the different modes. Furthermore, the experiment
shows that the cavity modes are coupled via the atomic ensemble contained in
the cavity. The experimental observations are supported by a detailed
theoretical analysis. The work paves the way to the use of interacting
degenerate modes in cavity-based quantum information processing, where qubits
corresponding to different cavity modes interact via an atom shared by the two
modes. Our results are also relevant to the experimental realization of quantum
spin glasses with ultracold atoms.
