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We develop a theory for the interaction of multi-level atoms with multi-modecavities yielding cavity-enhanced multi-photon resonances. The locations of theresonances are predicted from the use of effective two- and three-levelHamiltonians. As an application we show that quantum gates can be realised whenphotonic qubits are encoded on the cavity modes in arrangements where ancillaatoms transit the cavity. The fidelity of operations is increased byconditional measurements on the atom and by the use of a selected, dual-rail,Hilbert space. A universal set of gates is proposed, including the Fredkin gateand iSWAP operation; the system seems promising for scalability.

Multiphoton resonances for all-optical quantum logic with multiple cavities