Controlled-NOT gate for frequency-encoded qubits based on six-wave mixing

The possibility of implementing a controlled-NOT gate for frequency-encoded photonic qubits via six-wave mixing in a multilevel atomic system is shown. Starting from the Hamiltonian for six-wave mixing in a cavity, equations of motion for the amplitudes of the photonic states are derived and solved in the framework of input-output formalism. Analysis of these solutions show how to realize two-qubit gates including relevant matching conditions.