  We provide a theoretical scheme for realizing a Hadamard and a quantum
controlled-NOT logic gates operations in the anti-Jaynes-Cummings interaction
process. Standard Hadamard operation for a specified initial atomic state is
achieved by setting a specific sum frequency and photon number in the
anti-Jaynes-Cummings qubit state transition operation with the interaction
component of the anti-Jaynes-Cummings Hamiltonian generating the state
transitions. The quantum controlled-NOT logic gate is realized when a single
atomic qubit defined in a two-dimensional Hilbert space is the control qubit
and two non-degenerate and orthogonal polarized cavities defined in a
two-dimensional Hilbert space make the target qubit. With precise choice of
interaction time in the anti-Jaynes-Cummings qubit state transition operations
defined in the anti-Jaynes-Cummings sub-space spanned by normalized but
non-orthogonal basic qubit state vectors, we obtain ideal unit probabilities of
success in the quantum controlled-NOT operations.
