  We study one-step implementation of the Fredkin gate in a bi-modal cavity
under both resonant and large detuning conditions based on quantum Zeno
dynamics, which reduces the complexity of experiment operations. The influence
of cavity decay and atomic spontaneous emission is discussed by numerical
calculation. The results demonstrate that the fidelity and the success
probability are robust against cavity decay in both models and they are also
insensitive to atomic spontaneous emission in the large detuning model. In
addition, the interaction time is rather short in the resonant model compared
to the large detuning model.
