Preparation of Cluster States for Many Atoms in Cavity QED

We propose a scheme for the generation of the cluster states for many atoms in cavity QED. In our scheme, the atoms are sent through nonresonant cavity fields in the vacuum states. The cavity fields are only virtually excited and no quantum information will be transferred from the atoms to the cavity fields. The advantage is that the cavities are suppressed during the procedure. The scheme can also be generalized to the ion trap system.