Efficient Scheme for Greenberger--Horne--Zeilinger State and Cluster State with Trapped Ions

We propose a scheme to generate the Greenberger--Horne--Zeilinger (GHZ) states and the cluster states of many trapped ions. In the scheme, the ion is illuminated by a single laser tuned to the first lower vibrational sideband. The scheme only requires resonant interactions. Thus the scheme is very simple and the quantum dynamics operation can be realized at a high speed, which is important in view of decoherence.