Cluster States from Quantum Logic Gates with Trapped Ions in Thermal Motion

Following the recent proposal by Briegel et al. Phys. Rev. Lett. 86 (2001) 910, a procedure is proposed for one-step realizing quantum control phase gates with two trapped ions in thermal motion. It is shown
that the scheme can also be used to create a new special type of entangled states, i.e., cluster states of many trapped ions. In the scheme the two-trapped ions are simultaneously excited by a single laser beam and the frequency of the laser beam is slightly off resonance with the first lower vibration sideband of the trapped ions. The distinct advantage of the scheme is that it does not use the vibrational mode as the data bus. Furthermore, our scheme is insensitive to both the initial motional state and heating (or decay) as long as the system remains in the Lamb--Dicke regime.