Quantum logic gates with a two-level trapped ion in a high-finesse cavity beyond the Lamb-Dicke limit

In the system with a two-level ion confined both in a linear trap and in a high-Q single-mode cavity, we present a simple scheme to realize the basic two-qubit logic gates such as the quantum phase gate (QPG), the SWAP gate and the controlled-NOT (CNOT) gate beyond the Lamb-Dicke (LD) limit. We realize the three kinds of two-qubit quantum phase gates, i.e. QPG operation involving the cavity mode as well as the vibrational mode of the trapped ion, QPG operation involving the internal states as well as the vibrational mode of the trapped ion and QPG operation involving the internal states of the trapped ion as well as the cavity mode. The controlled-NOT gate can be implemented from a QPG operation through a rotation of the second qubit before and after the QPG operation. We can also perform the SWAP gate operation involving the ionic internal states of the trapped ion and the two-mode bosonic basis. The logic gates involving the cavity mode as well as the vibrational mode of the trapped ion are insensitive to spontaneous emission, and the logic gates involving the internal states as well as the vibrational mode of the trapped ion are insensitive to the decay of the cavity, which is an important feature for the practical implementation of quantum computing. Neither the LD approximation nor the auxiliary atomic level is needed in our scheme. Experimental feasibility for achieving our scheme is also discussed