A scheme of quantum phase gate for trapped ion

We propose a scheme to implement two-qubit controlled quantum phase gate(CQPG) via a single trapped two-level ion located in the standing wave field of a quantum cavity, in which the trap works beyond the Lamb--Dicke limit. When the light field is resonant with the atomic transition |g\rangle\leftrightarrow|e\rangle of the ion located at the antinode of the standing wave, we can perform CQPG between the internal and external states of the trapped ion; while the frequency of the light field is chosen to be resonant with the first red sideband of the collective vibrational mode of the ion located at the node of the standing wave, we can perform CQPG between the cavity mode and the collective vibrational mode of the trapped ion. Neither the Lamb--Dicke approximation nor the assistant classical laser is needed. Also we can generate a GHZ state if assisted with a classical laser.