How to control a geometric quantum gate

In this paper, we detail the theoretical ideas which are used to explain the mechanism of the laser controlling the geometric quantum gates introduced in the work by Ekert et al. (Ekert A, Ericsson M, Hayden P, Inanori H, Jones J A, Oi D K L and Vedral V 2000 J. Mod. Opt. 47 2501). We have introduced a two-level Hamiltonian system and directed to solve this system and then obtained the probability distribution of this two-level system. We also show the relationships between the external laser fields and the transition of the qubit in the two-qubit controlled-phase gate and how the transition of the qubit depends on the external laser fields and the states of the controlled qubit.