Implementation of holonomic quantum computation through engineering and manipulating the environment

We consider an atom-field coupled system in which two pairs of four-level atoms are respectively driven by laser fields and trapped in two distant cavities that are connected by an optical fiber. First, we show that an effective squeezing reservoir can be engineered under appropriate conditions. Then, we show that a two-qubit geometric controlled-PHASE (CPHASE) gate between the atoms in the two cavities can be implemented through adiabatically manipulating the engineered reservoir along a closed loop. This scheme that combines an engineering environment with decoherence-free space and geometric phase quantum computation together has the following remarkable feature: a CPHASE gate with arbitrary phase shift is implemented by simply changing the strength and relative phase of the driving fields.