Geometric Phases of Two Ising-interacting Spins in a Rotating Magnetic Field

We consider how to obtain a nontrivial two-qubit unitary transformation purely based on geometric phases of two spin-1/2's with Ising-like interaction in a magnetic field with a static z-component and a rotating xy-component. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. In previous approach, coupling of one of the qubit with the rotating component of field is ignored. By considering the exact two-spin geometric phases, we find that a nontrivial two-spin unitary transformation purely based on Berry phases can be obtained by using two consecutive cycles with opposite directions of the magnetic field and opposite signs of the interaction constant. In the nonadiabatic case, starting with a certain initial state, a cycle in the projected space of rays and thus Aharonov-Anandan phase can be achieved. The two-cycle scheme cancels the total phases, hence any unknown initial state evolves back to itself without a phase factor.