  Quantum mechanical methods for getting geometric phases for mixed states are
analyzed. Parallel transport equations for pure states are generalized to mixed
states by which dynamical phases are eliminated. The geometric phases of mixed
states are obtained as Pancharatnam phases which are valid also for open
cycles. The geometric phases are derived here by SU(2) transformations of mixed
thermal states which are different from those used in NMR and neutron
interference experiments. While the zeroth order Hamiltonian is given by the
interaction of a magnetic moment and constant magnetic field in the z
direction, the high order perturbations assumed in the present article are
composed of two oscillating magnetic fields in the same z direction. These
assumptions lead to a special form of the SU(2) unitary transformation of the
mixed thermal states by which results for geometric phase and for interference
intensity are derived.
