Off-diagonal geometric phases

We investigate the adiabatic evolution of a set of nondegenerate eigenstate s of a parametrized Hamiltonian. Their relative phase change can be related to geometric measurable quantities that extend the familiar concept of Ber ry phase to the evolution of more than one state. We present several physic al systems where these concepts can be applied, including an experiment on microwave cavities for which off-diagonal phases can be determined from pub lished data.