Interpretation of geometric phase via geometric distance and length during cyclic evolution

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Abstract
We cast the nonadiabatic geometric phase in terms of the geometric distance function and the geometric length of the curve for arbitrary cyclic evolution of the quantum states. An interpretation is given to the geometric phase as the value of the integral of the contracted length of the curve along which the system traverses. It is found that for arbitrary cyclic evolution of the quantum states the geometric phase β(C) acquired by the system cannot be greater than the total length of the curve l(C). We have argued that the geometric phase arises because of the fundamental inequality between the length of the curve and the distance function. Finally, we have illustrated the calculation of the geometric phase based on the geometric distance function and the geometric length of the curve.