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Pramana

On phases and length of curves in a cyclic quantum evolution

Arun Kumar Pati

Published Jun 1, 1994• 7 citations
Publisher
DOI
10.1007/bf02847127

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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 \ensuremath{\beta}(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.
The geometric phase and the geometric distance function are intimately related via length of the curve (a concept the author introduces) for any parametric evolution of the quantum system. He offers an interpretation of the nonadiabatic Berry phase as the integral over a difference between the differentials of two geometric quantities, which enables him to say that the geometric phase is just (half) the integral of the contracted length of the curve that the system traverses during a cyclic excursion.