Tight lower bounds on the time it takes to generate a geometric phase

Geometric phase is a concept of central importance in virtually every branch of physics. In this paper, we show that the evolution time of a cyclically evolving quantum system is restricted by the system's energy resources and the geometric phase acquired by the state. Specifically, we derive and examine three tight lower bounds on the time required to generate any prescribed Aharonov-Anandan geometric phase. The derivations are based on recent results on the geometric character of the Mandelstam-Tamm and Margolus-Levitin quantum speed limits.