Phase-fluctuation induced reduction of the kinetic energy at the superconducting transition

Recent reflectivity measurements provide evidence for a "violation" of the in-plane optical integral in the underdoped high-T_c compound Bi_2Sr_2CaCu_2O_{8+\delta} up to frequencies much higher than expected by standard BCS theory. The sum rule violation may be related to a loss of in-plane kinetic energy at the superconducting transition. Here, we show that a model based on phase fluctuations of the superconducting order parameter can account for this change of in-plane kinetic energy at T_c. The change is due to a transition from a phase-incoherent Cooper-pair motion in the pseudogap regime above T_c to a phase-coherent motion at T_c.