The first high confinement H-mode plasma has been obtained in the Experimental Advanced Superconducting Tokamak (EAST) with about 1 MW lower hybrid current drive after wall conditioning by lithium evaporation and real-time injection of Li powder. Following the L–H transition, a small-amplitude, low-frequency oscillation, termed a limit-cycle state, appears at the edge during the quiescent phase with good energy and particle confinement. Detailed measurements by edge Langmuir probes show modulation interaction and strong three-wave coupling between the low-frequency oscillations and high-frequency-broadband (80–500 kHz) turbulences that emerge after the L–H transition or in the inter-ELM phase. The potential fluctuations at the plasma edge are correlated with the limit-cycle oscillations, and the fluctuations in the floating potential signals at different toroidal, poloidal and radial locations are strongly correlated with each other, with nearly no phase differences poloidally and toroidally, and finite phase difference radially, thus providing strong evidence for zonal flows. The growth, saturation and disappearance of the zonal flows are strongly correlated with those of the high-frequency turbulence. And the measurements demonstrate that the energy gain of zonal flows is of the same order as the energy loss of turbulence. This strongly suggests the interactions between zonal flows and high-frequency turbulences at the pedestal during the limit-cycle state.