Phase-dependent properties for absorption and dispersion by spontaneously generated coherence in a four-level atomic system

A four-level atomic system with two closely lying upper levels driven by two coherent fields is considered. We show that, in the presence of a weak incoherent pump, steady-state gain can be achieved as a result of quantum interference from spontaneous emission. The gain depends on the relative phase between the two fields, so it can be modulated by control of the relative phase. Also, by controlling the relative phase, one can always obtain a large index of refraction with zero absorption.