Negative refraction without absorption via both coherent and incoherent fields in a four-level left-handed atomic system

This paper attempts a probe into negative refraction without absorption by means of an incoherent pump field and a strong coherent field coupling the dense four-level atomic system. With the application of the incoherent pump field to manipulate the populations in atomic levels and the variable strong coherent field to create quantum coherence, the constraint condition of two equal transition frequencies responding to the probe field in the atomic system isn't required. And these lead to the propagation transparency and strong magnetic response of the probe field, left-handedness with vanishing absorption in the atomic system. However, an excessive coherent field intensity would increase the absorption.