By investigating the third order nonlinearity of a four-level ladder-type atomic system, it is found that, with spontaneously generated coherence (SGC) present, the nonlinear absorption or refraction can be significantly enhanced with vanishing linear absorption. We attribute the enhancement of nonlinearity mainly to quantum interference in the two decay pathways from the two upper closely lying levels. With a standing-wave trigger field, absorption or phase grating, which effectively diffracts

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... a weak probe into high-order direction, can be induced by the SGC enhanced absorptive or refractive nonlinear modulation. In contrast to the schemes for enhancing nonlinearity, no additional coupling field is required. Moreover, the present gratings result from the nonlinear modulation which differs from the recent investigations based on linear modulation. PACS number(s): 42.50.Gy, 42.65.-k, 42.65.An I. INTRODUCTION Nonlinear optical phenomenon is well known as a fundamental process resulting from the interaction between light and matter. Especially the third order nonlinearity plays an important role and has many applications in polarization phase gates [1-3], optical switch [4, 5] , the generation of optical solitons [6, 7] , etc. However, the absence of sufficient nonlinearity and large absorption in conventional media become obstacles to its applications, therefore, giant third order nonlinearity at low light level is desirable in various optical devices. To avoid these disadvantages, coherently driven multilevel electromagnetically induced transparency (EIT) systems were proposed to enhance third order Kerr nonlinearity with vanishing linear absorption and then large self-or cross-phase modulation (SPM or XPM) can be achieved [8] [9] [10] [11] [12] [13] . Recently, giant Kerr nonlinearity was also obtained in active Raman gain (ARG) media [14] . The above studies on the enhancement of nonlinearity are all based on the laser induced atomic coherence in multilevel atoms, and it is crucial to have at least one coupling laser to create the necessary coherence. However, the atomic coherence can also occur in the process of spontaneous emission, in which the atom decays from closely placed upper levels to a single ground level. The quantum interference between the decay channels, which is called spontaneously generated coherence (SGC), can lead to many phenomena, such as amplification without inversion [15, 16] , narrowing and quenching of spontaneous emission [17-23], refractive index enhancement without absorption [24] , and giant SPM [25] . Another novel phenomenon caused by SGC is that it can induce dark state for a light-matter system and make a medium transparent to a short laser pulse [26, 27] . In contrast to the lossless propagation in three level EIT media, no coupling laser field is required. In this paper, we investigate the effect of SGC on the third order cross nonlinear interaction between the probe field and the trigger field in a four-level ladder-type system. When the trigger is resonant, the nonlinear two-photon absorption can be enhanced by constructive quantum interference, while the linear absorption is vanishing due to the destructive quantum interference between the two decay channels. We can also obtain enhanced nonlinear refraction with both linear and nonlinear absorption neglectable in the case of a large detuned trigger field. The