We present a theoretical investigation of cross-Kerr nonlinearities and associated four-wave mixing (FWM) in an $N$-type four-level coherent system. By pairing the probe field with another field, the system involves $\mathrm{\Lambda }$ and V types of electromagnetically induced transparency (EIT) which correspond to two cross-Kerr nonlinearities. Our analysis reveals that both the cross-Kerr nonlinearities and associated FWMs exhibit significant enhancement accompanied by compressed linewidths with increasing the detuning of the third field, though their enhancement behaviors differ remarkably. For the cross-Kerr nonlinearity and the associated FWM arising from the probe-coupling $\mathrm{\Lambda }$-type EIT, the positions of the enhanced nonlinearities remain essentially unchanged with increasing signal detuning, and their magnitudes grow monotonically with the signal detuning. In contrast, for those from the probe-signal $V$-type EIT, the positions of the nonlinearities shift with the increasing coupling detuning, forming a two-photon resonance between the probe and coupling fields, while their magnitudes first increase and then decrease with the coupling detuning.

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• Atomic gases
• Nonlinear optical materials
• Polarization
• Dipole approximation
• Perturbative methods