We present a theoretical investigation of third-order cross-nonlinearity in an $N$-type four-level system and a ladder-type three-level system, as a major extension of prior work [Phys. Rev. Lett. 91, 093601 (2003)]. The third-order cross-nonlinearity in the $N$-type four-level system can be enhanced with compressed linewidth under two-photon resonance between the probe and coupling fields, by reducing the coupling Rabi frequency or increasing the coupling detuning. The enhanced cross-nonlinearity can also be obtained in the ladder-type system and it shows that the cross-nonlinearity in the ladder-type system can be one order of magnitude larger than that in the $N$-type system under the same parameters. Furthermore, the cross-nonlinearity arising between the probe and coupling fields in the $N$-type system is also explored and the enhanced cross-nonlinearity with compressed linewidth can be achieved by reducing the signal Rabi frequency or increasing the signal detuning. Our study shows that this enhanced cross-nonlinearity stems from the $\mathrm{\Lambda }$-type system and is not larger than that of the $\mathrm{\Lambda }$-type system.

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• Light-matter interaction
• Nonlinear optics
• Quantum optics
• Kerr effect
• Perturbative methods
• Semiclassical methods