Yes — for Two-Photon Absorption (TPA) in semiconductor quantum systems, the third-order nonlinear susceptibility χ⁽³⁾ is indeed central, especially when modeled using the compact density matrix formalism.

A widely used general form of χ⁽³⁾ for TPA processes (in the dipole approximation) is:

χ^(3)(-ω; ω, ω, -ω) ∝ Σ_{i,j,k} [μ_gi · μ_ij · μ_jk · μ_kg] / [(ω_ig - ω - iΓ)(ω_jg - 2ω - iΓ)(ω_kg - ω - iΓ)]

Where:

• μᵢⱼ are the transition dipole moments
• ωᵢⱼ are the transition energies
• Γ is the damping factor (related to dephasing or linewidth)
• The summation runs over the relevant electronic states or subbands

This kind of expression can be derived from the density matrix approach (third-order perturbative expansion) under the rotating wave approximation.

For detailed references, I recommend:

• Boyd, R. W. – Nonlinear Optics, 3rd Edition, especially Chapters 4 and 5. http://lib.ysu.am/disciplines_bk/cce1226f312d5f73b4d8dd86e878c5b3.pdf
• Haug & Koch – Quantum Theory of the Optical and Electronic Properties of Semiconductors (for semiconductor-specific context). http://materias.df.uba.ar/cuanticaa2018c2/files/2012/07/haug_koch_theory_of_the_optical_and_electronic_properties_of_semiconductors.pdf
• You might also look into S. Hughes (2002), "Modified optical absorption and χ(3) in semiconductor quantum wells" — good for quantum well subband effects.

Hope this helps! Let me know if you’re working with a specific material system—some simplifications might apply.

Dear Saiful Miqdar thank you for your precious answer