Deduce the existence of the second-order susceptibility (2) in (7.6.1), which gives a response proportional to

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Deduce the existence of the second-order susceptibility χ(2) in (7.6.1), which gives a response proportional to the square of the electric field E2, following the procedure of Section 7.4 for P ∝ 〈ψt|x|ψt 〉, but using the second-order perturbation theory method introduced here to obtain 〈c|ψ(t)〉, for a transition from state |v〉 to state |c± via one intermediate state |m±. Assume that transitions between all three states are dipole-allowed (which means there cannot be inversion symmetry), and for simplicity, assume E = E0e−iωt.

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