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In Robert W. Boyd's book "Nonlinear Optics", the quantum treatment of the nonlinear optical susceptibility lead to the next expression, for the second order case:

[itex]\chi^{(2)}_{ijk}(\omega_{\sigma},\omega_q,\omega_p)=\frac{N}{\hbar^2} P_F\sum_{mn} \frac{\mu_{gn}^i\mu_{nm}^j\mu_{mg}^k}{(\omega_{ng}-\omega_{\sigma})(\omega_{mg}-\omega_p)}[/itex]

- At first time, I supposed [itex] \mu_{nm}=\langle{\phi_n} |\hat{\mu} |{\phi_m}\rangle[/itex] where [itex]\phi_n[/itex] and [itex]\phi_m[/itex] are eigenstates of nonperturbated system. But when I saw the illustration (the same that I show you here), the levels [itex]n[/itex] and [itex]m[/itex] are virtuals !

My question is: in concrete case, how to define and calculate the matrix element [itex] \mu_{nm}[/itex] ?

Thank you everybody.

* forgive me for my English.

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# Quantum treatment of nonlinear susceptibility

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