1.3: Third Order Response

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Since \(R^{(2)}\) orientationally averages to zero for isotropic systems, the third-order nonlinear response described the most widely used class of nonlinear spectroscopies.

\[ R^{(3)}\left(\tau_{1}, \tau_{2}, \tau_{3}\right)=\left(\frac{i}{\hbar}\right)^{3} \theta\left(\tau_{3}\right) \theta\left(\tau_{2}\right) \theta\left(\tau_{1}\right) \operatorname{Tr}\left\{\left[\mu_{I}\left(\tau_{1}+\tau_{2}+\tau_{3}\right), \mu_{I}\left(\tau_{1}+\tau_{2}\right)], \mu_{I}\left(\tau_{1}\right)], \mu_{I}(0)\right] \rho_{e q}\right\} \]

\[ R^{(3)}\left(\tau_{1}, \tau_{2}, \tau_{3}\right)=\left(\frac{i}{\hbar}\right)^{3} \theta\left(\tau_{3}\right) \theta\left(\tau_{2}\right) \theta\left(\tau_{1}\right) \sum_{\alpha=1}^{4}\left[R_\alpha \left(\tau_{1}+\tau_{2}+\tau_{3}\right) - R^*_\alpha \left(\tau_{1}+\tau_{2}+\tau_{3}\right)\right] \]

Here the convention for the time-ordered interactions with the density matrix is R₁ = ket / ket / ket ; R₂ = bra / ket / bra ; R₃ = bra / bra / ket ; and R₄ ⇒ ket / bra / bra . In the eigenstate representation, the individual correlation functions can be explicitly written in terms of a sum over all possible intermediate states (a,b,c,d)

\[\begin{array}{l}
R_{1}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}\right) \mu_{b a}(0)\right\rangle \\
R_{2}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}(0) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}\left(\tau_{1}\right)\right\rangle \\
R_{3}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}(0) \mu_{d c}\left(\tau_{1}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}\left(\tau_{1}+\tau_{2}\right)\right\rangle \\
R_{4}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}\left(\tau_{1}\right) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}(0)\right\rangle
\end{array} \]