1.3: Third Order Response
- Page ID
- 298951
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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)
\[\begin{array}{l}
R_{1}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}\right) \mu_{b a}(0)\right\rangle \\
R_{2}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}(0) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}\left(\tau_{1}\right)\right\rangle \\
R_{3}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}(0) \mu_{d c}\left(\tau_{1}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}\left(\tau_{1}+\tau_{2}\right)\right\rangle \\
R_{4}=\sum_{a, b, c, d} p_{a}\left\langle\mu_{a d}\left(\tau_{1}\right) \mu_{d c}\left(\tau_{1}+\tau_{2}\right) \mu_{c b}\left(\tau_{1}+\tau_{2}+\tau_{3}\right) \mu_{b a}(0)\right\rangle
\end{array} \]