1.4: Summary - General Expressions for nth Order Nonlinearity
- Page ID
- 302010
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For an nth-order nonlinear signal, there are n interactions with the incident electric field or fields that give rise to the radiated signal. Counting the radiated signal there are n+1 fields involved (n+1 light-matter interactions), so that nth order spectroscopy is at times referred to as (n+1)- wave mixing. The radiated nonlinear signal field is proportional to the nonlinear polarization:
\[P^{(n)}(t)=\int_0^{\infty}d\tau_n\dotsi\int_0^{\infty}d\tau_1R^{(n)}\left(\tau_1,\tau_2,\dotsc\tau_n\right)\bar E_1\left(t-\tau_n-\dotsb -\tau_1\right) \dotsm \bar E_n(t-\tau_n) \]
\[\begin{aligned}
R^{(n)}\left(\tau_{1}, \tau_{2}, \ldots \tau_{n}\right) &=\left(\frac{i}{\hbar}\right)^{n} \theta\left(\tau_{1}\right) \theta\left(\tau_{2}\right) \ldots \theta\left(\tau_{n}\right) \\
& \times \operatorname{Tr}\left\{\left[\left[\ldots\left[\mu_{I}\left(\tau_{n}+\tau_{n-1}+\ldots+\tau_{1}\right), \mu_{I}\left(\tau_{n-1}+\tau_{n}+\cdots \tau_{1}\right)\right], \ldots\right] \mu_{I}(0)\right] \rho_{e q}\right\}
\end{aligned} \label{2.4.2}\]
Here the interactions of the light and matter are expressed in terms of a sequence of consecutive time intervals \(\tau_1 \dotso \tau_n\) prior to observing 14 the system. For delta-function interactions, \(\bar E_i(t-t_0)=|\bar E_i|\delta(t-t_0)\), the polarization and response function are directly proportional
\[P^{(n)}(t)=R^{(n)}(\tau_1,\tau_2,\dotsc \tau_{n-1},t)|\bar E_1|\dotso|\bar E_n| \label{2.4.3}\]