1.4: Resumen - Expresiones generales para la no linealidad de orden n
- Page ID
- 73719
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Para una señal no lineal de orden n, hay n interacciones con el campo eléctrico incidente o campos que dan lugar a la señal radiada. Contando la señal radiada hay n +1 campos involucrados (n +1 interacciones luz-materia), de modo que la espectroscopia de orden n es en ocasiones referida como (n +1) - mezcla de ondas. El campo de señal no lineal radiada es proporcional a la polarización no lineal:
\[P^{(n)}(t)=\int_0^{\infty}d\tau_n\dotsi\int_0^{\infty}d\tau_1R^{(n)}\left(\tau_1,\tau_2,\dotsc\tau_n\right)\bar E_1\left(t-\tau_n-\dotsb -\tau_1\right) \dotsm \bar E_n(t-\tau_n) \]
\ [\ begin {alineada}
R^ {(n)}\ izquierda (\ tau_ {1},\ tau_ {2},\ ldots\ tau_ {n}\ derecha) &=\ izquierda (\ frac {i} {\ hbar}\ derecha) ^ {n}\ theta\ izquierda (\ tau_ {1}\ derecha)\ theta\ izquierda (\ tau_ {2} derecha)\ lpuntos\ theta\ izquierda (\ tau_ {n}\ derecha)\\
&\ veces\ nombreoperador {Tr}\ izquierda\ {\ izquierda [\ izquierda [\ ldots\ izquierda [\ mu_ {I}\ izquierda (\ tau_ {n} +\ tau_ {n-1} +\ ldots+\ tau_ {1}\ derecha),\ mu_ {I}\ izquierda (\ tau_ {n-1} +\ tau_ {n} +\ cdots\ tau_ {1}\ derecha)\ derecha],\ ldots\ derecha]\ mu_ {I} (0)\ derecha]\ rho_ {e q}\ derecho\}
\ final {alineado}\ etiqueta {2.4.2}\]
Aquí las interacciones de la luz y la materia se expresan en términos de una secuencia de intervalos de tiempo consecutivos\(\tau_1 \dotso \tau_n\) previos a la observación 14 del sistema. Para interacciones delta-función,\(\bar E_i(t-t_0)=|\bar E_i|\delta(t-t_0)\), la función de polarización y respuesta son directamente proporcionales
\[P^{(n)}(t)=R^{(n)}(\tau_1,\tau_2,\dotsc \tau_{n-1},t)|\bar E_1|\dotso|\bar E_n| \label{2.4.3}\]