deletions | additions       diff --git a/Intro.tex b/Intro.tex  index be3a61f..0c2b52b 100644  --- a/Intro.tex  +++ b/Intro.tex 
...
\[ \rho_n(t) = \frac{-i}{\hbar}\int_{0}^{t}{d\tau[H(\tau),\rho_{n-1}(\tau)]} \]  By differentiating both sides, we an equation of motion for the n\ts{th} order perturbation is recursively defined:  \[ \pd{\rho_n(t)}{t} = \frac{-i}{\hbar}[H(t),\rho_{n-1}(t)] \]  In nonlinear spectroscopy, the polarization is related to the perturbation of the density matrix as follows (see Mukamel):  \[ P_n(t) = \avg{\mu \rho_n(t)} \]  We wish to use a general form to relate our polarization measurement $P_3(t)$ with n\ts{th} order perturbations of our density matrix. In superoperator form, let us define $x$ as a concatenation of $\rho_n, 0\le n\le 3$:  \[ x(t) = \left( \begin{array}{cccc}  \rho_0(t) \\  \rho_1(t) \\  \rho_2(t) \\  \rho_3(t) \\  \end{array} \right)\]   The third-order polarization measurement $P_3(t)$ is therefore  \[ P_3(t) = \left(\begin{array}{c}0 & 0 & 0 & F\\\end{array}\right) x(t)\]  such that in operator form, $F \cdot = \avg{\mu \cdot}$. $x(t)$ is governed by the time evolution equation  \[ \pd{x(t)}{t} = \left( \begin{array}{cccc}  0 & 0 & 0 & 0 \\  A & 0 & 0 & 0 \\  0 & A & 0 & 0 \\  0 & 0 & A & 0 \\  \end{array} \right) x(t) \]   where