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\[ \pd{\rho_n(t)}{t} = \frac{-i}{\hbar}[H(t),\rho_{n-1}(t)] \]  In nonlinear spectroscopy,the  polarization is related to the perturbation of the density matrix as follows (see Mukamel): \[ P_n(t) = \avg{\mu \rho_n(t)} \]  I wish to use a general form to relate the polarization measurement $P_3(t)$ with n\ts{th} order perturbations of the density matrix. In superoperator form (where the elements of  $n\times n$ density matrices become concatenate to create  $1\times n^2$ vectors) , vectors),  let us define $x$ as a concatenation of $\rho_n, 0\le n\le 3$: \[ x(t) = \left( \begin{array}{cccc}  \rho_0(t) \\ 
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F \\  \end{array}\right) \cdot x(t)\]  such that applying $F$ (in operator form) behaves as follows:  $F \cdot = \avg{\mu \cdot}$, in operator form. \cdot}$. The $1\times 4n^2$ vector  $x(t)$ is governed by the time evolution equation \[ \pd{x(t)}{t} = \left( \begin{array}{cccc}  0 & 0 & 0 & 0 \\ 
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0 \\  0 \\  0 \\  \end{array} \right) $ and applying $A(t)$ (in operator form) behaves as follows:  $A(t) \cdot = \frac{-i}{\hbar}[H(t),\cdot]$, in operator form. \frac{-i}{\hbar}[H(t),\cdot]$.  This is a general form to describethe  polarization measurement $P_3(t)$ and perturbations of the density matrix.