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Here, I wish to motivate bilinear system identification techniques for 3\ts{rd} order spectroscopy.   The Von-Neumann equation describes time evolution of a density matrix $\rho = \sum_i p_i \ket{\psi_i}\bra{\psi_i}$ where $\{\ket{\psi_i}\}$ span the Hilbert space:  \[ \pd{\rho(t)}{t} = -\frac{i}{\hbar}[H(t),\rho(t)]\]  Integrating:  \[ \rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(t_1)]}\]  We can solve by iteratively plugging it into itself:  \[ \rho(t) = \rho(0) + \frac{-i}{\hbar}\int_{0}^{t}{dt_1[H(t_1),\rho(0)] + \left(\frac{-i}{\hbar}\right)^2 \int_{0}^{t}{dt_1 \int_{0}^{t_1}{dt_2[H(t_1),[H(t_2),\rho(t_2)]]}}}\]  And so on:  \[ \rho(t) = \rho(0) + \sum_{n=0}^{\infty}{} \]  \[ \frac{\hbar^2}{2m}\nabla^2\Psi + V(\mathbf{r})\Psi  = -i\hbar \pd{\Psi}{t} \]  You can get started by \textbf{double clicking} this text block and begin editing. You can also click the \textbf{Insert} button below to add new block elements. Or you can \textbf{drag and drop an image} right onto this text. Happy writing!