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-\usepackage[all]{hypcap}  % helps hyperref work properly
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-\begin{document}
-\pagenumbering{gobble}
-
-\begin{center}
-	\LARGE
-	
-	Supplementary Information
-
-	Global Analysis of Transient Grating and Transient Absorption \\ of PbSe Quantum Dots
-	
-	\normalsize
-	
-	\textit{Daniel D. Kohler, Blaise J. Thompson, John C. Wright*}
-
-	Department of Chemistry, University of Wisconsin--Madison\\
-	1101 University Ave., Madison, Wisconsin 53706
-\end{center}
-
-\vspace{\fill}
-
-*Corresponding Author \\
-\hspace*{2ex} email: wright@chem.wisc.edu \\
-\hspace*{2ex} phone: (608) 262-0351 \\
-\hspace*{2ex} fax: (608) 262-0381
-
-\pagebreak
-\renewcommand{\baselinestretch}{0.75}\normalsize
-\tableofcontents
-\renewcommand{\baselinestretch}{1.0}\normalsize
-
-\pagebreak
-\setcounter{page}{1}
-\pagenumbering{arabic}
-
-\section{Absorbance}  % ---------------------------------------------------------------------------
-
-\autoref{figure:absorbance} displays the absorbance spectra of the two batches considered in this
-work.
-The lower spectra are plotted relative to each batches 1S peak center,
-emphasizing the peak-shape differences around the 1S.
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{absorbance}
-  \label{figure:absorbance}
-  \caption{Normalized absorbance spectra of the two baches considered in this
-    work. In the upper plot, the spectra are plotted directly against energy. In
-  the lower plot the spectra are plotted relative to the 1S peak center.}
-\end{figure}
-
-To extract peak parameters from the rising continuum absorption, the data was fitted on the second
-derivative level, as described in the supplementary information of \textcite{Czech2015}.
-The script used to accomplish this fit, full parameter output, and additional figures showing the
-separate excitonic features and fit remainder are contained in the supplementary repository, as
-described in \autoref{section:repository}.
-
-Note that the aliquots used for each of the two Batch A experiments were at
-slightly different concentrations, a crucial detail for m-factor corrections
-(see \autoref{section:m-factors}). The two Batch B experiments were done using
-the same aliquot. The absorbance spectrum of each sample is kept in an
-associated ``cal'' directory in the supplementary repository (see \autoref{section:repository}).
-
-
-\pagebreak
-\section{Artifact correction}  % ------------------------------------------------------------------
-
-\subsection{Spectral delay correction}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{spectral_delay_correction}
-  \label{figure:spectral_delay_correction}
-  \caption{Frequency dependent delay calibration using CCl$_4$. (a) Measurement
-    of the pulse overlap position in $\tau_{21}$ space with respect to
-    $\omega_1$ ($\omega_2$ = 7500 cm$^{-1}$). The thick black line shows the
-    center of the temporal profile, as determined by Gaussian fits. (b) Same as
-    (a), but now $\omega_1$ is kept static while $\omega_2$ is scanned. (c) Same
-    as (a), but now active spectral delay corrections have been applied. (d)
-    Two-dimensional frequency-frequecy scan of CCl$_4$ with spectral delay
-    correction applied.}
-\end{figure}
-
-\pagebreak
-\subsection{Power factors}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{power_factors}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\pagebreak
-\subsection{m factors} \label{section:m-factors}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{m_factors}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\pagebreak
-\subsection{Processing}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{TG_artifacts}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{TA_artifacts}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-
-\pagebreak
-\section{Auger recombination dynamics}  % ---------------------------------------------------------
-
-%\begin{figure}[!htb]
-%	\centering
-%	\includegraphics[scale=0.5]{"fsb19-3"}
-%	\label{fig:matrix_flow_diagram}
-%	\caption{$S_{\mathsf{TG}}$ measured before and after multiexciton relaxation dynamics.}
-%\end{figure}
-
-Using a Poisson distribution to model the effects here: keep in mind that
-Poisson is only valid when excitation probability is "low". m
-Scholes thinks an equations of motion approach might be more fitting.
-Others have approached this by truncating the Poisson model so that dots are effectively "off" at
-high fluence (this is when pumping the continuum, so no SE contributions from the pump).
-
-According to the Poisson distribution, initial population created by pump is given by
-
-\begin{equation}
-P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}.
-\end{equation}
-
-Assumes all absorption events have equal probability.
-The absorption of the pumped sample will be proportional to
-
-\begin{eqnarray}
-a_{\mathsf{NL}} &=& a_0 \left(1-e^{-\lambda}\sum_{k=1}\frac{\lambda^k}{k!}\right)
-+ e^{-\lambda}\sum_{k=1}a_k\frac{\lambda^k}{k!} \\
-&=& a_0 - e^{-\lambda}\sum_{k=1} (a_0 - a_k)\frac{\lambda^k}{k!}.
-\end{eqnarray}
-
-So the difference in the absorption is
-
-\begin{equation}
-S(T=0) = a_{\mathsf{NL}} - a_0 = -e^{-\lambda}\sum_{k=1}(a_0-a_k)\frac{\lambda^k}{k!}.
-\end{equation}
-
-We will assume that absorption is proportional to the number of ground state excitons: $a_k = ck$
-for all $k$.
-Then
-
-\begin{eqnarray}
-S(T=0) &=& ce^{-\lambda}\sum_{k=1}k\frac{\lambda^k}{k!} \\
-&=& c\lambda e^{-\lambda}\sum_{k=0}\frac{\lambda^k}{k!} \\
-&=& c \lambda,
-\end{eqnarray}
-
-and the mean value corresponds to the response (as we expect when the relationship between
-occupation and signal is linear i.e. $<ck> = c\lambda$).
-
-After Auger recombination, the excited state distribution has homogenized to $k=1$.
-Signal is thus given by
-
-\begin{eqnarray}
-S &=& ce^{-\lambda}(a_0-a_1)\sum_{k=1}\frac{\lambda^k}{k!} \\
-&=& ce^{-\lambda}(e^\lambda-1) \\
-&=& c(1-e^{-\lambda}).
-\end{eqnarray}
-
-Previous work has analyzed this.
-
-Comparing the distribution theory with our results.
-
-The mean number of excitations should be proportional to our fluence: $\lambda = mI$.
-This predicts the linear scaling of intensity close to zero delay, and it also predicts the
-exponential saturation observed at longer delays.
-Both observations qualitatively agree with our results.
-Quantitatively, however, our two delays suggest different scaling constants with respect to pump
-fluence: the long-time $m$-value is roughly 40\% larger than the short time scaling.
-This means that our initial scaling underestimates how quickly the band saturates.
-
-Philosophically, there are two problems with this distribution: (1) I should use the equations of
-motion for degenerate pumping, and (2) the pump is filtered by $k$-vector conservation (two pumps).
-My strategy: come up with an expression for the distribution using coupled equations of motion.
-Assume the driven limit, so that a steady state is reached.
-We can account for these issues by utilizing the more general Conway-Maxwell-Poisson distribution.
-
-Estimate spot size as 300 um: 1 um ~ 1 mJ per cm squared.
-
-I think I should revisit the scaling of my exciton signal---I do not expect it to be the same as a
-Poisson distribution because of the stimulated emission channels.
-
-\begin{eqnarray}
-\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
-\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
-\frac{d \rho_{00}}{dt} &=& \cdots
-\end{eqnarray}
-
-\pagebreak
-\section{Supplementary repository} \label{section:repository}  % ----------------------------------
-
-All scripts and raw data used in this work have been uploaded to the Open Science Framework (OSF),
-a project of the Center for Open Science.
-These can be found at DOI: \href{http://dx.doi.org/10.17605/OSF.IO/N9CDP}{10.17605/OSF.IO/N9CDP}.
-
-To download the contents of this repository from your command line...  % TODO
-
-To completely reproduce this work, simply execute \texttt{./run.sh all} from your terminal.
-You will require the following:
-
-\begin{enumerate}
-  \item python 3.6
-  \item WrightTools VERSION TODO (and dependencies)
-  \item latex
-\end{enumerate}
-
-You can replace \texttt{all} with one of \texttt{data}, \texttt{simulations}, \texttt{figures},
-or \texttt{documents}.
-
-Otherwise, the OSF repository attempts to be generally self-explanatory.
-README files and comments are used to explain what was done.
-
-\pagebreak
-\printbibliography
-
-\end{document}