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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
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\tableofcontents
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\pagebreak
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\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
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\end{document}
