From 96d1fba00c2ae499056a81d430141d500d7695d8 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 14 May 2018 10:31:46 -0500
Subject: 2018-05-14 10:31

---
 mixed_domain/chapter.tex | 795 +++++++++++++++++++++++++----------------------
 1 file changed, 432 insertions(+), 363 deletions(-)

(limited to 'mixed_domain/chapter.tex')

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index 509faee..4c4445e 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -467,7 +467,6 @@ line-narrowing \cite{BesemannDanielM2004a, OudarJL1980a, CarlsonRogerJohn1990a,
   AgarwalRitesh2002a}.  %
 We simulate the effects of static inhomogeneous broadening by convolving the homogeneous response
 with a Gaussian distribution function.  %
-Further details of the convolution are in \autoref{mix:sec:convolution}.  %
 Dynamic broadening effects such as spectral diffusion are beyond the scope of this work.  %
 
 Here we describe how to transform the data of a single reference oscillator signal to that of an
@@ -588,113 +587,115 @@ There are three details of our simulation strategy that deserve more exposition:
 
 \subsection{Liouville pathway parameters}  % ------------------------------------------------------
 
-Table \ref{tab:table1} describes the relationship between our notation and the parameters that make
+Table \ref{mix:tab:table1} describes the relationship between our notation and the parameters that make
 up the 16 Liouville pathways.  %
 
-\begin{table}
-	\begin{tabular}{l c | c c c  r r r  r r r  c c c c}
-		$L$ & Liouville Pathway
-		& $x$ & $y$ & $z$
-		& $\lambda_1$ & $\lambda_2$ & $\lambda_3$
-		& $\kappa_1$ & $\kappa_2$ & $\kappa_3$
-		& $\mu_1$ & $\mu_2$ & $\mu_3$ & $\mu_4$ \\
-		\hline
-		I$\alpha$    & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$
-		& \multirow{3}{*}{$1$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$}
-		& 1 & -1 & -1
-		& \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1}
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
-		I$\beta$     & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 11$
-		& & &
-		& 1 & -1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
-		I$\gamma$    & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$
-		& & &
-		& 1 & 1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
-		\hline
-		II$\alpha$  &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$
-		& \multirow{2}{*}{$1$} & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$2$}
-		& 1 & 1 & 1
-		& \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1}
-		& $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\
-		II$\beta$   &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$
-		& & &
-		& 1 & 1 & -1
-		& & &
-		& $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\
-		\hline
-		III$\alpha$ &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$
-		& \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2^\prime$}
-		& -1 & 1 & -1
-		& \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1}
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
-		III$\beta$  &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$
-		& & &
-		& -1 & 1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
-		III$\gamma$ &  $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$
-		& & &
-		& -1 & -1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
-		\hline
-		IV$\alpha$  &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$
-		& \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2$}
-		& 1 & 1 & 1
-		& \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1}
-		& $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\
-		IV$\beta$   &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$
-		& & &
-		& 1 & 1 & -1
-		& & &
-		& $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\
-		\hline
-		V$\alpha$   &  $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$
-		& \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$1$}
-		& -1 & -1 & 1
-		& \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1}
-		& $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
-		V$\beta$    &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$
-		& & &
-		& -1 & 1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
-		V$\gamma$   &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$
-		& & &
-		& -1 & 1 & -1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
-		\hline
-		VI$\alpha$  &  $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$
-		& \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$}
-		& 1 & 1 & 1
-		& \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1}
-		& $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
-		VI$\beta$   &  $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 00$
-		& & &
-		& 1 & -1 & 1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
-		VI$\gamma$  &  $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$
-		& & &
-		& 1 & -1 & -1
-		& & &
-		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
-	\end{tabular}
- 	\caption{Parameters of each Liouville Pathway.}
-  \label{tab:table1}
-\end{table}
+\begin{landscape}
+  \begin{table}
+    \begin{tabular}{l c | c c c  r r r  r r r  c c c c}
+      $L$ & Liouville Pathway
+      & $x$ & $y$ & $z$
+      & $\lambda_1$ & $\lambda_2$ & $\lambda_3$
+      & $\kappa_1$ & $\kappa_2$ & $\kappa_3$
+      & $\mu_1$ & $\mu_2$ & $\mu_3$ & $\mu_4$ \\
+      \hline
+      I$\alpha$    & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$
+      & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$}
+      & 1 & -1 & -1
+      & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1}
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
+      I$\beta$     & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 11$
+      & & &
+      & 1 & -1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
+      I$\gamma$    & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$
+      & & &
+      & 1 & 1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
+      \hline
+      II$\alpha$  &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$
+      & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$2$}
+      & 1 & 1 & 1
+      & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1}
+      & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\
+      II$\beta$   &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$
+      & & &
+      & 1 & 1 & -1
+      & & &
+      & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\
+      \hline
+      III$\alpha$ &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$
+      & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2^\prime$}
+      & -1 & 1 & -1
+      & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1}
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
+      III$\beta$  &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$
+      & & &
+      & -1 & 1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
+      III$\gamma$ &  $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$
+      & & &
+      & -1 & -1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
+      \hline
+      IV$\alpha$  &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$
+      & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2$}
+      & 1 & 1 & 1
+      & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1}
+      & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\
+      IV$\beta$   &  $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$
+      & & &
+      & 1 & 1 & -1
+      & & &
+      & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\
+      \hline
+      V$\alpha$   &  $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$
+      & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$1$}
+      & -1 & -1 & 1
+      & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1}
+      & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
+      V$\beta$    &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$
+      & & &
+      & -1 & 1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
+      V$\gamma$   &  $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$
+      & & &
+      & -1 & 1 & -1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
+      \hline
+      VI$\alpha$  &  $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$
+      & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$}
+      & 1 & 1 & 1
+      & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1}
+      & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\
+      VI$\beta$   &  $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 00$
+      & & &
+      & 1 & -1 & 1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\
+      VI$\gamma$  &  $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$
+      & & &
+      & 1 & -1 & -1
+      & & &
+      & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
+    \end{tabular}
+    \caption{Parameters of each Liouville Pathway.}
+    \label{mix:tab:table1}
+  \end{table}
+\end{landscape}
 
 \subsection{Matrix formulation}  % ----------------------------------------------------------------
 
 \subsubsection{Generic single pathway}
 
 In this work we explicitly incorporate our time-dependent electric fields into the Liouville
-equations as done by \textcite{GelinMaximF2009.000}.  %
+equations as done by \textcite{GelinMaximF2009a}.  %
 
 Our third order experiment involves three successive light-matter interactions. In a generic
 Liouville pathway $\rho_0 \overset{x}\rightarrow \rho_1 \overset{y}\rightarrow \rho_2
@@ -723,21 +724,22 @@ Note that we do not include depletion due to light-matter interaction.
 Our three coupled differential equations can be cast into a matrix form:
 
 \begin{eqnarray}
-\frac{\mathrm{d}\overline{\rho}}{\mathrm{d}t} &=& \overline{\overline{Q}}(t)\overline{\rho} \label{eq:generic_diff} \\
-\overline{\rho} &\equiv&
-\begin{bmatrix}
-\tilde{\rho}_0 \\
-\tilde{\rho}_1 \\
-\tilde{\rho}_2 \\
-\tilde{\rho}_3
-\end{bmatrix} \\
-\overline{\overline{Q}}(t) &\equiv&
-\begin{bmatrix}
--\Gamma_0 & 0 & 0 & 0 \\
-S_1(t) & -\Gamma_1 & 0 & 0 \\
-0 & S_2(t) & -\Gamma_2 & 0 \\
-0 & 0 & S_3(t) & -\Gamma_3
-\end{bmatrix}
+  \frac{\mathrm{d}\overline{\rho}}{\mathrm{d}t} &=&
+  \overline{\overline{Q}}(t)\overline{\rho} \label{mix:eqn:generic_diff} \\
+  \overline{\rho} &\equiv&
+  \begin{bmatrix}
+  \tilde{\rho}_0 \\
+  \tilde{\rho}_1 \\
+  \tilde{\rho}_2 \\
+  \tilde{\rho}_3
+  \end{bmatrix} \\
+  \overline{\overline{Q}}(t) &\equiv&
+  \begin{bmatrix}
+  -\Gamma_0 & 0 & 0 & 0 \\
+  S_1(t) & -\Gamma_1 & 0 & 0 \\
+  0 & S_2(t) & -\Gamma_2 & 0 \\
+  0 & 0 & S_3(t) & -\Gamma_3
+  \end{bmatrix}
 \end{eqnarray}
 
 \autoref{mix:eqn:generic_diff} could be numerically integrated to determine $\tilde{\rho}_3$ for
@@ -745,8 +747,8 @@ this pathway.  %
 
 \subsubsection{Full formulation}  % ----------------------------------------------------------------
 
-As discussed throughout this chapter and shown in \textbf{TODO}, there are 6 unique time-orderings
-and 16 unique pathways to consider in this work.  %
+As discussed throughout this chapter and shown in \autoref{mix:fig:WMELs}, there are 6 unique
+time-orderings and 16 unique pathways to consider in this work.  %
 One could write a separate differential equation for each pathway---this would require tabulation
 of 48 density matrix elements.  %
 Instead, we capitalize on the symmetry of our pathways to create a single differential equation
@@ -764,8 +766,8 @@ that is computationally cheaper.  %
   }
 \end{figure}
 
-\autoref{fig:matrix_flow_diagram} diagrams our 16 Liouville pathways as a network of interconnected
-steps.  %
+\autoref{mix:fig:matrix_flow_diagram} diagrams our 16 Liouville pathways as a network of
+interconnected steps.  %
 Some density matrix elements, such as $\tilde{\rho}_{10}$, appear multiple times because there are
 several distinguishable pathways that involve that element.  %
 We do not include $\tilde{\rho}_{00}^{(1-2)}$ and $\tilde{\rho}_{00}^{(2^\prime-2)}$ because they
@@ -773,7 +775,8 @@ have exactly the same amplitude as $\tilde{\rho}_{11}^{(1-2)}$ and
 $\tilde{\rho}_{11}^{(2^\prime-2)}$ within our simulation. Pathways where $\rho_3$ is fed by
 $\tilde{\rho}_{00}$ are sign-flipped to account for this.  %
 
-First we define a state vector containing all nine elements in \autoref{fig:matrix_flow_diagram}:
+First we define a state vector containing all nine elements in
+\autoref{mix:fig:matrix_flow_diagram}:
 
 \begin{equation}
 \overline{\rho} \equiv
@@ -819,7 +822,9 @@ Each off-diagonal element in $\overline{\overline{Q}}$ has two influences determ
 \label{eq:single_Q}
 \end{equation}
 
-Note that this approach implicitly enforces our phase matching conditions and pathways. To simulate single time-orderings we simply remove elements from \ref{eq:single_Q}. For example, to isolate time ordering \RomanNumeral{1}:
+Note that this approach implicitly enforces our phase matching conditions and pathways.
+To simulate single time-orderings we simply remove elements from \ref{eq:single_Q}.
+For example, to isolate time ordering \RomanNumeral{1}:
 
 \begin{equation}
 \overline{\overline{Q}}_\RomanNumeral{1} \equiv
@@ -841,7 +846,7 @@ These equations are somewhat general to four wave mixing of systems like the one
 work.  %
 We have not taken all the simplifications that are possible in our specific system, such as
 $\omega_{10} = \omega_{21}$ and $\Gamma_{11} = 0$.  %
-See \autoref{section:scripts} for more details about our simulation, including where to find the
+See \autoref{mix:sec:scripts} for more details about our simulation, including where to find the
 scripts and raw output behind this work.  %
 
 \subsection{Heun method}
@@ -982,9 +987,15 @@ Our software packages are constantly being developed. The versions kept in this
 
 The raw simulation in this work was generated using NISE and the \texttt{NISE\_iterator.py} script found in the raw folder.
 
-In NISE, an \texttt{experiment} module is loaded to define the electric field variables and the experimental conditions, like the phase matching. In this case we used the \texttt{trive} module, which defines our normal, two-color four wave mixing experimental conditions.
+In NISE, an \texttt{experiment} module is loaded to define the electric field variables and the
+experimental conditions, like the phase matching.  %
+In this case we used the \texttt{trive} module, which defines our normal, two-color four wave
+mixing experimental conditions.  %
 
-Within this work we have represented our data in terms of dimensionless quantities like $\tau/\Delta_t$ and $(\omega-\omega_{10})/\Delta_\omega$. The simulation within NISE was done with choices for each parameter, as tabulated below. These quantities are necessary to fully understand the unprocessed arrays generated by NISE.
+Within this work we have represented our data in terms of dimensionless quantities like
+$\tau/\Delta_t$ and $(\omega-\omega_{10})/\Delta_\omega$.  %
+The simulation within NISE was done with choices for each parameter, as tabulated below.  %
+These quantities are necessary to fully understand the unprocessed arrays generated by NISE.  %
 
 \begin{table}
   \begin{tabular}{r l}
@@ -994,13 +1005,49 @@ Within this work we have represented our data in terms of dimensionless quantiti
 		$\Gamma_{11}$ & 0 $\mathrm{fs}^{-1}$ \\
 		$\Delta_t$ & 50 fs
 	\end{tabular}
+  \caption{
+    Parameters used in large NISE simulation.
+  }
 \end{table}
 
-$\Gamma_{10}$, $\Gamma_{21}$ and $\Gamma_{20}$ were kept equal. Their exact value for a given run of the simulation depends on the $\Gamma_{10}\Delta_t$ quantity as discussed in the paper. We use the term \texttt{dpr} (dephasing pulse ratio) which is the inverse of $\Gamma_{10}\Delta_t$.
+$\Gamma_{10}$, $\Gamma_{21}$ and $\Gamma_{20}$ were kept equal.
+Their exact value for a given run of the simulation depends on the $\Gamma_{10}\Delta_t$ quantity
+as discussed in the paper.
+We use the term \texttt{dpr} (dephasing pulse ratio) which is the inverse of $\Gamma_{10}\Delta_t$.
+
+In NISE the system parameters are contained within the \texttt{hamiltonian} module, in this case
+\texttt{H0}.
+The \texttt{Omega} object contains all of the system attributes you would expect, including the
+$\overline{\rho}(t)$ (\texttt{dm\_vector}) and $\overline{\overline{Q}}(t)$
+(\texttt{gen\_matrix()}) matrices as in \autoref{mix:eqn:generic_diff}.
+The matrix in the software does not account for relaxation and dephasing, that is accounted for
+directly during the integration.
+Within \texttt{H0} we actually define two $\overline{\overline{Q}}$ `permutations', one for
+pathways in which $E_1$ arrives first (\texttt{w1\_first == True}) and one for pathways in which
+$E_{2^\prime}$ arrives first (\texttt{w1\_first == False}).
+This separate permutation approach is mathematically identical to the single matrix approach in
+\autoref{eq:single_Q}, just slightly more computationally expensive.
+\texttt{H0.Omega} allows you to define which time-orderings are included using the \texttt{TOs}
+keyword argument.
+This directly affects how the propagator is assembled, as discussed in \autoref{eq:single_Q_pw1}.
+
+To generate the raw data we calculated the polarization at all coordinates within a
+four-dimensional experimental array:  %
+
+Arrays containing these points were assembled and passed into the \texttt{trive} module.
+These axes and \texttt{H0} were passed into the \texttt{scan} module for numerical integration.
+A single output array was saved for each scan.
+To keep the output array sizes reasonable a separate simulation was done for each
+$\Gamma_{10}\Delta_t$, time-ordering, and \texttt{d1} value.
+For each of these simulations we saved one five-dimensional array to an HDF5 file:
+
+The final index `timestep' contains the dependence of the output polarization on lab time.
+It changes from simulation to simulation to help with computation speed.
+For each simulation the timetep array is defined by a starting position (always 100 fs before the
+first pulse arrives) an ending position ($5 \times \tau_{10}$ fs after the last pulse arrives), and
+a step (4 fs for our longest dephasing time, 2 fs otherwise).
 
-In NISE the system parameters are contained within the \texttt{hamiltonian} module, in this case \texttt{H0}. The \texttt{Omega} object contains all of the system attributes you would expect, including the $\overline{\rho}(t)$ (\texttt{dm\_vector}) and $\overline{\overline{Q}}(t)$ (\texttt{gen\_matrix()}) matrices as in \autoref{eq:generic_diff}. The matrix in the software does not account for relaxation and dephasing, that is accounted for directly during the integration. Within \texttt{H0} we actually define two $\overline{\overline{Q}}$ `permutations', one for pathways in which $E_1$ arrives first (\texttt{w1\_first == True}) and one for pathways in which $E_{2^\prime}$ arrives first (\texttt{w1\_first == False}). This separate permutation approach is mathematically identical to the single matrix approach in \autoref{eq:single_Q}, just slightly more computationally expensive. \texttt{H0.Omega} allows you to define which time-orderings are included using the \texttt{TOs} keyword argument. This directly affects how the propagator is assembled, as discussed in \autoref{eq:single_Q_pw1}.
-
-To generate the raw data we calculated the polarization at all coordinates within a four-dimensional experimental array:
+The output polarization is kept as a complex array in the lab frame.
 
 \begin{table}
 	\begin{tabular}{c | c | c | c}
@@ -1010,10 +1057,11 @@ To generate the raw data we calculated the polarization at all coordinates withi
 		d1 & 0 & 400 & 21 \\
 		d2 & 0 & 400 & 21
 	\end{tabular}
+  \caption{
+    Description of four-dimensional simulation coordinates.
+  }
 \end{table}
 
-Arrays containing these points were assembled and passed into the \texttt{trive} module. These axes and \texttt{H0} were passed into the \texttt{scan} module for numerical integration. A single output array was saved for each scan. To keep the output array sizes reasonable a separate simulation was done for each $\Gamma_{10}\Delta_t$, time-ordering, and \texttt{d1} value. For each of these simulations we saved one five-dimensional array to an HDF5 file:
-
 \begin{table}
 	\begin{tabular}{c | c | c}
 		index & name & size \\ \hline
@@ -1023,15 +1071,33 @@ Arrays containing these points were assembled and passed into the \texttt{trive}
 		3 & permutation & 2 \\
 		4 & timestep & variable \\
 	\end{tabular}
+  \caption{
+    Simulation output format (polarization).
+  }
 \end{table}
 
-The final index `timestep' contains the dependence of the output polarization on lab time. It changes from simulation to simulation to help with computation speed. For each simulation the timetep array is defined by a starting position (always 100 fs before the first pulse arrives) an ending position ($5 \times \tau_{10}$ fs after the last pulse arrives), and a step (4 fs for our longest dephasing time, 2 fs otherwise).
+\subsubsection{Measured}
 
-The output polarization is kept as a complex array in the lab frame.
+As mentioned in the appendix, we introduce inhomogeneity by convolving the output with a
+distribution function on the intensity level: `smearing'.
+This is done in the measurement stage.
+We store a separate HDF5 file for each combination of \texttt{dpr}, time ordering, and
+$\Delta_{\text{inhom}}$.
+Each HDF5 file contains four arrays, shown in \autoref{mix:tab:measured}
 
-\subsubsection{Measured}
+The coordinate arrays are in their native units (fs, $\mathrm{cm}^{-1}$).
+The signal array is purely real, stored on the intensity level.
+
+We measured the entire simulation space twice, one with and one without the monochromator bandpass
+filter.  %
+
+\subsubsection{Representations}
 
-As mentioned in the appendix, we introduce inhomogeneity by convolving the output with a distribution function on the intensity level: `smearing'. This is done in the measurement stage. We store a separate HDF5 file for each combination of \texttt{dpr}, time ordering, and $\Delta_{\text{inhom}}$. Each HDF5 file contains four arrays:
+We present many representations of our simulated dataset in this work.
+The script used to create these figures is \texttt{figures.py}, found in this supplementary
+information.
+In some cases some small additional simulation or some pre-processing step was necessary, these can
+be found in the `precalculated' folder.
 
 \begin{table}
 	\begin{tabular}{c | c | c}
@@ -1042,16 +1108,12 @@ As mentioned in the appendix, we introduce inhomogeneity by convolving the outpu
 		\texttt{d2} & w2 & \texttt{(21,)} \\
 		\texttt{arr} & w1, w2, d1, d2 & \texttt{(41, 41, 21, 21)}
 	\end{tabular}
+  \caption{
+    Simulation output format (measured).
+  }
+  \label{mix:tab:measured}
 \end{table}
 
-The coordinate arrays are in their native units (fs, $\mathrm{cm}^{-1}$). The signal array is purely real, stored on the intensity level.
-
-We measured the entire simulation space twice, one with and one without the monochromator bandpass filter.
-
-\subsubsection{Representations}
-
-We present many representations of our simulated dataset in this work. The script used to create these figures is \texttt{figures.py}, found in this supplementary information. In some cases some small additional simulation or some pre-processing step was necessary, these can be found in the `precalculated' folder.
-
 \section{Results}  % ==============================================================================
 
 We now present portions of our simulated data that highlight the dependence of the spectral line
@@ -1060,19 +1122,6 @@ pulse delay times, and inhomogeneous broadening.  %
 
 \subsection{Evolution of single coherence} \label{mix:sec:evolution_SQC}  % -----------------------
 
-\begin{figure}
-	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
-	\caption[Relative importance of FID and driven response for a single quantum coherence.]{
-		The relative importance of FID and driven response for a single quantum coherence as a function
-    of the relative dephasing rate (values of $\Gamma_{10}\Delta_t$ are shown inset).
-		The black line shows the coherence amplitude profile, while the shaded color indicates the
-    instantaneous frequency (see colorbar).
-		For all cases, the pulsed excitation field (gray line, shown as electric field amplitude) is
-    slightly detuned (relative detuning, $\Omega_{fx}/\Delta_{\omega}=0.1$).
-	}
-	\label{mix:fig:fid_dpr}
-\end{figure}
-
 It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x}
 \rho_1$, under various excitation conditions.  %
 \autoref{mix:fig:fid_dpr} shows the temporal evolution of $\rho_1$ with various dephasing rates under
@@ -1108,56 +1157,11 @@ We note that our choices of $\Gamma_{10}\Delta_t=2.0, 1.0,$ and $0.5$ give coher
 mainly driven, roughly equal driven and FID parts, and mainly FID components, respectively.  %
 FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
 
-\begin{figure}
-	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs detuning"}
-	\caption[Pulsed excitation of a single quantum coherence and its dependance on pulse detuning.]{
-		Pulsed excitation of a single quantum coherence and its dependence on the pulse detuning.  In
-    all cases, relative dephasing is kept at $\Gamma_{10}\Delta_t = 1$.
-		(a) The relative importance of FID and driven response for a single quantum coherence as a
-    function of the detuning (values of relative detuning, $\Omega_{fx}/\Delta_{\omega}$, are shown
-    inset).
-		The color indicates the instantaneous frequency (scale bar on right), while the black line
-    shows the amplitude profile.  The gray line is the electric field amplitude.
-		%Comparison of the temporal evolution of single quantum coherences at different detunings
-    %(labeled inset).
-		(b)  The time-integrated coherence amplitude as a function of the detuning.  The integrated
-    amplitude is collected both with (teal) and without (magenta) a tracking monochromator that
-    isolates the driven frequency components.  %$\omega_m=\omega_\text{laser}$.
-		For comparison, the Green's function of the single quantum coherence is also shown (amplitude
-    is black, hashed; imaginary is black, solid).
-		In all plots, the gray line is the electric field amplitude.
-	}
-	\label{mix:fig:fid_detuning}
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/fid vs detuning with freq"}
-	\label{mix:fig:fid_vs_detuning_with_freq}
-	\caption[Frequency domain representation of a single quantum coherence vs pulse detuning.]{
-    Numerical simulation of a single quantum coherence under pulsed excitation
-    ($\Gamma_{10}\Delta_t=1$) at different detunings (labelled inset).
-    The coherence is shown in both the time (left column) and frequency (right column) domain.
-    The color indicates the frequency (scale on right), while the black line shows the amplitude
-    profile. The excitation profile is shown as a grey line.
-  }
-\end{figure}
-
-\begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"}
-	\caption[Time-gated amplitude of a single quantum coherence vs pulse detuning.]{
-    Amplitude of a single quantum coherence under pulsed excitation as a function of detuning (x
-    axis) and delay after excitation (line color, scale on right) for the three
-    $\Gamma_{10}\Delta_t$ values considered in this work.
-    For comparison, the excitation pulse lineshape (grey), absorptive material response (black,
-    narrow) and magnitude material response (black, wide) are also shown. Each peak is normalized
-    to its own maximum amplitude.
-  }
-	\label{mix:fig:sqc_vs_t}
-\end{figure}
 
 \autoref{mix:fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of
 $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.
-Figure \textbf{TODO} shows the Fourier domain representation of \autoref{mix:fig:fid_detuning}a.  %
+\autoref{mix:fig:fid_vs_detuning_with_freq} shows the Fourier domain representation of
+\autoref{mix:fig:fid_detuning}a.  %
 As detuning increases, total amplitude decreases, FID character vanishes, and $\rho_1$ assumes a
 more driven character, as expected.  %
 During the excitation, $\rho_1$ maintains a phase relationship with the input field (as seen by the
@@ -1172,8 +1176,8 @@ This spectral narrowing can be seen in \autoref{mix:fig:fid_detuning}a by compar
 amplitudes at $t=0$ (driven)  and at $t / \Delta_t=2$ (FID); amplitudes for all
 $\Omega_{fx}/\Delta_{\omega}$ values shown are comparable at $t=0$, but the lack of FID formation
 for $\Omega_{fx}/\Delta_{\omega}=\pm2$ manifests as a visibly disproportionate amplitude decay
-(\textbf{TODO} shows explicit plots of $\rho_1(\Gamma_{fx}/\Delta_\omega$ at discrete $t/\Delta_t$
-values).  %
+(\autoref{mix:fig:sqc_vs_t} shows explicit plots of $\rho_1(\Gamma_{fx}/\Delta_\omega$ at discrete
+$t/\Delta_t$ values).  %
 Many ultrafast spectroscopies take advantage of the latent FID to suppress non-resonant background,
 improving signal to noise. \cite{LagutchevAlexi2007a, LagutchevAlexi2010a,
   DonaldsonPaulMurray2010a, DonaldsonPaulMurray2008a}  %
@@ -1217,12 +1221,14 @@ isolating the driven frequency (tracking the monochromator with the excitation s
 The functional form of the measured line shape can be deduced by considering the frequency domain
 form of \autoref{mix:eqn:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$):
 \begin{equation} \label{mix:eqn:rho_f_int_freq}
-\tilde{\rho}_f (\omega) = \frac{i\lambda_f\mu_f}{2\sqrt{2\pi}} \cdot \frac{\mathcal{F}\left\{ c_x \right\}\left( \omega - \kappa_f\Omega_{fx} \right)}{\Gamma_f + i\omega},
+  \tilde{\rho}_f (\omega) =
+  \frac{i\lambda_f\mu_f}{2\sqrt{2\pi}} \cdot \frac{\mathcal{F}\left\{ c_x \right\}\left( \omega - \kappa_f\Omega_{fx} \right)}{\Gamma_f + i\omega},
 \end{equation}
 where $\mathcal{F}\left\{ c_x \right\}\left( \omega \right)$ denotes the Fourier transform of
 $c_x$, which in our case gives
 \begin{equation}
-\mathcal{F}\left\{ c_x \right\}\left( \omega \right) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(\Delta_t\omega)^2}{4\ln 2}}.
+  \mathcal{F}\left\{ c_x \right\}\left( \omega \right) =
+  \frac{1}{\sqrt{2\pi}} e^{-\frac{(\Delta_t\omega)^2}{4\ln 2}}.
 \end{equation}
 For squared-law detection of $\rho_f$, the importance of the tracking monochromator is highlighted
 by two limits of \autoref{mix:eqn:rho_f_int_freq}:
@@ -1241,34 +1247,68 @@ $\Omega_{fx}/\Delta_{\omega}$ values and the monochromator is not useful.  %
 \autoref{mix:fig:fid_detuning}b shows the various detection methods for the relative dephasing rate of
 $\Gamma_{10}\Delta_t=1$.  %
 
-\subsection{Evolution of single Liouville pathway}  % ---------------------------------------------
+\begin{figure}
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
+	\caption[Relative importance of FID and driven response for a single quantum coherence.]{
+		The relative importance of FID and driven response for a single quantum coherence as a function
+    of the relative dephasing rate (values of $\Gamma_{10}\Delta_t$ are shown inset).
+		The black line shows the coherence amplitude profile, while the shaded color indicates the
+    instantaneous frequency (see colorbar).
+		For all cases, the pulsed excitation field (gray line, shown as electric field amplitude) is
+    slightly detuned (relative detuning, $\Omega_{fx}/\Delta_{\omega}=0.1$).
+	}
+	\label{mix:fig:fid_dpr}
+\end{figure}
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"mixed_domain/pw1 lineshapes"}
-	\caption[2D frequency response of a single Liouville pathway at different delay values.]{
-		Changes to the 2D frequency response of a single Liouville pathway (I$\gamma$) at different
-    delay values.
-    The normalized dephasing rate is $\Gamma_{10}\Delta_t = 1$.
-		Left:  The 2D delay response of pathway I$\gamma$ at triple resonance.
-		Right: The 2D frequency response of pathway I$\gamma$ at different delay values.
-    The delays at which the 2D frequency plots are collected are indicated on the delay plot;
-    compare 2D spectrum frame color with dot color on 2D delay plot.
-		}
-	\label{mix:fig:pw1}
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs detuning"}
+	\caption[Pulsed excitation of a single quantum coherence and its dependance on pulse detuning.]{
+		Pulsed excitation of a single quantum coherence and its dependence on the pulse detuning.  In
+    all cases, relative dephasing is kept at $\Gamma_{10}\Delta_t = 1$.
+		(a) The relative importance of FID and driven response for a single quantum coherence as a
+    function of the detuning (values of relative detuning, $\Omega_{fx}/\Delta_{\omega}$, are shown
+    inset).
+		The color indicates the instantaneous frequency (scale bar on right), while the black line
+    shows the amplitude profile.  The gray line is the electric field amplitude.
+		%Comparison of the temporal evolution of single quantum coherences at different detunings
+    %(labeled inset).
+		(b)  The time-integrated coherence amplitude as a function of the detuning.  The integrated
+    amplitude is collected both with (teal) and without (magenta) a tracking monochromator that
+    isolates the driven frequency components.  %$\omega_m=\omega_\text{laser}$.
+		For comparison, the Green's function of the single quantum coherence is also shown (amplitude
+    is black, hashed; imaginary is black, solid).
+		In all plots, the gray line is the electric field amplitude.
+	}
+	\label{mix:fig:fid_detuning}
 \end{figure}
 
 \begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/pw1 lineshapes no mono"}
-	\caption[2D frequency response of a single Liouville pathway without a tracking monochromator.]{
-    Pathway \RomanNumeral{1}$\gamma$ temporal response in the 2D pulse delay space at triple
-    resonance (left) and the corresponding 2D frequency plots at different delay values.
-    The delays at which the 2D frequency plots are collected are indicated on the delay plot;
-    compare 2D spectrum frame color with dot color on 2D delay plot.
-    Unlike elsewhere in this work, signal here was not filtered by a tracking monochromator.
+	\includegraphics[width=\textwidth]{"mixed_domain/fid vs detuning with freq"}
+	\label{mix:fig:fid_vs_detuning_with_freq}
+	\caption[Frequency domain representation of a single quantum coherence vs pulse detuning.]{
+    Numerical simulation of a single quantum coherence under pulsed excitation
+    ($\Gamma_{10}\Delta_t=1$) at different detunings (labelled inset).
+    The coherence is shown in both the time (left column) and frequency (right column) domain.
+    The color indicates the frequency (scale on right), while the black line shows the amplitude
+    profile. The excitation profile is shown as a grey line.
   }
- 	\label{mix:fig:pw1_no_mono}
 \end{figure}
 
+\begin{figure}
+	\includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"}
+	\caption[Time-gated amplitude of a single quantum coherence vs pulse detuning.]{
+    Amplitude of a single quantum coherence under pulsed excitation as a function of detuning (x
+    axis) and delay after excitation (line color, scale on right) for the three
+    $\Gamma_{10}\Delta_t$ values considered in this work.
+    For comparison, the excitation pulse lineshape (grey), absorptive material response (black,
+    narrow) and magnitude material response (black, wide) are also shown. Each peak is normalized
+    to its own maximum amplitude.
+  }
+	\label{mix:fig:sqc_vs_t}
+\end{figure}
+
+\subsection{Evolution of single Liouville pathway}  % ---------------------------------------------
+
 We now consider the multidimensional response of a single Liouville pathway involving three pulse
 interactions.  %
 In a multi-pulse experiment, $\rho_1$ acts as a source term for $\rho_2$ (and subsequent
@@ -1303,27 +1343,8 @@ The prominence of FID signal can change the resonance conditions; \autoref{mix:t
 the changing resonance conditions for each of the four delay coordinates studied.  %
 Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromator must also be
 considered.  %
-\textbf{TODO} shows a simulation of \autoref{mix:fig:pw1} without monochromator frequency
-filtering.  %
-
-\begin{table}
-	\caption{\label{mix:tab:table2} Conditions for peak intensity at different pulse delays for pathway
-    I$\gamma$.}
-	\begin{tabular}{c c | c c c c}
-		\multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\
-		$\tau_{21}/\Delta_t$ & $\tau_{22^\prime}/\Delta_t$ & $\rho_0\xrightarrow{1}\rho_1$ &
-    $\rho_1\xrightarrow{2}\rho_2$ & $\rho_2\xrightarrow{2^\prime}\rho_3$ & $\rho_3\rightarrow$
-    detection at $\omega_m=\omega_1$ \\
-		\hline\hline
-		0   & 0    & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$    & $\omega_1=\omega_{10}$ & -- \\
-		0   & -2.4 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$    & $\omega_2=\omega_{10}$ &
-    $\omega_1=\omega_2$ \\
-		2.4 & 0    & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & --                     &
-    $\omega_1=\omega_{10}$ \\
-		2.4 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & $\omega_2=\omega_{10}$ &
-    $\omega_1=\omega_2$ \\
-	\end{tabular}
-\end{table}
+\autoref{mix:fig:pw1_no_mono} shows a simulation of \autoref{mix:fig:pw1} without monochromator
+frequency filtering.  %
 
 When the pulses are all overlapped ($\tau_{21}=\tau_{22^\prime}=0$, lower right, orange), all
 transitions in the Liouville pathway are simultaneously driven by the incident fields.  %
@@ -1336,7 +1357,6 @@ $\omega_2=\omega_{2^\prime}$, the first and third resonance conditions are ident
 making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical elongation along
 $\omega_1=\omega_{10}$.  %
 
-
 The other three spectra of \autoref{mix:fig:pw1} separate the pulse sequence over time so that not all
 interactions are driven.  %
 At $\tau_{21}=0$, $\tau_{22^\prime}=-2.4\Delta_t$ (lower left, pink), the first two resonances
@@ -1395,25 +1415,55 @@ show that time-gating itself causes significant line shape changes to the isolat
 The phenomenon of time-gating can cause frequency and delay axes to become functional of each other
 in unexpected ways.  %
 
-\subsection{Temporal pathway discrimination}  % ---------------------------------------------------
+\begin{figure}
+	\includegraphics[width=\linewidth]{"mixed_domain/pw1 lineshapes"}
+	\caption[2D frequency response of a single Liouville pathway at different delay values.]{
+		Changes to the 2D frequency response of a single Liouville pathway (I$\gamma$) at different
+    delay values.
+    The normalized dephasing rate is $\Gamma_{10}\Delta_t = 1$.
+		Left:  The 2D delay response of pathway I$\gamma$ at triple resonance.
+		Right: The 2D frequency response of pathway I$\gamma$ at different delay values.
+    The delays at which the 2D frequency plots are collected are indicated on the delay plot;
+    compare 2D spectrum frame color with dot color on 2D delay plot.
+		}
+	\label{mix:fig:pw1}
+\end{figure}
 
 \begin{figure}
-	\includegraphics[width=\linewidth]{"mixed_domain/delay space ratios"}
-	\caption[2D delay response for different relative dephasing rates.]{
-		Comparison of the 2D delay response for different relative dephasing rates (labeled atop each
-    column).
-		All pulses are tuned to exact resonance.
-		In each 2D delay plot, the signal amplitude is depicted by the colors.
-		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values
-    denoted on each contour.
-		The small plots above each 2D delay plot examine a $\tau_{22^\prime}$ slice of the delay
-    response ($\tau_{21}=0$).
-    The plot shows the total signal (black), as well as the component time-orderings VI (orange), V
-    (purple), and III or I (teal).
-	}
-	\label{mix:fig:delay_purity}
+	\includegraphics[width=\textwidth]{"mixed_domain/pw1 lineshapes no mono"}
+	\caption[2D frequency response of a single Liouville pathway without a tracking monochromator.]{
+    Pathway \RomanNumeral{1}$\gamma$ temporal response in the 2D pulse delay space at triple
+    resonance (left) and the corresponding 2D frequency plots at different delay values.
+    The delays at which the 2D frequency plots are collected are indicated on the delay plot;
+    compare 2D spectrum frame color with dot color on 2D delay plot.
+    Unlike elsewhere in this work, signal here was not filtered by a tracking monochromator.
+  }
+ 	\label{mix:fig:pw1_no_mono}
 \end{figure}
 
+\begin{table}
+	\begin{tabular}{c c | c c c c}
+		\multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\
+		$\tau_{21}/\Delta_t$ & $\tau_{22^\prime}/\Delta_t$ & $\rho_0\xrightarrow{1}\rho_1$ &
+    $\rho_1\xrightarrow{2}\rho_2$ & $\rho_2\xrightarrow{2^\prime}\rho_3$ & $\rho_3\rightarrow$
+    detection at $\omega_m=\omega_1$ \\
+		\hline\hline
+		0   & 0    & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$    & $\omega_1=\omega_{10}$ & -- \\
+		0   & -2.4 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$    & $\omega_2=\omega_{10}$ &
+    $\omega_1=\omega_2$ \\
+		2.4 & 0    & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & --                     &
+    $\omega_1=\omega_{10}$ \\
+		2.4 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & $\omega_2=\omega_{10}$ &
+    $\omega_1=\omega_2$ \\
+	\end{tabular}
+  \caption{
+    Conditions for peak intensity at different pulse delays for pathway I $\gamma$.
+  }
+  \label{mix:tab:table2}
+\end{table}
+
+\subsection{Temporal pathway discrimination}  % ---------------------------------------------------
+
 In the last section we showed how a single pathway's spectra can evolve with delay due to pulse
 effects and time gating.  %
 In real experiments, evolution with delay is further complicated by the six time-orderings/sixteen
@@ -1460,34 +1510,24 @@ where temporal discrimination is possible.  %
 As $\Delta_t\Gamma_{10}\rightarrow \infty$, however, such discrimination is only achieved at
 vanishing signal intensities; the contour of $P=0.99$ across our systems highlights this trend.  %
 
-\subsection{Multidimensional line shape dependence on pulse delay time}
-
 \begin{figure}
-	\includegraphics[width=\linewidth]{"mixed_domain/spectral evolution"}
-	\caption[Evolution of the 2D frequency response.]{
-		Evolution of the 2D frequency response as a function of $\tau_{21}$ (labeled inset) and the
-    influence of the relative dephasing rate ($\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and
-    $2.0$ (blue)).
-		In all plots $\tau_{22^\prime}=0$.  To ease comparison between different dephasing rates, the
-    colored line contours (showing the half-maximum) for all three relative dephasing rates are
-    overlaid.
-		The colored histograms below each 2D frequency plot show the relative weights of each
-    time-ordering for each relative dephasing rate.
-		Contributions from V and VI are grouped together because they have equal weights at
-    $\tau_{22^\prime}=0$.
+	\includegraphics[width=\linewidth]{"mixed_domain/delay space ratios"}
+	\caption[2D delay response for different relative dephasing rates.]{
+		Comparison of the 2D delay response for different relative dephasing rates (labeled atop each
+    column).
+		All pulses are tuned to exact resonance.
+		In each 2D delay plot, the signal amplitude is depicted by the colors.
+		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values
+    denoted on each contour.
+		The small plots above each 2D delay plot examine a $\tau_{22^\prime}$ slice of the delay
+    response ($\tau_{21}=0$).
+    The plot shows the total signal (black), as well as the component time-orderings VI (orange), V
+    (purple), and III or I (teal).
 	}
-	\label{mix:fig:hom_2d_spectra}
+	\label{mix:fig:delay_purity}
 \end{figure}
 
-\begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/spectral evolution full"}
-	\caption[Evolution of the 2D frequency response, with all contours shown.]{
-    Spectral evolution of the homogeneous exciton resonance as a function of $\tau_{21}$, with
-    $\tau_{22^\prime}=0$.
-    The 50\% contour is darkened to ease comparison with Figure 7.
-  }
-	\label{mix:fig:spectral_evolution_full}
-\end{figure}
+\subsection{Multidimensional line shape dependence on pulse delay time}  % ------------------------
 
 In the previous sections we showed how pathway spectra and weights evolve with delay.  %
 This section ties the two concepts together by exploring the evolution of the spectral line shape
@@ -1562,16 +1602,6 @@ only the absorptive line shape along $\omega_2$.  %
 This narrowing, however, is unresolvable when the pulse bandwidth becomes broader than that of the
 resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\Delta_t=0.5$.  %
 
-\begin{figure}
-	\includegraphics[width=\linewidth]{"mixed_domain/wigners full"}
-	\caption[Simulated Wigner spectra.]{
-    Mixed $\tau_{21}$, $\omega_1$ plots for each $\Gamma_{10}$ value simulated in this work.
-    For each plot, the corresponding $\omega_2$ value is shown as a gray vertical line.
-    Each plot is separately normalized.
-  }
-	\label{mix:fig:wigners}
-\end{figure}
-
 It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is
 frequency. \cite{KohlerDanielDavid2014a, AubockGerald2012a, CzechKyleJonathan2015a,
   PakoulevAndreiV2007a}  %
@@ -1587,48 +1617,45 @@ This representation also highlights the asymmetric broadening of the $\omega_1$
 pulse overlap when $\omega_2$ becomes non-resonant.  %
 Again, these features can resemble spectral diffusion even though our system is homogeneous.  %
 
-\subsection{Inhomogeneous broadening} \label{mix:sec:res_inhom}  % --------------------------------
-
 \begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/3PEPS"}
-	\label{mix:fig:3PEPS}
-	\caption[3PEPS tutorial.]{
-    Extraction of 3PEPS peak shifts from MR-CMDS delay space. Left-hand plot: thick colored lines
-    denote contours of constant $\tau$ for $T=0, 1, 2, 3$.
-    Dots indicate the fitted peak shift for each $\tau$ contour.
-    Right-hand plot: numerically simulated amplitude traces (solid), Gaussian fits (transparent)
-    and fit centers (vertical lines) for each $T$ (colors matched).
-  }
+	\includegraphics[width=\linewidth]{"mixed_domain/spectral evolution"}
+	\caption[Evolution of the 2D frequency response.]{
+		Evolution of the 2D frequency response as a function of $\tau_{21}$ (labeled inset) and the
+    influence of the relative dephasing rate ($\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and
+    $2.0$ (blue)).
+		In all plots $\tau_{22^\prime}=0$.  To ease comparison between different dephasing rates, the
+    colored line contours (showing the half-maximum) for all three relative dephasing rates are
+    overlaid.
+		The colored histograms below each 2D frequency plot show the relative weights of each
+    time-ordering for each relative dephasing rate.
+		Contributions from V and VI are grouped together because they have equal weights at
+    $\tau_{22^\prime}=0$.
+	}
+	\label{mix:fig:hom_2d_spectra}
 \end{figure}
 
 \begin{figure}
-	\includegraphics[width=0.5\linewidth]{"mixed_domain/inhom delay space ratios"}
-	\caption[2D delay response with inhomogeneity.]{
-		2D delay response for $\Gamma_{10}\Delta_t=1$ with sample inhomogeneity.  %
-		All pulses are tuned to exact resonance.  %
-		The colors depict the signal amplitude.  %
-		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values
-    denoted on each contour.  %
-		The thick yellow line denotes the peak amplitude position that is used for 3PEPS analysis.  %
-		The small plot above each 2D delay plot examines a $\tau_{22^\prime}$ slice of the delay
-    response ($\tau_{21}=0$).  %
-		The plot shows the total signal (black), as well as the component time-orderings VI (orange), V
-    (purple), III (teal, dashed), and I (teal, solid).  %
-	}
-	\label{mix:fig:delay_inhom}
+	\includegraphics[width=\textwidth]{"mixed_domain/spectral evolution full"}
+	\caption[Evolution of the 2D frequency response, with all contours shown.]{
+    Spectral evolution of the homogeneous exciton resonance as a function of $\tau_{21}$, with
+    $\tau_{22^\prime}=0$.
+    The 50\% contour is darkened to ease comparison with Figure 7.
+  }
+	\label{mix:fig:spectral_evolution_full}
 \end{figure}
 
 \begin{figure}
-	\includegraphics[width=\textwidth]{"mixed_domain/2D delays"}
-	\label{mix:fig:2D_delays}
-	\caption[2D delay response for all combinations of inhomogeneity, dephasing rate.]{
-    2D delay scans at $\omega_1=\omega_2=\omega_{10}$ for all 12 combinations of $\Gamma_{10}$
-    (rows) and $\Delta_{inhom}$ (columns) simulated in this work.
-    The 3PEPS shift trace is plotted in yellow, annotated to indicate the magnitude of the $\tau$
-    shift at $T=0$ and $T=4\Delta_t$.
+	\includegraphics[width=\linewidth]{"mixed_domain/wigners full"}
+	\caption[Simulated Wigner spectra.]{
+    Mixed $\tau_{21}$, $\omega_1$ plots for each $\Gamma_{10}$ value simulated in this work.
+    For each plot, the corresponding $\omega_2$ value is shown as a gray vertical line.
+    Each plot is separately normalized.
   }
+	\label{mix:fig:wigners}
 \end{figure}
 
+\subsection{Inhomogeneous broadening} \label{mix:sec:res_inhom}  % --------------------------------
+
 With the homogeneous system characterized, we can now consider the effect of inhomogeneity.  %
 For inhomogeneous systems, time-orderings III and V are enhanced because their final coherence will
 rephase to form a photon echo, whereas time-orderings I and VI will not.  %
@@ -1653,7 +1680,9 @@ the population (time-orderings V and VI).  %
 The trace runs parallel to the III-V time-ordering boundary (diagonal) if $\tau_{22^\prime}<0$ and
 runs parallel to the IV-VI time-ordering boundary (horizontal) if $\tau_{22^\prime}>0$, and both
 intersect at $\tau_{22^\prime}=0$; the $-\tau_{21}$ value at this intersection is $T$.  %
-In our 2D delay plots (\autoref{mix:fig:delay_purity}, \autoref{mix:fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.  %
+In our 2D delay plots (\autoref{mix:fig:delay_purity}, \autoref{mix:fig:delay_inhom}), the peak
+shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical
+line.  %
 \autoref{mix:fig:delay_inhom} highlights the peak shift profile as a function of population time with
 the yellow trace; it is easily verified that our static inhomogneous system exhibits a non-zero
 peak shift value for all population times.  %
@@ -1674,11 +1703,63 @@ This fact is easily illustrated by the dynamics of homogeneous system (Fig.
 \autoref{mix:fig:delay_purity}); even though the homogeneous system cannot rephase, there is a
 non-zero peak shift near $T=0$.  %
 The contamination of the 3PEPS measurement at pulse overlap is well-known and is described in some
-studies, \cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system properties on the
-distortion has not been investigated previously.  %
+studies, \cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system
+properties on the distortion has not been investigated previously.  %
 Peak-shifting due to pulse overlap is less important when $\omega_1\neq\omega_2$ because
 time-ordering III is decoupled by detuning.  %
 
+In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous
+broadening.  %
+\autoref{mix:fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous
+distribution.  %
+All systems are broadened by a distribution proportional to their dephasing bandwidth.  %
+As expected, the sequence again shows a gradual broadening along the $\omega_1$ axis, with a strong
+spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals.  %
+The anti-diagonal width at early delays (e.g. \autoref{mix:fig:inhom_2d_spectra},
+$\tau_{21}=2.0\Delta_t$) again depends on the pulse bandwidth and the monochromator slit width.  %
+At delay values that isolate time-orderings V and VI, however, the line shapes retain diagonal
+character, showing the characteristic balance of homogeneous and inhomogeneous width.  %
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{"mixed_domain/3PEPS"}
+	\label{mix:fig:3PEPS}
+	\caption[3PEPS tutorial.]{
+    Extraction of 3PEPS peak shifts from MR-CMDS delay space. Left-hand plot: thick colored lines
+    denote contours of constant $\tau$ for $T=0, 1, 2, 3$.
+    Dots indicate the fitted peak shift for each $\tau$ contour.
+    Right-hand plot: numerically simulated amplitude traces (solid), Gaussian fits (transparent)
+    and fit centers (vertical lines) for each $T$ (colors matched).
+  }
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/inhom delay space ratios"}
+	\caption[2D delay response with inhomogeneity.]{
+		2D delay response for $\Gamma_{10}\Delta_t=1$ with sample inhomogeneity.  %
+		All pulses are tuned to exact resonance.  %
+		The colors depict the signal amplitude.  %
+		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values
+    denoted on each contour.  %
+		The thick yellow line denotes the peak amplitude position that is used for 3PEPS analysis.  %
+		The small plot above each 2D delay plot examines a $\tau_{22^\prime}$ slice of the delay
+    response ($\tau_{21}=0$).  %
+		The plot shows the total signal (black), as well as the component time-orderings VI (orange), V
+    (purple), III (teal, dashed), and I (teal, solid).  %
+	}
+	\label{mix:fig:delay_inhom}
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=\textwidth]{"mixed_domain/2D delays"}
+	\label{mix:fig:2D_delays}
+	\caption[2D delay response for all combinations of inhomogeneity, dephasing rate.]{
+    2D delay scans at $\omega_1=\omega_2=\omega_{10}$ for all 12 combinations of $\Gamma_{10}$
+    (rows) and $\Delta_{inhom}$ (columns) simulated in this work.
+    The 3PEPS shift trace is plotted in yellow, annotated to indicate the magnitude of the $\tau$
+    shift at $T=0$ and $T=4\Delta_t$.
+  }
+\end{figure}
+
 \begin{figure}
 	\includegraphics[width=\linewidth]{"mixed_domain/inhom spectral evolution"}
 	\caption[Spectral evolution of an inhomogenious system.]{
@@ -1731,18 +1812,6 @@ time-ordering III is decoupled by detuning.  %
   }
 \end{figure}
 
-In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous
-broadening.  %
-\autoref{mix:fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous
-distribution.  %
-All systems are broadened by a distribution proportional to their dephasing bandwidth.  %
-As expected, the sequence again shows a gradual broadening along the $\omega_1$ axis, with a strong
-spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals.  %
-The anti-diagonal width at early delays (e.g. \autoref{mix:fig:inhom_2d_spectra},
-$\tau_{21}=2.0\Delta_t$) again depends on the pulse bandwidth and the monochromator slit width.  %
-At delay values that isolate time-orderings V and VI, however, the line shapes retain diagonal
-character, showing the characteristic balance of homogeneous and inhomogeneous width.  %
-
 \section{Discussion}  % ---------------------------------------------------------------------------
 
 \subsection{An intuitive picture of pulse effects}
@@ -1829,25 +1898,6 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
 
 \subsection{Extracting true material correlation}  % ----------------------------------------------
 
-\begin{figure}
-	\includegraphics[width=0.5\linewidth]{"mixed_domain/metrics"}
-	\caption[Metrics of correlation.]{
-		Temporal (3PEPS) and spectral (ellipticity) metrics of correlation and their relation to the
-    true system inhomogeneity.  %
-		The left column plots the relationship at pulse overlap ($T=0$) and the right column plots the
-    relationship at a delay where driven correlations are removed ($T=4\Delta_t$).  %
-    For the ellipticity measurements, $\tau_{22^\prime}=0$.  %
-		In each case, the two metrics are plotted directly against system inhomogeneity (top and middle
-    row) and against each other (bottom row).  %
-		Colored lines guide the eyes for systems with equal relative dephasing rates
-    ($\Gamma_{10}\Delta_t$, see upper legend), while the area of the data point marker indicates
-    the relative inhomogeneity ($\Delta_{\text{inhom}}/\Gamma_{10}$, see lower legend).  %
-		Gray lines indicate contours of constant relative inhomogeneity (scatter points with the same
-    area are connected).  %
-}
-	\label{mix:fig:metrics}
-\end{figure}
-
 We have shown that pulse effects mimic the qualitative signatures of inhomogeneity.  %
 Here we address how one can extract true system inhomogeneity in light of these effects.  %
 We focus on two ubiquitous metrics of inhomogeneity: 3PEPS for the time domain and
@@ -1925,6 +1975,25 @@ If a system has spectral diffusion, the mapping at late times will disagree with
 early times; both ellipticity and 3PEPS will be smaller at later times than predicted by the change
 in mappings alone.  %
 
+\begin{figure}
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/metrics"}
+	\caption[Metrics of correlation.]{
+		Temporal (3PEPS) and spectral (ellipticity) metrics of correlation and their relation to the
+    true system inhomogeneity.  %
+		The left column plots the relationship at pulse overlap ($T=0$) and the right column plots the
+    relationship at a delay where driven correlations are removed ($T=4\Delta_t$).  %
+    For the ellipticity measurements, $\tau_{22^\prime}=0$.  %
+		In each case, the two metrics are plotted directly against system inhomogeneity (top and middle
+    row) and against each other (bottom row).  %
+		Colored lines guide the eyes for systems with equal relative dephasing rates
+    ($\Gamma_{10}\Delta_t$, see upper legend), while the area of the data point marker indicates
+    the relative inhomogeneity ($\Delta_{\text{inhom}}/\Gamma_{10}$, see lower legend).  %
+		Gray lines indicate contours of constant relative inhomogeneity (scatter points with the same
+    area are connected).  %
+}
+	\label{mix:fig:metrics}
+\end{figure}
+
 \section{Conclusion}  % ---------------------------------------------------------------------------
 
 This study provides a framework to describe and disentangle the influence of the excitation pulses
-- 
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