2018-05-14 10:31

4 files changed, 450 insertions, 386 deletions

diff --git a/MX2/chapter.tex b/MX2/chapter.tex
index 6cfcdf8..01b2e15 100644
--- a/MX2/chapter.tex
+++ b/MX2/chapter.tex
@@ -465,7 +465,6 @@ features do not depend strongly on the $\omega_1$ frequency.
 
 The spectral features in Figures \ref{fig:Czech03}, \ref{fig:Czech04} and \ref{fig:Czech05} depend
 on the quantum mechanical interference effects caused by the different pathways.  %
-\autoref{fig:Czech06} shows all of the Liouville pathways required to understand the spectral
 features. \cite{PakoulevAndreiV2010a, PakoulevAndreiV2009a}  %
 These pathways correspond to the time orderings labeled V and VI in \autoref{fig:Czech02}b. The
 letters denote the density matrix elements, $\rho_{ij}$, where g representest the ground state and
@@ -540,8 +539,8 @@ For the AB peak, $\omega_2$ and $\omega_{2^\prime}$ generate two kinds of excito
 and (2) hot excitons in the A band.  %
 Relaxation to A may occur by interband transitions of B excitons or intraband transitions of hot A
 excitons.  %
-Either will lead to the GSB, SE, and ESA shown in the ee $\rightarrow$ e$^\prime$e$^\prime$
-population transfer pathways of \autoref{fig:Czech06}.  %
+Either will lead to the GSB, SE, and ESA of the ee $\rightarrow$ e$^\prime$e$^\prime$ population
+transfer pathways.  %
 This relaxation must occur on the time scale of our pulse-width since the cross-peak asymmetry is
 observed even during temporal overlap.  %
 We believe that intraband relaxation of hot A excitons is the main factor in breaking the symmetry
@@ -590,8 +589,6 @@ The pulse overlap region is complicated by the multiple Liouville pathways that
 considered.  %
 Additionally, interference between scattered light from the $\omega_1$ excitation beam and the
 output signal becomes a larger factor as the FWM signal decreases.  %
-\autoref{fig:Czech09} shows the $\omega_1$, $\omega_2$, and $\omega_{2^\prime}$ time ordered
-pathway that becomes and important consideration for positive $\tau_{21}$ delay times.  %
 Since $\tau_{22^\prime}=0$, the initial $\omega_1$ pulse creates an excited-state coherence, while
 the subsequent $\omega_2$ and $\omega_{2^\prime}$ pulses create the output coherence.  %
 The output signal is only important at short $\tau_{21}>0$ values because the initially excited
@@ -604,13 +601,11 @@ The resulting signal will therefore appear as the diagonal feature in \autoref{f
 (\textit{e.g.}, see the +40 fs 2D spectrum).  %
 In addition to the diagonal feature in \autoref{fig:Czech08}, there is also a vertical feature when
 $\omega_1$ is resonant with the A excitonic transition as well as the AB cross-peak.  %
-These features are attributed to the pathways in \autoref{fig:Czech06}.  %
 Although these pathways are depressed when $\tau_{21}>0$, there is sufficient temporal overlap
 between the $\omega_2$, $\omega_{2^\prime}$, and $\omega_1$ pulses to make their contribution
-comparable to those in \autoref{fig:Czech09}.  %
-More positie values of $\tau_{21}$ emphasize the \autoref{fig:Czech09} pathways over the
-\autoref{fig:Czech06} pathways, accounting for the increasing percentage of diagonal character at
-increasingly positive delays.  %
+comparable to V and VI.  %
+More positie values of $\tau_{21}$ emphasize the I, III pathways over the V, VI pathways,
+accounting for the increasing percentage of diagonal character at increasingly positive delays.  %
 
 \begin{figure}
   \includegraphics[width=\textwidth]{MX2/08}
diff --git a/PbSe_global_analysis/ta_vs_tg.pdf b/PbSe_global_analysis/ta_vs_tg.pdf
deleted file mode 100644
index 33897cd..0000000
--- a/PbSe_global_analysis/ta_vs_tg.pdf
+++ /dev/null
diff --git a/acquisition/chapter.tex b/acquisition/chapter.tex
index 7fbd1e7..0a7d7b7 100644
--- a/acquisition/chapter.tex
+++ b/acquisition/chapter.tex
@@ -220,10 +220,10 @@ Under ``Status'', the loop time and scan index help users gauge the progress of
 In this case, the displayed pixel (index 6, 40) took 2.448 seconds to acquire.  %

 

 The other tabs are for more advanced interactions, and will be discussed further in future

-sections.  % 

+sections.  %

 The ``Program'' tab contains various rarely-needed displays and inputs to configure PyCMDS.  %

 The ``Hardware'' tab is where the advanced menu for each hardware appears

-(\autoref{acq:sec:hardware}).  % 

+(\autoref{acq:sec:hardware}).  %

 The ``Devices'' tab is where sensor settings live (\autoref{acq:sec:sensors}).  %

 The ``Autonomic'' tab is where users configure the autonomic system for active correction

 (\autoref{acq:sec:autonomic}).  %

@@ -246,7 +246,7 @@ This functionality has not yet been implemented.  %
 \begin{figure}

   \includegraphics[width=9in]{"acquisition/screenshots/004"}

   \caption[PyCMDS queue.]{

-    PyCMDS queue while acquiring data, on the ps system.  % 

+    PyCMDS queue while acquiring data, on the ps system.  %

   }

   \label{acq:fig:pycmds_queue_screenshot}

 \end{figure}

@@ -256,7 +256,7 @@ This functionality has not yet been implemented.  %
 \begin{figure}

   \includegraphics[width=9in]{"acquisition/screenshots/000"}

   \caption[PyCMDS while scanning.]{

-    PyCMDS scan tab while acquiring data, on the ps system.  % 

+    PyCMDS scan tab while acquiring data, on the ps system.  %

   }

   \label{acq:fig:pycmds_screenshot_during_scan}

 \end{figure}

@@ -385,7 +385,7 @@ Hardware  # implements basic thread control
 \end{codefragment}

 The powerful thing about this strategy is that the three driver-specific classes

 (\python{Homemade}, \python{Thorlabs}, and \python{Newport}) need only implement minimal

-driver-specific code, typically \python{start}, \python{set} and \python{close}.  % 

+driver-specific code, typically \python{start}, \python{set} and \python{close}.  %

 This means that code is more maintainable and less repeated.  %

 For example, when I added the autonomic system to PyCMDS, I edited the parent \python{Delay} class

 to respect a new method \python{set_offset}.  %

@@ -462,7 +462,7 @@ string corresponding to the name of the method you wish to run in the worker thr
 The \python{q} instance will hold the instruction until the \python{Driver} instance is ready, at

 which point \python{Driver.dequeue} will be called in the worker thread.  %

 There is no way for the driver to command hardware to do something in the main thread, but the

-driver can trigger signals like \python{update_ui} and modify Mutexes.  % 

+driver can trigger signals like \python{update_ui} and modify Mutexes.  %

 

 \begin{figure}

 	\includepython{"acquisition/parent_hardware.py"}

@@ -476,7 +476,7 @@ driver can trigger signals like \python{update_ui} and modify Mutexes.  %
 

 \begin{figure}

 	\includepython{"acquisition/driver.py"}

-  \caption[TODO]{

+  \caption{

     Parent class of all drivers.  %

   }

   \label{acq:fig:parent_driver_class}

@@ -666,7 +666,7 @@ Spectrometer driver instances require the following:
   \item \python{get_grating_details}

   \item \python{set_turret}

 \end{ditemize}

- 

+

 \autoref{acq:fig:spectrometer_advanced} is a screenshot of the MicroHR advanced menu on the

 fs system.  %

 Nothing can be changed in the advanced menu except the label, and the offset can be seen.  %

@@ -952,7 +952,7 @@ each pixel in turn.  %
 Written simply, it does the following:

 \begin{codefragment}{python}

 def ndindex(shape):

-    rs = [range(s) for s in shape]  

+    rs = [range(s) for s in shape]

     pools = map(tuple, rs)

     result = [[]]

     for pool in pools:

@@ -984,9 +984,9 @@ for idx in ndindex(shape):
   for hardware in hardwares:

       hardware.wait_until_still()

   for sensor in sensors:

-      sensor.read()    

+      sensor.read()

   for sensor in sensors:

-      sensor.wait_until_done() 

+      sensor.wait_until_done()

 \end{codefragment}

 The pattern is simple: launch hardware, wait for still, launch sensors, wait for done.  %

 The simplicity of this central loop truly shows the power of abstraction in PyCMDS.  %

@@ -1021,7 +1021,7 @@ The TUNE TEST module does a simple thing: it sets a chosen OPA to each of the po
 curve and does a monochromator scan of set width about that setpoint.  %

 In this way the tune (output color) agreement between the curve and the OPA can be determined.  %

 A new point curve with remapped colors is automatically created.  %

-% TODO: link to place in tuning chapter...

+See \autoref{cha:opa} for more information.  %

 

 \subsubsection{MOTORTUNE}

 

@@ -1111,7 +1111,7 @@ to get as much use as possible out of the available time.  %
 

 Currently spectral delay correction is done ``manually''.  %

 Relevant Wigner scans are taken using the SCAN module, processed using WrightTools, and generated

-coset files are manually applied in the autonomic menu.  % 

+coset files are manually applied in the autonomic menu.  %

 Ideally, ``check'' Wigners are then taken to verify the veracity of the corrections.  %

 In the future, it would be preferable to have a dedicated SDC module that did all of these things

 automatically.  %

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