2018-01-31 18:37

12 files changed, 1113 insertions, 68 deletions

diff --git a/MX2/chapter.tex b/MX2/chapter.tex
index 77cfe6e..68e87b7 100644
--- a/MX2/chapter.tex
+++ b/MX2/chapter.tex
@@ -1,6 +1,5 @@
 \chapter{MX2}
 
-
 We report the first coherent multidimensional spectroscopy study of a MoS\textsubscript{2} film.  %
 A four-layer sample of MoS\textsubscript{2} was synthesized on a silica substrate by a simplified
 sulfidation reaction and characterized by absorption and Raman spectroscopy, atomic force
@@ -81,19 +80,23 @@ vector for each beam and the subscripts label the excitation frequencies.  %
 Multidimensional spectra result from measuring the output intensity dependence on frequency and
 delay times.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.5\textwidth]{MX2/01}
-  \caption[CMDS tutorial]{(a) Example delays of the $\omega_1$, $\omega_2$, and
-    $\omega_{2^\prime}$ excitation pulses. (b) Dependence of the output intensity on the
-    $\tau_{22^\prime}$ and $\tau_{21}$ time delays for $\omega_1=\omega_2$. The solid lines define
-    the regions for the six different time orderings of the $\omega_1$, $\omega_2$, and
-    $\omega_{2^\prime}$ excitation pulses. We have developed a convention for numbering these time
-    orderings, as shown. (c) Diagram of the band structure of MoS\textsubscript{2} at the $K$
-    point. The A and B exciton transitions are shown. (d) Two dimensional frequency-frequency plot
+  \caption[CMDS tutorial]{
+    (a) Example delays of the $\omega_1$, $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
+    (b) Dependence of the output intensity on the $\tau_{22^\prime}$ and $\tau_{21}$ time delays
+    for $\omega_1=\omega_2$.
+    The solid lines define the regions for the six different time orderings of the $\omega_1$,
+    $\omega_2$, and $\omega_{2^\prime}$ excitation pulses.
+    We have developed a convention for numbering these time orderings, as shown.
+    (c) Diagram of the band structure of MoS\textsubscript{2} at the $K$ point.
+    The A and B exciton transitions are shown. (d) Two dimensional frequency-frequency plot
     labeling two diagonal and cross-peak features for the A and B excitons.}
   \label{fig:Czech01}
 \end{figure}
+\clearpage}
 
 \autoref{fig:Czech01} introduces our conventions for representing multidimensional spectra.  %
 \autoref{fig:Czech01}b,d are simulated data.  %
@@ -141,12 +144,14 @@ B exciton states.  %
 
 \section{Methods}  % ------------------------------------------------------------------------------
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/S1}
   \caption{Schemiatic of the synthetic setup used for Mo thin film sulfidation reactions.}
   \label{fig:CzechS1}
 \end{figure}
+\clearpage}
 
 MoS\textsubscript{2} thin films were prepared \textit{via} a Mo film sulfidation reaction, similar
 to methods reported by \textcite{LaskarMasihhurR2013a}.  %
@@ -178,16 +183,20 @@ with DI water, and transferred to a Cu-mesh TEM grid.  %
 TEM experiments were performed on a FEI Titan aberration corrected (S)TEM under 200 kV accelerating
 voltage.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/10}
-  \caption[Mask and epi vs transmissive.]{(a) Mask. (b) 2D delay spectra at the BB diagonal
-    ($\omega_1=\omega_2\approx1.95$ eV) for transmissive and reflective geometries. Transmissive
-    signal is a mixture of MoS\textsubscript{2} signal and a large amount of driven signal from the
-    substrate that only appears in the pulse overlap region. Reflective signal is representative of
-    the pure MoS\textsubscript{2} response.}
+  \caption[Mask and epi vs transmissive.]{
+    (a) Mask.
+    (b) 2D delay spectra at the BB diagonal ($\omega_1=\omega_2\approx1.95$ eV) for transmissive
+    and reflective geometries.
+    Transmissive signal is a mixture of MoS\textsubscript{2} signal and a large amount of driven
+    signal from the substrate that only appears in the pulse overlap region. Reflective signal is
+    representative of the pure MoS\textsubscript{2} response.}
   \label{fig:Czech10}
 \end{figure}
+\clearpage}
 
 The coherent multidimensional spectroscopy system used a 35 fs seed pulse, centered at 800 nm and
 generated by a 1 kHz Tsunami Ti-sapphire oscillator.  %
@@ -200,12 +209,14 @@ Signal and idler were not filtered out, but played no role due to their low phot
 Pulse $\omega_2$ was split into pulses labeled $\omega_2$ and $\omega_{2^\prime}$ to create a total
 of three excitation pulses.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/S4}
   \caption{OPA outputs at each color explored.}
   \label{fig:CzechS4}
 \end{figure}
+\clearpage}
 
 In this experiment we use motorized OPAs which allow us to set the output color in software.  %
 OPA1 and OPA2 were used to create the $\omega_1$ and $\omega_2$ frequencies, respectively.  %
@@ -222,12 +233,14 @@ focused onto the sample surface by a 1 meter focal length spherical mirror in a
 geometry to form a 630, 580, and 580 $\mu$m FWHM spot sizes for $\omega_1$, $\omega_2$, and
 $\omega_{2^\prime}$, respectively.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/S5}
   \caption{Spectral delay correction.}
   \label{fig:CzechS5}
 \end{figure}
+\clearpage}
 
 \autoref{fig:CzechS5} represents delay corrections applied for each OPA.  %
 The corrections were experimentally determined using driven FWM output from fused silica.  %
@@ -262,25 +275,35 @@ signal id is the geometry chosen for this experiment.  %
 This discrimination between a film and the substrate was also seen in reflective and transmissive
 CARS microscopy experiments. \cite{VolkmerAndreas2001a}  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/11}
-  \caption[MoS\textsubscript{2} post processing.]{Visualization of data collection and processing.
+  \caption[MoS\textsubscript{2} post processing.]{
+    Visualization of data collection and processing.
     With the exception of (c), each subsequent pane represents an additional processing step on top
-  of previous processing. The color bar of each image is separate. (a) Voltages read by the
-  detector at teach color combination. The large vertical feature is $\omega_1$ scatter; the shape
-  is indicative of the power curve of the OPA. MoS\textsubscript{2} response can be barely seen
-  above this scatter. (b) Data after chopping and active background subtraction at the boxcar (100
-  shots). (c) The portion of chopped signal that is not material response. This portion is
-  extracted by averaging several collections at very positive $\tau_{21}$ values, where no material
-  response is present due to the short coherence times of MoS\textsubscript{2} electronic states.
-  The largest feature is $\omega_2$ scatter. Cross-talk between digital-to-analog channels can also
-  be seen as the negative portion that goes as $\omega_1$ intensity. (d) Signal after (c) is
-  subtracted. (e) Smoothed data. (f) Amplitude level (square root) data. This spectrum corresponds
-  to that at 0 delay in \autoref{fig:Czech03}. Note that the color bar's range is different than in
-  \autoref{fig:Czech03}.}
+    of previous processing.
+    The color bar of each image is separate.
+    (a) Voltages read by the detector at teach color combination.
+    The large vertical feature is $\omega_1$ scatter; the shape is indicative of the power curve of
+    the OPA.
+    MoS\textsubscript{2} response can be barely seen above this scatter.
+    (b) Data after chopping and active background subtraction at the boxcar (100 shots).
+    (c) The portion of chopped signal that is not material response.
+    This portion is extracted by averaging several collections at very positive $\tau_{21}$ values,
+    where no material response is present due to the short coherence times of MoS\textsubscript{2}
+    electronic states.
+    The largest feature is $\omega_2$ scatter.
+    Cross-talk between digital-to-analog channels can also be seen as the negative portion that
+    goes as $\omega_1$ intensity.
+    (d) Signal after (c) is subtracted.
+    (e) Smoothed data.
+    (f) Amplitude level (square root) data.
+    This spectrum corresponds to that at 0 delay in \autoref{fig:Czech03}.
+    Note that the color bar's range is different than in \autoref{fig:Czech03}.}
   \label{fig:Czech11}
 \end{figure}
+\clearpage}
 
 Once measured, the FWM signal was sent through a four-stage workup process to create the data set
 shown here.  %
@@ -303,19 +326,23 @@ processing and plotting in this work.
 
 \section{Results and discussion}  % ---------------------------------------------------------------
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.75\textwidth]{MX2/02}
-  \caption[Few-layer MoS\textsubscript{2} thin film characterization.]{Characterization of the
-    few-layer MoS\textsubscript{2} film studied in this work. Optical images of the
+  \caption[Few-layer MoS\textsubscript{2} thin film characterization.]{
+    Characterization of the few-layer MoS\textsubscript{2} film studied in this work.
+    Optical images of the
     MoS\textsubscript{2} thin film on fused silica substrate in (a) transmission and (b)
-    reflection. (c) Raman spectrum of the $E_{2g}^1$ and $A_{1g}$ vibrational
-    modes. (d) High-resolution TEM image and its corresponding FFT shown in the inset. (e)
-    Absorption (blue), photoluminescence (green), Gaussian fits to the A and B excitons, along with
-    the residules betwen the fits and absorbance (dotted), A and B exciton centers (dotted) and
-    representative excitation pulse shape (red).}
+    reflection.
+    (c) Raman spectrum of the $E_{2g}^1$ and $A_{1g}$ vibrational modes.
+    (d) High-resolution TEM image and its corresponding FFT shown in the inset.
+    (e) Absorption (blue), photoluminescence (green), Gaussian fits to the A and B excitons, along
+    with the residules betwen the fits and absorbance (dotted), A and B exciton centers (dotted)
+    and representative excitation pulse shape (red).}
   \label{fig:Czech02}
 \end{figure}
+\clearpage}
 
 The few-layer MoS\textsubscript{2} thin film sample studied in this work was prepared on a
 transparent fused silica substrate by a simple sufidation reaction of a Mo thin film using a
@@ -323,7 +350,7 @@ procedure modified from a recent report. \cite{LaskarMasihhurR2013a}  %
 \autoref{fig:Czech02}a and b show the homogeneous deposition and surface smoothness of the sample
 over the centimeter-sized fused silica substrate, respectively.  %
 The Raman spectrum shows the $E_{2g}^1$ and $A_{1g}$ vibrational modes (\autoref{fig:Czech02}c)
-that are characteristic of MoS\textsubscript{2}. \cite{LiSongLin2012a}  %
+that are characteristic of MoS\textsubscript{2}. \c ite{LiSongLin2012a}  %
 The transmission electron micrograph (TEM) in \autoref{fig:Czech02}d shows the lattice fringes of
 the film with an inset fast Fourier transform (FFT) of the TEM image indicative of the hexagonal
 crystal structure of the film corresponding to the 0001 plane of MoS\textsubscript{2}.
@@ -334,7 +361,8 @@ corresponds to approximately four monolayers.  %
 and B excitonic line shapes that were extracted from the absorption spectrum. A representative
 excitation pulse profile is also shown in red for comparison.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/S3}
   \caption[MoS\textsubscript{2} absorbance.]{Extraction of excitonic features from absorbance
@@ -343,6 +371,7 @@ excitation pulse profile is also shown in red for comparison.  %
     (black), Gaussian fits (blue and red), and remainder (black dotted).}
   \label{fig:CzechS3}
 \end{figure}
+\clearpage}
 
 Extracting the exciton absorbance spectrum is complicated by the large ``rising background'' signal
 from other MoS\textsubscript{2} bands.  %
@@ -366,7 +395,8 @@ In order to compare the FWM spectra with the absorption spectrum, the signal has
 the square root of the measured FWM signal since FWM depends quadratically on the sample
 concentration and path length.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/03}
   \caption[MoS\textsubscript{2} frequency-frequency slices.]{2D frequency-frequency spectra of the
@@ -380,6 +410,7 @@ concentration and path length.  %
     of the A and B excitons, as designated from the absorption spectrum.}
   \label{fig:Czech03}
 \end{figure}
+\clearpage}
 
 The main set of data presented in this work is an $\omega_1\omega_2\tau_{21}$ ``movie'' with
 $\tau_{22\prime}=0$. 
@@ -393,7 +424,8 @@ In contrast, we see no well-defined excitonic peaks along the $\omega_2$ ``pump'
 Instead, the signal amplitude increases toward bluer $\omega_2$ values.  %
 The decrease in FWM above 2.05 eV is caused by a drop in the $\omega_2$ OPA power.
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.75\textwidth]{MX2/04}
   \caption[MoS\textsubscript{2} $\omega_1$ Wigner progression.]{Mixed $\omega_1$---$\tau_{21}$
@@ -403,8 +435,10 @@ The decrease in FWM above 2.05 eV is caused by a drop in the $\omega_2$ OPA powe
     marked as dashed lines within each spectrum.}
   \label{fig:Czech04}
 \end{figure}
+\clearpage}
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.75\textwidth]{MX2/05}
   \caption[MoS\textsubscript{2} $\omega_2$ Wigner progression.]{Mixed $\omega_2$---$\tau_{21}$
@@ -414,6 +448,7 @@ The decrease in FWM above 2.05 eV is caused by a drop in the $\omega_2$ OPA powe
     marked as dashed lines within each spectrum.}
   \label{fig:Czech05}
 \end{figure}
+\clearpage}
 
 Figures \ref{fig:Czech04} and \ref{fig:Czech05} show representative 2D frequency-delay slices from
 this movie, where the absicissa is the $\omega_1$ or $\omega_2$ frequency, respectively, the
@@ -440,7 +475,8 @@ Both the line shapes and the dynamics of the spectral features are very similar.
 \autoref{fig:Czech05} is an excitation spectrum that shows that the dynamics of the spectral
 features do not depend strongly on the $\omega_1$ frequency.
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.5\textwidth]{MX2/06}
   \caption[Pathway V, VI liouville pathways.]{Liouville pathways for \autoref{fig:Czech04}. gg and
@@ -450,6 +486,7 @@ features do not depend strongly on the $\omega_1$ frequency.
     either A or B excitonic states.}
   \label{fig:Czech06}
 \end{figure}
+\clearpage}
 
 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.  %
@@ -541,15 +578,18 @@ $\omega_2$ is lower than the A exciton frequency (the top subplot).  %
 If population transfer of holes from the B to A valence bands occurred during temporal overlap, the
 B/A ratio would be independent of pump frequency at $\tau_21<0$.
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/07}
-  \caption[MoS\textsubscript{2} transients.]{Transients taken at the different $\omega_1$ and
-    $\omega_2$ frequencies indicated by the colored markers on the 2D spectrum. The dynamics are
-    assigned to a 680 fs fast time constant (black solid line) and a slow time constant represented
-    as an unchanging offset over this timescale (black dashed line).}
+  \caption[MoS\textsubscript{2} transients.]{
+    Transients taken at the different $\omega_1$ and $\omega_2$ frequencies indicated by the
+    colored markers on the 2D spectrum.
+    The dynamics are assigned to a 680 fs fast time constant (black solid line) and a slow time
+    constant represented as an unchanging offset over this timescale (black dashed line).}
   \label{fig:Czech07}
 \end{figure}
+\clearpage}
 
 \autoref{fig:Czech07} shows the delay transients at the different frequencies shown in the 2D
 spectrum.  %
@@ -565,7 +605,8 @@ offset that represents the long time decay.  %
 The 680 fs decay is similar to previously published pump-probe and transient absorption
 experiments. \cite{NieZhaogang2014a, SunDezheng2014a, DochertyCallumJ2014a}  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=\textwidth]{MX2/08}
   \caption[MoS\textsubscript{2} frequency-frequency slices near pulse overlap.]{2D
@@ -573,6 +614,7 @@ experiments. \cite{NieZhaogang2014a, SunDezheng2014a, DochertyCallumJ2014a}  %
     normalized to the brightest features in each spectrum.}
   \label{fig:Czech08}
 \end{figure}
+\clearpage}
 
 The spectral features change quantitatively for delay times near temporal overlap.  %
 \autoref{fig:Czech08} shows a series of 2D spectra for both positive and negative $\tau_{21}$ delay
@@ -584,7 +626,8 @@ The spectra also develop more diagonal character as the delay time moves from ne
 values.  %
 The AB cross-peak is also a strong feature in the spectrum at early times.  %
 
-\begin{figure}[!htb]
+\afterpage{
+\begin{figure}
   \centering
   \includegraphics[width=0.5\textwidth]{MX2/09}
   \caption[Pathways I, III Liouville pathways.]{Liouville pathways for the $\omega_1$, $\omega_2$,
@@ -592,6 +635,7 @@ The AB cross-peak is also a strong feature in the spectrum at early times.  %
     A or B excitonic states.}
   \label{fig:Czech09}
 \end{figure}
+\clearpage}
 
 The pulse overlap region is complicated by the multiple Liouville pathways that must be
 considered.  %
@@ -654,5 +698,4 @@ complex MoS\textsubscript{2} and other TMDC heterostructures with quantum-state
 The frequency domain based multiresonant CMDS methods described in this paper will play a central
 role in these measurements.  %
 They use longer, independently tunable pulses that provide state-selective excitation over a wide
-spectral range without the requirement for interferometric stability.  %
-
+spectral range without the requirement for interferometric stability.  %
\ No newline at end of file
diff --git a/dissertation.pdf b/dissertation.pdf
index db1e0e5..0f856c7 100644
--- a/dissertation.pdf
+++ b/dissertation.pdf
diff --git a/dissertation.syg b/dissertation.syg
index 4bf8c0d..7b1944a 100644
--- a/dissertation.syg
+++ b/dissertation.syg
@@ -1,3 +1,3 @@
 \glossaryentry{\ensuremath {N}?\glossentry{N}|setentrycounter[]{page}\glsnumberformat}{8}
 \glossaryentry{\ensuremath {N}?\glossentry{N}|setentrycounter[]{page}\glsnumberformat}{8}
-\glossaryentry{\ensuremath {\omega }?\glossentry{omega}|setentrycounter[]{page}\glsnumberformat}{46}
+\glossaryentry{\ensuremath {\omega }?\glossentry{omega}|setentrycounter[]{page}\glsnumberformat}{74}
diff --git a/dissertation.tex b/dissertation.tex
index b937a98..3bf6bb6 100644
--- a/dissertation.tex
+++ b/dissertation.tex
@@ -38,6 +38,7 @@
 \renewcommand{\familydefault}{\sfdefault}

 \newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}

 \usepackage{etoolbox}

+\usepackage{tabularx}

 \AtBeginEnvironment{verse}{\singlespacing}

 \AtBeginEnvironment{tabular}{\singlespacing}

 

@@ -97,7 +98,6 @@
 \makeglossaries

 \include{glossary}

 

-

 \usepackage{tocloft}

  

 \setlength\cftparskip{0pt}

@@ -189,8 +189,9 @@ This dissertation is approved by the following members of the Final Oral Committ
 

 % chapters ----------------------------------------------------------------------------------------

 

-\part{Background}

 \include{introduction/chapter}

+

+\part{Background}

 \include{spectroscopy/chapter}

 \include{materials/chapter}

 \include{mixed_domain/chapter}

diff --git a/mixed_domain/SQC lineshapes against t.pdf b/mixed_domain/SQC lineshapes against t.pdf
new file mode 100644
index 0000000..e932bca
--- /dev/null
+++ b/mixed_domain/SQC lineshapes against t.pdf
diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index 4d3ba1c..308146b 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -5,7 +5,7 @@ are similar to system dephasing times.  %
 In these experiments, expectations derived from the familiar driven and impulsive limits are not

 valid.  %

 This work simulates the mixed-domain Four Wave Mixing response of a model system to develop

-expectations for this more complex field-matter interaction.  %

+expectations for this more complex field-matter interaction. %

 We explore frequency and delay axes.  %

 We show that these line shapes are exquisitely sensitive to excitation pulse widths and delays.  %

 Near pulse overlap, the excitation pulses induce correlations which resemble signatures of dynamic

@@ -137,17 +137,20 @@ from these measurement artifacts.  %
 

 \section{Theory}

 

+\afterpage{

 \begin{figure}

   \centering

 	\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}

-	\caption{

+	\caption[Sixteen triply-resonant Liouville pathways.]{

 		The sixteen triply-resonant Liouville pathways for the third-order response of the system used

-    here.  Time flows from left to right.  Each excitation is labeled by the pulse stimulating the

-    transition; excitatons with $\omega_1$ are yellow, excitations with $\omega_2=\omega_{2'}$ are

-    purple, and the final emission is gray.

+    here.

+    Time flows from left to right.

+    Each excitation is labeled by the pulse stimulating the transition; excitatons with $\omega_1$

+    are yellow, excitations with $\omega_2=\omega_{2'}$ are purple, and the final emission is gray.

 	}

 	\label{fig:WMELs}

 \end{figure}

+\clearpage}

 

 We consider a simple three-level system (states $n=0,1,2$) that highlights the multidimensional

 line shape changes resulting from choices of the relative dephasing and detuning of the system and

@@ -231,9 +234,10 @@ this paper.  %
 The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discussed in Appendix

 \ref{sec:cw_imp}.  %

 

-\begin{figure*}

+\afterpage{

+\begin{figure}

 	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}

-	\caption{

+	\caption[Overview of the MR-CMDS simulation.]{

 		Overview of the MR-CMDS simulation. 

 		(a) The temporal profile of a coherence under pulsed excitation depends on how quickly the

     coherence dephases.  In all subsequent panes, the relative dephasing rate is kept constant at

@@ -249,7 +253,8 @@ The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discuss
     help introduce our delay convention.  

 	}

 	\label{fig:overview}

-\end{figure*}

+\end{figure}

+\clearpage}

 

 Fig. \ref{fig:overview} gives an overview of the simulations done in this work.  %

 Fig. \ref{fig:overview}a shows an excitation pulse (gray-shaded) and examples of a coherent

@@ -367,6 +372,214 @@ $S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ are recorded f
 Liouville pathway.  %

 Our simulations were done using the open-source SciPy library.\cite{Oliphant2007}  %

 

+\subsection{Characteristics of Driven and Impulsive Response}\label{sec:cw_imp}

+

+The changes in the spectral line shapes described in this work are best understood by examining the

+driven/continuous wave (CW) and impulsive limits of Equations \ref{eq:rho_f_int} and

+\ref{eq:E_L_full}.  %

+The driven limit is achieved when pulse durations are much longer than the response function

+dynamics: $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \gg 1$.  %

+In this limit, the system will adopt a steady state over excitation:  $d\rho / dt \approx 0$.  %

+Neglecting phase factors, the driven solution to Equation \ref{eq:rho_f_int} will be

+\begin{equation}\label{eq:sqc_driven}

+\tilde{\rho}_f(t) = \frac{\lambda_f \mu_f}{2} 

+\frac{c_x(t-\tau_x)e^{i\kappa_f \Omega_{fx}t}}{\kappa_f \Omega_{fx}} \tilde{\rho}_i(t).

+\end{equation}

+The frequency and temporal envelope of the excitation pulse controls the coherence time evolution,

+and the relative amplitude and phase of the coherence is directly related to detuning from

+resonance.  %

+

+The impulsive limit is achieved when the excitation pulses are much shorter than response function

+dynamics:  $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$.  %

+The full description of the temporal evolution has two separate expressions: one for times when the

+pulse is interacting with the system, and one for times after pulse interaction.  %

+Both expressions are important when describing CMDS experiments.  %

+

+For times after the pulse interaction, $t \gtrsim \tau_x + \Delta_t$, the field-matter coupling is

+negligible.  %

+The evolution for these times, on resonance, is given by

+\begin{equation}\label{eq:sqc_fid}

+\tilde{\rho}_f(t) =\frac{i \lambda_f\mu_f }{2} \tilde{\rho}_i(\tau_x)

+\int c_x(u) du \ e^{-\Gamma_f(t-\tau_x)}.

+\end{equation}

+This is classic free induction decay (FID) evolution:  the system evolves at its natural frequency

+and decays at rate $\Gamma_f$.  %

+It is important to note that, while this expression is explicitly derived from the impulsive limit,

+FID behavior is not exclusive to impulsive excitation, as we have defined it.  %

+A latent FID will form if the pulse vanishes at a fast rate relative to the system dynamics.  

+

+For evaluating times near pulse excitation, $t \lesssim \tau_x + \Delta_t$, we implement a Taylor

+expansion in the response function about zero: $e^{-(\Gamma_f+i\kappa_f\Omega_{fx})u} = 1 -

+(\Gamma_f+i\kappa_f\Omega_{fx})u+\cdots$.  %

+Our impulsive criterion requires that a low order expansion will suffice; it is instructive to

+consider the result of the first order expansion of Equation \ref{eq:rho_f_int}:  %

+\begin{equation}\label{eq:sqc_rise}

+\begin{split}

+\tilde{\rho}_f(t) =& \frac{i \lambda_f\mu_f}{2} e^{-i\kappa_f\omega_x\tau_x}e^{-i\kappa_f\Omega_{fx}t} \tilde{\rho}_i(\tau_x) \\

+& \times \bigg[ \left( 1-(\Gamma_f + i\kappa_f\Omega_{fx})(t-\tau_x) \right) \int_{-\infty}^{t-\tau_x} c_x(u) du \\

+& \quad +(\Gamma_f + i\kappa_f\Omega_{fx}) \int_{-\infty}^{t-\tau_x} c_x(u)u \ du \bigg].

+\end{split}

+\end{equation}

+During this time $\tilde{\rho}_f$ builds up roughly according to the integration of the pulse

+envelope.  %

+The build-up is integrated because the pulse transfers energy before appreciable dephasing or

+detuning occurs.  %

+Contrary to the expectation of impulsive evolution, the evolution of $\tilde{\rho}_f$ is explicitly

+affected by the pulse frequency, and the temporal profile evolves according to the pulse.  %

+

+It is important to recognize that the impulsive limit is defined not only by having slow relaxation

+relative to the pulse duration, but also by small detuning relative to the pulse bandwidth (as is

+stated in the inequality).  %

+As detuning increases, the higher orders of the response function Taylor expansion will be needed

+to describe the rise time, and the driven limit of Equation \ref{eq:sqc_driven} will become

+valid.  %

+The details of this build-up time can often be neglected in impulsive approximations because

+build-up contributions are often negligible in analysis; the period over which the initial

+excitation occurs is small in comparison to the free evolution of the system.  %

+The build-up behavior can be emphasized by the measurement, which makes Equation \ref{eq:sqc_rise}

+important.  %

+

+We now consider full Liouville pathways in the impulsive and driven limits of Equation

+\ref{eq:E_L_full}.  %

+For the driven limit, Equation \ref{eq:E_L_full} can be reduced to 

+\begin{equation}\label{eq:E_L_driven}

+\begin{split}

+E_L(t) =& \frac{1}{8} \lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 

+e^{-i(\kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z)} \\

+& \times e^{ i(\kappa_3\omega_z + \kappa_2\omega_y + \kappa_1\omega_x)t} \\

+& \times c_z(t-\tau_z)c_y(t-\tau_y)c_x(t-\tau_x) \\

+& \times \frac{1}{\kappa_1\Omega_{1x}-i\Gamma_1} \frac{1}{\kappa_1\Omega_{1x} + \kappa_2\Omega_{2y} - i\Gamma_2} \\

+& \times \frac{1}{\kappa_1\Omega_{1x} + \kappa_2 \Omega_{2y} + \kappa_3\Omega_{3z}-i\Gamma_3}.

+\end{split}

+\end{equation}

+It is important to note that the signal depends on the multiplication of all the fields; pathway

+discrimination based on pulse time-ordering is not achievable because polarizations exists only

+when all pulses are overlapped.  %

+This limit is the basis for frequency-domain techniques.  %

+Frequency axes, however, are not independent because the system is forced to the laser frequency

+and influences the resonance criterion for subsequent excitations.  %

+As an example, observe that the first two resonant terms in Equation \ref{eq:E_L_driven} are

+maximized when $\omega_x=\left|\omega_1\right|$ and $\omega_y=\left|\omega_2\right|$.  %

+If  $\omega_x$ is detuned by some value $\varepsilon$, however, the occurrence of the second

+resonance shifts to $\omega_y=\left|\omega_2\right|+\varepsilon$, effectively compensating for the

+$\omega_x$ detuning.  %

+This shifting of the resonance results in 2D line shape correlations.  %

+

+If the pulses do not temporally overlap $(\tau_x+\Delta_t \lesssim \tau_y +\Delta_t \lesssim \tau_z

++ \Delta_t \lesssim t)$, then the impulsive solution to the full Liouville pathway of Equation

+\ref{eq:E_L_full} is  %

+\begin{equation}\label{eq:E_L_impulsive}

+\begin{split}

+E_L(t) =& \frac{i}{8} \lambda_1\lambda_2\lambda_3\mu_1 \mu_2 \mu_3 \mu_4 e^{i(\omega_1 + \omega_2 + \omega_3)t} \\

+& \times \int c_x(w) dw \int c_y(v) dv \int c_z(u) du \\

+& \times e^{-\Gamma_1(\tau_y-\tau_x)} e^{-\Gamma_2(\tau_z-\tau_y)} e^{-\Gamma(t-\tau_z)}.

+\end{split}

+\end{equation}

+Pathway discrimination is demonstrated here because the signal is sensitive to the time-ordering of

+the pulses.  %

+This limit is suited for delay scanning techniques.  %

+The emitted signal frequency is determined by the system and can be resolved by scanning a

+monochromator.  %

+

+The driven and impulsive limits can qualitatively describe our simulated signals at certain

+frequency and delay combinations.  %

+Of the three expressions, the FID limit most resembles signal when pulses are near resonance and

+well-separated in time (so that build-up behavior is negligible).  %

+The build-up limit approximates well when pulses are near-resonant and arrive together (so that

+build-up behavior is emphasized).  %

+The driven limit holds for large detunings, regardless of delay.  %

+

+\subsection{Convolution Technique for Inhomogeneous Broadening}\label{sec:convolution}

+

+\afterpage{

+\begin{figure}

+	\includegraphics[width=\linewidth]{mixed_domain/convolve}

+  \caption[Convolution overview.]

+  {Overview of the convolution.

+    (a) The homogeneous line shape.

+    (b) The distribution function, $K$, mapped onto laser coordinates.

+    (c) The resulting ensemble line shape computed from the convolution.

+    The thick black line represents the FWHM of the distribution function.}

+	\label{fig:convolution}

+\end{figure}

+\clearpage}

+

+Here we describe how to transform the data of a single reference oscillator signal to that of an

+inhomogeneous distribution.  %

+The oscillators in the distribution are allowed have arbitrary energies for their states, which

+will cause frequency shifts in the resonances.  %

+To show this, we start with a modified, but equivalent, form of Equation \ref{eq:rho_f}:

+\begin{equation}\label{eq:rho_f_modified}

+\begin{split}

+\frac{d\tilde{\rho}_f}{dt} =& -\Gamma_f\tilde{\rho}_f + \frac{i}{2}\lambda_f\mu_f c_x(t-\tau_x) \\

+& \times e^{i\kappa_f\left( \vec{k}\cdot z + \omega_x \tau_x \right)} e^{-i\kappa_f\left( \omega_x-\left|\omega_f \right| \right)t}\tilde{\rho}_i(t).

+\end{split}

+\end{equation}

+

+We consider two oscillators with transition frequencies $\omega_f$ and $\omega_f^\prime=\omega_f +

+\delta$.  %

+So long as $\left| \delta \right| \leq \omega_f$ (so that $\left| \omega_f + \delta \right| =

+\left| \omega_f \right| + \delta$ and thus the rotating wave approximation does not change),

+Equation \ref{eq:rho_f_modified} shows that the two are related by  %

+\begin{equation}\label{eq:freq_translation}

+\frac{d\tilde{\rho}_f^\prime}{dt}(t;\omega_x) = \frac{d\tilde{\rho}_f}{dt}(t;\omega_x-\delta)e^{i\kappa_f \delta \tau_x}.

+\end{equation}

+

+Because both coherences are assumed to have the same initial conditions

+($\rho_0(-\infty)=\rho_0^\prime(-\infty)=0$), the equality also holds when both sides of the

+equation are integrated.  %

+The phase factor $e^{i\kappa_f\delta\tau_x}$ in the substitution arises from Equation \ref{eq:E_l},

+where the pulse carrier frequency maintains its phase within the pulse envelope for all delays.  %

+

+The resonance translation can be extended to higher order signals as well.  %

+For a third-order signal, we compare systems with transition frequencies

+$\omega_{10}^\prime=\omega_{10}+a$ and $\omega_{21}^\prime = \omega_{21}+b$.  %

+The extension of Equation \ref{eq:freq_translation} to pathway $V\beta$ gives  %

+\begin{equation}

+\begin{split}

+\tilde{\rho}_3^\prime(t;\omega_2, \omega_2^\prime, \omega_1) =& \tilde{\rho}_3(t;\omega_2-a,\omega_{2^\prime}-a,\omega_1-b) \\

+&\times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1}.

+\end{split}

+\end{equation}

+

+The translation of each laser coordinate depends on which transition is made (e.g. $a$ for

+transitions between $|0\rangle$ and $|1\rangle$ or $b$ for transitions between $|1\rangle$ and

+$|2\rangle$), so the exact translation relation differs between pathways.  %

+We can now compute the ensemble average of signal for pathway $V\beta$ as a convolution between the

+distribution function of the system, $K(a,b)$, and the single oscillator response:  %

+\begin{equation}

+\begin{split}

+\langle \tilde{\rho}_3 (t;\omega_2,\omega_{2^\prime},\omega_1) \rangle =& \iint K(a,b)\\

+& \times \tilde{\rho}_3 (t;\omega_2+a,\omega_{2^\prime}+a,\omega_1+b) \\

+& \times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1} da \ db.

+\end{split}

+\end{equation}

+For this work, we restrict ourselves to a simpler ensemble where all oscillators have equally

+spaced levels (i.e. $a=b$).  %

+This makes the translation identical for all pathways and reduces the dimensionality of the

+convolution.  %

+Since pathways follow the same convolution we may also perform the convolution on the total signal field:

+\begin{equation}

+\begin{split}

+\langle E_{\text{tot}}(t) \rangle =& \sum_L \mu_{4,L} \int K(a,a) \\

+& \times \tilde{\rho}_{3,L}(t;\omega_x-a,\omega_y-a\omega_z-a) \\

+& \times e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} da.

+\end{split}

+\end{equation}

+Furthermore, since $\kappa=-1$ for $E_1$ and $E_{2^\prime}$, while $\kappa=1$ for $E_2$, we have

+$e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} = e^{-ia\left( \tau_1 -

+    \tau_2 + \tau_{2^\prime} \right)}$ for all pathways.  %

+Equivalently, if the electric field is parameterized in terms of laser coordinates $\omega_1$ and $\omega_2$, the ensemble field can be calculated as

+\begin{equation}\label{eq:convolve_final}

+\begin{split}

+\langle E_{\text{tot}}(t;\omega_1,\omega_2) \rangle =& \int K(a,a)E_{\text{tot}}(t;\omega_1-a,\omega_2-a) \\

+&\times e^{-ia\left( \tau_1-\tau_2+\tau_{2^\prime} \right)} da.

+\end{split}

+\end{equation}

+which is a 1D convolution along the diagonal axis in frequency space.  %

+Fig. \ref{fig:convolution} demonstrates the use of Equation \ref{eq:convolve_final} on a

+homogeneous line shape.  %

+

 \section{Results}  % ------------------------------------------------------------------------------

 

 We now present portions of our simulated data that highlight the dependence of the spectral line

@@ -375,9 +588,11 @@ pulse delay times, and inhomogeneous broadening.  %
 

 \subsection{Evolution of single coherence}\label{sec:evolution_SQC}

 

+\afterpage{

 \begin{figure}

+  \centering

 	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}

-	\caption{

+	\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

@@ -387,6 +602,7 @@ pulse delay times, and inhomogeneous broadening.  %
 	}

 	\label{fig:fid_dpr}

 \end{figure}

+\clearpage}

 

 It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x}

 \rho_1$, under various excitation conditions.  %

@@ -423,3 +639,756 @@ 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 $\Gamma_{10}\Delta_t=2.0$.  %

 

+\afterpage{

+\begin{figure}

+  \centering

+	\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{fig:fid_detuning}

+\end{figure}

+\clearpage}

+

+Fig. \ref{fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.\footnote{

+	See Supplementary Fig. S3 for a Fourier domain representation of Fig. \ref{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

+instantaneous frequency in Fig. \ref{fig:fid_detuning}a).  %

+The coherence will persist beyond the pulse duration only if the pulse transfers energy into the

+system; FID evolution equates to absorption.  %

+The FID is therefore sensitive to the absorptive (imaginary) line shape of a transition, while the

+driven response is the composite of both absorptive and dispersive components.  %

+If the experiment isolates the latent FID response, there is consequently a narrower spectral

+response.  %

+This spectral narrowing can be seen in Fig. \ref{fig:fid_detuning}a by comparing the coherence 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.\footnote{

+	See Supplementary Fig. S4 for explicit plots of $\rho_1(\Omega_{fx}/\Delta_{\omega})$ 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{Lagutchev2007,Lagutchev2010,Donaldson2010,Donaldson2008}  %

+

+In driven experiments, the output frequency and line shape are fully constrained by the excitation

+beams.  %

+In such experiments, there is no additional information to be resolved in the output spectrum.  %

+The situation changes in the mixed domain, where $E_\text{tot}$ contains FID signal that lasts

+longer than the pulse duration.  %

+Fig. \ref{fig:fid_detuning}a provides insight on how frequency-resolved detection of coherent

+output can enhance resolution when pulses are spectrally broad.  %

+Without frequency-resolved detection, mixed-domain resonance enhancement occurs in two ways: (1)

+the peak amplitude increases, and (2) the coherence duration increases due to the FID transient.  %

+Frequency-resolved detection can further discriminate against detuning by requiring that the

+driving frequency agrees with latent FID.  %

+The implications of discrimination are most easily seen in Fig. \ref{fig:fid_detuning}a with

+$\Omega_{1x}/\Delta_{\omega}=\pm 1$, where the system frequency moves from the driving frequency to

+the FID frequency.  %

+When the excitation pulse frequency is scanned, the resonance will be more sensitive to detuning by

+isolating the driven frequency (tracking the monochromator with the excitation source).  %

+

+

+The functional form of the measured line shape can be deduced by considering the frequency domain form of Equation \ref{eq:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$):

+\begin{equation}\label{eq: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},

+\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}}.

+\end{equation}

+For squared-law detection of $\rho_f$, the importance of the tracking monochromator is highlighted

+by two limits of Equation \ref{eq:rho_f_int_freq}:

+\begin{itemize}

+	\item When the transient is not frequency resolved, $\text{sig} \approx \int{\left|

+        \tilde{\rho}_f(\omega) \right|^2 d\omega}$ and the measured line shape will be the

+    convolution of the pulse envelope and the intrinsic (Green's function) response (Fig.

+    \ref{fig:fid_detuning}b, magenta).

+	\item When the driven frequency is isolated, $\text{sig} \approx \left|

+      \tilde{\rho}_f(\kappa_f\Omega_{fx}) \right|^2$ and the measured line shape will give the

+    un-broadened Green's function (Fig. \ref{fig:fid_detuning}b, teal).

+\end{itemize}

+Monochromatic detection can remove broadening effects due to the pulse bandwidth.  %

+For large $\Gamma_{10}\Delta_t$ values, FID evolution is negligible at all

+$\Omega_{fx}/\Delta_{\omega}$ values and the monochromator is not useful.  %

+Fig. \ref{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}

+

+\afterpage{

+\begin{figure}

+  \centering

+	\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{fig:pw1}

+\end{figure}

+\clearpage}

+

+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

+excitations).  %

+The spectral and temporal features of $\rho_1$  that are transferred to $\rho_2$ depend on when the

+subsequent pulse arrives.  %

+Time-gating later in $\rho_1$ evolution will produce responses with FID behavior, while time-gating

+$\rho_1$ in the presence of the initial pulse will produce driven responses.  %

+An analogous relationship holds for $\rho_3$ with its source term $\rho_2$.  %

+As discussed above, signal that time-gates FID evolution gives narrower spectra than driven-gated

+signal.  %

+As a result, the spectra of even single Liouville pathways will change based on pulse delays.  %

+

+The final coherence will also be frequency-gated by the monochromator.  %

+The monochromator isolates signal at the fully driven frequency $\omega_\text{out} = \omega_1$.  %

+The monochromator will induce line-narrowing to the extent that FID takes place.  %

+It effectively enforces a frequency constraint that acts as an additional resonance condition,

+$\omega_\text{out}=\omega_1$.  %

+The driven frequency will be $\omega_1$ if $E_1$ is the last pulse interaction (time-orderings V

+and VI), and the monochromator tracks the coherence frequency effectively.  %

+If $E_1$ is not the last interaction, the output frequency may not be equal to the driven

+frequency, and the monochromator plays a more complex role.  %

+

+

+We demonstrate this delay dependence using the multidimensional response of the I$\gamma$ Liouville

+pathway as an example (see Fig. \ref{fig:WMELs}).  %

+Fig. \ref{fig:pw1} shows the resulting 2D delay profile of pathway I$\gamma$ signals for

+$\Gamma_{10}\Delta_t=1$ (left) and the corresponding $\omega_1, \omega_2$ 2D spectra at several

+pulse delay values (right).  %

+The spectral changes result from changes in the relative importance of driven and FID

+components.  %

+The prominence of FID signal can change the resonance conditions; Table \ref{tab:table2} summarizes

+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.\footnote{

+	See Supplementary Fig. S5 for a representation of Fig. 5 simulated without monochromator frequency filtering ($M(\omega-\omega_1)=1$ for Equation \ref{eq:S_tot}).

+}

+

+\begin{table*}

+  \centering

+	\caption{\label{tab:table2} Conditions for peak intensity at different pulse delays for pathway

+    I$\gamma$.}

+	\begin{tabularx}{0.7\linewidth}{c c | X X X X}

+		\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{tabularx}

+\end{table*}

+

+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.  %

+This spectrum strongly resembles the driven limit spectrum.  For this time-ordering, the first,

+second, and third density matrix elements have driven resonance conditions of

+$\omega_1=\omega_{10}$, $\omega_1-\omega_2=0$, and

+$\omega_1-\omega_2+\omega_{2^\prime}=\omega_{10}$, respectively.  %

+The second resonance condition causes elongation along the diagonal, and since

+$\omega_2=\omega_{2^\prime}$, the first and third resonance conditions are identical, effectively

+making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical elongation along

+$\omega_1=\omega_{10}$.  %

+

+

+The other three spectra of Fig. \ref{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

+remain the same as at pulse overlap (orange) but the last resonance is different.  %

+The final pulse, $E_{2^\prime}$, is latent and probes $\rho_2$ during its FID evolution after

+memory of the driven frequency is lost.  %

+There are two important consequences.  %

+Firstly, the third driven resonance condition is now approximated by

+$\omega_{2^\prime}=\omega_{10}$, which makes $\omega_1$ only singly resonant at

+$\omega_1=\omega_{10}$.  %

+Secondly, the driven portion of the signal frequency is determined only by the latent pulse:

+$\omega_{\text{out}}=\omega_{2^\prime}$.  %

+Since our monochromator gates $\omega_1$, we have the detection-induced correlation

+$\omega_1=\omega_{2^\prime}$.  %

+The net result is double resonance along $\omega_1=\omega_2$, and the vertical elongation of pulse

+overlap is strongly attenuated.  %

+

+At $\tau_{21}=2.4\Delta_t,\tau_{22^\prime}=0$ (upper right, purple), the first pulse $E_1$ precedes

+the latter two, which makes the two resonance conditions for the input fields

+$\omega_1=\omega_{10}$ and $\omega_2=\omega_{10}$.  %

+The signal depends on the FID conversion of $\rho_1$, which gives vertical elongation at

+$\omega_1=\omega_{10}$.  %

+Furthermore, $\rho_1$ has no memory of $\omega_1$ when $E_2$ interacts, which has two important

+implications.  %

+First, this means the second resonance condition $\omega_1=\omega_2$ and the associated diagonal

+elongation is now absent.  %

+Second, the final output polarization frequency content is no longer functional of $\omega_1$.

+Coupled with the fact that $E_2$ and $E_{2^\prime}$ are coincident, so that the final coherence can

+be approximated as driven by these two, we can approximate the final frequency as

+$\omega_{\text{out}} = \omega_{10}-\omega_2+\omega_{2^\prime} = \omega_{10}$.  %

+Surprisingly, the frequency content of the output is strongly independent of all pulse

+frequencies.  %

+The monochromator narrows the $\omega_1=\omega_{10}$ resonance.  %

+The $\omega_1=\omega_{10}$ resonance condition now depends on the monochromator slit width, the FID

+propagation of $\rho_1$, the spectral bandwidth of $\rho_3$; its spectral width is not easily

+related to material parameters.  %

+This resonance demonstrates the importance of the detection scheme for experiments and how the

+optimal detection can change depending on the pulse delay time.  %

+

+Finally, when all pulses are well-separated ($\tau_{21}=-\tau_{22^\prime}=2.4\Delta_t$, upper left,

+cyan), each resonance condition is independent and both $E_1$ and $E_2$ require FID buildup to

+produce final output.  %

+The resulting line shape is narrow in all directions.  %

+Again, the emitted frequency does not depend on $\omega_1$, yet the monochromator resolves the

+final coherence at frequency $\omega_1$.  %

+Since the driven part of the final interaction comes from $E_{2^\prime}$, and since the

+monochromator track $\omega_1$, the output signal will increase when

+$\omega_1=\omega_{2^\prime}$.  %

+As a result, the line shape acquires a diagonal character.  %

+

+The changes in line shape seen in Fig. \ref{fig:pw1} have significant ramifications for the

+interpretations and strategies of MR-CMDS in the mixed domain.  %

+Time-gating has been used to isolate the 2D spectra of a certain time-ordering\cite{Meyer2004,

+  Pakoulev2006,Donaldson2007}, but here we show that time-gating itself causes significant line

+shape changes to the isolated pathways.  %

+The phenomenon of time-gating can cause frequency and delay axes to become functional of each other

+in unexpected ways.  %

+

+\subsection{Temporal pathway discrimination}

+

+\afterpage{

+\begin{figure}

+  \centering

+	\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 Equation \ref{eq: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{fig:delay_purity}

+\end{figure}

+\clearpage}

+

+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

+pathways present in our three-beam experiment (see Fig. \ref{fig:WMELs}).  %

+Each time-ordering has different resonance conditions.  %

+When signal is collected near pulse overlap, multiple time-orderings contribute.  %

+To identify these effects, we start by considering how strongly time-orderings are isolated at each

+delay coordinate.  %

+

+While the general idea of using time delays to enhance certain time-ordered regions is widely

+applied, quantitation of this discrimination is rarely explored.  %

+Because the temporal profile of the signal is dependent on both the excitation pulse profile and

+the decay dynamics of the coherence itself, quantitation of pathway discrimination requires

+simulation.  %

+

+Fig. \ref{fig:delay_purity} shows the 2D delay space with all pathways present for

+$\omega_1,\omega_2=\omega_{10}$.  %

+It illustrates the interplay of pulse width and system decay rates on the isolation of time-ordered

+pathways.  %

+The color bar shows the signal amplitude.  %

+Signal is symmetric about the $\tau_{21}=\tau_{22^\prime}$ line because when $\omega_1=\omega_2$,

+$E_1$ and $E_{2^\prime}$ interactions are interchangeable:

+$S_\text{tot}(\tau_{21},\tau_{22^\prime})=S_\text{tot}(\tau_{22^\prime}, \tau_{21})$.  %

+The overlaid black contours represent signal ``purity,'' $P$, defined as the relative amount of

+signal that comes from the dominant pathway at that delay value:

+\begin{equation}\label{eq:P}

+P(\tau_{21},\tau_{22^\prime})=\frac{\max \left\{S_L\left( \tau_{21},\tau_{22^\prime} \right)\right\}}

+{\sum_L S_L\left( \tau_{21},\tau_{22^\prime} \right)}.

+\end{equation}

+The dominant pathway ($\max{\left\{ S_L \left( \tau_{21},\tau_{22^\prime} \right) \right\}}$) at

+given delays can be inferred by the time-ordered region defined in Fig. \ref{fig:overview}d.  %

+The contours of purity generally run parallel to the time-ordering boundaries with the exception of

+time-ordered regions II and IV, which involve the double quantum coherences that have been

+neglected.  %

+

+A commonly-employed metric for temporal selectivity is how definitively the pulses are ordered.  %

+This metric agrees with our simulations.  %

+The purity contours have a weak dependence on $\Delta_t \Gamma_{10}$ for

+$\left|\tau_{22^\prime}\right|/\Delta_t < 1$ or $\left|\tau_{21}\right|/\Delta_t < 1$ where there

+is significant pathway overlap and a stronger dependence at larger values where the pathways are

+well-isolated.  %

+Because responses decay exponentially, while pulses decay as Gaussians, there always exist delays

+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}

+

+\afterpage{

+\begin{figure}

+  \centering

+	\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{fig:hom_2d_spectra}

+\end{figure}

+\clearpage}

+

+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

+over a span of $\tau_{21}$ delay times that include all pathways.  %

+It is a common practice to explore spectral evolution against $\tau_{21}$ because this delay axis

+shows population evolution in a manner analogous to pump-probe spectroscopies.  %

+The $\vec{k}_2$ and $\vec{k}_{2^\prime}$ interactions correspond to the pump, and the $\vec{k}_1$

+interaction corresponds to the probe.  %

+Time-orderings V and VI are the normal pump-probe time-orderings, time-ordering III is a mixed

+pump-probe-pump ordering (so-called pump polarization coupling), and time-ordering I is the

+probe-pump ordering (so-called perturbed FID).  %

+Scanning $\tau_{21}$ through pulse overlap complicates interpretation of the line shape due to the

+changing nature and balance of the contributing time-orderings.  %

+At $\tau_{21}>0$, time-ordering I dominates; at $\tau_{21}=0$, all time-orderings contribute

+equally; at $\tau_{21}<0$ time-orderings V and VI dominate (Fig. \ref{fig:delay_purity}).  %

+Conventional pump-probe techniques recognized these complications long ago,\cite{BritoCruz1988,

+  Palfrey1985} but the extension of these effects to MR-CMDS has not previously been done.  %

+

+Fig. \ref{fig:hom_2d_spectra} shows the MR-CMDS spectra, as well as histograms of the pathway

+weights, while scanning $\tau_{21}$ through pulse overlap.  %

+The colored histogram bars and line shape contours correspond to different values of the relative

+dephasing rate, $\Gamma_{10}\Delta_t$.  %

+The contour is the half-maximum of the line shape.\footnote{Supplementary Fig. S6 shows fully

+  colored contour plots of each 2D frequency spectrum.}  The dependence of the line shape amplitude

+on $\tau_{21}$ can be inferred from Fig. \ref{fig:delay_purity}.  %

+

+The qualitative trend, as $\tau_{21}$ goes from positive to negative delays, is a change from

+diagonal/compressed line shapes to much broader resonances with no correlation ($\omega_1$ and

+$\omega_2$ interact with independent resonances).  %

+Such spectral changes could be misinterpreted as spectral diffusion, where the line shape changes

+from correlated to uncorrelated as population time increases due to system dynamics.  %

+The system dynamics included here, however, contain no structure that would allow for such

+diffusion.  %

+Rather, the spectral changes reflect the changes in the majority pathway contribution, starting

+with time-ordering I pathways, proceeding to an equal admixture of I, III, V, and VI, and finishing

+at an equal balance of V and VI when $E_1$ arrives well after $E_2$ and $E_{2^\prime}$.  %

+Time-orderings I and III both exhibit a spectral correlation in $\omega_1$ and $\omega_2$ when

+driven, but time-orderings V and VI do not.  %

+Moreover, such spectral correlation is forced near zero delay because the pulses time-gate the

+driven signals of the first two induced polarizations.  %

+The monochromator detection also plays a dynamic role, because time-orderings V always VI always

+emit a signal at the monochromator frequency, while in time-orderings I and III the emitted

+frequency is not defined by $\omega_1$, as discussed above.  %

+

+When we isolate time-orderings V and VI, we can maintain the proper scaling of FID bandwidth in the

+$\omega_1$ direction because our monochromator can gate the final coherence.  %

+This gating is not possible in time-orderings I and III because the final coherence frequency is

+determined by $\omega_{2^\prime}$ which is identical to $\omega_2$.  %

+

+There are differences in the line shapes for the different values of the relative dephasing rate,

+$\Gamma_{10}\Delta_t$.  %

+The spectral correlation at $\tau_{21}/\Delta_t=2$ decreases in strength as $\Gamma_{10}\Delta_t$

+decreases.  %

+As we illustrated in Fig. \ref{fig:pw1}, this spectral correlation is a signature of driven signal

+from temporal overlap of $E_1$ and $E_2$; the loss of spectral correlation reflects the increased

+prominence of FID in the first coherence as the field-matter interactions become more impulsive.  %

+This increased prominence of FID also reflects an increase in signal strength, as shown by

+$\tau_{21}$ traces in Fig. \ref{fig:delay_purity}.  %

+When all pulses are completely overlapped, ($\tau_{21}=0$), each of the line shapes exhibit

+spectral correlation.  %

+At $\tau_{21}/\Delta_t=-2$, the line shape shrinks as $\Gamma_{10}\Delta_t$ decreases, with the

+elongation direction changing from horizontal to vertical.  %

+The general shrinking reflects the narrowing homogeneous linewidth of the $\omega_{10}$

+resonance.  %

+In all cases, the horizontal line shape corresponds to the homogeneous linewidth because the narrow

+bandpass monochromator resolves the final $\omega_1$ resonance.  %

+The change in elongation direction is due to the resolving power of $\omega_2$.  %

+At $\Gamma_{10}\Delta_t=0.5$, the resonance is broader than our pulse bandwidth and is fully

+resolved vertically.  %

+It is narrower than the $\omega_1$ resonance because time-orderings V and VI interfere to isolate

+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 $\Gamma_{10}\Delta_t=0.5$.  %

+

+\afterpage{

+\begin{figure}

+  \centering

+	\includegraphics[width=\linewidth]{"mixed_domain/wigners"}

+	\caption[Wigners.]{

+		Transient ($\omega_1$) line shapes and their dependence on $\omega_2$ frequency.  

+		The relative dephasing rate is $\Gamma_{10}\Delta_t=1$ and $\tau_{22^\prime}=0$.  

+		For each plot, the corresponding $\omega_2$ value is shown as a light gray vertical line.}

+	\label{fig:wigners}

+\end{figure}

+\clearpage}

+

+It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is

+frequency.\cite{Kohler2014, Aubock2012,Czech2015,Pakoulev2007}  %

+In Fig. \ref{fig:wigners} we show five $\tau_{21},\omega_1$ plots for varying $\omega_2$ with

+$\tau_{22^\prime}=0$.\footnote{See Supplementary Fig. S8 for Wigner plots for all

+  $\Gamma_{10}\Delta_t$ values.}  %

+The plots are the analogue to the most common multidimensional experiment of Transient Absorption

+spectroscopy, where the non-linear probe spectrum is plotted as a function of the pump-probe

+delay.  %

+For each plot, the $\omega_2$ frequency is denoted by a vertical gray line.  %

+Each Wigner plot is scaled to its own dynamic range to emphasize the dependence on $\omega_2$.  %

+The dramatic line shape changes between positive and negative delays can be seen.  %

+This representation also highlights the asymmetric broadening of the $\omega_1$ line shape near

+pulse overlap when $\omega_2$ becomes non-resonant.  %

+Again, these features can resemble spectral diffusion even though our system is homogeneous.  %

+

+\subsection{Inhomogeneous broadening}

+

+\afterpage{

+\label{sec:res_inhom}

+\begin{figure}

+  \centering

+	\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 Equation \ref{eq: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{fig:delay_inhom}

+\end{figure}

+\clearpage}

+

+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.  %

+In delay space, this rephasing appears as a shift of signal to time-ordered regions III and V that

+persists for all population times.  %

+Fig. \ref{fig:delay_inhom} shows the calculated spectra for relative dephasing rate

+$\Gamma_{10}\Delta_t=1$ with a frequency broadening function of width

+$\Delta_{\text{inhom}}=0.441\Gamma_{10}$.  %

+The inhomogeneity makes it easier to temporally isolate the rephasing pathways and harder to

+isolate the non-rephasing pathways, as shown by the purity contours.  %

+

+A common metric of rephasing in delay space is the 3PEPS

+measurement.\cite{Weiner1985,Fleming1998,Boeij1998,Salvador2003}  %

+In 3PEPS, one measures the signal as the first coherence time, $\tau$, is scanned across both

+rephasing and non-rephasing pathways while keeping population time, $T$, constant.  %

+The position of the peak is measured; a peak shifted away from $\tau=0$ reflects the rephasing

+ability of the system.  %

+An inhomogeneous system will emit a photon echo in the rephasing pathway, enhancing signal in the

+rephasing time-ordering and creating the peak shift.  %

+In our 2D delay space, the $\tau$ trace can be defined if we assume $E_2$ and $E_{2^\prime}$ create

+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 $T$.\footnote{

+	See Supplementary Fig. S9 for an illustration of how 3PEPS shifts are measured from a 2D delay

+  plot.}  %

+In our 2D delay plots (Fig. \ref{fig:delay_purity}, Fig. \ref{fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.\footnote{

+	Supplementary Fig. S10 shows the 3PEPS measurements of all 12 combinations of

+  $\Gamma_{10}\Delta_t$ and $\Delta_{\text{inhom}}$, for every population delay surveyed.}  %

+Fig. \ref{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.  %

+

+The unanticipated feature of the 3PEPS analysis is the dependence on $T$.  % 

+Even though our inhomogeneity is static, the peak shift is maximal at $T=0$ and dissipates as $T$

+increases, mimicking spectral diffusion.  %

+This dynamic arises from signal overlap with time-ordering III, which uses $E_2$ and $E_1$ as the

+first two interactions ,and merely reflects $E_1$ and $E_2$ temporal overlap.  %

+At $T=0$, the $\tau$ trace gives two ways to make a rephasing pathway (time-orderings III and V)

+and only one way to make a non-rephasing pathway (time-ordering VI).  %

+This pathway asymmetry shifts signal away from $\tau=0$ into the rephasing direction.  %

+At large $T$ (large $\tau_{21}$), time-ordering III is not viable and pathway asymmetry

+disappears.  %

+Peak shifts imply inhomogeneity only when time-orderings V and III are minimally contaminated by

+each other i.e. at population times that exceed pulse overlap.  %

+This fact is easily illustrated by the dynamics of homogeneous system (Fig.

+\ref{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{DeBoeij1996,Agarwal2002} 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.  %

+

+\afterpage{

+\begin{figure}

+  \centering

+	\includegraphics[width=\linewidth]{"mixed_domain/inhom spectral evolution"}

+	\caption[Spectral evolution of an inhomogenious system.]{

+		Same as Fig. \ref{fig:hom_2d_spectra}, but each system has inhomogeneity

+    ($\Delta_{\text{inhom}}=0.441\Gamma_{10}$).

+		Relative dephasing rates are $\Gamma_{10}\Delta_t=0.5$ (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 shapes of all three

+    systems are overlaid.

+		Each 2D plot shows a single representative contour (half-maximum) for each

+    $\Gamma_{10}\Delta_t$ value.

+		The colored histograms below each 2D frequency plot show the relative weights of each

+    time-ordering for each 2D frequency plot.  

+		In contrast to Fig. \ref{fig:hom_2d_spectra}, inhomogeneity makes the relative contributions of

+    time-orderings V and VI unequal.

+	}

+	\label{fig:inhom_2d_spectra}

+\end{figure}

+\clearpage}

+

+In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous

+broadening.  %

+Fig. \ref{fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous

+distribution.\footnote{As in Fig. \ref{fig:hom_2d_spectra}, Fig. \ref{fig:inhom_2d_spectra} shows

+  only the contours at the half-maximum amplitude.  See Supplementary Fig. S7 for all contours.}  %

+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. Fig. \ref{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}

+

+Our chosen values of the relative dephasing time, $\Gamma_{10}\Delta_t$, describe experiments where

+neither the impulsive nor driven limit unilaterally applies.  %

+We have illustrated that in this intermediate regime, the multidimensional spectra contain

+attributes of both limits, and that it is possible to judge when these attributes apply.  %

+In our three-pulse experiment the second and third pulses time-gate coherences and populations

+produced by the previous pulse(s), and the monochromator frequency-gates the final coherence.  %

+Time-gating isolates different properties of the coherences and populations. Consequently, spectra

+evolve against delay.  %

+For any delay coordinate, one can develop qualitative line shape expectations by considering the following three principles:

+\begin{enumerate}

+	\item When time-gating during the pulse, the system pins to the driving frequency with a buildup efficiency determined by resonance.  

+	\item When time-gating after the pulse, the FID dominates the system response.

+	\item The emitted signal field contains both FID and driven components; the $\omega_{\text{out}} = \omega_1$ component is isolated by the tracking monochromator.

+\end{enumerate}

+Fig. \ref{fig:fid_dpr} illustrates principles 1 and 2 and Fig. \ref{fig:fid_detuning} illustrates

+principle 2 and 3.  %

+Fig. \ref{fig:pw1} provides a detailed example of the relationship between these principles and the

+multidimensional line shape changes for different delay times.  %

+

+The principles presented above apply to a single pathway.  %

+For rapidly dephasing systems it is difficult to achieve complete pathway discrimination, as shown

+in Fig. \ref{fig:delay_purity}.  %

+In such situations the interference between pathways must be considered to predict the line

+shape.  %

+The relative weight of each pathway to the interference can be approximated by the extent of pulse

+overlap.  %

+The pathway weights exchange when scanning across pulse overlap, which creates the dramatic line

+shape changes observed in Figs. \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}.  %

+

+\subsection{Conditional validity of the driven limit}

+

+We have shown that the driven limit misses details of the line shape if $\Gamma_{10} \Delta_t

+\approx 1$, but we have also reasoned that in certain conditions the driven limit can approximate

+the response well (see principle 1).  %

+Here we examine the line shape at delay values that demonstrate this agreement.  %  

+Fig. \ref{fig:steady_state} compares the results of our numerical simulation (third column) with

+the driven limit expressions for populations where $\Gamma_{11}\Delta_t=0$ (first column) or $1$

+(second column).  %

+The top and bottom rows compare the line shapes when $\left(\tau_{22^\prime},

+  \tau_{21}=(0,0)\right)$ and $(0,-4\Delta_t)$, respectively.  %

+The third column demonstrates the agreement between the driven limit approximations with the

+simulation by comparing the diagonal and anti-diagonal cross-sections of the 2D spectra.  %

+

+% TODO:  [ ] population resonance is not clear

+Note the very sharp diagonal feature that appears for $(\tau_{21},\tau_{22^\prime}) = (0,0)$ and $

+\Gamma_{11}=0$; this is due to population resonance in time-orderings I and III.  %

+This expression is inaccurate:  the narrow resonance is only observed when pulse durations are much

+longer than the coherence time.  %

+A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes (Yurs

+\textit{et al.}\cite{Yurs2011} and Kohler \textit{et al.}\cite{Kohler2014}, respectively)

+demonstrates this difference well.  %

+The driven equation fails to reproduce our numerical simulations here because resonant excitation

+of the population is impulsive; the experiment time-gates only the rise time of the population, yet

+driven theory predicts the resonance to be vanishingly narrow ($\Gamma_{11}=0$).  %

+In light of this, one can approximate this time-gating effect by substituting population lifetime

+with the pulse duration ($\Gamma_{11}\Delta_t=1$), which gives good agreement with the numerical

+simulation (third column).  %

+

+When $\tau_{22^\prime}=0$ and $\tau_{21}<\Delta_t$, signals can also be approximated by driven

+signal (Fig. \ref{fig:steady_state} bottom row).  %

+Only time-orderings V and VI are relevant.  %

+The intermediate population resonance is still impulsive but it depends on

+$\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %

+

+\afterpage{

+\begin{figure}

+  \centering

+	\includegraphics[width=\linewidth]{"mixed_domain/steady state"}

+	\caption[Conditional validity of the driven limit.]{

+		Comparing approximate expressions of the 2D frequency response with the directly integrated

+    response.  %

+		$\Gamma_{10}\Delta_t=1$.  %

+		The top row compares the 2D response of all time-orderings ($\tau_{21}=0$) and the bottom row

+    compares the response of time-orderings V and VI ($\tau_{21}=-4\Delta_t$).  %

+		First column:  The driven limit response.  Note the narrow diagonal resonance for $\tau_{21}=0$.

+		Second column:  Same as the first column, but with ad hoc substitution $\Gamma_{11}=\Delta_t$.  

+		Third column:  The directly integrated response.  %

+	}

+	\label{fig:steady_state}

+\end{figure}

+\clearpage}

+

+\subsection{Extracting true material correlation}

+

+\afterpage{

+\begin{figure}

+  \centering

+	\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{fig:metrics}

+\end{figure}

+\clearpage}

+

+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

+ellipticity\footnote{

+	There are many ways to characterize the ellipticity of a peak shape.  

+	We adopt the convention $\mathcal{E} = \left(a^2-b^2\right) / \left(a^2+b^2\right)$, where $a$ is the diagonal width and $b$ is the antidiagonal width.} 

+for the frequency domain\cite{Kwac2003,Okumura1999}.  %

+In the driven (impulsive) limit, ellipticity (3PEPS) corresponds to the frequency correlation

+function and uniquely extracts the inhomogeneity of the models presented here.  %

+In their respective limits, the metrics give values proportional to the inhomogeneity.  % 

+

+Fig. \ref{fig:metrics} shows the results of this characterization for all $\Delta_\text{inhom}$ and

+$\Gamma_{10}\Delta_t$ values explored in this work.  %

+We study how the correlations between the two metrics depend on the relative dephasing rate, $\Gamma_{10}\Delta_t$, the absolute inhomogeneity, $\Delta_\text{inhom} / \Delta_\omega$, the relative inhomogeneity $\Delta_\text{inhom} / \Gamma_{10}$, and the population time delay.\footnote{

+	The simulations for each value of the 3PEPS and ellipticity data in Fig. \ref{fig:metrics} appear in Supplementary Figs. S10-S12.}  

+The top row shows the correlations of the $\Delta_\text{3PEPS} / \Delta_t$ 3PEPS metric that

+represents the normalized coherence delay time required to reach the peak intensity.  %

+The upper right graph shows the correlations for a population time delay of $T = 4\Delta_t$ that

+isolates the V and VI time-orderings.  %

+For this time delay, the $\Delta_\text{3PEPS} / \Delta_t$ metric works well for all dephasing times

+of $\Gamma_{10}\Delta_t$ when the relative inhomogeneity is $\Delta_\text{inhom} / \Delta_\omega

+\ll 1$.  %

+It becomes independent of $\Delta_\text{inhom} / \Delta_\omega$ when $\Delta_\text{inhom} /

+\Delta_\omega > 1$.  %

+This saturation results because the frequency bandwidth of the excitation pulses becomes smaller

+than the inhomogeneous width and only a portion of the inhomogeneous ensemble contributes to the

+3PEPS experiment.\cite{Weiner1985}  %

+The corresponding graph for $T = 0$ shows a large peak shift occurs, even without inhomogeneity. 

+In this case, the peak shift depends on pathway overlap, as discussed in Section

+\ref{sec:res_inhom}.  %

+

+The middle row in Fig. \ref{fig:metrics} shows the ellipticity dependence on the relative dephasing

+rate and inhomogeneity assuming the measurement is performed when the first two pulses are

+temporally overlapped ($\tau_{22^\prime}=0$).  %

+For a $T=4\Delta_t$ population time, the ellipticity is proportional to the inhomogeneity until

+$\Delta_\text{inhom} / \Delta_\omega \ll 1$ where the excitation bandwidth is wide compared with

+the inhomogeneity.  %

+Unlike 3PEPS, saturation is not observed because pulse bandwidth does not limit the frequency range

+scanned.  %

+The 3PEPS and ellipticity metrics are therefore complementary since 3PEPS works well for

+$\Delta_\text{inhom} / \Delta_\omega \ll 1$ and ellipticity works well for $\Delta_\text{inhom} /

+\Delta_\omega \gg 1$.  %

+When all pulses are temporally overlapped at $T = 0$, the ellipticity is only weakly dependent on

+the inhomogeneity and dephasing rate.  %

+The ellipticity is instead dominated by the dependence on the excitation pulse frequency

+differences of time-orderings I and III that become important at pulse overlap.  %

+

+It is clear from the previous discussion that both metrics depend on the dephasing and

+inhomogeneity.  %

+The dephasing can be measured independently in the frequency or time domain, depending upon whether

+the dephasing is very fast or slow, respectively.  %

+In the mixed frequency/time domain, measurement of the dephasing becomes more difficult.  %

+One strategy to address this challenge is to use both the 3PEPS and ellipticity metrics.  %

+The bottom row in Fig. \ref{fig:metrics} plots 3PEPS against ellipticity to show how the

+relationship between the metrics changes for different amounts of dephasing and inhomogeneity.  %

+The anti-diagonal contours of constant relative inhomogeneity show that these metrics are

+complementary and can serve to extract the system correlation parameters.  %

+

+Importantly, the metrics are uniquely mapped both in the presence and absence of pulse-induced

+effects (demonstrated by $T = 0$ and $T = 4\Delta_t$, respectively).  %

+The combined metrics can be used to determine correlation at $T = 0$, but the correlation-inducing

+pulse effects give a mapping significantly different than at $T = 4\Delta_t$.  %

+At $T = 0$, 3PEPS is almost nonresponsive to inhomogeneity; instead, it is an almost independent

+characterization of the pure dephasing.  %

+In fact, the $T=0$ trace is equivalent to the original photon echo traces used to resolve pure

+dephasing rates.\cite{Aartsma1976}  %

+Both metrics are offset due to the pulse overlap effects.  %

+Accordingly, the region to the left of homogeneous contour is non-physical, because it represents

+observed correlations that are less than that given by pulse overlap effects.  %

+If the metrics are measured as a function of $T$, the mapping gradually changes from the left

+figure to the right figure in accordance with the pulse overlap.  %

+Both metrics will show a decrease, even with static inhomogeneity.  %

+If a system has spectral diffusion, the mapping at late times will disagree with the mapping at

+early times; both ellipticity and 3PEPS will be smaller at later times than predicted by the change

+in mappings alone.  %

+

+\section{Conclusion}  % ---------------------------------------------------------------------------

+

+This study provides a framework to describe and disentangle the influence of the excitation pulses

+in mixed-domain ultrafast spectroscopy.  %

+We analyzed the features of mixed domain spectroscopy through detailed simulations of MR-CMDS

+signals.  %

+When pulse durations are similar to coherence times, resolution is compromised by time-bandwidth

+uncertainty and the complex mixture of driven and FID response.  %

+The dimensionless quantity $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)$ captures the

+balance of driven and FID character in a single field-matter interaction.  %

+In the nonlinear experiment, with multiple field-matter interactions, this balance is also

+controlled by pulse delays and frequency-resolved detection.  %

+Our analysis shows how these effects can be intuitive.  %

+

+The dynamic nature of pulse effects can lead to misleading changes to spectra when delays are

+changed.  %

+When delays separate pulses, the spectral line shapes of individual pathways qualitatively change

+because the delays isolate FID contributions and de-emphasize driven response.  %

+When delays are scanned across pulse overlap, the weights of individual pathways change, further

+changing the line shapes.  %

+In a real system, these changes would all be present in addition to actual dynamics and spectral

+changes of the material.  %

+

+Finally, we find that, in either frequency or time domain, pulse effects mimic signatures of

+ultrafast inhomogeneity.  %

+Even homogeneous systems take on these signatures.  %

+For mixed domain experiments, pulse effects induce spectral ellipticity and photon echo signatures,

+even in homogeneous systems.  %

+Driven character gives rise to pathway overlap peak shifting in the 2D delay response, which

+artificially produces rephasing near pulse overlap.  %

+Driven character also produces resonances that depend on $\omega_1-\omega_2$ near pulse overlap.  %

+Determination of the homogeneous and inhomogeneous broadening at ultrashort times is only possible

+by performing correlation analysis in both the frequency and time domain.  %
\ No newline at end of file
diff --git a/mixed_domain/metrics.pdf b/mixed_domain/metrics.pdf
new file mode 100644
index 0000000..a2640c3
--- /dev/null
+++ b/mixed_domain/metrics.pdf
diff --git a/mixed_domain/spectral evolution full.pdf b/mixed_domain/spectral evolution full.pdf
new file mode 100644
index 0000000..f854286
--- /dev/null
+++ b/mixed_domain/spectral evolution full.pdf
diff --git a/mixed_domain/steady state.pdf b/mixed_domain/steady state.pdf
new file mode 100644
index 0000000..effbb82
--- /dev/null
+++ b/mixed_domain/steady state.pdf
diff --git a/mixed_domain/wigners full.pdf b/mixed_domain/wigners full.pdf
new file mode 100644
index 0000000..7816277
--- /dev/null
+++ b/mixed_domain/wigners full.pdf
diff --git a/mixed_domain/wigners.pdf b/mixed_domain/wigners.pdf
new file mode 100644
index 0000000..903abdf
--- /dev/null
+++ b/mixed_domain/wigners.pdf
diff --git a/procedures/chapter.tex b/procedures/chapter.tex
index 3cb42bf..4ac67ed 100644
--- a/procedures/chapter.tex
+++ b/procedures/chapter.tex
@@ -1 +1,33 @@
-\chapter{Procedures}
\ No newline at end of file
+\chapter{Procedures}
+
+\section{Aligining TOPAS-C}
+
+\section{Aligning Spitfire PRO}
+
+\section{Air Handling}
+
+\section{Six Month Maintenance}
+
+\section{Tuning MicroHR Monochromator}
+
+Visible Grating.  %
+
+Align the HeNe as perpendicular as possible to the monochromator entrance slit.  %
+
+Move the grating angle until the HeNe falls on the exit slit.  %
+
+Shine a flashlight through the entrance slit and observe the colour on the exit slit: if white,
+then you are at 0-order (0 nm), if red, then you are at 1st order (632.8 nm).  %
+
+Go to 0-order, narrow the slits, and slowly adjust the angle until the HeNe is going through the
+exit slit.  %
+
+Go to Jovin Yvon/utilities and find the motor configuration program.  %
+
+In the Gratings tab, select the 1st grating (1200 line density) and hit Calibrate.  %
+
+In theoretical wavelength, enter 0 nm.  %
+
+In experimental wavelength, enter the wavelength you observe from the control program.  %
+
+Hit set. %