From cd162fef9d9f3145c1e29c63439759636ba62c41 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 26 Feb 2018 17:08:07 -0600
Subject: 2018-02-26 17:07

---
 mixed_domain/chapter.tex | 100 +++++++++++++++--------------------------------
 1 file changed, 32 insertions(+), 68 deletions(-)

(limited to 'mixed_domain/chapter.tex')

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index 308146b..7f8a8b4 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -137,9 +137,7 @@ from these measurement artifacts.  %
 
 \section{Theory}
 
-\afterpage{
-\begin{figure}
-  \centering
+\begin{dfigure}
 	\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}
 	\caption[Sixteen triply-resonant Liouville pathways.]{
 		The sixteen triply-resonant Liouville pathways for the third-order response of the system used
@@ -149,8 +147,7 @@ from these measurement artifacts.  %
     are yellow, excitations with $\omega_2=\omega_{2'}$ are purple, and the final emission is gray.
 	}
 	\label{fig:WMELs}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 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
@@ -234,8 +231,7 @@ this paper.  %
 The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discussed in Appendix
 \ref{sec:cw_imp}.  %
 
-\afterpage{
-\begin{figure}
+\begin{dfigure}
 	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}
 	\caption[Overview of the MR-CMDS simulation.]{
 		Overview of the MR-CMDS simulation. 
@@ -253,8 +249,7 @@ 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}
-\clearpage}
+\end{dfigure}
 
 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
@@ -491,8 +486,7 @@ The driven limit holds for large detunings, regardless of delay.  %
 
 \subsection{Convolution Technique for Inhomogeneous Broadening}\label{sec:convolution}
 
-\afterpage{
-\begin{figure}
+\begin{dfigure}
 	\includegraphics[width=\linewidth]{mixed_domain/convolve}
   \caption[Convolution overview.]
   {Overview of the convolution.
@@ -501,8 +495,7 @@ The driven limit holds for large detunings, regardless of delay.  %
     (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}
+\end{dfigure}
 
 Here we describe how to transform the data of a single reference oscillator signal to that of an
 inhomogeneous distribution.  %
@@ -588,8 +581,7 @@ pulse delay times, and inhomogeneous broadening.  %
 
 \subsection{Evolution of single coherence}\label{sec:evolution_SQC}
 
-\afterpage{
-\begin{figure}
+\begin{dfigure}
   \centering
 	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
 	\caption[Relative importance of FID and driven response for a single quantum coherence.]{
@@ -601,8 +593,7 @@ pulse delay times, and inhomogeneous broadening.  %
     slightly detuned (relative detuning, $\Omega_{fx}/\Delta_{\omega}=0.1$).
 	}
 	\label{fig:fid_dpr}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x}
 \rho_1$, under various excitation conditions.  %
@@ -639,8 +630,7 @@ 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}
+\begin{dfigure}
   \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.]{
@@ -661,8 +651,7 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
 		In all plots, the gray line is the electric field amplitude. 
 	}
 	\label{fig:fid_detuning}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 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.
@@ -729,8 +718,7 @@ $\Gamma_{10}\Delta_t=1$.  %
 
 \subsection{Evolution of single Liouville pathway}
 
-\afterpage{
-\begin{figure}
+\begin{dfigure}
   \centering
 	\includegraphics[width=\linewidth]{"mixed_domain/pw1 lineshapes"}
 	\caption[2D frequency response of a single Liouville pathway at different delay values.]{
@@ -743,8 +731,7 @@ $\Gamma_{10}\Delta_t=1$.  %
     compare 2D spectrum frame color with dot color on 2D delay plot.
 		}
 	\label{fig:pw1}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 We now consider the multidimensional response of a single Liouville pathway involving three pulse
 interactions.  %
@@ -783,11 +770,10 @@ Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromat
 	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
+\begin{dtable}
 	\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}
+	\begin{tabular}{c c | c c c c}
 		\multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\
 		$\tau_{21}/\Delta_t$ & $\tau_{22^\prime}/\Delta_t$ & $\rho_0\xrightarrow{1}\rho_1$ &
     $\rho_1\xrightarrow{2}\rho_2$ & $\rho_2\xrightarrow{2^\prime}\rho_3$ & $\rho_3\rightarrow$
@@ -800,8 +786,8 @@ Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromat
     $\omega_1=\omega_{10}$ \\
 		2.4 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & $\omega_2=\omega_{10}$ &
     $\omega_1=\omega_2$ \\
-	\end{tabularx}
-\end{table*}
+	\end{tabular}
+\end{dtable}
 
 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.  %
@@ -875,9 +861,7 @@ in unexpected ways.  %
 
 \subsection{Temporal pathway discrimination}
 
-\afterpage{
-\begin{figure}
-  \centering
+\begin{dfigure}
 	\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
@@ -892,8 +876,7 @@ in unexpected ways.  %
     (purple), and III or I (teal).
 	}
 	\label{fig:delay_purity}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 In the last section we showed how a single pathway's spectra can evolve with delay due to pulse
 effects and time gating.  %
@@ -943,9 +926,7 @@ vanishing signal intensities; the contour of $P=0.99$ across our systems highlig
 
 \subsection{Multidimensional line shape dependence on pulse delay time}
 
-\afterpage{
-\begin{figure}
-  \centering
+\begin{dfigure}
 	\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
@@ -960,8 +941,7 @@ vanishing signal intensities; the contour of $P=0.99$ across our systems highlig
     $\tau_{22^\prime}=0$.
 	}
 	\label{fig:hom_2d_spectra}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 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
@@ -1036,17 +1016,14 @@ 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
+\begin{dfigure}
 	\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}
+\end{dfigure}
 
 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}  %
@@ -1063,12 +1040,9 @@ 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}
+\subsection{Inhomogeneous broadening} \label{sec:res_inhom}  % ------------------------------------
 
-\afterpage{
-\label{sec:res_inhom}
-\begin{figure}
-  \centering
+\begin{dfigure}
 	\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.  %
@@ -1083,8 +1057,7 @@ Again, these features can resemble spectral diffusion even though our system is
     (purple), III (teal, dashed), and I (teal, solid).  %
 	}
 	\label{fig:delay_inhom}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 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
@@ -1138,9 +1111,7 @@ 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
+\begin{dfigure}
 	\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
@@ -1157,8 +1128,7 @@ time-ordering III is decoupled by detuning.  %
     time-orderings V and VI unequal.
 	}
 	\label{fig:inhom_2d_spectra}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous
 broadening.  %
@@ -1241,9 +1211,7 @@ 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
+\begin{dfigure}
 	\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
@@ -1256,14 +1224,11 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
 		Third column:  The directly integrated response.  %
 	}
 	\label{fig:steady_state}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
-\subsection{Extracting true material correlation}
+\subsection{Extracting true material correlation}  % ----------------------------------------------
 
-\afterpage{
-\begin{figure}
-  \centering
+\begin{dfigure}
 	\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
@@ -1280,8 +1245,7 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
     area are connected).  %
 }
 	\label{fig:metrics}
-\end{figure}
-\clearpage}
+\end{dfigure}
 
 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.  %
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