From 10898eded280c3e7f072f5bd0fa881422b5b0733 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Thu, 10 May 2018 12:54:07 -0500
Subject: 2018-05-10 12:54

---
 mixed_domain/chapter.tex | 125 +++++++++++++++++++++++------------------------
 1 file changed, 61 insertions(+), 64 deletions(-)

(limited to 'mixed_domain/chapter.tex')

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index f95cf88..509faee 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -27,7 +27,7 @@ themselves. \cite{RentzepisPM1970a, MukamelShaul2000a}  %
 The ultrafast specta can be collected in the time domain or the frequency
 domain. \cite{ParkKisam1998a}  %
 
-Time-domain methods/ scan the pulse delays to resolve the free induction decay
+Time-domain methods scan the pulse delays to resolve the free induction decay
 (FID). \cite{GallagherSarahM1998a}   %
 The Fourier Transform of the FID gives the ultrafast spectrum.  %
 Ideally, these experiments are performed in the impulsive limit where FID dominates the
@@ -128,7 +128,7 @@ Here, the subscripts represent the excitation pulse frequencies, $\omega_1$ and
 \omega_{2'}$.  %
 These experimental conditions were recently used to explore line shapes of excitonic
 systems, \cite{KohlerDanielDavid2014a, CzechKyleJonathan2015a} and have been developed on
-vibrational states as well. \cite{MeyerKentA2004a}  %
+vibrational states as well \cite{MeyerKentA2004a}.  %
 Although MR-CMDS forms the context of this model, the treatment is quite general because the phase
 matching condition can describe any of the spectroscopies mentioned above with the exception of SFG
 and TRSF, for which the model can be easily extended.  %
@@ -146,18 +146,6 @@ from these measurement artifacts.  %
 
 \section{Theory}  % ===============================================================================
 
-\begin{figure}
-	\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
-    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{mix:fig:WMELs}
-\end{figure}
-
 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
 the temporal and spectral widths of the excitation pulses.  %
@@ -178,7 +166,6 @@ We neglect non-linear phase effects such as chirp so the FWHM of the frequency b
 transform limited:  $\Delta_{\omega}\Delta_t=4 \ln 2 \approx 2.77$, where $\Delta_{\omega}$ is the
 spectral FWHM (intensity scale).  %
 
-
 The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$:
 \begin{equation} \label{mix:eqn:LVN}
 \frac{d\bm{\rho}}{dt} = -\frac{i}{\hbar}\left[\bm{H_0} + \bm{\mu}\cdot \sum_{l=1,2,2^\prime} E_l(t), \bm{{\rho}}\right] + \bm{\Gamma \rho}.
@@ -240,26 +227,6 @@ this paper.  %
 % discussed in TODO
 % Appendix \ref{sec:cw_imp}.  %
 
-\begin{figure}
-	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}
-	\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
-    $\Gamma_{10}\Delta_t=1$.
-		(b) Simulated evolution of the density matrix elements of a third-order Liouville pathway
-    $V\gamma$ under fully resonant excitation. Pulses can be labeled both by their time of arrival
-    ($x$,$y$,$z$) and by the lab lasers used to stimulate the transitions ($2$,$2^\prime$,$1$). The
-    final coherence (teal) creates the output electric field.
-		(c) The frequency profile of the output electric field is filtered by a monochromator gating
-    function, $M(\omega)$, and the passed components (shaded) are measured.
-		(d-f) Signal is viewed  against two laser parameters, either as 2D delay (d), mixed
-    delay-frequency (e), or 2D frequency plots (f).  The six time-orderings are labeled in (d) to
-    help introduce our delay convention.
-	}
-	\label{mix:fig:overview}
-\end{figure}
-
 \autoref{mix:fig:overview} gives an overview of the simulations done in this work.  %
 \autoref{mix:fig:overview}a shows an excitation pulse (gray-shaded) and examples of a coherent
 transient for three different dephasing rates.  %
@@ -343,7 +310,39 @@ Table S1 summarizes the arguments for each Liouville pathway.  %
 \autoref{mix:fig:overview}f shows the 2D $(\omega_1, \omega_2)$ $S_{\text{tot}}$ spectrum resulting
 from the pulse delay times represented in \autoref{mix:fig:overview}b.  %
 
-\subsection{Characteristics of Driven and Impulsive Response} \label{mix:sec:cw_imp}  % -----------
+\begin{figure}
+	\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
+    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{mix:fig:WMELs}
+\end{figure}
+
+\begin{figure}
+	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}
+	\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
+    $\Gamma_{10}\Delta_t=1$.
+		(b) Simulated evolution of the density matrix elements of a third-order Liouville pathway
+    $V\gamma$ under fully resonant excitation. Pulses can be labeled both by their time of arrival
+    ($x$,$y$,$z$) and by the lab lasers used to stimulate the transitions ($2$,$2^\prime$,$1$). The
+    final coherence (teal) creates the output electric field.
+		(c) The frequency profile of [Eccentricity ]the output electric field is filtered by a
+    monochromator gating function, $M(\omega)$, and the passed components (shaded) are measured.
+		(d-f) Signal is viewed  against two laser parameters, either as 2D delay (d), mixed
+    delay-frequency (e), or 2D frequency plots (f).  The six time-orderings are labeled in (d) to
+    help introduce our delay convention.
+	}
+	\label{mix:fig:overview}
+\end{figure}
+
+\subsection{Characteristics of driven and impulsive response} \label{mix: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 \autoref{mix:eqn:rho_f_int} and
@@ -471,17 +470,6 @@ with a Gaussian distribution function.  %
 Further details of the convolution are in \autoref{mix:sec:convolution}.  %
 Dynamic broadening effects such as spectral diffusion are beyond the scope of this work.  %
 
-\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{mix:fig:convolution}
-\end{figure}
-
 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
@@ -558,6 +546,17 @@ which is a 1D convolution along the diagonal axis in frequency space.  %
 \autoref{mix:fig:convolution} demonstrates the use of \autoref{mix:eqn:convolve_final} on a
 homogeneous line shape.  %
 
+\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{mix:fig:convolution}
+\end{figure}
+
 \section{Methods}  % ==============================================================================
 
 A matrix representation of differential equations of the type in \autoref{mix:eqn:E_L_full} was
@@ -592,8 +591,7 @@ There are three details of our simulation strategy that deserve more exposition:
 Table \ref{tab:table1} describes the relationship between our notation and the parameters that make
 up the 16 Liouville pathways.  %
 
-\begin{table*}[!hb]
-	\caption{\label{tab:table1} Parameters of each Liouville Pathway.}
+\begin{table}
 	\begin{tabular}{l c | c c c  r r r  r r r  c c c c}
 		$L$ & Liouville Pathway
 		& $x$ & $y$ & $z$
@@ -687,7 +685,9 @@ up the 16 Liouville pathways.  %
 		& & &
 		& $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\
 	\end{tabular}
-\end{table*}
+ 	\caption{Parameters of each Liouville Pathway.}
+  \label{tab:table1}
+\end{table}
 
 \subsection{Matrix formulation}  % ----------------------------------------------------------------
 
@@ -986,15 +986,15 @@ In NISE, an \texttt{experiment} module is loaded to define the electric field va
 
 Within this work we have represented our data in terms of dimensionless quantities like $\tau/\Delta_t$ and $(\omega-\omega_{10})/\Delta_\omega$. The simulation within NISE was done with choices for each parameter, as tabulated below. These quantities are necessary to fully understand the unprocessed arrays generated by NISE.
 
-\begin{center}
-	\begin{tabular}{r l}
+\begin{table}
+  \begin{tabular}{r l}
 		$\omega_{10}$ & 7000 $\mathrm{cm}^{-1}$ \\
 		$\mu_{10}$ & 1 \\
 		$\mu_{21}$ & 1 \\
 		$\Gamma_{11}$ & 0 $\mathrm{fs}^{-1}$ \\
 		$\Delta_t$ & 50 fs
 	\end{tabular}
-\end{center}
+\end{table}
 
 $\Gamma_{10}$, $\Gamma_{21}$ and $\Gamma_{20}$ were kept equal. Their exact value for a given run of the simulation depends on the $\Gamma_{10}\Delta_t$ quantity as discussed in the paper. We use the term \texttt{dpr} (dephasing pulse ratio) which is the inverse of $\Gamma_{10}\Delta_t$.
 
@@ -1002,7 +1002,7 @@ In NISE the system parameters are contained within the \texttt{hamiltonian} modu
 
 To generate the raw data we calculated the polarization at all coordinates within a four-dimensional experimental array:
 
-\begin{center}
+\begin{table}
 	\begin{tabular}{c | c | c | c}
 		axis & center & full width & number of points \\ \hline
 		w1 & 7000 & 3000 & 41 \\
@@ -1010,11 +1010,11 @@ To generate the raw data we calculated the polarization at all coordinates withi
 		d1 & 0 & 400 & 21 \\
 		d2 & 0 & 400 & 21
 	\end{tabular}
-\end{center}
+\end{table}
 
 Arrays containing these points were assembled and passed into the \texttt{trive} module. These axes and \texttt{H0} were passed into the \texttt{scan} module for numerical integration. A single output array was saved for each scan. To keep the output array sizes reasonable a separate simulation was done for each $\Gamma_{10}\Delta_t$, time-ordering, and \texttt{d1} value. For each of these simulations we saved one five-dimensional array to an HDF5 file:
 
-\begin{center}
+\begin{table}
 	\begin{tabular}{c | c | c}
 		index & name & size \\ \hline
 		0 & w1 & 41 \\
@@ -1023,7 +1023,7 @@ Arrays containing these points were assembled and passed into the \texttt{trive}
 		3 & permutation & 2 \\
 		4 & timestep & variable \\
 	\end{tabular}
-\end{center}
+\end{table}
 
 The final index `timestep' contains the dependence of the output polarization on lab time. It changes from simulation to simulation to help with computation speed. For each simulation the timetep array is defined by a starting position (always 100 fs before the first pulse arrives) an ending position ($5 \times \tau_{10}$ fs after the last pulse arrives), and a step (4 fs for our longest dephasing time, 2 fs otherwise).
 
@@ -1033,7 +1033,7 @@ The output polarization is kept as a complex array in the lab frame.
 
 As mentioned in the appendix, we introduce inhomogeneity by convolving the output with a distribution function on the intensity level: `smearing'. This is done in the measurement stage. We store a separate HDF5 file for each combination of \texttt{dpr}, time ordering, and $\Delta_{\text{inhom}}$. Each HDF5 file contains four arrays:
 
-\begin{center}
+\begin{table}
 	\begin{tabular}{c | c | c}
 		alias & dimensions & shape \\ \hline
 		\texttt{w1} & w1 & \texttt{(41,)} \\
@@ -1042,7 +1042,7 @@ As mentioned in the appendix, we introduce inhomogeneity by convolving the outpu
 		\texttt{d2} & w2 & \texttt{(21,)} \\
 		\texttt{arr} & w1, w2, d1, d2 & \texttt{(41, 41, 21, 21)}
 	\end{tabular}
-\end{center}
+\end{table}
 
 The coordinate arrays are in their native units (fs, $\mathrm{cm}^{-1}$). The signal array is purely real, stored on the intensity level.
 
@@ -1061,7 +1061,6 @@ pulse delay times, and inhomogeneous broadening.  %
 \subsection{Evolution of single coherence} \label{mix:sec:evolution_SQC}  % -----------------------
 
 \begin{figure}
-  \centering
 	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
 	\caption[Relative importance of FID and driven response for a single quantum coherence.]{
 		The relative importance of FID and driven response for a single quantum coherence as a function
@@ -1110,7 +1109,6 @@ mainly driven, roughly equal driven and FID parts, and mainly FID components, re
 FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
 
 \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
@@ -1146,7 +1144,7 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
 
 \begin{figure}
 	\includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"}
-	\caption{
+	\caption[Time-gated amplitude of a single quantum coherence vs pulse detuning.]{
     Amplitude of a single quantum coherence under pulsed excitation as a function of detuning (x
     axis) and delay after excitation (line color, scale on right) for the three
     $\Gamma_{10}\Delta_t$ values considered in this work.
@@ -1246,7 +1244,6 @@ $\Gamma_{10}\Delta_t=1$.  %
 \subsection{Evolution of single Liouville pathway}  % ---------------------------------------------
 
 \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
@@ -1725,7 +1722,7 @@ time-ordering III is decoupled by detuning.  %
 
 \begin{figure}
 	\includegraphics[width=\textwidth]{"mixed_domain/2D frequences at -4"}
-	\label{mix:fig:2D_frequencies_at_-4}
+	\label{mix:fig:2D_frequencies_at large population time.at_-4}
 	\caption[Eccentricity at large population time.]{
     2D frequency scans at large $T$ ($\tau_{22^\prime}=0$, $\tau_{21}=-4\Delta_t$) for all 12
     combinations of $\Gamma_{10}$ (columns) and $\Delta_{inhom}$ (rows) simulated in this work.
-- 
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