2018-02-26 17:07

1 files changed, 32 insertions, 68 deletions

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.  %