From cd162fef9d9f3145c1e29c63439759636ba62c41 Mon Sep 17 00:00:00 2001 From: Blaise Thompson 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') 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. % -- cgit v1.2.3