From 798bd484137439bf1d0752ddde82bf49da4af6b7 Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 16 Apr 2018 13:32:20 -0500
Subject: 2018-04-16 13:32

---
 mixed_domain/chapter.tex | 37 +++++++++++++++++++------------------
 1 file changed, 19 insertions(+), 18 deletions(-)

(limited to 'mixed_domain/chapter.tex')

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index 60910c2..75e60ea 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -164,7 +164,7 @@ the temporal and spectral widths of the excitation pulses.  %
 For simplicity, we will ignore population relaxation effects:  $\Gamma_{11}=\Gamma_{00}=0$.  %
 
 The electric field pulses, $\left\{E_l \right\}$, are given by:
-\begin{equation}\label{eq:E_l}
+\begin{equation} \label{mix:eqn:E_l}
 E_l(t; \omega_l, \tau_l, \vec{k}_l \cdot z) = \frac{1}{2}\left[c_l(t-\tau_l)e^{i\vec{k}_l\cdot z}e^{-i\omega_l(t-\tau_l)} + c.c. \right],
 \end{equation}
 where $\omega_l$ is the field carrier frequency, $\vec{k}_l$ is the wavevector, $\tau_l$ is the
@@ -180,7 +180,7 @@ spectral FWHM (intensity scale).  %
 
 
 The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$:
-\begin{equation}\label{eq:LVN}
+\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}.
 \end{equation}
 Here $\bm{H_0}$ is the time-independent Hamiltonian, $\bm{\mu}$ is the dipole superoperator, and
@@ -202,7 +202,7 @@ $\rho_{ij}$ from $\rho_{ik}$ or $\rho_{kj}$.  %
 For brevity, we use $\hbar=1$ and abbreviate the initial and final density matrix elements as
 $\rho_i$ and $\rho_f$, respectively.  %
 Using the natural frequency rotating frame, $\tilde{\rho}_{ij}=\rho_{ij} e^{-i\omega_{ij}t}$, the formation of $\rho_f$ using pulse $x$ is written as
-\begin{equation}\label{eq:rho_f}
+\begin{equation} \label{mix:eqn:rho_f}
 \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)e^{i\kappa_f\left(\vec{k}_x\cdot z + \omega_x \tau_x \right)}e^{i\kappa_f\Omega_{fx}t}\tilde{\rho}_i(t),
@@ -223,7 +223,7 @@ In the following equations we neglect spatial dependence ($z=0$).  %
 It provides a general expression for arbitrary values of the dephasing rate and excitation pulse
 bandwidth.  %
 The integral solution is
-\begin{equation}\label{eq:rho_f_int}
+\begin{equation} \label{mix:eqn:rho_f_int}
 \begin{split}
 \tilde{\rho}_f(t) =& \frac{i}{2}\lambda_f \mu_f e^{i\kappa_f \omega_x \tau_x} e^{i\kappa_f \Omega_{fx} t} \\
 &\times \int_{-\infty}^{\infty} c_x(t-u-\tau_x)\tilde{\rho}_i(t-u)\Theta(u) \\
@@ -236,8 +236,8 @@ where $\Theta$ is the Heaviside step function.  %
 \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$.  %
 Both limits are important for understanding the multidimensional line shape changes discussed in
 this paper.  %
-The steady state and impulsive limits of Equation propagates \auotoref{mix:eqn:rho_f_int} are
-discussed in TODO
+%The steady state and impulsive limits of Equation propagates \auotoref{mix:eqn:rho_f_int} are
+% discussed in TODO
 % Appendix \ref{sec:cw_imp}.  %
 
 \begin{figure}
@@ -289,12 +289,12 @@ $2^\prime$ notation for the laser pulses with pathway $V\gamma$.
 
 The electric field emitted from a Liouville pathway is proportional to the polarization created by
 the third-order coherence:  %
-\begin{equation}\label{mix:eqn:E_L}
+\begin{equation} \label{mix:eqn:E_L}
 E_L(t) = i \mu_{4}\rho_{3}(t).
 \end{equation}
 \autoref{mix:eqn:E_L} assumes perfect phase-matching and no pulse distortions through propagation.
 \autoref{mix:eqn:rho_f_int} shows that the output field for this Liouville pathway is
-	\begin{gather}\label{mix:eqn:E_L_full}
+	\begin{gather} \label{mix:eqn:E_L_full}
 	\begin{split}
 	E_L(t) =& \frac{i}{8}\lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 
 	e^{i\left( \kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z \right)} 
@@ -380,7 +380,7 @@ Our simulations were done using the open-source SciPy library. \cite{OliphantTra
 
 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
-\ref{eq:E_L_full}.  %
+\ref{mix:eqn: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:  $d\rho / dt \approx 0$.  %
@@ -493,7 +493,7 @@ The build-up limit approximates well when pulses are near-resonant and arrive to
 build-up behavior is emphasized).  %
 The driven limit holds for large detunings, regardless of delay.  %
 
-\subsection{Convolution Technique for Inhomogeneous Broadening} \label{mix:sec:mixed_convolution}
+\subsection{Convolution technique for inhomogeneous broadening} \label{mix:sec:convolution}  % ----
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{mixed_domain/convolve}
@@ -712,7 +712,7 @@ $c_x$, which in our case gives
 \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 \autoref{eq:rho_f_int_freq}:
+by two limits of \autoref{mix:eqn: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
@@ -977,7 +977,7 @@ The colored histogram bars and line shape contours correspond to different value
 dephasing rate, $\Gamma_{10}\Delta_t$.  %
 The contour is the half-maximum of the line shape.  %
 The dependence of the line shape amplitude on $\tau_{21}$ can be inferred from
-\autoref{fig:delay_purity}.  %
+\autoref{mix: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
@@ -1006,7 +1006,7 @@ There are differences in the line shapes for the different values of the relativ
 $\Gamma_{10}\Delta_t$.  %
 The spectral correlation at $\tau_{21}/\Delta_t=2$ decreases in strength as $\Gamma_{10}\Delta_t$
 decreases.  %
-As we illustrated in \autoref{fig:pw1}, this spectral correlation is a signature of driven signal
+As we illustrated in \autoref{mix: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
@@ -1141,7 +1141,7 @@ time-ordering III is decoupled by detuning.  %
 
 In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous
 broadening.  %
-\autoref{fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous
+\autoref{mix:fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous
 distribution.  %
 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
@@ -1171,7 +1171,7 @@ For any delay coordinate, one can develop qualitative line shape expectations by
 \end{enumerate}
 \autoref{mix:fig:fid_dpr} illustrates principles 1 and 2 and \autoref{mix:fig:fid_detuning}
 illustrates principle 2 and 3.  %
-\autoref{fig:pw1} provides a detailed example of the relationship between these principles and the
+\autoref{mix: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.  %
@@ -1182,7 +1182,8 @@ 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 Figures \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}.  %
+shape changes observed in Figures \ref{mix:fig:hom_2d_spectra} and
+\ref{mix:fig:inhom_2d_spectra}.  %
 
 \subsection{Conditional validity of the driven limit}
 
@@ -1190,7 +1191,7 @@ We have shown that the driven limit misses details of the line shape if $\Gamma_
 \approx 1$, 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
+Fig. \ref{mix: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 $\left(\tau_{22^\prime},
@@ -1214,7 +1215,7 @@ with the pulse duration ($\Gamma_{11}\Delta_t=1$), which gives good agreement wi
 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).  %
+signal (\autoref{mix: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.  %
-- 
cgit v1.2.3

