2018-04-14 14:33

1 files changed, 153 insertions, 152 deletions

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index e7993c6..0880ecc 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -19,15 +19,15 @@ inhomogeneity.  %
 These simulations provide a foundation for interpretation of ultrafast experiments in the mixed

 domain.  %

 

-\section{Introduction}  % -------------------------------------------------------------------------

+\section{Introduction}  % =========================================================================

 

 Ultrafast spectroscopy is based on using nonlinear interactions, created by multiple ultrashort

-($10^{-9}-10^{-15}$s) pulses, to resolve spectral information on timescales as short as the pulses

+($10^{-9}-10^{-15}$ s) pulses, to resolve spectral information on timescales as short as the pulses

 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

@@ -127,8 +127,8 @@ $\vec{k}_\text{out} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$.  %
 Here, the subscripts represent the excitation pulse frequencies, $\omega_1$ and $\omega_2 =

 \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}  %

+systems, \cite{KohlerDanielDavid2014a, CzechKyleJonathan2015a} and have been developed on

+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.  %

@@ -144,7 +144,7 @@ evolution of this line shape can mimic spectral diffusion.  %
 We identify key signatures that can help differentiate true inhomogeneity and spectral diffusion

 from these measurement artifacts.  %

 

-\section{Theory}  % -------------------------------------------------------------------------------

+\section{Theory}  % ===============================================================================

 

 \begin{figure}

 	\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}

@@ -155,7 +155,7 @@ from these measurement artifacts.  %
     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{fig:WMELs}

+	\label{mix:fig:WMELs}

 \end{figure}

 

 We consider a simple three-level system (states $n=0,1,2$) that highlights the multidimensional

@@ -185,9 +185,8 @@ The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$:
 \end{equation}

 Here $\bm{H_0}$ is the time-independent Hamiltonian, $\bm{\mu}$ is the dipole superoperator, and

 $\bm{\Gamma}$ contains the pure dephasing rate of the system.  %

-We perform the standard perturbative expansion of Equation \ref{eq:LVN} to third order in the

-electric field

-interaction\cite{MukamelShaul1995a, YeeTK1978a, OudarJL1980a, ArmstrongJA1962a,

+We perform the standard perturbative expansion of \autoref{mix:eqn:LVN} to third order in the

+electric field interaction \cite{MukamelShaul1995a, YeeTK1978a, OudarJL1980a, ArmstrongJA1962a,

   SchweigertIgorV2008a} and restrict ourselves only to the terms that have the correct spatial wave

 vector $\vec{k}_{\text{out}}=\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$.  %

 This approximation narrows the scope to sets of three interactions, one from each field, that

@@ -195,8 +194,8 @@ result in the correct spatial dependence.  %
 The set of three interactions have $3!=6$ unique time-ordered sequences, and each time-ordering

 produces either two or three unique system-field interactions for our system, for a total of

 sixteen unique system-field interaction sequences, or Liouville pathways, to consider.  %

-Fig. \ref{fig:WMELs} shows these sixteen pathways as Wave Mixing Energy Level (WMEL)

-diagrams\cite{LeeDuckhwan1985a}.  %

+\autoref{mix:fig:WMELs} shows these sixteen pathways as Wave Mixing Energy Level (WMEL)

+diagrams \cite{LeeDuckhwan1985a}.  %

 

 We first focus on a single interaction in these sequences, where an excitation pulse, $x$, forms

 $\rho_{ij}$ from $\rho_{ik}$ or $\rho_{kj}$.  %

@@ -220,7 +219,7 @@ $\kappa_f$ also has a direct relationship to the phase matching relationship: fo
 $E_2$, $\kappa_f=1$, and for $E_1$ or $E_{2^\prime}$, $\kappa_f=-1$.  %

 In the following equations we neglect spatial dependence ($z=0$).  %

 

-Equation \ref{eq:rho_f} forms the basis for our simulations.  %

+\autoref{mix:eqn:rho_f} forms the basis for our simulations.  %

 It provides a general expression for arbitrary values of the dephasing rate and excitation pulse

 bandwidth.  %

 The integral solution is

@@ -232,12 +231,13 @@ The integral solution is
 \end{split}

 \end{equation}

 where $\Theta$ is the Heaviside step function.  %

-Equation \ref{eq:rho_f_int} becomes the steady state limit expression when $\Delta_t \left|\Gamma_f

+\autoref{mix:eqn:rho_f_int} becomes the steady state limit expression when $\Delta_t \left|\Gamma_f

   + i \kappa_f \Omega_{fx}\right| \gg 1$, and the impulsive limit expression results when $\Delta_t

 \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 \ref{eq: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}

@@ -257,26 +257,26 @@ The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discuss
     delay-frequency (e), or 2D frequency plots (f).  The six time-orderings are labeled in (d) to

     help introduce our delay convention.  

 	}

-	\label{fig:overview}

+	\label{mix:fig:overview}

 \end{figure}

 

-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

+\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.  %

-The color bindings to dephasing rates introduced in Fig. \ref{fig:overview}a will be used

+The color bindings to dephasing rates introduced in \autoref{mix:fig:overview}a will be used

 consistently throughout this work.  %

 Our simulations use systems with dephasing rates quantified relative to the pulse duration:

 $\Gamma_{10} \Delta_t = 0.5, 1$, or $2$.  %

 The temporal axes are normalized to the pulse duration, $\Delta_t$.  The $\Gamma_{10}\Delta_t=2$

 transient is mostly driven by the excitation pulse while $\Gamma_{10} \Delta_t = 0.5$ has a

 substantial free induction decay (FID) component at late times.  %

-Fig. \ref{fig:overview}b shows a pulse sequence (pulses are shaded orange and pink) and the

+\autoref{mix:fig:overview}b shows a pulse sequence (pulses are shaded orange and pink) and the

 resulting system evolution of pathway $V\gamma$ ($00 \xrightarrow{2} 01 \xrightarrow{2^\prime} 11

 \xrightarrow{1} 10 \xrightarrow{\text{out}} 00$) with $\Gamma_{10}\Delta_t=1$.  %

 The final polarization (teal) is responsible for the emitted signal, which is then passed through a

-frequency bandpass filter to emulate monochromator detection (Fig. \ref{fig:overview}c).  %

-The resulting signal is explored in 2D delay space (Fig. \ref{fig:overview}d), 2D frequency space

-(Fig. \ref{fig:overview}f), and hybrid delay-frequency space (Fig. \ref{fig:overview}e).  %

+frequency bandpass filter to emulate monochromator detection (\autoref{mix:fig:overview}c).  %

+The resulting signal is explored in 2D delay space (\autoref{mix:fig:overview}d), 2D frequency space

+(\autoref{mix:fig:overview}f), and hybrid delay-frequency space (\autoref{mix:fig:overview}e).  %

 The detuning frequency axes are also normalized by the pulse bandwidth, $\Delta_{\omega}$.  %

 

 We now consider the generalized Liouville pathway $L:\rho_0 \xrightarrow{x} \rho_1 \xrightarrow{y}

@@ -284,16 +284,17 @@ We now consider the generalized Liouville pathway $L:\rho_0 \xrightarrow{x} \rho
 properties of the first, second, and third pulse, respectively, and indices 0, 1, 2, 3, and 4

 define the properties of the ground state, first, second, third, and fourth density matrix

 elements, respectively.  %

-Fig. \ref{fig:overview}b demonstrates the correspondence between $x$, $y$, $z$ notation and 1, 2,

+\autoref{mix:fig:overview}b demonstrates the correspondence between $x$, $y$, $z$ notation and 1, 2,

 $2^\prime$ 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{eq:E_L}

+\begin{equation}\label{mix:eqn:E_L}

 E_L(t) = i \mu_{4}\rho_{3}(t).

 \end{equation}

-Equation \ref{eq:E_L} assumes perfect phase-matching and no pulse distortions through propagation.  Equation \ref{eq:rho_f_int} shows that the output field for this Liouville pathway is

-	\begin{gather}\label{eq:E_L_full}

+\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{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)} 

@@ -306,10 +307,10 @@ Equation \ref{eq:E_L} assumes perfect phase-matching and no pulse distortions th
 	\end{gather}

 where $R_L$ is the third-order response function for the Liouville pathway.  %

 The total electric field will be the superposition of all the Liouville pathways:

-\begin{equation}\label{eq:superposition}

+\begin{equation} \label{mix:eqn:superposition}

 E_{\text{tot}}= \sum_L E_L(t).

 \end{equation}

-For the superposition of Equation \ref{eq:superposition} to be non-canceling, certain symmetries

+For the superposition of \autoref{mix:eqn:superposition} to be non-canceling, certain symmetries

 between the pathways must be broken.  %

 In general, this requires one or more of the following inequalities:  $\Gamma_{10}\neq\Gamma_{21}$,

 $\omega_{10}\neq\omega_{21}$, and/or $\sqrt{2}\mu_{10}\neq\mu_{21}$.  %

@@ -323,41 +324,41 @@ Importantly, the dipole inequality does not break the symmetry of double quantum
 

 In MR-CMDS, a monochromator resolves the driven output frequency from other nonlinear output

 frequencies, which in our case is $\omega_m = \omega_1 - \omega_2 + \omega_{2'} = \omega_1$.  %

-The monochromator can also enhance spectral resolution, as we show in Section

-\ref{sec:evolution_SQC}.  %

+The monochromator can also enhance spectral resolution, as we show in

+\autoref{mix:sec:evolution_SQC}.  %

 In this simulation, the detection is emulated by transforming $E_{\text{tot}}(t)$ into the

 frequency domain, applying a narrow bandpass filter, $M(\omega)$, about $\omega_1$, and applying

 amplitude-scaled detection:

-\begin{equation}\label{eq:S_tot}

+\begin{equation} \label{mix:eqn:S_tot}

 S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime}) = 

 \sqrt{ \int\left| M(\omega-\omega_1) E_{\text{tot}}(\omega) \right|^2 d\omega},

 \end{equation}

-where $E_{\text{tot}}(\omega)$ denotes the Fourier transform of $E_{\text{tot}}(t)$ (see Fig.

-\ref{fig:overview}c).  %

+where $E_{\text{tot}}(\omega)$ denotes the Fourier transform of $E_{\text{tot}}(t)$ (see

+\autoref{mix:fig:overview}c).  %

 For $M$ we used a rectangular function of width $0.408\Delta_{\omega}$.  %

 The arguments of $S_{\text{tot}}$ refer to the \textit{experimental} degrees of freedom.  %

 The signal delay dependence is parameterized with the relative delays $\tau_{21}$ and

-$\tau_{22^\prime}$, where $\tau_{nm} = \tau_n - \tau_m$ (see Fig. \ref{fig:overview}b).  %

+$\tau_{22^\prime}$, where $\tau_{nm} = \tau_n - \tau_m$ (see \autoref{mix:fig:overview}b).  %

 Table S1 summarizes the arguments for each Liouville pathway.  %

-Fig. \ref{fig:overview}f shows the 2D $(\omega_1, \omega_2)$ $S_{\text{tot}}$ spectrum resulting

-from the pulse delay times represented in Fig. \ref{fig:overview}b.  %

+\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{Inhomogeneity}  % ---------------------------------------------------------------------

 

 Inhomogeneity is isolated in CMDS through both spectral signatures, such as

-line-narrowing\cite{BesemannDanielM2004a, OudarJL1980a, CarlsonRogerJ1990a, RiebeMichaelT1988a,

+line-narrowing \cite{BesemannDanielM2004a, OudarJL1980a, CarlsonRogerJ1990a, RiebeMichaelT1988a,

   SteehlerJK1985a}, and temporal signatures, such as photon echoes \cite{WeinerAM1985a,

   AgarwalRitesh2002a}.  % 

 We simulate the effects of static inhomogeneous broadening by convolving the homogeneous response

 with a Gaussian distribution function.  %

-Further details of the convolution are in Appendix \ref{sec:convolution}.  %

+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.  %

 

-\section{Methods}  % ------------------------------------------------------------------------------

+\section{Methods}  % ==============================================================================

 

-A matrix representation of differential equations of the type in Equation \ref{eq:E_L_full} was

+A matrix representation of differential equations of the type in \autoref{mix:eqn:E_L_full} was

 numerically integrated for parallel computation of Liouville elements (see SI for

-details).\cite{DickBernhard1983a, GelinMaximF2005a}  %

+details). \cite{DickBernhard1983a, GelinMaximF2005a}  %

 The lower bound of integration was $2\Delta_t$ before the first pulse, and the upper bound was

 $5\Gamma_{10}^{-1}$ after the last pulse, with step sizes much shorter than the pulse durations.  %

 Integration was performed in the FID rotating frame; the time steps were chosen so that both the

@@ -373,18 +374,18 @@ $\Delta_\text{inhom}/ \Delta_{\omega}=0,0.5,1,$ and $2$ for the three dephasing
 For all these dimensions, both $\rho_3(t;\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ and

 $S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ are recorded for each unique

 Liouville pathway.  %

-Our simulations were done using the open-source SciPy library.\cite{OliphantTravisE2007a}  %

+Our simulations were done using the open-source SciPy library. \cite{OliphantTravisE2007a}  %

 

-\subsection{Characteristics of Driven and Impulsive Response} \label{sec:cw_imp}  % ---------------

+\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 Equations \ref{eq:rho_f_int} and

+driven/continuous wave (CW) and impulsive limits of \autoref{mix:eqn:rho_f_int} and

 \ref{eq: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:  $d\rho / dt \approx 0$.  %

-Neglecting phase factors, the driven solution to Equation \ref{eq:rho_f_int} will be

-\begin{equation}\label{eq:sqc_driven}

+Neglecting phase factors, the driven solution to \autoref{mix:eqn:rho_f_int} will be

+\begin{equation} \label{mix:eqn:sqc_driven}

 \tilde{\rho}_f(t) = \frac{\lambda_f \mu_f}{2} 

 \frac{c_x(t-\tau_x)e^{i\kappa_f \Omega_{fx}t}}{\kappa_f \Omega_{fx}} \tilde{\rho}_i(t).

 \end{equation}

@@ -401,7 +402,7 @@ Both expressions are important when describing CMDS experiments.  %
 For times after the pulse interaction, $t \gtrsim \tau_x + \Delta_t$, the field-matter coupling is

 negligible.  %

 The evolution for these times, on resonance, is given by

-\begin{equation}\label{eq:sqc_fid}

+\begin{equation} \label{mix:eqn:sqc_fid}

 \tilde{\rho}_f(t) =\frac{i \lambda_f\mu_f }{2} \tilde{\rho}_i(\tau_x)

 \int c_x(u) du \ e^{-\Gamma_f(t-\tau_x)}.

 \end{equation}

@@ -415,8 +416,8 @@ For evaluating times near pulse excitation, $t \lesssim \tau_x + \Delta_t$, we i
 expansion in the response function about zero: $e^{-(\Gamma_f+i\kappa_f\Omega_{fx})u} = 1 -

 (\Gamma_f+i\kappa_f\Omega_{fx})u+\cdots$.  %

 Our impulsive criterion requires that a low order expansion will suffice; it is instructive to

-consider the result of the first order expansion of Equation \ref{eq:rho_f_int}:  %

-\begin{equation}\label{eq:sqc_rise}

+consider the result of the first order expansion of \autoref{mix:eqn:rho_f_int}:  %

+\begin{equation} \label{mix:eqn:sqc_rise}

 \begin{split}

 \tilde{\rho}_f(t) =& \frac{i \lambda_f\mu_f}{2} e^{-i\kappa_f\omega_x\tau_x}e^{-i\kappa_f\Omega_{fx}t} \tilde{\rho}_i(\tau_x) \\

 & \times \bigg[ \left( 1-(\Gamma_f + i\kappa_f\Omega_{fx})(t-\tau_x) \right) \int_{-\infty}^{t-\tau_x} c_x(u) du \\

@@ -434,18 +435,18 @@ It is important to recognize that the impulsive limit is defined not only by hav
 relative to the pulse duration, but also by small detuning relative to the pulse bandwidth (as is

 stated in the inequality).  %

 As detuning increases, the higher orders of the response function Taylor expansion will be needed

-to describe the rise time, and the driven limit of Equation \ref{eq:sqc_driven} will become

+to describe the rise time, and the driven limit of \autoref{mix:eqn:sqc_driven} will become

 valid.  %

 The details of this build-up time can often be neglected in impulsive approximations because

 build-up contributions are often negligible in analysis; the period over which the initial

 excitation occurs is small in comparison to the free evolution of the system.  %

-The build-up behavior can be emphasized by the measurement, which makes Equation \ref{eq:sqc_rise}

+The build-up behavior can be emphasized by the measurement, which makes \autoref{mix:eqn:sqc_rise}

 important.  %

 

-We now consider full Liouville pathways in the impulsive and driven limits of Equation

-\ref{eq:E_L_full}.  %

-For the driven limit, Equation \ref{eq:E_L_full} can be reduced to 

-\begin{equation}\label{eq:E_L_driven}

+We now consider full Liouville pathways in the impulsive and driven limits of

+\autoref{mix:eqn:E_L_full}.  %

+For the driven limit, \autoref{mix:eqn:E_L_full} can be reduced to 

+\begin{equation} \label{mix:eqn:E_L_driven}

 \begin{split}

 E_L(t) =& \frac{1}{8} \lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 

 e^{-i(\kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z)} \\

@@ -461,7 +462,7 @@ when all pulses are overlapped.  %
 This limit is the basis for frequency-domain techniques.  %

 Frequency axes, however, are not independent because the system is forced to the laser frequency

 and influences the resonance criterion for subsequent excitations.  %

-As an example, observe that the first two resonant terms in Equation \ref{eq:E_L_driven} are

+As an example, observe that the first two resonant terms in \autoref{mix:eqn:E_L_driven} are

 maximized when $\omega_x=\left|\omega_1\right|$ and $\omega_y=\left|\omega_2\right|$.  %

 If  $\omega_x$ is detuned by some value $\varepsilon$, however, the occurrence of the second

 resonance shifts to $\omega_y=\left|\omega_2\right|+\varepsilon$, effectively compensating for the

@@ -469,9 +470,9 @@ $\omega_x$ detuning.  %
 This shifting of the resonance results in 2D line shape correlations.  %

 

 If the pulses do not temporally overlap $(\tau_x+\Delta_t \lesssim \tau_y +\Delta_t \lesssim \tau_z

-+ \Delta_t \lesssim t)$, then the impulsive solution to the full Liouville pathway of Equation

-\ref{eq:E_L_full} is  %

-\begin{equation}\label{eq:E_L_impulsive}

++ \Delta_t \lesssim t)$, then the impulsive solution to the full Liouville pathway of

+\autoref{mix:eqn:E_L_full} is  %

+\begin{equation} \label{mix:eqn:E_L_impulsive}

 \begin{split}

 E_L(t) =& \frac{i}{8} \lambda_1\lambda_2\lambda_3\mu_1 \mu_2 \mu_3 \mu_4 e^{i(\omega_1 + \omega_2 + \omega_3)t} \\

 & \times \int c_x(w) dw \int c_y(v) dv \int c_z(u) du \\

@@ -492,7 +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{sec:mixed_convolution}  % --

+\subsection{Convolution Technique for Inhomogeneous Broadening} \label{mix:sec:mixed_convolution}

 

 \begin{figure}

 	\includegraphics[width=\linewidth]{mixed_domain/convolve}

@@ -502,15 +503,15 @@ The driven limit holds for large detunings, regardless of delay.  %
     (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{fig:convolution}

+	\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

 will cause frequency shifts in the resonances.  %

-To show this, we start with a modified, but equivalent, form of Equation \ref{eq:rho_f}:

-\begin{equation}\label{eq:rho_f_modified}

+To show this, we start with a modified, but equivalent, form of \autoref{mix:eqn:rho_f}:

+\begin{equation} \label{mix:eqn:rho_f_modified}

 \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) \\

 & \times e^{i\kappa_f\left( \vec{k}\cdot z + \omega_x \tau_x \right)} e^{-i\kappa_f\left( \omega_x-\left|\omega_f \right| \right)t}\tilde{\rho}_i(t).

@@ -521,21 +522,21 @@ We consider two oscillators with transition frequencies $\omega_f$ and $\omega_f
 \delta$.  %

 So long as $\left| \delta \right| \leq \omega_f$ (so that $\left| \omega_f + \delta \right| =

 \left| \omega_f \right| + \delta$ and thus the rotating wave approximation does not change),

-Equation \ref{eq:rho_f_modified} shows that the two are related by  %

-\begin{equation}\label{eq:freq_translation}

+\autoref{mix:eqn:rho_f_modified} shows that the two are related by  %

+\begin{equation} \label{mix:eqn:freq_translation}

 \frac{d\tilde{\rho}_f^\prime}{dt}(t;\omega_x) = \frac{d\tilde{\rho}_f}{dt}(t;\omega_x-\delta)e^{i\kappa_f \delta \tau_x}.

 \end{equation}

 

 Because both coherences are assumed to have the same initial conditions

 ($\rho_0(-\infty)=\rho_0^\prime(-\infty)=0$), the equality also holds when both sides of the

 equation are integrated.  %

-The phase factor $e^{i\kappa_f\delta\tau_x}$ in the substitution arises from Equation \ref{eq:E_l},

+The phase factor $e^{i\kappa_f\delta\tau_x}$ in the substitution arises from \autoref{mix:eqn:E_l},

 where the pulse carrier frequency maintains its phase within the pulse envelope for all delays.  %

 

 The resonance translation can be extended to higher order signals as well.  %

 For a third-order signal, we compare systems with transition frequencies

 $\omega_{10}^\prime=\omega_{10}+a$ and $\omega_{21}^\prime = \omega_{21}+b$.  %

-The extension of Equation \ref{eq:freq_translation} to pathway $V\beta$ gives  %

+The extension of \autoref{mix:eqn:freq_translation} to pathway $V\beta$ gives  %

 \begin{equation}

 \begin{split}

 \tilde{\rho}_3^\prime(t;\omega_2, \omega_2^\prime, \omega_1) =& \tilde{\rho}_3(t;\omega_2-a,\omega_{2^\prime}-a,\omega_1-b) \\

@@ -571,23 +572,23 @@ Furthermore, since $\kappa=-1$ for $E_1$ and $E_{2^\prime}$, while $\kappa=1$ fo
 $e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} = e^{-ia\left( \tau_1 -

     \tau_2 + \tau_{2^\prime} \right)}$ for all pathways.  %

 Equivalently, if the electric field is parameterized in terms of laser coordinates $\omega_1$ and $\omega_2$, the ensemble field can be calculated as

-\begin{equation}\label{eq:convolve_final}

+\begin{equation} \label{mix:eqn:convolve_final}

 \begin{split}

 \langle E_{\text{tot}}(t;\omega_1,\omega_2) \rangle =& \int K(a,a)E_{\text{tot}}(t;\omega_1-a,\omega_2-a) \\

 &\times e^{-ia\left( \tau_1-\tau_2+\tau_{2^\prime} \right)} da.

 \end{split}

 \end{equation}

 which is a 1D convolution along the diagonal axis in frequency space.  %

-Fig. \ref{fig:convolution} demonstrates the use of Equation \ref{eq:convolve_final} on a

+\autoref{mix:fig:convolution} demonstrates the use of \autoref{mix:eqn:convolve_final} on a

 homogeneous line shape.  %

 

-\section{Results}  % ------------------------------------------------------------------------------

+\section{Results}  % ==============================================================================

 

 We now present portions of our simulated data that highlight the dependence of the spectral line

 shapes and transients on excitation pulse width, the dephasing rate, detuning from resonance, the

 pulse delay times, and inhomogeneous broadening.  %

 

-\subsection{Evolution of single coherence}\label{sec:evolution_SQC}

+\subsection{Evolution of single coherence} \label{mix:sec:evolution_SQC}  % -----------------------

 

 \begin{figure}

   \centering

@@ -600,16 +601,16 @@ pulse delay times, and inhomogeneous broadening.  %
 		For all cases, the pulsed excitation field (gray line, shown as electric field amplitude) is

     slightly detuned (relative detuning, $\Omega_{fx}/\Delta_{\omega}=0.1$).

 	}

-	\label{fig:fid_dpr}

+	\label{mix:fig:fid_dpr}

 \end{figure}

 

 It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x}

 \rho_1$, under various excitation conditions.  %

-Fig. \ref{fig:fid_dpr} shows the temporal evolution of $\rho_1$ with various dephasing rates under

+\autoref{mix:fig:fid_dpr} shows the temporal evolution of $\rho_1$ with various dephasing rates under

 Gaussian excitation.  %

 The value of $\rho_1$ differs only by phase factors between various Liouville pathways (this can be

-verified by inspection of Equation \ref{eq:rho_f_int} under the various conditions in Table S1), so

-the profiles in Fig. \ref{fig:fid_dpr} apply for the first interaction of any pathway.  %

+verified by inspection of \ref{mix:eqn:rho_f_int} under the various conditions in Table S1), so

+the profiles in \autoref{mix:fig:fid_dpr} apply for the first interaction of any pathway.  %

 The pulse frequency was detuned from resonance so that frequency changes could be visualized by the

 color bar, but the detuning was kept slight so that it did not appreciably change the dimensionless

 product, $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)\approx \Gamma_{10}\Delta_t$.  %

@@ -620,20 +621,20 @@ The instantaneous frequency, $d\varphi/dt$, is defined as
 \frac{d\varphi}{dt} = \frac{d}{dt} \tan^{-1}\left( \frac{\text{Im}\left(\rho_1(t)\right)}{\text{Re}\left(\rho_1(t)\right)} \right).

 \end{equation}

 The cases of $\Gamma_{10}\Delta_t=0 (\infty)$ agree with the impulsive (driven) expressions derived

-in Appendix \ref{sec:cw_imp}.  %

+in \autoref{mix:sec:cw_imp}.  %

 For $\Gamma_{10}\Delta_t=0$, the signal rises as the integral of the pulse and has instantaneous

-frequency close to that of the pulse (Equation \ref{eq:sqc_rise}), but as the pulse vanishes, the

-signal adopts the natural system frequency and decay rate (Equation \ref{eq:sqc_fid}).  %

+frequency close to that of the pulse (\autoref{mix:eqn:sqc_rise}), but as the pulse vanishes, the

+signal adopts the natural system frequency and decay rate (\autoref{mix:eqn:sqc_fid}).  %

 For $\Gamma_{10}\Delta_t=\infty$, the signal follows the amplitude and frequency of the pulse for

-all times (the driven limit, Equation \ref{eq:sqc_driven}).  %

+all times (the driven limit, \autoref{mix:eqn:sqc_driven}).  %

 

 The other three cases show a smooth interpolation between limits.  %

 As $\Gamma_{10}\Delta_t$ increases from the impulsive limit, the coherence within the pulse region

 conforms less to a pulse integral profile and more to a pulse envelope profile.  %

 In accordance, the FID component after the pulse becomes less prominent, and the instantaneous

 frequency pins to the driving frequency more strongly through the course of evolution.  %

-The trends can be understood by considering the differential form of evolution (Equation

-\ref{eq:rho_f}), and the time-dependent balance of optical coupling and system relaxation.  %

+The trends can be understood by considering the differential form of evolution (

+\autoref{mix:eqn:rho_f}), and the time-dependent balance of optical coupling and system relaxation.  %

 We note that our choices of $\Gamma_{10}\Delta_t=2.0, 1.0,$ and $0.5$ give coherences that have

 mainly driven, roughly equal driven and FID parts, and mainly FID components, respectively.  %

 FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %

@@ -658,23 +659,23 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
     is black, hashed; imaginary is black, solid). 

 		In all plots, the gray line is the electric field amplitude. 

 	}

-	\label{fig:fid_detuning}

+	\label{mix:fig:fid_detuning}

 \end{figure}

 

-Fig. \ref{fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of

+\autoref{mix:fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of

 $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.

 

 As detuning increases, total amplitude decreases, FID character vanishes, and $\rho_1$ assumes a

 more driven character, as expected.  %

 During the excitation, $\rho_1$ maintains a phase relationship with the input field (as seen by the

-instantaneous frequency in Fig. \ref{fig:fid_detuning}a).  %

+instantaneous frequency in \autoref{mix:fig:fid_detuning}a).  %

 The coherence will persist beyond the pulse duration only if the pulse transfers energy into the

 system; FID evolution equates to absorption.  %

 The FID is therefore sensitive to the absorptive (imaginary) line shape of a transition, while the

 driven response is the composite of both absorptive and dispersive components.  %

 If the experiment isolates the latent FID response, there is consequently a narrower spectral

 response.  %

-This spectral narrowing can be seen in Fig. \ref{fig:fid_detuning}a by comparing the coherence

+This spectral narrowing can be seen in \autoref{mix:fig:fid_detuning}a by comparing the coherence

 amplitudes at $t=0$ (driven)  and at $t / \Delta_t=2$ (FID); amplitudes for all

 $\Omega_{fx}/\Delta_{\omega}$ values shown are comparable at $t=0$, but the lack of FID formation

 for $\Omega_{fx}/\Delta_{\omega}=\pm2$ manifests as a visibly disproportionate amplitude decay.  %

@@ -687,21 +688,22 @@ beams.  %
 In such experiments, there is no additional information to be resolved in the output spectrum.  %

 The situation changes in the mixed domain, where $E_\text{tot}$ contains FID signal that lasts

 longer than the pulse duration.  %

-Fig. \ref{fig:fid_detuning}a provides insight on how frequency-resolved detection of coherent

+\autoref{mix:fig:fid_detuning}a provides insight on how frequency-resolved detection of coherent

 output can enhance resolution when pulses are spectrally broad.  %

 Without frequency-resolved detection, mixed-domain resonance enhancement occurs in two ways: (1)

 the peak amplitude increases, and (2) the coherence duration increases due to the FID transient.  %

 Frequency-resolved detection can further discriminate against detuning by requiring that the

 driving frequency agrees with latent FID.  %

-The implications of discrimination are most easily seen in Fig. \ref{fig:fid_detuning}a with

+The implications of discrimination are most easily seen in \autoref{mix:fig:fid_detuning}a with

 $\Omega_{1x}/\Delta_{\omega}=\pm 1$, where the system frequency moves from the driving frequency to

 the FID frequency.  %

 When the excitation pulse frequency is scanned, the resonance will be more sensitive to detuning by

 isolating the driven frequency (tracking the monochromator with the excitation source).  %

 

 

-The functional form of the measured line shape can be deduced by considering the frequency domain form of Equation \ref{eq:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$):

-\begin{equation}\label{eq:rho_f_int_freq}

+The functional form of the measured line shape can be deduced by considering the frequency domain

+form of \autoref{mix:eqn:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$):

+\begin{equation} \label{mix:eqn:rho_f_int_freq}

 \tilde{\rho}_f (\omega) = \frac{i\lambda_f\mu_f}{2\sqrt{2\pi}} \cdot \frac{\mathcal{F}\left\{ c_x \right\}\left( \omega - \kappa_f\Omega_{fx} \right)}{\Gamma_f + i\omega},

 \end{equation}

 where $\mathcal{F}\left\{ c_x \right\}\left( \omega \right)$ denotes the Fourier transform of

@@ -710,23 +712,23 @@ $c_x$, 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 Equation \ref{eq:rho_f_int_freq}:

+by two limits of \autoref{eq: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

-    convolution of the pulse envelope and the intrinsic (Green's function) response (Fig.

-    \ref{fig:fid_detuning}b, magenta).

+    convolution of the pulse envelope and the intrinsic (Green's function) response

+    (\autoref{mix:fig:fid_detuning}b, magenta).

 	\item When the driven frequency is isolated, $\text{sig} \approx \left|

       \tilde{\rho}_f(\kappa_f\Omega_{fx}) \right|^2$ and the measured line shape will give the

-    un-broadened Green's function (Fig. \ref{fig:fid_detuning}b, teal).

+    un-broadened Green's function (\autoref{mix:fig:fid_detuning}b, teal).

 \end{itemize}

 Monochromatic detection can remove broadening effects due to the pulse bandwidth.  %

 For large $\Gamma_{10}\Delta_t$ values, FID evolution is negligible at all

 $\Omega_{fx}/\Delta_{\omega}$ values and the monochromator is not useful.  %

-Fig. \ref{fig:fid_detuning}b shows the various detection methods for the relative dephasing rate of

+\autoref{mix:fig:fid_detuning}b shows the various detection methods for the relative dephasing rate of

 $\Gamma_{10}\Delta_t=1$.  %

 

-\subsection{Evolution of single Liouville pathway}

+\subsection{Evolution of single Liouville pathway}  % ---------------------------------------------

 

 \begin{figure}

   \centering

@@ -740,7 +742,7 @@ $\Gamma_{10}\Delta_t=1$.  %
     The delays at which the 2D frequency plots are collected are indicated on the delay plot;

     compare 2D spectrum frame color with dot color on 2D delay plot.

 		}

-	\label{fig:pw1}

+	\label{mix:fig:pw1}

 \end{figure}

 

 We now consider the multidimensional response of a single Liouville pathway involving three pulse

@@ -768,19 +770,19 @@ frequency, and the monochromator plays a more complex role.  %
 

 

 We demonstrate this delay dependence using the multidimensional response of the I$\gamma$ Liouville

-pathway as an example (see Fig. \ref{fig:WMELs}).  %

-Fig. \ref{fig:pw1} shows the resulting 2D delay profile of pathway I$\gamma$ signals for

+pathway as an example (see \autoref{mix:fig:WMELs}).  %

+\autoref{mix:fig:pw1} shows the resulting 2D delay profile of pathway I$\gamma$ signals for

 $\Gamma_{10}\Delta_t=1$ (left) and the corresponding $\omega_1, \omega_2$ 2D spectra at several

 pulse delay values (right).  %

 The spectral changes result from changes in the relative importance of driven and FID

 components.  %

-The prominence of FID signal can change the resonance conditions; Table \ref{tab:table2} summarizes

+The prominence of FID signal can change the resonance conditions; \autoref{mix:tab:table2} summarizes

 the changing resonance conditions for each of the four delay coordinates studied.  %

 Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromator must also be

 considered.  %

 

 \begin{table}

-	\caption{\label{tab:table2} Conditions for peak intensity at different pulse delays for pathway

+	\caption{\label{mix:tab:table2} Conditions for peak intensity at different pulse delays for pathway

     I$\gamma$.}

 	\begin{tabular}{c c | c c c c}

 		\multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\

@@ -810,7 +812,7 @@ making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical
 $\omega_1=\omega_{10}$.  %

 

 

-The other three spectra of Fig. \ref{fig:pw1} separate the pulse sequence over time so that not all

+The other three spectra of \autoref{mix:fig:pw1} separate the pulse sequence over time so that not all

 interactions are driven.  % 

 At $\tau_{21}=0$, $\tau_{22^\prime}=-2.4\Delta_t$ (lower left, pink), the first two resonances

 remain the same as at pulse overlap (orange) but the last resonance is different.  %

@@ -860,10 +862,10 @@ monochromator track $\omega_1$, the output signal will increase when
 $\omega_1=\omega_{2^\prime}$.  %

 As a result, the line shape acquires a diagonal character.  %

 

-The changes in line shape seen in Fig. \ref{fig:pw1} have significant ramifications for the

+The changes in line shape seen in \autoref{mix:fig:pw1} have significant ramifications for the

 interpretations and strategies of MR-CMDS in the mixed domain.  %

 Time-gating has been used to isolate the 2D spectra of a certain

-time-ordering\cite{MeyerKentA2004a, PakoulevAndreiV2006a, DonaldsonPaulMurray2007b}, but here we

+time-ordering \cite{MeyerKentA2004a, PakoulevAndreiV2006a, DonaldsonPaulMurray2007b}, but here we

 show that time-gating itself causes significant line shape changes to the isolated pathways.  %

 The phenomenon of time-gating can cause frequency and delay axes to become functional of each other

 in unexpected ways.  %

@@ -877,20 +879,20 @@ in unexpected ways.  %
     column).

 		All pulses are tuned to exact resonance.  

 		In each 2D delay plot, the signal amplitude is depicted by the colors.  

-		The black contour lines show signal purity, $P$ (see Equation \ref{eq:P}), with purity values

+		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values

     denoted on each contour.  

 		The small plots above each 2D delay plot examine a $\tau_{22^\prime}$ slice of the delay

     response ($\tau_{21}=0$).

     The plot shows the total signal (black), as well as the component time-orderings VI (orange), V

     (purple), and III or I (teal).

 	}

-	\label{fig:delay_purity}

+	\label{mix:fig:delay_purity}

 \end{figure}

 

 In the last section we showed how a single pathway's spectra can evolve with delay due to pulse

 effects and time gating.  %

 In real experiments, evolution with delay is further complicated by the six time-orderings/sixteen

-pathways present in our three-beam experiment (see Fig. \ref{fig:WMELs}).  %

+pathways present in our three-beam experiment (see \autoref{mix:fig:WMELs}).  %

 Each time-ordering has different resonance conditions.  %

 When signal is collected near pulse overlap, multiple time-orderings contribute.  %

 To identify these effects, we start by considering how strongly time-orderings are isolated at each

@@ -902,7 +904,7 @@ Because the temporal profile of the signal is dependent on both the excitation p
 the decay dynamics of the coherence itself, quantitation of pathway discrimination requires

 simulation.  %

 

-Fig. \ref{fig:delay_purity} shows the 2D delay space with all pathways present for

+\autoref{mix:fig:delay_purity} shows the 2D delay space with all pathways present for

 $\omega_1,\omega_2=\omega_{10}$.  %

 It illustrates the interplay of pulse width and system decay rates on the isolation of time-ordered

 pathways.  %

@@ -912,12 +914,12 @@ $E_1$ and $E_{2^\prime}$ interactions are interchangeable:
 $S_\text{tot}(\tau_{21},\tau_{22^\prime})=S_\text{tot}(\tau_{22^\prime}, \tau_{21})$.  %

 The overlaid black contours represent signal ``purity,'' $P$, defined as the relative amount of

 signal that comes from the dominant pathway at that delay value:

-\begin{equation}\label{eq:P}

+\begin{equation} \label{mix:eqn:P}

 P(\tau_{21},\tau_{22^\prime})=\frac{\max \left\{S_L\left( \tau_{21},\tau_{22^\prime} \right)\right\}}

 {\sum_L S_L\left( \tau_{21},\tau_{22^\prime} \right)}.

 \end{equation}

 The dominant pathway ($\max{\left\{ S_L \left( \tau_{21},\tau_{22^\prime} \right) \right\}}$) at

-given delays can be inferred by the time-ordered region defined in Fig. \ref{fig:overview}d.  %

+given delays can be inferred by the time-ordered region defined in \autoref{mix:fig:overview}d.  %

 The contours of purity generally run parallel to the time-ordering boundaries with the exception of

 time-ordered regions II and IV, which involve the double quantum coherences that have been

 neglected.  %

@@ -949,7 +951,7 @@ vanishing signal intensities; the contour of $P=0.99$ across our systems highlig
 		Contributions from V and VI are grouped together because they have equal weights at

     $\tau_{22^\prime}=0$.

 	}

-	\label{fig:hom_2d_spectra}

+	\label{mix:fig:hom_2d_spectra}

 \end{figure}

 

 In the previous sections we showed how pathway spectra and weights evolve with delay.  %

@@ -965,17 +967,17 @@ probe-pump ordering (so-called perturbed FID).  %
 Scanning $\tau_{21}$ through pulse overlap complicates interpretation of the line shape due to the

 changing nature and balance of the contributing time-orderings.  %

 At $\tau_{21}>0$, time-ordering I dominates; at $\tau_{21}=0$, all time-orderings contribute

-equally; at $\tau_{21}<0$ time-orderings V and VI dominate (Fig. \ref{fig:delay_purity}).  %

-Conventional pump-probe techniques recognized these complications long ago,\cite{BritoCruzCH1988a,

+equally; at $\tau_{21}<0$ time-orderings V and VI dominate (\autoref{mix:fig:delay_purity}).  %

+Conventional pump-probe techniques recognized these complications long ago, \cite{BritoCruzCH1988a,

   PalfreySL1985a} but the extension of these effects to MR-CMDS has not previously been done.  %

 

-Fig. \ref{fig:hom_2d_spectra} shows the MR-CMDS spectra, as well as histograms of the pathway

+\autoref{mix:fig:hom_2d_spectra} shows the MR-CMDS spectra, as well as histograms of the pathway

 weights, while scanning $\tau_{21}$ through pulse overlap.  %

 The colored histogram bars and line shape contours correspond to different values of the relative

 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 Fig.

-\ref{fig:delay_purity}.  %

+The dependence of the line shape amplitude on $\tau_{21}$ can be inferred from

+\autoref{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

@@ -1004,11 +1006,11 @@ 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 Fig. \ref{fig:pw1}, this spectral correlation is a signature of driven signal

+As we illustrated in \autoref{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

-$\tau_{21}$ traces in Fig. \ref{fig:delay_purity}.  %

+$\tau_{21}$ traces in \autoref{mix:fig:delay_purity}.  %

 When all pulses are completely overlapped, ($\tau_{21}=0$), each of the line shapes exhibit

 spectral correlation.  %

 At $\tau_{21}/\Delta_t=-2$, the line shape shrinks as $\Gamma_{10}\Delta_t$ decreases, with the

@@ -1031,13 +1033,13 @@ resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\D
 		Transient ($\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}

+	\label{mix:fig:wigners}

 \end{figure}

 

 It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is

 frequency. \cite{KohlerDanielDavid2014a, AubockGerald2012a, CzechKyleJonathan2015a,

   PakoulevAndreiV2007a}  %

-In Fig. \ref{fig:wigners} we show five $\tau_{21},\omega_1$ plots for varying $\omega_2$ with

+In \autoref{mix:fig:wigners} we show five $\tau_{21},\omega_1$ plots for varying $\omega_2$ with

 $\tau_{22^\prime}=0$.  %

 The plots are the analogue to the most common multidimensional experiment of Transient Absorption

 spectroscopy, where the non-linear probe spectrum is plotted as a function of the pump-probe

@@ -1049,7 +1051,7 @@ 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} \label{sec:res_inhom}  % ------------------------------------

+\subsection{Inhomogeneous broadening} \label{mix:sec:res_inhom}  % --------------------------------

 

 \begin{figure}

 	\includegraphics[width=0.5\linewidth]{"mixed_domain/inhom delay space ratios"}

@@ -1057,7 +1059,7 @@ Again, these features can resemble spectral diffusion even though our system is
 		2D delay response for $\Gamma_{10}\Delta_t=1$ with sample inhomogeneity.  %

 		All pulses are tuned to exact resonance.  %

 		The colors depict the signal amplitude.  %

-		The black contour lines show signal purity, $P$ (see Equation \ref{eq:P}), with purity values

+		The black contour lines show signal purity, $P$ (see \autoref{mix:eqn:P}), with purity values

     denoted on each contour.  %

 		The thick yellow line denotes the peak amplitude position that is used for 3PEPS analysis.  %

 		The small plot above each 2D delay plot examines a $\tau_{22^\prime}$ slice of the delay

@@ -1065,7 +1067,7 @@ Again, these features can resemble spectral diffusion even though our system is
 		The plot shows the total signal (black), as well as the component time-orderings VI (orange), V

     (purple), III (teal, dashed), and I (teal, solid).  %

 	}

-	\label{fig:delay_inhom}

+	\label{mix:fig:delay_inhom}

 \end{figure}

 

 With the homogeneous system characterized, we can now consider the effect of inhomogeneity.  %

@@ -1073,14 +1075,14 @@ For inhomogeneous systems, time-orderings III and V are enhanced because their f
 rephase to form a photon echo, whereas time-orderings I and VI will not.  %

 In delay space, this rephasing appears as a shift of signal to time-ordered regions III and V that

 persists for all population times.  %

-Fig. \ref{fig:delay_inhom} shows the calculated spectra for relative dephasing rate

+\autoref{mix:fig:delay_inhom} shows the calculated spectra for relative dephasing rate

 $\Gamma_{10}\Delta_t=1$ with a frequency broadening function of width

 $\Delta_{\text{inhom}}=0.441\Gamma_{10}$.  %

 The inhomogeneity makes it easier to temporally isolate the rephasing pathways and harder to

 isolate the non-rephasing pathways, as shown by the purity contours.  %

 

 A common metric of rephasing in delay space is the 3PEPS

-measurement.\cite{WeinerAM1985a, FlemingGrahmR1998a, DeBoeijWimP1998a, SalvadorMayroseR2008a}  %

+measurement. \cite{WeinerAM1985a, FlemingGrahmR1998a, DeBoeijWimP1998a, SalvadorMayroseR2008a}  %

 In 3PEPS, one measures the signal as the first coherence time, $\tau$, is scanned across both

 rephasing and non-rephasing pathways while keeping population time, $T$, constant.  %

 The position of the peak is measured; a peak shifted away from $\tau=0$ reflects the rephasing

@@ -1092,8 +1094,8 @@ the population (time-orderings V and VI).  %
 The trace runs parallel to the III-V time-ordering boundary (diagonal) if $\tau_{22^\prime}<0$ and

 runs parallel to the IV-VI time-ordering boundary (horizontal) if $\tau_{22^\prime}>0$, and both

 intersect at $\tau_{22^\prime}=0$; the $-\tau_{21}$ value at this intersection is $T$.  %

-In our 2D delay plots (Fig. \ref{fig:delay_purity}, Fig. \ref{fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.  %

-Fig. \ref{fig:delay_inhom} highlights the peak shift profile as a function of population time with

+In our 2D delay plots (\autoref{mix:fig:delay_purity}, \autoref{mix:fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.  %

+\autoref{mix:fig:delay_inhom} highlights the peak shift profile as a function of population time with

 the yellow trace; it is easily verified that our static inhomogneous system exhibits a non-zero

 peak shift value for all population times.  %

 

@@ -1110,10 +1112,10 @@ disappears.  %
 Peak shifts imply inhomogeneity only when time-orderings V and III are minimally contaminated by

 each other i.e. at population times that exceed pulse overlap.  %

 This fact is easily illustrated by the dynamics of homogeneous system (Fig.

-\ref{fig:delay_purity}); even though the homogeneous system cannot rephase, there is a non-zero

-peak shift near $T=0$.  %

+\autoref{mix:fig:delay_purity}); even though the homogeneous system cannot rephase, there is a

+non-zero peak shift near $T=0$.  %

 The contamination of the 3PEPS measurement at pulse overlap is well-known and is described in some

-studies,\cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system properties on the

+studies, \cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system properties on the

 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.  %

@@ -1121,7 +1123,7 @@ time-ordering III is decoupled by detuning.  %
 \begin{figure}

 	\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

+		Same as \autoref{mix:fig:hom_2d_spectra}, but each system has inhomogeneity

     ($\Delta_{\text{inhom}}=0.441\Gamma_{10}$).

 		Relative dephasing rates are $\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and $2.0$ (blue).  

 		In all plots $\tau_{22^\prime}=0$.

@@ -1131,20 +1133,20 @@ time-ordering III is decoupled by detuning.  %
     $\Gamma_{10}\Delta_t$ value.

 		The colored histograms below each 2D frequency plot show the relative weights of each

     time-ordering for each 2D frequency plot.  

-		In contrast to Fig. \ref{fig:hom_2d_spectra}, inhomogeneity makes the relative contributions of

+		In contrast to \autoref{mix:fig:hom_2d_spectra}, inhomogeneity makes the relative contributions of

     time-orderings V and VI unequal.

 	}

-	\label{fig:inhom_2d_spectra}

+	\label{mix:fig:inhom_2d_spectra}

 \end{figure}

 

 In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous

 broadening.  %

-Fig. \ref{fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous

+\autoref{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

 spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals.  %

-The anti-diagonal width at early delays (e.g. Fig. \ref{fig:inhom_2d_spectra},

+The anti-diagonal width at early delays (e.g. \autoref{mix:fig:inhom_2d_spectra},

 $\tau_{21}=2.0\Delta_t$) again depends on the pulse bandwidth and the monochromator slit width.  %

 At delay values that isolate time-orderings V and VI, however, the line shapes retain diagonal

 character, showing the characteristic balance of homogeneous and inhomogeneous width.  %

@@ -1167,20 +1169,20 @@ For any delay coordinate, one can develop qualitative line shape expectations by
 	\item When time-gating after the pulse, the FID dominates the system response.

 	\item The emitted signal field contains both FID and driven components; the $\omega_{\text{out}} = \omega_1$ component is isolated by the tracking monochromator.

 \end{enumerate}

-Fig. \ref{fig:fid_dpr} illustrates principles 1 and 2 and Fig. \ref{fig:fid_detuning} illustrates

-principle 2 and 3.  %

-Fig. \ref{fig:pw1} provides a detailed example of the relationship between these principles and the

+\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

 multidimensional line shape changes for different delay times.  %

 

 The principles presented above apply to a single pathway.  %

 For rapidly dephasing systems it is difficult to achieve complete pathway discrimination, as shown

-in Fig. \ref{fig:delay_purity}.  %

+in \autoref{mix:fig:delay_purity}.  %

 In such situations the interference between pathways must be considered to predict the line

 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 Figs. \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}.  %

+shape changes observed in Figures \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}.  %

 

 \subsection{Conditional validity of the driven limit}

 

@@ -1201,10 +1203,9 @@ Note the very sharp diagonal feature that appears for $(\tau_{21},\tau_{22^\prim
 \Gamma_{11}=0$; this is due to population resonance in time-orderings I and III.  %

 This expression is inaccurate:  the narrow resonance is only observed when pulse durations are much

 longer than the coherence time.  %

-A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes (Yurs

-\textit{et al.}\cite{YursLenaA2011a} and \textcite{KohlerDanielDavid2014a},

-respectively)

-demonstrates this difference well.  %

+A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes

+(\textcite{YursLenaA2011a} and \textcite{KohlerDanielDavid2014a}, respectively) demonstrates this

+difference well.  %

 The driven equation fails to reproduce our numerical simulations here because resonant excitation

 of the population is impulsive; the experiment time-gates only the rise time of the population, yet

 driven theory predicts the resonance to be vanishingly narrow ($\Gamma_{11}=0$).  %

@@ -1230,7 +1231,7 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
 		Second column:  Same as the first column, but with ad hoc substitution $\Gamma_{11}=\Delta_t$.  

 		Third column:  The directly integrated response.  %

 	}

-	\label{fig:steady_state}

+	\label{mix:fig:steady_state}

 \end{figure}

 

 \subsection{Extracting true material correlation}  % ----------------------------------------------

@@ -1251,7 +1252,7 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
 		Gray lines indicate contours of constant relative inhomogeneity (scatter points with the same

     area are connected).  %

 }

-	\label{fig:metrics}

+	\label{mix:fig:metrics}

 \end{figure}

 

 We have shown that pulse effects mimic the qualitative signatures of inhomogeneity.  %

@@ -1265,7 +1266,7 @@ In the driven (impulsive) limit, ellipticity (3PEPS) corresponds to the frequenc
 function and uniquely extracts the inhomogeneity of the models presented here.  %

 In their respective limits, the metrics give values proportional to the inhomogeneity.  % 

 

-Fig. \ref{fig:metrics} shows the results of this characterization for all $\Delta_\text{inhom}$ and

+\autoref{mix:fig:metrics} shows the results of this characterization for all $\Delta_\text{inhom}$ and

 $\Gamma_{10}\Delta_t$ values explored in this work.  %

 We study how the correlations between the two metrics depend on the relative dephasing rate,

 $\Gamma_{10}\Delta_t$, the absolute inhomogeneity, $\Delta_\text{inhom} / \Delta_\omega$, the

@@ -1283,10 +1284,10 @@ This saturation results because the frequency bandwidth of the excitation pulses
 than the inhomogeneous width and only a portion of the inhomogeneous ensemble contributes to the

 3PEPS experiment. \cite{WeinerAM1985a}  %

 The corresponding graph for $T = 0$ shows a large peak shift occurs, even without inhomogeneity. 

-In this case, the peak shift depends on pathway overlap, as discussed in Section

-\ref{sec:res_inhom}.  %

+In this case, the peak shift depends on pathway overlap, as discussed in

+\autoref{mix:sec:res_inhom}.  %

 

-The middle row in Fig. \ref{fig:metrics} shows the ellipticity dependence on the relative dephasing

+The middle row in \autoref{mix:fig:metrics} shows the ellipticity dependence on the relative dephasing

 rate and inhomogeneity assuming the measurement is performed when the first two pulses are

 temporally overlapped ($\tau_{22^\prime}=0$).  %

 For a $T=4\Delta_t$ population time, the ellipticity is proportional to the inhomogeneity until

@@ -1308,7 +1309,7 @@ The dephasing can be measured independently in the frequency or time domain, dep
 the dephasing is very fast or slow, respectively.  %

 In the mixed frequency/time domain, measurement of the dephasing becomes more difficult.  %

 One strategy to address this challenge is to use both the 3PEPS and ellipticity metrics.  %

-The bottom row in Fig. \ref{fig:metrics} plots 3PEPS against ellipticity to show how the

+The bottom row in \autoref{mix:fig:metrics} plots 3PEPS against ellipticity to show how the

 relationship between the metrics changes for different amounts of dephasing and inhomogeneity.  %

 The anti-diagonal contours of constant relative inhomogeneity show that these metrics are

 complementary and can serve to extract the system correlation parameters.  %

@@ -1320,7 +1321,7 @@ pulse effects give a mapping significantly different than at $T = 4\Delta_t$.  %
 At $T = 0$, 3PEPS is almost nonresponsive to inhomogeneity; instead, it is an almost independent

 characterization of the pure dephasing.  %

 In fact, the $T=0$ trace is equivalent to the original photon echo traces used to resolve pure

-dephasing rates.\cite{AartsmaThijsJ1976a}  %

+dephasing rates. \cite{AartsmaThijsJ1976a}  %

 Both metrics are offset due to the pulse overlap effects.  %

 Accordingly, the region to the left of homogeneous contour is non-physical, because it represents

 observed correlations that are less than that given by pulse overlap effects.  %