From fa43cdc18d17b937d24abb6785c467364f14f2ad Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Mon, 29 Jan 2018 17:01:23 -0600
Subject: initial mixed domain

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+\chapter{Disentangling material and instrument response}
+
+Ultrafast spectroscopy is often collected in the mixed frequency/time domain, where pulse durations
+are similar to system dephasing times.  %
+In these experiments, expectations derived from the familiar driven and impulsive limits are not
+valid.  %
+This work simulates the mixed-domain Four Wave Mixing response of a model system to develop
+expectations for this more complex field-matter interaction.  %
+We explore frequency and delay axes.  %
+We show that these line shapes are exquisitely sensitive to excitation pulse widths and delays.  %
+Near pulse overlap, the excitation pulses induce correlations which resemble signatures of dynamic
+inhomogeneity.  %
+We describe these line shapes using an intuitive picture that connects to familiar field-matter
+expressions.  %
+We develop strategies for distinguishing pulse-induced correlations from true system
+inhomogeneity.  %
+These simulations provide a foundation for interpretation of ultrafast experiments in the mixed
+domain.  %
+
+\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
+themselves.\cite{Rentzepis1970,Mukamel2000}  % 
+The ultrafast specta can be collected in the time domain or the frequency domain.\cite{Park1998}  %
+
+Time-domain methods scan the pulse delays to resolve the free induction decay
+(FID).\cite{Gallagher1998}   %
+The Fourier Transform of the FID gives the ultrafast spectrum.  %
+Ideally, these experiments are performed in the impulsive limit where FID dominates the
+measurement.  % 
+FID occurs at the frequency of the transition that has been excited by a well-defined, time-ordered
+sequence of pulses.  % 
+Time-domain methods are compromised when the dynamics occur on faster time scales than the
+ultrafast excitation pulses.  %
+As the pulses temporally overlap, FID from other pulse time-orderings and emission driven by the
+excitation pulses both become important.  % 
+These factors are responsible for the complex ``coherent artifacts'' that are often ignored in
+pump-probe and related methods.\cite{Lebedev2007, Vardeny1981, Joffre1988, Pollard1992}  % 
+Dynamics faster than the pulse envelopes are best measured using line shapes in frequency domain
+methods.  %
+
+Frequency-domain methods scan pulse frequencies to resolve the ultrafast spectrum
+directly.\cite{Druet1979,Oudar1980}  % 
+Ideally, these experiments are performed in the driven limit where the steady state dominates the
+measurement.  %
+In the driven limit, all time-orderings of the pulse interactions are equally important and FID
+decay is negligible.  %
+The output signal is driven at the excitation pulse frequencies during the excitation pulse
+width.  %
+Frequency-domain methods are compromised when the spectral line shape is narrower than the
+frequency bandwidth of the excitation pulses.  % 
+Dynamics that are slower than the pulse envelopes can be measured in the time domain by resolving
+the phase oscillations of the output signal during the entire FID decay.  %
+
+There is also the hybrid mixed-time/frequency-domain approach, where pulse delays and pulse
+frequencies are both scanned to measure the system response.  %
+This approach is uniquely suited for experiments where the dephasing time is comparable to the
+pulse durations, so that neither frequency-domain nor time-domain approaches excel on their
+own.\cite{Oudar1980,Wright1997a,Wright1991}  %
+In this regime, both FID and driven processes are important.\cite{Pakoulev2006}  %
+Their relative importance depends on pulse frequencies and delays.  %
+Extracting the correct spectrum from the measurement then requires a more complex analysis that
+explicitly treats the excitation pulses and the different
+time-orderings.\cite{Pakoulev2007,Kohler2014,Gelin2009a}   %
+Despite these complications, mixed-domain methods have a practical advantage: the dual frequency-
+and delay-scanning capabilities allow these methods to address a wide variety of dephasing
+rates.  %
+
+The relative importance of FID and driven processes and the changing importance of different
+coherence pathways are important factors for understanding spectral features in all ultrafast
+methods.  %
+These methods include partially-coherent methods involving intermediate populations such as
+pump-probe,\cite{Hamm2000} transient grating,\cite{Salcedo1978,Fourkas1992,Fourkas1992a} transient
+absorption/reflection,\cite{Aubock2012,Bakker2002} photon
+echo,\cite{DeBoeij1996,Patterson1984,Tokmakoff1995} two dimensional-infrared spectroscopy
+(2D-IR),\cite{Hamm1999,Asplund2000,Zanni2001} 2D-electronic spectroscopy
+(2D-ES),\cite{Hybl2001a,Brixner2004} and three pulse photon echo peak shift
+(3PEPS)\cite{Emde1998,DeBoeij1996,DeBoeij1995,Cho1992,Passino1997} spectroscopies.  %
+These methods also include fully-coherent methods involving only coherences such as Stimulated
+Raman Spectroscopy (SRS),\cite{Yoon2005,McCamant2005} Doubly Vibrationally Enhanced
+(DOVE),\cite{Zhao1999,Zhao1999a,Zhao2000,Meyer2003,Donaldson2007,Donaldson2008,Fournier2008} Triply
+Resonant Sum Frequency (TRSF),\cite{Boyle2013a,Boyle2013,Boyle2014} Sum Frequency Generation
+(SFG)\cite{Lagutchev2007}, Coherent Anti-Stokes Raman Spectroscopy
+(CARS)\cite{Carlson1990b,Carlson1990a,Carlson1991} and other coherent Raman
+methods\cite{Steehler1985}.  %
+
+This paper focuses on understanding the nature of the spectral changes that occur in Coherent
+Multidimensional Spectroscopy (CMDS) as experiments transition between the two limits of frequency-
+and time-domain methods.  %
+CMDS is a family of spectroscopies that use multiple delay and/or frequency axes to extract
+homogeneous and inhomogeneous broadening, as well as detailed information about spectral diffusion
+and chemical changes.\cite{Kwac2003,Wright2016}  % 
+For time-domain CMDS (2D-IR, 2D-ES), the complications that occur when the impulsive approximation
+does not strictly hold has only recently been addressed.\cite{Erlik2017,Smallwood2016}  %
+
+Frequency-domain CMDS methods, referred to herein as multi-resonant CMDS (MR-CMDS), have similar
+capabilities for measuring homogeneous and inhomogeneous broadening.  %
+Although these experiments are typically described in the driven
+limit,\cite{Gallagher1998,Fourkas1992,Fourkas1992a} many of the experiments involve pulse widths
+that are comparable to the widths of the
+system.\cite{Meyer2003,Donaldson2007,Pakoulev2009,Zhao1999,Czech2015,Kohler2014}  % 
+MR-CMDS then becomes a mixed-domain experiment whereby resonances are characterized with marginal
+resolution in both frequency and time.  %
+For example, DOVE spectroscopy involves three different pathways\cite{Wright2003} whose relative
+importance depends on the relative importance of FID and driven responses.\cite{Donaldson2010}  %
+In the driven limit, the DOVE line shape depends on the difference between the first two pulse
+frequencies so the line shape has a diagonal character that mimics the effects of inhomogeneous
+broadening.  % 
+In the FID limit where the coherence frequencies are defined instead by the transition, the
+diagonal character is lost.  %  
+Understanding these effects is crucial for interpreting experiments, yet these effects have not
+been characterized for MR-CMDS.  %
+
+This work considers the third-order MR-CMDS response of a 3-level model system using three
+ultrafast excitation beams with the commonly used four-wave mixing (FWM) phase-matching condition,
+$\vec{k}_\text{out} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$.  % 
+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{Kohler2014,Czech2015} and have been developed on vibrational states as
+well.\cite{Meyer2004}  %
+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.  %
+We numerically simulate the MR-CMDS response with pulse durations at, above, and below the system
+coherence time.  %
+To highlight the role of pulse effects, we build an interpretation of the full MR-CMDS response by
+first showing how finite pulses affect the evolution of a coherence, and then how finite pulses
+affect an isolated third-order pathway.  %
+When considering the full MR-CMDS response, we show that spectral features change dramatically as a
+function of delay, even for a homogeneous system with elementary dynamics.  %
+Importantly, the line shape can exhibit correlations that mimic inhomogeneity, and the temporal
+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}
+
+\begin{figure}
+  \centering
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}
+	\caption{
+		The sixteen triply-resonant Liouville pathways for the third-order response of the system used
+    here.  Time flows from left to right.  Each excitation is labeled by the pulse stimulating the
+    transition; excitatons with $\omega_1$ are yellow, excitations with $\omega_2=\omega_{2'}$ are
+    purple, and the final emission is gray.
+	}
+	\label{fig:WMELs}
+\end{figure}
+
+We consider a simple three-level system (states $n=0,1,2$) that highlights the multidimensional
+line shape changes resulting from choices of the relative dephasing and detuning of the system and
+the temporal and spectral widths of the excitation pulses.  %  
+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}
+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
+pulse delay, and $c_l$ is a slowly varying envelope.  %
+In this work, we assume normalized (real-valued) Gaussian envelopes:  %
+\begin{equation}
+c_l(t) = \frac{1}{\Delta_t}\sqrt{\frac{2\ln 2}{2\pi}} \exp\left(-\ln 2 \left[\frac{t}{\Delta_t}\right]^2\right),
+\end{equation}
+where $\Delta_t$ is the temporal FWHM of the envelope intensity.  %
+We neglect non-linear phase effects such as chirp so the FWHM of the frequency bandwidth is
+transform limited:  $\Delta_{\omega}\Delta_t=4 \ln 2 \approx 2.77$, where $\Delta_{\omega}$ is the
+spectral FWHM (intensity scale).  %
+
+
+The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$:
+\begin{equation}\label{eq: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
+$\bm{\Gamma}$ 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{mukamel1995principles,Yee1978,Oudar1980,Armstrong1962,Schweigert2008} 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
+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{Lee1985}.  %
+
+We first focus on a single interaction in these sequences, where an excitation pulse, $x$, forms
+$\rho_{ij}$ 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{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),
+\end{split}
+\end{equation}
+where $\Omega_{fx}=\kappa_f^{-1}\omega_f - \omega_x (=\left|\omega_f\right| - \omega_x)$ is the
+detuning, $\omega_f$ is the transition frequency of the $i^{th}$ transition, $\mu_f$ is the
+transition dipole, and $\Gamma_f$ is the dephasing/relaxation rate for $\rho_f$.  %
+The $\lambda_f$ and $\kappa_f$ parameters describe the phases of the interaction:  $\lambda_f=+1$
+for ket-side transitions and -1 for bra-side transitions, and $\kappa_f$ depends on whether
+$\rho_f$ is formed via absorption ($\kappa_f= \lambda_f$) or emission
+($\kappa_f=-\lambda_f$).\footnote{$\kappa_f$ also has a direct relationship to the phase matching
+  relationship: for transitions with $E_2$, $\kappa_f=1$, 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.  %
+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{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) \\
+& \qquad \quad \ \ \times e^{-\left(\Gamma_f+i\kappa_f\Omega_{fx}\right)u}du, 
+\end{split}
+\end{equation}
+where $\Theta$ is the Heaviside step function.  %
+Equation \ref{eq:rho_f_int} becomes the steady state limit expression when $\Delta_t \left|\Gamma_f
+  + i \kappa_f \Omega_{fx}\right| \gg 1$, 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 Appendix
+\ref{sec:cw_imp}.  %
+
+\begin{figure*}
+	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}
+	\caption{
+		Overview of the MR-CMDS simulation. 
+		(a) The temporal profile of a coherence under pulsed excitation depends on how quickly the
+    coherence dephases.  In all subsequent panes, the relative dephasing rate is kept constant at
+    $\Gamma_{10}\Delta_t=1$.
+		(b) Simulated evolution of the density matrix elements of a third-order Liouville pathway
+    $V\gamma$ under fully resonant excitation. Pulses can be labeled both by their time of arrival
+    ($x$,$y$,$z$) and by the lab lasers used to stimulate the transitions ($2$,$2^\prime$,$1$). The
+    final coherence (teal) creates the output electric field.   
+		(c) The frequency profile of the output electric field is filtered by a monochromator gating
+    function, $M(\omega)$, and the passed components (shaded) are measured.   
+		(d-f) Signal is viewed  against two laser parameters, either as 2D delay (d), mixed
+    delay-frequency (e), or 2D frequency plots (f).  The six time-orderings are labeled in (d) to
+    help introduce our delay convention.  
+	}
+	\label{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
+transient for three different dephasing rates.  %
+The color bindings to dephasing rates introduced in Fig. \ref{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
+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).  %
+The detuning frequency axes are also normalized by the pulse bandwidth, $\Delta_{\omega}$.  %
+
+We now consider the generalized Liouville pathway $L:\rho_0 \xrightarrow{x} \rho_1 \xrightarrow{y}
+\rho_2 \xrightarrow{z} \rho_3 \xrightarrow{\text{out}} \rho_4$, where $x$, $y$, and $z$ denote
+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,
+$2^\prime$ notation for the laser pulses with pathway $V\gamma$.\footnote{For elucidation of the
+  relationship between the generalized Liouville pathway notation and the specific parameters for
+  each Liouville pathway, see Table S1 in the Supplementary Information.}  %
+
+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}
+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}
+	\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)} 
+	e^{-i\left( \kappa_3 \omega_z + \kappa_2 \omega_y + \kappa_1 \omega_x \right) t} \\
+	&\times \iiint_{-\infty}^{\infty} c_z(t-u-\tau_z) c_y(t-u-v-\tau_y) c_x(t-u-v-w-\tau_x) R_L(u,v,w) dw \ dv \ du	,	
+	\end{split}\\
+	R_L(u,v,w) = \Theta(w)e^{-\left(\Gamma_1 + i\kappa_1\Omega_{1x} \right)w}
+	\Theta(v)e^{-\left(\Gamma_2 + i\left[ \kappa_1\Omega_{1x}+\kappa_2\Omega_{2y} \right] \right)v}
+	\Theta(u)e^{-\left(\Gamma_3 + i\left[ \kappa_1\Omega_{1x}+\kappa_2\Omega_{2y}+\kappa_3\Omega_{3z} \right] \right)u},
+	\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}
+E_{\text{tot}}= \sum_L E_L(t).
+\end{equation}
+For the superposition of Equation \ref{eq: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 $\sqrt{2}\mu_{10}\neq\mu_{21}$.  %
+Our simulations use the last inequality, which is important in two-level systems ($\mu_{21}=0$) and
+in systems where state-filling dominates the non-linear response, such as in semiconductor
+excitons.  %
+The exact ratio between $\mu_{10}$ and $\mu_{21}$ affects the absolute amplitude of the field, but
+does not affect the multidimensional line shape.  %
+Importantly, the dipole inequality does not break the symmetry of double quantum coherence pathways
+(time-orderings II and IV), so such pathways are not present in our analysis.  %
+
+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}.  %
+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}
+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).  %
+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).  %
+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.  %
+
+\subsection{Inhomogeneity}
+
+Inhomogeneity is isolated in CMDS through both spectral signatures, such as
+line-narrowing\cite{Besemann2004,Oudar1980,Carlson1990,Riebe1988,Steehler1985}, and temporal
+signatures, such as photon echoes\cite{Weiner1985,Agarwal2002}.  %
+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}.  %
+Dynamic broadening effects such as spectral diffusion are beyond the scope of this work.  %
+
+\section{Methods}  % ------------------------------------------------------------------------------
+
+A matrix representation of differential equations of the type in Equation \ref{eq:E_L_full} was
+numerically integrated for parallel computation of Liouville elements (see SI for
+details).\cite{Dick1983,Gelin2005}  %
+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
+system-pulse difference frequencies and the pulse envelope were well-sampled.  %
+
+The following simulations explore the four-dimensional $(\omega_1, \omega_2, \tau_{21},
+\tau_{22^\prime})$ variable space.  %
+Both frequencies are scanned about the resonance, and both delays are scanned about pulse overlap.
+We explored the role of sample dephasing rate by calculating signal for systems with dephasing
+rates such that $\Gamma_{10}\Delta_t=0.5, 1,$ and $2$.  %
+Inhomogeneous broadening used a spectral FWHM, $\Delta_{\text{inhom}}$, that satisfied
+$\Delta_\text{inhom}/ \Delta_{\omega}=0,0.5,1,$ and $2$ for the three dephasing rates.  %
+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{Oliphant2007}  %
+
+\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}
+
+\begin{figure}
+	\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
+	\caption{
+		The relative importance of FID and driven response for a single quantum coherence as a function
+    of the relative dephasing rate (values of $\Gamma_{10}\Delta_t$ are shown inset).
+		The black line shows the coherence amplitude profile, while the shaded color indicates the
+    instantaneous frequency (see colorbar).
+		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}
+\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
+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.  %
+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, $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)\approx \Gamma_{10}\Delta_t$.  %
+In this case, the evolution demonstrates the maximum impulsive character the transient can
+achieve.  %
+The instantaneous frequency, $d\varphi/dt$, is defined as
+\begin{equation}
+\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}.  %
+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}).  %
+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}).  %
+
+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.  %
+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 $\Gamma_{10}\Delta_t=2.0$.  %
+
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