diff options
| author | Blaise Thompson <blaise@untzag.com> | 2018-01-31 18:37:43 -0600 |
|---|---|---|
| committer | Blaise Thompson <blaise@untzag.com> | 2018-01-31 18:37:43 -0600 |
| commit | f47e1abac351ed10110c9d982aea76ca73dd30a6 (patch) | |
| tree | 8ee65836db6a0c723941fbd7ee0c7dcf1697021f /mixed_domain | |
| parent | fa43cdc18d17b937d24abb6785c467364f14f2ad (diff) | |
2018-01-31 18:37
Diffstat (limited to 'mixed_domain')
| -rw-r--r-- | mixed_domain/SQC lineshapes against t.pdf | bin | 0 -> 171723 bytes | |||
| -rw-r--r-- | mixed_domain/chapter.tex | 987 | ||||
| -rw-r--r-- | mixed_domain/metrics.pdf | bin | 0 -> 141095 bytes | |||
| -rw-r--r-- | mixed_domain/spectral evolution full.pdf | bin | 0 -> 6702377 bytes | |||
| -rw-r--r-- | mixed_domain/steady state.pdf | bin | 0 -> 2112224 bytes | |||
| -rw-r--r-- | mixed_domain/wigners full.pdf | bin | 0 -> 6648960 bytes | |||
| -rw-r--r-- | mixed_domain/wigners.pdf | bin | 0 -> 2311832 bytes |
7 files changed, 978 insertions, 9 deletions
diff --git a/mixed_domain/SQC lineshapes against t.pdf b/mixed_domain/SQC lineshapes against t.pdf Binary files differnew file mode 100644 index 0000000..e932bca --- /dev/null +++ b/mixed_domain/SQC lineshapes against t.pdf diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 4d3ba1c..308146b 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -5,7 +5,7 @@ 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. %
+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
@@ -137,17 +137,20 @@ from these measurement artifacts. % \section{Theory}
+\afterpage{
\begin{figure}
\centering
\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}
- \caption{
+ \caption[Sixteen triply-resonant Liouville pathways.]{
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.
+ 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}
+\clearpage}
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
@@ -231,9 +234,10 @@ 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*}
+\afterpage{
+\begin{figure}
\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}
- \caption{
+ \caption[Overview of the MR-CMDS simulation.]{
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
@@ -249,7 +253,8 @@ The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discuss help introduce our delay convention.
}
\label{fig:overview}
-\end{figure*}
+\end{figure}
+\clearpage}
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
@@ -367,6 +372,214 @@ $S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ are recorded f Liouville pathway. %
Our simulations were done using the open-source SciPy library.\cite{Oliphant2007} %
+\subsection{Characteristics of Driven and Impulsive Response}\label{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
+\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}
+\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}
+The frequency and temporal envelope of the excitation pulse controls the coherence time evolution,
+and the relative amplitude and phase of the coherence is directly related to detuning from
+resonance. %
+
+The impulsive limit is achieved when the excitation pulses are much shorter than response function
+dynamics: $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$. %
+The full description of the temporal evolution has two separate expressions: one for times when the
+pulse is interacting with the system, and one for times after pulse interaction. %
+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}
+\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}
+This is classic free induction decay (FID) evolution: the system evolves at its natural frequency
+and decays at rate $\Gamma_f$. %
+It is important to note that, while this expression is explicitly derived from the impulsive limit,
+FID behavior is not exclusive to impulsive excitation, as we have defined it. %
+A latent FID will form if the pulse vanishes at a fast rate relative to the system dynamics.
+
+For evaluating times near pulse excitation, $t \lesssim \tau_x + \Delta_t$, we implement a Taylor
+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}
+\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 \\
+& \quad +(\Gamma_f + i\kappa_f\Omega_{fx}) \int_{-\infty}^{t-\tau_x} c_x(u)u \ du \bigg].
+\end{split}
+\end{equation}
+During this time $\tilde{\rho}_f$ builds up roughly according to the integration of the pulse
+envelope. %
+The build-up is integrated because the pulse transfers energy before appreciable dephasing or
+detuning occurs. %
+Contrary to the expectation of impulsive evolution, the evolution of $\tilde{\rho}_f$ is explicitly
+affected by the pulse frequency, and the temporal profile evolves according to the pulse. %
+
+It is important to recognize that the impulsive limit is defined not only by having slow relaxation
+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
+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}
+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}
+\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)} \\
+& \times e^{ i(\kappa_3\omega_z + \kappa_2\omega_y + \kappa_1\omega_x)t} \\
+& \times c_z(t-\tau_z)c_y(t-\tau_y)c_x(t-\tau_x) \\
+& \times \frac{1}{\kappa_1\Omega_{1x}-i\Gamma_1} \frac{1}{\kappa_1\Omega_{1x} + \kappa_2\Omega_{2y} - i\Gamma_2} \\
+& \times \frac{1}{\kappa_1\Omega_{1x} + \kappa_2 \Omega_{2y} + \kappa_3\Omega_{3z}-i\Gamma_3}.
+\end{split}
+\end{equation}
+It is important to note that the signal depends on the multiplication of all the fields; pathway
+discrimination based on pulse time-ordering is not achievable because polarizations exists only
+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
+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
+$\omega_x$ 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}
+\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 \\
+& \times e^{-\Gamma_1(\tau_y-\tau_x)} e^{-\Gamma_2(\tau_z-\tau_y)} e^{-\Gamma(t-\tau_z)}.
+\end{split}
+\end{equation}
+Pathway discrimination is demonstrated here because the signal is sensitive to the time-ordering of
+the pulses. %
+This limit is suited for delay scanning techniques. %
+The emitted signal frequency is determined by the system and can be resolved by scanning a
+monochromator. %
+
+The driven and impulsive limits can qualitatively describe our simulated signals at certain
+frequency and delay combinations. %
+Of the three expressions, the FID limit most resembles signal when pulses are near resonance and
+well-separated in time (so that build-up behavior is negligible). %
+The build-up limit approximates well when pulses are near-resonant and arrive together (so that
+build-up behavior is emphasized). %
+The driven limit holds for large detunings, regardless of delay. %
+
+\subsection{Convolution Technique for Inhomogeneous Broadening}\label{sec:convolution}
+
+\afterpage{
+\begin{figure}
+ \includegraphics[width=\linewidth]{mixed_domain/convolve}
+ \caption[Convolution overview.]
+ {Overview of the convolution.
+ (a) The homogeneous line shape.
+ (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}
+\end{figure}
+\clearpage}
+
+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}
+\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).
+\end{split}
+\end{equation}
+
+We consider two oscillators with transition frequencies $\omega_f$ and $\omega_f^\prime=\omega_f +
+\delta$. %
+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}
+\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},
+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 %
+\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) \\
+&\times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1}.
+\end{split}
+\end{equation}
+
+The translation of each laser coordinate depends on which transition is made (e.g. $a$ for
+transitions between $|0\rangle$ and $|1\rangle$ or $b$ for transitions between $|1\rangle$ and
+$|2\rangle$), so the exact translation relation differs between pathways. %
+We can now compute the ensemble average of signal for pathway $V\beta$ as a convolution between the
+distribution function of the system, $K(a,b)$, and the single oscillator response: %
+\begin{equation}
+\begin{split}
+\langle \tilde{\rho}_3 (t;\omega_2,\omega_{2^\prime},\omega_1) \rangle =& \iint K(a,b)\\
+& \times \tilde{\rho}_3 (t;\omega_2+a,\omega_{2^\prime}+a,\omega_1+b) \\
+& \times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1} da \ db.
+\end{split}
+\end{equation}
+For this work, we restrict ourselves to a simpler ensemble where all oscillators have equally
+spaced levels (i.e. $a=b$). %
+This makes the translation identical for all pathways and reduces the dimensionality of the
+convolution. %
+Since pathways follow the same convolution we may also perform the convolution on the total signal field:
+\begin{equation}
+\begin{split}
+\langle E_{\text{tot}}(t) \rangle =& \sum_L \mu_{4,L} \int K(a,a) \\
+& \times \tilde{\rho}_{3,L}(t;\omega_x-a,\omega_y-a\omega_z-a) \\
+& \times e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} da.
+\end{split}
+\end{equation}
+Furthermore, since $\kappa=-1$ for $E_1$ and $E_{2^\prime}$, while $\kappa=1$ for $E_2$, we have
+$e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} = e^{-ia\left( \tau_1 -
+ \tau_2 + \tau_{2^\prime} \right)}$ 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{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
+homogeneous line shape. %
+
\section{Results} % ------------------------------------------------------------------------------
We now present portions of our simulated data that highlight the dependence of the spectral line
@@ -375,9 +588,11 @@ pulse delay times, and inhomogeneous broadening. % \subsection{Evolution of single coherence}\label{sec:evolution_SQC}
+\afterpage{
\begin{figure}
+ \centering
\includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"}
- \caption{
+ \caption[Relative importance of FID and driven response for a single quantum coherence.]{
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
@@ -387,6 +602,7 @@ pulse delay times, and inhomogeneous broadening. % }
\label{fig:fid_dpr}
\end{figure}
+\clearpage}
It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x}
\rho_1$, under various excitation conditions. %
@@ -423,3 +639,756 @@ We note that our choices of $\Gamma_{10}\Delta_t=2.0, 1.0,$ and $0.5$ give coher mainly driven, roughly equal driven and FID parts, and mainly FID components, respectively. %
FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. %
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs detuning"}
+ \caption[Pulsed excitation of a single quantum coherence and its dependance on pulse detuning.]{
+ Pulsed excitation of a single quantum coherence and its dependence on the pulse detuning. In
+ all cases, relative dephasing is kept at $\Gamma_{10}\Delta_t = 1$.
+ (a) The relative importance of FID and driven response for a single quantum coherence as a
+ function of the detuning (values of relative detuning, $\Omega_{fx}/\Delta_{\omega}$, are shown
+ inset).
+ The color indicates the instantaneous frequency (scale bar on right), while the black line
+ shows the amplitude profile. The gray line is the electric field amplitude.
+ %Comparison of the temporal evolution of single quantum coherences at different detunings
+ %(labeled inset).
+ (b) The time-integrated coherence amplitude as a function of the detuning. The integrated
+ amplitude is collected both with (teal) and without (magenta) a tracking monochromator that
+ isolates the driven frequency components. %$\omega_m=\omega_\text{laser}$.
+ For comparison, the Green's function of the single quantum coherence is also shown (amplitude
+ is black, hashed; imaginary is black, solid).
+ In all plots, the gray line is the electric field amplitude.
+ }
+ \label{fig:fid_detuning}
+\end{figure}
+\clearpage}
+
+Fig. \ref{fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.\footnote{
+ See Supplementary Fig. S3 for a Fourier domain representation of Fig. \ref{fig:fid_detuning}a.
+}
+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). %
+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 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.\footnote{
+ See Supplementary Fig. S4 for explicit plots of $\rho_1(\Omega_{fx}/\Delta_{\omega})$ at discrete $t/\Delta_t$ values.
+} %
+Many ultrafast spectroscopies take advantage of the latent FID to suppress non-resonant background,
+improving signal to noise.\cite{Lagutchev2007,Lagutchev2010,Donaldson2010,Donaldson2008} %
+
+In driven experiments, the output frequency and line shape are fully constrained by the excitation
+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
+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
+$\Omega_{1x}/\Delta_{\omega}=\pm 1$, 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}
+\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
+$c_x$, which in our case gives
+\begin{equation}
+\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}:
+\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).
+ \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).
+\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
+$\Gamma_{10}\Delta_t=1$. %
+
+\subsection{Evolution of single Liouville pathway}
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=\linewidth]{"mixed_domain/pw1 lineshapes"}
+ \caption[2D frequency response of a single Liouville pathway at different delay values.]{
+ Changes to the 2D frequency response of a single Liouville pathway (I$\gamma$) at different
+ delay values.
+ The normalized dephasing rate is $\Gamma_{10}\Delta_t = 1$.
+ Left: The 2D delay response of pathway I$\gamma$ at triple resonance.
+ Right: The 2D frequency response of pathway I$\gamma$ at different delay values.
+ 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}
+\end{figure}
+\clearpage}
+
+We now consider the multidimensional response of a single Liouville pathway involving three pulse
+interactions. %
+In a multi-pulse experiment, $\rho_1$ acts as a source term for $\rho_2$ (and subsequent
+excitations). %
+The spectral and temporal features of $\rho_1$ that are transferred to $\rho_2$ depend on when the
+subsequent pulse arrives. %
+Time-gating later in $\rho_1$ evolution will produce responses with FID behavior, while time-gating
+$\rho_1$ in the presence of the initial pulse will produce driven responses. %
+An analogous relationship holds for $\rho_3$ with its source term $\rho_2$. %
+As discussed above, signal that time-gates FID evolution gives narrower spectra than driven-gated
+signal. %
+As a result, the spectra of even single Liouville pathways will change based on pulse delays. %
+
+The final coherence will also be frequency-gated by the monochromator. %
+The monochromator isolates signal at the fully driven frequency $\omega_\text{out} = \omega_1$. %
+The monochromator will induce line-narrowing to the extent that FID takes place. %
+It effectively enforces a frequency constraint that acts as an additional resonance condition,
+$\omega_\text{out}=\omega_1$. %
+The driven frequency will be $\omega_1$ if $E_1$ is the last pulse interaction (time-orderings V
+and VI), and the monochromator tracks the coherence frequency effectively. %
+If $E_1$ is not the last interaction, the output frequency may not be equal to the driven
+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
+$\Gamma_{10}\Delta_t=1$ (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 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.\footnote{
+ See Supplementary Fig. S5 for a representation of Fig. 5 simulated without monochromator frequency filtering ($M(\omega-\omega_1)=1$ for Equation \ref{eq:S_tot}).
+}
+
+\begin{table*}
+ \centering
+ \caption{\label{tab:table2} Conditions for peak intensity at different pulse delays for pathway
+ I$\gamma$.}
+ \begin{tabularx}{0.7\linewidth}{c c | X X X X}
+ \multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\
+ $\tau_{21}/\Delta_t$ & $\tau_{22^\prime}/\Delta_t$ & $\rho_0\xrightarrow{1}\rho_1$ &
+ $\rho_1\xrightarrow{2}\rho_2$ & $\rho_2\xrightarrow{2^\prime}\rho_3$ & $\rho_3\rightarrow$
+ detection at $\omega_m=\omega_1$ \\
+ \hline\hline
+ 0 & 0 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$ & $\omega_1=\omega_{10}$ & -- \\
+ 0 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$ & $\omega_2=\omega_{10}$ &
+ $\omega_1=\omega_2$ \\
+ 2.4 & 0 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & -- &
+ $\omega_1=\omega_{10}$ \\
+ 2.4 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & $\omega_2=\omega_{10}$ &
+ $\omega_1=\omega_2$ \\
+ \end{tabularx}
+\end{table*}
+
+When the pulses are all overlapped ($\tau_{21}=\tau_{22^\prime}=0$, lower right, orange), all
+transitions in the Liouville pathway are simultaneously driven by the incident fields. %
+This spectrum strongly resembles the driven limit spectrum. For this time-ordering, the first,
+second, and third density matrix elements have driven resonance conditions of
+$\omega_1=\omega_{10}$, $\omega_1-\omega_2=0$, and
+$\omega_1-\omega_2+\omega_{2^\prime}=\omega_{10}$, respectively. %
+The second resonance condition causes elongation along the diagonal, and since
+$\omega_2=\omega_{2^\prime}$, the first and third resonance conditions are identical, effectively
+making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical elongation along
+$\omega_1=\omega_{10}$. %
+
+
+The other three spectra of Fig. \ref{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. %
+The final pulse, $E_{2^\prime}$, is latent and probes $\rho_2$ during its FID evolution after
+memory of the driven frequency is lost. %
+There are two important consequences. %
+Firstly, the third driven resonance condition is now approximated by
+$\omega_{2^\prime}=\omega_{10}$, which makes $\omega_1$ only singly resonant at
+$\omega_1=\omega_{10}$. %
+Secondly, the driven portion of the signal frequency is determined only by the latent pulse:
+$\omega_{\text{out}}=\omega_{2^\prime}$. %
+Since our monochromator gates $\omega_1$, we have the detection-induced correlation
+$\omega_1=\omega_{2^\prime}$. %
+The net result is double resonance along $\omega_1=\omega_2$, and the vertical elongation of pulse
+overlap is strongly attenuated. %
+
+At $\tau_{21}=2.4\Delta_t,\tau_{22^\prime}=0$ (upper right, purple), the first pulse $E_1$ precedes
+the latter two, which makes the two resonance conditions for the input fields
+$\omega_1=\omega_{10}$ and $\omega_2=\omega_{10}$. %
+The signal depends on the FID conversion of $\rho_1$, which gives vertical elongation at
+$\omega_1=\omega_{10}$. %
+Furthermore, $\rho_1$ has no memory of $\omega_1$ when $E_2$ interacts, which has two important
+implications. %
+First, this means the second resonance condition $\omega_1=\omega_2$ and the associated diagonal
+elongation is now absent. %
+Second, the final output polarization frequency content is no longer functional of $\omega_1$.
+Coupled with the fact that $E_2$ and $E_{2^\prime}$ are coincident, so that the final coherence can
+be approximated as driven by these two, we can approximate the final frequency as
+$\omega_{\text{out}} = \omega_{10}-\omega_2+\omega_{2^\prime} = \omega_{10}$. %
+Surprisingly, the frequency content of the output is strongly independent of all pulse
+frequencies. %
+The monochromator narrows the $\omega_1=\omega_{10}$ resonance. %
+The $\omega_1=\omega_{10}$ resonance condition now depends on the monochromator slit width, the FID
+propagation of $\rho_1$, the spectral bandwidth of $\rho_3$; its spectral width is not easily
+related to material parameters. %
+This resonance demonstrates the importance of the detection scheme for experiments and how the
+optimal detection can change depending on the pulse delay time. %
+
+Finally, when all pulses are well-separated ($\tau_{21}=-\tau_{22^\prime}=2.4\Delta_t$, upper left,
+cyan), each resonance condition is independent and both $E_1$ and $E_2$ require FID buildup to
+produce final output. %
+The resulting line shape is narrow in all directions. %
+Again, the emitted frequency does not depend on $\omega_1$, yet the monochromator resolves the
+final coherence at frequency $\omega_1$. %
+Since the driven part of the final interaction comes from $E_{2^\prime}$, and since the
+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
+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{Meyer2004,
+ Pakoulev2006,Donaldson2007}, 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. %
+
+\subsection{Temporal pathway discrimination}
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=\linewidth]{"mixed_domain/delay space ratios"}
+ \caption[2D delay response for different relative dephasing rates.]{
+ Comparison of the 2D delay response for different relative dephasing rates (labeled atop each
+ 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
+ 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}
+\end{figure}
+\clearpage}
+
+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}). %
+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
+delay coordinate. %
+
+While the general idea of using time delays to enhance certain time-ordered regions is widely
+applied, quantitation of this discrimination is rarely explored. %
+Because the temporal profile of the signal is dependent on both the excitation pulse profile and
+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
+$\omega_1,\omega_2=\omega_{10}$. %
+It illustrates the interplay of pulse width and system decay rates on the isolation of time-ordered
+pathways. %
+The color bar shows the signal amplitude. %
+Signal is symmetric about the $\tau_{21}=\tau_{22^\prime}$ line because when $\omega_1=\omega_2$,
+$E_1$ 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}
+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. %
+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. %
+
+A commonly-employed metric for temporal selectivity is how definitively the pulses are ordered. %
+This metric agrees with our simulations. %
+The purity contours have a weak dependence on $\Delta_t \Gamma_{10}$ for
+$\left|\tau_{22^\prime}\right|/\Delta_t < 1$ or $\left|\tau_{21}\right|/\Delta_t < 1$ where there
+is significant pathway overlap and a stronger dependence at larger values where the pathways are
+well-isolated. %
+Because responses decay exponentially, while pulses decay as Gaussians, there always exist delays
+where temporal discrimination is possible. %
+As $\Delta_t\Gamma_{10}\rightarrow \infty$, however, such discrimination is only achieved at
+vanishing signal intensities; the contour of $P=0.99$ across our systems highlights this trend. %
+
+\subsection{Multidimensional line shape dependence on pulse delay time}
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=\linewidth]{"mixed_domain/spectral evolution"}
+ \caption[Evolution of the 2D frequency response.]{
+ Evolution of the 2D frequency response as a function of $\tau_{21}$ (labeled inset) and the
+ influence of the relative dephasing rate ($\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and
+ $2.0$ (blue)).
+ In all plots $\tau_{22^\prime}=0$. To ease comparison between different dephasing rates, the
+ colored line contours (showing the half-maximum) for all three relative dephasing rates are
+ overlaid.
+ The colored histograms below each 2D frequency plot show the relative weights of each
+ time-ordering for each relative dephasing rate.
+ Contributions from V and VI are grouped together because they have equal weights at
+ $\tau_{22^\prime}=0$.
+ }
+ \label{fig:hom_2d_spectra}
+\end{figure}
+\clearpage}
+
+In the previous sections we showed how pathway spectra and weights evolve with delay. %
+This section ties the two concepts together by exploring the evolution of the spectral line shape
+over a span of $\tau_{21}$ delay times that include all pathways. %
+It is a common practice to explore spectral evolution against $\tau_{21}$ because this delay axis
+shows population evolution in a manner analogous to pump-probe spectroscopies. %
+The $\vec{k}_2$ and $\vec{k}_{2^\prime}$ interactions correspond to the pump, and the $\vec{k}_1$
+interaction corresponds to the probe. %
+Time-orderings V and VI are the normal pump-probe time-orderings, time-ordering III is a mixed
+pump-probe-pump ordering (so-called pump polarization coupling), and time-ordering I is the
+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{BritoCruz1988,
+ Palfrey1985} 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
+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.\footnote{Supplementary Fig. S6 shows fully
+ colored contour plots of each 2D frequency spectrum.} The dependence of the line shape amplitude
+on $\tau_{21}$ can be inferred from Fig. \ref{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
+$\omega_2$ interact with independent resonances). %
+Such spectral changes could be misinterpreted as spectral diffusion, where the line shape changes
+from correlated to uncorrelated as population time increases due to system dynamics. %
+The system dynamics included here, however, contain no structure that would allow for such
+diffusion. %
+Rather, the spectral changes reflect the changes in the majority pathway contribution, starting
+with time-ordering I pathways, proceeding to an equal admixture of I, III, V, and VI, and finishing
+at an equal balance of V and VI when $E_1$ arrives well after $E_2$ and $E_{2^\prime}$. %
+Time-orderings I and III both exhibit a spectral correlation in $\omega_1$ and $\omega_2$ when
+driven, but time-orderings V and VI do not. %
+Moreover, such spectral correlation is forced near zero delay because the pulses time-gate the
+driven signals of the first two induced polarizations. %
+The monochromator detection also plays a dynamic role, because time-orderings V always VI always
+emit a signal at the monochromator frequency, while in time-orderings I and III the emitted
+frequency is not defined by $\omega_1$, as discussed above. %
+
+When we isolate time-orderings V and VI, we can maintain the proper scaling of FID bandwidth in the
+$\omega_1$ direction because our monochromator can gate the final coherence. %
+This gating is not possible in time-orderings I and III because the final coherence frequency is
+determined by $\omega_{2^\prime}$ which is identical to $\omega_2$. %
+
+There are differences in the line shapes for the different values of the relative dephasing rate,
+$\Gamma_{10}\Delta_t$. %
+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
+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}. %
+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
+elongation direction changing from horizontal to vertical. %
+The general shrinking reflects the narrowing homogeneous linewidth of the $\omega_{10}$
+resonance. %
+In all cases, the horizontal line shape corresponds to the homogeneous linewidth because the narrow
+bandpass monochromator resolves the final $\omega_1$ resonance. %
+The change in elongation direction is due to the resolving power of $\omega_2$. %
+At $\Gamma_{10}\Delta_t=0.5$, the resonance is broader than our pulse bandwidth and is fully
+resolved vertically. %
+It is narrower than the $\omega_1$ resonance because time-orderings V and VI interfere to isolate
+only the absorptive line shape along $\omega_2$. %
+This narrowing, however, is unresolvable when the pulse bandwidth becomes broader than that of the
+resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\Delta_t=0.5$. %
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=\linewidth]{"mixed_domain/wigners"}
+ \caption[Wigners.]{
+ Transient ($\omega_1$) line shapes and their dependence on $\omega_2$ frequency.
+ The relative dephasing rate is $\Gamma_{10}\Delta_t=1$ and $\tau_{22^\prime}=0$.
+ For each plot, the corresponding $\omega_2$ value is shown as a light gray vertical line.}
+ \label{fig:wigners}
+\end{figure}
+\clearpage}
+
+It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is
+frequency.\cite{Kohler2014, Aubock2012,Czech2015,Pakoulev2007} %
+In Fig. \ref{fig:wigners} we show five $\tau_{21},\omega_1$ plots for varying $\omega_2$ with
+$\tau_{22^\prime}=0$.\footnote{See Supplementary Fig. S8 for Wigner plots for all
+ $\Gamma_{10}\Delta_t$ values.} %
+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
+delay. %
+For each plot, the $\omega_2$ frequency is denoted by a vertical gray line. %
+Each Wigner plot is scaled to its own dynamic range to emphasize the dependence on $\omega_2$. %
+The dramatic line shape changes between positive and negative delays can be seen. %
+This representation also highlights the asymmetric broadening of the $\omega_1$ line shape near
+pulse overlap when $\omega_2$ becomes non-resonant. %
+Again, these features can resemble spectral diffusion even though our system is homogeneous. %
+
+\subsection{Inhomogeneous broadening}
+
+\afterpage{
+\label{sec:res_inhom}
+\begin{figure}
+ \centering
+ \includegraphics[width=0.5\linewidth]{"mixed_domain/inhom delay space ratios"}
+ \caption[2D delay response with inhomogeneity.]{
+ 2D delay response for $\Gamma_{10}\Delta_t=1$ with sample inhomogeneity. %
+ 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
+ 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
+ response ($\tau_{21}=0$). %
+ 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}
+\end{figure}
+\clearpage}
+
+With the homogeneous system characterized, we can now consider the effect of inhomogeneity. %
+For inhomogeneous systems, time-orderings III and V are enhanced because their final coherence will
+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
+$\Gamma_{10}\Delta_t=1$ 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{Weiner1985,Fleming1998,Boeij1998,Salvador2003} %
+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
+ability of the system. %
+An inhomogeneous system will emit a photon echo in the rephasing pathway, enhancing signal in the
+rephasing time-ordering and creating the peak shift. %
+In our 2D delay space, the $\tau$ trace can be defined if we assume $E_2$ and $E_{2^\prime}$ create
+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 $T$.\footnote{
+ See Supplementary Fig. S9 for an illustration of how 3PEPS shifts are measured from a 2D delay
+ plot.} %
+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.\footnote{
+ Supplementary Fig. S10 shows the 3PEPS measurements of all 12 combinations of
+ $\Gamma_{10}\Delta_t$ and $\Delta_{\text{inhom}}$, for every population delay surveyed.} %
+Fig. \ref{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. %
+
+The unanticipated feature of the 3PEPS analysis is the dependence on $T$. %
+Even though our inhomogeneity is static, the peak shift is maximal at $T=0$ and dissipates as $T$
+increases, mimicking spectral diffusion. %
+This dynamic arises from signal overlap with time-ordering III, which uses $E_2$ and $E_1$ as the
+first two interactions ,and merely reflects $E_1$ and $E_2$ temporal overlap. %
+At $T=0$, the $\tau$ trace gives two ways to make a rephasing pathway (time-orderings III and V)
+and only one way to make a non-rephasing pathway (time-ordering VI). %
+This pathway asymmetry shifts signal away from $\tau=0$ into the rephasing direction. %
+At large $T$ (large $\tau_{21}$), time-ordering III is not viable and pathway asymmetry
+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 $T=0$. %
+The contamination of the 3PEPS measurement at pulse overlap is well-known and is described in some
+studies,\cite{DeBoeij1996,Agarwal2002} 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. %
+
+\afterpage{
+\begin{figure}
+ \centering
+ \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
+ ($\Delta_{\text{inhom}}=0.441\Gamma_{10}$).
+ Relative dephasing rates are $\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and $2.0$ (blue).
+ In all plots $\tau_{22^\prime}=0$.
+ To ease comparison between different dephasing rates, the colored line shapes of all three
+ systems are overlaid.
+ Each 2D plot shows a single representative contour (half-maximum) for each
+ $\Gamma_{10}\Delta_t$ 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
+ time-orderings V and VI unequal.
+ }
+ \label{fig:inhom_2d_spectra}
+\end{figure}
+\clearpage}
+
+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
+distribution.\footnote{As in Fig. \ref{fig:hom_2d_spectra}, Fig. \ref{fig:inhom_2d_spectra} shows
+ only the contours at the half-maximum amplitude. See Supplementary Fig. S7 for all contours.} %
+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},
+$\tau_{21}=2.0\Delta_t$) 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. %
+
+\section{Discussion} % ---------------------------------------------------------------------------
+
+\subsection{An intuitive picture of pulse effects}
+
+Our chosen values of the relative dephasing time, $\Gamma_{10}\Delta_t$, describe experiments where
+neither the impulsive nor driven limit unilaterally applies. %
+We have illustrated that in this intermediate regime, the multidimensional spectra contain
+attributes of both limits, and that it is possible to judge when these attributes apply. %
+In our three-pulse experiment the second and third pulses time-gate coherences and populations
+produced by the previous pulse(s), and the monochromator frequency-gates the final coherence. %
+Time-gating isolates different properties of the coherences and populations. Consequently, spectra
+evolve against delay. %
+For any delay coordinate, one can develop qualitative line shape expectations by considering the following three principles:
+\begin{enumerate}
+ \item When time-gating during the pulse, the system pins to the driving frequency with a buildup efficiency determined by resonance.
+ \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
+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 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}. %
+
+\subsection{Conditional validity of the driven limit}
+
+We have shown that the driven limit misses details of the line shape if $\Gamma_{10} \Delta_t
+\approx 1$, 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
+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 $\left(\tau_{22^\prime},
+ \tau_{21}=(0,0)\right)$ and $(0,-4\Delta_t)$, respectively. %
+The third column demonstrates the agreement between the driven limit approximations with the
+simulation by comparing the diagonal and anti-diagonal cross-sections of the 2D spectra. %
+
+% TODO: [ ] population resonance is not clear
+Note the very sharp diagonal feature that appears for $(\tau_{21},\tau_{22^\prime}) = (0,0)$ and $
+\Gamma_{11}=0$; this is due to population resonance in time-orderings I and III. %
+This expression is inaccurate: the narrow resonance is only observed when pulse durations are much
+longer than the coherence time. %
+A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes (Yurs
+\textit{et al.}\cite{Yurs2011} and Kohler \textit{et al.}\cite{Kohler2014}, respectively)
+demonstrates this difference well. %
+The driven equation fails to reproduce our numerical simulations here because resonant excitation
+of the population is impulsive; the experiment time-gates only the rise time of the population, yet
+driven theory predicts the resonance to be vanishingly narrow ($\Gamma_{11}=0$). %
+In light of this, one can approximate this time-gating effect by substituting population lifetime
+with the pulse duration ($\Gamma_{11}\Delta_t=1$), which gives good agreement with the numerical
+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). %
+Only time-orderings V and VI are relevant. %
+The intermediate population resonance is still impulsive but it depends on
+$\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space. %
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=\linewidth]{"mixed_domain/steady state"}
+ \caption[Conditional validity of the driven limit.]{
+ Comparing approximate expressions of the 2D frequency response with the directly integrated
+ response. %
+ $\Gamma_{10}\Delta_t=1$. %
+ The top row compares the 2D response of all time-orderings ($\tau_{21}=0$) and the bottom row
+ compares the response of time-orderings V and VI ($\tau_{21}=-4\Delta_t$). %
+ First column: The driven limit response. Note the narrow diagonal resonance for $\tau_{21}=0$.
+ 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}
+\end{figure}
+\clearpage}
+
+\subsection{Extracting true material correlation}
+
+\afterpage{
+\begin{figure}
+ \centering
+ \includegraphics[width=0.5\linewidth]{"mixed_domain/metrics"}
+ \caption[Metrics of correlation.]{
+ Temporal (3PEPS) and spectral (ellipticity) metrics of correlation and their relation to the
+ true system inhomogeneity. %
+ The left column plots the relationship at pulse overlap ($T=0$) and the right column plots the
+ relationship at a delay where driven correlations are removed ($T=4\Delta_t$). %
+ For the ellipticity measurements, $\tau_{22^\prime}=0$. %
+ In each case, the two metrics are plotted directly against system inhomogeneity (top and middle
+ row) and against each other (bottom row). %
+ Colored lines guide the eyes for systems with equal relative dephasing rates
+ ($\Gamma_{10}\Delta_t$, see upper legend), while the area of the data point marker indicates
+ the relative inhomogeneity ($\Delta_{\text{inhom}}/\Gamma_{10}$, see lower legend). %
+ Gray lines indicate contours of constant relative inhomogeneity (scatter points with the same
+ area are connected). %
+}
+ \label{fig:metrics}
+\end{figure}
+\clearpage}
+
+We have shown that pulse effects mimic the qualitative signatures of inhomogeneity. %
+Here we address how one can extract true system inhomogeneity in light of these effects. %
+We focus on two ubiquitous metrics of inhomogeneity: 3PEPS for the time domain and
+ellipticity\footnote{
+ There are many ways to characterize the ellipticity of a peak shape.
+ We adopt the convention $\mathcal{E} = \left(a^2-b^2\right) / \left(a^2+b^2\right)$, where $a$ is the diagonal width and $b$ is the antidiagonal width.}
+for the frequency domain\cite{Kwac2003,Okumura1999}. %
+In the driven (impulsive) limit, ellipticity (3PEPS) corresponds to the frequency correlation
+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
+$\Gamma_{10}\Delta_t$ 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 relative inhomogeneity $\Delta_\text{inhom} / \Gamma_{10}$, and the population time delay.\footnote{
+ The simulations for each value of the 3PEPS and ellipticity data in Fig. \ref{fig:metrics} appear in Supplementary Figs. S10-S12.}
+The top row shows the correlations of the $\Delta_\text{3PEPS} / \Delta_t$ 3PEPS metric that
+represents the normalized coherence delay time required to reach the peak intensity. %
+The upper right graph shows the correlations for a population time delay of $T = 4\Delta_t$ that
+isolates the V and VI time-orderings. %
+For this time delay, the $\Delta_\text{3PEPS} / \Delta_t$ metric works well for all dephasing times
+of $\Gamma_{10}\Delta_t$ when the relative inhomogeneity is $\Delta_\text{inhom} / \Delta_\omega
+\ll 1$. %
+It becomes independent of $\Delta_\text{inhom} / \Delta_\omega$ when $\Delta_\text{inhom} /
+\Delta_\omega > 1$. %
+This saturation results because the frequency bandwidth of the excitation pulses becomes smaller
+than the inhomogeneous width and only a portion of the inhomogeneous ensemble contributes to the
+3PEPS experiment.\cite{Weiner1985} %
+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}. %
+
+The middle row in Fig. \ref{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
+$\Delta_\text{inhom} / \Delta_\omega \ll 1$ where the excitation bandwidth is wide compared with
+the inhomogeneity. %
+Unlike 3PEPS, saturation is not observed because pulse bandwidth does not limit the frequency range
+scanned. %
+The 3PEPS and ellipticity metrics are therefore complementary since 3PEPS works well for
+$\Delta_\text{inhom} / \Delta_\omega \ll 1$ and ellipticity works well for $\Delta_\text{inhom} /
+\Delta_\omega \gg 1$. %
+When all pulses are temporally overlapped at $T = 0$, the ellipticity is only weakly dependent on
+the inhomogeneity and dephasing rate. %
+The ellipticity is instead dominated by the dependence on the excitation pulse frequency
+differences of time-orderings I and III that become important at pulse overlap. %
+
+It is clear from the previous discussion that both metrics depend on the dephasing and
+inhomogeneity. %
+The dephasing can be measured independently in the frequency or time domain, depending upon whether
+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
+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. %
+
+Importantly, the metrics are uniquely mapped both in the presence and absence of pulse-induced
+effects (demonstrated by $T = 0$ and $T = 4\Delta_t$, respectively). %
+The combined metrics can be used to determine correlation at $T = 0$, but the correlation-inducing
+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{Aartsma1976} %
+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. %
+If the metrics are measured as a function of $T$, the mapping gradually changes from the left
+figure to the right figure in accordance with the pulse overlap. %
+Both metrics will show a decrease, even with static inhomogeneity. %
+If a system has spectral diffusion, the mapping at late times will disagree with the mapping at
+early times; both ellipticity and 3PEPS will be smaller at later times than predicted by the change
+in mappings alone. %
+
+\section{Conclusion} % ---------------------------------------------------------------------------
+
+This study provides a framework to describe and disentangle the influence of the excitation pulses
+in mixed-domain ultrafast spectroscopy. %
+We analyzed the features of mixed domain spectroscopy through detailed simulations of MR-CMDS
+signals. %
+When pulse durations are similar to coherence times, resolution is compromised by time-bandwidth
+uncertainty and the complex mixture of driven and FID response. %
+The dimensionless quantity $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)$ captures the
+balance of driven and FID character in a single field-matter interaction. %
+In the nonlinear experiment, with multiple field-matter interactions, this balance is also
+controlled by pulse delays and frequency-resolved detection. %
+Our analysis shows how these effects can be intuitive. %
+
+The dynamic nature of pulse effects can lead to misleading changes to spectra when delays are
+changed. %
+When delays separate pulses, the spectral line shapes of individual pathways qualitatively change
+because the delays isolate FID contributions and de-emphasize driven response. %
+When delays are scanned across pulse overlap, the weights of individual pathways change, further
+changing the line shapes. %
+In a real system, these changes would all be present in addition to actual dynamics and spectral
+changes of the material. %
+
+Finally, we find that, in either frequency or time domain, pulse effects mimic signatures of
+ultrafast inhomogeneity. %
+Even homogeneous systems take on these signatures. %
+For mixed domain experiments, pulse effects induce spectral ellipticity and photon echo signatures,
+even in homogeneous systems. %
+Driven character gives rise to pathway overlap peak shifting in the 2D delay response, which
+artificially produces rephasing near pulse overlap. %
+Driven character also produces resonances that depend on $\omega_1-\omega_2$ near pulse overlap. %
+Determination of the homogeneous and inhomogeneous broadening at ultrashort times is only possible
+by performing correlation analysis in both the frequency and time domain. %
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