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\subsection{Comparison of TA and TG line shapes}
We first attempted simple fits on a subset of the data to reduce the parameter complexity.
We consider our experimental data at $T=120$ fs to remove contributions from probe-pump and pump-probe-pump time-ordered processes.
By further restricting our considerations to a single probe slice ($\omega_2 = \omega_{\text{1S}}$), we discriminate against inhomogeneous broadening and thus neglect ensemble effects for initial considerations.
We fit our probe spectrum with Equation \ref{eq:chi3_lorentz} along with the added treatment of convolving the response with our monochromator instrumental function.
Proper accounting of all time-ordered pathways, finite pulse bandwidth, and inhomogeneity are treated later.
\begin{figure}
\includegraphics[scale=0.5]{kramers_kronig}
\caption{Kramers-Kronig analysis of TA spectra compared with TG spectra.}
\label{fig:kramers_kronig}
\end{figure}
We find that the TA spectra are more sensitive to the model parameters than TG, and that the parameter interplay necessary to reproduce the spectra can be easily described.
We note three features of the TA spectra that are crucial to reproduce in simulation: (1) the net bleach; (2) the photon energy of the bleach feature minimum is blue of the 1S absorption peak; (3) the probe spectrum decays quickly to zero away from the bleach resonance, leaving a minimal ESA feature to the red.
These features are consistent with the vast majority of published TA spectra of the 1S exciton,\cite{Trinh2013,Schins2009,Gesuele2012,Gdor2013a,Kraatz2014,DeGeyter2012} and can only be reproduced when all three of our nonlinearities (state-filling, excitation-induced broadening, and Coulombic coupling) are invoked (exposition of this result is found in supporting information).
The extracted fit parameters are listed in Table \ref{tab:fit1}.
\begin{table}[]
\centering
\caption{Parameters used in fitting experimental probe slices using Equation \ref{eq:chi3_lorentz}; $S=0.75$, $\omega_2 = \omega_\text{1S}$.}
\label{tab:fit1}
\begin{tabular}{l|cc}
& \multicolumn{2}{l}{Batch} \\
& A & B \\
\hline
$ \varepsilon_\text{Coul} \left(\text{cm}^{-1}\right)$ & 81 & 53 \\
$ \Gamma_{10} \left(\text{cm}^{-1}\right)$ & 380 & 200 \\
$ \xi $ & 1.35 & 1.39
\end{tabular}
\end{table}
With the 1S band TA well-characterized, we now consider applying the fitted parameters to the TG signal.
Transferring this simulation to the TG data poses technical challenges.
A critical factor is appropriately scaling the TG signals relative to TA signals.
The TA simulation can define the scaling through the Kramers-Kronig transform, which gives the transient refraction.
The computed transient refraction is unique to within an arbitrary offset; for a single resonant TA/TG feature, the transient refraction offset is zero.
We take the offset to be zero now and address this assumption later.
The transient refraction (Figure \ref{fig:cw_sim1}, third column) shows highly dispersive character with a node near resonance.
This means that there is a point in our spectrum at which $\left| \chi^{(3)} \right| = \left| \text{Im}\left[ \chi^{(3)} \right] \right|$.
Of course, we also have the constraint $ \left| \chi^{(3)} \right| \geq \left| \text{Im} \left[ \chi^{(3)} \right] \right|$ for every probe color.
These two constraints uniquely determine the appropriate scaling factor as the minimum scalar $c_0$ that satisfies $c_0 S_{\text{TG}} \geq \left| S_{\text{TA}} \right|$ for all probe colors.
%Such a scaling of the experimental data is consistent with our TA fit because the peak TA component is nearly equal to the peak TG amplitude (when the arbitrary offset of the KK-transform is zero).
As we alluded, the arbitrary offset of the Kramers-Kronig transform deserves special consideration.
%A single TA resonance should not cause an offset in the transient reflection spectrum, but it is conceivable that states outside our spectral range are strongly coupled to the 1S band and produce strong refractive signals at these colors.
%While the peaked TA line shape might seem to imply a dispersively shaped $\text{Re} \left[ \chi^{(3)} \right]$ with a node near the bleach center, this is not guaranteed by the Kramers-Kronig relations.
The physical origin for this offset would be coupling between the 1S band and states outside our spectral range.
If the coupling is sufficiently strong, the $\text{Re}\left[ \chi^{(3)} \right] $ offset may be large enough to remove the node, invalidating the minimum scaling factor method.
We believe such a large offset is not viable for several reasons.
From a physical standpoint, it seems very unlikely a non-resonant state would have coupling stronger coupling to the 1S band than the 1S band itself. % DK: elaborate?
Furthermore, TA studies with larger spectral ranges see no evidence of coupling features with strength comparable to the 1S bleach.\cite{Gdor2013a,Trinh2013}
Finally, a sufficiently large refractive offset would produce a dip in the $\left| \chi^{(3)} \right|$ line shape near the FHWM points; such features are definitively absent in the TG spectra.
% DK: a more direct topic sentence might be nice--jump to the fact that equation 4 is inadequate
%While we have confidence in relating the TG and TA measurements using the minimum scaling factor, we find two inconsistencies when fitting Equation \ref{eq:chi3_lorentz} with experiment.
%Firstly, the TA simulation gives a resonant bleach factor that is much greater than that predicted by state-filling alone: for instance, with $1-S=0.25$, we see $\phi>0.5$ for both batches (Figure \ref{fig:cw_sim1}, first column).
%While our parameters successfully recreate the features of the TA line shapes, the simulation grossly overshoots the magnitude of the non-linearity.
While we have confidence in relating the TG and TA measurements using the minimum scaling factor, Equation \ref{eq:chi3_lorentz} fails to accurately reproduce the TG spectrum (Figure \ref{fig:cw_sim1}, third column).
The errors are systematic: in both batches, our simulation misses the characteristic red skew of our experimental TG and instead skews signal to the blue.
Based on the excellent agreement with $S_{\text{TA}}$ (Figure \ref{fig:cw_sim1}, second column), it follows that the chief source of error in our simulation lies in $\text{Re} \left[ \chi^{(3)} \right]$.
The experimentally-consistent $\text{Re} \left[ \chi^{(3)} \right]$ can be calculated from Equation \ref{eq:chi_real} (Figure \ref{fig:cw_sim1}, fourth column).
The dark green curve highlights which of the two roots of Equation \ref{eq:chi_real} is closest to our simulation.
The choice of a negative non-linear refractive index value on resonance is in agreement with z-scan measurements.\cite{Moreels2006}
The discrepancy between the experimental and simulated real components is well-approximated by a constant offset.
\begin{figure}
\includegraphics[width=0.5\linewidth]{"CW_sim2"}
\caption{Top row: Global fits of $S_\text{TA}$ (blue), $S_\text{TG}$ (red), and the associated real projection (green) using Equation \ref{eq:offset_fit}. Light colors indicate the simulations and the darker lines indicate the experimental data. Bottom row: Final simulated absorption spectra for the excited state and the ground state.}
\label{fig:cw_sim2}
\end{figure}
The presence of this offset forced a re-evaluation of the model.
By revising our simulation to include a complex-valued offset term, $\Delta = |\Delta| e^{i\theta}$, so that
\begin{equation}\label{eq:offset_fit}
\chi^{(3)} = \text{Im} \left[ L_0(\omega_2) \right] \left[ SL_1(\omega_1) - L_0(\omega_1) + \Delta \right],
\end{equation}
the discrepancy between $S_{\text{TA}}$ and $S_{\text{TG}}$ can be resolved.
It was found, however, that minimizing error between Equation \ref{eq:offset_fit} and the two datasets alone does not confine all variables uniquely.
Specifically, the effects of the EID non-linearity ($\xi$) on the line shapes were highly correlated with those of the broadband photoinduced absorption (PA) term, $\text{Im}\left[ \Delta \right]$, so that getting a unique parameter combination was not possible.
The fitting routine was robust,\footnote{
Robustness here is defined as the ability to permute the fitting parameter order when minimizing the residual. For example, $\tau_{10}$ can be fit either before or after $\Delta$ is fit without significantly changing the resulting parameters.
} however, when the resonant bleach magnitude was pinned to the state-filling: $\phi \approx 1-S$.
The resulting parameters are shown in Table \ref{tab:fit2}, and the results of the fit are shown in Figure \ref{fig:cw_sim2}.
As both $\phi=0.25$ and $\phi_{\text{int}}=0.25$ have been measured, this added constraint has a reasonable precedence.
As mentioned earlier, EID and Coulombic coupling prevent this equality (as in equation \ref{eq:chi3_lorentz}), but PA can counteract these terms and keep the resonant bleach near $1-S$.
In addition, since EID and PA are correlated, decreasing the resonant bleach also decreases the need for EID to influence the line shape, which results in a reduced EID non-linearity in this fit (compare $\xi$ in Table \ref{tab:fit1} and Table \ref{tab:fit2}).
\begin{table}[]
\centering
\caption{Parameters of the simulated $\chi^{(3)}$ response extracted by global fits of TA and TG at $T=120$ fs using Equation \ref{eq:offset_fit} and with $S=0.75$. Numbers in parentheses refer to fits at $T=300$ fs}
\label{tab:fit2}
\begin{tabular}{l|cc}
Batch & A & B \\
\hline
$ \Gamma_{10} \left( \text{cm}^{-1} \right) $ & 340 (320) & 210 (210) \\
$ \xi $ & 1.07 (1.04) & 1.05 (1.02) \\
$ \epsilon_\text{Coul} \left( \text{cm}^{-1} \right)$ & 54 (46) & 28 (26) \\
$ \left|\Delta \right| / \text{Im}\left[ L_0(\omega_\text{1S}) \right] $ & 0.07 (0.06) & 0.06 (0.06) \\
$ \theta \left( \text{deg} \right)$ & 151 (156) & 146 (148)
\end{tabular}
\end{table}
The nature of $\Delta$ may be associated with resonant and non-resonant transitions from the 1S excitonic state to the continuum of intraband states involving the electron and/or hole.
The magnitude and phase of this contribution would then depend on the ensemble average from all transitions.
This contribution has been identified in previous TA studies.
DeGeyter et.al. isolated a net absorption at sub-bandgap probe
frequencies.\cite{DeGeyter2012}
Geigerat et.al. found an absorptive contribution was needed to explain the fluence dependence of the 1S-resonant bleach.\cite{Geiregat2014}
The absorptive contribution was $5\%$ of the 1S absorbance, a value that agrees with this work (see Table \ref{tab:fit2}).
Our data unifies both observations by showing that additional contribution persists at both bandgap and sub-bandgap frequencies.
In addition, our data provides the spectral phase of the contribution.
It also shows that the red skew of the TG line shape is very sensitive to the relative importance of the 1S resonance and the additional contribution.
There may also be a relationship to studies on CdSe quantum dots where an additional broad ESA feature was observed for $\omega_1 < \omega_{\text{1S}}$.
The feature had separate narrow and broad components.
The narrow component closest to the band edge bleach corresponded to the Coulombically shifted biexciton transition.
Since the broad component correlated with inadequate surface passivation, it was attributed to the surface inducing ESA transitions to the broad band of continuum states that would normally be forbidden.
In addition to creating additional ESA transitions, it also created a short-lived transient that was similar to the transients attributed to multiexciton relaxation and multiexcion generation.
\subsection{Determination of State Filling Factor}
% Given the
%We measured the peak $chi^{(3)}$ hyperpolarizability of Batch B via standard additions (SI) and found good agreement with the TA hyperpolarizability from Equation \ref{eq:gamma3_state_filling} (for $\phi=0.25$).
%A offset that would remove the node would create a difference between the peak $\chi^{(3)}$ values measured from TA and TG.
%To check this possibility we measured the absolute susceptibility of the TG response and compared it to the susceptibility due to TA.
Our results show that the peak susceptibility is almost entirely imaginary, which means we can attribute our TG peak $\left|\chi^{(3)}\right|$ susceptibility to the TA state-filling factor.
A standard addition method was used to extract the peak TG hyperpolarizability of $\left| \gamma^{(3)} \right|=7\pm 2 \cdot 10^{-31} \text{cm}^6 \ \text{erg}^{-1}$ for Batch A, while Equation \ref{eq:gamma3_state_filling} predicts a peak imaginary hyperpolarizability of $\text{Im}\left[ \gamma^{(3)} \right] =2.1\phi \cdot 10^{-30} \text{cm}^6 \ \text{erg}^{-1}$.
The two values are similar for $\phi=0.25$, while $\phi=0.125$ gives an imaginary component that is much smaller than the total susceptibility.
We conclude that only the $\phi=0.25$ bleach factor is consistent with our observations. % DK: also discuss what S likely is
\subsection{Inhomogeneity and the Pulse Overlap Response}
Our parameter extraction above gives plausible parameters to explain the observed photophysics of a small slice of our multidimensional data.
We now apply a more rigorous simulation of the model system to address the entire dataset and consider the broader experimental space.
This rigorous simulation is meant to account for the complex signals that arise at temporal pulse overlap, the pulsed nature of our excitation beams, and sample inhomogeneity.
We calculate signal through numerical integration techniques. % DK: cite paper 1
The homogeneous and inhomogeneous broadening were constrained to compensate each other so that the total ensemble line shape was kept constant and equal to that extracted from absorption measurements (Table \ref{tab:QD_abs}).\footnote{
For a Lorentzian of FWHM $2\Gamma_{10}$ and a Gaussian line shape of standard deviation $\sigma_{\text{inhom}}$, the resulting Voigt line shape has a FWHM well-approximated by $\text{FWHM}_{\text{tot}} \left[ \text{cm}^{-1} \right] \approx 5672 \Gamma_{10}\left[ \text{fs}^{-1} \right] + \sqrt{2298 \Gamma_{10}\left[ \text{fs}^{-1} \right] + 8 \ln 2 \left(\sigma_{\text{inhom}}\left[ \text{cm}^{-1} \right]\right) }$.\cite{Olivero1977}
}
\begin{table}[]
\centering
\caption{}
\label{tab:fit3}
\begin{tabular}{l|cc}
Batch & A & B \\
\hline
$ \Gamma_{10} \left( \text{cm}^{-1} \right) $ & 220 & 130 \\
$\text{FWHM}_\text{inhom} \left( \text{cm}^{-1} \right)$ & 520 & 360
\end{tabular}
\end{table}
Rather than a complete global fit of all parameters, we accept the non-linear parameters extracted earlier (Table \ref{tab:fit2}) and optimize the inhomogeneous-homogeneous ratio only, using the ellipticity of the 2D peak shape\cite{Okumura1999} at late population times as the figure of merit.
Table \ref{tab:fit3} shows the degree of inhomogeneity in the sample from matching the ellipticity of the peak shape.
As expected, Batch B fits to a smaller Gaussian distribution of transition energies than Batch A, but it also exhibits a longer homogeneous dephasing time, suggesting different system-bath coupling strengths for both samples.
Previous research on the coherent dynamics of PbSe 1S exciton feature noted homogeneous lifetimes are a significant source of broadening on the 1S exciton;\cite{Kohler2014} our results demonstrate that the relationship between exciton size distribution and 1S exciton linewidth is further complicated by sample-dependent system-bath coupling.
\begin{figure}
\includegraphics[width=\linewidth]{"movies_fitted"}
\caption{
Global simulation using numerical integration and comparison with experiment.
Batches A (left block) and B (right block) are shown, with the TG experimental (top), the simulated TG (2nd row), the experimental TA (3rd row), and the simulated TA (bottom row) data.
Pump probe delay times of $T=0$, and $120$ fs are shown in each case (see
column labels). For each pair, the colors are globally normalized and the
contours are locally normalized.}
\label{fig:nise_fits}
\end{figure}
The results of this final simulation are compared with the experimental data in Figure \ref{fig:nise_fits}.
It is important to note that the simulations get many details of the rise-time spectra correct.
Particularly, the transition from narrow, diagonal line shape to a broad, less diagonal line shape is reproduced very well in both TA and TG simulations.
Such behavior is expected for responses from excitonic peaks of material systems; the rise time behavior for such systems was studied in detail previously.\cite{Kohler2017}
%One qualitative disagreement between the experiment and the simulation is the amount of excited-state absorption in ππTA at $\omega_1 < $ππ1 < ππ1S, ππ=120fs.
%We attribute this to the aforementioned broadening of the signal from CW simulations, from which most parameters are taken.
Because these simulations do not account for hot-exciton creation from the pump, simulations differ from experiment increasingly as the pump becomes bluer than the 1S center.
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