\subsection{Pump-Probe 3D acquisitions for TA and TG}

\begin{figure}
	\includegraphics[scale=0.5]{"movies_combined"}
	\caption{$S_{\text{TG}}$ (left) and $S_{\text{TA}}$ 2D spectra (see colorbar
    labels) of Batch A (top) and Batch B (bottom) as a function of T delay. The
    colors of each 2D spectrum are normalized to the global maximum of the 3D
    acquisition, while the contour lines are normalized to each particular 2D
    spectrum.}
	\label{fig:movies}
\end{figure}

For both samples, 2D spectra were collected for increments along the population rise time. 
For these acquisitions, concentrated samples ($\text{OD}_{\text{1S}} \sim 0.6, 0.8$) were used to minimize contributions from non-resonant background. 
Both samples maintained constant signal amplitude for at least hundreds of picoseconds after initial excitation, indicating multiexcitons and trapping were negligible effects in these studies. 
The TA and TG results for both batches are shown in Figure \ref{fig:movies}. For $T<0$ (probe arrives before pump), both collections show spectral line-narrowing in the anti-diagonal direction. 
This highly correlated line shape is indicative of an inhomogeneous distribution, but the correlation is enhanced by pulse overlap effects. When the probe arrives before or at the same time as the pump, the typical pump-probe pathways are suppressed and more unconventional pathways with probe-pump and pump-probe-pump pulse orderings are enhanced.
Such pathways exhibit resonant enhancement when $\omega_1=\omega_2$, even in the absence of inhomogeneity. 
The pulse overlap effect is well-understood in both TA\cite{BritoCruz1988} and TG\cite{Kohler2017} experiments. 

After the initial excitation rise time ($T > 50$ fs), the signal reaches a maximum, followed by a slight loss of signal ($\sim 10\%$) over the course of ~150 fs, after which the signal converges to a line shape that remains static over the dynamic range of our experiment ($200$ ps). 
This signal loss occurs in both samples in both TA and TG; in TA measurements, the loss of amplitude occurred on both the ESA feature and the bleach feature, so that the band integral\cite{Gdor2013a} did not appreciably change. We do not know the cause of this loss, but speculate it could be a signature of bandgap renormalization.

The static line shape distinguishes the homogeneous and inhomogeneous contributions to the 1S band. 
The elongation of the peak along the diagonal, relative to the antidiagonal, demonstrates a persistent correlation between the pumped state and excited state; we attribute this correlation to the size distribution of the synthesized quantum dots. 
The diagonal elongation is much more noticeable in the TA spectrum; the TG spectra is much more elongated along the $\omega_1$ axis, which makes discerning the antidiagonal and diagonal widths more difficult. 
The TG spectrum is elongated along $\omega_1$ because it measures both the absorptive and refractive components of the probe spectrum, while it is sensitive only to the absorptive components along the pump axis. 
At all delays, Batch A exhibits a much broader diagonal line shape than that of Batch B, indicative of its larger size distribution.

Our spectra show that the 2D line shape of the 1S exciton is significantly distorted by contributions from hot carrier excitation just above the 1S state. 
These hot carriers arise from transitions between the 1S and 1P resonances, which have been attributed to either the “rising edge” of the continuum or the pseudo-forbidden 1S-1P exciton transition\cite{Schins2009,Peterson2007}. 
Contributions from these hot carriers distort the 1S 2D line shape for $\omega_2 > \omega_{\text{1S}}$, resulting in a bleach feature centered at $\omega_1=\omega_{\text{1S}}$ and containing bleach contributions from the unresolved ensemble. 
The rise time of this feature is indistinguishable from the 1S rise time, indicating either extremely fast ($\leq 50$ fs) relaxation or direct excitation of a hot 1S exciton. 
Since the ensemble is inhomogeneous, these hot exciton contributions are presumably also present within the 1S band due to the larger (lower energy bandgap) members of the ensemble. 
Such contributions would not be recognized or resolved without scanning the pump frequency.

\subsection{The skewed TG probe spectrum}
The most surprising spectral feature presented here is the skew of the TG probe spectrum towards the red of $\omega_1=\omega_{\text{1S}}$. 
If 1S state-filling completely describes the nonlinear response, the TG signal will mimic the absorptive bleach behavior of TA and show a line shape symmetric about $\omega_1$. 
Although the spectral range of our experimental system limits the measurement of the red skew of Batch A, this feature was reproducible across many batches and system alignments. 
We find no grounds to discount the red skew based on our experimental procedures or sample reproducibility issues.

As $T$ is scanned, the skewed part rises in concert with the 1S-resonant signal that has the pump-probe pulse sequence. 
We therefore explain the skewness as either an instantaneous spectral signature of the photoexcited population or a feature with dynamics much faster than our pulses. 
For all pump colors, the skew maintains a magnitude of $30-40\%$ of maximum TG signal for each probe slice.  % BJT: we should show this in the SI
In contrast, TA signal red of the 1S exction is no more than $10\%$ of the maximum amplitude of the bleach. 
The difference in prominence shows that the redshifted feature is primarily refractive in character.
