\chapter{Global Analysis of Transient Grating and Transient Absorption of PbSe Quantum
  Dots} \label{cha:psg}

\textit{This Chapter borrows extensively from a work-in-progress publication. The authors are:
  \begin{denumerate}
    \item Daniel D. Kohler
    \item Blaise J. Thompson
    \item John C. Wright
  \end{denumerate}
}

We examine the non-linear response of PbSe quantum dots about the 1S exciton using two-dimensional
transient absorption and transient grating techniques.  %
The combined analysis of both methods provides the complete amplitude and phase of the non-linear
susceptibility.  %
The phased spectra reconcile questions about the relationships between the PbSe quantum dot
electronic states and the nature of nonlinearities measured by two-dimensional absorption and
transient grating methods.  %
The fits of the combined dataset reveal and quantify the presence of continuum transitions.  %

\clearpage

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

Lead chalcogenide nanocrystals are among the simplest manifestations of quantum
confinement \cite{Wise2000} and provide a foundation for the rational design of nano-engineered
photovoltaic materials.  %
The time and frequency resolution capabilities of the different types of ultrafast pump-probe
methods have provided the most detailed understanding of quantum dot (QD) photophysics.  %
Transient absorption (TA) studies have dominated the literature.  %
In a typical TA experiment, the pump pulse induces a change in the transmission of the medium that
is measured by a subsequent probe pulse.  %
The change in transmission is described by the change in the dissipative (imaginary) part of the
complex refractive index, which is linked to the dynamics and structure of photoexcited species.  %
TA does not provide information on the real-valued refractive index changes.  %
Although the real component is less important for photovoltaic performance, it is an equal
indicator of underlying structure and dynamics.  %
In practice, having both real and imaginary components is often helpful.  
For example, the fully-phased response is crucial for correctly interpreting spectroscopy when
interfaces are important, which is common in evaluation of materials. \cite{Price2015, Yang2015,
  Yang2017}  %
The real and imaginary responses are directly related by the Kramers-Kronig relation, but it is
experimentally difficult to measure the ultrafast response over the range of frequencies required
for a Hilbert transform.  % 
Interferometric methods, such as two-dimensional eletronic spectroscopy (2DES), can resolve both
components, but they are demanding methods and not commonly used.  %
% note that they often use TA to phase spectra


Transient grating (TG) is a pump-probe method closely related to TA.
Figures \ref{fig:tg_vs_ta} illustrates both methods. 
In TG, two pulsed and independently tunable excitation fields, $E_1$ and $E_2$, are incident on a
sample.  %
The TG experiment modulates the optical properties of the sample by creating a population grating
from the interference between the two crossed beams, $E_2$ and $E_{2^\prime}$.  % 
The grating diffracts the $E_1$ probe field into a new direction defined by the phase matching
condition $\vec{k}_{\text{sig}} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2^\prime}$.  %
In contrast, the TA experiment creates a spatially uniform excited population, but temporally
modulates the ground and excited state populations with a chopper.  % 
TA can be seen as a special case of a TG experiment in which the grating fringes
become infinitely spaced ($\vec{k}_2-\vec{k}_{2^\prime} \rightarrow \vec{0}$)
and, instead of being diffracted, the nonlinear field overlaps and interferes with the probe
beam.  %

Like TA, TG does not fully characterize the non-linear response.  %
Both imaginary and real parts of the refractive index spatially modulate in the TG experiment. 
The diffracted probe is sensitive only to the total grating contrast (the response
\textit{amplitude}), and not the phase relationships of the grating.  %
Since both techniques are sensitive to different components of the non-linear response, however,
the combination of both TA and TG can solve the fully-phased response.  %

Here we report the results of dual 2DTA-2DTG experiments of PbSe quantum dots at the 1S exciton
transition.  %
We explore the three-dimensional experimental space of pump color, probe color, and population
delay time.  %
We define the important experimental factors that must be taken into account for accurate
comparison of the two methods.  %
We show that both methods exhibit reproducible spectra across different batches of different
exciton sizes.  %
Finally, we show that the methods can be used to construct a phased third-order response spectrum.
Both experiments can be reproduced via simulations using the standard theory of PbSe excitons.  %
Interestingly, the combined information reveals broadband contributions to the quantum dots
non-linearity, barely distinguishable with transient absorption spectra alone.  %
This work demonstrates TG and TA serve as complementary methods for the study of exciton structure
and dynamics.  %

\begin{figure}
	\includegraphics[width=\linewidth]{"PbSe_global_analysis/ta_vs_tg"}
	\caption{The similarities between transient grating and transient absorption measurements. 
		Both signals are derived from creating a population difference in the sample. 
			(a) A transient grating experiment crosses two pump beams of the same optical frequency
      ($E_2$, $E_{2^\prime}$) to create an intensity grating roughly perpendicular to the direction
      of propagation. 
			(b) The intensity grating consequently spatially modulates the balance of ground state and
      excited state in the sample.  
				The probe beam ($E_1$) is diffracted, and the diffracted intensity is measured. 
			In transient absorption (c), the probe creates a monolithic population difference, which
      changes the attenuation the probe beam experiences through the sample. 
			(d) The pump is modulated by a chopper, which facilitates measurement of the population
      difference.}
	\label{psg:fig:tg_vs_ta}
\end{figure}

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

[FIGURE]

The optical non-linearity of near-bandgap QD excitons has been extensively investigated. [CITE]  %
The response derives largely from state-filling and depends strongly on the exciton occupancy of
the dots.  %
In a PbSe quantum dot, the 1S peak is composed of transitions between 8 1S electrons and 8 1S
holes. \cite{Kang1997}  %
Figure \ref{fig:model_system}a shows the ground state configuration for a PbSe quantum dot.  %
The 8-fold degeneracy means there are $8 \times 8 = 64$ states in the single-exciton ($|1\rangle$)
manifold and $7 \times 7 = 49$ states in the biexciton ($|2\rangle$) manifold, so $1/4$ of optical
transitions are lost upon single exciton creation.  %

Figure \ref{fig:model_system} shows the model system used in this study and the parameters that
control the third-order response.  %

We assign all electron-hole transitions the same dipole moment, $\mu_{eh}$, so that the total
cross-section between manifolds, $N_i \mu_{eh}^2$, is determined by the number optically active
transitions available, $N_i$.  %
% BJT: state more correctly about what we are doing--there is the assumption that all dipoles are
% the same, and there is the observable that cross-sections correspond to the number of optically
% active transitions.
Although this assumption has come under scrutiny \cite{Karki2013,Gdor2015} it remains valid for the
perturbative fluence used in this study.  %
This model corresponds to the weakly interacting boson model, used to describe the four-wave mixing
response of quantum wells,\cite{Svirko1999} in the limit of small quantum well area.  %

With this excitonic structure, we now describe the resulting non-linear polarization.  %
We restrict ourselves to field-matter interactions in which the pump, $E_2$, precedes the probe,
$E_1$ (the ``true'' pump-probe time-ordering).  %
Note that $E_2$ and $E_{2^\prime}$ represent the same pulse in TA, but they are distinct fields
($\vec{k}_2 \neq \vec{k}_{2^\prime}$) in TG.  %
For brevity, we will write equations assuming these pulse parameters are interchangeable.  %
We consider the limit of low pump fluence, so that only single absorption events need be
considered: $\text{Tr}\left[ \rho \right]  = (1-\bar{n})\rho_{00} + \bar{n} \rho_{11}$, where
$\bar{n}\ll 1$ is the (average) fractional conversion of population.  %
In this limit, $\bar{n} = \frac{\sigma_0(\omega_2)}{\hbar \omega_2} \int{I_2(t) dt}$, where
$I_2(t)$ is the pump intensity and $\sigma_0$ is the ground state absorptive cross-section.  %
%In TA, $I_2(t) = \frac{nc}{8\pi}\left| E_2(t) \right|^2$, while in TG, the average pump intensity is $I_2(t) = \frac{nc}{4\pi}\left| E_2(t) \right|^2$
%The pump induces a non-equilibrium population difference closely approximated by the Poisson distribution; in the case of low fluence ($\hbar \omega_2 \ll \sigma_{QD} \int{I_2 dt}$), 
%The expected population conversion is $\langle n \rangle = \frac{\sigma_0(\omega_2)}{\hbar \omega_2} \int{I_2(t) dt}$, where $ I_2(t) = \frac{nc}{8\pi}\left| E_2(t) \right|^2 $ and $\sigma_0$ is the ground state absorptive cross-section.  
For a Gaussian temporal profile of standard deviation $\Delta_t$, $I_2(t) = I_{\text{2,peak}} \exp
\left( -t^2 / 	2\Delta_t^2 \right)$, the exciton population is
\begin{equation}\label{psg:eq:n}
	\bar{n} = \frac{\sigma_0(\omega_2)\sqrt{2\pi}}{\hbar \omega_2} \Delta_t I_{\text{2,peak}}.
\end{equation}
When the probe interrogates this ensemble; each population will interact linearly:
\begin{equation}\label{psg:eq:ptot}
\begin{split}
	P_{\text{tot}} &= \left( 1-\bar{n} \right)\chi_0^{(1)}(\omega_1)E_1 + \bar{n} \chi_1^{(1)} (\omega_1)E_1 \\ 
		&= \chi_0^{(1)}E_1 + \bar{n}\left( \chi_1^{(1)}(\omega_1) - \chi_0^{(1)}(\omega_1) \right)E_1.
\end{split}
\end{equation}
Here $\chi_0^{(1)}$ ($\chi_1^{(1)}$) denotes the linear susceptibility of the pure state $|0\rangle$ ($|1\rangle$). The third-order field scales as $I_2 E_1$, so from Equation \ref{eq:ptot}
\begin{equation}\label{psg:eq:chi3}
	\chi^{(3)} \propto \sigma_0 (\omega_2) \left( \chi_1^{(1)} - \chi_0^{(1)} \right).
\end{equation}
This expression accounts for the familiar population-level pathways such as excited state absorption/emission and ground state depletion. 
Conforming the linear susceptibilities to our model, the non-linear portion of Equation \ref{eq:chi3} can be written as:
\begin{gather}
	\chi^{(3)} \propto \mu_{eh}^4\text{Im}\left[L_0(\omega_2)\right]\left[ SL_1(\omega) -
    L_0(\omega_1) \right], \label{psg:eq:chi3_lorentz} \\
	L_0(\omega) = \frac{1}{\omega - \omega_{10} + i\Gamma_{10}} ,\\
	L_1(\omega) = \frac{1}{\omega - (\omega_{10} - \epsilon) + i\xi\Gamma_{10}} ,
\end{gather}
where $S = \frac{N_2}{N_1 + 1}$. The denominator of $S$ is larger than $N_1$ due to the
contribution of stimulated emission; this contribution is often neglected.  % 
From Equation \ref{eq:chi3_lorentz} we can see that a finite response can result from three
conditions: $S\neq 1$, $\xi \neq 1$, and/or $\epsilon \neq 0$.  %
The first inequality is the model's manifestation of state-filling, $S < 1$.  %
If we assume that all 64 ground state transitions are optically active, then $S = 0.75$. 
The second condition is met by exciton-induced dephasing (EID), $\xi > 1$, 
% EID has also been attributed to stark splitting of exciton states
and the third from the net attractive Coulombic coupling of excitons, $ \epsilon > 0 $. 
The finite bandwidth of the monochromator can be accounted for by convolving equation
\ref{eq:chi3_lorentz} with the monochromator instrumental function.  %

\subsection{The Bleach Nonlinearity}  % -----------------------------------------------------------

The bleach of the 1S band is the most studied nonlinear signature of the PbSe quantum dots. 
Experiments keep track of the bleach factor, $\phi$, which is the proportionality factor that
relates the relative change in the absorption coefficient at the exciton resonance,
$\alpha_\text{FWM}(\omega_{10})$, with the average exciton occupation:  %
\begin{equation}\label{psg:eq:bleach_factor}
	\frac{\alpha_\text{FWM}(\omega_{10})}{\alpha_0(\omega_{10})} = -\phi \bar{n}
\end{equation}
where $\alpha_0$ is the linear absorption coefficient.  
If QDs are completely bleached by the creation of a single exciton, then $\phi = 1$; if QDs are
unperturbed by the exciton, then $\phi=0$.  %
For PbSe, 1S-resonant values of $\phi=0.25$ and $0.125$ have been reported in literature
\cite{Gdor2013a, Schaller2003, Nootz2011, Omari2012, Geiregat2014}, each with supporting theories
on how state-filling should behave in an 8-fold degenerate system.  %
Inspection of Equation \ref{eq:chi3_lorentz} shows that $\phi = \frac{\text{Im} \left[ L_0 - SL_1
  \right]}{L_0}$; if the 1S nonlinearity is dominated by state-filling ($\xi=1$ and $\epsilon=0$),
then the bleach fraction has perfect correspondence with the change in the number of optically
active states: $\phi = 1-S$.  %
Because Coulombic shifts and EID act to decrease the resonant absorption of the excited state, we
have the strict relation $\phi \geq 1-S$.  %

More recently, a bleach factor metric has been adopted\cite{Trinh2008, Trinh2013} as the
proportionality between the spectrally integrated probe and the carrier concentration:  %
\begin{equation}\label{psg:eq:bleach_factor_int}
	\frac{\int{\alpha_\text{FWM}(\omega) d\omega}}{\int{\alpha_0(\omega)d\omega}} =
  -\phi_{\text{int}} \bar{n}.
\end{equation}
This metric is a more robust description of state filling, because it is unaffected by Coulomb
shifts or EID: Equation \ref{eq:chi3_lorentz} gives $\phi_{\text{int}}=1-S$ regardless of $\xi$ and
$\epsilon$.  %
An experimental value of $\phi_{\text{int}}=0.25$ has been reported\cite{Trinh2013} which
consequently supports the measurement of $\phi = 0.25$.  %

\subsection{TG/TA scaling}  % ---------------------------------------------------------------------

TG is often quantified in the framework of non-linear spectroscopy, which measures the non-linear
susceptibility, $\chi^{(3)}$, while TA often uses metrics of absorption spectroscopy.  %
The global study of both TA and TG requires relating the typical metrics of both experiments.  % 
Here we outline how the measured signals from both methods compare. We assume perfect phase
matching and collinear beams, and we neglect frequency dispersion of the linear refractive
index.  %

When absorption is important ($\alpha_0 \ell \gg 0$), the spatial dependence of the electric field
amplitudes must be considered.  %
For TG, the polarization modulated in the phase-matched direction is given by
\begin{equation}
	P_{\text{TG}}(z) = E_2^* E_{2^\prime} e^{-\alpha_0(\omega_2)z} E_1 e^{-\alpha_0(\omega_1)z / 2} \chi^{(3)}
\end{equation}
The TG electric field propagation can be solved using the slowly varying envelope approximation,
which yields an output intensity of \cite{Carlson1989}
\begin{gather}
	I_{\text{TG}} \propto \omega_1^2 M_{\text{TG}}^2 I_1 I_2 I_{2^\prime} \left| \chi^{(3)} \right|^2 \ell^2 \\
	M_{\text{TG}}(\omega_1, \omega_2) = e^{-\alpha_0(\omega_1)\ell / 2}
  \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell}.
\end{gather}
%$A_0(\omega)= \alpha_0 \ell \ln 10$ is the absorbance spectrum of the sample with path length $\ell$. 
This motivates the following metric for TG:
\begin{equation}
\begin{split} \label{psg:eq:S_TG}
	S_{\text{TG}} &\equiv \frac{1}{M_{\text{TG}}\omega_1}\sqrt{\frac{N_{\text{TG}}}{I_1 I_2^2}} \\
	&\propto \left| \chi^{(3)}\right|
\end{split}
\end{equation}
Here $N_{\text{TG}}$ is the measured four-wave mixing signal from a squared-law detector
($~I_{\text{TG}} / \omega_1$).  %
Again, the third-order response amplitude is extracted from this measurement.  %

We now derive a comparable metric for TA measurements.  %
Because the third-order field is spatially overlapped with the probe field, $E_1$, the relevant
polarization includes the first- and third-order susceptibility:  %
\begin{equation}
	P_{\text{TA}} = \left( \chi^{(3)} (z;\omega_1) + e^{-\alpha_0(\omega_2)z} \left| E_2 \right|^2 \chi^{(3)}(z;\bm{\omega})\right)E_1 .
\end{equation}
Maxwell's equations show that the imaginary component of this polarization changes the intensity of
the $E_1$ beam with a total absorption coefficient that is a composite of the linear and non-linear
propagation:  %
\begin{equation}
\begin{split}
	\alpha_{\text{tot}} &= \frac{2\omega_1}{c} 
	\text{Im}\left[\sqrt{
		1 + 4\pi \left( 
			\chi^{(1)} + \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell} \left| E_2 \right|^2 \chi^{(3)} 
		\right) 
	} \right] \\
	& \approx \frac{4\pi\omega_1}{c} \left( \text{Im}\left[ \chi^{(1)} \right]  + 
		\left[ \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2)\ell} \left( \frac{8\pi}{nc}I_2 \right) \right] \text{Im}\left[ \chi^{(3)} \right] \right)
\end{split}
\end{equation}
The $\chi^{(3)}$ response term can be isolated by chopping the pump beam, which yields a measured
transmittance indicative of the nominal absorption, $T_0 = I_1 e^{-\alpha_0(\omega_1)\ell}$, and
the change in the absorption $T = I_1 e^{-\alpha_{\text{tot}}\ell}$.  %
We quantify the non-linear absorption by $\alpha_\text{FWM}=1/\ell \ln \frac{T-T_0}{T_0}= \alpha_0
- \alpha_{\text{tot}}$, which can now be written as  %
\begin{gather}
	\alpha_\text{FWM}(\omega_1; \omega_2) = \frac{32\pi^2 \omega_1}{nc^2}M_{\text{TA}}(\omega_2) I_2 \text{Im} \left[\chi^{(3)}\right], \label{psg:eq:alpha_fwm} \\
	M_{\text{TA}}(\omega_2) = \frac{1-e^{-\alpha_0(\omega_2)\ell}}{\alpha_0(\omega_2) \ell}.
\end{gather}
Similar to TG propagation, $\alpha_\text{FWM}$ suffers from absorption effects that distort the
proportional relationship to $\text{Im} \left[ \chi^{(3)} \right] $.  %
It is notable that in this case distortions are only from the pump beam.  % 
The signal field heterodynes with the probe, which takes the absorption losses into account
automatically.  %
Note that $M_{\text{TA}}$ simply accounts for the fraction of pump light absorbed in the sample,
and consequently is closely related to the average exciton occupation across the entire path length
of the sample, $\bar{n}_\ell = \ell^{-1} \int_0^\ell \bar{n}(z)dz$, which can be written using
Equation \ref{eq:n} as:  %
\begin{equation} \label{psg:eq:n_tot}
	\bar{n}_\ell = \frac{\sqrt{2} \sigma_0 (\omega_2) \Delta_t}{\hbar \omega_2} I_{\text{2,peak}} M_{\text{TA}}
\end{equation}
We define an experimental metric that isolates the $\chi^{(3)}$ tensor:
\begin{equation}
\begin{split}
	S_{\text{TA}} &= \frac{\alpha_\text{FWM}}{\omega_1 I_2 M_{\text{TA}}} \\
		&\propto \text{Im} \left[ \chi^{(3)} \right]
\end{split}
\end{equation}
For the general $\chi^{(3)}$ response, the imaginary and real components of the spectrum satisfy
complicated relations owing to the causality of all three laser interactions.  %
For the pump-probe time-ordered processes, the probe causality is separable from the pump
excitation event, which makes the causality relation of the pump and probe separable.
\cite{Hutchings1992}  %
The causality relations of the probe spectrum are the famous Kramers-Kronig equations that relate
ground state absorption to the index of refraction.  %
This relation is foundational in analysis of TG \cite{Hogemann1996} and TA measurements.
% DK: need better citations for this

Theoretically, TA probe spectra alone could be transformed to generate the real spectrum. 
In practice, such a transform is difficult because the spectral breadth needed to accurately
calculate the integral is experimentally difficult to achieve.  % 
When both the absolute value and the imaginary projection of $\chi^{(3)}$ are known, however, the
real part can also be defined by the much simpler relation:  %
\begin{equation} \label{psg:eq:chi_real}
	\text{Re} \left[ \chi^{(3)} \right] =
  \pm \sqrt{\left| \chi^{(3)} \right|^2 - \text{Im} \left[ \chi^{(3)} \right]^2}
\end{equation}
% DK: concluding sentence

\subsection{The Absorptive Third-Order Susceptibility}  % -----------------------------------------

Though the bleach factor is defined within the context of absorptive measurements, it can be
converted into the form of a third-order susceptibility as well.  %
Equations \ref{eq:ptot} and \ref{eq:bleach_factor} motivate alternative expressions for
differential absorptivity of the probe:  %
\begin{equation}\label{psg:eq:alpha_fwm_to_bleach1}
\begin{split}
	\alpha_\text{FWM} &= \bar{n} \left( \alpha_1(\omega_1) - \alpha_0(\omega_1) \right) \\
		& =-\phi \bar{n} \alpha_0(\omega_1).
\end{split}
\end{equation}
Substituting Equation \ref{eq:n_tot} into \ref{eq:alpha_fwm_to_bleach1}, we can write the non-linear absorption as
\begin{equation}
	\alpha_\text{FWM}(\omega_1) = -\phi \frac{\sqrt{2\pi}}{\hbar \omega_2} \alpha_0(\omega_1)\alpha_0(\omega_2) \Delta_t I_{\text{2,peak}} M_{\text{TA}}
\end{equation}
By Equation \ref{eq:alpha_fwm}, $\chi^{(3)}$ and the molecular hyperpolarizability, $\gamma_{\text{QD}}^{(3)} = \chi_{\text{QD}}^{(3)} / F^4 N_{\text{QD}}$, where $F$ is the local field factor, can also be related to the bleach fraction:
\begin{gather}\label{psg:eq:chi3_state_filling}
	\text{Im}\left[ \chi^{(3)}_{\text{QD}} \right] = - \phi \frac{\sqrt{2\pi} n c^2 \Delta_t N_{\text{QD}}}{32\pi^2 \hbar \omega_1 \omega_2} \sigma_0(\omega_2)\sigma_0(\omega_1) \\
	\text{Im}\left[ \gamma^{(3)}_{\text{QD}} \right] = - \phi \frac{\sqrt{2\pi} n c^2 \Delta_t }{32\pi^2 \hbar \omega_1 \omega_2 F^4} \sigma_0(\omega_2)\sigma_0(\omega_1). \label{psg:eq:gamma3_state_filling} 
\end{gather}
Because this formula only predicts the imaginary component of the signal, its magnitude gives an
approximate lower limit for the peak susceptibility and hyperpolarizability.  %
Absorptive cross-sections have been experimentally determined for PbSe QDs.  %
\cite{Dai2009, Moreels2007}  %

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

Quantum dot samples used in this study were synthesized using the hot injection
method. \cite{Wehrenberg2002}  %
Samples were kept in a glovebox after synthesis and exposure to visible and UV light was minimized.  
These conditions preserved the dots for several months.  
Two samples, Batch A and Batch B, are presented in this study, in an effort to show the robustness
of the results.  %
Properties of their optical characterization are shown in Table \ref{tab:QD_abs}.  
The 1S band of Batch A is broader than Batch B, an effect which is usually attributed to a wider
size distribution and therefore greater inhomogeneous broadening.  %

The experimental system for the TG experiment has been previously
explained. \cite{Kohler2014, Czech2015}   %
Briefly, two independently tunable OPAs are used to make pulses $E_1$ and $E_2$ with colors
$\omega_1$ and $\omega_2$.   %
The third beam, $E_{2^\prime}$, is split off from $E_2$. The TG experiment utilized here uses
temporally overlapped $E_2$ and $E_{2^\prime}$.  %
Previous ultrafast TG work has characterized the delay of $E_1$ as $\tau_{21}=\tau_2-\tau_1$; to
connect the experimental space with the TA measurements, we will report the population delay time
between the probe and the pump as $T(=-\tau_{21})$.  %
Pulse timing is controlled by a motorized stage that adjusts the arrival time of $E_1$ relative to
$E_2$ and $E_{2^\prime}$.  %

All three beams are focused onto the sample in a BOXCARS geometry and the direction
$\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$ is isolated and sent to a monochromator to isolate the
$\omega_1$ frequency with $\sim 120 \text{cm}^{-1}$ detection bandwidth.  %
The signal, $N_{\text{TG}}$, was detected with an InSb photodiode. Reflective neutral density
filters (Inconel) limit the pulse fluence to avoid multi-photon absorption.  % 
To control for frequency-dependent changes in pulse arrival time due to the OPAs and the neutral
density, a calibration table was established to assign a correct zero delay for each color
combination (see supporting information for more details).  %

The TA experiments were designed to minimally change the TG experimental conditions.  %
The $E_{2^\prime}$ beam was blocked and signal in the $\vec{k}_1$ direction was measured.  %
$E_2$ was chopped and the differential signal and the average signal were measured to define $T_0$
and $T$ needed to compute $\Delta A$.  %  
Just as in TG experiments, the excitation frequencies were scanned while the monochromator was
locked at $\omega_m=\omega_1$.  %
% DK: perhaps leave this part out
%Finally, fluence studies resonant with the 1S band were performed to test for indications of
%intensity-dependent relaxation.   
%These studies showed no indication of accelerated Auger recombination rates (see supporting info).

\begin{table}[]
	\centering
	\caption{Batch Parameters extracted from absorption spectra.  $\langle d \rangle$: average QD
    diameter, as inferred by the 1S transition energy.}
	\label{psg:tab:QD_abs}
	\begin{tabular}{l|cc}
		 													  & A    & B \\
		\hline
		$ \omega_{10} \left( \text{cm}^{-1} \right)$		  & 7570 & 6620 \\
		$ \text{FWHM} \left(\text{cm}^{-1}\right) $  		  & 780  & 540 \\
		$ \langle d \rangle  \left(\text{nm}\right)$ 		  & 4    & 4.8 \\
		$ \sigma_0 \left( \times 10^{16} \text{cm}^2 \right)$ & 1.7  & 2.9
	\end{tabular}
\end{table}

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

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

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.  %

\begin{figure}
	\includegraphics[scale=0.5]{"PbSe_global_analysis/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{psg:fig:movies}
\end{figure}

\section{Discussion}  % ===========================================================================

\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]{"PbSe_global_analysis/kramers_kronig"}
	\caption{Kramers-Kronig analysis of TA spectra compared with TG spectra.}
	\label{psg: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{psg: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=\textwidth]{"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{psg: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{psg: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, however, when the resonant bleach magnitude was pinned to the
state-filling: $\phi \approx 1-S$.  %
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.  %
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}[]
	\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}
  \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{psg:tab:fit2}
\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}).  %
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{psg: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]{"PbSe_global_analysis/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{psg: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} 
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.

\section{Conclusion}  % ===========================================================================

By combining TA and TG measurements, we have described the complex third-order, 2D susceptibility
of the 1S resonance of PbSe quantum dots.  %
We have demonstrated a parameter extraction procedure that is reproducible for different quantum
dot samples, and that some of the parameters, such as the pure dephasing time, are batch
dependent.  %
Inhomogeneity, exciton-induced broadening, exciton-exciton coulombic coupling shifts, and intraband
absorption are all required to reconcile both datasets.  % 
TA features about 1S exciton band are not exclusively assigned as 1S transitions, which can have
important consequences for interpreting the evolution of the 1S bleach.  %

While the TA spectra show prominent 1S-resonant features, the intraband absorption and its
associated refractive index signature are most visible in the TG dataset, so that disentangling the
1S resonant response and the broadband response is a more well-defined problem when both datasets
are used together.  % 
This approach is thus useful for characterization of non-linear signals in spectrally congested
systems.  %
