\subsection{Nonlinear Band Edge Response}

\begin{figure}
	\includegraphics[width=\linewidth]{"model_system"}
	\caption{
		Model system for the 1S band of PbSe quantum dots. 
			(a) The ground state shown in the electron-hole basis. 
				All electrons (holes) are in the valence (conduction) band. 
				There are two electrons and holes in each of the four degenerate $L$ points. 
			(b) The excitonic basis and the transitions accessible in this experiment. 
				The arrows illustrate the available absorptive or emissive transitions that take place in the $\chi^{(3)}$ experiment, and are labeled by parameters that control the cross-sectional strength (arrow width qualitatively indicates transition strength).
			}
	\label{fig:model_system}
\end{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}  
%In PbSe, near-bandgap excitons arise from confinement of direct transitions at the four $L$-points of the FCC lattice, yielding an 8-fold degeneracy within the 1S band.\cite{Kang1997}  
%Both the electron states and hole states are split by exchange and Coulombic coupling but these splittings are small.  
Figure \ref{fig:model_system}a shows the ground state configuration for a PbSe quantum dot.  The energy levels
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.    

Occupancy reduces the number of available transitions and 
%A microscopic description of the optical properties of each state is outside the scope of this work.  

%The 8-fold degenerate lead chalcogenide 1S exciton peak is composed of 8 electrons and 8 holes, which gives 64 states in the single exciton ($|1\rangle$) manifold and 49 states in the biexciton ($|2\rangle$) manifold.  
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, $N_i$. 
Although this assumption % 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.  
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).
\footnote{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{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{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{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{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{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{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{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{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{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{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{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{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{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}