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-\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}