2018-04-12 21:00

13 files changed, 708 insertions, 2195 deletions

diff --git a/PbSe_global_analysis/README.txt b/PbSe_global_analysis/README.txt
deleted file mode 100644
index 22ecdb5..0000000
--- a/PbSe_global_analysis/README.txt
+++ /dev/null
@@ -1,4 +0,0 @@
-To replicate data processing you will need Python 3, WrightTools 2.13.7 and dependencies thereof.
-See http://wright.tools/en/master/install.html for more information.
-
-To run everything, simply call "./run.sh all".
diff --git a/PbSe_global_analysis/SI.tex b/PbSe_global_analysis/SI.tex
deleted file mode 100644
index d3cb783..0000000
--- a/PbSe_global_analysis/SI.tex
+++ /dev/null
@@ -1,279 +0,0 @@
-% document
-\documentclass[11pt, full]{article}
-\usepackage[letterpaper, portrait, margin=0.75in]{geometry}
-\usepackage{setspace}
-\usepackage{color}
-
-% text
-\usepackage[utf8]{inputenc}
-\setlength\parindent{0pt}
-\setlength{\parskip}{1em}
-\usepackage{enumitem}
-\renewcommand{\familydefault}{\sfdefault}
-\newcommand{\RomanNumeral}[1]{\textrm{\uppercase\expandafter{\romannumeral #1\relax}}}
-
-% math
-\usepackage{amssymb}
-\usepackage{amsmath}
-\usepackage[cm]{sfmath}
-\usepackage{commath}
-\usepackage{multirow}
-\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}
-
-% graphics
-\usepackage{graphics}
-\usepackage{graphicx}
-\usepackage{epsfig}
-\usepackage{epstopdf}
-\usepackage{xpatch}
-\graphicspath{{./figures/}}
-
-% "S" prefix
-\renewcommand{\theequation}{S\arabic{equation}}
-\renewcommand{\thefigure}{S\arabic{figure}}
-\renewcommand{\thetable}{S\arabic{table}}
-
-% bibliography
-\usepackage[backend=biber, natbib=true, url=false, sorting=none, maxbibnames=99]{biblatex}
-\bibliography{mybib}
-
-% hyperref
-\usepackage[colorlinks=true, linkcolor=black, urlcolor=blue, citecolor=black, anchorcolor=black]{hyperref}
-\usepackage[all]{hypcap}  % helps hyperref work properly
-
-\begin{document}
-\pagenumbering{gobble}
-
-\begin{center}
-	\LARGE
-	
-	Supplementary Information
-
-	Global Analysis of Transient Grating and Transient Absorption \\ of PbSe Quantum Dots
-	
-	\normalsize
-	
-	\textit{Daniel D. Kohler, Blaise J. Thompson, John C. Wright*}
-
-	Department of Chemistry, University of Wisconsin--Madison\\
-	1101 University Ave., Madison, Wisconsin 53706
-\end{center}
-
-\vspace{\fill}
-
-*Corresponding Author \\
-\hspace*{2ex} email: wright@chem.wisc.edu \\
-\hspace*{2ex} phone: (608) 262-0351 \\
-\hspace*{2ex} fax: (608) 262-0381
-
-\pagebreak
-\renewcommand{\baselinestretch}{0.75}\normalsize
-\tableofcontents
-\renewcommand{\baselinestretch}{1.0}\normalsize
-
-\pagebreak
-\setcounter{page}{1}
-\pagenumbering{arabic}
-
-\section{Absorbance}  % ---------------------------------------------------------------------------
-
-\autoref{figure:absorbance} displays the absorbance spectra of the two batches considered in this
-work.
-The lower spectra are plotted relative to each batches 1S peak center,
-emphasizing the peak-shape differences around the 1S.
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{absorbance}
-  \label{figure:absorbance}
-  \caption{Normalized absorbance spectra of the two baches considered in this
-    work. In the upper plot, the spectra are plotted directly against energy. In
-  the lower plot the spectra are plotted relative to the 1S peak center.}
-\end{figure}
-
-To extract peak parameters from the rising continuum absorption, the data was fitted on the second
-derivative level, as described in the supplementary information of \textcite{Czech2015}.
-The script used to accomplish this fit, full parameter output, and additional figures showing the
-separate excitonic features and fit remainder are contained in the supplementary repository, as
-described in \autoref{section:repository}.
-
-Note that the aliquots used for each of the two Batch A experiments were at
-slightly different concentrations, a crucial detail for m-factor corrections
-(see \autoref{section:m-factors}). The two Batch B experiments were done using
-the same aliquot. The absorbance spectrum of each sample is kept in an
-associated ``cal'' directory in the supplementary repository (see \autoref{section:repository}).
-
-
-\pagebreak
-\section{Artifact correction}  % ------------------------------------------------------------------
-
-\subsection{Spectral delay correction}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{spectral_delay_correction}
-  \label{figure:spectral_delay_correction}
-  \caption{Frequency dependent delay calibration using CCl$_4$. (a) Measurement
-    of the pulse overlap position in $\tau_{21}$ space with respect to
-    $\omega_1$ ($\omega_2$ = 7500 cm$^{-1}$). The thick black line shows the
-    center of the temporal profile, as determined by Gaussian fits. (b) Same as
-    (a), but now $\omega_1$ is kept static while $\omega_2$ is scanned. (c) Same
-    as (a), but now active spectral delay corrections have been applied. (d)
-    Two-dimensional frequency-frequecy scan of CCl$_4$ with spectral delay
-    correction applied.}
-\end{figure}
-
-\pagebreak
-\subsection{Power factors}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{power_factors}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\pagebreak
-\subsection{m factors} \label{section:m-factors}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{m_factors}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\pagebreak
-\subsection{Processing}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{TG_artifacts}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-\begin{figure}[!htb]
-  \centering
-  \includegraphics[scale=0.5]{TA_artifacts}
-  \label{figure:power_factors}
-  \caption{TODO}
-\end{figure}
-
-
-\pagebreak
-\section{Auger recombination dynamics}  % ---------------------------------------------------------
-
-%\begin{figure}[!htb]
-%	\centering
-%	\includegraphics[scale=0.5]{"fsb19-3"}
-%	\label{fig:matrix_flow_diagram}
-%	\caption{$S_{\mathsf{TG}}$ measured before and after multiexciton relaxation dynamics.}
-%\end{figure}
-
-Using a Poisson distribution to model the effects here: keep in mind that
-Poisson is only valid when excitation probability is "low". m
-Scholes thinks an equations of motion approach might be more fitting.
-Others have approached this by truncating the Poisson model so that dots are effectively "off" at
-high fluence (this is when pumping the continuum, so no SE contributions from the pump).
-
-According to the Poisson distribution, initial population created by pump is given by
-
-\begin{equation}
-P(k;\lambda) = \frac{\lambda^k e^{-\lambda}}{k!}.
-\end{equation}
-
-Assumes all absorption events have equal probability.
-The absorption of the pumped sample will be proportional to
-
-\begin{eqnarray}
-a_{\mathsf{NL}} &=& a_0 \left(1-e^{-\lambda}\sum_{k=1}\frac{\lambda^k}{k!}\right)
-+ e^{-\lambda}\sum_{k=1}a_k\frac{\lambda^k}{k!} \\
-&=& a_0 - e^{-\lambda}\sum_{k=1} (a_0 - a_k)\frac{\lambda^k}{k!}.
-\end{eqnarray}
-
-So the difference in the absorption is
-
-\begin{equation}
-S(T=0) = a_{\mathsf{NL}} - a_0 = -e^{-\lambda}\sum_{k=1}(a_0-a_k)\frac{\lambda^k}{k!}.
-\end{equation}
-
-We will assume that absorption is proportional to the number of ground state excitons: $a_k = ck$
-for all $k$.
-Then
-
-\begin{eqnarray}
-S(T=0) &=& ce^{-\lambda}\sum_{k=1}k\frac{\lambda^k}{k!} \\
-&=& c\lambda e^{-\lambda}\sum_{k=0}\frac{\lambda^k}{k!} \\
-&=& c \lambda,
-\end{eqnarray}
-
-and the mean value corresponds to the response (as we expect when the relationship between
-occupation and signal is linear i.e. $<ck> = c\lambda$).
-
-After Auger recombination, the excited state distribution has homogenized to $k=1$.
-Signal is thus given by
-
-\begin{eqnarray}
-S &=& ce^{-\lambda}(a_0-a_1)\sum_{k=1}\frac{\lambda^k}{k!} \\
-&=& ce^{-\lambda}(e^\lambda-1) \\
-&=& c(1-e^{-\lambda}).
-\end{eqnarray}
-
-Previous work has analyzed this.
-
-Comparing the distribution theory with our results.
-
-The mean number of excitations should be proportional to our fluence: $\lambda = mI$.
-This predicts the linear scaling of intensity close to zero delay, and it also predicts the
-exponential saturation observed at longer delays.
-Both observations qualitatively agree with our results.
-Quantitatively, however, our two delays suggest different scaling constants with respect to pump
-fluence: the long-time $m$-value is roughly 40\% larger than the short time scaling.
-This means that our initial scaling underestimates how quickly the band saturates.
-
-Philosophically, there are two problems with this distribution: (1) I should use the equations of
-motion for degenerate pumping, and (2) the pump is filtered by $k$-vector conservation (two pumps).
-My strategy: come up with an expression for the distribution using coupled equations of motion.
-Assume the driven limit, so that a steady state is reached.
-We can account for these issues by utilizing the more general Conway-Maxwell-Poisson distribution.
-
-Estimate spot size as 300 um: 1 um ~ 1 mJ per cm squared.
-
-I think I should revisit the scaling of my exciton signal---I do not expect it to be the same as a
-Poisson distribution because of the stimulated emission channels.
-
-\begin{eqnarray}
-\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
-\frac{d \rho_{00}}{dt} &=& \frac{i}{\hbar} E \rho_{01} + \Gamma\rho_{11} \\
-\frac{d \rho_{00}}{dt} &=& \cdots
-\end{eqnarray}
-
-\pagebreak
-\section{Supplementary repository} \label{section:repository}  % ----------------------------------
-
-All scripts and raw data used in this work have been uploaded to the Open Science Framework (OSF),
-a project of the Center for Open Science.
-These can be found at DOI: \href{http://dx.doi.org/10.17605/OSF.IO/N9CDP}{10.17605/OSF.IO/N9CDP}.
-
-To download the contents of this repository from your command line...  % TODO
-
-To completely reproduce this work, simply execute \texttt{./run.sh all} from your terminal.
-You will require the following:
-
-\begin{enumerate}
-  \item python 3.6
-  \item WrightTools VERSION TODO (and dependencies)
-  \item latex
-\end{enumerate}
-
-You can replace \texttt{all} with one of \texttt{data}, \texttt{simulations}, \texttt{figures},
-or \texttt{documents}.
-
-Otherwise, the OSF repository attempts to be generally self-explanatory.
-README files and comments are used to explain what was done.
-
-\pagebreak
-\printbibliography
-
-\end{document}
diff --git a/PbSe_global_analysis/__init__.py b/PbSe_global_analysis/__init__.py
deleted file mode 100644
index e69de29..0000000
--- a/PbSe_global_analysis/__init__.py
+++ /dev/null
diff --git a/PbSe_global_analysis/chapter.tex b/PbSe_global_analysis/chapter.tex
index 2a20876..9805c7c 100644
--- a/PbSe_global_analysis/chapter.tex
+++ b/PbSe_global_analysis/chapter.tex
@@ -1,5 +1,710 @@
-\chapter{PbSe} \label{cha:psg}
+\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.}
+\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}
+}
 
-\clearpage
\ No newline at end of file
+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.  %
diff --git a/PbSe_global_analysis/conclusion.tex b/PbSe_global_analysis/conclusion.tex
deleted file mode 100644
index e7491d1..0000000
--- a/PbSe_global_analysis/conclusion.tex
+++ /dev/null
@@ -1,7 +0,0 @@
-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.

diff --git a/PbSe_global_analysis/discussion.tex b/PbSe_global_analysis/discussion.tex
deleted file mode 100644
index 97ea143..0000000
--- a/PbSe_global_analysis/discussion.tex
+++ /dev/null
@@ -1,176 +0,0 @@
-\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 𝑆𝑆TA at $\omega_1 < $𝜔𝜔1 < 𝜔𝜔1S, 𝑇𝑇=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.

diff --git a/PbSe_global_analysis/introduction.tex b/PbSe_global_analysis/introduction.tex
deleted file mode 100644
index 4fd37cb..0000000
--- a/PbSe_global_analysis/introduction.tex
+++ /dev/null
@@ -1,50 +0,0 @@
-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.

-% BJT: we might consider introducing TA first, since it is more familiar

-

-\begin{figure}

-	\includegraphics[width=\linewidth]{"tg vs ta"}

-	\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{fig:tg_vs_ta}

-\end{figure}

-

-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.  

-%A local oscillator beam can act as a phase-sensitive reference and is often used to provide that resolution in time-domain techniques. 

-%In this paper, we demonstrate that one can discern the complete, fully-phased optically-induced refractive from frequency domain techniques.

-

-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.

diff --git a/PbSe_global_analysis/main.tex b/PbSe_global_analysis/main.tex
deleted file mode 100644
index f7f5e83..0000000
--- a/PbSe_global_analysis/main.tex
+++ /dev/null
@@ -1,60 +0,0 @@
-\documentclass[journal=jpccck]{achemso}

-%\usepackage[version=3]{mhchem} % Formula subscripts using \ce{}

-%\usepackage[T1]{fontenc}       % Use modern font encodings

-

-\usepackage{amsmath}

-\usepackage{amsfonts}

-\usepackage{amssymb}

-\usepackage{graphicx}

-%\usepackage{multirow}

-\usepackage{bm} % bold mathtype

-%\usepackage{tabularx}

-

-\graphicspath{{"figures/"}}

-

-\title{Global Analysis of Transient Grating and Transient Absorption of PbSe Quantum Dots}

-\author{Daniel D. Kohler}

-\author{Blaise J. Thompson}

-\author{John C. Wright}

-\affiliation{Department of Chemistry,

-	           University of Wisconsin-Madison,

-           	 1101 University Ave, 

-	           Madison, WI 53706, United States}

-\keywords{CMDS, FWM}  % BJT: we need more / better keywords

-

-\date{\today}

-\begin{document}

-

-\begin{abstract}

-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.

-\end{abstract}

-

-\maketitle

-\section{Introduction}

-	\input{introduction}

-\section{Theory}

-	\input{theory}

-\section{Methods}

-	\input{methods}

-\section{Results}

-	\input{results}

-\section{Discussion}

-	\input{discussion}

-\section{Conclusion}

-	\input{conclusion}

-

-\begin{acknowledgement}

-The authors grateful acknowledge staff scientist Andrei Pakoulev (1960-2014) for insightful conversations and devoted tutelage. We thank Qi Ding for her TEM characterizations of the quantum dot samples.

-\end{acknowledgement}

-

-\begin{suppinfo}

-lots of data is online

-\end{suppinfo}

-

-%\bibliographystyle{achemso}

-\bibliography{mybib}

-

-\end{document}

diff --git a/PbSe_global_analysis/methods.tex b/PbSe_global_analysis/methods.tex
deleted file mode 100644
index a62d590..0000000
--- a/PbSe_global_analysis/methods.tex
+++ /dev/null
@@ -1,38 +0,0 @@
-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.

-

-\begin{table}[]

-	\centering

-	\caption{Batch Parameters extracted from absorption spectra.  $\langle d \rangle$: average QD diameter, as inferred by the 1S transition energy. }

-	\label{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}

-

-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).

diff --git a/PbSe_global_analysis/mybib.bib b/PbSe_global_analysis/mybib.bib
deleted file mode 100644
index b195cce..0000000
--- a/PbSe_global_analysis/mybib.bib
+++ /dev/null
@@ -1,1268 +0,0 @@
-@article{Besemann2004,
-  author =       {Besemann, Daniel M. and Meyer, Kent A. and Wright, John C.},
-  title =        {{Spectroscopic Characteristics of Triply Vibrationally
-                  Enhanced Four-Wave Mixing Spectroscopy †}},
-  journal =      {The Journal of Physical Chemistry B},
-  volume =       108,
-  number =       29,
-  pages =        {10493--10504},
-  year =         2004,
-  doi =          {10.1021/jp049597l},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp049597l},
-  abstract =     {Triply vibrationally enhanced (TRIVE) four-wave mixing is a
-                  fully resonant, frequency domain spectroscopy that is capable
-                  of coherent multidimensional vibrational spectroscopy. TRIVE
-                  has 12 different coherence pathways that differ in their time
-                  ordering and resonances. The pathways are the coherent
-                  analogue to two-color pump-probe pathways. Specific pathways
-                  or sets of pathways can be chosen by appropriate selection of
-                  time delays and resonance conditions. The pathways have
-                  characteristic positions and line shapes in three-dimensional
-                  frequency space and their coherent interference has
-                  consequences in interpreting the spectra. The line shapes and
-                  the relative intensities of different pathways are dependent
-                  on the population relaxation and dephasing rates. The
-                  different pathways also have different capabilities for
-                  line-narrowing inhomogeneously broadened transitions. The
-                  narrowing is controlled by the interference between pathways
-                  and the quantum level interference between different parts of
-                  the inhomogeneously broadened envelope. We also show that
-                  selection of the output frequency in two-color TRIVE methods
-                  constrains the selection rules that control the relative
-                  transition probabilities of the four transitions.},
-  file =         {::},
-  issn =         {1520-6106},
-  month =        {jul},
-}
-
-@article{Block2012,
-  author =       {Block, Stephen B. and Yurs, Lena a. and Pakoulev, Andrei V.
-                  and Selinsky, Rachel Sarah and Jin, Song and Wright, John C.},
-  title =        {{Multiresonant Multidimensional Spectroscopy of
-                  Surface-Trapped Excitons in PbSe Quantum Dots}},
-  journal =      {The Journal of Physical Chemistry Letters},
-  volume =       3,
-  number =       18,
-  pages =        {2707--2712},
-  year =         2012,
-  doi =          {10.1021/jz300599b},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jz300599b},
-  file =         {::},
-  issn =         {1948-7185},
-  month =        {sep},
-}
-
-@article{BritoCruz1988,
-  author =       {{Brito Cruz}, C.H. and Gordon, J.P. and Becker, P.C. and Fork,
-                  R.L. and Shank, C.V.},
-  title =        {{Dynamics of spectral hole burning}},
-  journal =      {IEEE Journal of Quantum Electronics},
-  volume =       24,
-  number =       2,
-  pages =        {261--269},
-  year =         1988,
-  doi =          {10.1109/3.122},
-  url =          {http://ieeexplore.ieee.org/document/122/},
-  file =         {::},
-  issn =         {0018-9197},
-  month =        {feb},
-}
-
-@article{Carlson1989,
-  author =       {Carlson, Roger J. and Wright, John C.},
-  title =        {{Absorption and Coherent Interference Effects in Multiply
-                  Resonant Four-Wave Mixing Spectroscopy}},
-  journal =      {Applied Spectroscopy},
-  volume =       43,
-  number =       7,
-  pages =        {1195--1208},
-  year =         1989,
-  doi =          {10.1366/0003702894203408},
-  url =
-                  {http://openurl.ingenta.com/content/xref?genre=article{\&}issn=0003-7028{\&}volume=43{\&}issue=7{\&}spage=1195},
-  abstract =     {The effects of coherent interference between resonant and
-                  nonresonant signals and the effects of absorption on
-                  multiresonant four-wave mixing spectra are investigated both
-                  theoretically and experimentally in azu- iene-doped
-                  naphthalene crystals at 2 K. Both effects can strongly alter
-                  line shapes and intensities. Line splittings and negative
-                  spectral features are demonstrated. Coherent interference,
-                  although measurable in this system, is weak. Absorption,
-                  however, plays an important role, and its effects on phase
-                  matching, peak shapes, and peak intensities are pre- dicted
-                  theoretically and are confirmed by the experimental measure-
-                  ments. A new expression for the four-wave mixing is derived in
-                  terms of sample absorption coefficients and absorption cross
-                  sections, and gen- eral conditions for maximum efficiency are
-                  determined.},
-  file =         {::},
-  issn =         00037028,
-  keywords =     {M-factors},
-  mendeley-tags ={M-factors},
-  month =        {sep},
-}
-
-@article{Czech2015,
-  author =       {Czech, Kyle J. and Thompson, Blaise J. and Kain, Schuyler and
-                  Ding, Qi and Shearer, Melinda J. and Hamers, Robert J. and
-                  Jin, Song and Wright, John C.},
-  title =        {{Measurement of Ultrafast Excitonic Dynamics of Few-Layer MoS
-                  2 Using State-Selective Coherent Multidimensional
-                  Spectroscopy}},
-  journal =      {ACS Nano},
-  volume =       9,
-  number =       12,
-  pages =        {12146--12157},
-  year =         2015,
-  doi =          {10.1021/acsnano.5b05198},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/acsnano.5b05198},
-  abstract =     {We report the first coherent multidimensional spectroscopy
-                  study of a MoS2 film. A four-layer sample of MoS2 was
-                  synthesized on a silica substrate by a simplified sulfidation
-                  reaction and characterized by absorption and Raman
-                  spectroscopy, atomic force microscopy, and transmission
-                  electron microscopy. State-selective coherent multidimensional
-                  spectroscopy (CMDS) on the as-prepared MoS2 film resolved the
-                  dynamics of a series of diagonal and cross-peak features
-                  involving the spin-orbit split A and B excitonic states and
-                  continuum states. The spectra are characterized by striped
-                  features that are similar to those observed in CMDS studies of
-                  quantum wells where the continuum states contribute strongly
-                  to the initial excitation of both the diagonal and cross-peak
-                  features, while the A and B excitonic states contributed
-                  strongly to the final output signal. The strong contribution
-                  from the continuum states to the initial excitation shows that
-                  the continuum states are coupled to the A and B excitonic
-                  states and that fast intraband relaxation is occurring on a
-                  sub-70 fs time scale. A comparison of the CMDS excitation
-                  signal and the absorption spectrum shows that the relative
-                  importance of the continuum states is determined primarily by
-                  their absorption strength. Diagonal and cross-peak features
-                  decay with a 680 fs time constant characteristic of exciton
-                  recombination and/or trapping. The short time dynamics are
-                  complicated by coherent and partially coherent pathways that
-                  become important when the excitation pulses are temporally
-                  overlapped. In this region, the coherent dynamics create
-                  diagonal features involving both the excitonic states and
-                  continuum states, while the partially coherent pathways
-                  contribute to cross-peak features.},
-  file =         {::},
-  issn =         {1936-0851},
-  keywords =     {2D,2d,are layered,molybdenum sul fi de,molybdenum
-                  sulfide,multidimensional,nonlinear,ransition metal
-                  dichalcogenides,semiconductors with strong spin,such as mos
-                  2,tmdcs,transition metal dichalcogenides,ultrafast
-                  dynamics,{\`{a}}},
-  month =        {dec},
-  pmid =         26525496,
-}
-
-@article{Dai2009,
-  author =       {Dai, Quanqin and Wang, Yingnan Yiding and Li, Xinbi and Zhang,
-                  Yu and Pellegrino, Donald J. and Zhao, Muxun and Zou, Bo and
-                  Seo, JaeTae and Wang, Yingnan Yiding and Yu, William W.},
-  title =        {{Size-dependent composition and molar extinction coefficient
-                  of PbSe semiconductor nanocrystals.}},
-  journal =      {ACS nano},
-  volume =       3,
-  number =       6,
-  pages =        {1518--24},
-  year =         2009,
-  doi =          {10.1021/nn9001616},
-  url =          {http://www.ncbi.nlm.nih.gov/pubmed/19702314},
-  abstract =     {Atomic compositions and molar extinction coefficients of PbSe
-                  semiconductor nanocrystals were determined by atomic
-                  absorption spectrometry, UV-vis-NIR spectrophotometry, and
-                  transmission electron microscopy. The Pb/Se atomic ratio was
-                  found to be size-dependent with a systematic excess of Pb
-                  atoms in the PbSe nanocrystal system. Experimental results
-                  indicated that the individual PbSe nanocrystal was
-                  nonstoichiometric, consisting of a PbSe core and an extra
-                  layer of Pb atoms. For these nonstoichiometric PbSe
-                  semiconductor nanocrystals, we proposed a new computational
-                  approach to calculate the total number of Pb and Se atoms in
-                  different sized particles. This calculation played a key role
-                  on the accurate determination of the strongly size-dependent
-                  extinction coefficient, which followed a power law with an
-                  exponent of approximately 2.5.},
-  file =         {::},
-  isbn =         {1936-0851},
-  issn =         {1936-086X},
-  keywords =     {Composition,Molar extinction coefficient,PbSe,Semiconductor
-                  nanocrystal,Size dependence,informative,size-exciton
-                  correlation},
-  mendeley-tags ={informative},
-  month =        {jun},
-  pmid =         19435305,
-}
-
-@article{DeGeyter2012,
-  author =       {{De Geyter}, Bram and Geiregat, Pieter and Gao, Yunan and {Ten
-                  Cate}, Sybren and Houtepen, Arjan J. and Schins, Juleon M. and
-                  {Van Thourhout}, Dries and Siebbeles, Laurens D A and Hens,
-                  Zeger},
-  title =        {{Broadband and picosecond intraband absorption in lead based
-                  colloidal quantum dots}},
-  journal =      {ACS Nano},
-  number =       7,
-  pages =        {6067--6074},
-  year =         2012,
-  doi =          {10.1109/ICTON.2012.6254469},
-  abstract =     {Using femtosecond transient absorption spectroscopy we
-                  demonstrate that lead chalcogenide nanocrystals show
-                  considerable, photoinduced absorption (PA) in a broad
-                  wavelength range just below the bandgap. The time-dependent
-                  decay of the PA signal correlates with the recovery of the
-                  band gap absorption, indicating that the same carriers are
-                  involved. Based on this, we assign this PA signal to intraband
-                  absorption, i.e., the excitation of photogenerated carriers
-                  from the bottom of the conduction band or the top of the
-                  valence band to higher energy levels in the conduction and
-                  valence band continuum. We confirm our experiments with
-                  tight-binding calculations. This broadband response in the
-                  commercially interesting near to mid-infrared range is very
-                  relevant for ultra high speed all optical signal processing.
-                  We benchmark the performance with bulk-Si and
-                  Si-nanocrystals.},
-  file =         {::},
-  isbn =         9781467322270,
-  issn =         21627339,
-  keywords =     {colloidal nanocrystals,free carrier absorption,intraband
-                  absorption,lead chalcogenide,optical signal
-                  processing,tight-binding,transient absorption spectroscopy},
-  pmid =         22686663,
-}
-
-@article{DelCoso2004,
-  author =       {del Coso, Ra{\'{u}}l and Solis, Javier},
-  title =        {{Relation between nonlinear refractive index and third-order
-                  susceptibility in absorbing media}},
-  journal =      {Journal of the Optical Society of America B},
-  volume =       21,
-  number =       3,
-  pages =        640,
-  year =         2004,
-  doi =          {10.1364/JOSAB.21.000640},
-  url =          {http://www.opticsinfobase.org/abstract.cfm?URI=JOSAB-21-3-640
-                  https://www.osapublishing.org/abstract.cfm?URI=josab-21-3-640},
-  abstract =     {Expressions relating complex third-order optical
-                  susceptibility ($\chi$(3)=$\chi$R(3)+i$\chi$I(3)) with
-                  nonlinear refractive index (n2) and nonlinear absorption
-                  coefficient ($\beta$) have been formulated that eliminate the
-                  commonly used approximation of a negligible linear absorption
-                  coefficient. The resulting equations do not show the
-                  conventional linear dependence of $\chi$R(3) with n2 and
-                  $\chi$I(3) with $\beta$. Nonlinear refraction and absorption
-                  result instead from the interplay between the real and
-                  imaginary parts of the first- and third-order susceptibilities
-                  of the material. This effect is illustrated in the case of a
-                  metal–dielectric nanocomposite for which n2 and $\beta$ values
-                  were experimentally obtained by Z-scan measurements and for
-                  which the use of the new formulas for $\chi$R(3) and
-                  $\chi$I(3) yield a large correction and a sign reversal for
-                  $\chi$I(3).},
-  file =         {::},
-  issn =         {0740-3224},
-  month =        {mar},
-  pmid =         220057300022,
-}
-
-@article{Gdor2012,
-  author =       {Gdor, Itay and Sachs, Hanan and Roitblat, Avishy and
-                  Strasfeld, David B. and Bawendi, Moungi G. and Ruhman,
-                  Sanford},
-  title =        {{Exploring exciton relaxation and multiexciton generation in
-                  PbSe nanocrystals using hyperspectral near-IR probing.}},
-  journal =      {ACS nano},
-  volume =       6,
-  number =       4,
-  pages =        {3269--77},
-  year =         2012,
-  doi =          {10.1021/nn300184n},
-  url =          {http://www.ncbi.nlm.nih.gov/pubmed/22390473},
-  abstract =     {Hyperspectral femtosecond transient absorption spectroscopy is
-                  employed to record exciton relaxation and recombination in
-                  colloidal lead selenide (PbSe) nanocrystals in unprecedented
-                  detail. Results obtained with different pump wavelengths and
-                  fluences are scrutinized with regard to three issues: (1)
-                  early subpicosecond spectral features due to "hot" excitons
-                  are analyzed in terms of suggested underlying mechanisms; (2)
-                  global kinetic analysis facilitates separation of the
-                  transient difference spectra into single, double, and triple
-                  exciton state contributions, from which individual band
-                  assignments can be tested; and (3) the transient spectra are
-                  screened for signatures of multiexciton generation (MEG) by
-                  comparing experiments with excitation pulses both below and
-                  well above the theoretical threshold for multiplication. For
-                  the latter, a recently devised ultrafast pump-probe
-                  spectroscopic approach is employed. Scaling sample
-                  concentrations and pump pulse intensities inversely with the
-                  extinction coefficient at each excitation wavelength overcomes
-                  ambiguities due to direct multiphoton excitation,
-                  uncertainties of absolute absorption cross sections, and low
-                  signal levels. As observed in a recent application of this
-                  method to InAs core/shell/shell nanodots, no sign of MEG was
-                  detected in this sample up to photon energy 3.7 times the band
-                  gap. Accordingly, numerous reports of efficient MEG in other
-                  samples of PbSe suggest that the efficiency of this process
-                  varies from sample to sample and depends on factors yet to be
-                  determined.},
-  file =         {::},
-  isbn =         {1936-086X (Electronic)$\backslash$r1936-0851 (Linking)},
-  issn =         {1936-086X},
-  keywords =     {SADS,carrier multiplication,ciency in nanocrystals has,exciton
-                  cooling,in accord,led,meg effi-,multiexciton
-                  generation,quantum dots,the apparent enhancement of,ultrafast
-                  spectroscopy},
-  mendeley-tags ={SADS},
-  month =        {apr},
-  pmid =         22390473,
-}
-
-@article{Gdor2013a,
-  author =       {Gdor, Itay and Yang, Chunfan and Yanover, Diana and Sachs,
-                  Hanan and Lifshitz, Efrat and Ruhman, Sanford},
-  title =        {{Novel Spectral Decay Dynamics of Hot Excitons in PbSe
-                  Nanocrystals: A Tunable Femtosecond Pump–Hyperspectral Probe
-                  Study}},
-  journal =      {The Journal of Physical Chemistry C},
-  volume =       117,
-  number =       49,
-  pages =        {26342--26350},
-  year =         2013,
-  doi =          {10.1021/jp409530z},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp409530z},
-  abstract =     {Ultrafast exciton cooling in highly monodisperse PbSe
-                  nanocrystals is followed with tunable pump?hyperspectral
-                  near-IR probe spectroscopy. Unexpected kinetic and spectral
-                  correlations between induced bleach and absorption features
-                  are revealed, which are incompatible with standard models for
-                  excited nanocrystal absorption. Interband optical excitation
-                  immediately generates a sharp bleach feature near the 1Sh1Se
-                  transition which is unchanged during exciton thermalization,
-                  while pumping well above the band edge induces an intense
-                  absorption at frequencies just below the band edge which
-                  decays concurrently with a buildup of renewed absorbance at
-                  the 1Ph1Pe peak during exciton cooling. Transient spectra of
-                  hot single and double excitons are nearly indistinguishable,
-                  arguing against the controversial involvement of Auger cooling
-                  in the rapid dissipation of excess energy in excited PbSe QDs.
-                  Finally, quantitative signal analysis shows no signs of
-                  multiexciton generation up to photon energies four times the
-                  sample band gap. Ultrafast exciton cooling in highly
-                  monodisperse PbSe nanocrystals is followed with tunable
-                  pump?hyperspectral near-IR probe spectroscopy. Unexpected
-                  kinetic and spectral correlations between induced bleach and
-                  absorption features are revealed, which are incompatible with
-                  standard models for excited nanocrystal absorption. Interband
-                  optical excitation immediately generates a sharp bleach
-                  feature near the 1Sh1Se transition which is unchanged during
-                  exciton thermalization, while pumping well above the band edge
-                  induces an intense absorption at frequencies just below the
-                  band edge which decays concurrently with a buildup of renewed
-                  absorbance at the 1Ph1Pe peak during exciton cooling.
-                  Transient spectra of hot single and double excitons are nearly
-                  indistinguishable, arguing against the controversial
-                  involvement of Auger cooling in the rapid dissipation of
-                  excess energy in excited PbSe QDs. Finally, quantitative
-                  signal analysis shows no signs of multiexciton generation up
-                  to photon energies four times the sample band gap.},
-  file =         {::},
-  isbn =         1557522790,
-  issn =         {1932-7447},
-  month =        {dec},
-}
-
-@article{Gdor2015,
-  author =       {Gdor, Itay and Shapiro, Arthur and Yang, Chunfan and Yanover,
-                  Diana and Lifshitz, Efrat and Ruhman, Sanford},
-  title =        {{Three-pulse femtosecond spectroscopy of PbSe nanocrystals: 1S
-                  bleach nonlinearity and sub-band-edge excited-state absorption
-                  assignment}},
-  journal =      {ACS Nano},
-  volume =       9,
-  number =       2,
-  pages =        {2138--2147},
-  year =         2015,
-  doi =          {10.1021/nn5074868},
-  abstract =     {Above band-edge photo-excitation of PbSe nanocrystals induces
-                  strong below band-gap absorption as well as a multi-phased
-                  buildup of bleaching in the 1Se1Sh transition. The amplitudes
-                  and kinetics of these features deviate from expectations based
-                  on bi-exciton shifts and state filling which are the
-                  mechanisms usually evoked to explain them. To clarify these
-                  discrepancies, the same transitions are investigated here by
-                  double-pump probe spectroscopy. Re-exciting in the below
-                  band-gap induced absorption characteristic of hot excitons is
-                  shown to produce additional excitons with high probability. In
-                  addition, pump-probe experiments on a sample saturated with
-                  single relaxed excitons proves that the resulting 1Se1Sh
-                  bleach is not linear with the number of excitons per
-                  nanocrystal. This finding holds for two samples differing
-                  significantly in size, demonstrating its generality. Analysis
-                  of the results suggest that below band edge induced absorption
-                  in hot exciton states is due to excited state absorption and
-                  not to shifted absorption of cold carriers, and that 1Se1Sh
-                  bleach signals are not an accurate counter of sample excitons
-                  when their distribution includes multi-exciton states.},
-  file =         {::},
-  isbn =         {1936-0851},
-  issn =         {1936086X},
-  keywords =     {exciton cooling,multiexciton generation,nanocrystals,quantum
-                  dots,ultrafast spectroscopy},
-}
-
-@article{Gdor2015a,
-  author =       {Gdor, Itay and Yanover, Dianna and Yang, Chunfan and Shapiro,
-                  Arthur and Lifshitz, Efrat and Ruhman, Sanford},
-  title =        {{Three Pulse Femtosecond Spectroscopy of PbSe Nano-Crystals ;
-                  1S Bleach Nonlinearity and Sub Band-Edge Excited State
-                  Absorption Assignment Three Pulse Femtosecond Spectroscopy of
-                  PbSe nano-Crystals ; 1S Bleach Nonlinearity and Sub Band-edge
-                  Excited State A}},
-  journal =      {ACS Nano},
-  pages =        {2--5},
-  year =         2015,
-  doi =          {10.1021/nn5074868},
-  file =         {::},
-}
-
-@article{Geiregat2014,
-  author =       {Geiregat, Pieter and Houtepen, Arjan J. and Justo, Yolanda and
-                  Grozema, Ferdinand C. and {Van Thourhout}, Dries and Hens,
-                  Zeger},
-  title =        {{Coulomb Shifts upon Exciton Addition to Photoexcited PbS
-                  Colloidal Quantum Dots}},
-  journal =      {The Journal of Physical Chemistry C},
-  volume =       118,
-  number =       38,
-  pages =        {22284--22290},
-  year =         2014,
-  doi =          {10.1021/jp505530k},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp505530k},
-  abstract =     {Using ultrafast hyperspectral transient absorption (TA)
-                  spectroscopy, we determine the biexciton addition energies in
-                  PbS quantum dots (QDs) with different sizes when either a cold
-                  or a hot electron−hole pair is added to a QD already
-                  containing a cooled exciton. The observed dependence of this
-                  so-called biexciton addition energy on the QD diameter and the
-                  exciton energy can be rationalized by interpreting the
-                  addition energies as the result of an imbalance in the Coulomb
-                  interactions between the newly created carriers and the
-                  carriers already present in a QD. The obtained results are
-                  therefore relevant from both a fundamental and practical point
-                  of view. They provide experimental data on Coulomb interaction
-                  between charge carriers in confined semiconductors that can be
-                  compared with theoretical estimates. Moreover, understanding
-                  the way hot−cold biexciton addition energies influence the
-                  transient absorption spectrum adds a new element to the
-                  transient absorption toolbox for the optoelectronic properties
-                  of colloidal QDs. ■ INTRODUCTION Colloidal semiconductor
-                  nanocrystals or quantum dots (QDs) are an increasingly
-                  prominent class of low-dimensional nanomaterials that combine
-                  size-tunable electronic and optical properties with a
-                  suitability for solution-based processing. Starting from
-                  fundamental research and theoretical modeling on their unique
-                  physical properties, QDs are now applied in a variety of
-                  domains such as solar energy harvesting, photo-detection, and
-                  light-emitting diodes or displays. 1−5 These applications
-                  typically rely on the linear optical properties of QDs, i.e.,
-                  light absorption by unexcited QDs and light emission by
-                  radiative recombination in excited QDs. On the other hand,
-                  various studies have shown that the spectral and
-                  time-dependent properties of excited QDs can strongly enhance
-                  the performance of QD-based devices in the above mention
-                  applications or enable QDs to be used in completely different
-                  applications. Quantum dots excited with photons having
-                  energies exceeding twice that of the QD bandgap transition
-                  can, for example, dissipate their excess energy by forming
-                  biexcitons in a process called multiple exciton generation
-                  (MEG) that can considerably enhance the short circuit current
-                  of single junction, QD-based solar cells. 2,6−8 Controlling
-                  the recombination rate of biexcitons by nonradiative Auger
-                  processes allowed for the formation of blinking-free QDs and
-                  facilitated the formation of QD-based lasers. 9,10 Moreover,
-                  it was proven that excited QDs exhibit a broadband and
-                  ultrafast photoinduced absorption related to intraband
-                  transitions of either the excited electron or hole, which
-                  could be used for optical modulation. 11,12},
-  file =         {::;::},
-  issn =         {1932-7447},
-  month =        {sep},
-}
-
-@article{Gesuele2012,
-  author =       {Gesuele, F and Sfeir, M Y and Murray, C B and Heinz, T F and
-                  Wong, C W},
-  title =        {{Biexcitonic Effects in Excited Ultrafast Supercontinuum
-                  S{\aa}ectroscopy of Carrier Multiplication and Biexctionic
-                  Effects in Excited States of PbS Quantum Dots}},
-  journal =      {Nano letters},
-  volume =       12,
-  pages =        2658,
-  year =         2012,
-  file =         {::},
-  keywords =     {1,2 the strong spatial,based-materials for third-generation
-                  photovol-,carrier multiplication,confinement of electronic
-                  wave,here is great interest,in the properties
-                  of,multiple-exciton generation,qd,quantum dot,solar
-                  cells,taics,ultrafast spectroscopy},
-}
-
-@article{Hogemann1996,
-  author =       {H{\"{o}}gemann, Claudia and Pauchard, Marc and Vauthey, Eric},
-  title =        {{Picosecond transient grating spectroscopy: The nature of the
-                  diffracted spectrum}},
-  journal =      {Review of Scientific Instruments},
-  volume =       67,
-  number =       10,
-  pages =        {3449--3453},
-  year =         1996,
-  doi =          {10.1063/1.1147157},
-  url =          {http://link.aip.org/link/RSINAK/v67/i10/p3449/s1{\&}Agg=doi
-                  http://aip.scitation.org/doi/10.1063/1.1147157},
-  file =         {::},
-  issn =         {0034-6748},
-  month =        {oct},
-}
-
-@article{Hutchings1992,
-  author =       {Hutchings, D C and Sheik-Bahae, M and Hagan, D J and {Van
-                  Stryland}, E W},
-  title =        {{Kramers-Kronig relations in nonlinear optics}},
-  journal =      {Optical and Quantum Electronics},
-  volume =       24,
-  number =       1,
-  pages =        {1--30},
-  year =         1992,
-  doi =          {10.1007/BF01234275},
-  url =
-                  {http://dx.doi.org/10.1007/BF01234275{\%}5Cnhttp://www.springerlink.com/index/10.1007/BF01234275},
-  abstract =     {We review dispersion relations, which relate the real part of
-                  the optical susceptibility (refraction) to the imaginary part
-                  (absorption). We derive and discuss these relations as applied
-                  to nonlinear optical systems. It is shown that in the
-                  nonlinear case, for self-action effects the correct form for
-                  such dispersion relations is nondegenerate, i.e. it is
-                  necessary to use multiple frequency arguments. Nonlinear
-                  dispersion relations have been shown to be very useful as they
-                  usually only require integration over a limited frequency
-                  range (corresponding to frequencies at which the absorption
-                  changes), unlike the conventional linear Kramers-Kr{\"{o}}nig
-                  relation which requires integration over all absorbing
-                  frequencies. Furthermore, calculation of refractive index
-                  changes using dispersion relations is easier than a direct
-                  calculation of the susceptibility, as transition rates (which
-                  give absorption coefficients) are, in general, far easier to
-                  calculate than the expectation value of the optical
-                  polarization. Both resonant (generation of some excitation
-                  that is long lived compared with an optical period) and
-                  nonresonant ‘instantaneous' optical nonlinearities are
-                  discussed, and it is shown that the nonlinear dispersion
-                  relation has a common form and can be understood in terms of
-                  the linear Kramers-Kr{\"{o}}nig relation applied to a new
-                  system consisting of the material plus some ‘perturbation'. We
-                  present several examples of the form of this external
-                  perturbation, which can be viewed as the pump in a pump-probe
-                  experiment. We discuss the two-level saturated atom model and
-                  bandfilling in semiconductors among others for the resonant
-                  case. For the nonresonant case some recent work is included
-                  where the electronic nonlinear refractive coefficient,n2, is
-                  determined from the nonlinear absorption processes of
-                  two-photon absorption, Raman transitions and the a.c. Stark
-                  effect. We also review how the dispersion relations can be
-                  extended to give alternative forms for frequency summation
-                  which, for example, allows the real and imaginary parts of?(2)
-                  to be related.},
-  file =         {::;::},
-  isbn =         {0306-8919},
-  issn =         03068919,
-}
-
-@article{Kang1997,
-  author =       {Kang, Inuk and Wise, Frank W.},
-  title =        {{Electronic structure and optical properties of PbS and PbSe
-                  quantum dots}},
-  journal =      {Journal of the Optical Society of America B},
-  volume =       14,
-  number =       7,
-  pages =        1632,
-  year =         1997,
-  doi =          {10.1364/JOSAB.14.001632},
-  url =
-                  {http://www.opticsinfobase.org/abstract.cfm?URI=josab-14-7-1632},
-  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
-                  Desktop/Downloaded/Kang, Wise - 1997 - Electronic structure
-                  and optical properties of PbS and PbSe quantum dots.pdf:pdf},
-  issn =         {0740-3224},
-  month =        {jul},
-}
-
-@article{Karki2013,
-  author =       {Karki, Khadga J and Ma, Fei and Zheng, Kaibo and Zidek, Karel
-                  and Mousa, Abdelrazek and Abdellah, Mohamed and Messing, Maria
-                  E and Wallenberg, L Reine and Yartsev, Arkadi and Pullerits,
-                  T{\~{o}}nu},
-  title =        {{Multiple exciton generation in nano-crystals revisited:
-                  consistent calculation of the yield based on pump-probe
-                  spectroscopy.}},
-  journal =      {Scientific reports},
-  volume =       3,
-  pages =        2287,
-  year =         2013,
-  doi =          {10.1038/srep02287},
-  url =
-                  {http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3724175{\&}tool=pmcentrez{\&}rendertype=abstract},
-  abstract =     {Multiple exciton generation (MEG) is a process in which more
-                  than one exciton is generated upon the absorption of a high
-                  energy photon, typically higher than two times the band gap,
-                  in semiconductor nanocrystals. It can be observed
-                  experimentally using time resolved spectroscopy such as the
-                  transient absorption measurements. Quantification of the MEG
-                  yield is usually done by assuming that the bi-exciton signal
-                  is twice the signal from a single exciton. Herein we show that
-                  this assumption is not always justified and may lead to
-                  significant errors in the estimated MEG yields. We develop a
-                  methodology to determine proper scaling factors to the signals
-                  from the transient absorption experiments. Using the
-                  methodology we find modest MEG yields in lead chalcogenide
-                  nanocrystals including the nanorods.},
-  file =         {::},
-  issn =         {2045-2322},
-  month =        {jan},
-  pmid =         23887181,
-}
-
-@article{Kohler2014,
-  author =       {Kohler, Daniel D. and Block, Stephen B. and Kain, Schuyler and
-                  Pakoulev, Andrei V. and Wright, John C.},
-  title =        {{Ultrafast Dynamics within the 1S Exciton Band of Colloidal
-                  PbSe Quantum Dots Using Multiresonant Coherent
-                  Multidimensional Spectroscopy}},
-  journal =      {The Journal of Physical Chemistry C},
-  volume =       118,
-  number =       9,
-  pages =        {5020--5031},
-  year =         2014,
-  doi =          {10.1021/jp412058u},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp412058u},
-  file =         {::},
-  issn =         {1932-7447},
-  month =        {mar},
-}
-
-@article{Kohler2017,
-  author =       {Kohler, Daniel D and Thompson, Blaise J and Wright, John C},
-  title =        {{Frequency-domain coherent multidimensional spectroscopy when
-                  dephasing rivals pulsewidth: Disentangling material and
-                  instrument response}},
-  journal =      {The Journal of Chemical Physics},
-  volume =       147,
-  number =       8,
-  pages =        084202,
-  year =         2017,
-  doi =          {10.1063/1.4986069},
-  url =          {http://dx.doi.org/10.1063/1.4986069
-                  http://aip.scitation.org/toc/jcp/147/8
-                  http://aip.scitation.org/doi/10.1063/1.4986069},
-  abstract =     {Ultrafast spectroscopy is often collected in the mixed
-                  frequency/time domain, where pulse durations are similar to
-                  system dephasing times. In these experiments, expectations
-                  derived from the familiar driven and impulsive limits are not
-                  valid. This work simulates the mixed-domain four-wave mixing
-                  response of a model system to develop expectations for this
-                  more complex field-matter interaction. We explore frequency
-                  and delay axes. We show that these line shapes are exquisitely
-                  sensitive to excitation pulse widths and delays. Near pulse
-                  overlap, the excitation pulses induce correlations that
-                  resemble signatures of dynamic inhomogeneity. We describe
-                  these line shapes using an intuitive picture that connects to
-                  familiar field-matter expressions. We develop strategies for
-                  distinguishing pulse-induced correlations from true system
-                  inhomogeneity. These simulations provide a founda-tion for
-                  interpretation of ultrafast experiments in the mixed domain.},
-  file =         {::},
-  issn =         {0021-9606},
-  month =        {aug},
-}
-
-@article{Kraatz2014,
-  author =       {Kraatz, Ingvar T and Booth, Matthew and Whitaker, Benjamin J
-                  and Nix, Michael G D and Critchley, Kevin},
-  title =        {{Sub-Bandgap Emission and Intraband Defect-Related
-                  Excited-State Dynamics in Colloidal CuInS 2 /ZnS Quantum Dots
-                  Revealed by Femtosecond Pump − Dump − Probe Spectroscopy}},
-  year =         {2014},
-  file =         {::},
-}
-
-@article{Lucarini2008,
-  author =       {Lucarini, Valerio},
-  title =        {{Response theory for equilibrium and non-equilibrium
-                  statistical mechanics: Causality and generalized
-                  kramers-kronig relations}},
-  journal =      {Journal of Statistical Physics},
-  volume =       131,
-  number =       3,
-  pages =        {543--558},
-  year =         2008,
-  doi =          {10.1007/s10955-008-9498-y},
-  abstract =     {We consider the general response theory proposed by Ruelle for
-                  describing the impact of small perturbations to the
-                  non-equilibrium steady states resulting from Axiom A dynamical
-                  systems. We show that the causality of the response functions
-                  allows for writing a set of Kramers-Kronig relations for the
-                  corresponding susceptibilities at all orders of nonlinearity.
-                  Nonetheless, only a special class of observable
-                  susceptibilities obey Kramers-Kronig relations. Specific
-                  results are provided for arbitrary order harmonic response,
-                  which allows for a very comprehensive Kramers-Kronig analysis
-                  and the establishment of sum rules connecting the asymptotic
-                  behavior of the susceptibility to the short-time response of
-                  the system. These results generalize previous findings on
-                  optical Hamiltonian systems and simple mechanical models, and
-                  shed light on the general impact of considering the principle
-                  of causality for testing self-consistency: the described
-                  dispersion relations constitute unavoidable benchmarks for any
-                  experimental and model generated dataset. In order to connect
-                  the response theory for equilibrium and non equilibrium
-                  systems, we rewrite the classical results by Kubo so that
-                  response functions formally identical to those proposed by
-                  Ruelle, apart from the measure involved in the phase space
-                  integration, are obtained. We briefly discuss how these
-                  results, taking into account the chaotic hypothesis, might be
-                  relevant for climate research. In particular, whereas the
-                  fluctuation-dissipation theorem does not work for
-                  non-equilibrium systems, because of the non-equivalence
-                  between internal and external fluctuations, Kramers-Kronig
-                  relations might be more robust tools for the definition of a
-                  self-consistent theory of climate change.},
-  archivePrefix ={arXiv},
-  arxivId =      {0710.0958},
-  eprint =       {0710.0958},
-  file =         {::},
-  issn =         00224715,
-  keywords =     {Axiom A dynamical systems,Chaotic hypothesis,Climate,Harmonic
-                  generation,Kramers-Kronig relations,Kubo response
-                  theory,Non-equilibrium steady states,Ruelle response
-                  theory,SRB measure},
-}
-
-@article{Moreels2006,
-  author =       {Moreels, I. and Hens, Z. and Kockaert, P. and Loicq, J. and
-                  {Van Thourhout}, D.},
-  title =        {{Spectroscopy of the nonlinear refractive index of colloidal
-                  PbSe nanocrystals}},
-  journal =      {Applied Physics Letters},
-  volume =       89,
-  number =       19,
-  pages =        {0--4},
-  year =         2006,
-  doi =          {10.1063/1.2385658},
-  abstract =     {A spectroscopic study of the optical nonlinearity of PbSe
-                  colloidal solutions was performed with the Z-scan technique at
-                  wavelength intervals of 1200-1350 and 1540-1750 nm. While
-                  nonlinear absorption remains below the detection threshold,
-                  the third order nonlinear refractive index n(2) shows clear
-                  resonances, somewhat blueshifted relative to the exciton
-                  transitions in the absorbance spectrum. The occurrence of
-                  thermal effects is ruled out by time-resolved measurements. At
-                  1.55 mu m, measured (resonant) n(2) values exceed typical bulk
-                  semiconductor values by two orders of magnitude. At high
-                  optical intensity, the refractive index change saturates,
-                  indicating that statefilling lies at the origin of the
-                  observed effect.},
-  file =         {::},
-  isbn =         {0003-6951},
-  issn =         00036951,
-}
-
-@article{Moreels2007,
-  author =       {Moreels, Iwan and Lambert, Karel and {De Muynck}, David and
-                  Vanhaecke, Frank and Poelman, Dirk and Martins, Jos{\'{e}} C.
-                  and Allan, Guy and Hens, Zeger},
-  title =        {{Composition and Size-Dependent Extinction Coefficient of
-                  Colloidal PbSe Quantum Dots}},
-  journal =      {Chemistry of Materials},
-  volume =       19,
-  number =       25,
-  pages =        {6101--6106},
-  year =         2007,
-  doi =          {10.1021/cm071410q},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/cm071410q},
-  file =         {::},
-  issn =         {0897-4756},
-  keywords =     {reference,size-exciton correlation},
-  mendeley-tags ={reference},
-  month =        {dec},
-}
-
-@article{Moreels2008,
-  author =       {Moreels, Iwan and Hens, Zeger},
-  title =        {{On the interpretation of colloidal quantum-dot absorption
-                  spectra.}},
-  journal =      {Small (Weinheim an der Bergstrasse, Germany)},
-  volume =       4,
-  number =       11,
-  pages =        {1866--8; author reply 1869--70},
-  year =         2008,
-  doi =          {10.1002/smll.200800068},
-  url =          {http://www.ncbi.nlm.nih.gov/pubmed/18855972},
-  file =         {::},
-  issn =         {1613-6829},
-  keywords =     {Absorption,Colloids,Colloids: chemistry,Quantum Dots,Spectrum
-                  Analysis,Spectrum Analysis: methods},
-  month =        {nov},
-  pmid =         18855972,
-}
-
-@article{Moreels2009,
-  author =       {Moreels, Iwan and Lambert, Karel and Muynck, David De and
-                  Vanhaecke, Frank and Poelman, Dirk and Martins, Jos{\'{e}} C
-                  and Allan, Guy and Hens, Zeger},
-  title =        {{Size-Dependent Optical Properties of Colloidal {\{}PbS{\}}
-                  Quantum Dots}},
-  journal =      {ACS Nano},
-  volume =       3,
-  number =       10,
-  pages =        {3023--3030},
-  year =         2009,
-  doi =          {10.1021/nn900863a},
-  abstract =     {We quantitatively investigate the size-dependent optical
-                  properties of colloidal {\{}PbS{\}} nanocrystals or quantum
-                  dots (Qdots), by combining the Qdot absorbance spectra with
-                  detailed elemental analysis of the Qdot suspensions. At high
-                  energies, the molar extinction coefficient ?? increases with
-                  the Qdot volume d3 and agrees with theoretical calculations
-                  using the Maxwell???Garnett effective medium theory and bulk
-                  values for the Qdot dielectric function. This demonstrates
-                  that quantum confinement has no influence on ?? in this
-                  spectral range, and it provides an accurate method to
-                  calculate the Qdot concentration. Around the band gap, ?? only
-                  increases with d1.3, and values are comparable to the ?? of
-                  {\{}PbSe{\}} Qdots. The data are related to the oscillator
-                  strength fif of the band gap transition and results agree well
-                  with theoretical tight-binding calculations, predicting a
-                  linear dependence of fif on d. For both {\{}PbS{\}} and
-                  {\{}PbSe{\}} Qdots, the exciton lifetime ?? is calculated from
-                  fif. We find values ranging between 1 and 3 ??s, in agreement
-                  with experimental literature data from time-resolved
-                  luminescence spectroscopy. Our results provide a thorough
-                  general framework to calculate and understand the optical
-                  properties of suspended colloidal quantum dots. Most
-                  importantly, it highlights the significance of the local field
-                  factor in these systems.},
-  file =         {::},
-  issn =         {1936-0851},
-  keywords =     {exciton lifetime,extinction coefficient,lead
-                  chalcogenide,molar,oscillator strength,pbse,semiconductor
-                  nanocrystals},
-}
-
-@article{Nootz2011,
-  author =       {Nootz, Gero and Padilha, Lazaro A. and Levina, Larissa and
-                  Sukhovatkin, Vlad and Webster, Scott and Brzozowski, Lukasz
-                  and Sargent, Edward H. and Hagan, David J. and {Van Stryland},
-                  Eric W.},
-  title =        {{Size dependence of carrier dynamics and carrier
-                  multiplication in PbS quantum dots}},
-  journal =      {Physical Review B - Condensed Matter and Materials Physics},
-  volume =       83,
-  number =       15,
-  pages =        {1--7},
-  year =         2011,
-  doi =          {10.1103/PhysRevB.83.155302},
-  abstract =     {The time dynamics of the photoexcited carriers and
-                  carrier-multiplication efficiencies in PbS quantum dots (QDs)
-                  are investigated. In particular, we report on the carrier
-                  dynamics, including carrier multiplication, as a function of
-                  QD size and compare them to the bulk value. We show that the
-                  intraband 1P -{\textgreater} 1S decay becomes faster for
-                  smaller QDs, in agreement with the absence of a phonon
-                  bottleneck. Furthermore, as the size of the QDs decreases, the
-                  energy threshold for carrier multiplication shifts from the
-                  bulk value to higher energies. However, the energy threshold
-                  shift is smaller than the band-gap shift and, therefore, for
-                  the smallest QDs, the threshold approaches 2.35 E(g), which is
-                  close to the theoretical energy conservation limit of twice
-                  the band gap. We also show that the carrier-multiplication
-                  energy efficiency increases with decreasing QD size. By
-                  comparing to theoretical models, our results suggest that
-                  impact ionization is sufficient to explain carrier
-                  multiplication in QDs.},
-  file =         {::},
-  isbn =         {1098-0121},
-  issn =         10980121,
-}
-
-@article{Okumura1999,
-  author =       {Okumura, Ko and Tokmakoff, Andrei and Tanimura, Yoshitaka},
-  title =        {{Two-dimensional line-shape analysis of photon-echo signal}},
-  journal =      {Chemical Physics Letters},
-  volume =       314,
-  number =       {5-6},
-  pages =        {488--495},
-  year =         1999,
-  doi =          {10.1016/S0009-2614(99)01173-2},
-  url =          {http://linkinghub.elsevier.com/retrieve/pii/S0009261499011732},
-  abstract =     {We analyze the two-dimensional (2D) line shape obtained by 2D
-                  Fourier transforming the time-domain response of a photon-echo
-                  signal as a function of the two coherence periods, t(1) and
-                  t(3). The line shape obtained for a two-level system with
-                  homogeneous and inhomogeneous broadening is shown to be
-                  sensitive to the magnitude of both of these line-broadening
-                  mechanisms. It is shown that the ellipticity of the 2D line
-                  shape can be related to the ratio of homogeneous to
-                  inhomogeneous broadening. (C) 1999 Elsevier Science B.V. All
-                  rights reserved.},
-  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
-                  Desktop/Downloaded/Okumura, Tokmakoff, Tanimura - 1999 -
-                  Two-dimensional line-shape analysis of photon-echo
-                  signal.pdf:pdf},
-  isbn =         {0009-2614},
-  issn =         00092614,
-  keywords =     {3rd-order,dynamics,glasses,phase,probes,spectroscopy},
-  month =        {dec},
-}
-
-@article{Olivero1977,
-  author =       {Olivero, J.J. and Longbothum, R.L.},
-  title =        {{Empirical fits to the voigt line width: A brief review}},
-  journal =      {J. Quant. Spectrosc. Radiat. Transfer},
-  volume =       17,
-  pages =        {233--236},
-  year =         1977,
-  file =         {::},
-}
-
-@article{Omari2012,
-  author =       {Omari, Abdoulghafar and Moreels, Iwan and Masia, Francesco and
-                  Langbein, Wolfgang and Borri, Paola and {Van Thourhout}, Dries
-                  and Kockaert, Pascal and Hens, Zeger},
-  title =        {{Role of interband and photoinduced absorption in the
-                  nonlinear refraction and absorption of resonantly excited PbS
-                  quantum dots around 1550 nm}},
-  journal =      {Physical Review B},
-  volume =       85,
-  number =       11,
-  pages =        115318,
-  year =         2012,
-  doi =          {10.1103/PhysRevB.85.115318},
-  url =          {http://link.aps.org/doi/10.1103/PhysRevB.85.115318},
-  file =         {::},
-  issn =         {1098-0121},
-  month =        {mar},
-}
-
-@article{Pang1991,
-  author =       {Pang, Yang and Samoc, Marek and Prasad, Paras N.},
-  title =        {{Third‐order nonlinearity and two‐photon‐induced molecular
-                  dynamics: Femtosecond time‐resolved transient absorption, Kerr
-                  gate, and degenerate four‐wave mixing studies in poly ( p
-                  ‐phenylene vinylene)/sol‐gel silica film}},
-  journal =      {The Journal of Chemical Physics},
-  volume =       94,
-  number =       8,
-  pages =        {5282--5290},
-  year =         1991,
-  doi =          {10.1063/1.460512},
-  url =
-                  {http://scitation.aip.org/content/aip/journal/jcp/94/8/10.1063/1.460512
-                  http://aip.scitation.org/doi/10.1063/1.460512},
-  abstract =     {Femtosecond response and relaxation of the third-order optical
-                  nonlinearity in a newly developed poly (p-phenylene
-                  vinylene)/sol-gel silica composite are investigated by time-
-                  resolved forward wave degenerate four-wave mixing, Kerr gate,
-                  and transient absorption techniques using 60 fs pulses at 620
-                  nm. Using a theoretical description of two- and four-wave
-                  mixing in optically nonlinear media, it is shown that the
-                  results obtained from simultaneous use of these techniques
-                  yield valuable information on the real and imaginary
-                  components of the third-order susceptibility. In the composite
-                  material investigated here, the imaginary component is derived
-                  from the presence of a two-photon resonance at the wavelength
-                  of 620 nm used for the present study. This two-photon
-                  resonance is observed as transient absorption of the probe
-                  beam induced by the presence of a strong pump beam. It also
-                  provides fifth-order nonlinear response both in transient
-                  absorption and in degenerate four-wave mixing. The fifth-
-                  order contributions are derived from the two-photon generated
-                  excited species which can absorb at the measurement wavelength
-                  and therefore modify both the absorption coefficient and the
-                  refractive index of the medium.},
-  file =         {::},
-  issn =         {0021-9606},
-  month =        {apr},
-}
-
-@article{Peterson2007,
-  author =       {Peterson, JJ and Huang, Libai and Delerue, C. and Allan, Guy},
-  title =        {{Uncovering forbidden optical transitions in PbSe
-                  nanocrystals}},
-  journal =      {nano Letters},
-  year =         2007,
-  url =          {http://pubs.acs.org/doi/abs/10.1021/nl072487g},
-  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
-                  Desktop/Downloaded/Peterson et al. - 2007 - Uncovering
-                  forbidden optical transitions in PbSe nanocrystals.pdf:pdf},
-  keywords =     {1P exciton,2 photon absorption,new,relevant},
-  mendeley-tags ={new,relevant},
-}
-
-@article{Price2015,
-  author =       {Price, Michael B. and Butkus, Justinas and Jellicoe, Tom C.
-                  and Sadhanala, Aditya and Briane, Anouk and Halpert, Jonathan
-                  E. and Broch, Katharina and Hodgkiss, Justin M. and Friend,
-                  Richard H. and Deschler, Felix},
-  title =        {{Hot-carrier cooling and photoinduced refractive index changes
-                  in organic–inorganic lead halide perovskites}},
-  journal =      {Nature Communications},
-  volume =       6,
-  number =       {May},
-  pages =        8420,
-  year =         2015,
-  doi =          {10.1038/ncomms9420},
-  url =          {http://dx.doi.org/10.1038/ncomms9420
-                  http://www.nature.com/doifinder/10.1038/ncomms9420},
-  archivePrefix ={arXiv},
-  arxivId =      {arXiv:1504.07508},
-  eprint =       {arXiv:1504.07508},
-  file =         {::},
-  issn =         {2041-1723},
-  publisher =    {Nature Publishing Group},
-}
-
-@article{Schaller2003,
-  author =       {Schaller, Richard D and Petruska, M. a. and Klimov, Victor I.},
-  title =        {{Tunable Near-Infrared Optical Gain and Amplified Spontaneous
-                  Emission Using PbSe Nanocrystals}},
-  journal =      {The Journal of Physical Chemistry B},
-  volume =       107,
-  number =       50,
-  pages =        {13765--13768},
-  year =         2003,
-  doi =          {10.1021/jp0311660},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp0311660},
-  file =         {::;::},
-  issn =         {1520-6106},
-  month =        {dec},
-}
-
-@article{Schins2009,
-  author =       {Schins, Juleon and Trinh, M. and Houtepen, Arjan and
-                  Siebbeles, Laurens},
-  title =        {{Probing formally forbidden optical transitions in PbSe
-                  nanocrystals by time- and energy-resolved transient absorption
-                  spectroscopy}},
-  journal =      {Physical Review B},
-  volume =       80,
-  number =       3,
-  pages =        035323,
-  year =         2009,
-  doi =          {10.1103/PhysRevB.80.035323},
-  url =          {http://link.aps.org/doi/10.1103/PhysRevB.80.035323},
-  file =         {::},
-  issn =         {1098-0121},
-  month =        {jul},
-}
-
-@article{Svirko1999,
-  author =       {{P. Svirko}, Yuri and Shirane, Masayuki and Suzuura, Hidekatsu
-                  and Kuwata-Gonokami, Makoto},
-  title =        {{Four-Wave Mixing Theory at the Excitonic Resonance: Weakly
-                  Interacting Boson Model}},
-  journal =      {Journal of the Physical Society of Japan},
-  volume =       68,
-  number =       2,
-  pages =        {674--682},
-  year =         1999,
-  doi =          {10.1143/JPSJ.68.674},
-  url =          {http://journals.jps.jp/doi/10.1143/JPSJ.68.674},
-  file =         {::},
-  issn =         {0031-9015},
-  keywords =     {Exciton,Four-particle correlation,Four-wave mixing,Normal mode
-                  splitting,Semiconductor microcavity},
-  month =        {feb},
-}
-
-@article{Trinh2008,
-  author =       {Trinh, M Tuan and Houtepen, Arjan J and Schins, Juleon M and
-                  Piris, Jorge and Siebbeles, Laurens D a},
-  title =        {{Nature of the second optical transition in PbSe
-                  nanocrystals.}},
-  journal =      {Nano letters},
-  volume =       8,
-  number =       7,
-  pages =        {2112--7},
-  year =         2008,
-  doi =          {10.1021/nl8010963},
-  url =          {http://www.ncbi.nlm.nih.gov/pubmed/18510369},
-  abstract =     {The second peak in the optical absorption spectrum of PbSe
-                  nanocrystals is arguably the most discussed optical transition
-                  in semiconductor nanocrystals. Ten years of scientific debate
-                  have produced many theoretical and experimental claims for the
-                  assignment of this feature as the 1P e1P h as well as the 1S
-                  h,e1P e,h transitions. We studied the nature of this
-                  absorption feature by pump-probe spectroscopy, exactly
-                  controlling the occupation of the states involved, and present
-                  conclusive evidence that the optical transition involves
-                  neither 1S e nor 1S h states. This suggests that it is the 1P
-                  h1P e transition that gives rise to the second peak in the
-                  absorption spectrum of PbSe nanocrystals.},
-  file =         {::},
-  issn =         {1530-6984},
-  month =        {jul},
-  pmid =         18510369,
-}
-
-@article{Trinh2013,
-  author =       {Trinh, M. Tuan and Sfeir, Matthew Y. and Choi, Joshua J. and
-                  Owen, Jonathan S. and Zhu, Xiaoyang},
-  title =        {{A Hot Electron–Hole Pair Breaks the Symmetry of a
-                  Semiconductor Quantum Dot}},
-  journal =      {Nano Letters},
-  volume =       13,
-  number =       12,
-  pages =        {6091--6097},
-  year =         2013,
-  doi =          {10.1021/nl403368y},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/nl403368y},
-  file =         {::},
-  issn =         {1530-6984},
-  keywords =     {google the phrase,hot carriers,nanocrystals or,near the
-                  bandgap is,now well-,one,ptical excitation of
-                  semiconductor,qds,quantum dots,selection rules,stark
-                  effect,symmetry breaking,transient absorption,understood},
-  month =        {dec},
-}
-
-@article{Wehrenberg2002,
-  author =       {Wehrenberg, Brian L. and Wang, Congjun and Guyot-Sionnest,
-                  Philippe},
-  title =        {{Interband and Intraband Optical Studies of PbSe Colloidal
-                  Quantum Dots}},
-  journal =      {The Journal of Physical Chemistry B},
-  volume =       106,
-  number =       41,
-  pages =        {10634--10640},
-  year =         2002,
-  doi =          {10.1021/jp021187e},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp021187e},
-  file =         {::},
-  issn =         {1520-6106},
-  month =        {oct},
-}
-
-@article{Wise2000,
-  author =       {Wise, Frank W.},
-  title =        {{Lead salt quantum dots: the limit of strong quantum
-                  confinement.}},
-  journal =      {Accounts of chemical research},
-  volume =       33,
-  number =       11,
-  pages =        {773--80},
-  year =         2000,
-  url =          {http://www.ncbi.nlm.nih.gov/pubmed/11087314},
-  abstract =     {Nanocrystals or quantum dots of the IV-VI semiconductors PbS,
-                  PbSe, and PbTe provide unique properties for investigating the
-                  effects of strong confinement on electrons and phonons. The
-                  degree of confinement of charge carriers can be many times
-                  stronger than in most II-VI and III-V semiconductors, and lead
-                  salt nanostructures may be the only materials in which the
-                  electronic energies are determined primarily by quantum
-                  confinement. This Account briefly reviews recent research on
-                  lead salt quantum dots.},
-  file =         {:C$\backslash$:/Users/Dan/AppData/Local/Mendeley Ltd./Mendeley
-                  Desktop/Downloaded/Wise - 2000 - Lead salt quantum dots the
-                  limit of strong quantum confinement.pdf:pdf},
-  issn =         {0001-4842},
-  keywords =     {Crystallization,Electrochemistry,Lead,Lead:
-                  chemistry,Semiconductors,Temperature},
-  month =        {nov},
-  pmid =         11087314,
-}
-
-@article{Yang2015,
-  author =       {Yang, Ye and Yan, Yong and Yang, Mengjin and Choi, Sukgeun and
-                  Zhu, Kai and Luther, Joseph M and Beard, Matthew C},
-  title =        {{Low surface recombination velocity in}},
-  journal =      {Nature Communications},
-  volume =       6,
-  pages =        {1--6},
-  year =         2015,
-  doi =          {10.1038/ncomms8961},
-  url =          {http://dx.doi.org/10.1038/ncomms8961},
-  file =         {::},
-  publisher =    {Nature Publishing Group},
-}
-
-@article{Yang2017,
-  author =       {Yang, Ye and Yang, Mengjin and Moore, David?T. and Yan, Yong
-                  and Miller, Elisa?M. and Zhu, Kai and Beard, Matthew?C.},
-  title =        {{Top and bottom surfaces limit carrier lifetime in lead iodide
-                  perovskite films}},
-  journal =      {Nature Energy},
-  volume =       2,
-  number =       2,
-  pages =        16207,
-  year =         2017,
-  doi =          {10.1038/nenergy.2016.207},
-  url =          {http://www.nature.com/articles/nenergy2016207},
-  abstract =     {Carrier recombination at defects is detrimental to the
-                  performance of solar energy conversion systems, including
-                  solar cells and photoelectrochemical devices. Point defects
-                  are localized within the bulk crystal while extended defects
-                  occur at surfaces and grain boundaries. If not properly
-                  managed, surfaces can be a large source of carrier
-                  recombination. Separating surface carrier dynamics from bulk
-                  and/or grain-boundary recombination in thin films is
-                  challenging. Here, we employ transient reflection spectroscopy
-                  to measure the surface carrier dynamics in methylammonium lead
-                  iodide perovskite polycrystalline films. We find that surface
-                  recombination limits the total carrier lifetime in perovskite
-                  polycrystalline thin films, meaning that recombination inside
-                  grains and/or at grain boundaries is less important than top
-                  and bottom surface recombination. The surface recombination
-                  velocity in polycrystalline films is nearly an order of
-                  magnitude smaller than that in single crystals, possibly due
-                  to unintended surface passivation of the films during
-                  synthesis.},
-  file =         {::},
-  issn =         {2058-7546},
-  month =        {jan},
-}
-
-@article{Yurs2011,
-  author =       {Yurs, Lena A and Block, Stephen B and Pakoulev, Andrei V. and
-                  Selinsky, Rachel S. and Jin, Song and Wright, John},
-  title =        {{Multiresonant Coherent Multidimensional Electronic
-                  Spectroscopy of Colloidal PbSe Quantum Dots}},
-  journal =      {The Journal of Physical Chemistry C},
-  volume =       115,
-  number =       46,
-  pages =        {22833--22844},
-  year =         2011,
-  doi =          {10.1021/jp207273x},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp207273x},
-  file =         {::},
-  issn =         {1932-7447},
-  month =        {nov},
-}
-
-@article{Yurs2012,
-  author =       {Yurs, Lena A and Block, Stephen B and Pakoulev, Andrei V. and
-                  Selinsky, Rachel S. and Jin, Song and Wright, John},
-  title =        {{Spectral Isolation and Measurement of Surface-Trapped State
-                  Multidimensional Nonlinear Susceptibility in Colloidal Quantum
-                  Dots}},
-  journal =      {The Journal of Physical Chemistry C},
-  volume =       116,
-  number =       9,
-  pages =        {5546--5553},
-  year =         2012,
-  doi =          {10.1021/jp3014139},
-  url =          {http://pubs.acs.org/doi/abs/10.1021/jp3014139},
-  file =         {::},
-  issn =         {1932-7447},
-  month =        {mar},
-}
\ No newline at end of file
diff --git a/PbSe_global_analysis/results.tex b/PbSe_global_analysis/results.tex
deleted file mode 100644
index 0bb0f03..0000000
--- a/PbSe_global_analysis/results.tex
+++ /dev/null
@@ -1,47 +0,0 @@
-\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.

diff --git a/PbSe_global_analysis/run.sh b/PbSe_global_analysis/run.sh
deleted file mode 100755
index a38e6ff..0000000
--- a/PbSe_global_analysis/run.sh
+++ /dev/null
@@ -1,75 +0,0 @@
-set -e  # force exit upon error
-
-function printColor {
-  YELLOW='\033[0;33m'
-  NC='\033[0m' # No Color
-  printf "${YELLOW}$*${NC}\n"
-}
-
-function printLine {
-  printColor "%`tput cols`s"|tr ' ' '#'
-}
-
-function printThenPython {
-  printColor "python $*"
-  python "$*"
-}
-
-if [[ $# -eq 0 ]] ; then
-    echo 'please provide an argument in [data, simulations, figures, documents, all]'
-    exit 1
-fi
-
-# process data
-if [[ "$1" = "all" ]] || [[ "$1" = "data" ]] ; then
-  printLine
-  printColor data
-  printThenPython "data/TA A.py"
-  printThenPython "data/TA B.py"
-  printThenPython "data/TG A.py"
-  printThenPython "data/TG B.py"  
-fi
-
-# run simulations
-if [[ "$1" = "all" ]] || [[ "$1" = "simulations" ]] ; then
-  printLine
-  printColor simulations
-  #printThenPython "simulations/fit fsb19.py"
-  #printThenPython "simulations/fit fsb25.py"
-fi
-
-# make figures
-if [[ "$1" = "all" ]] || [[ "$1" = "figures" ]] ; then
-  printLine
-  printColor figures
-  printThenPython "figures/absorbance.py"
-  printThenPython "figures/kramers_kronig.py"
-  printThenPython "figures/m_factors.py"
-  #printThenPython "figures/movies_fitted.py"
-  printThenPython "figures/movies_combined.py"
-  printThenPython "figures/power_factors.py"
-  #printThenPython "figures/driven_initial.py"
-  printThenPython "figures/TA_artifacts.py"
-  printThenPython "figures/TG_artifacts.py"
-  printThenPython "figures/ta_vs_tg.py"
-fi
-
-# render documents
-if [[ "$1" = "all" ]] || [[ "$1" = "documents" ]] ; then
-  printLine
-  printColor documents
-  # main
-  printColor main
-  pdflatex --interaction=nonstopmode main
-  bibtex main
-  pdflatex --interaction=nonstopmode main
-  pdflatex --interaction=nonstopmode main
-  # SI
-  printColor SI
-  pdflatex --interaction=nonstopmode SI
-  biber SI
-  pdflatex --interaction=nonstopmode SI
-  pdflatex --interaction=nonstopmode SI
-fi
-
-printColor finished
diff --git a/PbSe_global_analysis/theory.tex b/PbSe_global_analysis/theory.tex
deleted file mode 100644
index d20d844..0000000
--- a/PbSe_global_analysis/theory.tex
+++ /dev/null
@@ -1,188 +0,0 @@
-\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}