\chapter{Resonant third-order susceptibility of PbSe quantum dots determined by standard dilution
  and transient grating spectroscopy} \label{cha:pss}

\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}
}

Here we detail the extraction of quantitative information from ultrafast multiresonant CMDS
spectra.  %
PbSe quantum dots are used as a model system.  %
We investigate the severity of pulse propagation effects, as well as the increased prominence of
solvent background contributions to signals.  %
We characterize ultrafast Four-Wave Mixing signals of colloidal quantum dots using non-resonant
solvent response as a standard.  %
We show that dilution studies can be quantitatively described, and apply a robust, few-parameter
fit to extract the peak susceptibility.  %  
These method should be generally applicable to solution phase systems and other CMDS
techniques.    %

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

Coherent multidimensional spectroscopy (CMDS) provides a wealth of information on the structure,
energetics, and dynamics of solution phase systems.  %
By using multiple excitations to probe resonances, CMDS elucidates correlations and couplings
between electronic, vibrational, and vibronic states.  %
It is now commonplace to interrogate such structures with femtosecond time resolution.  %  
The time resolved nature of the measurement in this regime allows characterization of transient
states that are unresolvable in steady-state methods.  %
Multiresonant CMDS (MR-CMDS) is a frequency-domain technique, whereby tunable lasers are scanned to
obtain multidimensional spectra.  %
It has been demonstrated on material systems with femtosecond pulses. \cite{KohlerDanielDavid2014a,
  CzechKyleJonathan2015a}

Traditionally CMDS employs line shape analysis (peak center, width, sign, homogeneous and
inhomogeneous line width) and dynamics analysis (time constants, amplitudes) to give insight into
the material of interest.  %
The magnitudes of optical non-linearities, though commonly ignored, are intrinsic properties that
also inform on microscopic properties.  %
The microscopic mechanisms for optical nonlinearities are determined by a sequence of field-matter
interactions (Liouville pathways) that depend on \textit{linear} properties (cross-sections, etc.)
with each interaction.  %
Some methods (TSF) isolate single Liouville pathways so that the magnitude of the non-linearity is
a simple product of cross-sections.  %
Some cross sections are easily measured with conventional experiments (Raman, absorption), but
others depend on exotic/inaccessible transitions.  %
Techniques like CMDS must be employed to measure the cross-sections of these inaccessible
transitions.  %

Other methods (2DES, TG, TA) have multiple similar Liouville pathways, such that the nonlinearity
arises from imperfect cancellation between them.  %
This lack of cancellation also is connected to the microscopic properties. 
For example, saturation effects, such as state-filling, have to do with the degeneracy of a
transition.  %
CMDS can explicitly measure the strengths of these saturation effects.  
This paper describes the measurement of state-filling.  

Absolute susceptibility is not an observable with most experimental configurations.  
It is uncommon for CMDS spectra to obtain absolute units of susceptibility in the spectra they
report.  %
Measurements such as the $z$-scan \cite{SheikBahaeMansoor1989a, SheikBahaeMansoor1990a} and
transient absorption, specialize in quantifying optical non-linearities, but these methods are
limited in the multidimensional space they can explore.  %
TA cannot do 3-color non-linearities, and $z$-scan cannot interrogate dynamics.  %

Internal standards are a convenient means to quantify the non-linearity magnitude.
\cite{LevensonMD1974a}  % 
The method has also been demonstrated on MR-CMDS of quantum dots under picosecond excitation
pulses. \cite{YursLenaA2012a}  %
These methods typically require a full characterization (phase and amplitude) of analyte spectral
properties in order to relate the two signals.  %
For CMDS methods that possess a multitude of Liouville pathways, this characterization requires a
large number of parameters and can be quite complex to solve.  %

This work details a few-parameter extraction of the third-order susceptibility of the 1S band of
PbSe quantum dots (QDs).  %
We utilize standard additions for characterizing the absolute third-order susceptibility of
resonant signals.  %
Applying justified approximations, we extract the absolute susceptibility without explicitly
modeling Liouville pathways of an excitonic manifold.  %
We connect the common phenomenologies of optical non-linearities to the theory of state-filling.  %
Finally, we compare our measurements with the diverse and numerous values of literature.  
Once pulse-length factors and propagation distortions are accounted for, we find that the measured
non-linear susceptibility of these femtosecond experiments is in good agreement with previously
published values.  %

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

These experiments consider the CMDS signal resulting from a chromophore resonance in a transparent
solvent.  %
We first formulate the CMDS intensity in terms of the separate contributions of solvent and
solute.  %
We then connect the well-known theory of optical bleaching of the 1S band to our measurements.  %

\subsection{Extraction of susceptibility}  % ------------------------------------------------------

In the Maker-Terhune convention, the relevant third-order polarization, $P^{(3)}$, is related to
the non-linear susceptibility, $\chi^{(3)}$, by \textcite{MakerPD1965a}  %
\begin{equation} \label{pss:eq:Maker_Terhune}
  P^{(3)}(z, \omega) = D \chi^{(3)}(\omega; \omega_1, -\omega_2,	\omega_{2^\prime})
  \times E_1(z, \omega_1) E_2(z, -\omega_2) E_{2^\prime}(z, \omega_{2^\prime}),
\end{equation}
where $z$ is the optical axis coordinate (the experiment is approximately collinear), $E_i$ is the
real-valued electric field of pulse $i$, and $\omega_i$ is the frequency of pulse $i$.  %  
The degeneracy factor $D = 3! / (3 - n)!$ accounts for the permutation symmetry that arises from
the interference of $n$ distinguishable excitation fields.  %
$D = 6$ for transient absorption and transient grating, and $D = 3$ for $z$-scan.  %
Permutation symmetry reflects the strength of the excitation fields and not the intrinsic
non-linearity of the sample.  %
Including $D$ in our convention makes $\chi^{(3)}$ invariant to different beam geometries.  %

\autoref{pss:eq:Maker_Terhune} is the steady-state solution to the Liouville Equation, valid when
excitation fields are greatly detuned from resonance and/or much longer than coherence times.  %
This convention is invalid for impulsive excitation, where $\chi^{(3)}$ will be sensitive to pulse
duration.  %

The non-linear polarization launches an output field.
The intensity of this output depends on the accumulation of polarization throughout the sample.  
For a homogeneous material, the output intensity, $I$, is proportional
to \cite{CarlsonRogerJohn1989a}  %
\begin{equation} \label{pss:eq:fwm_intensity}
	\begin{split}
		I &\propto \left| \int P^{(3)} (z, \omega) dz \right|^2 \\
		  &\propto \left| M P^{(3)}(0, \omega) \ell \right|^2 \\
		  &= |M D \chi^{(3)} E_1(0,\omega_1) E_2(0, -\omega_2) E_3(0, \omega_{2^\prime})|^2. 
	\end{split}
\end{equation}
Here $\ell$ is the sample length and $M$ is a frequency-dependent factor that accounts for phase
mismatch and absorption effects.  %
Phase mismatch is negligible in these experiments.  %
For purely absorptive effects, $M$ may be written as \cite{CarlsonRogerJohn1989a, YursLenaA2011a}
\begin{equation}
	M(\omega_1, \omega_2) = \frac{e^{-\alpha_1 \ell /2}\left(1 - e^{-\alpha_2 \ell} \right)}{\alpha_2 \ell}
\end{equation}
where $\alpha_i = \sigma_i N_\text{QD}$ is the absorptivity of the sample at frequency
$\omega_i$.  %
Absorption effects disrupt the proportional relationship between $I$ and $\chi^{(3)}$.  %
\autoref{pss:eq:fwm_intensity} shows that we can derive spectra free of pulse propagation effects
by normalizing the output intensity by $M^2$.  %
The distortions incurred by optically thick samples are well-known and have been treated in similar
CMDS experiments \cite{SpencerAustinP2015a, YetzbacherMichaelK2007a, ChoByungmoon2009a,
  KeustersDorine2004a}.  %

For cuvettes, the sample solution is sandwiched between two transparent windows.  
Rather than \autoref{pss:eq:fwm_intensity}, the total polarization has three distinct homogeneous regions:  the front window, the solution, and the back window.
The windows each have the same thickness, $\ell_\text{w}$, and susceptibility, $\chi_\text{w}^{(3)}$.
The (absorption-corrected) output intensity is proportional to:
\begin{equation} \label{pss:eq:fwm_intensity2}
	\frac{I}{I_1 I_2 I_{2^\prime} M^2} \propto 
		\left| \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2 \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} 
			+ \chi_\text{sol}^{(3)} + \chi_\text{QD}^{(3)} 
		\right|^2
\end{equation}
where $\chi^{(3)}_\text{QD}$ is the QD susceptibility and the $\chi^{(3)}_\text{sol}$ is the solvent susceptibility.
Each susceptibility depends on the chromophore number density and local field enhancements for each wave:
\begin{equation} \label{pss:eq:hyperpolarizability}
	\chi_i^{(3)} = 
		f(\omega_1)^2 f(\omega_2)^2 N_i \gamma_i^{(3)},
\end{equation}
where $\gamma_i^{(3)}$ is the intrinsic (per-QD/per-molecule, \textit{in vacuo}) hyperpolarizability, $N_i$ is the number density of species $i$, and $f(\omega)$ is the local field enhancement factor.
Since QDs constitute a negligible number/volume fraction of the solution, the field enhancement is derived entirely from the solvent:  $f(\omega) = \left( \frac{n_\text{sol}(\omega)^2 + 2}{3} \right)$, where $n_\text{sol}$ is the solvent refractive index.  
Both $n$ and $f$ are frequency dependent, but both vary small amounts ($\sim 0.1 \%$) over the frequency ranges considered here.  
We approximate both as constants, and remove the frequency argument from further equations.


\autoref{pss:eq:fwm_intensity2} can be expressed as the classic signal-local oscillator interference, 
\begin{equation} \label{pss:eq:LO}
	I \propto \left| E_\text{LO} \right|^2 + N_\text{QD}^2 f^8 \left| \gamma_\text{QD} \right|^2 + 2 N_\text{QD} f^4 \text{Re}\left[ E_\text{LO} \gamma_\text{QD}^* \right]
\end{equation}
where we have used the substitutions $E_\text{LO} = \alpha_2 \ell_\text{w} \frac{1 + e^{-\alpha_2
    \ell_s}}{1 - e^{-\alpha_2 \ell_s}} \chi_\text{w}^{(3)} + \chi_\text{sol}^{(3)}$.  %
The character of the interference depends both on the amplitude of the QD field and on the phase
relationship between the two fields.  %
The QD field amplitude can be controlled by $N_\text{QD}$.  %
At low concentrations there is a linear dependence on $N_\text{QD}$, but this changes at high
optical densities due to an $\alpha_2$ dependence on the window contribution.  %
The phase relationship cannot be externally controlled and is frequency dependent:  it is defined
by the resonant character of each material.  %
The phase is defined by electronic resonances in QD and by Raman resonances in the solvent and the
windows.  %
The local oscillator and signal fields are non-additive unless the phase difference is $\pm \pi /
2$.  %  

\subsection{Optical bleaching and dependencies on experimental conditions}  % ---------------------

Most non-linear experiments on QDs extract pulse propagation parameters, such as the non-linear
absorptivity, $\beta$ or non-linear index of refraction, $n_2$.  %
These parameters are connected to the third-order susceptibility (in the cgs unit system) by
\begin{gather} \label{pss:eq:beta_to_chi}
	\beta = \frac{32 \pi^2 D \omega}{n_0^2 c^2} \text{Im}\left[ \chi^{(3)} \right]
  \\	\label{pss:eq:n2_to_chi}
	n_2 = \frac{16 \pi^2 D }{n_0^2 c} \text{Re}\left[ \chi^{(3)} \right].
\end{gather}

At the band edge, the non-linear absorptivity of semiconductor QDs is dominated by state-filling
\cite{Klimov2000}, where absorption is reduced in proportion to the average number of excitons
pumped into the band, $\langle n \rangle$.  %
Due to an 8-fold degeneracy, lead chalcogenide (PbX) QDs are bleached fractionally by the presence
of an exciton.  %
An 8-fold degeneracy predicts a bleach fraction of $\phi = 0.25$.
\cite{Trinh2008,Trinh2013,Gdor2013a,Spoor,Schaller2003}  %
For a Gaussian pump pulse of peak intensity $I$, frequency $\omega$, and full-width at half-maximum
(FWHM) of $\Delta_t$, $\langle n \rangle = \frac{\sqrt{2 \pi} \sigma}{\hbar \omega} \Delta_t I$
where $\sigma$ is the QD absorptive cross-section at frequency $\omega$.  %
We can then write the non-linear change in absorptivity as
\begin{equation} \label{pss:eq:Delta_alpha1}
\begin{split}
	\beta I_2 &= -\phi \langle n \rangle \alpha \\
		&= - \phi N_\text{QD} \frac{\sqrt{2 \pi} \sigma_1 \sigma_2}{\hbar \omega} \Delta_t I_2 
\end{split}
\end{equation}
where the indexes $1$ and $2$ denote properties of the probe and pump fields, respectively.
In some techniques (e.g. $z$-scan), both probe and pump fields are the same, in which case the subscripts become unnecessary.

By combining Equations \ref{pss:eq:hyperpolarizability}, \ref{pss:eq:beta_to_chi}, and
\ref{pss:eq:Delta_alpha1} we can relate the bleach factor directly to the hyperpolarizability:
\begin{equation} \label{pss:eq:gamma_to_phi}
	\text{Im}\left[ \gamma^{(3)} \right] = 
		-\phi \frac{\sqrt{2\pi} n^2 c^2}{32 \pi^2 D f^4 \hbar \omega_1 \omega_2} \sigma_1 \sigma_2 \Delta_t.
\end{equation}
\autoref{pss:eq:gamma_to_phi} will be useful for benchmarking our results because it connects our
observable, $\gamma_\text{QD}$, with the nonlinearity of the microscopic model, $\phi$.  %

\section{Experimental}  % =========================================================================

\subsubsection{Sample preparation and characterization}  % ----------------------------------------

QDs were created using a standard solution-phase technique. \cite{WehrenbergBrianL2002a}  %
QDs were washed in ethanol-toluene before being immersed in carbon tetrachloride (CCl$_4$) and
stored in a nitrogen-pumped glovebox.  %
Successive dilutions created the aliquots of different concentration used here.  % 
Aliquots were stored in 1 mm path length fused silica cuvettes with 1.25 mm thick windows.  %

Each aliquot was characterized by absorption spectroscopy (JASCO).  %
The spectra are consistent between all dilutions (no agglomeration, see Supplementary Info).  %
The 1S feature peaks at 0.937 eV and has a FWHM of 92 meV.  %
Concentrations were extracted using Beer's law and published cross-sections.
\cite{Moreels2007,Dai2009}  %
The peak ODs range from 0.06 to 0.86 (QD densities of $10^{16} - 10^{17} \ \text{cm}^{-3}$). 

\autoref{pss:fig:absorbance}, top, shows the absorption spectra gathered for each of the aliquots
used in this study.  %
All spectra were baseline-subtracted to account for reflection of the cuvette.  %
Differences in spectral properties between aliquots can be observed by normalizing each spectra to
the 1S band (\autoref{pss:fig:absorbance}, middle).  %
No significant differences were observed; the small differences between aliquots near $\hbar\omega
\approx 0.88$ eV is attributed to a small absorption feature of the fused silica cuvette, typically
attributed to OH stretches).  %
For computation of absorptive losses, the effective absorptivity of the pules were computed.  
The effective absorptivity spectrum is achieved by convolving the absorption spectra with the pulse
bandwidth.  %
\autoref{pss:fig:absorbance}, bottom, shows the differences between the effective absorptivity and
the absorptivity of the darkest aliquot used.  %

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/absorbance"}
	\caption[Absorption spectra of QD aliquots used in this study.]{
		The absorption spectra of the QD aliquots used in this study.  
		Top:  the raw absorbance spectra of each aliquot used.
		Middle:  the absorbance spectra normalized by the 1S peak.  
		Bottom:  Sample absorptivity (solid) and the effective absorption (assuming 50 meV FWHM pulse
    bandwidth).
	}
	\label{pss:fig:absorbance}
\end{figure}

\subsubsection{Four-wave Mixing}

An ultrafast oscillator (Tsunami, Spectra-Physics) produced a 80 MHz train of 35 fs pulses, which
were amplified (Spitifire Pro XP, Spectra-Physics, 1kHz) and split to pump two independently
tunable OPAs (TOPAS-C, Light Conversion): OPA1 and OPA2.  %
The frequency-dependent OPA power output was measured (407-A Thermopile, Spectra-Physics) and used
to normalize the non-linear spectra.  %
Pulses from OPA2 were split again, for a total of three excitation pulses:  $E_1$, $E_2$, and
$E_{2^\prime}$.  %
These pulses were passed through motorized (Newport MFA-CC) retroreflectors to control their
relative delays, defined as $\tau_{21} = \tau_2 - \tau_1$ and $\tau_{22^\prime} = \tau_2 -
\tau_{2^\prime}$.  %
 
The three excitation pulses were focused (1m FL spherical mirror) into the sample using a BOXCARS
geometry ($\sim 1 \deg$ angle of incidence for all beams).  %
All input fields were co-polarized.  %
The coherent output at $k_1 - k_2 + k_{2^\prime}$ was isolated using apertures and passed into a
monochromator, with an InSb photodiode (Teledyne-Judson) at the exit slit.  %
The monochromator was scanned with the FWM output frequency:  $\omega_m = \omega_1$.  %

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

In this section we describe the extraction of the QD susceptibility through standard dilution.
First, we examine the window and solvent response, which will be our local oscillator, in frequency
and time.  % 
Next, we isolate the pure QD response, using temporal discrimination, to validate the correction
factors used to account for concentration dependence.  %
Finally, we consider the interference between the solvent and QDs at pulse overlap, extracting the
QD susceptibility by ratio.  %

\subsection{Solvent and window response}  % -------------------------------------------------------

Carbon tetrachloride is an ideal solvent because of the high QD solubility, transparency in the
near infra-red, and its well-studied non-linear properties.  %
The FWM response of transparent solvents, like carbon tetrachloride, has components from nuclear
and electronic non-linearities. \cite{HellwarthRW1971a, HellwarthRW1977a}  
The electronic perturbations renormalize nearly instantaneously and thus are only present with
pulse overlap.  %
The nuclear response depends on the vibrational dephasing times (ps and longer). \cite{HoPP1979a,
  MatsuoShigeki1997a}  %
Vibrational features appear in the 2D spectra when stimulated Raman pathways resonantly enhance the
FWM at constant ($\omega_1 - \omega_2$) frequencies.  %

\autoref{pss:fig:ccl4} summarizes the nonlinear measurements performed on neat CCl$_4$.
In general, our results corroborate with impulsive stimulated Raman experiments.
\cite{MatsuoShigeki1997a, VoehringerPeter1995a}  %
When all pulses are overlapped (\autoref{pss:fig:ccl4}a), the electronic response creates a
featureless 2D spectrum.  %
The horizontal and vertical structure observed in \autoref{pss:fig:ccl4}a is believed to reflect the
power levels of our OPAs, which were not accounted for in these scans.  %
The weak diagonal enhancement observed may result from overdamped nuclear libration.  
The broad spectrum tracks with temporal pulse overlap, quickly disappearing at finite delays.  
If pulses $E_1$ and $E_2$ are kept overlapped and the $E_{2^\prime}$ is delayed
(\autoref{pss:fig:ccl4}b), the contributions from the Raman resonances can be resolved.  %
These ``TRIVE-Raman'' \cite{MeyerKentA2004a} resonances have been observed in carbon tetrachloride
previously. \cite{KohlerDanielDavid2014a}  %
The bright mode seen at approximately $\omega_1 - \omega_2 = \pm 50 \ \text{meV}$ is the $\nu1$
symmetric stretch ($459 \ \text{cm}^{-1}$). \cite{ShimanouchiT}

Characterization of the solvent response at pulse overlap can be simplified if Raman resonances are
negligible.  %
If Raman resonances are important, their spectral phase needs to be characterized and included in
modeling. \cite{YursLenaA2012a}  %
To estimate the relative magnitude of Raman components at pulse overlap, we consider a delay trace.  
\autoref{pss:fig:ccl4}c shows the signal dependence on $\tau_{22^\prime}$ with pulse frequencies
resonant with the large $\nu 1$ resonance.  %
The transient was fit to two components:  a fast Gaussian (electronic) component and an exponential
decay (Raman) component.  %
The oscillations in the exponential decay are quantum beating between Raman modes of CCl$_4$ and
are well-understood. \cite{KohlerDanielDavid2014a}  %
We determined the fast (non-resonant) component to be $4.0 \pm 0.7$ times larger than the long
(Raman) contributions (amplitude level).  %
At most colors, the ratio will be much less (confer \autoref{pss:fig:ccl4}b).
Since the Raman features are small in magnitude and spectrally sparse, we assume the CCl$_4$
spectrum near pulse overlap is well-approximated by non-resonant response ($\gamma_\text{sol}$ is
constant and real-valued).  %
This simplifies \autoref{pss:eq:fwm_intensity2} because the dispersion of the interference term is
determined completely by the real component of quantum dot response:  $\text{Re} \left[ E_\text{LO}
  \gamma_\text{QD}^* \right]  = E_\text{LO} \text{Re} \left[ \gamma_\text{QD} \right]$.  %

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/ccl4_raman"}
	\caption[CMDS amplitude of neat carbon tetrachloride.]{
		CMDS amplitude of neat CCl$_4$.  In all plots, $E_1$ and $E_2$ are coincident ($\tau_{21}= 0$ fs).
		Spectra are not normalized by the frequency-dependent OPA input powers.  
		(a)  The 2D frequency response at pulse overlap ($\tau_{21}=\tau_{22^\prime} = 0$).  
		(b)  Same as (a), but $E_{2^\prime}$ is latent by 200 fs.  
		(c)  The $\tau_{22^\prime}$ dependence on CMDS amplitude (thin blue line) is tracked at
    $\left(\hbar\omega_1, \hbar\omega_2 \right) = \left( .905, 0.955 \right)$ eV, so that the $\nu
    1$ Raman mode is resonantly excited.  %  
		The fit to the measured transient (thick blue line) is described further in the text.  
		The $\omega_1, \omega_2$ frequency combination is represented in (a) and (b) as a blue dot.
	}
	\label{pss:fig:ccl4}
\end{figure}

\subsection{Concentration-dependent corrections}  % ------------------------------------------------

It is important to address concentration effects on the CMDS output intensity because the resulting
absorption dependence can dramatically change the signal features.  %
\autoref{pss:fig:mfactors} shows $\omega_1$ spectra gathered at the QD concentrations explored in
this work.  %
All spectra were gathered at delay values $\tau_{21} = -200$ fs, $\tau_{22^\prime} = 0$ fs, and
$\omega_2$ is tuned to the exciton resonance.  %
The pulse delays are chosen to remove all solvent and window contributions; the signal is due
entirely to QDs ($\chi_\text{w},\chi_\text{solvent}=0$ in \autoref{pss:eq:fwm_intensity2}).  %
Power-normalized output amplitudes (\autoref{pss:fig:mfactors}a) are positively correlated with QD
concentration.  %
Density-normalized ($N_\text{QD}$) output amplitudes (\autoref{pss:fig:mfactors}b) are negatively
correlated with concentration because of absorption effects.  %
This normalization is adopted because the QD intensity term remains constant for any dilution
level.  %
This loss in efficiency is due to both absorption of the unresolved $\omega_2$ axis (loss across
all $\omega_1$ values) and the plotted $\omega_1$ axis (losses correlate to sample absorbance,
thick grey line).  %
After normalizing by $M$ (\autoref{pss:fig:mfactors}c), the density-normalized output amplitudes
agree for all QD concentrations.  %
The robustness of these corrections (derived from accurate absorption spectra) implies that data
can be taken at large concentrations and corrected to reveal clean signal with large dynamic
range.  %
The nature of the corrected line shape, including the tail to lower energies, will be addressed in
\autoref{cha:psg}.  %

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/mfactors_check"}
	\caption[Absorption effects in QD dilution study.]{
		The three panels show the changes in the FWM spectra of the five QD concentrations when
    corrected for concentration and absorption effects.
		The legend at the top identifies each QD loading level by the number density (units of $10^{16}
    \ \text{cm}^{-3}$).
		In all plots a representative QD absorption spectrum is overlaid (gray).
		Top:  $I / I_1 I_2 I_{2^\prime}$ spectra (intensity level).  
		Middle:  FWM amplitude spectra after normalizing by the carrier concentration ($\sqrt{I /
      \left( I_1 I_2 I_{2^\prime} N_\text{QD}^2 \right)}$).
		Bottom:  same as middle, but with the additional normalization by the absorptive correction
    factor ($M$).  
	}
	\label{pss:fig:mfactors}
\end{figure}

\subsection{Quantum dot response}  % --------------------------------------------------------------

We now consider the behavior at pulse overlap, where solvent and window contributions are important.  
\autoref{pss:fig:dilution_integral}a shows the (absorption-corrected) spectra for all samples at
zero delay.  %
The spectrum changes qualitatively with concentration, from peaked symmetric at high concentration
(purple), to dispersed and antisymmetric at low concentration (yellow).  %  
This behavior contrasts with the signals at finite $\tau_{21}$ delays, where the sample spectra are
independent of concentration (\autoref{pss:fig:mfactors}c).  %
Pulse overlap is complicated by the interference of multiple time-orderings and pulse effects.
\cite{KohlerDanielDavid2017a, BritoCruzCH1988a, JoffreM1988a}  %
These line shapes are not easily related to material properties, such as inhomogeneous broadening
and pure dephasing.  %

The concentration dependence in \autoref{pss:fig:dilution_integral} can be understood with our
knowledge of the solvent/window character and \autoref{pss:eq:LO}.  %
We approximate the solvent and window susceptibilities as real and constant, such that the
frequency dependence of the interference is solely from the real projecton of the QD
nonlinearity.  %
As a consequence, the interference term will be the Kramers-Kronig counterpart of the peaked
transient absorption spectrum.  % 
This explains the observed antisymmetric, dispersed line shape at low concentrations.  %
We analyze these spectra through two different methods:  spectral integration and global line shape
fitting.  %

\subsubsection{Spectral integration}

If we integrate \autoref{pss:eq:fwm_intensity2}, the integral of the solvent-QD interference term
disappears and the contributions are additive again.  %
We can write
\begin{equation} \label{pss:eq:fit_integral}
  \int_a^{a+\Delta} 
	\frac{I}{I_1 I_2 I_{2^\prime} M^2} \ d\omega_1	
	=  A \Delta \left( 1 + 
	\frac{\chi_\text{w}^{(3)}}{\chi_\text{sol}^{(3)}} f(N_\text{QD}) \right)^2
	+ \frac{A N_\text{QD}^2}{\gamma_\text{sol}^2 N_\text{sol}^2} \int_a^{a+\Delta} |\gamma_\text{QD}^{(3)}|^2  \ d\omega_1
\end{equation}
where $A$ is a proportionality factor and $f(N_\text{QD}) = \sigma_2 N_\text{QD} \ell_\text{w}
\frac{1 + e^{-\sigma_2 N_\text{QD} \ell_s}}{1 - e^{-\sigma_2 N_\text{QD} \ell_s}}$.  %
Care must be taken when choosing integral bounds $a$ and $a + \Delta$ so that the odd character of
the interference is adequately destroyed.  %

\autoref{pss:fig:dilution_integral}b shows the integral values for all five concentrations
considered in this work (colored circles).  %
At high concentrations the QD intensity dominates and we see quadratic scaling with $N_\text{QD}$.  
The lower intensities converge to a fixed offset due to the solvent and window contributions.  
Our data fit well to \autoref{pss:eq:fit_integral} (black dashed line).

Notably, our fit fails to distinguish between window and solvent contributions.
The solvent integral is invariant to $N_\text{QD}$, while the window contribution changes only
moderately over the concentrations studied ($f(N_\text{QD})$ varies by $\sim 0.3x$).  %
In contrast, the QD integral will change by $\sim 100x$ over these concentration ranges,
overwhelming the changes in window behavior.  %
The approximation of $f(N_\text{QD})$ as constant produces equally good fits.  %

In order to distinguish between window and solvent contributions, we take literature values from
Kerr lensing $z$-scan measurements of $\chi_\text{w} / \chi_\text{sol} \approx 0.13$.
\cite{Rau2008}  % 
This value agrees with our comparisons of FWM in cuvette windows and CCl$_4$-filled cuvettes (data
not shown).  %

The peak QD susceptibility can now be determined by assuming a line shape function;
if $\gamma_\text{QD}$ is a causal Lorentzian with half-width at half-maximum (HWHM) $\Gamma$,
\begin{equation}
	\gamma_\text{QD} = \gamma_\text{QD,peak} \frac{\Gamma}{\omega_\text{1S} - \omega_1 - i \Gamma},
\end{equation}
then $\gamma_\text{QD, peak} = 2 \sqrt{\Gamma^{-1} \pi^{-1} \int |\gamma_\text{QD}|^2 d\omega_1}$.
The peak width can be inferred, for instance, from spectra with high QD concentration ($\sim 25$
meV HWHM).  %
This gives a peak hyperpolarizability of $|\gamma_\text{QD, peak}| = 1.2 \times 10^{6}
\gamma_\text{sol}$.  %

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/dilution_integral"}
	\caption[FWM at pulse overlap, and integral thereof.]{
		FWM at temporal pulse overlap ($\tau_{21}=\tau_{22^\prime} = 0$ fs), with $\omega_2 =
    \omega_\text{1S}$.
		(a) Absorption-corrected $\omega_1$ spectra for each of the concentrations, offset for clarity. 
			Yellow is most dilute, purple is most concentrated. 
			Each spectrum is individually normalized (amplification factors are shown by each spectrum).  
		(b) The integrals of the FWM line shapes in part (a) are plotted against the QD concentration.  
			The dashed black line is the result of a linear fit (the $x$-axis is logarithmic).
	}
	\label{pss:fig:dilution_integral}
\end{figure}

\subsubsection{Global line shape fitting}

The integration approach provides a simple means to separate the contributions to the non-linearity, 
but it relies on QDs having a purely resonant line shape.  %
This approximation may not be appropriate for PbX QDs.  %
Many studies have reported a broadband contribution, attributed to excited state absorption of
excitons, in addition to the narrow 1S bleach feature. \cite{YursLenaA2012a, GeiregatPieter2014a,
  DeGeyterBram2012a}  %
To account for this feature, we perform a global fit of \autoref{pss:eq:fwm_intensity2} with the QD
line shape definition  %
\begin{equation} \label{pss:eq:fit_lineshape}
	\gamma_\text{QD}^{(3)} = \gamma_\text{QD,peak}^{(3)} \frac{\Gamma}{\omega_\text{1S} - \omega_1 - i \Gamma} + B,
\end{equation}
where $\Gamma$ is a line width parameter and $B$ is the broadband QD contribution.  %
The results of the fit are overlaid with our data in \autoref{pss:fig:dilution2}.
The data is normalized by $N_\text{QD}^2$ (as in \autoref{pss:fig:mfactors}c) so that least-squares
fitting weighs all samples on similar scales.  %
The fit parameters are listed in \autoref{pss:tab:lineshape_fit}.  
Again, we use a literature value for $\chi_\text{w} / \chi_\text{sol}$.
The extracted value of $\gamma_\text{QD}$ is $\sim 35\%$ smaller than in the integral analysis
because the integral method did not distinguish between the broadband contribution and the 1S
state-filling feature.  %

Unlike the integral method, the line shape fit also gives signed information:  we find that the
sign of $\gamma_\text{QD}$ is in fact negative, consistent with a photobleach.  %
The broadband contribution has a positive imaginary component, consistent with excited state
absorption.  %

\subsubsection{Choice of $\gamma_\text{sol}$}

To translate our result into an absolute susceptibility, $|\gamma_\text{sol}|$ must be known.
There are numerous measurements in the literature, \cite{HellwarthRW1971a, LevensonMD1974a,
  Levine1975, Cherlow1976, HoPP1979a, Thalhammer1983, Etchepare1985, Nibbering1995, Rau2004} but
unfortunately, the variation between values is quite large ($\pm 50\% $) for quantitative
analysis.  % 
This is the largest uncertainty in the determination of $\gamma_\text{QD}^{(3)}$.  %
With this concern noted, we adopt the median susceptibility of $\gamma_\text{sol} = 4 \cdot
10^{-37} \frac{\text{cm}^{6}}{\text{erg}}$ as our value to give comparisons to literature.  %
This yields a peak QD hyperpolarizability of $3 \cdot 10^{-31}  \text{cm}^6 / \text{erg}$.  

\begin{figure}
	\includegraphics[width=0.5\linewidth]{"PbSe_susceptibility/dilution_fits"}
	\caption[CMDS signal with different concentrations of PbSe.]{
		CMDS signal with different concentrations of PbSe.  
		In all spectra $\omega_2 = \omega_\text{1S}$.
		Calculated $\gamma^{(3)}$ spectra (\autoref{pss:eq:fwm_intensity2}) for the different QD
    concentrations.  
		The thick, lighter lines are the result of a global fit.  
	}
	\label{pss:fig:dilution2}
\end{figure}

\begin{table}
  \begin{tabular}{l|c}
		variable & value \\
		\hline
		$N_\text{sol} \ (\text{cm}^{-3})$ & $6.23 \times 10^{21}$ \\
		$\hbar \omega_\text{1S} \ (\text{eV})$ & 0.945 \\
		$\sigma_2 \ (\text{cm}^2)$ & $1.47 \times 10^{-16}$ \cite{Moreels2007, Dai2009} \\
		$\chi_\text{w} / \chi_\text{sol}$ & 0.13 \cite{Rau2008} \\
		$\mathbf{ \Gamma \ (\text{meV})}$ & 25 \\
		$\mathbf{\gamma_\text{QD,peak} / \gamma_\text{sol}}$ & $-7.7 \times 10^5$  \\
		$\mathbf{B / \gamma_\text{QD}}$ & $0.10 - 0.13i$ \\
	\end{tabular}
  \caption[Parameters and extracted values from the gloabl line shape fit.]{
    Parameters and extracted values from the global line shape fit using Equations
    \ref{pss:eq:fwm_intensity2} and \ref{pss:eq:fit_lineshape}.
    Bold items were extracted by least squares minimization.  All other values were fixed
    parameters.
  }
	\label{pss:tab:lineshape_fit}
\end{table}

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

We now consider the agreement of our non-linearity with those of literature. 
Comparison between different measured non-linearities is difficult because the effects of the
excitation sources are often intertwined with the non-linear response.
\cite{KohlerDanielDavid2017a}  %  
\autoref{pss:eq:gamma_to_phi} shows that the metrics of non-linear response, $\phi$ and
$\gamma^{(3)}$, are connected not just by intrinsic properties, but by experimental parameters $D$
and $\Delta_t$.  %

Since $\gamma^{(3)}$ assumes the driven limit for field-matter interactions (see
\autoref{pss:eq:Maker_Terhune}), signal scales with pulse intensity and not fluence.  %
The third-order susceptibility will be proportional to the pulse duration of the experiment,
$\Delta_t$.  %
Conversely, $\gamma^{(3)}$ factors out the degeneracy of the experiment, but non-linear
absorptivity does not (\autoref{pss:eq:beta_to_chi}).  %
Since $\phi$ is defined by the non-linear absorptivity, it is also proportional to $D$.  

\autoref{pss:eq:Delta_alpha1} is valid for a temporally separated pump and probe, so that the
probe sees the entire population created by the pump.  %  
Our experiments examine the non-linearity for temporally overlapped pump and probe pulses. 
The differences due to these effects can be calculated under reasonable assumptions (see the
Supplementary Materials); we find the population seen at temporal overlap about $80\%$ that of the
excited state probed after the pump.  %
This factor is needed for comparisons between our measurements and transient absorption with well
separated pulses.  %
This correction factor is small compared to our uncertainty, so we neglect it.  
It may be important in more precise measurements.

The most direct comparison of our measurements with literature is \textcite{YursLenaA2012a},
who performed the picosecond pulse analogue of the work herein.  %
Both studies used CCl$_4$ as an internal standard, so we can compare the ratios directly to avoid
uncertainty from the value of $\gamma_\text{sol}$.  %
Note, however, that the picosecond study does not account for window contributions, which could
mean their reported ratios are under-reported (the solvent field is actually the solvent and window
fields).  %
The values are shown in \autoref{pss:tab:gamma_ratio}. 
Both Raman and 1S band susceptibilities differ by approximately the ratio of pulse durations,
consistent with a intermediate state with lifetime longer than 1 ps (a Raman coherence and a 1S
population, respectively).  %

The broadband QD hyperpolarizability ($B$) is similar with both pulse durations, indicating that
this contribution originates not from a 1S population, but something very fast (driven limit).  %
Possible explanations are double/zero quantum coherences, ultrafast relaxation, or simply a
non-resonant polarization.  %
This broadband feature may be different from that observed in transient absorption because temporal
pulse overlap isolates the fastest observable features (most TA features are analyzed at finite
delays from pulse overlap).  %

\autoref{pss:tab:litcompare} compares various non-linear quantities for this work,
\textcite{YursLenaA2012a}, and a PbS experiment \cite{OmariAbdoulghafar2012a}.  %
We will continue to refer to this table for the rest of this discussion.
Note that $\gamma_\text{QD} / \Delta_t$ is similar between Yurs and our measurement, as expected
from the pulse duration dependence.  %

The sample studied by Yurs et. al. was significantly degraded, and the authors described their QD
spectra using mechanisms other than state-filling.  %
The relative similarity of the absolute susceptibility, given such extraordinary spectral
differences, is noteworthy.  %
Yurs et. al. also uses a different $\gamma_\text{sol}$ value to calculate their absolute
susceptibility, which gives more disagreement in reported values than the literature suggests.  %

\textcite{OmariAbdoulghafar2012a} performed $z$-scan measurements of PbS QDs to quantify the
non-linear parameters (see right-hand column of \autoref{pss:tab:litcompare}).  %
In contrast to our measurements, their degenerate susceptibility is primarily real in character and
much larger than that reported here or in Yurs.  %
While we cannot reconcile the real component, the imaginary component agrees with the standard
bleach theory ($\phi = 0.15$).  %
Omari et. al. report that their results do not agree with the $\phi = 0.25$ bleach theory of
transient absorption, but we note that their observed bleach fractions is actually in great
agreement once the experimental degeneracy is accounted for (a transient absorption measurement of
their sample would give $\phi = 0.3$).  %

We now turn our focus to comparison between our measurement and $\phi$.
There is some variance in the value of $\phi$ reported for PbX quantum dots.
The commonly accepted value for the bleach fraction of $\phi = 0.25$ is approximate, and runs
counter to saturated absorption studies, where a fully inverted 1S band reduces by a factor of 1/8
after Auger recombination yields single-exciton species. \cite{NootzGero2011a, Istrate2008} 
Only a few transient absorption studies address the photobleach magnitude explicitly, rather than
the more common state-filling analysis via the A:B ratio.  %

We can check our measured susceptibility with the accepted $\phi$ value using
\autoref{pss:eq:gamma_to_phi}.  %
If the peak susceptibility is mostly imaginary, we can attribute our TG peak
$\left|\chi^{(3)}\right|$ susceptibility to the TA state-filling factor.  %
Our peak TG hyperpolarizability measurements give values of $\left| \gamma^{(3)} \right| = 3 \pm 2
\cdot 10^{-31} \text{cm}^6 \ \text{erg}^{-1}$, while \autoref{pss:eq:gamma_to_phi} predicts
$\text{Im}\left[ \gamma^{(3)} \right] = -1.6 \cdot 10^{-31} \text{cm}^6 \text{erg}^{-1}$ for $\phi
= 0.25$.
Our method gives agreement with the $\phi=0.25$ bleach factor.  %

\begin{table}
	\begin{tabular}{r|ccc}
		&  this work &  Yurs et. al. & ratio \\ \hline
		$ |\gamma_\text{QD,peak}| / |\gamma_\text{sol}| $ & $7.3 \cdot 10^5$ & $1.1 \cdot 10^7$ & 15 \\
		$ |B| / |\gamma_\text{sol}| $ & $1.3 \cdot 10^5$ & $1.6 \cdot 10^5$ & 1.3 \\
		$ |\gamma_{\nu1}| / |\gamma_\text{sol}| $ & $0.25 \pm 0.04 $ & $5.1$ & 20.4 \\
	\end{tabular}
  \caption[Non-linear parameters relative to carbon tetrachloride hyperpolarizability.]{
    Non-linear parameters relative to CCl$_4$ hyperpolarizability.  $\gamma_{\nu 1}$:
    hyperpolarizability of the $\nu_1$ Raman transition.}
	\label{pss:tab:gamma_ratio}
\end{table}

\begin{table}
  \begin{tabular}{l|ccc}
		&  this work   &  Yurs et. al. & Omari et. al.\footnote{samples D (imaginary) and K (real)} \\
		QD & PbSe & PbSe & PbS \\
		measurement & $|\gamma|$ & $|\gamma|$ & $\gamma$ \\
		\hline
		$ \Delta_t \left[ \text{fs} \right]$
		& $\sim 50 $ & $\sim 1250 $	& $\sim 2500$ \\
		$ \left| \gamma_\text{QD}^{(3)} \right| \left[ 10^{-30} \frac{\text{cm}^6}{\text{erg}} \right]$
		& 0.2 & 8.8 & $-(1 + .03i) \cdot 10^4$ \\
		$ \left| \gamma_\text{QD}^{(3)} / \Delta_t \right| \left[ 10^{-18} \frac{\text{cm}^6}{\text{erg s}} \right]$
		& 4 & 7 & 120 \\
		$\phi$ (\autoref{pss:eq:gamma_to_phi}) & 0.3 & 0.6 & 0.15 \\
	\end{tabular}
  \caption[Comparison with PbX literature measurements.]{
    Comparison of these measurements with PbX measurements in literature.
    $\gamma_\text{raman}^{(3)}$ refers to the 465 $\text{cm}^{-1}$ (symmetric stretch) mode of
    $\text{CCl}_4$}
	\label{pss:tab:litcompare}
\end{table}

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

We have shown that ultrafast CMDS can isolate the non-linearities of resonant signal and background
solvent in manner similar to classic three-wave mixing analyses of the past.  %
At pulse overlap, featureless solvent contributions can be especially large.  %
The resonant Raman contributions decrease in prominence when pulses are shorter than the Raman
lifetimes.  %
Solvent contributions can also be suppressed by using large sample concentrations, in which case
spectra have to be corrected by well-defined absorptive correction factors.  %
The solute and solvent interference can be separated using simple, few parameter models, as we have
demonstrated here with quantum dots.  %
We have employed simple, few-parameter fits to easily disentangle the role of solvent and solute.  
These methods should be applicable to other CMDS spectroscopies, but the description of the solvent
may change, especially when non-co-polarized excitations are used. \cite{DeegFW1989a}  %  
In fact, the polarization behavior provides a useful way to alter the balance of solvent and solute
contributions in a predictable way, and is likely a viable method for separating solvent and solute
contributions.  %

Solvent may be used as an internal standard to measure the solute non-linearity, but there are
still large uncertainties in the non-linear susceptibility that propagate to the solute optical
constants.  %  
More work should be done to reduce this uncertainty and characterize the dispersion of non-linear
susceptibility of solvents.  %

Absolute nonlinearities are an important property to study in material systems because their
relation to linear susceptibilities informs on the underlying physics.  %
For MR-CMDS, it is important to identify how to extract these non-linearities.  %  
We have demonstrated ultrafast MR-CMDS as a viable method to extract the absolute non-linear
susceptibility by using CCl$_4$ as an internal standard.  %
The extraction requires accounting for the impulsive population creation, as well as absorptive
propagation effects within the sample.  %
These absorptive effects are also crucial factors for general analysis of MR-CMDS spectra.  %