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-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.