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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.
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