2018-02-27 23:58

1 files changed, 99 insertions, 91 deletions

diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex
index 7f8a8b4..aad2be1 100644
--- a/mixed_domain/chapter.tex
+++ b/mixed_domain/chapter.tex
@@ -1,5 +1,7 @@
 \chapter{Disentangling material and instrument response}

 

+\textit{This Chapter borrows extensively from \textcite{KohlerDanielDavid2017a}.}

+

 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

@@ -17,15 +19,16 @@ inhomogeneity.  %
 These simulations provide a foundation for interpretation of ultrafast experiments in the mixed

 domain.  %

 

-\section{Introduction}

+\section{Introduction}  % -------------------------------------------------------------------------

 

 Ultrafast spectroscopy is based on using nonlinear interactions, created by multiple ultrashort

 ($10^{-9}-10^{-15}$s) pulses, to resolve spectral information on timescales as short as the pulses

-themselves.\cite{Rentzepis1970,Mukamel2000}  % 

-The ultrafast specta can be collected in the time domain or the frequency domain.\cite{Park1998}  %

+themselves. \cite{RentzepisPM1970a, MukamelShaul2000a}  % 

+The ultrafast specta can be collected in the time domain or the frequency

+domain. \cite{ParkKisam1998a}  %

 

 Time-domain methods scan the pulse delays to resolve the free induction decay

-(FID).\cite{Gallagher1998}   %

+(FID). \cite{GallagherSarahM1998a}   %

 The Fourier Transform of the FID gives the ultrafast spectrum.  %

 Ideally, these experiments are performed in the impulsive limit where FID dominates the

 measurement.  % 

@@ -36,12 +39,13 @@ ultrafast excitation pulses.  %
 As the pulses temporally overlap, FID from other pulse time-orderings and emission driven by the

 excitation pulses both become important.  % 

 These factors are responsible for the complex ``coherent artifacts'' that are often ignored in

-pump-probe and related methods.\cite{Lebedev2007, Vardeny1981, Joffre1988, Pollard1992}  % 

+pump-probe and related methods. \cite{LebedevMV2007a, VardenyZ1981a, JoffreM1988a,

+  PollardW1992a}  % 

 Dynamics faster than the pulse envelopes are best measured using line shapes in frequency domain

 methods.  %

 

 Frequency-domain methods scan pulse frequencies to resolve the ultrafast spectrum

-directly.\cite{Druet1979,Oudar1980}  % 

+directly. \cite{DruetSAJ1979a, OudarJL1980a}  % 

 Ideally, these experiments are performed in the driven limit where the steady state dominates the

 measurement.  %

 In the driven limit, all time-orderings of the pulse interactions are equally important and FID

@@ -57,12 +61,12 @@ There is also the hybrid mixed-time/frequency-domain approach, where pulse delay
 frequencies are both scanned to measure the system response.  %

 This approach is uniquely suited for experiments where the dephasing time is comparable to the

 pulse durations, so that neither frequency-domain nor time-domain approaches excel on their

-own.\cite{Oudar1980,Wright1997a,Wright1991}  %

-In this regime, both FID and driven processes are important.\cite{Pakoulev2006}  %

+own. \cite{OudarJL1980a, WrightJohnCurtis1997b, WrightJohnCurtis1991a}  %

+In this regime, both FID and driven processes are important. \cite{PakoulevAndreiV2006a}  %

 Their relative importance depends on pulse frequencies and delays.  %

 Extracting the correct spectrum from the measurement then requires a more complex analysis that

 explicitly treats the excitation pulses and the different

-time-orderings.\cite{Pakoulev2007,Kohler2014,Gelin2009a}   %

+time-orderings. \cite{PakoulevAndreiV2007a, KohlerDanielDavid2014a, GelinMaximF2009b}   %

 Despite these complications, mixed-domain methods have a practical advantage: the dual frequency-

 and delay-scanning capabilities allow these methods to address a wide variety of dephasing

 rates.  %

@@ -71,39 +75,44 @@ The relative importance of FID and driven processes and the changing importance
 coherence pathways are important factors for understanding spectral features in all ultrafast

 methods.  %

 These methods include partially-coherent methods involving intermediate populations such as

-pump-probe,\cite{Hamm2000} transient grating,\cite{Salcedo1978,Fourkas1992,Fourkas1992a} transient

-absorption/reflection,\cite{Aubock2012,Bakker2002} photon

-echo,\cite{DeBoeij1996,Patterson1984,Tokmakoff1995} two dimensional-infrared spectroscopy

-(2D-IR),\cite{Hamm1999,Asplund2000,Zanni2001} 2D-electronic spectroscopy

-(2D-ES),\cite{Hybl2001a,Brixner2004} and three pulse photon echo peak shift

-(3PEPS)\cite{Emde1998,DeBoeij1996,DeBoeij1995,Cho1992,Passino1997} spectroscopies.  %

+pump-probe \cite{HammPeter2000a}, transient grating \cite{SalcedoJR1978a, FourkasJohnT1992a,

+  FourkasJohnT1992b}, transient absorption/reflection \cite{AubockGerald2012a, BakkerHJ2002a}, photon

+echo \cite{DeBoeijWimP1996a, PattersonFG1984a, TokmakoffAndrei1995a}, two dimensional-infrared

+spectroscopy (2D-IR) \cite{HammPeter1999a, AsplundMC2000a, ZanniMartinT2001a}, 2D-electronic

+spectroscopy (2D-ES) \cite{HyblJohnD2001b, BrixnerTobias2004a}, and three pulse photon echo peak

+shift (3PEPS) \cite{EmdeMichelF1998a, DeBoeijWimP1996a, DeBoeijWimP1995a, ChoMinhaeng1992a,

+  PassinoSeanA1997a} spectroscopies.  %

 These methods also include fully-coherent methods involving only coherences such as Stimulated

-Raman Spectroscopy (SRS),\cite{Yoon2005,McCamant2005} Doubly Vibrationally Enhanced

-(DOVE),\cite{Zhao1999,Zhao1999a,Zhao2000,Meyer2003,Donaldson2007,Donaldson2008,Fournier2008} Triply

-Resonant Sum Frequency (TRSF),\cite{Boyle2013a,Boyle2013,Boyle2014} Sum Frequency Generation

-(SFG)\cite{Lagutchev2007}, Coherent Anti-Stokes Raman Spectroscopy

-(CARS)\cite{Carlson1990b,Carlson1990a,Carlson1991} and other coherent Raman

-methods\cite{Steehler1985}.  %

+Raman Spectroscopy (SRS) \cite{YoonSangwoon2005a, McCamantDavidW2005a}, Doubly Vibrationally

+Enhanced (DOVE) \cite{ZhaoWei1999a, ZhaoWei1999b, ZhaoWei2000a, MeyerKentA2003a,

+  DonaldsonPaulM2007a, DonaldsonPaulM2008a, FournierFrederic2008a},

+Triply Resonant Sum Frequency (TRSF) \cite{BoyleErinSelene2013a, BoyleErinSelene2013b,

+  BoyleErinSelene2014a}, Sum Frequency Generation (SFG) \cite{LagutchevAlexi2007a}, Coherent

+Anti-Stokes Raman Spectroscopy (CARS) \cite{CarlsonRogerJ1990b, CarlsonRogerJ1990c,

+  CarlsonRogerJ1991a}, and other coherent Raman methods \cite{SteehlerJK1985a}.  %

 

 This paper focuses on understanding the nature of the spectral changes that occur in Coherent

 Multidimensional Spectroscopy (CMDS) as experiments transition between the two limits of frequency-

 and time-domain methods.  %

 CMDS is a family of spectroscopies that use multiple delay and/or frequency axes to extract

 homogeneous and inhomogeneous broadening, as well as detailed information about spectral diffusion

-and chemical changes.\cite{Kwac2003,Wright2016}  % 

+and chemical changes. \cite{KwacKijeong2003a, WrightJohnCurtis2016a}  % 

 For time-domain CMDS (2D-IR, 2D-ES), the complications that occur when the impulsive approximation

-does not strictly hold has only recently been addressed.\cite{Erlik2017,Smallwood2016}  %

+does not strictly hold has only recently been addressed. \cite{PerlikVaclav2017a,

+  SmallwoodChristopherL2016a}  %

 

 Frequency-domain CMDS methods, referred to herein as multi-resonant CMDS (MR-CMDS), have similar

 capabilities for measuring homogeneous and inhomogeneous broadening.  %

 Although these experiments are typically described in the driven

-limit,\cite{Gallagher1998,Fourkas1992,Fourkas1992a} many of the experiments involve pulse widths

-that are comparable to the widths of the

-system.\cite{Meyer2003,Donaldson2007,Pakoulev2009,Zhao1999,Czech2015,Kohler2014}  % 

+limit, \cite{GallagherSarahM1998a, FourkasJohnT1992a, FourkasJohnT1992b} many of the experiments

+involve pulse widths that are comparable to the widths of the system. \cite{MeyerKentA2003a,

+  DonaldsonPaulM2007a,  PakoulevAndreiV2009a, ZhaoWei1999a, CzechKyleJonathan2015a,

+  KohlerDanielDavid2014a}  %

 MR-CMDS then becomes a mixed-domain experiment whereby resonances are characterized with marginal

 resolution in both frequency and time.  %

-For example, DOVE spectroscopy involves three different pathways\cite{Wright2003} whose relative

-importance depends on the relative importance of FID and driven responses.\cite{Donaldson2010}  %

+For example, DOVE spectroscopy involves three different pathways \cite{WrightJohnCurtis2003a} whose

+relative importance depends on the relative importance of FID and driven

+responses. \cite{DonaldsonPaulM2010a}  %

 In the driven limit, the DOVE line shape depends on the difference between the first two pulse

 frequencies so the line shape has a diagonal character that mimics the effects of inhomogeneous

 broadening.  % 

@@ -118,8 +127,8 @@ $\vec{k}_\text{out} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$.  %
 Here, the subscripts represent the excitation pulse frequencies, $\omega_1$ and $\omega_2 =

 \omega_{2'}$.  %

 These experimental conditions were recently used to explore line shapes of excitonic

-systems,\cite{Kohler2014,Czech2015} and have been developed on vibrational states as

-well.\cite{Meyer2004}  %

+systems,\cite{KohlerDanielDavid2014a, CzechKyleJonathan2015a} and have been developed on

+vibrational states as well.\cite{MeyerKentA2004a}  %

 Although MR-CMDS forms the context of this model, the treatment is quite general because the phase

 matching condition can describe any of the spectroscopies mentioned above with the exception of SFG

 and TRSF, for which the model can be easily extended.  %

@@ -135,7 +144,7 @@ evolution of this line shape can mimic spectral diffusion.  %
 We identify key signatures that can help differentiate true inhomogeneity and spectral diffusion

 from these measurement artifacts.  %

 

-\section{Theory}

+\section{Theory}  % -------------------------------------------------------------------------------

 

 \begin{dfigure}

 	\includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"}

@@ -178,16 +187,16 @@ Here $\bm{H_0}$ is the time-independent Hamiltonian, $\bm{\mu}$ is the dipole su
 $\bm{\Gamma}$ contains the pure dephasing rate of the system.  %

 We perform the standard perturbative expansion of Equation \ref{eq:LVN} to third order in the

 electric field

-interaction\cite{mukamel1995principles,Yee1978,Oudar1980,Armstrong1962,Schweigert2008} and restrict

-ourselves only to the terms that have the correct spatial wave vector

-$\vec{k}_{\text{out}}=\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$.  %

+interaction\cite{MukamelShaul1995a, YeeTK1978a, OudarJL1980a, ArmstrongJA1962a,

+  SchweigertIgorV2008a} and restrict ourselves only to the terms that have the correct spatial wave

+vector $\vec{k}_{\text{out}}=\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$.  %

 This approximation narrows the scope to sets of three interactions, one from each field, that

 result in the correct spatial dependence.  %

 The set of three interactions have $3!=6$ unique time-ordered sequences, and each time-ordering

 produces either two or three unique system-field interactions for our system, for a total of

 sixteen unique system-field interaction sequences, or Liouville pathways, to consider.  %

 Fig. \ref{fig:WMELs} shows these sixteen pathways as Wave Mixing Energy Level (WMEL)

-diagrams\cite{Lee1985}.  %

+diagrams\cite{LeeDuckhwan1985a}.  %

 

 We first focus on a single interaction in these sequences, where an excitation pulse, $x$, forms

 $\rho_{ij}$ from $\rho_{ik}$ or $\rho_{kj}$.  %

@@ -206,9 +215,9 @@ transition dipole, and $\Gamma_f$ is the dephasing/relaxation rate for $\rho_f$.
 The $\lambda_f$ and $\kappa_f$ parameters describe the phases of the interaction:  $\lambda_f=+1$

 for ket-side transitions and -1 for bra-side transitions, and $\kappa_f$ depends on whether

 $\rho_f$ is formed via absorption ($\kappa_f= \lambda_f$) or emission

-($\kappa_f=-\lambda_f$).\footnote{$\kappa_f$ also has a direct relationship to the phase matching

-  relationship: for transitions with $E_2$, $\kappa_f=1$, and for $E_1$ or $E_{2^\prime}$,

-  $\kappa_f=-1$.}  %

+($\kappa_f=-\lambda_f$).  %

+$\kappa_f$ also has a direct relationship to the phase matching relationship: for transitions with

+$E_2$, $\kappa_f=1$, and for $E_1$ or $E_{2^\prime}$, $\kappa_f=-1$.  %

 In the following equations we neglect spatial dependence ($z=0$).  %

 

 Equation \ref{eq:rho_f} forms the basis for our simulations.  %

@@ -228,8 +237,8 @@ Equation \ref{eq:rho_f_int} becomes the steady state limit expression when $\Del
 \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$.  %

 Both limits are important for understanding the multidimensional line shape changes discussed in

 this paper.  %

-The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discussed in Appendix

-\ref{sec:cw_imp}.  %

+The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discussed in TODO

+% Appendix \ref{sec:cw_imp}.  %

 

 \begin{dfigure}

 	\includegraphics[width=\linewidth]{"mixed_domain/simulation overview"}

@@ -276,9 +285,7 @@ properties of the first, second, and third pulse, respectively, and indices 0, 1
 define the properties of the ground state, first, second, third, and fourth density matrix

 elements, respectively.  %

 Fig. \ref{fig:overview}b demonstrates the correspondence between $x$, $y$, $z$ notation and 1, 2,

-$2^\prime$ notation for the laser pulses with pathway $V\gamma$.\footnote{For elucidation of the

-  relationship between the generalized Liouville pathway notation and the specific parameters for

-  each Liouville pathway, see Table S1 in the Supplementary Information.}  %

+$2^\prime$ notation for the laser pulses with pathway $V\gamma$.

 

 The electric field emitted from a Liouville pathway is proportional to the polarization created by

 the third-order coherence:  %

@@ -335,11 +342,12 @@ Table S1 summarizes the arguments for each Liouville pathway.  %
 Fig. \ref{fig:overview}f shows the 2D $(\omega_1, \omega_2)$ $S_{\text{tot}}$ spectrum resulting

 from the pulse delay times represented in Fig. \ref{fig:overview}b.  %

 

-\subsection{Inhomogeneity}

+\subsection{Inhomogeneity}  % ---------------------------------------------------------------------

 

 Inhomogeneity is isolated in CMDS through both spectral signatures, such as

-line-narrowing\cite{Besemann2004,Oudar1980,Carlson1990,Riebe1988,Steehler1985}, and temporal

-signatures, such as photon echoes\cite{Weiner1985,Agarwal2002}.  %

+line-narrowing\cite{BesemannDanielM2004a, OudarJL1980a, CarlsonRogerJ1990a, RiebeMichaelT1988a,

+  SteehlerJK1985a}, and temporal signatures, such as photon echoes \cite{WeinerAM1985a,

+  AgarwalRitesh2002a}.  % 

 We simulate the effects of static inhomogeneous broadening by convolving the homogeneous response

 with a Gaussian distribution function.  %

 Further details of the convolution are in Appendix \ref{sec:convolution}.  %

@@ -349,7 +357,7 @@ Dynamic broadening effects such as spectral diffusion are beyond the scope of th
 

 A matrix representation of differential equations of the type in Equation \ref{eq:E_L_full} was

 numerically integrated for parallel computation of Liouville elements (see SI for

-details).\cite{Dick1983,Gelin2005}  %

+details).\cite{DickBernhard1983a, GelinMaximF2005a}  %

 The lower bound of integration was $2\Delta_t$ before the first pulse, and the upper bound was

 $5\Gamma_{10}^{-1}$ after the last pulse, with step sizes much shorter than the pulse durations.  %

 Integration was performed in the FID rotating frame; the time steps were chosen so that both the

@@ -365,9 +373,9 @@ $\Delta_\text{inhom}/ \Delta_{\omega}=0,0.5,1,$ and $2$ for the three dephasing
 For all these dimensions, both $\rho_3(t;\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ and

 $S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ are recorded for each unique

 Liouville pathway.  %

-Our simulations were done using the open-source SciPy library.\cite{Oliphant2007}  %

+Our simulations were done using the open-source SciPy library.\cite{OliphantTravisE2007a}  %

 

-\subsection{Characteristics of Driven and Impulsive Response}\label{sec:cw_imp}

+\subsection{Characteristics of Driven and Impulsive Response} \label{sec:cw_imp}  % ---------------

 

 The changes in the spectral line shapes described in this work are best understood by examining the

 driven/continuous wave (CW) and impulsive limits of Equations \ref{eq:rho_f_int} and

@@ -484,7 +492,7 @@ The build-up limit approximates well when pulses are near-resonant and arrive to
 build-up behavior is emphasized).  %

 The driven limit holds for large detunings, regardless of delay.  %

 

-\subsection{Convolution Technique for Inhomogeneous Broadening}\label{sec:convolution}

+\subsection{Convolution Technique for Inhomogeneous Broadening} \label{sec:mixed_convolution}  % --

 

 \begin{dfigure}

 	\includegraphics[width=\linewidth]{mixed_domain/convolve}

@@ -653,9 +661,9 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$.  %
 	\label{fig:fid_detuning}

 \end{dfigure}

 

-Fig. \ref{fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.\footnote{

-	See Supplementary Fig. S3 for a Fourier domain representation of Fig. \ref{fig:fid_detuning}a.

-}

+Fig. \ref{fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of

+$\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$.

+

 As detuning increases, total amplitude decreases, FID character vanishes, and $\rho_1$ assumes a

 more driven character, as expected.  %

 During the excitation, $\rho_1$ maintains a phase relationship with the input field (as seen by the

@@ -666,11 +674,13 @@ The FID is therefore sensitive to the absorptive (imaginary) line shape of a tra
 driven response is the composite of both absorptive and dispersive components.  %

 If the experiment isolates the latent FID response, there is consequently a narrower spectral

 response.  %

-This spectral narrowing can be seen in Fig. \ref{fig:fid_detuning}a by comparing the coherence amplitudes at $t=0$ (driven)  and at $t / \Delta_t=2$ (FID); amplitudes for all $\Omega_{fx}/\Delta_{\omega}$ values shown are comparable at $t=0$, but the lack of FID formation for $\Omega_{fx}/\Delta_{\omega}=\pm2$ manifests as a visibly disproportionate amplitude decay.\footnote{

-	See Supplementary Fig. S4 for explicit plots of $\rho_1(\Omega_{fx}/\Delta_{\omega})$ at discrete $t/\Delta_t$ values.

-}  %

+This spectral narrowing can be seen in Fig. \ref{fig:fid_detuning}a by comparing the coherence

+amplitudes at $t=0$ (driven)  and at $t / \Delta_t=2$ (FID); amplitudes for all

+$\Omega_{fx}/\Delta_{\omega}$ values shown are comparable at $t=0$, but the lack of FID formation

+for $\Omega_{fx}/\Delta_{\omega}=\pm2$ manifests as a visibly disproportionate amplitude decay.  %

 Many ultrafast spectroscopies take advantage of the latent FID to suppress non-resonant background,

-improving signal to noise.\cite{Lagutchev2007,Lagutchev2010,Donaldson2010,Donaldson2008}  %

+improving signal to noise. \cite{LagutchevAlexi2007a, LagutchevAlexi2010a, DonaldsonPaulM2010a,

+  DonaldsonPaulM2008a}  %

 

 In driven experiments, the output frequency and line shape are fully constrained by the excitation

 beams.  %

@@ -766,9 +776,8 @@ The spectral changes result from changes in the relative importance of driven an
 components.  %

 The prominence of FID signal can change the resonance conditions; Table \ref{tab:table2} summarizes

 the changing resonance conditions for each of the four delay coordinates studied.  %

-Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromator must also be considered.\footnote{

-	See Supplementary Fig. S5 for a representation of Fig. 5 simulated without monochromator frequency filtering ($M(\omega-\omega_1)=1$ for Equation \ref{eq:S_tot}).

-}

+Since $E_1$ is not the last pulse in pathway I$\gamma$, the tracking monochromator must also be

+considered.  %

 

 \begin{dtable}

 	\caption{\label{tab:table2} Conditions for peak intensity at different pulse delays for pathway

@@ -853,13 +862,13 @@ As a result, the line shape acquires a diagonal character.  %
 

 The changes in line shape seen in Fig. \ref{fig:pw1} have significant ramifications for the

 interpretations and strategies of MR-CMDS in the mixed domain.  %

-Time-gating has been used to isolate the 2D spectra of a certain time-ordering\cite{Meyer2004,

-  Pakoulev2006,Donaldson2007}, but here we show that time-gating itself causes significant line

-shape changes to the isolated pathways.  %

+Time-gating has been used to isolate the 2D spectra of a certain

+time-ordering\cite{MeyerKentA2004a, PakoulevAndreiV2006a, DonaldsonPaulM2007a}, but here we show that

+time-gating itself causes significant line shape changes to the isolated pathways.  %

 The phenomenon of time-gating can cause frequency and delay axes to become functional of each other

 in unexpected ways.  %

 

-\subsection{Temporal pathway discrimination}

+\subsection{Temporal pathway discrimination}  % ---------------------------------------------------

 

 \begin{dfigure}

 	\includegraphics[width=\linewidth]{"mixed_domain/delay space ratios"}

@@ -957,16 +966,16 @@ Scanning $\tau_{21}$ through pulse overlap complicates interpretation of the lin
 changing nature and balance of the contributing time-orderings.  %

 At $\tau_{21}>0$, time-ordering I dominates; at $\tau_{21}=0$, all time-orderings contribute

 equally; at $\tau_{21}<0$ time-orderings V and VI dominate (Fig. \ref{fig:delay_purity}).  %

-Conventional pump-probe techniques recognized these complications long ago,\cite{BritoCruz1988,

-  Palfrey1985} but the extension of these effects to MR-CMDS has not previously been done.  %

+Conventional pump-probe techniques recognized these complications long ago,\cite{BritoCruzCH1988a,

+  PalfreySL1985a} but the extension of these effects to MR-CMDS has not previously been done.  %

 

 Fig. \ref{fig:hom_2d_spectra} shows the MR-CMDS spectra, as well as histograms of the pathway

 weights, while scanning $\tau_{21}$ through pulse overlap.  %

 The colored histogram bars and line shape contours correspond to different values of the relative

 dephasing rate, $\Gamma_{10}\Delta_t$.  %

-The contour is the half-maximum of the line shape.\footnote{Supplementary Fig. S6 shows fully

-  colored contour plots of each 2D frequency spectrum.}  The dependence of the line shape amplitude

-on $\tau_{21}$ can be inferred from Fig. \ref{fig:delay_purity}.  %

+The contour is the half-maximum of the line shape.  %

+The dependence of the line shape amplitude on $\tau_{21}$ can be inferred from Fig.

+\ref{fig:delay_purity}.  %

 

 The qualitative trend, as $\tau_{21}$ goes from positive to negative delays, is a change from

 diagonal/compressed line shapes to much broader resonances with no correlation ($\omega_1$ and

@@ -1026,10 +1035,10 @@ resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\D
 \end{dfigure}

 

 It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is

-frequency.\cite{Kohler2014, Aubock2012,Czech2015,Pakoulev2007}  %

+frequency. \cite{KohlerDanielDavid2014a, AubockGerald2012a, CzechKyleJonathan2015a,

+  PakoulevAndreiV2007a}  %

 In Fig. \ref{fig:wigners} we show five $\tau_{21},\omega_1$ plots for varying $\omega_2$ with

-$\tau_{22^\prime}=0$.\footnote{See Supplementary Fig. S8 for Wigner plots for all

-  $\Gamma_{10}\Delta_t$ values.}  %

+$\tau_{22^\prime}=0$.  %

 The plots are the analogue to the most common multidimensional experiment of Transient Absorption

 spectroscopy, where the non-linear probe spectrum is plotted as a function of the pump-probe

 delay.  %

@@ -1071,7 +1080,7 @@ The inhomogeneity makes it easier to temporally isolate the rephasing pathways a
 isolate the non-rephasing pathways, as shown by the purity contours.  %

 

 A common metric of rephasing in delay space is the 3PEPS

-measurement.\cite{Weiner1985,Fleming1998,Boeij1998,Salvador2003}  %

+measurement.\cite{WeinerAM1985a, FlemingGrahmR1998a, DeBoeijWimP1998a, SalvadorMayroseR2008a}  %

 In 3PEPS, one measures the signal as the first coherence time, $\tau$, is scanned across both

 rephasing and non-rephasing pathways while keeping population time, $T$, constant.  %

 The position of the peak is measured; a peak shifted away from $\tau=0$ reflects the rephasing

@@ -1080,12 +1089,10 @@ An inhomogeneous system will emit a photon echo in the rephasing pathway, enhanc
 rephasing time-ordering and creating the peak shift.  %

 In our 2D delay space, the $\tau$ trace can be defined if we assume $E_2$ and $E_{2^\prime}$ create

 the population (time-orderings V and VI).  %

-The trace runs parallel to the III-V time-ordering boundary (diagonal) if $\tau_{22^\prime}<0$ and runs parallel to the IV-VI time-ordering boundary (horizontal) if $\tau_{22^\prime}>0$, and both intersect at $\tau_{22^\prime}=0$; the $-\tau_{21}$ value at this intersection is $T$.\footnote{

-	See Supplementary Fig. S9 for an illustration of how 3PEPS shifts are measured from a 2D delay

-  plot.}  %

-In our 2D delay plots (Fig. \ref{fig:delay_purity}, Fig. \ref{fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.\footnote{

-	Supplementary Fig. S10 shows the 3PEPS measurements of all 12 combinations of

-  $\Gamma_{10}\Delta_t$ and $\Delta_{\text{inhom}}$, for every population delay surveyed.}  %

+The trace runs parallel to the III-V time-ordering boundary (diagonal) if $\tau_{22^\prime}<0$ and

+runs parallel to the IV-VI time-ordering boundary (horizontal) if $\tau_{22^\prime}>0$, and both

+intersect at $\tau_{22^\prime}=0$; the $-\tau_{21}$ value at this intersection is $T$.  %

+In our 2D delay plots (Fig. \ref{fig:delay_purity}, Fig. \ref{fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line.  %

 Fig. \ref{fig:delay_inhom} highlights the peak shift profile as a function of population time with

 the yellow trace; it is easily verified that our static inhomogneous system exhibits a non-zero

 peak shift value for all population times.  %

@@ -1106,7 +1113,7 @@ This fact is easily illustrated by the dynamics of homogeneous system (Fig.
 \ref{fig:delay_purity}); even though the homogeneous system cannot rephase, there is a non-zero

 peak shift near $T=0$.  %

 The contamination of the 3PEPS measurement at pulse overlap is well-known and is described in some

-studies,\cite{DeBoeij1996,Agarwal2002} but the dependence of pulse and system properties on the

+studies,\cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system properties on the

 distortion has not been investigated previously.  %

 Peak-shifting due to pulse overlap is less important when $\omega_1\neq\omega_2$ because

 time-ordering III is decoupled by detuning.  %

@@ -1133,8 +1140,7 @@ time-ordering III is decoupled by detuning.  %
 In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous

 broadening.  %

 Fig. \ref{fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous

-distribution.\footnote{As in Fig. \ref{fig:hom_2d_spectra}, Fig. \ref{fig:inhom_2d_spectra} shows

-  only the contours at the half-maximum amplitude.  See Supplementary Fig. S7 for all contours.}  %

+distribution.  %

 All systems are broadened by a distribution proportional to their dephasing bandwidth.  %

 As expected, the sequence again shows a gradual broadening along the $\omega_1$ axis, with a strong

 spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals.  %

@@ -1196,7 +1202,8 @@ Note the very sharp diagonal feature that appears for $(\tau_{21},\tau_{22^\prim
 This expression is inaccurate:  the narrow resonance is only observed when pulse durations are much

 longer than the coherence time.  %

 A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes (Yurs

-\textit{et al.}\cite{Yurs2011} and Kohler \textit{et al.}\cite{Kohler2014}, respectively)

+\textit{et al.}\cite{YursLenaA2011a} and \textcite{KohlerDanielDavid2014a},

+respectively)

 demonstrates this difference well.  %

 The driven equation fails to reproduce our numerical simulations here because resonant excitation

 of the population is impulsive; the experiment time-gates only the rise time of the population, yet

@@ -1250,18 +1257,19 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space.  %
 We have shown that pulse effects mimic the qualitative signatures of inhomogeneity.  %

 Here we address how one can extract true system inhomogeneity in light of these effects.  %

 We focus on two ubiquitous metrics of inhomogeneity: 3PEPS for the time domain and

-ellipticity\footnote{

-	There are many ways to characterize the ellipticity of a peak shape.  

-	We adopt the convention $\mathcal{E} = \left(a^2-b^2\right) / \left(a^2+b^2\right)$, where $a$ is the diagonal width and $b$ is the antidiagonal width.} 

-for the frequency domain\cite{Kwac2003,Okumura1999}.  %

+ellipticity for the frequency domain \cite{KwacKijeong2003a, OkumuraKo1999a}.  %

+There are many ways to characterize the ellipticity of a peak shape.  %

+We adopt the convention $\mathcal{E} = \left(a^2-b^2\right) / \left(a^2+b^2\right)$, where $a$ is

+the diagonal width and $b$ is the antidiagonal width.  %

 In the driven (impulsive) limit, ellipticity (3PEPS) corresponds to the frequency correlation

 function and uniquely extracts the inhomogeneity of the models presented here.  %

 In their respective limits, the metrics give values proportional to the inhomogeneity.  % 

 

 Fig. \ref{fig:metrics} shows the results of this characterization for all $\Delta_\text{inhom}$ and

 $\Gamma_{10}\Delta_t$ values explored in this work.  %

-We study how the correlations between the two metrics depend on the relative dephasing rate, $\Gamma_{10}\Delta_t$, the absolute inhomogeneity, $\Delta_\text{inhom} / \Delta_\omega$, the relative inhomogeneity $\Delta_\text{inhom} / \Gamma_{10}$, and the population time delay.\footnote{

-	The simulations for each value of the 3PEPS and ellipticity data in Fig. \ref{fig:metrics} appear in Supplementary Figs. S10-S12.}  

+We study how the correlations between the two metrics depend on the relative dephasing rate,

+$\Gamma_{10}\Delta_t$, the absolute inhomogeneity, $\Delta_\text{inhom} / \Delta_\omega$, the

+relative inhomogeneity $\Delta_\text{inhom} / \Gamma_{10}$, and the population time delay.  %

 The top row shows the correlations of the $\Delta_\text{3PEPS} / \Delta_t$ 3PEPS metric that

 represents the normalized coherence delay time required to reach the peak intensity.  %

 The upper right graph shows the correlations for a population time delay of $T = 4\Delta_t$ that

@@ -1273,7 +1281,7 @@ It becomes independent of $\Delta_\text{inhom} / \Delta_\omega$ when $\Delta_\te
 \Delta_\omega > 1$.  %

 This saturation results because the frequency bandwidth of the excitation pulses becomes smaller

 than the inhomogeneous width and only a portion of the inhomogeneous ensemble contributes to the

-3PEPS experiment.\cite{Weiner1985}  %

+3PEPS experiment. \cite{WeinerAM1985a}  %

 The corresponding graph for $T = 0$ shows a large peak shift occurs, even without inhomogeneity. 

 In this case, the peak shift depends on pathway overlap, as discussed in Section

 \ref{sec:res_inhom}.  %

@@ -1312,7 +1320,7 @@ pulse effects give a mapping significantly different than at $T = 4\Delta_t$.  %
 At $T = 0$, 3PEPS is almost nonresponsive to inhomogeneity; instead, it is an almost independent

 characterization of the pure dephasing.  %

 In fact, the $T=0$ trace is equivalent to the original photon echo traces used to resolve pure

-dephasing rates.\cite{Aartsma1976}  %

+dephasing rates.\cite{AartsmaThijsJ1976a}  %

 Both metrics are offset due to the pulse overlap effects.  %

 Accordingly, the region to the left of homogeneous contour is non-physical, because it represents

 observed correlations that are less than that given by pulse overlap effects.  %