From f47e1abac351ed10110c9d982aea76ca73dd30a6 Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Wed, 31 Jan 2018 18:37:43 -0600 Subject: 2018-01-31 18:37 --- mixed_domain/SQC lineshapes against t.pdf | Bin 0 -> 171723 bytes mixed_domain/chapter.tex | 987 +++++++++++++++++++++++++++++- mixed_domain/metrics.pdf | Bin 0 -> 141095 bytes mixed_domain/spectral evolution full.pdf | Bin 0 -> 6702377 bytes mixed_domain/steady state.pdf | Bin 0 -> 2112224 bytes mixed_domain/wigners full.pdf | Bin 0 -> 6648960 bytes mixed_domain/wigners.pdf | Bin 0 -> 2311832 bytes 7 files changed, 978 insertions(+), 9 deletions(-) create mode 100644 mixed_domain/SQC lineshapes against t.pdf create mode 100644 mixed_domain/metrics.pdf create mode 100644 mixed_domain/spectral evolution full.pdf create mode 100644 mixed_domain/steady state.pdf create mode 100644 mixed_domain/wigners full.pdf create mode 100644 mixed_domain/wigners.pdf (limited to 'mixed_domain') diff --git a/mixed_domain/SQC lineshapes against t.pdf b/mixed_domain/SQC lineshapes against t.pdf new file mode 100644 index 0000000..e932bca Binary files /dev/null and b/mixed_domain/SQC lineshapes against t.pdf differ diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 4d3ba1c..308146b 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -5,7 +5,7 @@ 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. % +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 which resemble signatures of dynamic @@ -137,17 +137,20 @@ from these measurement artifacts. % \section{Theory} +\afterpage{ \begin{figure} \centering \includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"} - \caption{ + \caption[Sixteen triply-resonant Liouville pathways.]{ The sixteen triply-resonant Liouville pathways for the third-order response of the system used - here. Time flows from left to right. Each excitation is labeled by the pulse stimulating the - transition; excitatons with $\omega_1$ are yellow, excitations with $\omega_2=\omega_{2'}$ are - purple, and the final emission is gray. + here. + Time flows from left to right. + Each excitation is labeled by the pulse stimulating the transition; excitatons with $\omega_1$ + are yellow, excitations with $\omega_2=\omega_{2'}$ are purple, and the final emission is gray. } \label{fig:WMELs} \end{figure} +\clearpage} We consider a simple three-level system (states $n=0,1,2$) that highlights the multidimensional line shape changes resulting from choices of the relative dephasing and detuning of the system and @@ -231,9 +234,10 @@ this paper. % The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discussed in Appendix \ref{sec:cw_imp}. % -\begin{figure*} +\afterpage{ +\begin{figure} \includegraphics[width=\linewidth]{"mixed_domain/simulation overview"} - \caption{ + \caption[Overview of the MR-CMDS simulation.]{ Overview of the MR-CMDS simulation. (a) The temporal profile of a coherence under pulsed excitation depends on how quickly the coherence dephases. In all subsequent panes, the relative dephasing rate is kept constant at @@ -249,7 +253,8 @@ The steady state and impulsive limits of Equation \ref{eq:rho_f_int} are discuss help introduce our delay convention. } \label{fig:overview} -\end{figure*} +\end{figure} +\clearpage} Fig. \ref{fig:overview} gives an overview of the simulations done in this work. % Fig. \ref{fig:overview}a shows an excitation pulse (gray-shaded) and examples of a coherent @@ -367,6 +372,214 @@ $S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime})$ are recorded f Liouville pathway. % Our simulations were done using the open-source SciPy library.\cite{Oliphant2007} % +\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 +\ref{eq:E_L_full}. % +The driven limit is achieved when pulse durations are much longer than the response function +dynamics: $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \gg 1$. % +In this limit, the system will adopt a steady state over excitation: $d\rho / dt \approx 0$. % +Neglecting phase factors, the driven solution to Equation \ref{eq:rho_f_int} will be +\begin{equation}\label{eq:sqc_driven} +\tilde{\rho}_f(t) = \frac{\lambda_f \mu_f}{2} +\frac{c_x(t-\tau_x)e^{i\kappa_f \Omega_{fx}t}}{\kappa_f \Omega_{fx}} \tilde{\rho}_i(t). +\end{equation} +The frequency and temporal envelope of the excitation pulse controls the coherence time evolution, +and the relative amplitude and phase of the coherence is directly related to detuning from +resonance. % + +The impulsive limit is achieved when the excitation pulses are much shorter than response function +dynamics: $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$. % +The full description of the temporal evolution has two separate expressions: one for times when the +pulse is interacting with the system, and one for times after pulse interaction. % +Both expressions are important when describing CMDS experiments. % + +For times after the pulse interaction, $t \gtrsim \tau_x + \Delta_t$, the field-matter coupling is +negligible. % +The evolution for these times, on resonance, is given by +\begin{equation}\label{eq:sqc_fid} +\tilde{\rho}_f(t) =\frac{i \lambda_f\mu_f }{2} \tilde{\rho}_i(\tau_x) +\int c_x(u) du \ e^{-\Gamma_f(t-\tau_x)}. +\end{equation} +This is classic free induction decay (FID) evolution: the system evolves at its natural frequency +and decays at rate $\Gamma_f$. % +It is important to note that, while this expression is explicitly derived from the impulsive limit, +FID behavior is not exclusive to impulsive excitation, as we have defined it. % +A latent FID will form if the pulse vanishes at a fast rate relative to the system dynamics. + +For evaluating times near pulse excitation, $t \lesssim \tau_x + \Delta_t$, we implement a Taylor +expansion in the response function about zero: $e^{-(\Gamma_f+i\kappa_f\Omega_{fx})u} = 1 - +(\Gamma_f+i\kappa_f\Omega_{fx})u+\cdots$. % +Our impulsive criterion requires that a low order expansion will suffice; it is instructive to +consider the result of the first order expansion of Equation \ref{eq:rho_f_int}: % +\begin{equation}\label{eq:sqc_rise} +\begin{split} +\tilde{\rho}_f(t) =& \frac{i \lambda_f\mu_f}{2} e^{-i\kappa_f\omega_x\tau_x}e^{-i\kappa_f\Omega_{fx}t} \tilde{\rho}_i(\tau_x) \\ +& \times \bigg[ \left( 1-(\Gamma_f + i\kappa_f\Omega_{fx})(t-\tau_x) \right) \int_{-\infty}^{t-\tau_x} c_x(u) du \\ +& \quad +(\Gamma_f + i\kappa_f\Omega_{fx}) \int_{-\infty}^{t-\tau_x} c_x(u)u \ du \bigg]. +\end{split} +\end{equation} +During this time $\tilde{\rho}_f$ builds up roughly according to the integration of the pulse +envelope. % +The build-up is integrated because the pulse transfers energy before appreciable dephasing or +detuning occurs. % +Contrary to the expectation of impulsive evolution, the evolution of $\tilde{\rho}_f$ is explicitly +affected by the pulse frequency, and the temporal profile evolves according to the pulse. % + +It is important to recognize that the impulsive limit is defined not only by having slow relaxation +relative to the pulse duration, but also by small detuning relative to the pulse bandwidth (as is +stated in the inequality). % +As detuning increases, the higher orders of the response function Taylor expansion will be needed +to describe the rise time, and the driven limit of Equation \ref{eq:sqc_driven} will become +valid. % +The details of this build-up time can often be neglected in impulsive approximations because +build-up contributions are often negligible in analysis; the period over which the initial +excitation occurs is small in comparison to the free evolution of the system. % +The build-up behavior can be emphasized by the measurement, which makes Equation \ref{eq:sqc_rise} +important. % + +We now consider full Liouville pathways in the impulsive and driven limits of Equation +\ref{eq:E_L_full}. % +For the driven limit, Equation \ref{eq:E_L_full} can be reduced to +\begin{equation}\label{eq:E_L_driven} +\begin{split} +E_L(t) =& \frac{1}{8} \lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 +e^{-i(\kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z)} \\ +& \times e^{ i(\kappa_3\omega_z + \kappa_2\omega_y + \kappa_1\omega_x)t} \\ +& \times c_z(t-\tau_z)c_y(t-\tau_y)c_x(t-\tau_x) \\ +& \times \frac{1}{\kappa_1\Omega_{1x}-i\Gamma_1} \frac{1}{\kappa_1\Omega_{1x} + \kappa_2\Omega_{2y} - i\Gamma_2} \\ +& \times \frac{1}{\kappa_1\Omega_{1x} + \kappa_2 \Omega_{2y} + \kappa_3\Omega_{3z}-i\Gamma_3}. +\end{split} +\end{equation} +It is important to note that the signal depends on the multiplication of all the fields; pathway +discrimination based on pulse time-ordering is not achievable because polarizations exists only +when all pulses are overlapped. % +This limit is the basis for frequency-domain techniques. % +Frequency axes, however, are not independent because the system is forced to the laser frequency +and influences the resonance criterion for subsequent excitations. % +As an example, observe that the first two resonant terms in Equation \ref{eq:E_L_driven} are +maximized when $\omega_x=\left|\omega_1\right|$ and $\omega_y=\left|\omega_2\right|$. % +If $\omega_x$ is detuned by some value $\varepsilon$, however, the occurrence of the second +resonance shifts to $\omega_y=\left|\omega_2\right|+\varepsilon$, effectively compensating for the +$\omega_x$ detuning. % +This shifting of the resonance results in 2D line shape correlations. % + +If the pulses do not temporally overlap $(\tau_x+\Delta_t \lesssim \tau_y +\Delta_t \lesssim \tau_z ++ \Delta_t \lesssim t)$, then the impulsive solution to the full Liouville pathway of Equation +\ref{eq:E_L_full} is % +\begin{equation}\label{eq:E_L_impulsive} +\begin{split} +E_L(t) =& \frac{i}{8} \lambda_1\lambda_2\lambda_3\mu_1 \mu_2 \mu_3 \mu_4 e^{i(\omega_1 + \omega_2 + \omega_3)t} \\ +& \times \int c_x(w) dw \int c_y(v) dv \int c_z(u) du \\ +& \times e^{-\Gamma_1(\tau_y-\tau_x)} e^{-\Gamma_2(\tau_z-\tau_y)} e^{-\Gamma(t-\tau_z)}. +\end{split} +\end{equation} +Pathway discrimination is demonstrated here because the signal is sensitive to the time-ordering of +the pulses. % +This limit is suited for delay scanning techniques. % +The emitted signal frequency is determined by the system and can be resolved by scanning a +monochromator. % + +The driven and impulsive limits can qualitatively describe our simulated signals at certain +frequency and delay combinations. % +Of the three expressions, the FID limit most resembles signal when pulses are near resonance and +well-separated in time (so that build-up behavior is negligible). % +The build-up limit approximates well when pulses are near-resonant and arrive together (so that +build-up behavior is emphasized). % +The driven limit holds for large detunings, regardless of delay. % + +\subsection{Convolution Technique for Inhomogeneous Broadening}\label{sec:convolution} + +\afterpage{ +\begin{figure} + \includegraphics[width=\linewidth]{mixed_domain/convolve} + \caption[Convolution overview.] + {Overview of the convolution. + (a) The homogeneous line shape. + (b) The distribution function, $K$, mapped onto laser coordinates. + (c) The resulting ensemble line shape computed from the convolution. + The thick black line represents the FWHM of the distribution function.} + \label{fig:convolution} +\end{figure} +\clearpage} + +Here we describe how to transform the data of a single reference oscillator signal to that of an +inhomogeneous distribution. % +The oscillators in the distribution are allowed have arbitrary energies for their states, which +will cause frequency shifts in the resonances. % +To show this, we start with a modified, but equivalent, form of Equation \ref{eq:rho_f}: +\begin{equation}\label{eq:rho_f_modified} +\begin{split} +\frac{d\tilde{\rho}_f}{dt} =& -\Gamma_f\tilde{\rho}_f + \frac{i}{2}\lambda_f\mu_f c_x(t-\tau_x) \\ +& \times e^{i\kappa_f\left( \vec{k}\cdot z + \omega_x \tau_x \right)} e^{-i\kappa_f\left( \omega_x-\left|\omega_f \right| \right)t}\tilde{\rho}_i(t). +\end{split} +\end{equation} + +We consider two oscillators with transition frequencies $\omega_f$ and $\omega_f^\prime=\omega_f + +\delta$. % +So long as $\left| \delta \right| \leq \omega_f$ (so that $\left| \omega_f + \delta \right| = +\left| \omega_f \right| + \delta$ and thus the rotating wave approximation does not change), +Equation \ref{eq:rho_f_modified} shows that the two are related by % +\begin{equation}\label{eq:freq_translation} +\frac{d\tilde{\rho}_f^\prime}{dt}(t;\omega_x) = \frac{d\tilde{\rho}_f}{dt}(t;\omega_x-\delta)e^{i\kappa_f \delta \tau_x}. +\end{equation} + +Because both coherences are assumed to have the same initial conditions +($\rho_0(-\infty)=\rho_0^\prime(-\infty)=0$), the equality also holds when both sides of the +equation are integrated. % +The phase factor $e^{i\kappa_f\delta\tau_x}$ in the substitution arises from Equation \ref{eq:E_l}, +where the pulse carrier frequency maintains its phase within the pulse envelope for all delays. % + +The resonance translation can be extended to higher order signals as well. % +For a third-order signal, we compare systems with transition frequencies +$\omega_{10}^\prime=\omega_{10}+a$ and $\omega_{21}^\prime = \omega_{21}+b$. % +The extension of Equation \ref{eq:freq_translation} to pathway $V\beta$ gives % +\begin{equation} +\begin{split} +\tilde{\rho}_3^\prime(t;\omega_2, \omega_2^\prime, \omega_1) =& \tilde{\rho}_3(t;\omega_2-a,\omega_{2^\prime}-a,\omega_1-b) \\ +&\times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1}. +\end{split} +\end{equation} + +The translation of each laser coordinate depends on which transition is made (e.g. $a$ for +transitions between $|0\rangle$ and $|1\rangle$ or $b$ for transitions between $|1\rangle$ and +$|2\rangle$), so the exact translation relation differs between pathways. % +We can now compute the ensemble average of signal for pathway $V\beta$ as a convolution between the +distribution function of the system, $K(a,b)$, and the single oscillator response: % +\begin{equation} +\begin{split} +\langle \tilde{\rho}_3 (t;\omega_2,\omega_{2^\prime},\omega_1) \rangle =& \iint K(a,b)\\ +& \times \tilde{\rho}_3 (t;\omega_2+a,\omega_{2^\prime}+a,\omega_1+b) \\ +& \times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1} da \ db. +\end{split} +\end{equation} +For this work, we restrict ourselves to a simpler ensemble where all oscillators have equally +spaced levels (i.e. $a=b$). % +This makes the translation identical for all pathways and reduces the dimensionality of the +convolution. % +Since pathways follow the same convolution we may also perform the convolution on the total signal field: +\begin{equation} +\begin{split} +\langle E_{\text{tot}}(t) \rangle =& \sum_L \mu_{4,L} \int K(a,a) \\ +& \times \tilde{\rho}_{3,L}(t;\omega_x-a,\omega_y-a\omega_z-a) \\ +& \times e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} da. +\end{split} +\end{equation} +Furthermore, since $\kappa=-1$ for $E_1$ and $E_{2^\prime}$, while $\kappa=1$ for $E_2$, we have +$e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} = e^{-ia\left( \tau_1 - + \tau_2 + \tau_{2^\prime} \right)}$ for all pathways. % +Equivalently, if the electric field is parameterized in terms of laser coordinates $\omega_1$ and $\omega_2$, the ensemble field can be calculated as +\begin{equation}\label{eq:convolve_final} +\begin{split} +\langle E_{\text{tot}}(t;\omega_1,\omega_2) \rangle =& \int K(a,a)E_{\text{tot}}(t;\omega_1-a,\omega_2-a) \\ +&\times e^{-ia\left( \tau_1-\tau_2+\tau_{2^\prime} \right)} da. +\end{split} +\end{equation} +which is a 1D convolution along the diagonal axis in frequency space. % +Fig. \ref{fig:convolution} demonstrates the use of Equation \ref{eq:convolve_final} on a +homogeneous line shape. % + \section{Results} % ------------------------------------------------------------------------------ We now present portions of our simulated data that highlight the dependence of the spectral line @@ -375,9 +588,11 @@ pulse delay times, and inhomogeneous broadening. % \subsection{Evolution of single coherence}\label{sec:evolution_SQC} +\afterpage{ \begin{figure} + \centering \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"} - \caption{ + \caption[Relative importance of FID and driven response for a single quantum coherence.]{ The relative importance of FID and driven response for a single quantum coherence as a function of the relative dephasing rate (values of $\Gamma_{10}\Delta_t$ are shown inset). The black line shows the coherence amplitude profile, while the shaded color indicates the @@ -387,6 +602,7 @@ pulse delay times, and inhomogeneous broadening. % } \label{fig:fid_dpr} \end{figure} +\clearpage} It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x} \rho_1$, under various excitation conditions. % @@ -423,3 +639,756 @@ We note that our choices of $\Gamma_{10}\Delta_t=2.0, 1.0,$ and $0.5$ give coher mainly driven, roughly equal driven and FID parts, and mainly FID components, respectively. % FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. % +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs detuning"} + \caption[Pulsed excitation of a single quantum coherence and its dependance on pulse detuning.]{ + Pulsed excitation of a single quantum coherence and its dependence on the pulse detuning. In + all cases, relative dephasing is kept at $\Gamma_{10}\Delta_t = 1$. + (a) The relative importance of FID and driven response for a single quantum coherence as a + function of the detuning (values of relative detuning, $\Omega_{fx}/\Delta_{\omega}$, are shown + inset). + The color indicates the instantaneous frequency (scale bar on right), while the black line + shows the amplitude profile. The gray line is the electric field amplitude. + %Comparison of the temporal evolution of single quantum coherences at different detunings + %(labeled inset). + (b) The time-integrated coherence amplitude as a function of the detuning. The integrated + amplitude is collected both with (teal) and without (magenta) a tracking monochromator that + isolates the driven frequency components. %$\omega_m=\omega_\text{laser}$. + For comparison, the Green's function of the single quantum coherence is also shown (amplitude + is black, hashed; imaginary is black, solid). + In all plots, the gray line is the electric field amplitude. + } + \label{fig:fid_detuning} +\end{figure} +\clearpage} + +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. +} +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 +instantaneous frequency in Fig. \ref{fig:fid_detuning}a). % +The coherence will persist beyond the pulse duration only if the pulse transfers energy into the +system; FID evolution equates to absorption. % +The FID is therefore sensitive to the absorptive (imaginary) line shape of a transition, while the +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. +} % +Many ultrafast spectroscopies take advantage of the latent FID to suppress non-resonant background, +improving signal to noise.\cite{Lagutchev2007,Lagutchev2010,Donaldson2010,Donaldson2008} % + +In driven experiments, the output frequency and line shape are fully constrained by the excitation +beams. % +In such experiments, there is no additional information to be resolved in the output spectrum. % +The situation changes in the mixed domain, where $E_\text{tot}$ contains FID signal that lasts +longer than the pulse duration. % +Fig. \ref{fig:fid_detuning}a provides insight on how frequency-resolved detection of coherent +output can enhance resolution when pulses are spectrally broad. % +Without frequency-resolved detection, mixed-domain resonance enhancement occurs in two ways: (1) +the peak amplitude increases, and (2) the coherence duration increases due to the FID transient. % +Frequency-resolved detection can further discriminate against detuning by requiring that the +driving frequency agrees with latent FID. % +The implications of discrimination are most easily seen in Fig. \ref{fig:fid_detuning}a with +$\Omega_{1x}/\Delta_{\omega}=\pm 1$, where the system frequency moves from the driving frequency to +the FID frequency. % +When the excitation pulse frequency is scanned, the resonance will be more sensitive to detuning by +isolating the driven frequency (tracking the monochromator with the excitation source). % + + +The functional form of the measured line shape can be deduced by considering the frequency domain form of Equation \ref{eq:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$): +\begin{equation}\label{eq:rho_f_int_freq} +\tilde{\rho}_f (\omega) = \frac{i\lambda_f\mu_f}{2\sqrt{2\pi}} \cdot \frac{\mathcal{F}\left\{ c_x \right\}\left( \omega - \kappa_f\Omega_{fx} \right)}{\Gamma_f + i\omega}, +\end{equation} +where $\mathcal{F}\left\{ c_x \right\}\left( \omega \right)$ denotes the Fourier transform of +$c_x$, which in our case gives +\begin{equation} +\mathcal{F}\left\{ c_x \right\}\left( \omega \right) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(\Delta_t\omega)^2}{4\ln 2}}. +\end{equation} +For squared-law detection of $\rho_f$, the importance of the tracking monochromator is highlighted +by two limits of Equation \ref{eq:rho_f_int_freq}: +\begin{itemize} + \item When the transient is not frequency resolved, $\text{sig} \approx \int{\left| + \tilde{\rho}_f(\omega) \right|^2 d\omega}$ and the measured line shape will be the + convolution of the pulse envelope and the intrinsic (Green's function) response (Fig. + \ref{fig:fid_detuning}b, magenta). + \item When the driven frequency is isolated, $\text{sig} \approx \left| + \tilde{\rho}_f(\kappa_f\Omega_{fx}) \right|^2$ and the measured line shape will give the + un-broadened Green's function (Fig. \ref{fig:fid_detuning}b, teal). +\end{itemize} +Monochromatic detection can remove broadening effects due to the pulse bandwidth. % +For large $\Gamma_{10}\Delta_t$ values, FID evolution is negligible at all +$\Omega_{fx}/\Delta_{\omega}$ values and the monochromator is not useful. % +Fig. \ref{fig:fid_detuning}b shows the various detection methods for the relative dephasing rate of +$\Gamma_{10}\Delta_t=1$. % + +\subsection{Evolution of single Liouville pathway} + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/pw1 lineshapes"} + \caption[2D frequency response of a single Liouville pathway at different delay values.]{ + Changes to the 2D frequency response of a single Liouville pathway (I$\gamma$) at different + delay values. + The normalized dephasing rate is $\Gamma_{10}\Delta_t = 1$. + Left: The 2D delay response of pathway I$\gamma$ at triple resonance. + Right: The 2D frequency response of pathway I$\gamma$ at different delay values. + The delays at which the 2D frequency plots are collected are indicated on the delay plot; + compare 2D spectrum frame color with dot color on 2D delay plot. + } + \label{fig:pw1} +\end{figure} +\clearpage} + +We now consider the multidimensional response of a single Liouville pathway involving three pulse +interactions. % +In a multi-pulse experiment, $\rho_1$ acts as a source term for $\rho_2$ (and subsequent +excitations). % +The spectral and temporal features of $\rho_1$ that are transferred to $\rho_2$ depend on when the +subsequent pulse arrives. % +Time-gating later in $\rho_1$ evolution will produce responses with FID behavior, while time-gating +$\rho_1$ in the presence of the initial pulse will produce driven responses. % +An analogous relationship holds for $\rho_3$ with its source term $\rho_2$. % +As discussed above, signal that time-gates FID evolution gives narrower spectra than driven-gated +signal. % +As a result, the spectra of even single Liouville pathways will change based on pulse delays. % + +The final coherence will also be frequency-gated by the monochromator. % +The monochromator isolates signal at the fully driven frequency $\omega_\text{out} = \omega_1$. % +The monochromator will induce line-narrowing to the extent that FID takes place. % +It effectively enforces a frequency constraint that acts as an additional resonance condition, +$\omega_\text{out}=\omega_1$. % +The driven frequency will be $\omega_1$ if $E_1$ is the last pulse interaction (time-orderings V +and VI), and the monochromator tracks the coherence frequency effectively. % +If $E_1$ is not the last interaction, the output frequency may not be equal to the driven +frequency, and the monochromator plays a more complex role. % + + +We demonstrate this delay dependence using the multidimensional response of the I$\gamma$ Liouville +pathway as an example (see Fig. \ref{fig:WMELs}). % +Fig. \ref{fig:pw1} shows the resulting 2D delay profile of pathway I$\gamma$ signals for +$\Gamma_{10}\Delta_t=1$ (left) and the corresponding $\omega_1, \omega_2$ 2D spectra at several +pulse delay values (right). % +The spectral changes result from changes in the relative importance of driven and FID +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}). +} + +\begin{table*} + \centering + \caption{\label{tab:table2} Conditions for peak intensity at different pulse delays for pathway + I$\gamma$.} + \begin{tabularx}{0.7\linewidth}{c c | X X X X} + \multicolumn{2}{c}{Delay} & \multicolumn{4}{|c}{Approximate Resonance Conditions} \\ + $\tau_{21}/\Delta_t$ & $\tau_{22^\prime}/\Delta_t$ & $\rho_0\xrightarrow{1}\rho_1$ & + $\rho_1\xrightarrow{2}\rho_2$ & $\rho_2\xrightarrow{2^\prime}\rho_3$ & $\rho_3\rightarrow$ + detection at $\omega_m=\omega_1$ \\ + \hline\hline + 0 & 0 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$ & $\omega_1=\omega_{10}$ & -- \\ + 0 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_1=\omega_2$ & $\omega_2=\omega_{10}$ & + $\omega_1=\omega_2$ \\ + 2.4 & 0 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & -- & + $\omega_1=\omega_{10}$ \\ + 2.4 & -2.4 & $\omega_1=\omega_{10}$ & $\omega_2=\omega_{10}$ & $\omega_2=\omega_{10}$ & + $\omega_1=\omega_2$ \\ + \end{tabularx} +\end{table*} + +When the pulses are all overlapped ($\tau_{21}=\tau_{22^\prime}=0$, lower right, orange), all +transitions in the Liouville pathway are simultaneously driven by the incident fields. % +This spectrum strongly resembles the driven limit spectrum. For this time-ordering, the first, +second, and third density matrix elements have driven resonance conditions of +$\omega_1=\omega_{10}$, $\omega_1-\omega_2=0$, and +$\omega_1-\omega_2+\omega_{2^\prime}=\omega_{10}$, respectively. % +The second resonance condition causes elongation along the diagonal, and since +$\omega_2=\omega_{2^\prime}$, the first and third resonance conditions are identical, effectively +making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical elongation along +$\omega_1=\omega_{10}$. % + + +The other three spectra of Fig. \ref{fig:pw1} separate the pulse sequence over time so that not all +interactions are driven. % +At $\tau_{21}=0$, $\tau_{22^\prime}=-2.4\Delta_t$ (lower left, pink), the first two resonances +remain the same as at pulse overlap (orange) but the last resonance is different. % +The final pulse, $E_{2^\prime}$, is latent and probes $\rho_2$ during its FID evolution after +memory of the driven frequency is lost. % +There are two important consequences. % +Firstly, the third driven resonance condition is now approximated by +$\omega_{2^\prime}=\omega_{10}$, which makes $\omega_1$ only singly resonant at +$\omega_1=\omega_{10}$. % +Secondly, the driven portion of the signal frequency is determined only by the latent pulse: +$\omega_{\text{out}}=\omega_{2^\prime}$. % +Since our monochromator gates $\omega_1$, we have the detection-induced correlation +$\omega_1=\omega_{2^\prime}$. % +The net result is double resonance along $\omega_1=\omega_2$, and the vertical elongation of pulse +overlap is strongly attenuated. % + +At $\tau_{21}=2.4\Delta_t,\tau_{22^\prime}=0$ (upper right, purple), the first pulse $E_1$ precedes +the latter two, which makes the two resonance conditions for the input fields +$\omega_1=\omega_{10}$ and $\omega_2=\omega_{10}$. % +The signal depends on the FID conversion of $\rho_1$, which gives vertical elongation at +$\omega_1=\omega_{10}$. % +Furthermore, $\rho_1$ has no memory of $\omega_1$ when $E_2$ interacts, which has two important +implications. % +First, this means the second resonance condition $\omega_1=\omega_2$ and the associated diagonal +elongation is now absent. % +Second, the final output polarization frequency content is no longer functional of $\omega_1$. +Coupled with the fact that $E_2$ and $E_{2^\prime}$ are coincident, so that the final coherence can +be approximated as driven by these two, we can approximate the final frequency as +$\omega_{\text{out}} = \omega_{10}-\omega_2+\omega_{2^\prime} = \omega_{10}$. % +Surprisingly, the frequency content of the output is strongly independent of all pulse +frequencies. % +The monochromator narrows the $\omega_1=\omega_{10}$ resonance. % +The $\omega_1=\omega_{10}$ resonance condition now depends on the monochromator slit width, the FID +propagation of $\rho_1$, the spectral bandwidth of $\rho_3$; its spectral width is not easily +related to material parameters. % +This resonance demonstrates the importance of the detection scheme for experiments and how the +optimal detection can change depending on the pulse delay time. % + +Finally, when all pulses are well-separated ($\tau_{21}=-\tau_{22^\prime}=2.4\Delta_t$, upper left, +cyan), each resonance condition is independent and both $E_1$ and $E_2$ require FID buildup to +produce final output. % +The resulting line shape is narrow in all directions. % +Again, the emitted frequency does not depend on $\omega_1$, yet the monochromator resolves the +final coherence at frequency $\omega_1$. % +Since the driven part of the final interaction comes from $E_{2^\prime}$, and since the +monochromator track $\omega_1$, the output signal will increase when +$\omega_1=\omega_{2^\prime}$. % +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. % +The phenomenon of time-gating can cause frequency and delay axes to become functional of each other +in unexpected ways. % + +\subsection{Temporal pathway discrimination} + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/delay space ratios"} + \caption[2D delay response for different relative dephasing rates.]{ + Comparison of the 2D delay response for different relative dephasing rates (labeled atop each + column). + All pulses are tuned to exact resonance. + In each 2D delay plot, the signal amplitude is depicted by the colors. + The black contour lines show signal purity, $P$ (see Equation \ref{eq:P}), with purity values + denoted on each contour. + The small plots above each 2D delay plot examine a $\tau_{22^\prime}$ slice of the delay + response ($\tau_{21}=0$). + The plot shows the total signal (black), as well as the component time-orderings VI (orange), V + (purple), and III or I (teal). + } + \label{fig:delay_purity} +\end{figure} +\clearpage} + +In the last section we showed how a single pathway's spectra can evolve with delay due to pulse +effects and time gating. % +In real experiments, evolution with delay is further complicated by the six time-orderings/sixteen +pathways present in our three-beam experiment (see Fig. \ref{fig:WMELs}). % +Each time-ordering has different resonance conditions. % +When signal is collected near pulse overlap, multiple time-orderings contribute. % +To identify these effects, we start by considering how strongly time-orderings are isolated at each +delay coordinate. % + +While the general idea of using time delays to enhance certain time-ordered regions is widely +applied, quantitation of this discrimination is rarely explored. % +Because the temporal profile of the signal is dependent on both the excitation pulse profile and +the decay dynamics of the coherence itself, quantitation of pathway discrimination requires +simulation. % + +Fig. \ref{fig:delay_purity} shows the 2D delay space with all pathways present for +$\omega_1,\omega_2=\omega_{10}$. % +It illustrates the interplay of pulse width and system decay rates on the isolation of time-ordered +pathways. % +The color bar shows the signal amplitude. % +Signal is symmetric about the $\tau_{21}=\tau_{22^\prime}$ line because when $\omega_1=\omega_2$, +$E_1$ and $E_{2^\prime}$ interactions are interchangeable: +$S_\text{tot}(\tau_{21},\tau_{22^\prime})=S_\text{tot}(\tau_{22^\prime}, \tau_{21})$. % +The overlaid black contours represent signal ``purity,'' $P$, defined as the relative amount of +signal that comes from the dominant pathway at that delay value: +\begin{equation}\label{eq:P} +P(\tau_{21},\tau_{22^\prime})=\frac{\max \left\{S_L\left( \tau_{21},\tau_{22^\prime} \right)\right\}} +{\sum_L S_L\left( \tau_{21},\tau_{22^\prime} \right)}. +\end{equation} +The dominant pathway ($\max{\left\{ S_L \left( \tau_{21},\tau_{22^\prime} \right) \right\}}$) at +given delays can be inferred by the time-ordered region defined in Fig. \ref{fig:overview}d. % +The contours of purity generally run parallel to the time-ordering boundaries with the exception of +time-ordered regions II and IV, which involve the double quantum coherences that have been +neglected. % + +A commonly-employed metric for temporal selectivity is how definitively the pulses are ordered. % +This metric agrees with our simulations. % +The purity contours have a weak dependence on $\Delta_t \Gamma_{10}$ for +$\left|\tau_{22^\prime}\right|/\Delta_t < 1$ or $\left|\tau_{21}\right|/\Delta_t < 1$ where there +is significant pathway overlap and a stronger dependence at larger values where the pathways are +well-isolated. % +Because responses decay exponentially, while pulses decay as Gaussians, there always exist delays +where temporal discrimination is possible. % +As $\Delta_t\Gamma_{10}\rightarrow \infty$, however, such discrimination is only achieved at +vanishing signal intensities; the contour of $P=0.99$ across our systems highlights this trend. % + +\subsection{Multidimensional line shape dependence on pulse delay time} + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/spectral evolution"} + \caption[Evolution of the 2D frequency response.]{ + Evolution of the 2D frequency response as a function of $\tau_{21}$ (labeled inset) and the + influence of the relative dephasing rate ($\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and + $2.0$ (blue)). + In all plots $\tau_{22^\prime}=0$. To ease comparison between different dephasing rates, the + colored line contours (showing the half-maximum) for all three relative dephasing rates are + overlaid. + The colored histograms below each 2D frequency plot show the relative weights of each + time-ordering for each relative dephasing rate. + Contributions from V and VI are grouped together because they have equal weights at + $\tau_{22^\prime}=0$. + } + \label{fig:hom_2d_spectra} +\end{figure} +\clearpage} + +In the previous sections we showed how pathway spectra and weights evolve with delay. % +This section ties the two concepts together by exploring the evolution of the spectral line shape +over a span of $\tau_{21}$ delay times that include all pathways. % +It is a common practice to explore spectral evolution against $\tau_{21}$ because this delay axis +shows population evolution in a manner analogous to pump-probe spectroscopies. % +The $\vec{k}_2$ and $\vec{k}_{2^\prime}$ interactions correspond to the pump, and the $\vec{k}_1$ +interaction corresponds to the probe. % +Time-orderings V and VI are the normal pump-probe time-orderings, time-ordering III is a mixed +pump-probe-pump ordering (so-called pump polarization coupling), and time-ordering I is the +probe-pump ordering (so-called perturbed FID). % +Scanning $\tau_{21}$ through pulse overlap complicates interpretation of the line shape due to the +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. % + +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 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 +$\omega_2$ interact with independent resonances). % +Such spectral changes could be misinterpreted as spectral diffusion, where the line shape changes +from correlated to uncorrelated as population time increases due to system dynamics. % +The system dynamics included here, however, contain no structure that would allow for such +diffusion. % +Rather, the spectral changes reflect the changes in the majority pathway contribution, starting +with time-ordering I pathways, proceeding to an equal admixture of I, III, V, and VI, and finishing +at an equal balance of V and VI when $E_1$ arrives well after $E_2$ and $E_{2^\prime}$. % +Time-orderings I and III both exhibit a spectral correlation in $\omega_1$ and $\omega_2$ when +driven, but time-orderings V and VI do not. % +Moreover, such spectral correlation is forced near zero delay because the pulses time-gate the +driven signals of the first two induced polarizations. % +The monochromator detection also plays a dynamic role, because time-orderings V always VI always +emit a signal at the monochromator frequency, while in time-orderings I and III the emitted +frequency is not defined by $\omega_1$, as discussed above. % + +When we isolate time-orderings V and VI, we can maintain the proper scaling of FID bandwidth in the +$\omega_1$ direction because our monochromator can gate the final coherence. % +This gating is not possible in time-orderings I and III because the final coherence frequency is +determined by $\omega_{2^\prime}$ which is identical to $\omega_2$. % + +There are differences in the line shapes for the different values of the relative dephasing rate, +$\Gamma_{10}\Delta_t$. % +The spectral correlation at $\tau_{21}/\Delta_t=2$ decreases in strength as $\Gamma_{10}\Delta_t$ +decreases. % +As we illustrated in Fig. \ref{fig:pw1}, this spectral correlation is a signature of driven signal +from temporal overlap of $E_1$ and $E_2$; the loss of spectral correlation reflects the increased +prominence of FID in the first coherence as the field-matter interactions become more impulsive. % +This increased prominence of FID also reflects an increase in signal strength, as shown by +$\tau_{21}$ traces in Fig. \ref{fig:delay_purity}. % +When all pulses are completely overlapped, ($\tau_{21}=0$), each of the line shapes exhibit +spectral correlation. % +At $\tau_{21}/\Delta_t=-2$, the line shape shrinks as $\Gamma_{10}\Delta_t$ decreases, with the +elongation direction changing from horizontal to vertical. % +The general shrinking reflects the narrowing homogeneous linewidth of the $\omega_{10}$ +resonance. % +In all cases, the horizontal line shape corresponds to the homogeneous linewidth because the narrow +bandpass monochromator resolves the final $\omega_1$ resonance. % +The change in elongation direction is due to the resolving power of $\omega_2$. % +At $\Gamma_{10}\Delta_t=0.5$, the resonance is broader than our pulse bandwidth and is fully +resolved vertically. % +It is narrower than the $\omega_1$ resonance because time-orderings V and VI interfere to isolate +only the absorptive line shape along $\omega_2$. % +This narrowing, however, is unresolvable when the pulse bandwidth becomes broader than that of the +resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\Delta_t=0.5$. % + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/wigners"} + \caption[Wigners.]{ + Transient ($\omega_1$) line shapes and their dependence on $\omega_2$ frequency. + The relative dephasing rate is $\Gamma_{10}\Delta_t=1$ and $\tau_{22^\prime}=0$. + For each plot, the corresponding $\omega_2$ value is shown as a light gray vertical line.} + \label{fig:wigners} +\end{figure} +\clearpage} + +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} % +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.} % +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. % +For each plot, the $\omega_2$ frequency is denoted by a vertical gray line. % +Each Wigner plot is scaled to its own dynamic range to emphasize the dependence on $\omega_2$. % +The dramatic line shape changes between positive and negative delays can be seen. % +This representation also highlights the asymmetric broadening of the $\omega_1$ line shape near +pulse overlap when $\omega_2$ becomes non-resonant. % +Again, these features can resemble spectral diffusion even though our system is homogeneous. % + +\subsection{Inhomogeneous broadening} + +\afterpage{ +\label{sec:res_inhom} +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth]{"mixed_domain/inhom delay space ratios"} + \caption[2D delay response with inhomogeneity.]{ + 2D delay response for $\Gamma_{10}\Delta_t=1$ with sample inhomogeneity. % + All pulses are tuned to exact resonance. % + The colors depict the signal amplitude. % + The black contour lines show signal purity, $P$ (see Equation \ref{eq:P}), with purity values + denoted on each contour. % + The thick yellow line denotes the peak amplitude position that is used for 3PEPS analysis. % + The small plot above each 2D delay plot examines a $\tau_{22^\prime}$ slice of the delay + response ($\tau_{21}=0$). % + The plot shows the total signal (black), as well as the component time-orderings VI (orange), V + (purple), III (teal, dashed), and I (teal, solid). % + } + \label{fig:delay_inhom} +\end{figure} +\clearpage} + +With the homogeneous system characterized, we can now consider the effect of inhomogeneity. % +For inhomogeneous systems, time-orderings III and V are enhanced because their final coherence will +rephase to form a photon echo, whereas time-orderings I and VI will not. % +In delay space, this rephasing appears as a shift of signal to time-ordered regions III and V that +persists for all population times. % +Fig. \ref{fig:delay_inhom} shows the calculated spectra for relative dephasing rate +$\Gamma_{10}\Delta_t=1$ with a frequency broadening function of width +$\Delta_{\text{inhom}}=0.441\Gamma_{10}$. % +The inhomogeneity makes it easier to temporally isolate the rephasing pathways and harder to +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} % +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 +ability of the system. % +An inhomogeneous system will emit a photon echo in the rephasing pathway, enhancing signal in the +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.} % +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. % + +The unanticipated feature of the 3PEPS analysis is the dependence on $T$. % +Even though our inhomogeneity is static, the peak shift is maximal at $T=0$ and dissipates as $T$ +increases, mimicking spectral diffusion. % +This dynamic arises from signal overlap with time-ordering III, which uses $E_2$ and $E_1$ as the +first two interactions ,and merely reflects $E_1$ and $E_2$ temporal overlap. % +At $T=0$, the $\tau$ trace gives two ways to make a rephasing pathway (time-orderings III and V) +and only one way to make a non-rephasing pathway (time-ordering VI). % +This pathway asymmetry shifts signal away from $\tau=0$ into the rephasing direction. % +At large $T$ (large $\tau_{21}$), time-ordering III is not viable and pathway asymmetry +disappears. % +Peak shifts imply inhomogeneity only when time-orderings V and III are minimally contaminated by +each other i.e. at population times that exceed pulse overlap. % +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 +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. % + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/inhom spectral evolution"} + \caption[Spectral evolution of an inhomogenious system.]{ + Same as Fig. \ref{fig:hom_2d_spectra}, but each system has inhomogeneity + ($\Delta_{\text{inhom}}=0.441\Gamma_{10}$). + Relative dephasing rates are $\Gamma_{10}\Delta_t=0.5$ (red), $1.0$ (green), and $2.0$ (blue). + In all plots $\tau_{22^\prime}=0$. + To ease comparison between different dephasing rates, the colored line shapes of all three + systems are overlaid. + Each 2D plot shows a single representative contour (half-maximum) for each + $\Gamma_{10}\Delta_t$ value. + The colored histograms below each 2D frequency plot show the relative weights of each + time-ordering for each 2D frequency plot. + In contrast to Fig. \ref{fig:hom_2d_spectra}, inhomogeneity makes the relative contributions of + time-orderings V and VI unequal. + } + \label{fig:inhom_2d_spectra} +\end{figure} +\clearpage} + +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.} % +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. % +The anti-diagonal width at early delays (e.g. Fig. \ref{fig:inhom_2d_spectra}, +$\tau_{21}=2.0\Delta_t$) again depends on the pulse bandwidth and the monochromator slit width. % +At delay values that isolate time-orderings V and VI, however, the line shapes retain diagonal +character, showing the characteristic balance of homogeneous and inhomogeneous width. % + +\section{Discussion} % --------------------------------------------------------------------------- + +\subsection{An intuitive picture of pulse effects} + +Our chosen values of the relative dephasing time, $\Gamma_{10}\Delta_t$, describe experiments where +neither the impulsive nor driven limit unilaterally applies. % +We have illustrated that in this intermediate regime, the multidimensional spectra contain +attributes of both limits, and that it is possible to judge when these attributes apply. % +In our three-pulse experiment the second and third pulses time-gate coherences and populations +produced by the previous pulse(s), and the monochromator frequency-gates the final coherence. % +Time-gating isolates different properties of the coherences and populations. Consequently, spectra +evolve against delay. % +For any delay coordinate, one can develop qualitative line shape expectations by considering the following three principles: +\begin{enumerate} + \item When time-gating during the pulse, the system pins to the driving frequency with a buildup efficiency determined by resonance. + \item When time-gating after the pulse, the FID dominates the system response. + \item The emitted signal field contains both FID and driven components; the $\omega_{\text{out}} = \omega_1$ component is isolated by the tracking monochromator. +\end{enumerate} +Fig. \ref{fig:fid_dpr} illustrates principles 1 and 2 and Fig. \ref{fig:fid_detuning} illustrates +principle 2 and 3. % +Fig. \ref{fig:pw1} provides a detailed example of the relationship between these principles and the +multidimensional line shape changes for different delay times. % + +The principles presented above apply to a single pathway. % +For rapidly dephasing systems it is difficult to achieve complete pathway discrimination, as shown +in Fig. \ref{fig:delay_purity}. % +In such situations the interference between pathways must be considered to predict the line +shape. % +The relative weight of each pathway to the interference can be approximated by the extent of pulse +overlap. % +The pathway weights exchange when scanning across pulse overlap, which creates the dramatic line +shape changes observed in Figs. \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}. % + +\subsection{Conditional validity of the driven limit} + +We have shown that the driven limit misses details of the line shape if $\Gamma_{10} \Delta_t +\approx 1$, but we have also reasoned that in certain conditions the driven limit can approximate +the response well (see principle 1). % +Here we examine the line shape at delay values that demonstrate this agreement. % +Fig. \ref{fig:steady_state} compares the results of our numerical simulation (third column) with +the driven limit expressions for populations where $\Gamma_{11}\Delta_t=0$ (first column) or $1$ +(second column). % +The top and bottom rows compare the line shapes when $\left(\tau_{22^\prime}, + \tau_{21}=(0,0)\right)$ and $(0,-4\Delta_t)$, respectively. % +The third column demonstrates the agreement between the driven limit approximations with the +simulation by comparing the diagonal and anti-diagonal cross-sections of the 2D spectra. % + +% TODO: [ ] population resonance is not clear +Note the very sharp diagonal feature that appears for $(\tau_{21},\tau_{22^\prime}) = (0,0)$ and $ +\Gamma_{11}=0$; this is due to population resonance in time-orderings I and III. % +This expression is inaccurate: the narrow resonance is only observed when pulse durations are much +longer than the coherence time. % +A comparison of picosecond and femtosecond studies of quantum dot exciton line shapes (Yurs +\textit{et al.}\cite{Yurs2011} and Kohler \textit{et al.}\cite{Kohler2014}, respectively) +demonstrates this difference well. % +The driven equation fails to reproduce our numerical simulations here because resonant excitation +of the population is impulsive; the experiment time-gates only the rise time of the population, yet +driven theory predicts the resonance to be vanishingly narrow ($\Gamma_{11}=0$). % +In light of this, one can approximate this time-gating effect by substituting population lifetime +with the pulse duration ($\Gamma_{11}\Delta_t=1$), which gives good agreement with the numerical +simulation (third column). % + +When $\tau_{22^\prime}=0$ and $\tau_{21}<\Delta_t$, signals can also be approximated by driven +signal (Fig. \ref{fig:steady_state} bottom row). % +Only time-orderings V and VI are relevant. % +The intermediate population resonance is still impulsive but it depends on +$\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space. % + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=\linewidth]{"mixed_domain/steady state"} + \caption[Conditional validity of the driven limit.]{ + Comparing approximate expressions of the 2D frequency response with the directly integrated + response. % + $\Gamma_{10}\Delta_t=1$. % + The top row compares the 2D response of all time-orderings ($\tau_{21}=0$) and the bottom row + compares the response of time-orderings V and VI ($\tau_{21}=-4\Delta_t$). % + First column: The driven limit response. Note the narrow diagonal resonance for $\tau_{21}=0$. + Second column: Same as the first column, but with ad hoc substitution $\Gamma_{11}=\Delta_t$. + Third column: The directly integrated response. % + } + \label{fig:steady_state} +\end{figure} +\clearpage} + +\subsection{Extracting true material correlation} + +\afterpage{ +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth]{"mixed_domain/metrics"} + \caption[Metrics of correlation.]{ + Temporal (3PEPS) and spectral (ellipticity) metrics of correlation and their relation to the + true system inhomogeneity. % + The left column plots the relationship at pulse overlap ($T=0$) and the right column plots the + relationship at a delay where driven correlations are removed ($T=4\Delta_t$). % + For the ellipticity measurements, $\tau_{22^\prime}=0$. % + In each case, the two metrics are plotted directly against system inhomogeneity (top and middle + row) and against each other (bottom row). % + Colored lines guide the eyes for systems with equal relative dephasing rates + ($\Gamma_{10}\Delta_t$, see upper legend), while the area of the data point marker indicates + the relative inhomogeneity ($\Delta_{\text{inhom}}/\Gamma_{10}$, see lower legend). % + Gray lines indicate contours of constant relative inhomogeneity (scatter points with the same + area are connected). % +} + \label{fig:metrics} +\end{figure} +\clearpage} + +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}. % +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.} +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 +isolates the V and VI time-orderings. % +For this time delay, the $\Delta_\text{3PEPS} / \Delta_t$ metric works well for all dephasing times +of $\Gamma_{10}\Delta_t$ when the relative inhomogeneity is $\Delta_\text{inhom} / \Delta_\omega +\ll 1$. % +It becomes independent of $\Delta_\text{inhom} / \Delta_\omega$ when $\Delta_\text{inhom} / +\Delta_\omega > 1$. % +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} % +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}. % + +The middle row in Fig. \ref{fig:metrics} shows the ellipticity dependence on the relative dephasing +rate and inhomogeneity assuming the measurement is performed when the first two pulses are +temporally overlapped ($\tau_{22^\prime}=0$). % +For a $T=4\Delta_t$ population time, the ellipticity is proportional to the inhomogeneity until +$\Delta_\text{inhom} / \Delta_\omega \ll 1$ where the excitation bandwidth is wide compared with +the inhomogeneity. % +Unlike 3PEPS, saturation is not observed because pulse bandwidth does not limit the frequency range +scanned. % +The 3PEPS and ellipticity metrics are therefore complementary since 3PEPS works well for +$\Delta_\text{inhom} / \Delta_\omega \ll 1$ and ellipticity works well for $\Delta_\text{inhom} / +\Delta_\omega \gg 1$. % +When all pulses are temporally overlapped at $T = 0$, the ellipticity is only weakly dependent on +the inhomogeneity and dephasing rate. % +The ellipticity is instead dominated by the dependence on the excitation pulse frequency +differences of time-orderings I and III that become important at pulse overlap. % + +It is clear from the previous discussion that both metrics depend on the dephasing and +inhomogeneity. % +The dephasing can be measured independently in the frequency or time domain, depending upon whether +the dephasing is very fast or slow, respectively. % +In the mixed frequency/time domain, measurement of the dephasing becomes more difficult. % +One strategy to address this challenge is to use both the 3PEPS and ellipticity metrics. % +The bottom row in Fig. \ref{fig:metrics} plots 3PEPS against ellipticity to show how the +relationship between the metrics changes for different amounts of dephasing and inhomogeneity. % +The anti-diagonal contours of constant relative inhomogeneity show that these metrics are +complementary and can serve to extract the system correlation parameters. % + +Importantly, the metrics are uniquely mapped both in the presence and absence of pulse-induced +effects (demonstrated by $T = 0$ and $T = 4\Delta_t$, respectively). % +The combined metrics can be used to determine correlation at $T = 0$, but the correlation-inducing +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} % +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. % +If the metrics are measured as a function of $T$, the mapping gradually changes from the left +figure to the right figure in accordance with the pulse overlap. % +Both metrics will show a decrease, even with static inhomogeneity. % +If a system has spectral diffusion, the mapping at late times will disagree with the mapping at +early times; both ellipticity and 3PEPS will be smaller at later times than predicted by the change +in mappings alone. % + +\section{Conclusion} % --------------------------------------------------------------------------- + +This study provides a framework to describe and disentangle the influence of the excitation pulses +in mixed-domain ultrafast spectroscopy. % +We analyzed the features of mixed domain spectroscopy through detailed simulations of MR-CMDS +signals. % +When pulse durations are similar to coherence times, resolution is compromised by time-bandwidth +uncertainty and the complex mixture of driven and FID response. % +The dimensionless quantity $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)$ captures the +balance of driven and FID character in a single field-matter interaction. % +In the nonlinear experiment, with multiple field-matter interactions, this balance is also +controlled by pulse delays and frequency-resolved detection. % +Our analysis shows how these effects can be intuitive. % + +The dynamic nature of pulse effects can lead to misleading changes to spectra when delays are +changed. % +When delays separate pulses, the spectral line shapes of individual pathways qualitatively change +because the delays isolate FID contributions and de-emphasize driven response. % +When delays are scanned across pulse overlap, the weights of individual pathways change, further +changing the line shapes. % +In a real system, these changes would all be present in addition to actual dynamics and spectral +changes of the material. % + +Finally, we find that, in either frequency or time domain, pulse effects mimic signatures of +ultrafast inhomogeneity. % +Even homogeneous systems take on these signatures. % +For mixed domain experiments, pulse effects induce spectral ellipticity and photon echo signatures, +even in homogeneous systems. % +Driven character gives rise to pathway overlap peak shifting in the 2D delay response, which +artificially produces rephasing near pulse overlap. % +Driven character also produces resonances that depend on $\omega_1-\omega_2$ near pulse overlap. % +Determination of the homogeneous and inhomogeneous broadening at ultrashort times is only possible +by performing correlation analysis in both the frequency and time domain. % \ No newline at end of file diff --git a/mixed_domain/metrics.pdf b/mixed_domain/metrics.pdf new file mode 100644 index 0000000..a2640c3 Binary files /dev/null and b/mixed_domain/metrics.pdf differ diff --git a/mixed_domain/spectral evolution full.pdf b/mixed_domain/spectral evolution full.pdf new file mode 100644 index 0000000..f854286 Binary files /dev/null and b/mixed_domain/spectral evolution full.pdf differ diff --git a/mixed_domain/steady state.pdf b/mixed_domain/steady state.pdf new file mode 100644 index 0000000..effbb82 Binary files /dev/null and b/mixed_domain/steady state.pdf differ diff --git a/mixed_domain/wigners full.pdf b/mixed_domain/wigners full.pdf new file mode 100644 index 0000000..7816277 Binary files /dev/null and b/mixed_domain/wigners full.pdf differ diff --git a/mixed_domain/wigners.pdf b/mixed_domain/wigners.pdf new file mode 100644 index 0000000..903abdf Binary files /dev/null and b/mixed_domain/wigners.pdf differ -- cgit v1.2.3