From fa43cdc18d17b937d24abb6785c467364f14f2ad Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Mon, 29 Jan 2018 17:01:23 -0600 Subject: initial mixed domain --- mixed_domain/chapter.tex | 425 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 425 insertions(+) create mode 100644 mixed_domain/chapter.tex (limited to 'mixed_domain/chapter.tex') diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex new file mode 100644 index 0000000..4d3ba1c --- /dev/null +++ b/mixed_domain/chapter.tex @@ -0,0 +1,425 @@ +\chapter{Disentangling material and instrument response} + +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 +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. % +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 +inhomogeneity. % +We describe these line shapes using an intuitive picture that connects to familiar field-matter +expressions. % +We develop strategies for distinguishing pulse-induced correlations from true system +inhomogeneity. % +These simulations provide a foundation for interpretation of ultrafast experiments in the mixed +domain. % + +\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} % + +Time-domain methods scan the pulse delays to resolve the free induction decay +(FID).\cite{Gallagher1998} % +The Fourier Transform of the FID gives the ultrafast spectrum. % +Ideally, these experiments are performed in the impulsive limit where FID dominates the +measurement. % +FID occurs at the frequency of the transition that has been excited by a well-defined, time-ordered +sequence of pulses. % +Time-domain methods are compromised when the dynamics occur on faster time scales than the +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} % +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} % +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 +decay is negligible. % +The output signal is driven at the excitation pulse frequencies during the excitation pulse +width. % +Frequency-domain methods are compromised when the spectral line shape is narrower than the +frequency bandwidth of the excitation pulses. % +Dynamics that are slower than the pulse envelopes can be measured in the time domain by resolving +the phase oscillations of the output signal during the entire FID decay. % + +There is also the hybrid mixed-time/frequency-domain approach, where pulse delays and pulse +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} % +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} % +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. % + +The relative importance of FID and driven processes and the changing importance of different +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. % +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}. % + +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} % +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} % + +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} % +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} % +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. % +In the FID limit where the coherence frequencies are defined instead by the transition, the +diagonal character is lost. % +Understanding these effects is crucial for interpreting experiments, yet these effects have not +been characterized for MR-CMDS. % + +This work considers the third-order MR-CMDS response of a 3-level model system using three +ultrafast excitation beams with the commonly used four-wave mixing (FWM) phase-matching condition, +$\vec{k}_\text{out} = \vec{k}_1 - \vec{k}_2 + \vec{k}_{2'}$. % +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} % +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. % +We numerically simulate the MR-CMDS response with pulse durations at, above, and below the system +coherence time. % +To highlight the role of pulse effects, we build an interpretation of the full MR-CMDS response by +first showing how finite pulses affect the evolution of a coherence, and then how finite pulses +affect an isolated third-order pathway. % +When considering the full MR-CMDS response, we show that spectral features change dramatically as a +function of delay, even for a homogeneous system with elementary dynamics. % +Importantly, the line shape can exhibit correlations that mimic inhomogeneity, and the temporal +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} + +\begin{figure} + \centering + \includegraphics[width=0.5\linewidth]{"mixed_domain/WMELs"} + \caption{ + 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. + } + \label{fig:WMELs} +\end{figure} + +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 +the temporal and spectral widths of the excitation pulses. % +For simplicity, we will ignore population relaxation effects: $\Gamma_{11}=\Gamma_{00}=0$. % + +The electric field pulses, $\left\{E_l \right\}$, are given by: +\begin{equation}\label{eq:E_l} +E_l(t; \omega_l, \tau_l, \vec{k}_l \cdot z) = \frac{1}{2}\left[c_l(t-\tau_l)e^{i\vec{k}_l\cdot z}e^{-i\omega_l(t-\tau_l)} + c.c. \right], +\end{equation} +where $\omega_l$ is the field carrier frequency, $\vec{k}_l$ is the wavevector, $\tau_l$ is the +pulse delay, and $c_l$ is a slowly varying envelope. % +In this work, we assume normalized (real-valued) Gaussian envelopes: % +\begin{equation} +c_l(t) = \frac{1}{\Delta_t}\sqrt{\frac{2\ln 2}{2\pi}} \exp\left(-\ln 2 \left[\frac{t}{\Delta_t}\right]^2\right), +\end{equation} +where $\Delta_t$ is the temporal FWHM of the envelope intensity. % +We neglect non-linear phase effects such as chirp so the FWHM of the frequency bandwidth is +transform limited: $\Delta_{\omega}\Delta_t=4 \ln 2 \approx 2.77$, where $\Delta_{\omega}$ is the +spectral FWHM (intensity scale). % + + +The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$: +\begin{equation}\label{eq:LVN} +\frac{d\bm{\rho}}{dt} = -\frac{i}{\hbar}\left[\bm{H_0} + \bm{\mu}\cdot \sum_{l=1,2,2^\prime} E_l(t), \bm{{\rho}}\right] + \bm{\Gamma \rho}. +\end{equation} +Here $\bm{H_0}$ is the time-independent Hamiltonian, $\bm{\mu}$ is the dipole superoperator, and +$\bm{\Gamma}$ 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 +$\vec{k}_{\text{out}}=\vec{k}_1-\vec{k}_2+\vec{k}_{2^\prime}$. % +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}. % + +We first focus on a single interaction in these sequences, where an excitation pulse, $x$, forms +$\rho_{ij}$ from $\rho_{ik}$ or $\rho_{kj}$. % +For brevity, we use $\hbar=1$ and abbreviate the initial and final density matrix elements as +$\rho_i$ and $\rho_f$, respectively. % +Using the natural frequency rotating frame, $\tilde{\rho}_{ij}=\rho_{ij} e^{-i\omega_{ij}t}$, the formation of $\rho_f$ using pulse $x$ is written as +\begin{equation}\label{eq:rho_f} +\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)e^{i\kappa_f\left(\vec{k}_x\cdot z + \omega_x \tau_x \right)}e^{i\kappa_f\Omega_{fx}t}\tilde{\rho}_i(t), +\end{split} +\end{equation} +where $\Omega_{fx}=\kappa_f^{-1}\omega_f - \omega_x (=\left|\omega_f\right| - \omega_x)$ is the +detuning, $\omega_f$ is the transition frequency of the $i^{th}$ transition, $\mu_f$ is the +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 $E_{2^\prime}$, + $\kappa_f=-1$.} % +In the following equations we neglect spatial dependence ($z=0$). % + +Equation \ref{eq:rho_f} forms the basis for our simulations. % +It provides a general expression for arbitrary values of the dephasing rate and excitation pulse +bandwidth. % +The integral solution is +\begin{equation}\label{eq:rho_f_int} +\begin{split} +\tilde{\rho}_f(t) =& \frac{i}{2}\lambda_f \mu_f e^{i\kappa_f \omega_x \tau_x} e^{i\kappa_f \Omega_{fx} t} \\ +&\times \int_{-\infty}^{\infty} c_x(t-u-\tau_x)\tilde{\rho}_i(t-u)\Theta(u) \\ +& \qquad \quad \ \ \times e^{-\left(\Gamma_f+i\kappa_f\Omega_{fx}\right)u}du, +\end{split} +\end{equation} +where $\Theta$ is the Heaviside step function. % +Equation \ref{eq:rho_f_int} becomes the steady state limit expression when $\Delta_t \left|\Gamma_f + + i \kappa_f \Omega_{fx}\right| \gg 1$, and the impulsive limit expression results when $\Delta_t +\left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$. % +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}. % + +\begin{figure*} + \includegraphics[width=\linewidth]{"mixed_domain/simulation overview"} + \caption{ + 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 + $\Gamma_{10}\Delta_t=1$. + (b) Simulated evolution of the density matrix elements of a third-order Liouville pathway + $V\gamma$ under fully resonant excitation. Pulses can be labeled both by their time of arrival + ($x$,$y$,$z$) and by the lab lasers used to stimulate the transitions ($2$,$2^\prime$,$1$). The + final coherence (teal) creates the output electric field. + (c) The frequency profile of the output electric field is filtered by a monochromator gating + function, $M(\omega)$, and the passed components (shaded) are measured. + (d-f) Signal is viewed against two laser parameters, either as 2D delay (d), mixed + delay-frequency (e), or 2D frequency plots (f). The six time-orderings are labeled in (d) to + help introduce our delay convention. + } + \label{fig:overview} +\end{figure*} + +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 +transient for three different dephasing rates. % +The color bindings to dephasing rates introduced in Fig. \ref{fig:overview}a will be used +consistently throughout this work. % +Our simulations use systems with dephasing rates quantified relative to the pulse duration: +$\Gamma_{10} \Delta_t = 0.5, 1$, or $2$. % +The temporal axes are normalized to the pulse duration, $\Delta_t$. The $\Gamma_{10}\Delta_t=2$ +transient is mostly driven by the excitation pulse while $\Gamma_{10} \Delta_t = 0.5$ has a +substantial free induction decay (FID) component at late times. % +Fig. \ref{fig:overview}b shows a pulse sequence (pulses are shaded orange and pink) and the +resulting system evolution of pathway $V\gamma$ ($00 \xrightarrow{2} 01 \xrightarrow{2^\prime} 11 +\xrightarrow{1} 10 \xrightarrow{\text{out}} 00$) with $\Gamma_{10}\Delta_t=1$. % +The final polarization (teal) is responsible for the emitted signal, which is then passed through a +frequency bandpass filter to emulate monochromator detection (Fig. \ref{fig:overview}c). % +The resulting signal is explored in 2D delay space (Fig. \ref{fig:overview}d), 2D frequency space +(Fig. \ref{fig:overview}f), and hybrid delay-frequency space (Fig. \ref{fig:overview}e). % +The detuning frequency axes are also normalized by the pulse bandwidth, $\Delta_{\omega}$. % + +We now consider the generalized Liouville pathway $L:\rho_0 \xrightarrow{x} \rho_1 \xrightarrow{y} +\rho_2 \xrightarrow{z} \rho_3 \xrightarrow{\text{out}} \rho_4$, where $x$, $y$, and $z$ denote +properties of the first, second, and third pulse, respectively, and indices 0, 1, 2, 3, and 4 +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.} % + +The electric field emitted from a Liouville pathway is proportional to the polarization created by +the third-order coherence: % +\begin{equation}\label{eq:E_L} +E_L(t) = i \mu_{4}\rho_{3}(t). +\end{equation} +Equation \ref{eq:E_L} assumes perfect phase-matching and no pulse distortions through propagation. Equation \ref{eq:rho_f_int} shows that the output field for this Liouville pathway is + \begin{gather}\label{eq:E_L_full} + \begin{split} + E_L(t) =& \frac{i}{8}\lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 + e^{i\left( \kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z \right)} + e^{-i\left( \kappa_3 \omega_z + \kappa_2 \omega_y + \kappa_1 \omega_x \right) t} \\ + &\times \iiint_{-\infty}^{\infty} c_z(t-u-\tau_z) c_y(t-u-v-\tau_y) c_x(t-u-v-w-\tau_x) R_L(u,v,w) dw \ dv \ du , + \end{split}\\ + R_L(u,v,w) = \Theta(w)e^{-\left(\Gamma_1 + i\kappa_1\Omega_{1x} \right)w} + \Theta(v)e^{-\left(\Gamma_2 + i\left[ \kappa_1\Omega_{1x}+\kappa_2\Omega_{2y} \right] \right)v} + \Theta(u)e^{-\left(\Gamma_3 + i\left[ \kappa_1\Omega_{1x}+\kappa_2\Omega_{2y}+\kappa_3\Omega_{3z} \right] \right)u}, + \end{gather} +where $R_L$ is the third-order response function for the Liouville pathway. % +The total electric field will be the superposition of all the Liouville pathways: +\begin{equation}\label{eq:superposition} +E_{\text{tot}}= \sum_L E_L(t). +\end{equation} +For the superposition of Equation \ref{eq:superposition} to be non-canceling, certain symmetries +between the pathways must be broken. % +In general, this requires one or more of the following inequalities: $\Gamma_{10}\neq\Gamma_{21}$, +$\omega_{10}\neq\omega_{21}$, and/or $\sqrt{2}\mu_{10}\neq\mu_{21}$. % +Our simulations use the last inequality, which is important in two-level systems ($\mu_{21}=0$) and +in systems where state-filling dominates the non-linear response, such as in semiconductor +excitons. % +The exact ratio between $\mu_{10}$ and $\mu_{21}$ affects the absolute amplitude of the field, but +does not affect the multidimensional line shape. % +Importantly, the dipole inequality does not break the symmetry of double quantum coherence pathways +(time-orderings II and IV), so such pathways are not present in our analysis. % + +In MR-CMDS, a monochromator resolves the driven output frequency from other nonlinear output +frequencies, which in our case is $\omega_m = \omega_1 - \omega_2 + \omega_{2'} = \omega_1$. % +The monochromator can also enhance spectral resolution, as we show in Section +\ref{sec:evolution_SQC}. % +In this simulation, the detection is emulated by transforming $E_{\text{tot}}(t)$ into the +frequency domain, applying a narrow bandpass filter, $M(\omega)$, about $\omega_1$, and applying +amplitude-scaled detection: +\begin{equation}\label{eq:S_tot} +S_{\text{tot}}(\omega_1, \omega_2, \tau_{21}, \tau_{22^\prime}) = +\sqrt{ \int\left| M(\omega-\omega_1) E_{\text{tot}}(\omega) \right|^2 d\omega}, +\end{equation} +where $E_{\text{tot}}(\omega)$ denotes the Fourier transform of $E_{\text{tot}}(t)$ (see Fig. +\ref{fig:overview}c). % +For $M$ we used a rectangular function of width $0.408\Delta_{\omega}$. % +The arguments of $S_{\text{tot}}$ refer to the \textit{experimental} degrees of freedom. % +The signal delay dependence is parameterized with the relative delays $\tau_{21}$ and +$\tau_{22^\prime}$, where $\tau_{nm} = \tau_n - \tau_m$ (see Fig. \ref{fig:overview}b). % +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} + +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}. % +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}. % +Dynamic broadening effects such as spectral diffusion are beyond the scope of this work. % + +\section{Methods} % ------------------------------------------------------------------------------ + +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} % +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 +system-pulse difference frequencies and the pulse envelope were well-sampled. % + +The following simulations explore the four-dimensional $(\omega_1, \omega_2, \tau_{21}, +\tau_{22^\prime})$ variable space. % +Both frequencies are scanned about the resonance, and both delays are scanned about pulse overlap. +We explored the role of sample dephasing rate by calculating signal for systems with dephasing +rates such that $\Gamma_{10}\Delta_t=0.5, 1,$ and $2$. % +Inhomogeneous broadening used a spectral FWHM, $\Delta_{\text{inhom}}$, that satisfied +$\Delta_\text{inhom}/ \Delta_{\omega}=0,0.5,1,$ and $2$ for the three dephasing rates. % +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} % + +\section{Results} % ------------------------------------------------------------------------------ + +We now present portions of our simulated data that highlight the dependence of the spectral line +shapes and transients on excitation pulse width, the dephasing rate, detuning from resonance, the +pulse delay times, and inhomogeneous broadening. % + +\subsection{Evolution of single coherence}\label{sec:evolution_SQC} + +\begin{figure} + \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"} + \caption{ + 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 + instantaneous frequency (see colorbar). + For all cases, the pulsed excitation field (gray line, shown as electric field amplitude) is + slightly detuned (relative detuning, $\Omega_{fx}/\Delta_{\omega}=0.1$). + } + \label{fig:fid_dpr} +\end{figure} + +It is illustrative to first consider the evolution of single coherences, $\rho_0 \xrightarrow{x} +\rho_1$, under various excitation conditions. % +Fig. \ref{fig:fid_dpr} shows the temporal evolution of $\rho_1$ with various dephasing rates under +Gaussian excitation. % +The value of $\rho_1$ differs only by phase factors between various Liouville pathways (this can be +verified by inspection of Equation \ref{eq:rho_f_int} under the various conditions in Table S1), so +the profiles in Fig. \ref{fig:fid_dpr} apply for the first interaction of any pathway. % +The pulse frequency was detuned from resonance so that frequency changes could be visualized by the +color bar, but the detuning was kept slight so that it did not appreciably change the dimensionless +product, $\Delta_t \left(\Gamma_f + i\kappa_f \Omega_{fx}\right)\approx \Gamma_{10}\Delta_t$. % +In this case, the evolution demonstrates the maximum impulsive character the transient can +achieve. % +The instantaneous frequency, $d\varphi/dt$, is defined as +\begin{equation} +\frac{d\varphi}{dt} = \frac{d}{dt} \tan^{-1}\left( \frac{\text{Im}\left(\rho_1(t)\right)}{\text{Re}\left(\rho_1(t)\right)} \right). +\end{equation} +The cases of $\Gamma_{10}\Delta_t=0 (\infty)$ agree with the impulsive (driven) expressions derived +in Appendix \ref{sec:cw_imp}. % +For $\Gamma_{10}\Delta_t=0$, the signal rises as the integral of the pulse and has instantaneous +frequency close to that of the pulse (Equation \ref{eq:sqc_rise}), but as the pulse vanishes, the +signal adopts the natural system frequency and decay rate (Equation \ref{eq:sqc_fid}). % +For $\Gamma_{10}\Delta_t=\infty$, the signal follows the amplitude and frequency of the pulse for +all times (the driven limit, Equation \ref{eq:sqc_driven}). % + +The other three cases show a smooth interpolation between limits. % +As $\Gamma_{10}\Delta_t$ increases from the impulsive limit, the coherence within the pulse region +conforms less to a pulse integral profile and more to a pulse envelope profile. % +In accordance, the FID component after the pulse becomes less prominent, and the instantaneous +frequency pins to the driving frequency more strongly through the course of evolution. % +The trends can be understood by considering the differential form of evolution (Equation +\ref{eq:rho_f}), and the time-dependent balance of optical coupling and system relaxation. % +We note that our choices of $\Gamma_{10}\Delta_t=2.0, 1.0,$ and $0.5$ give coherences that have +mainly driven, roughly equal driven and FID parts, and mainly FID components, respectively. % +FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. % + -- cgit v1.2.3