From 96d1fba00c2ae499056a81d430141d500d7695d8 Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Mon, 14 May 2018 10:31:46 -0500 Subject: 2018-05-14 10:31 --- mixed_domain/chapter.tex | 795 +++++++++++++++++++++++++---------------------- 1 file changed, 432 insertions(+), 363 deletions(-) (limited to 'mixed_domain') diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 509faee..4c4445e 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -467,7 +467,6 @@ line-narrowing \cite{BesemannDanielM2004a, OudarJL1980a, CarlsonRogerJohn1990a, AgarwalRitesh2002a}. % We simulate the effects of static inhomogeneous broadening by convolving the homogeneous response with a Gaussian distribution function. % -Further details of the convolution are in \autoref{mix:sec:convolution}. % Dynamic broadening effects such as spectral diffusion are beyond the scope of this work. % Here we describe how to transform the data of a single reference oscillator signal to that of an @@ -588,113 +587,115 @@ There are three details of our simulation strategy that deserve more exposition: \subsection{Liouville pathway parameters} % ------------------------------------------------------ -Table \ref{tab:table1} describes the relationship between our notation and the parameters that make +Table \ref{mix:tab:table1} describes the relationship between our notation and the parameters that make up the 16 Liouville pathways. % -\begin{table} - \begin{tabular}{l c | c c c r r r r r r c c c c} - $L$ & Liouville Pathway - & $x$ & $y$ & $z$ - & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ - & $\kappa_1$ & $\kappa_2$ & $\kappa_3$ - & $\mu_1$ & $\mu_2$ & $\mu_3$ & $\mu_4$ \\ - \hline - I$\alpha$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$ - & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} - & 1 & -1 & -1 - & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1} - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ - I$\beta$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 11$ - & & & - & 1 & -1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ - I$\gamma$ & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$ - & & & - & 1 & 1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ - \hline - II$\alpha$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$ - & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$2$} - & 1 & 1 & 1 - & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1} - & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\ - II$\beta$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$ - & & & - & 1 & 1 & -1 - & & & - & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\ - \hline - III$\alpha$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$ - & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2^\prime$} - & -1 & 1 & -1 - & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1} - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ - III$\beta$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$ - & & & - & -1 & 1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ - III$\gamma$ & $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$ - & & & - & -1 & -1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ - \hline - IV$\alpha$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$ - & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2$} - & 1 & 1 & 1 - & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1} - & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\ - IV$\beta$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$ - & & & - & 1 & 1 & -1 - & & & - & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\ - \hline - V$\alpha$ & $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$ - & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$1$} - & -1 & -1 & 1 - & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1} - & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ - V$\beta$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$ - & & & - & -1 & 1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ - V$\gamma$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$ - & & & - & -1 & 1 & -1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ - \hline - VI$\alpha$ & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$ - & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} - & 1 & 1 & 1 - & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1} - & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ - VI$\beta$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 00$ - & & & - & 1 & -1 & 1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ - VI$\gamma$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$ - & & & - & 1 & -1 & -1 - & & & - & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ - \end{tabular} - \caption{Parameters of each Liouville Pathway.} - \label{tab:table1} -\end{table} +\begin{landscape} + \begin{table} + \begin{tabular}{l c | c c c r r r r r r c c c c} + $L$ & Liouville Pathway + & $x$ & $y$ & $z$ + & $\lambda_1$ & $\lambda_2$ & $\lambda_3$ + & $\kappa_1$ & $\kappa_2$ & $\kappa_3$ + & $\mu_1$ & $\mu_2$ & $\mu_3$ & $\mu_4$ \\ + \hline + I$\alpha$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$ + & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} + & 1 & -1 & -1 + & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1} + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ + I$\beta$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 11$ + & & & + & 1 & -1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ + I$\gamma$ & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$ + & & & + & 1 & 1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ + \hline + II$\alpha$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$ + & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$2$} + & 1 & 1 & 1 + & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1} + & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\ + II$\beta$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$ + & & & + & 1 & 1 & -1 + & & & + & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\ + \hline + III$\alpha$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$ + & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} & \multirow{3}{*}{$2^\prime$} + & -1 & 1 & -1 + & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1} + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ + III$\beta$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$ + & & & + & -1 & 1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ + III$\gamma$ & $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$ + & & & + & -1 & -1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ + \hline + IV$\alpha$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 10 \rightarrow 00$ + & \multirow{2}{*}{$2^\prime$} & \multirow{2}{*}{$1$} & \multirow{2}{*}{$2$} + & 1 & 1 & 1 + & \multirow{2}{*}{-1} & \multirow{2}{*}{-1} & \multirow{2}{*}{1} + & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ & $\mu_{10}^*$ \\ + IV$\beta$ & $00 \rightarrow 10 \rightarrow 20 \rightarrow 21 \rightarrow 11$ + & & & + & 1 & 1 & -1 + & & & + & $\mu_{10}$ & $\mu_{21}$ & $\mu_{10}$ & $\mu_{21}^*$ \\ + \hline + V$\alpha$ & $00 \rightarrow 01 \rightarrow 00 \rightarrow 10 \rightarrow 00$ + & \multirow{3}{*}{$2$} & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$1$} + & -1 & -1 & 1 + & \multirow{3}{*}{1} & \multirow{3}{*}{-1} & \multirow{3}{*}{-1} + & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ + V$\beta$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 21 \rightarrow 11$ + & & & + & -1 & 1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ + V$\gamma$ & $00 \rightarrow 01 \rightarrow 11 \rightarrow 10 \rightarrow 00$ + & & & + & -1 & 1 & -1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ + \hline + VI$\alpha$ & $00 \rightarrow 10 \rightarrow 00 \rightarrow 10 \rightarrow 00$ + & \multirow{3}{*}{$2^\prime$} & \multirow{3}{*}{$2$} & \multirow{3}{*}{$1$} + & 1 & 1 & 1 + & \multirow{3}{*}{-1} & \multirow{3}{*}{1} & \multirow{3}{*}{-1} + & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}$ & $\mu_{10}^*$ \\ + VI$\beta$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 21 \rightarrow 00$ + & & & + & 1 & -1 & 1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{21}$ & $\mu_{21}^*$ \\ + VI$\gamma$ & $00 \rightarrow 10 \rightarrow 11 \rightarrow 10 \rightarrow 00$ + & & & + & 1 & -1 & -1 + & & & + & $\mu_{10}$ & $\mu_{10}$ & $\mu_{10}^*$ & $\mu_{10}^*$ \\ + \end{tabular} + \caption{Parameters of each Liouville Pathway.} + \label{mix:tab:table1} + \end{table} +\end{landscape} \subsection{Matrix formulation} % ---------------------------------------------------------------- \subsubsection{Generic single pathway} In this work we explicitly incorporate our time-dependent electric fields into the Liouville -equations as done by \textcite{GelinMaximF2009.000}. % +equations as done by \textcite{GelinMaximF2009a}. % Our third order experiment involves three successive light-matter interactions. In a generic Liouville pathway $\rho_0 \overset{x}\rightarrow \rho_1 \overset{y}\rightarrow \rho_2 @@ -723,21 +724,22 @@ Note that we do not include depletion due to light-matter interaction. Our three coupled differential equations can be cast into a matrix form: \begin{eqnarray} -\frac{\mathrm{d}\overline{\rho}}{\mathrm{d}t} &=& \overline{\overline{Q}}(t)\overline{\rho} \label{eq:generic_diff} \\ -\overline{\rho} &\equiv& -\begin{bmatrix} -\tilde{\rho}_0 \\ -\tilde{\rho}_1 \\ -\tilde{\rho}_2 \\ -\tilde{\rho}_3 -\end{bmatrix} \\ -\overline{\overline{Q}}(t) &\equiv& -\begin{bmatrix} --\Gamma_0 & 0 & 0 & 0 \\ -S_1(t) & -\Gamma_1 & 0 & 0 \\ -0 & S_2(t) & -\Gamma_2 & 0 \\ -0 & 0 & S_3(t) & -\Gamma_3 -\end{bmatrix} + \frac{\mathrm{d}\overline{\rho}}{\mathrm{d}t} &=& + \overline{\overline{Q}}(t)\overline{\rho} \label{mix:eqn:generic_diff} \\ + \overline{\rho} &\equiv& + \begin{bmatrix} + \tilde{\rho}_0 \\ + \tilde{\rho}_1 \\ + \tilde{\rho}_2 \\ + \tilde{\rho}_3 + \end{bmatrix} \\ + \overline{\overline{Q}}(t) &\equiv& + \begin{bmatrix} + -\Gamma_0 & 0 & 0 & 0 \\ + S_1(t) & -\Gamma_1 & 0 & 0 \\ + 0 & S_2(t) & -\Gamma_2 & 0 \\ + 0 & 0 & S_3(t) & -\Gamma_3 + \end{bmatrix} \end{eqnarray} \autoref{mix:eqn:generic_diff} could be numerically integrated to determine $\tilde{\rho}_3$ for @@ -745,8 +747,8 @@ this pathway. % \subsubsection{Full formulation} % ---------------------------------------------------------------- -As discussed throughout this chapter and shown in \textbf{TODO}, there are 6 unique time-orderings -and 16 unique pathways to consider in this work. % +As discussed throughout this chapter and shown in \autoref{mix:fig:WMELs}, there are 6 unique +time-orderings and 16 unique pathways to consider in this work. % One could write a separate differential equation for each pathway---this would require tabulation of 48 density matrix elements. % Instead, we capitalize on the symmetry of our pathways to create a single differential equation @@ -764,8 +766,8 @@ that is computationally cheaper. % } \end{figure} -\autoref{fig:matrix_flow_diagram} diagrams our 16 Liouville pathways as a network of interconnected -steps. % +\autoref{mix:fig:matrix_flow_diagram} diagrams our 16 Liouville pathways as a network of +interconnected steps. % Some density matrix elements, such as $\tilde{\rho}_{10}$, appear multiple times because there are several distinguishable pathways that involve that element. % We do not include $\tilde{\rho}_{00}^{(1-2)}$ and $\tilde{\rho}_{00}^{(2^\prime-2)}$ because they @@ -773,7 +775,8 @@ have exactly the same amplitude as $\tilde{\rho}_{11}^{(1-2)}$ and $\tilde{\rho}_{11}^{(2^\prime-2)}$ within our simulation. Pathways where $\rho_3$ is fed by $\tilde{\rho}_{00}$ are sign-flipped to account for this. % -First we define a state vector containing all nine elements in \autoref{fig:matrix_flow_diagram}: +First we define a state vector containing all nine elements in +\autoref{mix:fig:matrix_flow_diagram}: \begin{equation} \overline{\rho} \equiv @@ -819,7 +822,9 @@ Each off-diagonal element in $\overline{\overline{Q}}$ has two influences determ \label{eq:single_Q} \end{equation} -Note that this approach implicitly enforces our phase matching conditions and pathways. To simulate single time-orderings we simply remove elements from \ref{eq:single_Q}. For example, to isolate time ordering \RomanNumeral{1}: +Note that this approach implicitly enforces our phase matching conditions and pathways. +To simulate single time-orderings we simply remove elements from \ref{eq:single_Q}. +For example, to isolate time ordering \RomanNumeral{1}: \begin{equation} \overline{\overline{Q}}_\RomanNumeral{1} \equiv @@ -841,7 +846,7 @@ These equations are somewhat general to four wave mixing of systems like the one work. % We have not taken all the simplifications that are possible in our specific system, such as $\omega_{10} = \omega_{21}$ and $\Gamma_{11} = 0$. % -See \autoref{section:scripts} for more details about our simulation, including where to find the +See \autoref{mix:sec:scripts} for more details about our simulation, including where to find the scripts and raw output behind this work. % \subsection{Heun method} @@ -982,9 +987,15 @@ Our software packages are constantly being developed. The versions kept in this The raw simulation in this work was generated using NISE and the \texttt{NISE\_iterator.py} script found in the raw folder. -In NISE, an \texttt{experiment} module is loaded to define the electric field variables and the experimental conditions, like the phase matching. In this case we used the \texttt{trive} module, which defines our normal, two-color four wave mixing experimental conditions. +In NISE, an \texttt{experiment} module is loaded to define the electric field variables and the +experimental conditions, like the phase matching. % +In this case we used the \texttt{trive} module, which defines our normal, two-color four wave +mixing experimental conditions. % -Within this work we have represented our data in terms of dimensionless quantities like $\tau/\Delta_t$ and $(\omega-\omega_{10})/\Delta_\omega$. The simulation within NISE was done with choices for each parameter, as tabulated below. These quantities are necessary to fully understand the unprocessed arrays generated by NISE. +Within this work we have represented our data in terms of dimensionless quantities like +$\tau/\Delta_t$ and $(\omega-\omega_{10})/\Delta_\omega$. % +The simulation within NISE was done with choices for each parameter, as tabulated below. % +These quantities are necessary to fully understand the unprocessed arrays generated by NISE. % \begin{table} \begin{tabular}{r l} @@ -994,13 +1005,49 @@ Within this work we have represented our data in terms of dimensionless quantiti $\Gamma_{11}$ & 0 $\mathrm{fs}^{-1}$ \\ $\Delta_t$ & 50 fs \end{tabular} + \caption{ + Parameters used in large NISE simulation. + } \end{table} -$\Gamma_{10}$, $\Gamma_{21}$ and $\Gamma_{20}$ were kept equal. Their exact value for a given run of the simulation depends on the $\Gamma_{10}\Delta_t$ quantity as discussed in the paper. We use the term \texttt{dpr} (dephasing pulse ratio) which is the inverse of $\Gamma_{10}\Delta_t$. +$\Gamma_{10}$, $\Gamma_{21}$ and $\Gamma_{20}$ were kept equal. +Their exact value for a given run of the simulation depends on the $\Gamma_{10}\Delta_t$ quantity +as discussed in the paper. +We use the term \texttt{dpr} (dephasing pulse ratio) which is the inverse of $\Gamma_{10}\Delta_t$. + +In NISE the system parameters are contained within the \texttt{hamiltonian} module, in this case +\texttt{H0}. +The \texttt{Omega} object contains all of the system attributes you would expect, including the +$\overline{\rho}(t)$ (\texttt{dm\_vector}) and $\overline{\overline{Q}}(t)$ +(\texttt{gen\_matrix()}) matrices as in \autoref{mix:eqn:generic_diff}. +The matrix in the software does not account for relaxation and dephasing, that is accounted for +directly during the integration. +Within \texttt{H0} we actually define two $\overline{\overline{Q}}$ `permutations', one for +pathways in which $E_1$ arrives first (\texttt{w1\_first == True}) and one for pathways in which +$E_{2^\prime}$ arrives first (\texttt{w1\_first == False}). +This separate permutation approach is mathematically identical to the single matrix approach in +\autoref{eq:single_Q}, just slightly more computationally expensive. +\texttt{H0.Omega} allows you to define which time-orderings are included using the \texttt{TOs} +keyword argument. +This directly affects how the propagator is assembled, as discussed in \autoref{eq:single_Q_pw1}. + +To generate the raw data we calculated the polarization at all coordinates within a +four-dimensional experimental array: % + +Arrays containing these points were assembled and passed into the \texttt{trive} module. +These axes and \texttt{H0} were passed into the \texttt{scan} module for numerical integration. +A single output array was saved for each scan. +To keep the output array sizes reasonable a separate simulation was done for each +$\Gamma_{10}\Delta_t$, time-ordering, and \texttt{d1} value. +For each of these simulations we saved one five-dimensional array to an HDF5 file: + +The final index `timestep' contains the dependence of the output polarization on lab time. +It changes from simulation to simulation to help with computation speed. +For each simulation the timetep array is defined by a starting position (always 100 fs before the +first pulse arrives) an ending position ($5 \times \tau_{10}$ fs after the last pulse arrives), and +a step (4 fs for our longest dephasing time, 2 fs otherwise). -In NISE the system parameters are contained within the \texttt{hamiltonian} module, in this case \texttt{H0}. The \texttt{Omega} object contains all of the system attributes you would expect, including the $\overline{\rho}(t)$ (\texttt{dm\_vector}) and $\overline{\overline{Q}}(t)$ (\texttt{gen\_matrix()}) matrices as in \autoref{eq:generic_diff}. The matrix in the software does not account for relaxation and dephasing, that is accounted for directly during the integration. Within \texttt{H0} we actually define two $\overline{\overline{Q}}$ `permutations', one for pathways in which $E_1$ arrives first (\texttt{w1\_first == True}) and one for pathways in which $E_{2^\prime}$ arrives first (\texttt{w1\_first == False}). This separate permutation approach is mathematically identical to the single matrix approach in \autoref{eq:single_Q}, just slightly more computationally expensive. \texttt{H0.Omega} allows you to define which time-orderings are included using the \texttt{TOs} keyword argument. This directly affects how the propagator is assembled, as discussed in \autoref{eq:single_Q_pw1}. - -To generate the raw data we calculated the polarization at all coordinates within a four-dimensional experimental array: +The output polarization is kept as a complex array in the lab frame. \begin{table} \begin{tabular}{c | c | c | c} @@ -1010,10 +1057,11 @@ To generate the raw data we calculated the polarization at all coordinates withi d1 & 0 & 400 & 21 \\ d2 & 0 & 400 & 21 \end{tabular} + \caption{ + Description of four-dimensional simulation coordinates. + } \end{table} -Arrays containing these points were assembled and passed into the \texttt{trive} module. These axes and \texttt{H0} were passed into the \texttt{scan} module for numerical integration. A single output array was saved for each scan. To keep the output array sizes reasonable a separate simulation was done for each $\Gamma_{10}\Delta_t$, time-ordering, and \texttt{d1} value. For each of these simulations we saved one five-dimensional array to an HDF5 file: - \begin{table} \begin{tabular}{c | c | c} index & name & size \\ \hline @@ -1023,15 +1071,33 @@ Arrays containing these points were assembled and passed into the \texttt{trive} 3 & permutation & 2 \\ 4 & timestep & variable \\ \end{tabular} + \caption{ + Simulation output format (polarization). + } \end{table} -The final index `timestep' contains the dependence of the output polarization on lab time. It changes from simulation to simulation to help with computation speed. For each simulation the timetep array is defined by a starting position (always 100 fs before the first pulse arrives) an ending position ($5 \times \tau_{10}$ fs after the last pulse arrives), and a step (4 fs for our longest dephasing time, 2 fs otherwise). +\subsubsection{Measured} -The output polarization is kept as a complex array in the lab frame. +As mentioned in the appendix, we introduce inhomogeneity by convolving the output with a +distribution function on the intensity level: `smearing'. +This is done in the measurement stage. +We store a separate HDF5 file for each combination of \texttt{dpr}, time ordering, and +$\Delta_{\text{inhom}}$. +Each HDF5 file contains four arrays, shown in \autoref{mix:tab:measured} -\subsubsection{Measured} +The coordinate arrays are in their native units (fs, $\mathrm{cm}^{-1}$). +The signal array is purely real, stored on the intensity level. + +We measured the entire simulation space twice, one with and one without the monochromator bandpass +filter. % + +\subsubsection{Representations} -As mentioned in the appendix, we introduce inhomogeneity by convolving the output with a distribution function on the intensity level: `smearing'. This is done in the measurement stage. We store a separate HDF5 file for each combination of \texttt{dpr}, time ordering, and $\Delta_{\text{inhom}}$. Each HDF5 file contains four arrays: +We present many representations of our simulated dataset in this work. +The script used to create these figures is \texttt{figures.py}, found in this supplementary +information. +In some cases some small additional simulation or some pre-processing step was necessary, these can +be found in the `precalculated' folder. \begin{table} \begin{tabular}{c | c | c} @@ -1042,16 +1108,12 @@ As mentioned in the appendix, we introduce inhomogeneity by convolving the outpu \texttt{d2} & w2 & \texttt{(21,)} \\ \texttt{arr} & w1, w2, d1, d2 & \texttt{(41, 41, 21, 21)} \end{tabular} + \caption{ + Simulation output format (measured). + } + \label{mix:tab:measured} \end{table} -The coordinate arrays are in their native units (fs, $\mathrm{cm}^{-1}$). The signal array is purely real, stored on the intensity level. - -We measured the entire simulation space twice, one with and one without the monochromator bandpass filter. - -\subsubsection{Representations} - -We present many representations of our simulated dataset in this work. The script used to create these figures is \texttt{figures.py}, found in this supplementary information. In some cases some small additional simulation or some pre-processing step was necessary, these can be found in the `precalculated' folder. - \section{Results} % ============================================================================== We now present portions of our simulated data that highlight the dependence of the spectral line @@ -1060,19 +1122,6 @@ pulse delay times, and inhomogeneous broadening. % \subsection{Evolution of single coherence} \label{mix:sec:evolution_SQC} % ----------------------- -\begin{figure} - \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"} - \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 - 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{mix: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. % \autoref{mix:fig:fid_dpr} shows the temporal evolution of $\rho_1$ with various dephasing rates under @@ -1108,56 +1157,11 @@ 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 $\Gamma_{10}\Delta_t=2.0$. % -\begin{figure} - \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{mix:fig:fid_detuning} -\end{figure} - -\begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/fid vs detuning with freq"} - \label{mix:fig:fid_vs_detuning_with_freq} - \caption[Frequency domain representation of a single quantum coherence vs pulse detuning.]{ - Numerical simulation of a single quantum coherence under pulsed excitation - ($\Gamma_{10}\Delta_t=1$) at different detunings (labelled inset). - The coherence is shown in both the time (left column) and frequency (right column) domain. - The color indicates the frequency (scale on right), while the black line shows the amplitude - profile. The excitation profile is shown as a grey line. - } -\end{figure} - -\begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"} - \caption[Time-gated amplitude of a single quantum coherence vs pulse detuning.]{ - Amplitude of a single quantum coherence under pulsed excitation as a function of detuning (x - axis) and delay after excitation (line color, scale on right) for the three - $\Gamma_{10}\Delta_t$ values considered in this work. - For comparison, the excitation pulse lineshape (grey), absorptive material response (black, - narrow) and magnitude material response (black, wide) are also shown. Each peak is normalized - to its own maximum amplitude. - } - \label{mix:fig:sqc_vs_t} -\end{figure} \autoref{mix:fig:fid_detuning}a shows the temporal evolution of $\rho_1$ at several values of $\Omega_{1x}/\Delta_{\omega}$ with $\Gamma_{10}\Delta_t=1$. -Figure \textbf{TODO} shows the Fourier domain representation of \autoref{mix:fig:fid_detuning}a. % +\autoref{mix:fig:fid_vs_detuning_with_freq} shows the Fourier domain representation of +\autoref{mix: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 @@ -1172,8 +1176,8 @@ This spectral narrowing can be seen in \autoref{mix:fig:fid_detuning}a by compar 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 -(\textbf{TODO} shows explicit plots of $\rho_1(\Gamma_{fx}/\Delta_\omega$ at discrete $t/\Delta_t$ -values). % +(\autoref{mix:fig:sqc_vs_t} shows explicit plots of $\rho_1(\Gamma_{fx}/\Delta_\omega$ 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{LagutchevAlexi2007a, LagutchevAlexi2010a, DonaldsonPaulMurray2010a, DonaldsonPaulMurray2008a} % @@ -1217,12 +1221,14 @@ isolating the driven frequency (tracking the monochromator with the excitation s The functional form of the measured line shape can be deduced by considering the frequency domain form of \autoref{mix:eqn:rho_f_int} (assume $\rho_i=1$ and $\tau_x=0$): \begin{equation} \label{mix:eqn: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}, + \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}}. + \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 \autoref{mix:eqn:rho_f_int_freq}: @@ -1241,34 +1247,68 @@ $\Omega_{fx}/\Delta_{\omega}$ values and the monochromator is not useful. % \autoref{mix: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} % --------------------------------------------- +\begin{figure} + \includegraphics[width=0.5\linewidth]{"mixed_domain/fid vs dpr"} + \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 + 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{mix:fig:fid_dpr} +\end{figure} \begin{figure} - \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{mix:fig:pw1} + \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{mix:fig:fid_detuning} \end{figure} \begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/pw1 lineshapes no mono"} - \caption[2D frequency response of a single Liouville pathway without a tracking monochromator.]{ - Pathway \RomanNumeral{1}$\gamma$ temporal response in the 2D pulse delay space at triple - resonance (left) and the corresponding 2D frequency plots 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. - Unlike elsewhere in this work, signal here was not filtered by a tracking monochromator. + \includegraphics[width=\textwidth]{"mixed_domain/fid vs detuning with freq"} + \label{mix:fig:fid_vs_detuning_with_freq} + \caption[Frequency domain representation of a single quantum coherence vs pulse detuning.]{ + Numerical simulation of a single quantum coherence under pulsed excitation + ($\Gamma_{10}\Delta_t=1$) at different detunings (labelled inset). + The coherence is shown in both the time (left column) and frequency (right column) domain. + The color indicates the frequency (scale on right), while the black line shows the amplitude + profile. The excitation profile is shown as a grey line. } - \label{mix:fig:pw1_no_mono} \end{figure} +\begin{figure} + \includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"} + \caption[Time-gated amplitude of a single quantum coherence vs pulse detuning.]{ + Amplitude of a single quantum coherence under pulsed excitation as a function of detuning (x + axis) and delay after excitation (line color, scale on right) for the three + $\Gamma_{10}\Delta_t$ values considered in this work. + For comparison, the excitation pulse lineshape (grey), absorptive material response (black, + narrow) and magnitude material response (black, wide) are also shown. Each peak is normalized + to its own maximum amplitude. + } + \label{mix:fig:sqc_vs_t} +\end{figure} + +\subsection{Evolution of single Liouville pathway} % --------------------------------------------- + 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 @@ -1303,27 +1343,8 @@ The prominence of FID signal can change the resonance conditions; \autoref{mix:t 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. % -\textbf{TODO} shows a simulation of \autoref{mix:fig:pw1} without monochromator frequency -filtering. % - -\begin{table} - \caption{\label{mix:tab:table2} Conditions for peak intensity at different pulse delays for pathway - I$\gamma$.} - \begin{tabular}{c c | c c c c} - \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{tabular} -\end{table} +\autoref{mix:fig:pw1_no_mono} shows a simulation of \autoref{mix:fig:pw1} without monochromator +frequency filtering. % 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. % @@ -1336,7 +1357,6 @@ $\omega_2=\omega_{2^\prime}$, the first and third resonance conditions are ident making $\omega_1$ doubly resonant at $\omega_{10}$ and resulting in the vertical elongation along $\omega_1=\omega_{10}$. % - The other three spectra of \autoref{mix: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 @@ -1395,25 +1415,55 @@ show that time-gating itself causes significant line shape changes to the isolat The phenomenon of time-gating can cause frequency and delay axes to become functional of each other in unexpected ways. % -\subsection{Temporal pathway discrimination} % --------------------------------------------------- +\begin{figure} + \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{mix:fig:pw1} +\end{figure} \begin{figure} - \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 \autoref{mix:eqn: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{mix:fig:delay_purity} + \includegraphics[width=\textwidth]{"mixed_domain/pw1 lineshapes no mono"} + \caption[2D frequency response of a single Liouville pathway without a tracking monochromator.]{ + Pathway \RomanNumeral{1}$\gamma$ temporal response in the 2D pulse delay space at triple + resonance (left) and the corresponding 2D frequency plots 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. + Unlike elsewhere in this work, signal here was not filtered by a tracking monochromator. + } + \label{mix:fig:pw1_no_mono} \end{figure} +\begin{table} + \begin{tabular}{c c | c c c c} + \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{tabular} + \caption{ + Conditions for peak intensity at different pulse delays for pathway I $\gamma$. + } + \label{mix:tab:table2} +\end{table} + +\subsection{Temporal pathway discrimination} % --------------------------------------------------- + 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 @@ -1460,34 +1510,24 @@ 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} - \begin{figure} - \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 - $\tau_{22^\prime}=0$. + \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 \autoref{mix:eqn: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{mix:fig:hom_2d_spectra} + \label{mix:fig:delay_purity} \end{figure} -\begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/spectral evolution full"} - \caption[Evolution of the 2D frequency response, with all contours shown.]{ - Spectral evolution of the homogeneous exciton resonance as a function of $\tau_{21}$, with - $\tau_{22^\prime}=0$. - The 50\% contour is darkened to ease comparison with Figure 7. - } - \label{mix:fig:spectral_evolution_full} -\end{figure} +\subsection{Multidimensional line shape dependence on pulse delay time} % ------------------------ 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 @@ -1562,16 +1602,6 @@ 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 $\Gamma_{10}\Delta_t=0.5$. % -\begin{figure} - \includegraphics[width=\linewidth]{"mixed_domain/wigners full"} - \caption[Simulated Wigner spectra.]{ - Mixed $\tau_{21}$, $\omega_1$ plots for each $\Gamma_{10}$ value simulated in this work. - For each plot, the corresponding $\omega_2$ value is shown as a gray vertical line. - Each plot is separately normalized. - } - \label{mix:fig:wigners} -\end{figure} - It is also common to represent data as ``Wigner plots,'' where one axis is delay and the other is frequency. \cite{KohlerDanielDavid2014a, AubockGerald2012a, CzechKyleJonathan2015a, PakoulevAndreiV2007a} % @@ -1587,48 +1617,45 @@ This representation also highlights the asymmetric broadening of the $\omega_1$ pulse overlap when $\omega_2$ becomes non-resonant. % Again, these features can resemble spectral diffusion even though our system is homogeneous. % -\subsection{Inhomogeneous broadening} \label{mix:sec:res_inhom} % -------------------------------- - \begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/3PEPS"} - \label{mix:fig:3PEPS} - \caption[3PEPS tutorial.]{ - Extraction of 3PEPS peak shifts from MR-CMDS delay space. Left-hand plot: thick colored lines - denote contours of constant $\tau$ for $T=0, 1, 2, 3$. - Dots indicate the fitted peak shift for each $\tau$ contour. - Right-hand plot: numerically simulated amplitude traces (solid), Gaussian fits (transparent) - and fit centers (vertical lines) for each $T$ (colors matched). - } + \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{mix:fig:hom_2d_spectra} \end{figure} \begin{figure} - \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 \autoref{mix:eqn: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{mix:fig:delay_inhom} + \includegraphics[width=\textwidth]{"mixed_domain/spectral evolution full"} + \caption[Evolution of the 2D frequency response, with all contours shown.]{ + Spectral evolution of the homogeneous exciton resonance as a function of $\tau_{21}$, with + $\tau_{22^\prime}=0$. + The 50\% contour is darkened to ease comparison with Figure 7. + } + \label{mix:fig:spectral_evolution_full} \end{figure} \begin{figure} - \includegraphics[width=\textwidth]{"mixed_domain/2D delays"} - \label{mix:fig:2D_delays} - \caption[2D delay response for all combinations of inhomogeneity, dephasing rate.]{ - 2D delay scans at $\omega_1=\omega_2=\omega_{10}$ for all 12 combinations of $\Gamma_{10}$ - (rows) and $\Delta_{inhom}$ (columns) simulated in this work. - The 3PEPS shift trace is plotted in yellow, annotated to indicate the magnitude of the $\tau$ - shift at $T=0$ and $T=4\Delta_t$. + \includegraphics[width=\linewidth]{"mixed_domain/wigners full"} + \caption[Simulated Wigner spectra.]{ + Mixed $\tau_{21}$, $\omega_1$ plots for each $\Gamma_{10}$ value simulated in this work. + For each plot, the corresponding $\omega_2$ value is shown as a gray vertical line. + Each plot is separately normalized. } + \label{mix:fig:wigners} \end{figure} +\subsection{Inhomogeneous broadening} \label{mix:sec:res_inhom} % -------------------------------- + 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. % @@ -1653,7 +1680,9 @@ 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 $T$. % -In our 2D delay plots (\autoref{mix:fig:delay_purity}, \autoref{mix:fig:delay_inhom}), the peak shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical line. % +In our 2D delay plots (\autoref{mix:fig:delay_purity}, \autoref{mix:fig:delay_inhom}), the peak +shift is seen as the diagonal displacement of the signal peak from the $\tau_{21}=0$ vertical +line. % \autoref{mix: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. % @@ -1674,11 +1703,63 @@ This fact is easily illustrated by the dynamics of homogeneous system (Fig. \autoref{mix: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{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system properties on the -distortion has not been investigated previously. % +studies, \cite{DeBoeijWimP1996a, AgarwalRitesh2002a} but the dependence of pulse and system +properties on the distortion has not been investigated previously. % Peak-shifting due to pulse overlap is less important when $\omega_1\neq\omega_2$ because time-ordering III is decoupled by detuning. % +In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous +broadening. % +\autoref{mix:fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous +distribution. % +All systems are broadened by a distribution proportional to their dephasing bandwidth. % +As expected, the sequence again shows a gradual broadening along the $\omega_1$ axis, with a strong +spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals. % +The anti-diagonal width at early delays (e.g. \autoref{mix: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. % + +\begin{figure} + \includegraphics[width=\textwidth]{"mixed_domain/3PEPS"} + \label{mix:fig:3PEPS} + \caption[3PEPS tutorial.]{ + Extraction of 3PEPS peak shifts from MR-CMDS delay space. Left-hand plot: thick colored lines + denote contours of constant $\tau$ for $T=0, 1, 2, 3$. + Dots indicate the fitted peak shift for each $\tau$ contour. + Right-hand plot: numerically simulated amplitude traces (solid), Gaussian fits (transparent) + and fit centers (vertical lines) for each $T$ (colors matched). + } +\end{figure} + +\begin{figure} + \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 \autoref{mix:eqn: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{mix:fig:delay_inhom} +\end{figure} + +\begin{figure} + \includegraphics[width=\textwidth]{"mixed_domain/2D delays"} + \label{mix:fig:2D_delays} + \caption[2D delay response for all combinations of inhomogeneity, dephasing rate.]{ + 2D delay scans at $\omega_1=\omega_2=\omega_{10}$ for all 12 combinations of $\Gamma_{10}$ + (rows) and $\Delta_{inhom}$ (columns) simulated in this work. + The 3PEPS shift trace is plotted in yellow, annotated to indicate the magnitude of the $\tau$ + shift at $T=0$ and $T=4\Delta_t$. + } +\end{figure} + \begin{figure} \includegraphics[width=\linewidth]{"mixed_domain/inhom spectral evolution"} \caption[Spectral evolution of an inhomogenious system.]{ @@ -1731,18 +1812,6 @@ time-ordering III is decoupled by detuning. % } \end{figure} -In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous -broadening. % -\autoref{mix:fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous -distribution. % -All systems are broadened by a distribution proportional to their dephasing bandwidth. % -As expected, the sequence again shows a gradual broadening along the $\omega_1$ axis, with a strong -spectral correlation at early delays ($\tau_{21}>0$) for the more driven signals. % -The anti-diagonal width at early delays (e.g. \autoref{mix: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} @@ -1829,25 +1898,6 @@ $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space. % \subsection{Extracting true material correlation} % ---------------------------------------------- -\begin{figure} - \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{mix:fig:metrics} -\end{figure} - 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 @@ -1925,6 +1975,25 @@ If a system has spectral diffusion, the mapping at late times will disagree with early times; both ellipticity and 3PEPS will be smaller at later times than predicted by the change in mappings alone. % +\begin{figure} + \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{mix:fig:metrics} +\end{figure} + \section{Conclusion} % --------------------------------------------------------------------------- This study provides a framework to describe and disentangle the influence of the excitation pulses -- cgit v1.2.3