diff options
| author | Blaise Thompson <blaise@untzag.com> | 2018-05-14 10:31:46 -0500 |
|---|---|---|
| committer | Blaise Thompson <blaise@untzag.com> | 2018-05-14 10:31:46 -0500 |
| commit | 96d1fba00c2ae499056a81d430141d500d7695d8 (patch) | |
| tree | 31720a4c6091a915d3e363328066164eb18ae492 /mixed_domain | |
| parent | 10898eded280c3e7f072f5bd0fa881422b5b0733 (diff) | |
2018-05-14 10:31
Diffstat (limited to 'mixed_domain')
| -rw-r--r-- | mixed_domain/chapter.tex | 795 |
1 files changed, 432 insertions, 363 deletions
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
|