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| -rw-r--r-- | mixed_domain/chapter.tex | 115 |
1 files changed, 14 insertions, 101 deletions
diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 8128ba5..f95cf88 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -558,93 +558,6 @@ which is a 1D convolution along the diagonal axis in frequency space. % \autoref{mix:fig:convolution} demonstrates the use of \autoref{mix:eqn:convolve_final} on a
homogeneous line shape. %
-\begin{figure}
- \includegraphics[width=\linewidth]{mixed_domain/convolve}
- \caption[Convolution overview.]
- {Overview of the convolution.
- (a) The homogeneous line shape.
- (b) The distribution function, $K$, mapped onto laser coordinates.
- (c) The resulting ensemble line shape computed from the convolution.
- The thick black line represents the FWHM of the distribution function.}
- \label{mix:fig:convolution}
-\end{figure}
-
-Here we describe how to transform the data of a single reference oscillator signal to that of an
-inhomogeneous distribution. %
-The oscillators in the distribution are allowed have arbitrary energies for their states, which
-will cause frequency shifts in the resonances. %
-To show this, we start with a modified, but equivalent, form of \autoref{mix:eqn:rho_f}:
-\begin{equation} \label{mix:eqn:rho_f_modified}
-\begin{split}
-\frac{d\tilde{\rho}_f}{dt} =& -\Gamma_f\tilde{\rho}_f + \frac{i}{2}\lambda_f\mu_f c_x(t-\tau_x) \\
-& \times e^{i\kappa_f\left( \vec{k}\cdot z + \omega_x \tau_x \right)} e^{-i\kappa_f\left( \omega_x-\left|\omega_f \right| \right)t}\tilde{\rho}_i(t).
-\end{split}
-\end{equation}
-
-We consider two oscillators with transition frequencies $\omega_f$ and $\omega_f^\prime=\omega_f +
-\delta$. %
-So long as $\left| \delta \right| \leq \omega_f$ (so that $\left| \omega_f + \delta \right| =
-\left| \omega_f \right| + \delta$ and thus the rotating wave approximation does not change),
-\autoref{mix:eqn:rho_f_modified} shows that the two are related by %
-\begin{equation} \label{mix:eqn:freq_translation}
-\frac{d\tilde{\rho}_f^\prime}{dt}(t;\omega_x) = \frac{d\tilde{\rho}_f}{dt}(t;\omega_x-\delta)e^{i\kappa_f \delta \tau_x}.
-\end{equation}
-
-Because both coherences are assumed to have the same initial conditions
-($\rho_0(-\infty)=\rho_0^\prime(-\infty)=0$), the equality also holds when both sides of the
-equation are integrated. %
-The phase factor $e^{i\kappa_f\delta\tau_x}$ in the substitution arises from \autoref{mix:eqn:E_l},
-where the pulse carrier frequency maintains its phase within the pulse envelope for all delays. %
-
-The resonance translation can be extended to higher order signals as well. %
-For a third-order signal, we compare systems with transition frequencies
-$\omega_{10}^\prime=\omega_{10}+a$ and $\omega_{21}^\prime = \omega_{21}+b$. %
-The extension of \autoref{mix:eqn:freq_translation} to pathway $V\beta$ gives %
-\begin{equation}
-\begin{split}
-\tilde{\rho}_3^\prime(t;\omega_2, \omega_2^\prime, \omega_1) =& \tilde{\rho}_3(t;\omega_2-a,\omega_{2^\prime}-a,\omega_1-b) \\
-&\times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1}.
-\end{split}
-\end{equation}
-
-The translation of each laser coordinate depends on which transition is made (e.g. $a$ for
-transitions between $|0\rangle$ and $|1\rangle$ or $b$ for transitions between $|1\rangle$ and
-$|2\rangle$), so the exact translation relation differs between pathways. %
-We can now compute the ensemble average of signal for pathway $V\beta$ as a convolution between the
-distribution function of the system, $K(a,b)$, and the single oscillator response: %
-\begin{equation}
-\begin{split}
-\langle \tilde{\rho}_3 (t;\omega_2,\omega_{2^\prime},\omega_1) \rangle =& \iint K(a,b)\\
-& \times \tilde{\rho}_3 (t;\omega_2+a,\omega_{2^\prime}+a,\omega_1+b) \\
-& \times e^{i\kappa_2 a \tau_2} e^{i\kappa_{2^\prime} a \tau_{2^\prime}} e^{i\kappa_1 b \tau_1} da \ db.
-\end{split}
-\end{equation}
-For this work, we restrict ourselves to a simpler ensemble where all oscillators have equally
-spaced levels (i.e. $a=b$). %
-This makes the translation identical for all pathways and reduces the dimensionality of the
-convolution. %
-Since pathways follow the same convolution we may also perform the convolution on the total signal field:
-\begin{equation}
-\begin{split}
-\langle E_{\text{tot}}(t) \rangle =& \sum_L \mu_{4,L} \int K(a,a) \\
-& \times \tilde{\rho}_{3,L}(t;\omega_x-a,\omega_y-a\omega_z-a) \\
-& \times e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} da.
-\end{split}
-\end{equation}
-Furthermore, since $\kappa=-1$ for $E_1$ and $E_{2^\prime}$, while $\kappa=1$ for $E_2$, we have
-$e^{ia\left( \kappa_x\tau_x + \kappa_y\tau_y + \kappa_z\tau_z \right)} = e^{-ia\left( \tau_1 -
- \tau_2 + \tau_{2^\prime} \right)}$ for all pathways. %
-Equivalently, if the electric field is parameterized in terms of laser coordinates $\omega_1$ and $\omega_2$, the ensemble field can be calculated as
-\begin{equation} \label{mix:eqn:convolve_final}
-\begin{split}
-\langle E_{\text{tot}}(t;\omega_1,\omega_2) \rangle =& \int K(a,a)E_{\text{tot}}(t;\omega_1-a,\omega_2-a) \\
-&\times e^{-ia\left( \tau_1-\tau_2+\tau_{2^\prime} \right)} da.
-\end{split}
-\end{equation}
-which is a 1D convolution along the diagonal axis in frequency space. %
-\autoref{mix:fig:convolution} demonstrates the use of \autoref{mix:eqn:convolve_final} on a
-homogeneous line shape. %
-
\section{Methods} % ==============================================================================
A matrix representation of differential equations of the type in \autoref{mix:eqn:E_L_full} was
@@ -842,7 +755,7 @@ that is computationally cheaper. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/matrix flow diagram"}
\label{mix:fig:matrix_flow_diagram}
- \caption{
+ \caption[Liouville pathway network.]{
Network of Liouville pathways in this work.
Superscripts denote the field interactions that have occurred to make the density matrix
element.
@@ -1025,7 +938,7 @@ this time. \begin{figure}
\includegraphics[scale=0.5]{"mixed_domain/heun"}
\label{mix:fig:heun}
- \caption{
+ \caption[Integration technique comparison.]{
Comparison of numerical integration techniques for a Liouville pathway signal with different
number of time steps.
}
@@ -1222,7 +1135,7 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/fid vs detuning with freq"}
\label{mix:fig:fid_vs_detuning_with_freq}
- \caption{
+ \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.
@@ -1233,7 +1146,6 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/SQC lineshapes against t"}
- \label{fig:sqc_vs_t}
\caption{
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
@@ -1242,6 +1154,7 @@ FID character is difficult to isolate when $\Gamma_{10}\Delta_t=2.0$. % 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
@@ -1349,14 +1262,14 @@ $\Gamma_{10}\Delta_t=1$. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/pw1 lineshapes no mono"}
- \label{mix:fig:pw1_no_mono}
- \caption{
+ \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}
We now consider the multidimensional response of a single Liouville pathway involving three pulse
@@ -1571,12 +1484,12 @@ vanishing signal intensities; the contour of $P=0.99$ across our systems highlig \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/spectral evolution full"}
- \label{mix:fig:spectral_evolution_full}
- \caption{
+ \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}
In the previous sections we showed how pathway spectra and weights evolve with delay. %
@@ -1654,7 +1567,7 @@ resonance, which gives rise to a vertically elongated signal when $\Gamma_{10}\D \begin{figure}
\includegraphics[width=\linewidth]{"mixed_domain/wigners full"}
- \caption{
+ \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.
@@ -1682,7 +1595,7 @@ Again, these features can resemble spectral diffusion even though our system is \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/3PEPS"}
\label{mix:fig:3PEPS}
- \caption{
+ \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.
@@ -1711,7 +1624,7 @@ Again, these features can resemble spectral diffusion even though our system is \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/2D delays"}
\label{mix:fig:2D_delays}
- \caption{
+ \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 $\tau$
@@ -1791,7 +1704,7 @@ time-ordering III is decoupled by detuning. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/inhom spectral evolution full"}
\label{mix:fig:inhom_spectral_evolution_full}
- \caption{
+ \caption[Spectral evolution of an inhomogeneous system, with all contours shown.]{
Spectral evolution of the exciton resonance as a function of $\tau_{21}$, with
$\tau_{22^\prime}=0$.
For each system $\Delta_{inhom}=0.441\Gamma_{10}$.
@@ -1802,7 +1715,7 @@ time-ordering III is decoupled by detuning. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/2D frequences at zero"}
\label{mix:fig:2D_frequencies_at_zero}
- \caption{
+ \caption[Eccentricity at zero delay.]{
2D frequency scans at $\tau_{21}=\tau_{22^\prime}=0$ for all 12 combinations of $\Gamma_{10}$
(columns) and $\Delta_{inhom}$ (rows) simulated in this work.
The eccentricity of each spectrum is inset and represented by the yellow ellipse (50\%
@@ -1813,7 +1726,7 @@ time-ordering III is decoupled by detuning. % \begin{figure}
\includegraphics[width=\textwidth]{"mixed_domain/2D frequences at -4"}
\label{mix:fig:2D_frequencies_at_-4}
- \caption{
+ \caption[Eccentricity at large population time.]{
2D frequency scans at large $T$ ($\tau_{22^\prime}=0$, $\tau_{21}=-4\Delta_t$) for all 12
combinations of $\Gamma_{10}$ (columns) and $\Delta_{inhom}$ (rows) simulated in this work.
The eccentricity of each spectrum is inset and represented by the yellow ellipse (50\%
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