From 798bd484137439bf1d0752ddde82bf49da4af6b7 Mon Sep 17 00:00:00 2001 From: Blaise Thompson Date: Mon, 16 Apr 2018 13:32:20 -0500 Subject: 2018-04-16 13:32 --- mixed_domain/chapter.tex | 37 +++++++++++++++++++------------------ 1 file changed, 19 insertions(+), 18 deletions(-) (limited to 'mixed_domain') diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 60910c2..75e60ea 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -164,7 +164,7 @@ the temporal and spectral widths of the excitation pulses. % For simplicity, we will ignore population relaxation effects: $\Gamma_{11}=\Gamma_{00}=0$. % The electric field pulses, $\left\{E_l \right\}$, are given by: -\begin{equation}\label{eq:E_l} +\begin{equation} \label{mix:eqn:E_l} E_l(t; \omega_l, \tau_l, \vec{k}_l \cdot z) = \frac{1}{2}\left[c_l(t-\tau_l)e^{i\vec{k}_l\cdot z}e^{-i\omega_l(t-\tau_l)} + c.c. \right], \end{equation} where $\omega_l$ is the field carrier frequency, $\vec{k}_l$ is the wavevector, $\tau_l$ is the @@ -180,7 +180,7 @@ spectral FWHM (intensity scale). % The Liouville-von Neumann Equation propagates the density matrix, $\bm{\rho}$: -\begin{equation}\label{eq:LVN} +\begin{equation} \label{mix:eqn:LVN} \frac{d\bm{\rho}}{dt} = -\frac{i}{\hbar}\left[\bm{H_0} + \bm{\mu}\cdot \sum_{l=1,2,2^\prime} E_l(t), \bm{{\rho}}\right] + \bm{\Gamma \rho}. \end{equation} Here $\bm{H_0}$ is the time-independent Hamiltonian, $\bm{\mu}$ is the dipole superoperator, and @@ -202,7 +202,7 @@ $\rho_{ij}$ from $\rho_{ik}$ or $\rho_{kj}$. % For brevity, we use $\hbar=1$ and abbreviate the initial and final density matrix elements as $\rho_i$ and $\rho_f$, respectively. % Using the natural frequency rotating frame, $\tilde{\rho}_{ij}=\rho_{ij} e^{-i\omega_{ij}t}$, the formation of $\rho_f$ using pulse $x$ is written as -\begin{equation}\label{eq:rho_f} +\begin{equation} \label{mix:eqn:rho_f} \begin{split} \frac{d\tilde{\rho}_f}{dt} =& -\Gamma_f\tilde{\rho}_f \\ &+ \frac{i}{2} \lambda_f \mu_f c_x(t-\tau_x)e^{i\kappa_f\left(\vec{k}_x\cdot z + \omega_x \tau_x \right)}e^{i\kappa_f\Omega_{fx}t}\tilde{\rho}_i(t), @@ -223,7 +223,7 @@ In the following equations we neglect spatial dependence ($z=0$). % It provides a general expression for arbitrary values of the dephasing rate and excitation pulse bandwidth. % The integral solution is -\begin{equation}\label{eq:rho_f_int} +\begin{equation} \label{mix:eqn:rho_f_int} \begin{split} \tilde{\rho}_f(t) =& \frac{i}{2}\lambda_f \mu_f e^{i\kappa_f \omega_x \tau_x} e^{i\kappa_f \Omega_{fx} t} \\ &\times \int_{-\infty}^{\infty} c_x(t-u-\tau_x)\tilde{\rho}_i(t-u)\Theta(u) \\ @@ -236,8 +236,8 @@ where $\Theta$ is the Heaviside step function. % \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \ll 1$. % Both limits are important for understanding the multidimensional line shape changes discussed in this paper. % -The steady state and impulsive limits of Equation propagates \auotoref{mix:eqn:rho_f_int} are -discussed in TODO +%The steady state and impulsive limits of Equation propagates \auotoref{mix:eqn:rho_f_int} are +% discussed in TODO % Appendix \ref{sec:cw_imp}. % \begin{figure} @@ -289,12 +289,12 @@ $2^\prime$ notation for the laser pulses with pathway $V\gamma$. The electric field emitted from a Liouville pathway is proportional to the polarization created by the third-order coherence: % -\begin{equation}\label{mix:eqn:E_L} +\begin{equation} \label{mix:eqn:E_L} E_L(t) = i \mu_{4}\rho_{3}(t). \end{equation} \autoref{mix:eqn:E_L} assumes perfect phase-matching and no pulse distortions through propagation. \autoref{mix:eqn:rho_f_int} shows that the output field for this Liouville pathway is - \begin{gather}\label{mix:eqn:E_L_full} + \begin{gather} \label{mix:eqn:E_L_full} \begin{split} E_L(t) =& \frac{i}{8}\lambda_1\lambda_2\lambda_3\mu_1\mu_2\mu_3\mu_4 e^{i\left( \kappa_1\omega_x\tau_x + \kappa_2\omega_y\tau_y + \kappa_3\omega_z\tau_z \right)} @@ -380,7 +380,7 @@ Our simulations were done using the open-source SciPy library. \cite{OliphantTra The changes in the spectral line shapes described in this work are best understood by examining the driven/continuous wave (CW) and impulsive limits of \autoref{mix:eqn:rho_f_int} and -\ref{eq:E_L_full}. % +\ref{mix:eqn:E_L_full}. % The driven limit is achieved when pulse durations are much longer than the response function dynamics: $\Delta_t \left|\Gamma_f + i \kappa_f \Omega_{fx}\right| \gg 1$. % In this limit, the system will adopt a steady state over excitation: $d\rho / dt \approx 0$. % @@ -493,7 +493,7 @@ The build-up limit approximates well when pulses are near-resonant and arrive to build-up behavior is emphasized). % The driven limit holds for large detunings, regardless of delay. % -\subsection{Convolution Technique for Inhomogeneous Broadening} \label{mix:sec:mixed_convolution} +\subsection{Convolution technique for inhomogeneous broadening} \label{mix:sec:convolution} % ---- \begin{figure} \includegraphics[width=\linewidth]{mixed_domain/convolve} @@ -712,7 +712,7 @@ $c_x$, which in our case gives \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{eq:rho_f_int_freq}: +by two limits of \autoref{mix:eqn:rho_f_int_freq}: \begin{itemize} \item When the transient is not frequency resolved, $\text{sig} \approx \int{\left| \tilde{\rho}_f(\omega) \right|^2 d\omega}$ and the measured line shape will be the @@ -977,7 +977,7 @@ The colored histogram bars and line shape contours correspond to different value dephasing rate, $\Gamma_{10}\Delta_t$. % The contour is the half-maximum of the line shape. % The dependence of the line shape amplitude on $\tau_{21}$ can be inferred from -\autoref{fig:delay_purity}. % +\autoref{mix:fig:delay_purity}. % The qualitative trend, as $\tau_{21}$ goes from positive to negative delays, is a change from diagonal/compressed line shapes to much broader resonances with no correlation ($\omega_1$ and @@ -1006,7 +1006,7 @@ There are differences in the line shapes for the different values of the relativ $\Gamma_{10}\Delta_t$. % The spectral correlation at $\tau_{21}/\Delta_t=2$ decreases in strength as $\Gamma_{10}\Delta_t$ decreases. % -As we illustrated in \autoref{fig:pw1}, this spectral correlation is a signature of driven signal +As we illustrated in \autoref{mix:fig:pw1}, this spectral correlation is a signature of driven signal from temporal overlap of $E_1$ and $E_2$; the loss of spectral correlation reflects the increased prominence of FID in the first coherence as the field-matter interactions become more impulsive. % This increased prominence of FID also reflects an increase in signal strength, as shown by @@ -1141,7 +1141,7 @@ time-ordering III is decoupled by detuning. % In frequency space, spectral elongation along the diagonal is the signature of inhomogeneous broadening. % -\autoref{fig:inhom_2d_spectra} shows the line shape changes of a Gaussian inhomogeneous +\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 @@ -1171,7 +1171,7 @@ For any delay coordinate, one can develop qualitative line shape expectations by \end{enumerate} \autoref{mix:fig:fid_dpr} illustrates principles 1 and 2 and \autoref{mix:fig:fid_detuning} illustrates principle 2 and 3. % -\autoref{fig:pw1} provides a detailed example of the relationship between these principles and the +\autoref{mix:fig:pw1} provides a detailed example of the relationship between these principles and the multidimensional line shape changes for different delay times. % The principles presented above apply to a single pathway. % @@ -1182,7 +1182,8 @@ shape. % The relative weight of each pathway to the interference can be approximated by the extent of pulse overlap. % The pathway weights exchange when scanning across pulse overlap, which creates the dramatic line -shape changes observed in Figures \ref{fig:hom_2d_spectra} and \ref{fig:inhom_2d_spectra}. % +shape changes observed in Figures \ref{mix:fig:hom_2d_spectra} and +\ref{mix:fig:inhom_2d_spectra}. % \subsection{Conditional validity of the driven limit} @@ -1190,7 +1191,7 @@ We have shown that the driven limit misses details of the line shape if $\Gamma_ \approx 1$, but we have also reasoned that in certain conditions the driven limit can approximate the response well (see principle 1). % Here we examine the line shape at delay values that demonstrate this agreement. % -Fig. \ref{fig:steady_state} compares the results of our numerical simulation (third column) with +Fig. \ref{mix:fig:steady_state} compares the results of our numerical simulation (third column) with the driven limit expressions for populations where $\Gamma_{11}\Delta_t=0$ (first column) or $1$ (second column). % The top and bottom rows compare the line shapes when $\left(\tau_{22^\prime}, @@ -1214,7 +1215,7 @@ with the pulse duration ($\Gamma_{11}\Delta_t=1$), which gives good agreement wi simulation (third column). % When $\tau_{22^\prime}=0$ and $\tau_{21}<\Delta_t$, signals can also be approximated by driven -signal (Fig. \ref{fig:steady_state} bottom row). % +signal (\autoref{mix:fig:steady_state} bottom row). % Only time-orderings V and VI are relevant. % The intermediate population resonance is still impulsive but it depends on $\omega_{2^\prime}-\omega_2$ which is not explored in our 2D frequency space. % -- cgit v1.2.3