From ee6453ad4a44984bd354ff5220c32336a4df3c3a Mon Sep 17 00:00:00 2001
From: Blaise Thompson <blaise@untzag.com>
Date: Sat, 31 Mar 2018 17:56:55 -0500
Subject: 2018-03-31 17:56

---
 spectroscopy/chapter.tex | 228 +++++++++++++++++++++++++++++++----------------
 1 file changed, 151 insertions(+), 77 deletions(-)

(limited to 'spectroscopy/chapter.tex')

diff --git a/spectroscopy/chapter.tex b/spectroscopy/chapter.tex
index fe10f60..9062e89 100644
--- a/spectroscopy/chapter.tex
+++ b/spectroscopy/chapter.tex
@@ -10,7 +10,7 @@
 % TODO: https://pubs.acs.org/doi/abs/10.1021/acs.jpcb.7b02693
 % TODO: http://journals.sagepub.com/doi/10.1177/0003702816669730
 
-\chapter{Spectroscopy}
+\chapter{Spectroscopy} \label{cha:spc}
 
 \begin{dquote}
   A hundred years ago, Auguste Comte, … a great philosopher, said that humans will never be able to
@@ -26,48 +26,113 @@
   
 \clearpage
 
-In this chapter I lay out the foundations of spectroscopy.
+In this chapter I lay out the foundations of spectroscopy as relevant to this dissertation.  %
+Spectroscopy is the study of the interaction of light (electromagnetic radiation) and matter
+(molecules, crystals, solids, liquids etc).  %
+
+\section{Light-matter interaction}  % =============================================================
+
+As scientists, light is perhaps the most useful tool we have for interrogating materials.  %
+Light is relatively easy to create and control, and light-matter interaction tells us a lot about
+the microscopic physics of the material under investigation.  %
+Spectroscopists use light-matter interaction as an analytical tool.  %
+For the purposes of this document, light can be treated as a classical electromagnetic wave and
+matter can be treated in the quantum mechanical density matrix formalism.  %
+More complete treatments which also take the quantum-mechanical nature of light into account are
+possible (see: ``quantum optics'', ``quantum electrodynamics''), but beyond the scope of this
+dissertation.  %
+This classical treatment still captures the full richness of the wave-nature of light, including
+interference effects.  \cite{HuygensChristiaan1913a}  %
+It merely ignores the quantitization of the electric field---a valid assumption in the limit of
+many photons.  %
+
+% TODO: language from 'how a photon is created or destroyed'
+
+For simplicity, consider a two state system: ``a'' and ``b''.  %
+These two states might be the inital and final states in a transition.  %
+The wavefunction for this system can be written as a sum of the stationary states (eigenstates)
+with appropriate scaling coefficients:
+\begin{equation}
+  \Psi(r, t) = c_a(t)\psi_a(r) + c_b(t)\psi_b(r)
+\end{equation}
+The time dependence lies in the $c_a$ and $c_b$ coefficients, and the spatial dependence lies in
+the $\psi_a$ and $\psi_b$ eigienstates.  %
 
-\section{Light}
+Now we will expose this two-state system to an electric field:
+\begin{equation}
+  E = E^{\circ}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
+\end{equation}
 
-% TODO: add reference to HuygensChristiaan1913.000
+For simplicity, we consider a single transition dipole, $\mu$.  %
 
-% TODO: add reference to MaimanTheodore.000
+The Hamiltonian which controls the coupling of or simple system to the electric field described in
+...:
+\begin{equation}
+  H = H_{\circ} - \mu \dot E
+\end{equation}
 
-\section{Light-Matter Interaction}
+Solving for the time-dependent coefficients, then:
+\begin{eqnarray}
+  c_a(t) &=& \cos{\frac{\Omega t}{2}} \me^{-i\omega_at} \\
+  c_b(5) &=& \sin{\frac{\Omega t}{2}} \me^{-i\omega_bt}
+\end{eqnarray}
+Fast and slow parts...
+Bohr and Rabi freuencies...
 
-Spectroscopic experiments all derive from the interaction of light and matter. Many material
-properties can be deduced by measuring the nature of this interaction.  %
+Where $\Omega$ is the \emph{Rabi frequency}:  %
+\begin{equation}
+  \Omega \equiv \frac{\mu E^\circ}{\hbar}
+\end{equation}
 
-Nonlinear spectroscopy relies upon higher-order terms in the light-matter interaction. In a generic
-system, each term is roughly ten times smaller than the last.  % TODO: cite?
+In Dirac notation \cite{DiracPaulAdrienMaurice1939a}., an observable (such as $\mu(t)$) can be written simply:  %
+\begin{equation}
+  \mu(t) = \left< c_aa + c_bb \left| \hat{\mu} \right| c_aa + c_bb \right>
+\end{equation}
+The complex wavefunction is called a \emph{ket}, represented $|b>$.  %
+The complex conjugate is called a \emph{bra}, represented $<a|$.  %
+When expanded,
+\begin{equation}
+  \mu(t) = c_a^2\mu_a + c_b^2\mu_b + \left< c_aa \left| \hat{mu} \right| c_bb \right> +
+  \left<c_bb \left| \hat{mu} \right| c_aa \right>
+\end{equation}
+The first two terms are populations and the final two terms are coherences.  %
+The coherent terms will evolve with the rapid Bohr oscillations, coupling the dipole observable
+with the time-dependent electric field.  %
 
-% TODO: Discuss dephasing induced resonance. Example: florescence
+We commonly represent quantum mechanical systems using density matrices, where diagonal elements
+are populations and off-diagonal elements are coherences.  %
+Each density matrix element has the form $\rho_{kb}$, where $k$ is the ket and $b$ is the bra.  %
+% TODO: 4 member density matrix representing system above
+A more complete discussion of the formalism we use to describe light-matter interaction is
+presented in \autoref{cha:mix}.  %
 
-\subsection{Representations}
+% TODO: homogeneous line-width
+
+Spectroscopic experiments are typically performed on an ensemble of states.  %
+In such circumstances, inhomogeneous broadening becomes relevant.  %
+Inhomogeneous broadening arises from permanent differences between different oscillators in the
+ensemble.  %
+% TODO: why is inhomogeneous broadening important?
 
 Many strategies have been introduced for diagrammatically representing the interaction of multiple
 electric fields in an experiment.  %
+Spectroscopists have used diagrams to represent nonlinear optical phenomena since 1965.
+\cite{WardJF1965a}  %
+Several competing strategies have been defined over the years.  %
+In 1978, \textcite{YeeTK1978a} defined the ``circle diagram'' convention.  %
+Since then, the more popular  ``closed-time path-loop'' \cite{MarxChristophA2008a,
+  RoslyakOleksiy2009a} and ``double-sided Feynman'' diagrams \cite{MukamelShaul1995a} (also known
+as Mukamel diagrams) were introduced.  %
+\textcite{BiggsJasonD2012a} have written a paper which does an excellent job defining and comparing
+these two strategies.  %
+In their seminal 1985 work, \emph{A Unified View of Raman, Resonance Raman, and Fluorescence
+  Spectroscopy}, \textcite{LeeDuckhwan1985a} defined the conventions for a ``wave-mixing energy
+level'' (WMEL) diagram.  %
+Today, double-sided Feynman diagrams are probably most popular, but WMELs will be used in this
+document due to author preference.  %
 
-\subsubsection{Circle Diagrams}
-
-% TODO: add reference to YeeTK1978.000
-
-% TODO: Discuss circle diagrams from a historical perspective
-
-\subsubsection{Double-sided Feynman Diagrams}
-
-% TODO: Discuss double-sided Feynman diagrams from a historical perspective
-
-\subsubsection{WMEL Diagrams}
-
-So-called wave mixing energy level (\gls{WMEL}) diagrams are the most familiar way of representing
-spectroscopy for Wright group members.  %
-\gls{WMEL} diagrams were first proposed by Lee and Albrecht in an appendix to their seminal work
-\emph{A Unified View of Raman, Resonance Raman, and Fluorescence Spectroscopy}
-\cite{LeeDuckhwan1985a}.  %
 \gls{WMEL} diagrams are drawn using the following rules.  %
-\begin{enumerate}
+\begin{denumerate}
 	\item The energy ladder is represented with horizontal lines - solid for real states and dashed
     for virtual states.
 	\item Individual electric field interactions are represented as vertical arrows. The arrows span
@@ -77,15 +142,13 @@ spectroscopy for Wright group members.  %
 	\item Ket-side interactions are represented with solid arrows.
 	\item Bra-side interactions are represented with dashed arrows.
 	\item Output is represented as a solid wavy line.
-\end{enumerate}
-
-\subsubsection{Mukamel Diagrams}
+\end{denumerate}
 
-% TODO: Discuss Mukamel diagrams from a historical perspective
+% TODO: representative WMEL?
 
-\section{Linear Spectroscopy}
+\section{Types of spectroscopy}  % ================================================================
 
-\subsection{Reflectivity}
+\subsection{Linear vs multidimensional}  % --------------------------------------------------------
 
 This derivation adapted from \textit{Optical Processes in Semiconductors} by Jacques I. Pankove
 \cite{PankoveJacques1975a}.  %
@@ -99,30 +162,14 @@ Further derivation adapted from \cite{KumarNardeep2013a}.  %
 To extend reflectivity to a differential measurement
 % TODO: finish derivation
 
-\section{Coherent Multidimensional Spectroscopy}
-
 % TODO: (maybe) include discussion of photon echo famously discovered in 1979 in Groningen
 
-\gls{multiresonant coherent multidimensional spectroscopy}
-
-
-\subsection{Three Wave}
-
-\subsection{Four Wave}
-
-Fluorescence
-
-Raman
-
-\subsection{Five Wave}
+% TODO: spectral congestion figure
 
-\subsection{Six Wave}
-
-\gls{multiple population-period transient spectroscopy} (\Gls{MUPPETS})
-
-\section{Strategies for CMDS}
+Nonlinear spectroscopy relies upon higher-order terms in the light-matter interaction. In a generic
+system, each term is roughly ten times smaller than the last.  % TODO: cite?
 
-\subsection{Homodyne vs. Heterodyne Detection}
+\subsection{Homodyne vs heterodyne}  % ------------------------------------------------------------
 
 Two kinds of spectroscopies: 1) \gls{heterodyne} 2) \gls{homodyne}.
 Heterodyne techniques may be \gls{self heterodyne} or explicitly heterodyned with a local
@@ -134,9 +181,7 @@ This literally means that homodyne signals go as the square of heterodyne signal
 mean when we say that homodyne signals are intensity level and heterodyne signals are amplitude
 level.
 
-\Gls{transient absorption}, \gls{TA}
-
-\subsection{Frequency vs. Time Domain}
+\subsection{Frequency vs time domain}  % ----------------------------------------------------------
 
 Time domain techniques become more and more difficult when large frequency bandwidths are
 needed.  %
@@ -157,7 +202,7 @@ Since time-domain pulses in-fact possess all colors in them they cannot be trust
 perturbative fluence.  %
 See that paper that Natalia presented...  %
 
-\subsection{Triply Electronically Enhanced Spectroscopy}
+\subsection{Transient grating}  % -----------------------------------------------------------------
 
 Triply Electronically Enhanced (TrEE) spectroscopy has become the workhorse homodyne-detected 4WM
 experiment in the Wright Group.  %
@@ -166,7 +211,34 @@ experiment in the Wright Group.  %
 
 % TODO: Discussion of old and current delay space
 
-\subsection{Transient Absorbance Spectroscopy} 
+% TODO: discuss current delay space physical conventions (see inbox)
+
+\begin{figure}
+  \includegraphics[scale=1]{"spectroscopy/wmels/trive_on_diagonal"}
+  \caption[CAPTION TODO]{
+    CAPTION TODO
+  }
+  \label{spc:fig:trive_on_diagonal}
+\end{figure}
+
+
+\begin{figure}
+  \includegraphics[scale=1]{"spectroscopy/wmels/trive_off_diagonal"}
+  \caption[CAPTION TODO]{
+    CAPTION TODO
+  }
+  \label{spc:fig:trive_off_diagonal}
+\end{figure}
+
+\begin{figure}
+  \includegraphics[scale=1]{"spectroscopy/wmels/trive_population_transfer"}
+  \caption[CAPTION TODO]{
+    CAPTION TODO
+  }
+  \label{spc:fig:trive_population_transfer}
+\end{figure}
+
+\subsection{Transient absorbance}  % --------------------------------------------------------------
 
 \Gls{transient absorption} (\gls{TA})
 
@@ -247,13 +319,23 @@ expression for $\Delta A$ that only depends on my eight measurables.  %
 \Delta A = - \log_{10} \left(\frac{C_\mathrm{T}(V_\mathrm{T} + V_{\Delta\mathrm{T}}) + C_\mathrm{R}(V_\mathrm{R} + V_{\Delta\mathrm{R}})}{C_\mathrm{T} V_\mathrm{T} + C_\mathrm{R} V_\mathrm{R}}\right)
 \end{equation}
 
-\subsection{Cross Polarized TrEE}
+\subsection{Pump CMDS-probe}  % -------------------------------------------------------------------
+
+\clearpage
+\section{Instrumentation}  % ======================================================================
+
 
-\subsection{Pump-TrEE-Probe}
+\subsection{LASER}  % -----------------------------------------------------------------------------
 
-\Gls{pump TrEE probe} (\gls{PTP}).
+% TODO: add reference to MaimanTheodore.000 (ruby laser)
 
-\section{Instrumental Response Function}
+\subsection{Optical parametric amplifiers}  % -----------------------------------------------------
+
+\subsection{Delay stages}  % ----------------------------------------------------------------------
+
+\subsection{Spectrometers}  % ---------------------------------------------------------------------
+
+\subsection{Instrumental response function}  % ----------------------------------------------------
 
 The instrumental response function (IRF) is a classic concept in analytical science.  %
 Defining IRF becomes complex with instruments as complex as these, but it is still useful to
@@ -261,7 +343,7 @@ attempt.  %
 
 It is particularly useful to define bandwidth.
  
-\subsection{Time Domain}
+\subsubsection{Time Domain}
 
 I will use four wave mixing to extract the time-domain pulse-width.   %
 I use a driven signal \textit{e.g.} near infrared carbon tetrachloride response.  %
@@ -310,23 +392,15 @@ Finally, since we measure $\sigma_P$ and wish to extract $\sigma$:
 
 Again, all of these widths are on the \textit{intensity} level.
 
-\subsection{Frequency Domain}
+\subsubsection{Frequency Domain}
 
 We can directly measure $\sigma$ (the width on the intensity-level) in the frequency domain using a
 spectrometer.  %
 A tune test contains this information.  %
 
-\subsection{Time-Bandwidth Product}
+\subsubsection{Time-Bandwidth Product}
 
 For a Gaussian, approximately 0.441
 
 % TODO: find reference
-% TODO: number defined on INTENSITY level!
-
-
-
-
-
-
-
-
+% TODO: number defined on INTENSITY level!
\ No newline at end of file
-- 
cgit v1.2.3

