\chapter{Spectroscopy} \label{cha:spc}

\begin{dquote}
  A hundred years ago, Auguste Comte, … a great philosopher, said that humans will never be able to
  visit the stars, that we will never know what stars are made out of, that that's the one thing
  that science will never ever understand, because they're so far away.  %
  And then, just a few years later, scientists took starlight, ran it through a prism, looked at
  the rainbow coming from the starlight, and said: ``Hydrogen!''  %
  Just a few years after this very rational, very reasonable, very scientific prediction was made,
  that we'll never know what stars are made of.  %

  \dsignature{Michio Kaku}
\end{dquote}
  
\clearpage

%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'

The basics of light matter interaction have been covered in many texts.  %
For a beginners introduction I recommend ``How a Photon is Created or Absorbed'' by
\textcite{HendersonGiles1994a}.  %
Here I present a very minimal overview.  %

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.  %

Now we will expose this two-state system to an electric field:
\begin{equation} \label{spc:eqn:E}
  E = E^{0}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
\end{equation}

For simplicity, we consider a single transition dipole, $\mu$.  %

The Hamiltonian which controls the coupling of or simple system to the electric field described in
\autoref{spc:eqn:E} can be written.  %
\begin{eqnarray}
  H &=& H_{0} + \mu \cdot E \\
  &=& H_{0} + \mu \cdot \frac{E^0}{2}\left[ \me^{i(kz-\omega t)} + \me^{-i(kz-\omega t)} \right]
\end{eqnarray}

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}
Where $\omega_a$ and $\omega_b$ are the fast (and familiar) Bohr frequencies and $\Omega$ is the
\emph{Rabi frequency}:  %
\begin{equation}
  \Omega \equiv \frac{\mu E^\circ}{\hbar}
\end{equation}

In Dirac notation \cite{DiracPaulAdrienMaurice1939a}, an observable (such as $\mu(t)$) can be
written simply:  %
\begin{equation}
  \mu(t) = |c_a(t)|^2 \langle \phi_a | \mu | \phi_a \rangle + |c_b(t)|^2 \langle \phi_b | \mu | \phi_b \rangle +  c_a(t) c_b^*(t) \langle \phi_b | \mu | \phi_a \rangle  +
  c_b(t) c_a^*(t) \langle \phi_a | \mu | \phi_b \rangle 
\end{equation}
The complex wavefunction is called a \emph{ket}, represented $\left|b\right>$.  %
The complex conjugate is called a \emph{bra}, represented $\left<a\right|$.  %
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.  %

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.  %
A more complete discussion of the formalism we use to describe light-matter interaction is
presented in \autoref{cha:mix}.  %

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.  %

WMEL diagrams are drawn using the following rules.  %
\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
    the distance between the initial and final state in the energy ladder.
	\item The time ordering of the interactions is represented by the ordering of arrows, from left
    to right.
	\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{denumerate}
WMELs can be found throughout this dissertation.  %

\section{Types of spectroscopy}  % ================================================================

Scientists have come up with many ways of exploiting light-matter interaction for measurement
purposes.  %
This section discusses several of these strategies.  %
I take a compare-and-contrast approach, rather than getting too caught up in the infinite diversity
of possible spectroscopic strategies.  %

\subsection{Linear vs multidimensional}  % --------------------------------------------------------

Most familiar spectroscopic experiments are linear.  %
That is to say, they have just one frequency axis, and they interrogate just one resonance
condition.  %
These are workhorse experiments, like absorbance, reflectance, FTIR, UV-Vis, and common old
ordinary Raman spectroscopy (COORS).  %
These experiments are incredibly robust, and are typically performed using easy to use commercial
desktop instruments.  %
There are now even handheld Raman spectrometers for use in industrial settings. \cite{nanoram}  %

Multidimensional spectroscopy contains a lot more information about the material under
investigation.  %
In this work, by ``multidimensional'' I mean higher-order spectroscopy.  %
I ignore ``correlation spectroscopy'' (otherwise known as covariance spectroscopy), which tracks
linear spectral features against non-spectral dimensions like lab time, pressure, and
temperature.  %
So, in the context of this dissertation, multidimensional spectroscopy is synonymous with nonlinear
spectroscopy.  %

Nonlinear spectroscopy relies upon higher-order terms in the light-matter interaction. In a generic
system, a rule of thumb states that each term is roughly ten times smaller than the last.  %
This means that nonlinear spectroscopy is typically very weak.  %
Still, nonlinear signals are fairly easy to isolate and measure using modern instrumentation, as
this dissertation describes.  %

The most obvious advantage of multidimensional spectroscopy comes directly from the dimensionality
itself.  %
Multidimensional spectroscopy can \emph{decongest} spectra with overlapping peaks by isolating
peaks in a multidimensional resonance landscape.  %
This decongestion arises directly from the multiple resonances the multidimensional spectroscopy
demands of a material.  %

\subsection{Frequency vs time domain}  % ----------------------------------------------------------

Broadly, there are two ways to collect nonlinear spectroscopic signals: frequency and time
domain.  %
Both techniques involve exciting a sample with multiple pulses of light and measuring the output
signal.  %
The techniques differ in how they resolve the multiple frequency axes of interest.  %

Frequency domain is probably the more intuitive strategy: frequency axes are resolved directly by
iteratively tuning the frequency of excitation pulses against each-other.  %
This relies on pulsed light sources with tunable frequencies.  %

Time domain experiments use an interferometric technique to resolve frequency axes.  %
Broadband excitation pulses which contain all of the necessary frequencies are used to excite the
sample.  %
The delay (phase) between pulses is scanned, and the resonances along that axis are resolved
through Fourier transform of the resulting interferogram.  %
In modern experiments, pulse shapers are used to control the delay between pulses in a very
precise, fast, and reproducible way.  %
The time domain strategy is by-far the most popular technique in multidimensional spectroscopy
because these technologies allow for rapid, robust data collection.  %

This dissertation focuses on less-popular frequency domain strategies, so some discussion of the
advantages of frequency domain when compared to time domain are warranted.  %

One of the biggest instrumental limitations of multidimensional spectroscopy is bandwidth.  %
It is easy to get absorbance spectra over the entire visible spectrum, and even into the
ultraviolet and near infrared.  %
Not so for multidimensional spectroscopy.  %
Multidimensional spectroscopy is limited by the bandwidth of our (tunable) light sources.  %
For frequency domain techniques, this limitation is incidental: sources with greater tunability
will be easy to incorporate into these instruments, and creating such sources is only a matter of
more optomechanical engineering---more moving parts.  %
Time domain techniques, on the other hand, have a more fundamental issue with bandwidth.  %
Time domain requires that all of the desired frequencies be present within the single excitation
pulse, and pulses with very large frequency bandwidth (very short in time) become very hard to use
and control.   %
With short, broad pulses:
\begin{ditemize}
	\item Non-resonant signal becomes brighter relative to resonant signal (the resonance advantage
    is lost). \cite{ChengJixin2001a}
	\item Pulse distortions become essentially unavoidable. \cite{SpencerAustinP2015a}
\end{ditemize}

Time domain experiments require a phase-locked, independently controlled local oscillator in order
to collect the interferogram at the heart of such techniques.  %
This local oscillator enhances the information-gathering power of time domain because it allows the
experiment to explicitly collect nonlinear spectra with full phase information.  %
At the same time, the local oscillator requirement limits the flexability of the time-domain
because it essentially requires that the output frequency must be the same as one of the inputs.  %
Novel, often fully coherent, experiments cannot be accomplished under this limitation.  %

%Another idea in defense of frequency domain is for the case of power studies.  %
%Since time-domain pulses in-fact possess all colors in them they cannot be trusted as much at
%perturbative fluence.  %
%See that paper that Natalia presented...  %

\subsection{Homodyne vs heterodyne}  % ------------------------------------------------------------

Within frequency domain multidimensional spectroscopy, one is free to use or forgo a local
oscillator.  %
That is to say, frequency domain spectroscopy can be collected in a heterodyne or homodyne
technique.  %
As discussed in the previous section, use of a local oscillator means that more useful phase
information can be extracted from the spectrum.  %
At the same time, generation of a phase locked, controllable local oscillator can be cumbersome,
limiting the flexibility of possible experiments.  %

Note that heterodyne techniques may be self heterodyned (as in transient absorption) or
``explicitly'' heterodyned with a local oscillator.  %

Besides the aforementioned phase information, probably the biggest difference between heterodyne
and homodyne-detected experiments is their scaling with oscillator number density, $N$.  %
In all heterodyne spectroscopies, signal goes linearly, as $N$.  %
If the number of oscillators is doubled, the signal doubles.  %
In all homodyne spectroscopies, signal goes as $N^2$.  % 
If the number of oscillators is doubled, the signal goes up by four times.  %
This is what we mean when we say that homodyne signals are ``intensity level'' and heterodyne
signals are ``amplitude level''.  %

Recently we have been taking to representing homodyne-detected multidimensional experiments on the
``amplitude level'' by plotting the square root of the collected signal.  %
Many of the figures in this dissertation are plotted in this way.  %
In my opinion, this strategy makes interpretation of spectra easier.  %
Certainly it eases comparison with other experiments, like absorbance and COORS, which go as
$N$.  %

One easy-to-miss consequence of homodyne collected experiments is the behavior of signals in delay
space.  %
Since signal goes as $N^2$, signal decays much faster in homodyne-collected experiments.  %
If signal decays as a single exponential, the extracted decay is twice as fast for homodyne vs
heterodyne-detected data.  %
This fact is often forgotten, and papers have been corrected for forgetting this factor of
two. \cite{KoivistoinenJuha2017a}  %

% \section{Instrumentation}  % ======================================================================

% In this section I introduce the key components of the MR-CMDS instrument.  %
% This also serves to introduce the reader to the particular components used in my research.  %

% \subsection{LASER}  % -----------------------------------------------------------------------------

% Light Amplified by Stimulated Emission of Radiation (LASER) light sources are absolutely crucial
% components of the modern MR-CMDS instrument.  %
% The first laser was built in 1960 by \textcite{MaimanTheodore1960a}, and pulsed lasers were
% invented soon after.  %
% Today, ultrafast light sources are relatively cheap and reliable.  %
% Our SpectraPhysics ``Tsunami'' oscillator uses passive Kerr-lens mode-locking to generate $\sim$35
% fs seed pulses at $\sim$80 MHz (one pulse every 12.5 nanoseconds). \cite{Tsunami}  %
% This seed is split and fed into two 1 KHz amplifiers, a picosecond ``Spitfire Ace''
% \cite{SpitfireAce} and a femtosecond ``Spitfire Pro'' \cite{SpitfirePro}.  %
% These amplifiers each output several watts of ultrafast pulses at 1 KHz (one pulse per
% millisecond).  %

% \subsection{Optical parametric amplifiers}  % -----------------------------------------------------

% Optical Parametric Amplifiers (OPAs) are arguably the most crucial component of modern MR-CMDS, as
% they provide the frequency tunable light sources that we require. \cite{CerulloGiulio2003a}  %
% OPAs provide tunability through three-wave sum and difference frequency generation processes.  %
% ``Fundamental'' tunability is achieved by splitting the 800 nm photons into two lower energy
% photons, with a splitting ratio determined by motorized optics.  %
% These split photons are called ``signal'' and ``idler'', with signal being the higher energy and
% idler the lower energy photon.  %

% % TODO: paragraph about phase matching conditions and polarization

% Signal and idler are then either used directly, or amplified through sum or difference frequency
% processes to provide broadband tuneability.  %
% All available optical processes for the TOPAS-C OPAs \cite{TOPAS-C} used on the femtosecond table
% are shown in \autoref{spc:fig:ranges}.  %
% Amazingly, these OPAs offer tunability from the mid-infrared to the ultraviolet.  %
% Currently we do not have proper automated filters to make it possible to continuously scan between
% regions, but such things are possible.  %

% On the picosecond table we have three separate kinds of OPAs, including one TOPAS-800
% \cite{TOPAS-800} and two OPA-800 models that have been modified with precision micro control
% \cite{PMC} servo motors to provide automated tunability.  %

% \subsection{Delay stages}  % ----------------------------------------------------------------------

% Delay stages are simple, one-motor devices which are used to control the relative arrival time of
% pulses at the sample.  %
% The Wright Group currently owns four models of delay stage: 1. Newport MFA-CC [CITE], 2. Aerotech
% ... [CITE], 3. Thorlabs LTS300 [CITE].  %
% The fourth ``model'' are actually homemade stages that are driven using PMC motors.  %

% \subsection{Spectrometers}  % ---------------------------------------------------------------------

% Spectrometers...

% \begin{figure}
%   \includegraphics[width=\linewidth]{spectroscopy/ranges}
%   \caption{
%     CAPTION TODO
%   }
%   \label{spc:fig:ranges}
% \end{figure}