Update on chapter 2

Chapter1/chapter1.tex

@@ -2,7 +2,7 @@
 %*********************************** First Chapter *****************************
 %*******************************************************************************
 
-\chapter{Chapter title}  %Title of the First Chapter
+\chapter{Introduction}  %Title of the First Chapter
 
 \ifpdf
     \graphicspath{{Chapter1/Figs/Raster/}{Chapter1/Figs/PDF/}{Chapter1/Figs/}}

Chapter2/chapter2.tex

@@ -44,7 +44,184 @@ from a small number of sites with a large filling factor (e.g.~BECs
 trapped in a double-well potential) to a an extended multi-site
 lattice with a low filling factor (e.g.~a system with one atom per
 site will exhibit the Mott insulator to superfluid quantum phase
-transition). \mynote{extra fermion citations, Piazza?}
+transition). \mynote{extra fermion citations, Piazza? Look up Gabi's
+  AF paper.}
 
+\mynote{Potentially some more crap, but come to think of it the
+  content will strongly depend on what was included in the preceding
+  section on plain ultracold bosons}
 
+As we have seen in the previous section, an optical lattice can be
+formed with classical light beams that form standing waves. Depending
+on the detuning with respect to the atomic resonance, the nodes or
+antinodes form the lattice sites in which atoms accumulate. As shown
+in Fig. \ref{fig:LatticeDiagram} the trapped bosons (green) are
+illuminated with a coherent probe beam (red) and scatter light into a
+different mode (blue) which is then measured with a detector. The most
+straightforward measurement is to simply count the number of photons
+with a photodetector, but it is also possible to perform a quadrature
+measurement by using a homodyne detection scheme. The experiment can
+be performed in free space where light can scatter in any
+direction. The atoms can also be placed inside a cavity which has the
+advantage of being able to enhance light scattering in a particular
+direction. Furthermore, cavities allow for the formation of a fully
+quantum potential in contrast to the classical lattice trap.
 
+\begin{figure}[htbp!]
+  \centering
+  \includegraphics[width=1.0\textwidth]{LatticeDiagram}
+  \caption[LatticeDiagram]{Atoms (green) trapped in an optical lattice
+    are illuminated by a coherent probe beam (red). The light scatters
+    (blue) in free space or into a cavity and is measured by a
+    detector. If the experiment is in free space light can scatter in
+    any direction. A cavity on the other hand enhances scattering in
+    one particular direction.}
+  \label{fig:LatticeDiagram}
+\end{figure}
+
+For simplicity, we will be considering one-dimensional lattices most
+of the time. However, the model itself is derived for any number of
+dimensions and since none of our arguments will ever rely on
+dimensionality our results straightforwardly generalise to 2- and 3-D
+systems. This simplification allows us to present a much simpler
+picture of the physical setup where we only need to concern ourselves
+with a single angle for each optical mode. As shown in
+Fig. \ref{fig:LatticeDiagram} the angle between the normal to the
+lattice and the probe and detected beam are denoted by $\theta_0$ and
+$\theta_1$ respectively. We will consider these angles to be tunable
+although the same effect can be achieved by varying the wavelength of
+the light modes. However, it is much more intuitive to consider
+variable angles in our model as this lends itself to a simpler
+geometrical representation.
+
+\subsection{Derivation of the Hamiltonian}
+
+A general many-body Hamiltonian coupled to a quantized light field in
+second quantized can be separated into three parts,
+\begin{equation}
+\label{eq:TwoH}
+  \H = \H_f + \H_a + \H_{fa}.
+\end{equation}
+The term $\H_f$ represents the optical part of the Hamiltonian,
+\begin{equation}
+\label{eq:Hf}
+  \H_f = \sum_l \hbar \omega_l \ad_l \a_l -
+  i \hbar \sum_l \left( \eta_l^* \a_l - \eta_l \ad \right).
+\end{equation}
+The operators $\a_l$ ($\ad$) are the annihilation (creation) operators
+of light modes with frequencies $\omega_l$, wave vectors $\b{k}_l$,
+and mode functions $u_l(\b{r})$, which can be pumped by coherent
+fields with amplitudes $\eta_l$. The second part of the Hamiltonian,
+$\H_a$, is the matter-field component given by
+\begin{equation}
+\label{eq:Ha}
+  \H_a = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,a}
+  \Psi(\b{r}) + \frac{2 \pi a_s \hbar^2}{m} \int \mathrm{d}^3 \b{r}
+  \Psi^\dagger(\b{r}) \Psi^\dagger(\b{r}) \Psi(\b{r}) \Psi(\b{r}).
+\end{equation}
+Here, $\Psi(\b{r})$ ($\Psi^\dagger(\b{r})$) are the matter-field
+operators that annihilate (create) an atom at position $\b{r}$, $a_s$
+is the $s$-wave scattering length characterising the interatomic
+interaction, and $\H_{1,a}$ is the atomic part of the single-particle
+Hamiltonian $\H_1$. The final component of the total Hamiltonian is
+the interaction given by 
+\begin{equation}
+  \label{eq:Hfa}
+  \H_{fa} = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,fa}
+  \Psi(\b{r}),
+\end{equation}
+where $\H_{1,fa}$ is the interaction part of the single-particle
+Hamiltonian, $\H_1$.
+
+The single-particle Hamiltonian in the rotating-wave and dipole
+approximation is given by
+\begin{equation}
+  \H_1 = \H_f + \H_{1,a} + \H_{1,fa},
+\end{equation}
+\begin{equation}
+  \H_{1,a} = \frac{\b{p}^2} {2 m_a} + \frac{\hbar \omega_a}{2} \sigma_z,
+\end{equation}
+\begin{equation}
+  \H_{1,fa} =  - i \hbar \sum_l \left[ \sigma^+ g_l \a_l u_l(\b{r}) - \sigma^- g^*_l
+    \ad_l u^*_l(\b{r}) \right].
+\end{equation}
+In the equations above, $\b{p}$ and $\b{r}$ are the momentum and
+position operators of an atom of mass $m_a$ and resonance frequency
+$\omega_a$. The operators $\sigma^+ = |g \rangle \langle e|$,
+$\sigma^- = |e \rangle \langle g|$, and
+$\sigma_z = |e \rangle \langle e| - |g \rangle \langle g|$ are the
+atomic raising, lowering and population difference operators, where
+$|g \rangle$ and $| e \rangle$ denote the ground and excited states of
+the two-level atom respectively. $g_l$ are the atom-light coupling
+constants for each mode. It is the inclusion of the interaction of the
+boson with quantized light that distinguishes our work from the
+typical approach to ultracold atoms where all the optical fields,
+including the trapping potentials, are treated classically.
+
+We will now simplify the single-particle Hamiltonian by adiabatically
+eliminating the upper excited level of the atom. The equations of
+motion for the time evolution of operator $\hat{A}$ in the Heisenberg
+picture are given by
+\begin{equation}
+  \dot{\hat{A}} = \frac{i}{\hbar} \left[\H, \hat{A} \right].
+\end{equation}
+Therefore, the Heisenberg equation for the lowering operator of a
+single particle is
+\begin{equation}
+  \dot{\sigma}^- = \frac{i}{\hbar} \left[\H_1, \hat{\sigma}^- \right]
+  = \hbar \omega_a \sigma^- + i \hbar \sum_l \sigma_z g_l \a_l u_l(\b{r}).
+\end{equation}
+We will consider nonresonant interactions between light and atoms
+where the detunings between the light fields and the atomic resonance,
+$\Delta_{la} = \omega_l - \omega_a$, are much larger than the
+spontaneous emission rate and Rabi frequencies $g_l \a_l$. Therefore,
+the atom will be predominantly found in the ground state and we can
+set $\sigma_z = -1$ which is also known as the linear dipole
+approximation as the dipoles respond linearly to the light amplitude
+when the excited state has negligible population. Moreover, we can
+adiabatically eliminate the polarization $\sigma^-$. Firstly we will
+re-write its equation of motion in a frame rotating at $\omega_p$, the
+external probe frequency, such that
+$\sigma^- = \tilde{\sigma}^- \exp(i \omega_p t)$, and similarly for
+$\tilde{\a}_l$. The resulting equation is given by
+\begin{equation}
+  \dot{\tilde{\sigma}}^- = - \hbar \Delta_a \tilde{\sigma}^- - i \hbar
+  \sum_l g_l \tilde{\a}_l u_l(\b{r}),
+\end{equation}
+where $\Delta_a = \omega_p - \omega_a$ is the atom-probe
+detuning. Within this rotating frame we will take
+$\dot{\tilde{\sigma}}^- \approx 0$ and thus obtain the following
+equation for the lowering operator
+\begin{equation}
+  \sigma^- = - \frac{i}{\Delta_a} \sum_l g_l \a_l u_l(\b{r}).
+\end{equation}
+Therefore, by inserting this expression into the Heisenberg equation
+for the light mode $m$ given by
+\begin{equation}
+  \dot{\a}_m = - \sigma^- g^*_m u^*_m(\b{r})
+\end{equation}
+we get the following equation of motion
+\begin{equation}
+  \dot{\a}_m = \frac{i}{\Delta_a} \sum_l g_l g^*_m u_l(\b{r})
+  u^*_m(\b{r}) \a_l.
+\end{equation}
+An effective Hamiltonian which results in the same optical equations
+of motion can be written as
+$\H^\mathrm{eff}_1 = \H_f + \H^\mathrm{eff}_{1,a} +
+\H^\mathrm{eff}_{1,fa}$. The effective atomic and interaction
+Hamiltonians  are
+\begin{equation}
+  \H^\mathrm{eff}_{1,a} = \frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}),
+\end{equation}
+\begin{equation}
+  \H^\mathrm{eff}_{1,fa} = \frac{\hbar}{\Delta_a} \sum_{l,m}
+  u_l^*(\b{r}) u_m(\b{r}) g_l g_m \ad_l \a_m,
+\end{equation}
+where we have explicitly extracted
+$V_\mathrm{cl}(\b{r}) = \hbar g_\mathrm{cl}^2 |a_\mathrm{cl}
+u_\mathrm{cl}(\b{r})|^2 / \Delta_{\mathrm{cl},a}$, the classical
+trapping potential, from the interaction terms. However, we consider
+the trapping beam to be sufficiently detuned from the other light
+modes that we can neglect any scattering between them. However, a
+later inclusion of this scattered light would not be difficult due to
+the linearity of the dipoles we assumed.

Preamble/preamble.tex

@@ -200,3 +200,10 @@
 \fi
 
 % Example todo: \mynote{Hey! I have a note}
+
+% ***************************** Shorthand operator notation ********************
+
+\renewcommand{\H}{\hat{H}}
+\newcommand{\ad}{a^\dagger}
+\renewcommand{\a}{a} % in case we decide to put hats on
+\renewcommand{\b}[1]{\mathbf{#1}}

thesis.tex

@@ -1,7 +1,7 @@
 % ******************************* PhD Thesis Template **************************
 % Please have a look at the README.md file for info on how to use the template
 
-\documentclass[a4paper,12pt,times,numbered,print,index,draft]{Classes/PhDThesisPSnPDF}
+\documentclass[a4paper,12pt,times,numbered,print,index,draft,chapter]{Classes/PhDThesisPSnPDF}
 
 % ******************************************************************************
 % ******************************* Class Options ********************************
@@ -101,7 +101,7 @@
 % To use choose `chapter' option in the document class
 
 \ifdefineChapter
- \includeonly{Chapter3/chapter3}
+ \includeonly{Chapter2/chapter2}
 \fi
 
 % ******************************** Front Matter ********************************