diff --git a/Chapter2/Figs/BDiagram.pdf b/Chapter2/Figs/BDiagram.pdf
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diff --git a/Chapter2/Figs/Wanniers.mat b/Chapter2/Figs/Wanniers.mat
new file mode 100644
index 0000000..b798206
Binary files /dev/null and b/Chapter2/Figs/Wanniers.mat differ
diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex
index 0cf5f5d..976fe73 100644
--- a/Chapter2/chapter2.tex
+++ b/Chapter2/chapter2.tex
@@ -27,7 +27,11 @@ general Hamiltonian that describes the coupling of atoms with
 far-detuned optical beams \cite{mekhov2012}. This will serve as the
 basis from which we explore the system in different parameter regimes,
 such as nondestructive measurement in free space or quantum
-measurement backaction in a cavity.
+measurement backaction in a cavity. Another interesting direction for
+this field of research are quantum optical lattices where the trapping
+potential is treated quantum mechanically. However this is beyond the
+scope of this work.
+\mynote{insert our paper citations here}
 
 We consider $N$ two-level atoms in an optical lattice with $M$
 sites. For simplicity we will restrict our attention to spinless
@@ -43,14 +47,10 @@ capable of describing a range of different experimental setups ranging
 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
+site which will exhibit the Mott insulator to superfluid quantum phase
 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
@@ -83,9 +83,9 @@ 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
+systems. This simplification allows us to present a much more
+intuitive 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
@@ -199,11 +199,11 @@ equation for the lowering operator
 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})
+  \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})
+  \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
@@ -311,30 +311,49 @@ This contribution can be separated into two parts, one which couples
 directly to the on-site atomic density and one that couples to the
 tunnelling operators. We will define the operator
 \begin{equation}
+  \label{eq:F}
   \hat{F}_{l,m} = \hat{D}_{l,m} + \hat{B}_{l,m},
 \end{equation}
 where $\hat{D}_{l,m}$ is the direct coupling to atomic density
 \begin{equation}
+  \label{eq:D}
   \hat{D}_{l,m} = \sum_{i}^K J^{l,m}_{i,i} \hat{n}_i,
 \end{equation}
 and $\hat{B}_{l,m}$ couples to the matter-field via the
 nearest-neighbour tunnelling operators
 \begin{equation}
+  \label{eq:B}
   \hat{B}_{l,m} = \sum_{\langle i, j \rangle}^K J^{l,m}_{i,j} \bd_i b_j,
 \end{equation}
-and we neglect couplings beyond nearest neighbours for the same reason
-as before when deriving the matter Hamiltonian.
+where $K$ denotes a sum over the illuminated sites and we neglect
+couplings beyond nearest neighbours for the same reason as before when
+deriving the matter Hamiltonian.
+
+\mynote{make sure all group papers are cited here} These equations
+encapsulate the simplicity and flexibility of the measurement scheme
+that we are proposing. The operators given above are entirely
+determined by the values of the $J^{l,m}_{i,j}$ coefficients and
+despite its simplicity, this is sufficient to give rise to a host of
+interesting phenomena via measurement back-action such as the
+generation of multipartite entangled spatial modes in an optical
+lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the appearance
+of long-range correlated tunnelling capable of entangling distant
+lattice sites, and in the case of fermions, the break-up and
+protection of strongly interacting pairs \cite{mazzucchi2016,
+  kozlowski2016zeno}. Additionally, these coefficients are easy to
+manipulate experimentally by adjusting the optical geometry via the
+light mode functions $u_l(\b{r})$.
 
 It is important to note that we are considering a situation where the
 contribution of quantized light is much weaker than that of the
 classical trapping potential. If that was not the case, it would be
-necessary to determine thw Wannier functions in a self-consistent way
+necessary to determine the Wannier functions in a self-consistent way
 which takes into account the depth of the quantum poterntial generated
 by the quantized light modes. This significantly complicates the
 treatment, but can lead to interesting physics. Amongst other things,
 the atomic tunnelling and interaction coefficients will now depend on
-the quantum state of light.
-\mynote{cite Santiago's papers and Maschler/Igor EPJD}
+the quantum state of light.  \mynote{cite Santiago's papers and
+  Maschler/Igor EPJD}
 
 Therefore, combining these final simplifications we finally arrive at
 our quantum light-matter Hamiltonian
@@ -346,18 +365,25 @@ our quantum light-matter Hamiltonian
 \end{equation}
 where we have phenomologically included the cavity decay rates
 $\kappa_l$ of the modes $a_l$. A crucial observation about the
-structure of this Hamiltonian is that in the last term, the light
-modes $a_l$ couple to the matter in a global way. Instead of
+structure of this Hamiltonian is that in the interaction term, the
+light modes $a_l$ couple to the matter in a global way. Instead of
 considering individual coupling to each site, the optical field
 couples to the global state of the atoms within the illuminated region
 via the operator $\hat{F}_{l,m}$. This will have important
 implications for the system and is one of the leading factors
-responsible for many-body behaviour beyong the Bose-Hubbard
-Hamiltonian paradigm.
+responsible for many-body behaviour beyond the Bose-Hubbard
+Hamiltonian paradigm. 
+
+Furthermore, it is also vital to note that light couples to the matter
+via an operator, namely $\hat{F}_{l,m}$, which makes it sensitive to
+the quantum state of matter. This is a key element of our treatment of
+the ultimate quantum regime of light-matter interaction that goes
+beyond previous treatments.
 
 \subsection{Scattered light behaviour}
+\label{sec:a}
 
-Having derived the full quantum light-matter Hamiltonian we will no
+Having derived the full quantum light-matter Hamiltonian we will now
 look at the behaviour of the scattered light. We begin by looking at
 the equations of motion in the Heisenberg picture
 \begin{equation}
@@ -391,6 +417,7 @@ pumping $\eta_l$. We also limit ourselves to only a single scattered
 mode, $a_1$. This leads to a simple linear relationship between the
 light mode and the atomic operator $\hat{F}_{1,0}$
 \begin{equation}
+  \label{eq:a}
   \a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} =
   C \hat{F},
 \end{equation}
@@ -400,19 +427,45 @@ cavity. Furthermore, since there is no longer any ambiguity in the
 indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv
 \Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv
 \hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv
-\hat{D}$ and $\hat{B}_{1,0} \equiv \hat{B}$.
+\hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and the coefficients
+$J^{1,0}_{i,j} \equiv J_{i,j}$.
 
-Whilst the light amplitude itself is only linear in atomic operators,
-we can easily have access to higher moments by simply simply
-considering higher moments of the $\a_1$ such as the photon number
-$\ad_1 \a_1$. Additionally, even though we only consider a single
-scattered mode, this model can be applied to free space by simply
-varying the direction of the scattered light mode if multiple
-scattering events can be neglected. This is likely to be the case
-since the interactions will be dominated by photons scattering from
-the much larger coherent probe.
+The operator $\a_1$ itself is not an observable. However, it is
+possible to combine the outgoing light field with a stronger local
+oscillator beam in order to measure the light quadrature
+\begin{equation}
+  \hat{X}_\phi = \frac{1}{2} \left( \a_1 e^{-i \phi} + \ad_1 e^{i \phi} \right),
+\end{equation}
+which in turn can expressed via the quadrature of $\hat{F}$,
+$\hat{X}^F_\beta$, as
+\begin{equation}
+  \hat{X}_\phi = |C| \hat{X}_\beta^F = \frac{|C|}{2} \left( \hat{F}
+    e^{-i \beta} + \hat{F}^\dagger e^{i \beta} \right),
+\end{equation}
+where $\beta = \phi - \phi_C$, $C = |C| \exp(i \phi_C)$, and $\phi$ is
+the local oscillator phase.
+
+Whilst the light amplitude and the quadratures are only linear in
+atomic operators, we can easily have access to higher moments via
+related quantities such as the photon number
+$\ad_1 \a_1 = |C|^2 \hat{F}^\dagger \hat{F}$ or the quadrature
+variance
+\begin{equation}
+  \label{eq:Xvar}
+  ( \Delta X_\phi )^2 = \langle \hat{X}_\phi^2 \rangle - \langle
+  \hat{X}_\phi \rangle^2 = \frac{1}{4} + |C|^2 (\Delta X^F_\beta)^2,
+\end{equation}
+which reflect atomic correlations and fluctuations.
+
+Finally, even though we only consider a single scattered mode, this
+model can be applied to free space by simply varying the direction of
+the scattered light mode if multiple scattering events can be
+neglected. This is likely to be the case since the interactions will
+be dominated by photons scattering from the much larger coherent
+probe.
 
 \subsection{Density and Phase Observables}
+\label{sec:B}
 
 Light scatters due to its interactions with the dipole moment of the
 atoms which for off-resonant light, like the type that we consider,
@@ -430,21 +483,22 @@ condensates even though the two components have initially well-defined
 atom numbers which is phase's conjugate variable.
 
 In our model light couples to the operator $\hat{F}$ which consists of
-a density opertor part, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a
-phase operator part, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j}
-\bd_i b_j$. Most of the time the density component dominates, $\hat{D}
-\gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However, it is
-possible to engineer an optical geometry in which $\hat{D} = 0$
+a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase
+component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i
+b_j$. In general, the density component dominates,
+$\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$.  However,
+it is possible to engineer an optical geometry in which $\hat{D} = 0$
 leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is
 fundamentally different from the aforementioned double-well proposals
 as it directly couples to the interference terms caused by atoms
 tunnelling rather than combining light scattered from different
 sources.
 
-For clarity we will consider a 1D lattice with lattice spacing $d$
-along the $x$-axis direction, but the results can be applied and
-generalised to higher dimensions. Central to engineering the $\hat{F}$
-operator are the coefficients $J_{i,j}$ given by
+For clarity we will consider a 1D lattice as shown in
+Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the
+$x$-axis direction, but the results can be applied and generalised to
+higher dimensions. Central to engineering the $\hat{F}$ operator are
+the coefficients $J_{i,j}$ given by
 \begin{equation}
   \label{eq:Jcoeff}
   J_{i,j} = \int \mathrm{d} x \,\,\, w(x - x_i) u_1^*(x) u_0(x) w(x - x_j).
@@ -453,156 +507,139 @@ The operators $\hat{B}$ and $\hat{D}$ depend on the values of
 $J_{i,i+1}$ and $J_{i,i}$ respectively. These coefficients are
 determined by the convolution of the coupling strength between the
 probe and scattered light modes, $u_1^*(x)u_0(x)$, with the relevant
-Wannier function overlap shown in Fig. \ref{fig:WannierOverlaps}. For
+Wannier function overlap shown in Fig. \ref{fig:WannierProducts}a. For
 the $\hat{B}$ operator we calculate the convolution with the nearest
-neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$ shown in
-Fig. \ref{fig:WannierOverlaps}c, and for the $\hat{D}$ operator we
-calculate the convolution with the square of the Wannier function at a
-single site, $W_0(x) \equiv w^2(x)$ shown in
-Fig. \ref{fig:WannierOverlaps}b. Therefore, in order to enhance the
-$\hat{B}$ term we need to maximise the overlap between the light modes
-and the nearest neighbour Wannier overlap, $W_1(x)$. This can be
-achieved by concentrating the light between the sites rather than at
-the positions of the atoms. Ideally, one could measure between two
-sites similarly to single-site addressing, which would measure a
-single term $\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$, e.g.~by
-superposing a deeper optical lattice shifted by $d/2$ with respect to
-the original one, catching and measuring the atoms in the new lattice
-sites. A single-shot success rate of atom detection will be small. As
-single-site addressing is challenging, we proceed with the global
-scattering.
+neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$, and for the
+$\hat{D}$ operator we calculate the convolution with the square of the
+Wannier function at a single site, $W_0(x) \equiv w^2(x)$. Therefore,
+in order to enhance the $\hat{B}$ term we need to maximise the overlap
+between the light modes and the nearest neighbour Wannier overlap,
+$W_1(x)$. This can be achieved by concentrating the light between the
+sites rather than at the positions of the atoms. Ideally, one could
+measure between two sites similarly to single-site addressing
+\cite{greiner2009, bloch2011}, which would measure a single term
+$\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$. This could be achieved,
+for example, by superposing a deeper optical lattice shifted by $d/2$
+with respect to the original one, catching and measuring the atoms in
+the new lattice sites. A single-shot success rate of atom detection
+will be small. As single-site addressing is challenging, we proceed
+with the global scattering.
 
-\mynote{Fix labels in this figure}
-\begin{figure}[htbp!]
-  \centering
-  \includegraphics[width=1.0\textwidth]{Wannier1}
-  \includegraphics[width=1.0\textwidth]{Wannier2}
-  \caption[Wannier Function Overlaps]{(a) The Wannier functions
-    corresponding to four neighbouring sites in a 1D
-    lattice. $\lambda$ is the wavelength of the trapping beams, thus
-    lattice sites occur every $\lambda/2$. (Bottom Left) The square of
-    a single Wannier function - this quantity is used when evaluating
-    $\hat{D}$. It's much larger than the overlap between two
-    neighbouring Wannier functions, but it is localised to the
-    position of the lattice site it belongs to. (Bottom Right) The
-    overlap of two neighbouring Wannier functions - this quantity is
-    used when evaluating $\hat{B}$. It is much smaller than the square
-    of a Wannier function, but since it's localised in between the
-    sites, thus $\hat{B}$ can be maximised while $\hat{D}$ minimised
-    by focusing the light in between the sites.}
-  \label{fig:WannierOverlaps}
-\end{figure}
-
-\mynote{show the expansion into an FT}
+\mynote{Potentially expand details of the derivation of these equations}
 In order to calculate the $J_{i,j}$ coefficients we perform numerical
-calculations using realistic Wannier functions. However, it is
-possible to gain some analytic insight into the behaviour of these
-values by looking at the Fourier transforms of the Wannier function
-overlaps, $\mathcal{F}[W_{0,1}](k)$, shown in Fig.
-\ref{fig:WannierFT}b. This is because the light mode product,
-$u_1^*(x) u_0(x)$, can be in general decomposed into a sum of
-oscillating exponentials of the form $e^{i k x}$ making the integral
-in Eq. \eqref{eq:Jcoeff} a sum of Fourier transforms of
-$W_{0,1}(x)$. We consider both the detected and probe beam to be
-standing waves which gives the following expressions for the $\hat{D}$
-and $\hat{B}$ operators
-\begin{eqnarray}
+calculations using realistic Wannier functions
+\cite{walters2013}. However, it is possible to gain some analytic
+insight into the behaviour of these values by looking at the Fourier
+transforms of the Wannier function overlaps,
+$\mathcal{F}[W_{0,1}](k)$, shown in Fig.
+\ref{fig:WannierProducts}b. This is because the for plane and standing
+wave light modes the product $u_1^*(x) u_0(x)$ can be in general
+decomposed into a sum of oscillating exponentials of the form
+$e^{i k x}$ making the integral in Eq. \eqref{eq:Jcoeff} a sum of
+Fourier transforms of $W_{0,1}(x)$. We consider both the detected and
+probe beam to be standing waves which gives the following expressions
+for the $\hat{D}$ and $\hat{B}$ operators
+\begin{align}
   \label{eq:FTs}
-  \hat{D} =
-  \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_- x_m
-    +\varphi_-)
-    \nonumber\\ +\mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+
-    x_m +\varphi_+)], \nonumber\\ \hat{B} =
-  \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m
-    +\frac{k_-d}{2}+\varphi_-)
-    \nonumber\\ +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
-    x_m +\frac{k_+d}{2}+\varphi_+)],
-\end{eqnarray}
-where $k_\pm = k_{0x} \pm k_{1x}$, $k_{(0,1)x} = k_{0,1}
-\sin(\theta_{0,1}$), $\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and
+  \hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_-
+              x_m +\varphi_-) \nonumber\\ 
+            & + \mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+ x_m +\varphi_+)],
+              \nonumber\\ 
+  \hat{B} = & \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m
+              +\frac{k_-d}{2}+\varphi_-) \nonumber\\ 
+            & +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
+              x_m +\frac{k_+d}{2}+\varphi_+)],
+\end{align}
+where $k_\pm = k_{0x} \pm k_{1x}$,
+$k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$),
+$\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and
 $\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the
 $\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the
 $\hat{D}$ operator since it depends on the amplitude of light in
-between the lattice sites and not at the positions of the atoms,
+between the lattice sites and not at the positions of the atoms
 allowing to decouple them at specific angles.
 
-\begin{figure}[htbp!]
-  \begin{center}
-    \includegraphics[width=\linewidth]{WF_S}
-  \end{center}
-  \caption[Wannier Function Fourier Transforms]{The Wannier function
-    products: (a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed
-    line, left axis) and their (b) Fourier transforms
-    $\mathcal{F}[W_{0,1}]$. The Density $J_{i,i}$ and
-    matter-interference $J_{i,i+1}$ coefficients in diffraction
-    maximum (c) and minimum (d) as are shown as functions of standing
-    wave shifts $\varphi$ or, if one were to measure the quadrature
-    variance $(\Delta X^F_\beta)^2$, the local oscillator phase
-    $\beta$. The black points indicate the positions, where light
-    measures matter interference $\hat{B} \ne 0$, and the density-term
-    is suppressed, $\hat{D} = 0$. The trapping potential depth is
-    approximately 5 recoil energies.}
-  \label{fig:WannierFT}
+\begin{figure}[hbtp!]
+  \centering
+  \includegraphics[width=0.8\linewidth]{BDiagram}
+  \caption[Maximising Light-Matter Coupling between Lattice
+  Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
+    between lattice sites. The thin black line indicates the trapping
+    potential (not to scale). (a) Arrangement for the uniform pattern
+    $J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
+    $J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
+    is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
 \end{figure}
 
-The simplest case is to find a diffraction maximum where $J_{i,i+1} =
-J_B$. This can be achieved by crossing the light modes such that
-$\theta_0 = -\theta_1$ and $k_{0x} = k_{1x} = \pi/d$ and choosing the
-light mode phases such that $\varphi_+ = 0$. Fig. \ref{fig:WannierFT}c
-shows the value of the $J_{i,j}$ coefficients under these
-circumstances. In order to make the $\hat{B}$ contribution to light
-scattering dominant we need to set $\hat{D} = 0$ which from
-Eq. \eqref{eq:FTs} we see is possible if $\varphi_0 = -\varphi_1 =
-\arccos[-\mathcal{F}[W_0](2\pi/d)/\mathcal{F}[W_0](0)]/2$. This
-arrangement of light modes maximizes the interference signal,
+\begin{figure}[hbtp!]
+  \centering
+  \includegraphics[width=\linewidth]{WF_S}
+  \caption[Wannier Function Products]{The Wannier function products:
+    (a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
+    axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
+    Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
+    in diffraction maximum (c) and minimum (d) as are shown as
+    functions of standing wave shifts $\varphi$ or, if one were to
+    measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
+    oscillator phase $\beta$. The black points indicate the positions,
+    where light measures matter interference $\hat{B} \ne 0$, and the
+    density-term is suppressed, $\hat{D} = 0$. The trapping potential
+    depth is approximately 5 recoil energies.}
+  \label{fig:WannierProducts}
+\end{figure}
+
+The simplest case is to find a diffraction maximum where
+$J_{i,i+1} = J_1$, where $J_1$ is a constant. This can be achieved by
+crossing the light modes such that $\theta_0 = -\theta_1$ and
+$k_{0x} = k_{1x} = \pi/d$ and choosing the light mode phases such that
+$\varphi_+ = 0$. Fig. \ref{fig:BDiagram}a shows the resulting light
+mode functions and their product along the lattice and
+Fig. \ref{fig:WannierProducts}c shows the value of the $J_{i,j}$
+coefficients under these circumstances. In order to make the $\hat{B}$
+contribution to light scattering dominant we need to set $\hat{D} = 0$
+which from Eq. \eqref{eq:FTs} we see is possible if
+\begin{equation}
+  \xi \equiv \varphi_0 = -\varphi_1 =
+  \frac{1}{2}\arccos[-\mathcal{F}[W_0]\left(\frac{2\pi}{d}\right)/\mathcal{F}[W_0](0)].
+\end{equation}
+This arrangement of light modes maximizes the interference signal,
 $\hat{B}$, by suppressing the density signal, $\hat{D}$, via
-interference compensating for the spreading of the Wannier
-functions. 
+interference compensating for the spreading of the Wannier functions.
 
 Another possibility is to obtain an alternating pattern similar
-corresponding to a classical diffraction minimum. We consider an
-arrangement where the beams are arranged such that $k_{0x} = 0$ and
+corresponding to a diffraction minimum. We consider an arrangement
+where the beams are arranged such that $k_{0x} = 0$ and
 $k_{1x} = \pi/d$ which gives the following expressions for the density
 and interference terms
-\begin{eqnarray}
+\begin{align}
   \label{eq:DMin}
-	\hat{D} = \mathcal{F}[W_0](\pi/d) \sum_m (-1)^m \hat{n}_m
-        \cos(\varphi_0) \cos(\varphi_1) \nonumber \\ \hat{B} =
-        -\mathcal{F}[W_1](\pi/d) \sum_m (-1)^m \hat{B}_m
-        \cos(\varphi_0) \sin(\varphi_1).
-\end{eqnarray}
-The corresponding $J_{i,j}$ coefficients are shown in
-Fig. \ref{fig:WannierFT}d for $\varphi_0=0$. It is clear that for
-$\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is intuitive as this
-places the lattice sites at the nodes of the mode $u_1(x)$. This is a
-diffraction minimum as the light amplitude is also zero, $\langle
-\hat{B} \rangle = 0$, because contributions from alternating
-inter-site regions interfere destructively. However, the intensity
-$\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$ is
-proportional to the variance of $\hat{B}$ and is non-zero. 
+  \hat{D} = & \mathcal{F}[W_0](\pi/d) \sum_m (-1)^m \hat{n}_m
+              \cos(\varphi_0) \cos(\varphi_1) \nonumber \\ 
+  \hat{B} = & -\mathcal{F}[W_1](\pi/d) \sum_m (-1)^m \hat{B}_m
+              \cos(\varphi_0) \sin(\varphi_1).
+\end{align}
+The corresponding $J_{i,j}$ coefficients are given by
+$J_{i,i+1} = -(-1)^i J_2$, where $J_2$ is a constant, and are shown in
+Fig. \ref{fig:WannierProducts}d for $\varphi_0=0$. The light mode
+coupling along the lattice is shown in Fig. \ref{fig:BDiagram}b. It is
+clear that for $\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is
+intuitive as this places the lattice sites at the nodes of the mode
+$u_1(x)$. This is a diffraction minimum as the light amplitude is also
+zero, $\langle \hat{B} \rangle = 0$, because contributions from
+alternating inter-site regions interfere destructively. However, the
+intensity $\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$
+is proportional to the variance of $\hat{B}$ and is non-zero.
 
-\mynote{explain quadrature}
 Alternatively, one can use the arrangement for a diffraction minimum
 described above, but use travelling instead of standing waves for the
-probe and detected beams and measure the light quadrature variance. In
-this case $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B}
-\sin(\beta)$ and by varying the local oscillator phase, one can choose
-which conjugate operator to measure.
-
-\mynote{fix labels}
-\begin{figure}[hbtp!]
-	\includegraphics[width=\linewidth]{BDiagram}
-	\caption[Maximising Light-Matter Coupling between Lattice
-          Sites]{Light field arrangements which maximise coupling,
-          $u_1^*u_0$, between lattice sites. The thin black line
-          indicates the trapping potential (not to scale). (a)
-          Arrangement for the uniform pattern $J_{m,m+1} = J_1$. (b)
-          Arrangement for spatially varying pattern $J_{m,m+1}=(-1)^m
-          J_2$; here $u_0=1$ so it is not shown and $u_1$ is real thus
-          $u_1^*u_0=u_1$. \label{fig:BDiagram}}
-\end{figure}
+probe and detected beams and measure the light quadrature variance.
+In this case
+$\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ and by
+varying the local oscillator phase, one can choose which conjugate
+operator to measure.
 
 \subsection{Electric Field Stength}
+\label{sec:Efield}
 
 The Electric field operator at position $\b{r}$ and at time $t$ is
 usually written in terms of its positive and negative components:
@@ -634,7 +671,7 @@ atom located at $\b{r}^\prime$ at the observation point $\b{r}$ is
 given by
 \begin{equation}
   \label{eq:Ep}
-  \b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d \sin \eta}{4 \pi
+  \b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d_A \sin \eta}{4 \pi
     \epsilon_0 c^2 |\b{r} - \b{r}^\prime|} \hat{\epsilon} \sigma^-
   \left( \b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right),
 \end{equation}
@@ -644,10 +681,9 @@ in the far field, $\eta$ is the angle the dipole makes with
 $\b{r} - \b{r}^\prime$, $\hat{\epsilon}$ is the polarization vector
 which is perpendicular to $\b{r} - \b{r}^\prime$ and lies in the plane
 defined by $\b{r} - \b{r}^\prime$ and the dipole, $\omega_a$ is the
-atomic transition frequency, and $d$ is the dipole matrix element
+atomic transition frequency, and $d_A$ is the dipole matrix element
 between the two levels, and $c$ is the speed of light in vacuum.
 
-
 We have already derived an expression for the atomic lowering
 operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by
 \begin{equation}
diff --git a/Chapter3/Figs/Ep1.pdf b/Chapter3/Figs/Ep1.pdf
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diff --git a/Chapter3/Figs/WF_S.pdf b/Chapter3/Figs/WF_S.pdf
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diff --git a/Chapter3/Figs/oph11.pdf b/Chapter3/Figs/oph11.pdf
new file mode 100644
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diff --git a/Chapter3/Figs/oph22.pdf b/Chapter3/Figs/oph22.pdf
new file mode 100644
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diff --git a/Chapter3/chapter3.tex b/Chapter3/chapter3.tex
index 42b2a5e..f65d7ee 100644
--- a/Chapter3/chapter3.tex
+++ b/Chapter3/chapter3.tex
@@ -2,8 +2,8 @@
 %*********************************** Third Chapter *****************************
 %*******************************************************************************
 
-\chapter{Probing Matter-Field and Atom-Number Correlations in Optical Lattices
-  by Global Nondestructive Addressing}  %Title of the Third Chapter
+\chapter{Probing Correlations by Global 
+Nondestructive Addressing} %Title of the Third Chapter
 
 \ifpdf
     \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}}
@@ -13,3 +13,481 @@
 
 
 %********************************** %First Section  **************************************
+
+\section{Introduction}
+
+Having developed the basic theoretical framework within which we can
+treat the fully quantum regime of light-matter interactions we now
+consider possible applications. There are three prominent directions
+in which we can apply our model: nondestructive probing, quantum
+measurement backaction and quantum optical lattices. Here, we deal
+with the first of the three options.
+
+\mynote{adjust for the fact that the derivation of B operator has been moved}
+\mynote{update some outdated mentions to previous experiments}
+In this chapter we develop a method to measure properties of ultracold
+gases in optical lattices by light scattering. We show that such
+measurements can reveal not only density correlations, but also
+matter-field interference.  Recent quantum non-demolition (QND)
+schemes \cite{rogers2014, mekhov2007prl, eckert2008} probe density
+fluctuations and thus inevitably destroy information about phase,
+i.e.~the conjugate variable, and as a consequence destroy matter-field
+coherence. In contrast, we focus on probing the atom interference
+between lattice sites. Our scheme is nondestructive in contrast to
+totally destructive methods such as time-of-flight measurements. It
+enables in-situ probing of the matter-field coherence at its shortest
+possible distance in an optical lattice, i.e. the lattice period,
+which defines key processes such as tunnelling, currents, phase
+gradients, etc. This is achieved by concentrating light between the
+sites. By contrast, standard destructive time-of-flight measurements
+deal with far-field interference and a relatively near-field one was
+used in Ref. \cite{miyake2011}. Such a counter-intuitive configuration
+may affect works on quantum gases trapped in quantum potentials
+\cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013,
+  ivanov2014, caballero2015} and quantum measurement-induced
+preparation of many-body atomic states \cite{mazzucchi2016,
+  mekhov2009prl, pedersen2014, elliott2015}. Within the mean-field
+treatment, this enables measurements of the order parameter,
+matter-field quadratures and squeezing. This can have an impact on
+atom-wave metrology and information processing in areas where quantum
+optics already made progress, e.g., quantum imaging with pixellized
+sources of non-classical light \cite{golubev2010, kolobov1999}, as an
+optical lattice is a natural source of multimode nonclassical matter
+waves.
+
+Furthermore, the scattering angular distribution is nontrivial, even
+when classical diffraction is forbidden and we derive generalized
+Bragg conditions for this situation. The method works beyond
+mean-field, which we demonstrate by distinguishing all three phases in
+the Mott insulator - superfluid - Bose glass phase transition in a 1D
+disordered optical lattice. We underline that transitions in 1D are
+much more visible when changing an atomic density rather than for
+fixed-density scattering. It was only recently that an experiment
+distinguished a Mott insulator from a Bose glass \cite{derrico2014}.
+
+\section{Global Nondestructive Measurement}
+
+As we have seen in section \ref{sec:a} under certain approximations
+the scattered light mode, $\a_1$, is linked to the quantum state of
+matter via 
+\begin{equation}
+  \label{eq:a-3}
+  \a_1 = C \hat{F} = C \left(\hat{D} + \hat{B} \right),
+\end{equation}
+where the atomic operators $\hat{D}$ and $\hat{B}$, given by
+Eq. \eqref{eq:D} and Eq. \eqref{eq:B}, are responsible for the
+coupling to on-site density and inter-site interference
+respectively. It crucial to note that light couples to the bosons via
+an operator as this makes it sensitive to the quantum state of the
+matter.
+
+Here, we will use this fact that the light is sensitive to the atomic
+quantum state due to the coupling of the optical and matter fields via
+operators in order to develop a method to probe the properties of an
+ultracold gas. Therefore, we neglect the measurement back-action and
+we will only consider expectation values of light observables. Since
+the scheme is nondestructive (in some cases, it even satisfies the
+stricter requirements for a QND measurement \cite{mekhov2007pra,
+  mekhov2012}) and the measurement only weakly perturbs the system,
+many consecutive measurements can be carried out with the same atoms
+without preparing a new sample. Again, we will show how the extreme
+flexibility of the the measurement operator $\hat{F}$ allows us to
+probe a variety of different atomic properties in-situ ranging from
+density correlations to matter-field interference.
+
+\subsection{On-site density measurements}
+
+Typically, the dominant term in $\hat{F}$ is the density term
+$\hat{D}$, rather than inter-site matter-field interference $\hat{B}$
+\cite{mekhov2007pra, rist2010, lakomy2009, ruostekoski2009,
+  LP2009}. However, before we move onto probing the interference
+terms, $\hat{B}$, we will first discuss typical light scattering. We
+start with a simpler case when scattering is faster than tunneling and
+$\hat{F} = \hat{D}$. This corresponds to a QND scheme
+\cite{mekhov2007prl, mekhov2007pra, eckert2008, rogers2014}. The
+density-related measurement destroys some matter-phase coherence in
+the conjugate variable \cite{mekhov2009pra, LP2010, LP2011}
+$\bd_i b_{i+1}$, but this term is neglected. For this purpose we will
+define an auxiliary quantity,
+\begin{equation}
+  \label{eq:R}
+  R = \langle \ad_1 \a_1 \rangle - | \langle \a_1 \rangle |^2,
+\end{equation}
+which we will call the ``quantum addition'' to light scattering. $R$
+is simply the full light intensity minus the classical field intensity
+and thus it faithfully represents the new contribution from the
+quantum light-matter interaction to the diffraction pattern.
+
+\begin{figure}[htbp!]
+  \centering
+  \includegraphics[width=\linewidth]{Ep1}
+  \caption[Light Scattering Angular Distribution]{Light intensity
+    scattered into a standing wave mode from a superfluid in a 3D
+    lattice (units of $R/N_K$). Arrows denote incoming travelling wave
+    probes. The Bragg condition, $\Delta \b{k} = \b{G}$, is not
+    fulfilled, so there is no classical diffraction, but intensity
+    still shows multiple peaks, whose heights are tunable by simple
+    phase shifts of the optical beams: (a) $\varphi_1=0$; (b)
+    $\varphi_1=\pi/2$. Interestingly, there is also a significant
+    uniform background level of scattering which does not occur in its
+    classical counterpart. }
+  \label{fig:Scattering}
+\end{figure}
+
+In a deep lattice,
+\begin{equation} 
+  \hat{D}=\sum_i^K u_1^*({\bf r}_i) u_0({\bf r}_i) \hat{n}_i,
+\end{equation} 
+which for travelling
+[$u_l(\b{r})=\exp(i \b{k}_l \cdot \b{r}+i\varphi_l)$] or standing
+[$u_l(\b{r})=\cos(\b{k}_l \cdot \b{r}+\varphi_l)$] waves is just a
+density Fourier transform at one or several wave vectors
+$\pm(\b{k}_1 \pm \b{k}_0)$. The quadrature, as defined in section
+\ref{sec:a}, for two travelling waves is reduced to
+\begin{equation} 
+  \hat{X}^F_\beta = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot
+  \b{r}_i - \beta].
+\end{equation} 
+Note that different light quadratures are differently coupled to the
+atom distribution, hence varying local oscillator phase and detection
+angle, one scans the coupling from maximal to zero. An identical
+expression exists for $\hat{D}$ for a standing wave, where $\beta$ is
+replaced by $\varphi_l$, and scanning is achieved by varying the
+position of the wave with respect to atoms. Thus, variance
+$(\Delta X^F_\beta)^2$ and quantum addition $R$, have a non-trivial
+angular dependence, showing more peaks than classical diffraction and
+the peaks can be tuned by the light-atom coupling.
+
+Fig. \ref{fig:Scattering} shows the angular dependence of $R$ for
+standing and travelling waves scattering from bosons in a 3D optical
+lattice. The isotropic background gives the density fluctuations
+[$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$ in
+mean-field with inter-site correlations neglected]. The radius of the
+sphere changes from zero, when it is a Mott insulator with suppressed
+fluctuations, to half the atom number at $K$ sites, $N_K/2$, in the
+deep superfluid. There exist peaks at angles different than the
+classical Bragg ones and thus, can be observed without being masked by
+classical diffraction. Interestingly, even if 3D diffraction
+\cite{miyake2011} is forbidden as seen in Fig. \ref{fig:Scattering},
+the peaks are still present. As $(\Delta X^F_\beta)^2$ and $R$ are
+quadratic variables, the generalized Bragg conditions for the peaks
+are $2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves,
+where $\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal
+lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and
+travelling $\a_0$, which is clearly different from the classical Bragg
+condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
+local oscillator phase or standing wave shift as seen in Fig.
+\ref{fig:Scattering}b.
+
+In section \ref{sec:Efield} we have estimated the mean photon
+scattering rates integrated over the solid angle for the only two
+experiments so far on light diffraction from truly ultracold bosons
+where the measurement object was light
+\begin{equation} 
+  n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8}
+  (\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2).
+\end{equation} 
+The background signal should reach $n_\Phi \approx 10^6$ s$^{-1}$ in
+Ref. \cite{weitenberg2011} (150 atoms in 2D), and
+$n_\Phi \approx 10^{11}$ s$^{-1}$ in Ref. \cite{miyake2011} ($10^5$
+atoms in 3D). These numbers show that the diffraction patterns we have
+seen due to the ``quantum addition'' should be visible using currently
+available technology, especially since the most prominent features,
+such as Bragg diffraction peaks, do not coincide at all with the
+classical diffraction pattern.
+
+\subsection{Matter-field interference measurements}
+
+We now focus on enhancing the interference term $\hat{B}$ in the
+operator $\hat{F}$. 
+
+Firstly, we will use this result to show how one can probe
+$\langle \hat{B} \rangle$ which in MF gives information about the
+matter-field amplitude, $\Phi = \langle b \rangle$. 
+
+Hence, by measuring the light quadrature we probe the kinetic energy
+and, in MF, the matter-field amplitude (order parameter) $\Phi$:
+$\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2
+\mathcal{F}[W_1](2\pi/d) (K-1)$.
+
+Secondly, we show that it is also possible to access the fluctuations
+of matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd
+e^{i\alpha})/2$, which in MF can be probed by measuring the variance
+of $\hat{B}$. Across the phase transition, the matter field changes
+its state from Fock (in MI) to coherent (deep SF) through an
+amplitude-squeezed state as shown in Fig. \ref{Quads}(a,b). 
+
+Assuming $\Phi$ is real in MF:
+\begin{equation}
+  \label{intensity} 
+  \langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1)\mathcal{F}^2[W_1](\frac{\pi}{d})
+  \times [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ]
+\end{equation} 
+and it is shown as a function of $U/(zJ^\text{cl})$ in
+Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum
+yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the
+intensity. This quantity is of great interest as it gives us access to
+the quadrature variances of the matter-field
+\begin{equation} 
+  (\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm
+  (\langle b^2 \rangle - \Phi^2)]/2,
+\end{equation} 
+where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density.
+
+\begin{figure}[htbp!]
+  \centering
+  \includegraphics[width=\linewidth]{Quads}
+  \captionsetup{justification=centerlast,font=small}
+  \caption[Mean-Field Matter Quadratures]{Photon number scattered in a
+    diffraction minimum, given by Eq. (\ref{intensity}), where
+    $\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$.  More
+    light is scattered from a MI than a SF due to the large
+    uncertainty in phase in the insulator. (a) The variances of
+    quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$
+    (dashed) of the matter field across the phase transition. Level
+    1/4 is the minimal (Heisenberg) uncertainty. There are three
+    important points along the phase transition: the coherent state
+    (SF) at A, the amplitude-squeezed state at B, and the Fock state
+    (MI) at C. (b) The uncertainties plotted in phase space.}
+	\label{Quads}
+\end{figure}
+
+Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations
+with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring
+the photon number variance, which is standard in quantum optics, will
+lead up to 8-point correlations similar to 4-point density
+correlations \cite{mekhov2007pra}. These are of significant interest,
+because it has been shown that there are quantum entangled states that
+manifest themselves only in high-order correlations
+\cite{kaszlikowski2008}.
+
+Surprisingly, inter-site terms scatter more light from a Mott
+insulator than a superfluid Eq. \eqref{intensity}, as shown in
+Fig. \eqref{Quads}, although the mean inter-site density
+$\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a
+fundamental effect of the boson interference in Fock states. It indeed
+happens between two sites, but as the phase is uncertain, it results
+in the large variance of $\hat{n}(\b{r})$ captured by light as shown
+in Eq. \eqref{intensity}. The interference between two macroscopic
+BECs has been observed and studied theoretically
+\cite{horak1999}. When two BECs in Fock states interfere a phase
+difference is established between them and an interference pattern is
+observed which disappears when the results are averaged over a large
+number of experimental realizations. This reflects the large
+shot-to-shot phase fluctuations corresponding to a large inter-site
+variance of $\hat{n}(\b{r})$. By contrast, our method enables the
+observation of such phase uncertainty in a Fock state directly between
+lattice sites on the microscopic scale in-situ.
+
+\subsection{Mapping the quantum phase diagram}
+
+\begin{figure}[htbp!]  
+  \centering
+  \includegraphics[width=\linewidth]{oph11}
+  \caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular
+    dependence of scattered light $R$ for a superfluid (thin black,
+    left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green,
+    right scale, $U/2J^\text{cl} =10$). The two phases differ in both
+    their value of $R_\text{max}$ as well as $W_R$ showing that
+    density correlations in the two phases differ in magnitude as well
+    as extent. Light scattering maximum $R_\text{max}$ is shown in (b,
+    d) and the width $W_R$ in (c, e).  It is very clear that varying
+    chemical potential $\mu$ or density $\langle n\rangle$ sharply
+    identifies the superfluid-Mott insulator transition in both
+    quantities. (b) and (c) are cross-sections of the phase diagrams
+    (d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple),
+    and 4 (dashed blue). Insets show density dependencies for the
+    $U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.}
+	\label{fig:SFMI}
+\end{figure}
+
+We have shown how in mean-field, we can track the order parameter,
+$\Phi$, by probing the matter-field interference using the coupling of
+light to the $\hat{B}$ operator. In this case, it is very easy to
+follow the superfluid to Mott insulator quantum phase transition since
+we have direct access to the order parameter which goes to zero in the
+insulating phase. In fact, if we're only interested in the critical
+point, we only need access to any quantity that yields information
+about density fluctuations which also go to zero in the MI phase and
+this can be obtained by measuring
+$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many
+situations where the mean-field approximation is not a valid
+description of the physics. A prominent example is the Bose-Hubbard
+model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012,
+  pino2013}. Observing the transition in 1D by light at fixed density
+was considered to be difficult \cite{rogers2014} or even impossible
+\cite{roth2003}. By contrast, here we propose varying the density or
+chemical potential, which sharply identifies the transition. We
+perform these calculations numerically by calculating the ground state
+using DMRG methods \cite{tnt} from which we can compute all the
+necessary atomic observables. Experiments typically use an additional
+harmonic confining potential on top of the optical lattice to keep the
+atoms in place which means that the chemical potential will vary in
+space. However, with careful consideration of the full
+($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
+Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
+system \cite{batrouni2002}.
+
+The 1D phase transition is best understood in terms of two-point
+correlations as a function of their separation \cite{giamarchi}. In
+the Mott insulating phase, the two-point correlations
+$\langle \bd_i b_j \rangle$ and
+$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
+($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
+exponentially with $|i-j|$. On the other hand the superfluid will
+exhibit long-range order which in dimensions higher than one,
+manifests itself with an infinite correlation length. However, in 1D
+only pseudo long-range order happens and both the matter-field and
+density fluctuation correlations decay algebraically \cite{giamarchi}.
+
+The method we propose gives us direct access to the structure factor,
+which is a function of the two-point correlation $\langle \delta
+\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light
+intensity. For two travelling waves maximally coupled to the density
+(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the
+quantum addition is given by
+\begin{equation} 
+  R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
+  (\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
+  \hat{n}_j \rangle,
+\end{equation}
+
+The angular dependence of $R$ for a Mott insulator and a superfluid is
+shown in Fig. \ref{fig:SFMI}a, and there are two variables
+distinguishing the states. Firstly, maximal $R$,
+$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
+probes the fluctuations and compressibility $\kappa'$
+($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
+\rangle$).  The Mott insulator is incompressible and thus will have
+very small on-site fluctuations and it will scatter little light
+leading to a small $R_\text{max}$. The deeper the system is in the MI
+phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller
+these values will be until ultimately it will scatter no light at all
+in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can
+be seen in the value of the peak in $R$. The value $R_\text{max}$ in
+the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI
+phase ($U/2J^\text{cl} = 10$) by a factor of
+$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
+$R_\text{max}$ changes across the phase transition. We see that the
+transition shows up very sharply as $\mu$ is varied.
+
+Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$
+is a direct measure of the correlation length $l$, $W_R \propto
+1/l$. The Mott insulator being an insulating phase is characterised by
+exponentially decaying correlations and as such it will have a very
+large $W_R$. However, the superfluid in 1D exhibits pseudo long-range
+order which manifests itself in algebraically decaying two-point
+correlations \cite{giamarchi} which significantly reduces the dip in
+the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see
+that this identifies the phase transition very sharply as $\mu$ is
+varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with
+experimentally measuring $W_R$ is that it might be obstructed by the
+classical diffraction maxima which appear at angles corresponding to
+the minima in $R$. However, the width of such a peak is much smaller
+as its width is proportional to $1/M$.
+
+It is also possible to analyse the phase transition quantitatively
+using our method. Unlike in higher dimensions where an order parameter
+can be easily defined within the MF approximation there is no such
+quantity in 1D. However, a valid description of the relevant 1D low
+energy physics is provided by Luttinger liquid theory
+\cite{giamarchi}. In this model correlations in the supefluid phase as
+well as the superfluid density itself are characterised by the
+Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
+the phase transition in the thermodynamic limit at $K_b = 1/2$. This
+quantity can be extracted from various correlation functions and in
+our case it can be extracted directly from $R$ \cite{ejima2011}. By
+extracting this parameter from $R$ for various lattice lengths from
+numerical DMRG calculations it was even possible to give a theoretical
+estimate of the critical point for commensurate filling, $N = M$, in
+the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$
+\cite{ejima2011}. Our proposal provides a method to directly measure
+$R$ in a lab which can then be used to experimentally determine the
+location of the critical point in 1D.
+
+So far both variables we considered, $R_\text{max}$ and $W_R$, provide
+similar information. Next, we present a case where it is very
+different. The Bose glass is a localized insulating phase with
+exponentially decaying correlations but large compressibility and
+on-site fluctuations in a disordered optical lattice. Therefore,
+measuring both $R_\text{max}$ and $W_R$ will distinguish all the
+phases. In a Bose glass we have finite compressibility, but
+exponentially decaying correlations. This gives a large $R_\text{max}$
+and a large $W_R$. A Mott insulator will also have exponentially
+decaying correlations since it is an insulator, but it will be
+incompressible. Thus, it will scatter light with a small
+$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
+range correlations and large compressibility which results in a large
+$R_\text{max}$ and a small $W_R$.
+
+\begin{figure}[htbp!]  
+  \centering
+  \includegraphics[width=\linewidth]{oph22}
+  \caption[Mapping the Disoredered Phase Diagram]{The
+    Mott-superfluid-glass phase diagrams for light scattering maximum
+    $R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both
+    quantities distinguish all three phases. Transition lines are
+    shifted due to finite size effects \cite{roux2008}, but it is
+    possible to apply well known numerical methods to extract these
+    transition lines from such experimental data extracted from $R$
+    \cite{ejima2011}. $K=M=N=35$.}
+  \label{fig:BG}
+\end{figure}
+
+We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of
+superlattice- to trapping lattice-period $r\approx 0.77$ for various
+disorder strengths $V$ \cite{roux2008}. Here, we only consider
+calculations for a fixed density, because the usual interpretation of
+the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane
+for a fixed ratio $V/U$ becomes complicated due to the presence of
+multiple compressible and incompressible phases between successive MI
+lobes \cite{roux2008}. This way, we have limited our parameter space
+to the three phases we are interested in: superfluid, Mott insulator,
+and Bose glass. From Fig. \ref{fig:BG} we see that all three phases
+can indeed be distinguished. In the 1D BHM there is no sharp MI-SF
+phase transition in 1D at a fixed density \cite{cazalilla2011,
+  ejima2011, kuhner2000, pino2012, pino2013} just like in
+Figs. \ref{fig:SFMI}(d,e) if we follow the transition through the tip
+of the lobe which corresponds to a line of unit density. However,
+despite the lack of an easily distinguishable critical point it is
+possible to quantitatively extract the location of the transition
+lines by extracting the Tomonaga-Luttinger parameter from the
+scattered light, $R$, in the same way it was done for an unperturbed
+BHM \cite{ejima2011}.
+
+Only recently \cite{derrico2014} a Bose glass phase was studied by
+combined measurements of coherence, transport, and excitation spectra,
+all of which are destructive techniques. Our method is simpler as it
+only requires measurement of the quantity $R$ and additionally, it is
+nondestructive.
+
+\section{Conclusions}
+
+In summary, we proposed a nondestructive method to probe quantum gases
+in an optical lattice. Firstly, we showed that the density-term in
+scattering has an angular distribution richer than classical
+diffraction, derived generalized Bragg conditions, and estimated
+parameters for the only two relevant experiments to date
+\cite{weitenberg2011, miyake2011}. Secondly, we proposed how to
+measure the matter-field interference by concentrating light between
+the sites. This corresponds to interference at the shortest possible
+distance in an optical lattice. By contrast, standard destructive
+time-of-flight measurements deal with far-field interference and a
+relatively near-field one was used in Ref. \cite{miyake2011}. This
+defines most processes in optical lattices. E.g. matter-field phase
+changes may happen not only due to external gradients, but also due to
+intriguing effects such quantum jumps leading to phase flips at
+neighbouring sites and sudden cancellation of tunneling
+\cite{vukics2007}, which should be accessible by our method. In
+mean-field, one can measure the matter-field amplitude (order
+parameter), quadratures and squeezing. This can link atom optics to
+areas where quantum optics has already made progress, e.g., quantum
+imaging \cite{golubev2010, kolobov1999}, using an optical lattice as
+an array of multimode nonclassical matter-field sources with a high
+degree of entanglement for quantum information processing. Thirdly, we
+demonstrated how the method accesses effects beyond mean-field and
+distinguishes all the phases in the Mott-superfluid-glass transition,
+which is currently a challenge \cite{derrico2014}. Based on
+off-resonant scattering, and thus being insensitive to a detailed
+atomic level structure, the method can be extended to molecules
+\cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.
diff --git a/Preamble/preamble.tex b/Preamble/preamble.tex
index 8bd434a..c4319ea 100644
--- a/Preamble/preamble.tex
+++ b/Preamble/preamble.tex
@@ -92,7 +92,7 @@
 % Choose linespacing as appropriate. Default is one-half line spacing as per the
 % University guidelines
 
-% \doublespacing
+ \doublespacing
 % \onehalfspacing
 % \singlespacing
 
diff --git a/References/references.bib b/References/references.bib
index 07b8e33..add4bb6 100644
--- a/References/references.bib
+++ b/References/references.bib
@@ -1,6 +1,258 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Books, theses, reference material
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+@book{foot,
+  author = {Foot, C. J.},
+  title = {{Atomic Physics}},
+  publisher = {Oxford University Press},
+  year = {2005}
+}
+@book{giamarchi,
+  author = {Giamarchi, T.},
+  title = {{Quantum Physics in One Dimension}},
+  publisher = {Clarendon Press, Oxford},
+  year = {2003}
+}
+@phdthesis{weitenbergThesis,
+  author = {Weitenberg, Christof},
+  number = {April},
+  title = {{Single-Atom Resolved Imaging and Manipulation in an Atomic
+                  Mott Insulator}},
+  year = {2011}
+}
+@article{steck,
+  author = {Steck, Daniel Adam},
+  title = {{Rubidium 87 D Line Data Author contact information :}},
+  url = {http://steck.us/alkalidata}
+}
+@inbook{Scully,
+  author = {Scully, M. and Zubairy, S.},
+  title = {{Quantum Optics}},
+  publisher = {Cambridge University Press},
+  chapter = {10.1},
+  pages = {293},
+  year = {1997}
+}
+@misc{tnt,
+  howpublished="\url{http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/}"
+}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Igor's original papers
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+@article{mekhov2007prl,
+  title={Cavity-enhanced light scattering in optical lattices to probe
+                  atomic quantum statistics},
+  author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
+  journal={Physical review letters},
+  volume={98},
+  number={10},
+  pages={100402},
+  year={2007},
+  publisher={APS}
+}
+@article{mekhov2007pra,
+  title={Light scattering from ultracold atoms in optical lattices as
+                  an optical probe of quantum statistics},
+  author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
+  journal={Physical Review A},
+  volume={76},
+  number={5},
+  pages={053618},
+  year={2007},
+  publisher={APS}
+}
+@article{mekhov2008,
+  title = {Dicke quantum phase transition with a superfluid gas in an
+                  optical cavity},
+  author = {Maschler, C. and Mekhov, I. B. and Ritsch, H.},
+  journal = {Eur. Phys. J. D},
+  volume = {146},
+  pages = {545},
+  year = {2008},
+}
+@article{mekhov2009prl,
+  author = {Mekhov, Igor B and Ritsch, Helmut},
+  doi = {10.1103/PhysRevLett.102.020403},
+  journal = {Phys. Rev. Lett.},
+  month = jan,
+  number = {2},
+  pages = {020403},
+  title = {{Quantum Nondemolition Measurements and State Preparation
+                  in Quantum Gases by Light Detection}},
+  volume = {102},
+  year = {2009}
+}
+@article{mekhov2009pra,
+  author = {Mekhov, Igor B and Ritsch, Helmut},
+  doi = {10.1103/PhysRevA.80.013604},
+  journal = {Phys. Rev. A},
+  month = jul,
+  number = {1},
+  pages = {013604},
+  title = {{Quantum optics with quantum gases: Controlled state
+                  reduction by designed light scattering}},
+  volume = {80},
+  year = {2009}
+}
+@article{LP2009,
+  title={Quantum optics with quantum gases},
+  author={Mekhov, Igor B and Ritsch, Helmut},
+  journal={Laser physics},
+  volume={19},
+  number={4},
+  pages={610--615},
+  year={2009},
+  publisher={Springer}
+}
+@article{LP2010,
+  author = {Mekhov, Igor B and Ritsch, Helmut},
+  doi = {10.1134/S1054660X10050105},
+  journal = {Laser Phys.},
+  volume = {20},
+  pages = {694},
+  title = {Quantum Optical Measurements in Ultracold Gases:
+                  Macroscopic Bose–Einstein Condensates},
+  year = {2010}
+}
+@article{LP2011,
+  author = {Mekhov, Igor B and Ritsch, Helmut},
+  doi = {10.1134/S1054660X11150163},
+  journal = {Laser Phys.},
+  volume = {21},
+  pages = {1486},
+  title = {Atom State Evolution and Collapse in Ultracold Gases during
+                  Light Scattering into a Cavity},
+  year = {2011}
+}
+@article{LP2013,
+  author={Igor B Mekhov},
+  title={Quantum non-demolition detection of polar molecule complexes:
+                  dimers, trimers, tetramers},
+  journal={Laser Phys.},
+  volume={23},
+  number={1},
+  pages={015501},
+  year={2013},
+}
+@article{mekhov2012,
+  title={Quantum optics with ultracold quantum gases: towards the full
+                  quantum regime of the light--matter interaction},
+  author={Mekhov, Igor B and Ritsch, Helmut},
+  journal={Journal of Physics B: Atomic, Molecular and Optical
+                  Physics},
+  volume={45},
+  number={10},
+  pages={102001},
+  year={2012},
+  publisher={IOP Publishing}
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Group papers
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+@article{elliott2015,
+  author = {Elliott, T. J. and Kozlowski, W. and Caballero-Benitez,
+                  S. F. and Mekhov, I. B.},
+  doi = {10.1103/PhysRevLett.114.113604},
+  journal = {Phys. Rev. Lett.},
+  pages = {113604},
+  title = {{Multipartite Entangled Spatial Modes of Ultracold Atoms
+                  Generated and Controlled by Quantum Measurement}},
+  volume = {114},
+  year = {2015}
+}
+@article{atoms2015,
+  title={Probing and Manipulating Fermionic and Bosonic Quantum Gases
+                  with Quantum Light},
+  author={Elliott, Thomas J and Mazzucchi, Gabriel and Kozlowski,
+                  Wojciech and Caballero-Benitez, Santiago F and
+                  Mekhov, Igor B},
+  journal={Atoms},
+  volume={3},
+  number={3},
+  pages={392--406},
+  year={2015},
+  publisher={Multidisciplinary Digital Publishing Institute}
+}
+@article{mazzucchi2016,
+  title = {Quantum measurement-induced dynamics of many-body ultracold
+                  bosonic and fermionic systems in optical lattices},
+  author = {Mazzucchi, Gabriel and Kozlowski, Wojciech and
+                  Caballero-Benitez, Santiago F. and Elliott, Thomas
+                  J. and Mekhov, Igor B.},
+  journal = {Physical Review A},
+  volume = {93},
+  issue = {2},
+  pages = {023632},
+  numpages = {12},
+  year = {2016},
+  month = {Feb},
+  publisher = {American Physical Society}
+}
+@article{kozlowski2016zeno,
+  title={Non-hermitian dynamics in the quantum zeno limit},
+  author={Kozlowski, Wojciech and Caballero-Benitez, Santiago F and Mekhov, Igor B},
+  journal={arXiv preprint arXiv:1510.04857},
+  year={2015}
+}
+@article{mazzucchi2016af,
+  title={Quantum measurement-induced antiferromagnetic order and
+                  density modulations in ultracold Fermi gases in
+                  optical lattices},
+  author={Mazzucchi, Gabriel and Caballero-Benitez, Santiago F and
+                  Mekhov, Igor B},
+  journal={arXiv preprint arXiv:1510.04883},
+  year={2015}
+}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% Other papers
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+@article{walters2013,
+title = {Ab initio derivation of Hubbard models for cold atoms in
+                  optical lattices},
+author = {Walters, R. and Cotugno, G. and Johnson, T. H. and Clark,
+                  S. R. and Jaksch, D.},
+journal = {Phys. Rev. A},
+volume = {87},
+pages = {043613},
+year = {2013},
+month = {Apr},
+doi = {10.1103/PhysRevA.87.043613},
+}
+@article{bloch2011,
+  title = {Single-spin addressing in an atomic Mott insulator},
+  author = {Weitenberg, C. and Endres, M. and Sherson, J. F. and
+                  Cheneau, M. and Schauss, P. and Fukuhara, T. and
+                  Bloch, I. and Kuhr, S.},
+  journal = {Nature},
+  volume = {471},
+  pages = {319--324},
+  year = {2011},
+  doi = {10.1038/nature09827},
+}
+@article{greiner2009,
+  title = {A quantum gas microscope for detecting single atoms in a
+                  Hubbard-regime optical lattice},
+  author = {Bakr, W. S. and Gillen, J. I. and Peng, A. and Folling,
+                  S. and Greiner, M.},
+  journal = {Nature},
+  volume = {462},
+  pages = {74--77},
+  year = {2009},
+  doi = {10.1038/nature08482},
+}
 @article{cirac1996,
-  title={Continuous observation of interference fringes from Bose condensates},
-  author={Cirac, J I and Gardiner, C W and Naraschewski, M and Zoller, P},
+  title={Continuous observation of interference fringes from Bose
+                  condensates},
+  author={Cirac, J I and Gardiner, C W and Naraschewski, M and Zoller,
+                  P},
   journal={Physical Review A},
   volume={54},
   number={5},
@@ -19,7 +271,8 @@
   publisher={APS}
 }
 @article{ruostekoski1997,
-  title={Nondestructive optical measurement of relative phase between two Bose-Einstein condensates},
+  title={Nondestructive optical measurement of relative phase between
+                  two Bose-Einstein condensates},
   author={Ruostekoski, Janne and Walls, Dan F},
   journal={Physical Review A},
   volume={56},
@@ -30,7 +283,8 @@
 }
 @article{ruostekoski1998,
   title={Macroscopic superpositions of Bose-Einstein condensates},
-  author={Ruostekoski, Janne and Collett, M J and Graham, Robert and Walls, Dan F},
+  author={Ruostekoski, Janne and Collett, M J and Graham, Robert and
+                  Walls, Dan F},
   journal={Physical Review A},
   volume={57},
   number={1},
@@ -39,7 +293,8 @@
   publisher={APS}
 }
 @article{ashida2015,
-  title={Diffraction-Unlimited Position Measurement of Ultracold Atoms in an Optical Lattice},
+  title={Diffraction-Unlimited Position Measurement of Ultracold Atoms
+                  in an Optical Lattice},
   author={Ashida, Yuto and Ueda, Masahito},
   journal={Physical review letters},
   volume={115},
@@ -49,14 +304,17 @@
   publisher={APS}
 }
 @article{ashida2015a,
-  title={Multi-Particle Quantum Dynamics under Continuous Observation},
+  title={Multi-Particle Quantum Dynamics under Continuous
+                  Observation},
   author={Ashida, Yuto and Ueda, Masahito},
   journal={arXiv preprint arXiv:1510.04001},
   year={2015}
 }
 @article{rogers2014,
-  title={Characterization of Bose-Hubbard models with quantum nondemolition measurements},
-  author={Rogers, B and Paternostro, M and Sherson, J F and De Chiara, G},
+  title={Characterization of Bose-Hubbard models with quantum
+                  nondemolition measurements},
+  author={Rogers, B and Paternostro, M and Sherson, J F and De Chiara,
+                  G},
   journal={Physical Review A},
   volume={90},
   number={4},
@@ -64,16 +322,6 @@
   year={2014},
   publisher={APS}
 }
-@article{LP2009,
-  title={Quantum optics with quantum gases},
-  author={Mekhov, Igor B and Ritsch, Helmut},
-  journal={Laser physics},
-  volume={19},
-  number={4},
-  pages={610--615},
-  year={2009},
-  publisher={Springer}
-}
 @article{rist2012,
   title={Homodyne detection of matter-wave fields},
   author={Rist, Stefan and Morigi, Giovanna},
@@ -84,26 +332,12 @@
   year={2012},
   publisher={APS}
 }
-@book{foot,
-  author = {Foot, C. J.},
-  title = {{Atomic Physics}},
-  publisher = {Oxford University Press},
-  year = {2005}
-}
-@phdthesis{weitenbergThesis,
-author = {Weitenberg, Christof},
-number = {April},
-title = {{Single-Atom Resolved Imaging and Manipulation in an Atomic Mott Insulator}},
-year = {2011}
-}
-@article{steck,
-author = {Steck, Daniel Adam},
-title = {{Rubidium 87 D Line Data Author contact information :}},
-url = {http://steck.us/alkalidata}
-}
 @article{weitenberg2011,
-  title={Coherent light scattering from a two-dimensional Mott insulator},
-  author={Weitenberg, Christof and Schau{\ss}, Peter and Fukuhara, Takeshi and Cheneau, Marc and Endres, Manuel and Bloch, Immanuel and Kuhr, Stefan},
+  title={Coherent light scattering from a two-dimensional Mott
+                  insulator},
+  author={Weitenberg, Christof and Schau{\ss}, Peter and Fukuhara,
+                  Takeshi and Cheneau, Marc and Endres, Manuel and
+                  Bloch, Immanuel and Kuhr, Stefan},
   journal={Physical review letters},
   volume={106},
   number={21},
@@ -112,8 +346,11 @@ url = {http://steck.us/alkalidata}
   publisher={APS}
 }
 @article{miyake2011,
-  title={Bragg scattering as a probe of atomic wave functions and quantum phase transitions in optical lattices},
-  author={Miyake, Hirokazu and Siviloglou, Georgios A and Puentes, Graciana and Pritchard, David E and Ketterle, Wolfgang and Weld, David M},
+  title={Bragg scattering as a probe of atomic wave functions and
+                  quantum phase transitions in optical lattices},
+  author={Miyake, Hirokazu and Siviloglou, Georgios A and Puentes,
+                  Graciana and Pritchard, David E and Ketterle,
+                  Wolfgang and Weld, David M},
   journal={Physical review letters},
   volume={107},
   number={17},
@@ -121,69 +358,303 @@ url = {http://steck.us/alkalidata}
   year={2011},
   publisher={APS}
 }
-@article{mekhov2012,
-  title={Quantum optics with ultracold quantum gases: towards the full quantum regime of the light--matter interaction},
-  author={Mekhov, Igor B and Ritsch, Helmut},
-  journal={Journal of Physics B: Atomic, Molecular and Optical Physics},
-  volume={45},
-  number={10},
-  pages={102001},
-  year={2012},
-  publisher={IOP Publishing}
+@article{eckert2008,
+  title = {Dicke quantum phase transition with a superfluid gas in an optical cavity},
+  author = {Eckert, K. and Romero-Isart, O. and Rodriguez, M. and
+                  Lewenstein, M. and Polzik, E. S. and Sanpera, A.},
+  journal = {Nat. Phys.},
+  volume = {4},
+  pages = {50},
+  year = {2008},
 }
-@article{mazzucchi2016,
-  title = {Quantum measurement-induced dynamics of many-body ultracold bosonic and fermionic systems in optical lattices},
-  author = {Mazzucchi, Gabriel and Kozlowski, Wojciech and Caballero-Benitez, Santiago F. and Elliott, Thomas J. and Mekhov, Igor B.},
-  journal = {Physical Review A},
-  volume = {93},
-  issue = {2},
-  pages = {023632},
-  numpages = {12},
-  year = {2016},
+@article{larson2008,
+  title = {Mott-Insulator States of Ultracold Atoms in Optical
+                  Resonators},
+  author = {Larson, Jonas and Damski, Bogdan and Morigi, Giovanna and
+                  Lewenstein, Maciej},
+  journal = {Phys. Rev. Lett.},
+  volume = {100},
+  issue = {5},
+  pages = {050401},
+  numpages = {4},
+  year = {2008},
   month = {Feb},
-  publisher = {American Physical Society}
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevLett.100.050401},
 }
-@article{atoms2015,
-  title={Probing and Manipulating Fermionic and Bosonic Quantum Gases with Quantum Light},
-  author={Elliott, Thomas J and Mazzucchi, Gabriel and Kozlowski, Wojciech and Caballero-Benitez, Santiago F and Mekhov, Igor B},
-  journal={Atoms},
-  volume={3},
-  number={3},
-  pages={392--406},
-  year={2015},
-  publisher={Multidisciplinary Digital Publishing Institute}
+@article{chen2009,
+  title = {Bistable Mott-insulator\char21{}to\char21{}superfluid phase
+                  transition in cavity optomechanics},
+  author = {Chen, W. and Zhang, K. and Goldbaum, D. S. and
+                  Bhattacharya, M. and Meystre, P.},
+  journal = {Phys. Rev. A},
+  volume = {80},
+  issue = {1},
+  pages = {011801},
+  numpages = {4},
+  year = {2009},
+  month = {Jul},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevA.80.011801},
 }
-@article{mazzucchi2016af,
-  title={Quantum measurement-induced antiferromagnetic order and density modulations in ultracold Fermi gases in optical lattices},
-  author={Mazzucchi, Gabriel and Caballero-Benitez, Santiago F and Mekhov, Igor B},
-  journal={arXiv preprint arXiv:1510.04883},
-  year={2015}
+@article{habibian2013,
+  title = {Bose-Glass Phases of Ultracold Atoms due to Cavity
+                  Backaction},
+  author = {Habibian, Hessam and Winter, Andr\'e and Paganelli, Simone
+                  and Rieger, Heiko and Morigi, Giovanna},
+  journal = {Phys. Rev. Lett.},
+  volume = {110},
+  issue = {7},
+  pages = {075304},
+  numpages = {5},
+  year = {2013},
+  month = {Feb},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevLett.110.075304},
 }
-@inbook{Scully,
-author = {Scully, M. and Zubairy, S.},
-title = {{Quantum Optics}},
-publisher = {Cambridge University Press},
-chapter = {10.1},
-pages = {293},
-year = {1997}
+@article{ivanov2014,
+  title={Feedback-enhanced self-organization of atoms in an optical cavity},
+  author={Ivanov, DA and Ivanova, T Yu},
+  journal={JETP letters},
+  volume={100},
+  number={7},
+  pages={481--485},
+  year={2014},
+  publisher={Springer}
 }
-@article{mekhov2007prl,
-  title={Cavity-enhanced light scattering in optical lattices to probe atomic quantum statistics},
-  author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
+@article{caballero2015,
+  title={Quantum optical lattices for emergent many-body phases of
+                  ultracold atoms},
+  author={Caballero-Benitez, Santiago F and Mekhov, Igor B},
   journal={Physical review letters},
-  volume={98},
-  number={10},
-  pages={100402},
-  year={2007},
+  volume={115},
+  number={24},
+  pages={243604},
+  year={2015},
   publisher={APS}
 }
-@article{mekhov2007pra,
-  title={Light scattering from ultracold atoms in optical lattices as an optical probe of quantum statistics},
-  author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
-  journal={Physical Review A},
-  volume={76},
-  number={5},
-  pages={053618},
-  year={2007},
-  publisher={APS}
-}
\ No newline at end of file
+@article{pedersen2014,
+  author={Mads Kock Pedersen and Jens Jakob W H Sorensen and Malte C
+                  Tichy and Jacob F Sherson},
+  title={{Many-body state engineering using measurements and fixed
+                  unitary dynamics}},
+  journal={New J. Phys.},
+  volume={16},
+  number={11},
+  pages={113038},
+  year={2014}
+}
+@article{kolobov1999,
+  title = {The spatial behavior of nonclassical light},
+  author = {Kolobov, Mikhail I.},
+  journal = {Rev. Mod. Phys.},
+  volume = {71},
+  issue = {5},
+  pages = {1539--1589},
+  numpages = {0},
+  year = {1999},
+  month = {Oct},
+  publisher = {American Physical Society},
+  doi = {10.1103/RevModPhys.71.1539},
+}
+@article{golubev2010,
+  title = {Entanglement measurement of the quadrature components
+                  without homodyne detection in the bright, spatially
+                  multimode far field},
+  author = {Golubeva, T. and Golubev, Yu. and Samburskaya, K. and
+                  Fabre, C. and Treps, N. and Kolobov, M.},
+  journal = {Phys. Rev. A},
+  volume = {81},
+  issue = {1},
+  pages = {013831},
+  numpages = {8},
+  year = {2010},
+  month = {Jan},
+  publisher = {American Physical Society},
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diff --git a/thesis.tex b/thesis.tex
index 6648dc4..0e6eb48 100644
--- a/thesis.tex
+++ b/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]{Classes/PhDThesisPSnPDF}
 
 % ******************************************************************************
 % ******************************* Class Options ********************************