Finished chapter 6 and added section on the Bose-Hubbard model to chapter 2

Chapter1/chapter1.tex

@@ -12,3 +12,4 @@
 
 
 %********************************** %First Section  **************************************
+

Chapter2/chapter2.tex

@@ -380,6 +380,265 @@ beyond previous treatments.
 
 \section{The Bose-Hubbard Hamiltonian}
 
+\subsection{The Model}
+
+The original Hubbard model was developed as a model for the motion of
+electrons within transition metals \cite{hubbard1963}. The key
+elements were the confinement of the motion to a lattice and the
+approximation of the Coulomb screening interaction as a simple on-site
+interaction. Despite these enormous simplifications the model was very
+succesful \cite{leggett}. The Bose-Hubbard model is an even simpler
+variation where instead of fermions we consider spinless bosons. It
+was originally devised as a toy model, but in 1998 it was shown by
+Jaksch \emph{et.~al.} that it can be realised with ultracold atoms in
+an optical lattice \cite{jaksch1998}. Shortly afterwards it was
+obtained in a ground-breaking experiment \cite{greiner2002}. The model
+has been the subject of intense research since then, because despite
+its simplicity it possesses highly nontrivial properties such as the
+superfluid to Mott insulator quantum phase transition. Furthermore, it
+is one of the most controllable quantum many-body systems thus
+providing a solid basis for new experiments and technologies.
+
+The model we have derived is essentially an extension of the
+well-known Bose-Hubbard model that also includes interactions with
+quantised light. Therefore, it should come as no surprise that if we
+eliminate all the quantized fields from the Hamiltonian we obtain
+exactly the Bose-Hubbard model
+\begin{equation}
+  \H_a = -J \sum_{\langle i,j \rangle}^M \bd_i b_j + 
+  \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1).
+\end{equation}
+The first term is the kinetic energy term and denotes the hopping of
+atoms between neighbouring sites at a hopping rate given by $J$. The
+second term describes the on-site repulsion which acts mainly to
+suppress multiple occupancies of any site, but at the same time it
+lies at the heart of strongly correlated phenomena in the Bose-Hubbard
+model. Unfortunately, despite its simplicity, this system is not
+solvable for any finite value $U/J$ using known techniques in finite
+dimensions \cite{krauth1991, kolovsky2004, calzetta2006}. However,
+exact solutions exist for the ground state in the two limits of
+$U = 0$ and $J = 0$ which already provide some insight as they
+correspond to the two different phases of the quantum phase transition
+that is present in the model. In higher dimensions (e.g.~two or three)
+the system is reasonably well described by a mean-field approximation
+which provides us with a good understanding of the full quantum phase
+diagram.
+
+\subsection{The Superfluid and Mott Insulator Ground States}
+
+Before trying to understand the quantum phase transition itself we
+will look at the ground states of the two extremal cases first and for
+simplicity we will consider only homogenous systems with periodic
+boundary conditions here. We will begin with the $U = 0$ limit where
+the Hamiltonian's kinetic term can be diagonalised by transforming to
+momentum space defined by the annihilation operator
+\begin{equation}
+  b_\b{k} = \frac{1} {\sqrt{M}} \sum_m b_m e^{i \b{k} \cdot \b{r}_m},
+\end{equation}
+where $\b{k}$ denotes the wavevector running over the first Brillouin
+zone. The Hamiltonian then is given by
+\begin{equation}
+  \H_a = \sum_\b{k} \epsilon_\b{k} \bd_\b{k} b_\b{k},
+\end{equation}
+where the free particle dispersion is
+\begin{equation}
+  \epsilon_\b{k} = -2 J \sum_{d = 1}^D \cos(k_d a),
+\end{equation}
+and $d$ denotes the dimension out of a total of $D$ dimensions. It
+should now be obvious that the ground state is simply the state with
+all the atoms in the $\b{k} = 0$ state and thus it is given by
+\begin{equation}
+  \label{eq:GSSF}
+  | \Psi_\mathrm{SF} \rangle = \frac{ ( \bd_{\b{k} = 0} )^N } {\sqrt{N!}} | 0
+  \rangle = \frac{1} {\sqrt{N!}} \left( \frac{1} {\sqrt{M}} \sum_m^M
+    \bd_m \right)^N | 0 \rangle,
+\end{equation}
+where $| 0 \rangle$ denotes the vacuum state. This state is called a
+superfluid (SF) due to its viscous-free flow properties
+\cite{leggett1999}. The macroscopic occupancy of the single-particle
+ground state is the signature and defining property of a
+Bose-Einstein condensate. Note that the zero momentum states consist of a
+superposition of an atom at all the sites of the optical lattice
+highlighting the fact that kinetic energy is minimised by delocalising
+the particles. This in turn implies that a superfluid state will have
+infinite range correlations. It is also important to note that in the
+thermodynamic limit the superfluid state is gapless, i.e.~it takes
+zero energy to excite the system.
+
+At the other end of the spectrum, i.e.~$J = 0$, the kinetic term
+vanishes and the system is dominated by the on-site interactions. This
+has the effect of decoupling the Hamiltonian into a sum of identical
+single-site Hamiltonians which are already diagonal in the atom number
+basis. Therefore, the ground state for $N = gM$, where $g$ is an
+integer, is given by
+\begin{equation}
+  \label{eq:GSMI}
+  | \Psi_\mathrm{MI} \rangle = \prod_m^M \frac{1} {\sqrt{g!}}
+  (\bd_m)^g | 0 \rangle.
+\end{equation}
+This state has precisely the same number of bosons, $g$, at each site
+and is commonly known as the Mott insulator (MI). Unlike in the
+superfluid state, the atoms are all localised to a specific lattice
+site and thus not only are there no long-range correlations, there are
+actually no two-site correlations in this state at all. Additionally,
+this state is gapped, i.e.~it costs a finite amount of energy to
+excite the system into its lowest excited state even in the
+thermodynamic limit. In fact, the existance of the gap is the defining
+property that distinguishes the Mott insulator from the superfluid in
+the Bose-Hubbard model. Note that we have only considered a
+commensurate case (number of atoms an integer multiple of the number
+of sites). If we were to insert another atom on top of the Mott
+insulator, the energy of the system would not depend on the lattice
+site in which it is inserted. This implies that as soon as $J$ becomes
+finite, the ground state would prefer this atom be delocalised and
+effectively becomes gapless which means the system is a superfluid and
+it will not exhibit a quantum phase transition.
+
+\subsection{Mean-Field Theory}
+
+Now that we have a basic understanding of the two limiting cases we
+can now consider the model in between these two extremes. We have
+already mentioned that an exact solution is not known, but fotunately
+a very good mean-field approximation exists \cite{fisher1989}. In this
+approach the interaction term is treated exactly, but the kinetic
+energy term is decoupled as
+\begin{equation}
+  \bd_m b_n = \langle \bd_m \rangle b_n + \bd_m \langle b_n \rangle -
+  \langle \bd_m \rangle \langle b_n \rangle = \Phi^* b_n + \Phi \bd_m
+  - | \Phi |^2,
+\end{equation}
+where the expectation values are for the $T = 0$ ground state. Note
+that we have assumed that $\Phi = \langle b_m \rangle $ is non-zero
+and homogenous. $\Phi$ describes the influence of hopping between
+neighbouring sites and is the mean -field order parameter. This
+decoupling effect means that we can write the Hamiltonian as $\hat{H}_a
+= \sum_m^M \hat{h}_a$, where
+\begin{equation}
+  \hat{h}_a = -z J ( \Phi \bd \Phi^* b) + z J | \Phi |^2 + \frac{U}{2}
+  \n (\n - 1) - \mu \n,
+\end{equation}
+and $z$ is the coordination number, i.e. the number of nearest
+neighbours of a single site. We have also introduced the chemical
+potential $\mu$ since we now consider the grand canonical ensemble as
+the Hamiltonian no longer conserves the total atom number. The ground
+state of the $| \Psi_0 \rangle$ of the overall system will be a
+site-wise product of the individual ground states of
+$\hat{h}_a$. These can be found very easily using standard
+diagonalisation techniques. $\Phi$ is then self-consistently
+determined by minimising the energy of the ground state with respect
+to $\Phi$.
+
+The main advantage of the mean-field treatment is that it lets us
+study the quantum phase transition between the sueprfluid and Mott
+insulator phases discussed in the previous sections. The phase
+boundaries can be obtained from second-order perturbation theory as 
+\begin{equation}
+  \left( \frac{\mu} {U} \right)_\pm = \frac{1}{2} \left[ 2g - 1 -
+    \left( \frac{zJ}{U} \right) \pm \sqrt{ 1 - 2 (2g + 1) \left( \frac{zJ}{U}
+        \right) + \left( \frac{zJ}{U} \right)^2} \right].
+\end{equation}
+This yields a phase diagram with multiple Mott insulating lobes as
+seen in Fig. \ref{fig:BHPhase} where each lobe represents a Mott insualtor with a
+different number of atoms per site. The tip of the lobe which
+corresponds to the quantum phase transition along a line of fixed
+density is obtained by equating the two branches of the above equation
+to get
+\begin{equation}
+  \left( \frac{J} {U} \right) = \frac{1} {z (2g + 1 + \sqrt{4g^2 + 4g} )}.
+\end{equation}
+It is important to note that the fact that this formalism predicts a
+phase transition is in contrast to other mean-field theories such as
+the Bogoliubov approximation \cite{PitaevskiiStringari}. This
+highlights the fact that the interactions, which are treated exactly
+here, are the dominant driver leading to this phase transition and
+other strongly correlated effects in the Bose-Hubbard model.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.8\linewidth]{BHPhase}
+  \caption[Mean-Field Bose-Hubbard Phase Diagram]{Mean-field phase
+    diagram of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
+    Ref. \cite{StephenThesis}. The shaded regions are the Mott
+    insulator lobes and each lobe corresponds to a different on-site
+    filling labeled by $n$. The rest of the space corresponds to the
+    superfluid phase. The dashed lines are the phase boundaries
+    obtained from first-order perturbation theory. The solid lines are
+    are lines of constant density in the superfluid
+    phase. \label{fig:BHPhase}}
+\end{figure}
+
+\subsection{The Bose-Hubbard Model in One Dimension}
+
+The mean-field theory in the previous section is very useful tool for
+studying the quantum phase transition in the Bose-Hubbard
+model. However, it is effectively an infinite-dimensional theory and
+in practice it only works in two dimensions or more. The phase
+transition in 1D is poorly described, because it actually belongs to a
+different universality class. This is clearly seen from the one
+dimensional phase diagram shown in Fig. \ref{fig:1DPhase}.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.8\linewidth]{1DPhase}
+  \caption[Exact 1D Bose-Hubbard Phase Diagram]{Exact phase diagram
+    of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
+    Ref. \cite{StephenThesis}. The shaded region corresponds to the
+    $n = 1$ Mott insulator lobe. The sharp tip distinguishes it from
+    the mean-field phase diagram. The red dotted line denotes the
+    reentrance phase transition which occurs for a fixed $\mu$.The
+    critical point for the fixed density $n = 1$ transition is denoted
+    by an 'x'. \label{fig:1DPhase}}
+\end{figure}
+
+Some general conclusions can be obtained by looking at Haldane's
+prescription for Luttinger liquids \cite{haldane1981,
+  giamarchi}. Without a periodic potential the low-energy physics of
+the system is described by the Hamiltonian
+\begin{equation}
+  \hat{H}_a = \frac{1}{2 \pi} \int \mathrm{d} x \left\{ v K [ \hat{\Pi}(x)
+    ]^2 + \frac{v} {K} [\partial_x \hat{\Phi}(x) ]^2 \right\},
+\end{equation}
+where we have expressed the bosonic field operators in terms of a
+density operator $\hat{\rho}(x)$ and a phase operator $\hat{\Phi}(x)$
+as $\hat{\Psi}(x) = \sqrt{\hat{\rho}(x)} e^{i \hat{\hat{\Phi}(x)}}$
+and $\hat{\Pi}(x)$ is the density fluctuation operator. Provided the
+parameters $v$ and $K$ can be correctly determined this Hamiltonian
+gives the correct description of the gapless superfluid phase of the
+Bose-Hubbard model. Most importantly it gives an expression for the
+spatial correlation functions such as
+\begin{equation}
+  \langle \bd_m b_n \rangle = A \left( \frac{\alpha} {|m - n|} \right)^{K/2},
+\end{equation}
+where $A$ is some amplitude and $\alpha$ is a necessary cutoff to
+regularise the theory at short distances. Unlike the superfluid ground
+state in Eq. \eqref{eq:GSSF} this state does not have infinite range
+correlations. They decay according to a power law. However, for
+non-interacting systems $K = 0$ and long-range order is re-established
+as before though it is important to note that in higher dimensions
+this long-range order persists in the whole superfluid phase even with
+interactions present. In order to describe the phase transition and
+the Mott insulating phase it is necessary to introduce a periodic
+lattice potential. It can be shown that this system exhibits at
+$T = 0$ a Berezinskii-Kosterlitz-Thouless phase transition as the
+parameter $K$ is varied with a critical point at $K = \frac{1}{2}$
+where $K < \frac{1}{2}$ is a superfluid. Above $K = \frac{1}{2}$ the
+value of $K$ jumps discontinuously to $K \rightarrow \infty$ producing
+the Mott insulator phase. Unlike the gapless superfluid phase the
+spatial correlations decay exponentially as
+\begin{equation}
+  \langle \bd_m b_n \rangle = B e^{ - |m - n|/\xi},
+\end{equation}
+where $B$ is some constant  and the correlation length is given by
+$\xi = v / \Delta$ where $\Delta$ is the energy gap.
+
+Using advanced numerical methods such as density matrix
+renormalisation group (DMRG) calculations it is possible to identify
+the critical point by fitting the power-law decay correlations in
+order to obtain $K$. The resulting phase transition is shown in
+Fig. \ref{fig:1DPhase} and the critical point was shown to be $(U/zJ) = 1.68$. Note
+that unusually the phase diagram exhibits a reentrance phase
+transition for a fixed $\mu$.
+
 \section{Scattered light behaviour}
 \label{sec:a}
 
@@ -832,6 +1091,7 @@ Therefore, we can now express the quantity $n_{\Phi}$ as
 \begin{equation}
   n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
 \end{equation}
+
 Estimates of the scattering rate using real experimental parameters
 are given in Table \ref{tab:photons}. Rubidium atom data has been
 taken from Ref. \cite{steck}. Miyake \emph{et al.} experimental
@@ -866,7 +1126,7 @@ $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
 (1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz
 per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
 
-\begin{table}[!htbp]
+\begin{table}
   \centering
   \begin{tabular}{l c c}
     \toprule

Chapter3/chapter3.tex

@@ -4,6 +4,7 @@
 
 \chapter{Probing Correlations by Global 
 Nondestructive Addressing} %Title of the Third Chapter
+\label{chap:qnd}
 
 \ifpdf
     \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}}

Chapter4/chapter4.tex

@@ -3,6 +3,7 @@
 %*******************************************************************************
 
 \chapter{Quantum Measurement Backaction}  
+\label{chap:backaction}
 % Title of the Fourth Chapter
 
 \ifpdf
@@ -244,6 +245,7 @@ between fields. Therefore, the final form of the Hamiltonian
 Eq. \eqref{eq:fullH} that we will be using in the following chapters
 is
 \begin{equation}
+  \label{eq:backaction}
   \hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F}
 \end{equation}
 \begin{equation}
@@ -277,7 +279,26 @@ nonlocal way and most importantly not only will it not degrade
 long-range coherence, it will in fact lead to such long-range
 correlations itself.
 
-\mynote{introduce citations from PRX/PRA above}
+In Chapter \ref{chap:qnd} we used highly efficient DMRG methods
+\cite{tnt} to calculate the ground state of the Bose-Hubbard
+Hamiltonian. Related techniques such as Time-Evolving Block Decimation
+(TEBD) or t-DMRG are often used for numerical calculations of time
+evolution. However, despite the fact our Hamiltonian in
+Eq. \eqref{eq:backaction} is simply the Bose-Hubbard model with a
+non-Hermitian term added due to measurement it is actually difficult
+to apply these methods to our system. The problem lies in the fact
+that Matrix Product methods we mentioned are only efficient for
+one-dimensional systems that obey the area law for entanglement
+entropy, i.e.~systems with only short-range quantum
+correlations. Unfortunately, the global nature of the measurement we
+consider violates the assumptions made in deriving the area law and,
+as we shall see in the following chapters, leads to long-range
+correlations regardless of coupling strength. Therefore, we resort to
+using exact methods such as exact diagonalisation which we solve with
+well-known ordinary differential equation solvers. This means that we
+can at most simulate a few atoms, but as we shall see it is the
+geometry of the measurement that matters the most and these effects
+are already visible in smaller systems.
 
 \section{The Master Equation}
 \label{sec:master}

Chapter5/chapter5.tex

@@ -1692,7 +1692,7 @@ obtained by cooling or projecting from an initial ground state. The
 combination of tunnelling with measurement is necessary.
 
 \begin{figure}[hbtp!]
-	\includegraphics[width=\linewidth]{figure3}
+	\includegraphics[width=\linewidth]{steady}
 	\caption[Non-Hermitian Steady State]{A trajectory simulation
           for eight atoms in eight sites, initially in
           $|1,1,1,1,1,1,1,1 \rangle$, with periodic boundary

Chapter6/chapter6.tex

@@ -14,10 +14,616 @@
 
 \section{Introduction}
 
+Light scatters due to its interaction with the dipole moment of the
+atoms which for off-resonant light results in an effective coupling
+with atomic density, not the matter-wave amplitude. Therefore, it is
+challenging to couple light to the phase of the matter-field, as is
+typical in quantum optics for optical fields. In the previous chapter
+we only considered measurement that couples directly to atomic density
+operators just most of the existing work \cite{LP2009, rogers2014,
+  mekhov2012, ashida2015, ashida2015a}. However, we have shown in
+section \ref{sec:B} that it is possible to couple to the the relative
+phase differences between sites in an optical lattice by illuminating
+the bonds between them. Furthermore, we have also shown how it can be
+applied to probe the Bose Hubbard order parameter or even matter-field
+quadratures in Chapter \ref{chap:qnd}. This concept has also been
+applied to the study of quantum optical potentials formed in a cavity
+and shown to lead to a host of interesting quantum phase diagrams
+\cite{caballero2015, caballero2015njp, caballero2016,
+  caballero2016a}. This is a multi-site generalisation of previous
+double-well schemes \cite{cirac1996, castin1997, ruostekoski1997,
+  ruostekoski1998, rist2012}, although the physical mechanism is
+fundametally different as it involves direct coupling to the
+interference terms caused by atoms tunnelling rather than combining
+light scattered from different sources.
+
+We have already mentioned that there are three primary avenues in
+which the field of quantum optics with ultracold gases can be taken:
+nondestructive measurement, quantum measurement backaction, and
+quantum optical potentials. All three have been covered in the context
+of density-based measurement either here or in other works. However,
+coupling to phase observables in lattices has only been proposed and
+considered in the context of nondestructive measurements (see Chapter
+\ref{chap:qnd}) and quantum optical potentials. \cite{caballero2015,
+  caballero2015njp, caballero2016, caballero2016a}. In this chapter,
+we go in a new direction by considering the effect of measurement
+backaction on the atomic gas that results from such coupling.  We
+investigate this mechanism using light scattered from these
+phase-related observables. The novel combination of measurement
+backaction as the physical mechanism driving the dynamics and phase
+coherence as the observable to which the optical fields couple to
+provides a completely new opportunity to affect and manipulate the
+quantum state. We first show how this scheme enables us to prepare
+energy eigenstates of the lattice Hamiltonian. Furthermore, in the
+second part we also demonstrate a novel type of a projection due to
+measurement which occurs even when there is significant competition
+with the Hamiltonian dynamics. This projection is fundamentally
+different to the standard formulation of the Copenhagen postulate
+projection or the quantum Zeno effect \cite{misra1977, facchi2008,
+  raimond2010, raimond2012, signoles2014} thus providing an extension
+of the measurement postulate to dynamical systems subject to weak
+measurement.
+
+
 \section{Diffraction Maximum and Energy Eigenstates}
 
+In this chapter we will only consider measurement backaction when
+$\a_1 = C \hat{B}$ and $\c = \sqrt{2 \kappa} \a_1$ as introduced in
+section \ref{sec:B}. We have seen that there are two ways of
+engineering the spatial profile of the measurement such that the
+density contribution from $\hat{D}$ is suppressed. We will first
+consider the case when the profile is uniform, i.e.~the diffraction
+maximum of scattered light, when our measurement operator is given by
+\begin{equation}
+  \B = \Bmax = J^B_\mathrm{max} \sum_j^K \left( \bd_j b_{j+1} b_j
+    \bd_{j+1} \right)
+  = 2 J^B_\mathrm{max} \sum_k \bd_k b_k \cos(ka),
+\end{equation}
+where the second equality follows from converting to momentum space,
+denoted by index $k$, via
+$b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates
+an atom with momentum $k$ in the range
+$\{ \frac{(M-1) \pi} {M}, \frac{(M-2) \pi} {M}, ..., \pi \}$. Note
+that this operator is diagonal in momentum space which means that its
+eigenstates are simply Fock momentum Fock states. We have also seen in
+Chapter \ref{chap:backaction} how the global nature of the jump
+operators introduces a nonlocal quadratic term to the Hamiltonian,
+$\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the
+competition between tunnelling and measurement backaction we again
+consider non-interacting atoms, $U = 0$. Therefore, $\B$ is
+proportional to the Hamiltonian and both operators have the same
+eigenstates.
+
+The combined Hamiltonian for the non-interacting gas subject to
+measurement of the from $\a_1 = C \B$ is thus
+\begin{equation}
+  \hat{H} = -J \sum_j \left( \bd_j b_{j+1} + b_j \bd_{j+1} \right) - i
+  \gamma \Bmax^\dagger \Bmax.
+\end{equation} 
+Furthermore, whenever a photon is detected, the operator
+$\c = \sqrt{2 \kappa} C \Bmax$ is applied to the wavefunction. We can
+rewrite the above equation as
+\begin{equation}
+  \hat{H} = -\frac{J}{J^B_\mathrm{max}} \Bmax - i \gamma \Bmax^\dagger \Bmax.
+\end{equation}
+The eigenstates of this operator will be exactly the same as of the
+isolated Hamiltonian which we will label as $| h_l \rangle$, where
+$h_l$ denotes the corresponding eigenvalue. Therefore, for an initial
+state
+\begin{equation}
+  | \Psi \rangle = \sum_l z_l^0 | h_l \rangle
+\end{equation}
+it is easy to show that the state after $m$ photocounts and time $t$
+is given by
+\begin{equation}
+  | \Psi (m,t) \rangle = \frac{1}{\sqrt{F(t)}} \sum_l h_l^m z_l^0 e^{[i h_l -
+    \gamma(J^B_\mathrm{max}/J)^2 h_l^2]t} | h_l \rangle,
+\end{equation}
+where $\sqrt{F(t)}$ is the normalisation factor. Therefore, the probability
+of being in state $| h_l \rangle$ at time $t$ after $m$ photocounts is
+given by
+\begin{equation}
+  p(h_l, m, t) = \frac{h_l^{2m}} {F(t)} \exp\left[ - 2 \gamma \left(
+    \frac{J^B_\mathrm{max}} {J} \right)^2 h_l^2 t \right] |z_l^0|^2.
+\end{equation}
+As a lot of eigenstates are degenerate, we are actually more
+interested in the probability of being in eigenspace with the
+eigenvalue $h_l = (J/J^B_\mathrm{max})B_\mathrm{max}$. This probability
+is given by
+\begin{equation}
+  \label{eq:bmax}
+  p(B_\mathrm{max}, m, t) = \frac{B_\mathrm{max}^{2m}} {F(t)} \exp\left[ - 2
+    \gamma B_\mathrm{max}^2 t \right] p_0 (B_\mathrm{max}),
+\end{equation}
+where
+$p_0(B_\mathrm{max}) = \sum_{J_\mathrm{max} h_l = J B_\mathrm{max}}
+|z_l^0|^2$. This distribution will have two distinct peaks at
+$B_\mathrm{max} = \pm \sqrt{m/2\kappa |C|^2 t}$ and an initially broad
+distribution will narrow down around these two peaks with successive
+photocounts. The final state is in a superposition, because we measure
+the photon number, $\ad_1 \a_1$ and not field amplitude. Therefore,
+the measurement is insensitive to the phase of $\a_1 = C \B$ and we
+get a superposition of $\pm B_\mathrm{max}$. This is exactly the same
+situation that we saw for the macroscopic oscillations of two distinct
+components when the atom number difference between two modes is
+measured as seen in Fig. \ref{fig:oscillations}(b). However, this
+means that the matter is still entangled with the light as the two
+states scatter light with different phase which the photocount
+detector cannot distinguish. Fortunately, this is easily mitigated at
+the end of the experiment by switching off the probe beam and allowing
+the cavity to empty out or by measuring the light phase (quadrature)
+to isolate one of the components \cite{mekhov2009pra, mekhov2012,
+  atoms2015}.
+
+Unusually, we do not have to worry about the timing of the quantum
+jumps, because the measurement operator commutes with the
+Hamiltonian. This highlights an important feature of this measurement
+- it does not compete with atomic tunnelling, and represents a quantum
+nondemolition (QND) measurement of the phase-related observable
+\cite{brune1992}. Eq. \eqref{eq:bmax} shows that regardless of the
+initial state or the photocount trajectory the system will project
+onto a superposition of eigenstates of the $\Bmax$ operator. In fact,
+the final state probability distribution would be exactly the same if
+we were to simply employ a projective measurement of the same
+operator. What is unusual in our case is that this has been achieved
+with weak measurement in the presence of significant atomic
+dynamics. The conditions for such a measurement have only recently
+been rigorously derived in Ref. \cite{weinberg2016} and here we
+provide a practical realisation using in ultracold bosonic gases.
+
+This projective behaviour is in contrast to conventional density based
+measurements which squeeze the atom number in the illuminated region
+and thus are in direct competition with the atom dynamics (which
+spreads the atoms), effectively requiring strong couplings for a
+projection as seen in the previous chapter. Here a projection is
+achieved at any measurement strength which allows for a weaker probe
+and thus effectively less heating and a longer experimental
+lifetime. Furthermore. This is in contrast to the quantum Zeno effect
+which requires a very strong probe to compete effectively with the
+Hamiltonian \cite{misra1977, facchi2008, raimond2010, raimond2012,
+  signoles2014}. Interestingly, the $\Bmax$ measurement will even
+establish phase coherence across the lattice,
+$\langle \bd_j b_j \rangle \ne 0$, in contrast to density based
+measurements where the opposite is true: Fock states with no
+coherences are favoured.
+
 \section{General Model for Weak Measurement Projection}
 
-\section{Determining the Projection Subspace}
+\subsection{Projections for Incompatible Dynamics and Measurement}
+
+In section \ref{sec:B} we have also shown that it is possible to
+achieve a more complicated spatial profile of the
+$\B$-measurement. The optical geometry can be adjusted such that each
+bond scatters light in anti-phase with its neighbours leading to a
+diffraction minimum where the expectation value of the amplitude is
+zero. In this case the $\B$ operator is given by
+\begin{equation}
+  \B = \Bmin = J^B_\mathrm{min} \sum_m^K (-1)^m \hat{B}_m
+  = 2 i J^B_\mathrm{min} \sum_k \bd_k b_{k - \pi/a} \sin(ka).
+\end{equation}
+Note how the measurement operator now couples the momentum mode $k$
+with mode $k - \pi/a$. This measurement operator no longer commutes
+with the Hamiltonian so it is no longer QND and we do not expect there
+to be a steady state as before. In order to understand the measurement
+it will be easier to work in a basis in which it is diagonal. We
+perform the transformation
+$\beta_k = \frac{1}{\sqrt{2}} \left( b_k + i b_{k - \pi/a} \right)$,
+$\tilde{\beta}_k = \frac{1}{\sqrt{2}} \left( b_k - i b_{k - \pi/a}
+\right)$, which yields the following forms of the measurement operator
+and the Hamiltonian:
+\begin{equation}
+  \label{eq:BminBeta}
+  \Bmin = 2 J^B_\mathrm{min} \sum_{\mathrm{RBZ}} \sin(ka) \left( \beta^\dagger_k
+    \beta_k - \tilde{\beta}_k^\dagger \tilde{\beta}_k \right),
+\end{equation}
+\begin{equation}
+  \label{eq:H0Beta}
+  \hat{H}_0 = - 2 J \sum_{\mathrm{RBZ}} \cos(ka) \left( \beta_k^\dagger
+    \tilde{\beta}_k + \tilde{\beta}^\dagger_k \beta_k \right),
+\end{equation}
+where the summations are performed over the reduced Brilluoin Zone
+(RBZ), $0 < k \le \pi/a$, to ensure the transformation is
+canonical. We see that the measurement operator now consists of two
+types of modes, $\beta_k$ and $\tilde{\beta_k}$, which are
+superpositions of two momentum states, $k$ and $k - \pi/a$. Note how a
+spatial pattern with a period of two sites leads to a basis with two
+modes whilst a uniform pattern had only one mode, $b_k$. Furthermore,
+note the similarities to
+$\D = \Delta \hat{N} = \hat{N}_\mathrm{even} - \hat{N}_\mathrm{odd}$
+which is the density measurement operator obtained by illuminated the
+alttice such that neighbouring sites scatter light in anti-phase. This
+further highlights the importance of geometry for global measurement.
+
+Trajectory simulations confirm that there is no steady state. However,
+unexpectedly, for each trajectory we observe that the dynamics always
+ends up confined to some subspace as seen in
+Fig. \ref{fig:projections}. In general, this subspace is neither an
+eigenspace of the measurement operator or the Hamiltonian. In
+Fig. \ref{fig:projections}(b) it in fact clearly consists of multiple
+measurement eigenspaces. This clearly distinguishes it from the
+fundamental projections predicted by the Copenhagen postulates. It is
+also not the quantum Zeno effect which predicts that strong
+measurement can confine the evolution of a system as this subspace
+must be an eigenspace of the measurement operator \cite{misra1977,
+  facchi2008, raimond2010, raimond2012, signoles2014}. Furthermore,
+the projection we see in Fig. \ref{fig:projections} occurs for even
+weak measurement strengths compared to the Hamiltonian's own
+evolution, a regime in which the quantum Zeno effect does not
+happen. It is also possible to dissipatively prepare quantum states in
+an eigenstate of a Hamiltonian provided it is also a dark state of the
+jump operator, $\c | \Psi \rangle = 0$~\cite{diehl2008}. However,
+this is also clearly not the case as the final state in
+Fig. \ref{fig:projections}(c) is not only not confined to a single
+measurement operator eigenspace, it also spans multiple Hamiltonian
+eigenspaces. Therefore, the dynamics induced by $\a_1 = C\Bmin$ project
+the system into some subspace, but since this does not happen via any
+of the mechanisms described above it is not immediately obvious what
+this subspace is.
+
+\begin{figure}[hbtp!]
+	\includegraphics[width=\linewidth]{Projections}
+	\caption[Projections for Non-Commuting Observable and
+        Hamiltonian] {Subspace projections. Projection to a
+          $\mathcal{P}_M$ space for four atoms on eight sites with
+          periodic boundary conditions. The parameters used are $J=1$,
+          $U=0$, $\gamma = 0.1$, and the initial state was
+          $| 0,0,1,1,1,1,0,0 \rangle$. (a) The
+          $\langle \hat{O}_k \rangle = \langle \n_k + \n_{k - \pi/a}
+          \rangle$ distribution reaches a steady state at
+          $Jt \approx 8$ indicating the system has been projected. (b)
+          Populations of the $\Bmin$ eigenspaces. (c) Population of
+          the $\hat{H}_0$ eigenspaces. Once the projection is achieved
+          at $Jt\approx8$ we can see from (b-c) that the system is not
+          in an eigenspace of either $\Bmin$ or $\hat{H}_0$, but it
+          becomes confined to some subspace. The system has been
+          projected onto a subspace, but it is neither that of the
+          measurement operator or the
+          Hamiltonian. \label{fig:projections}}
+\end{figure}
+
+To understand this dynamics we will look at the master equation for
+open systems described by the density matrix, $\hat{\rho}$,
+\begin{equation}
+  \dot{\hat{\rho}} = -i \left[\hat{H}_0, \hat{\rho} \right] + 
+  \left[ \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho}
+      \cd \c \right) \right].
+\end{equation}
+The following results will not depend on the nature or exact form of
+the jump operator $\c$. However, whenever we refer to our simulations
+or our model we will be considering $\c = \sqrt{2 \kappa} C(\D + \B)$
+as before, but the results are more general and can be applied to ther
+setups. This equation describes the state of the system if we discard
+all knowledge of the outcome. The commutator describes coherent
+dynamics due to the isolated Hamiltonian and the remaining terms are
+due to measurement. This is a convenient way to find features of the
+dynamics common to every measurement trajectory.
+
+Just like in the preceding chapter we define the projectors of the
+measurement eigenspaces, $P_m$, which have no effect on any of the
+(possibly degenerate) eigenstates of $\c$ with eigenvalue $c_m$, but
+annihilate everything else, thus
+$P_m = \sum_{c_n = c_m} | c_n \rangle \langle c_n |,$ where
+$| c_n \rangle$ is an eigenstate of $\c$ with eigenvalue $c_n$. Note
+that in the specific case of our quantum gas model
+$\c = \sqrt{2 \kappa} C(\D + \B)$ so these projectors act on the
+matter state. Recall from the previous chapter that this allows us to
+decompose the master equation in terms of the measurement basis as a
+series of equations $P_m \dot{\hat{\rho}} P_n$. We have seen that for
+$m = n$ we obtain decoherence free subspaces,
+$P_m \dot{\hat{\rho}} P_m = -i P_m \left[\hat{H}_0, \hat{\rho} \right]
+P_m$, where the measurement terms disappear which shows that a state
+in a single eigenspace is unaffected by observation. On the other
+hand, for $m \ne n$ the Hamiltonian evolution actively competes
+against measurement. In general, if $\c$ does not commute with the
+Hamiltonian then a projection to a single eigenspace $P_m$ is
+impossible unless the measurement is strong enough for the quantum
+Zeno effect to occur.
+
+We now go beyond what we previously did and define a new type of
+projector $\mathcal{P}_M = \sum_{m \in M} P_m$, such that
+$\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M$ and
+$\sum_M \mathcal{P}_M = \hat{1}$ where $M$ denotes some arbitrary
+subspace. The first equation implies that the subspaces can be built
+from $P_m$ whilst the second and third equation are properties of
+projectors and specify that these projectors do not overlap and that
+they cover the whole Hilbert space. Furthermore, we will also require
+that $[\mathcal{P}_M, \hat{H}_0 ] = 0$ and $[\mathcal{P}_M, \c] =
+0$. The second commutator simply follows from the definition
+$\mathcal{P}_M = \sum_{m \in M} P_m$, but the first one is
+non-trivial. However, if we can show that
+$\mathcal{P}_M = \sum_{m \in M} | h_m \rangle \langle h_m |$, where
+$| h_m \rangle$ is an eigenstate of $\hat{H}_0$ then the commutator is
+guaranteed to be zero. This is a complex set of requirements and it is
+unclear if it is possible to satisfy all of them at once. However, we
+note that there exists a trivial case where all these conditions are
+satisfied and that is when there is only one such projector
+$\mathcal{P}_M = \hat{1}$.
+
+Assuming that it is possible to have non-trivial cases where
+$\mathcal{P}_M \ne \hat{1}$ we can write the master equation as
+\begin{equation} 
+  \label{eq:bigP}
+  \mathcal{P}_M \dot{\hat{\rho}} \mathcal{P}_N = -i
+  \left[\hat{H}_0, \mathcal{P}_M \hat{\rho} \mathcal{P}_N \right] +
+  \left[ \c \mathcal{P}_M \hat{\rho}
+  \mathcal{P}_N \cd - \frac{1}{2} \left( \cd \c \mathcal{P}_M \hat{\rho}
+  \mathcal{P}_N + \mathcal{P}_M \hat{\rho} \mathcal{P}_N \cd \c \right)
+  \right].
+\end{equation} 
+Crucially, thanks to the commutation relations we were able to divide
+the density matrix in such a way that each submatrix's time evolution
+depends only on itself. Every submatrix
+$\mathcal{P}_M \dot{\hat{\rho}} \mathcal{P}_N$ depends only on the
+current state of itself and its evolution is ignorant of anything else
+in the total density matrix. This is in contrast to the partitioning
+we achieved with $P_m$. Previously we only identitified subspaces that
+were decoherence free, i.e.~unaffected by measurement. However, those
+submatrices could still couple with the rest of the density matrix via
+the coherent term $P_m [\hat{H}_0, \hat{\rho}] P_n$.
+
+We note that when $M = N$ the equations for
+$\mathcal{P}_M \hat{\rho} \mathcal{P}_M$ will include decoherence free
+subspaces, i.e.~$P_m \hat{\rho} P_m$. Therefore, parts of the
+$\mathcal{P}_M \hat{\rho} \mathcal{P}_M$ submatrices will also remain
+unaffected by measurement. However, the submatrices
+$\mathcal{P}_M \hat{\rho} \mathcal{P}_N$, for which $M \ne N$, are
+guaranteed to not contain any of these measurement free subspaces
+thanks to the orthogonality of $\mathcal{P}_M$. Therefore, for
+$M \ne N$ all elements of $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$
+will experience a non-zero measurement term whose effect is always
+dissipative/lossy. Furthermore, these coherence submatrices
+$\mathcal{P}_M \hat{\rho} \mathcal{P}_N$ are not coupled to any other
+part of the density matrix as seen from Eq. \eqref{eq:bigP} and so
+they can never increase in magnitude; the remaining coherent evolution
+is unable to counteract the dissipative term without an `external
+pump' from other parts of the density matrix. The combined effect is
+such that all $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$ for which
+$M \ne N$ will always go to zero due to dissipation.
+
+When all these cross-terms vanish, we are left with a density matrix
+that is a mixed state of the form
+$\hat{\rho} = \sum_M \mathcal{P}_M \hat{\rho} \mathcal{P}_M$. Since
+there are no coherences, $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$,
+this state contains only classical uncertainty about which subspace,
+$\mathcal{P}_M$, is occupied - there are no quantum superpositions
+between different $\mathcal{P}_M$ spaces. Therefore, in a single
+measurement run we are guaranteed to have a state that lies entirely
+within a subspace defined by $\mathcal{P}_M$. This is once again
+analogous to the qubit example from section \ref{sec:master}.
+
+Such a non-trivial case is indeed possible for our $\hat{H}_0$ and
+$\a_1 = C\Bmin$ and we can see the effect in
+Fig. \ref{fig:projections}. We can clearly see how a state that was
+initially a superposition of a large number of eigenstates of both
+operators becomes confined to some small subspace that is neither an
+eigenspace of $\a_1$ or $\hat{H}_0$. In this case the projective spaces,
+$\mathcal{P}_M$, are defined by the parities (odd or even) of the
+combined number of atoms in the $\beta_k$ and $\tilde{\beta}_k$ modes
+for different momenta $0 < k < \pi/a$ that are distinguishable to
+$\Bmin$. The explanation requires careful consideration of where the
+eigenstates of the two operators overlap.
+
+\subsection{Determining the Projection Subspace}
+
+To find $\mathcal{P}_M$ we need to identify the subspaces $M$ which
+satisfy the following relation
+$\sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle \langle h_m
+|$. This can be done iteratively by (i) selecting some $P_m$, (ii)
+identifying the $| h_m \rangle$ which overlap with this subspace,
+(iii) identifying any other $P_m$ which also overlap with these
+$| h_m \rangle$ from step (ii). We repeat (ii)-(iii) for all the $P_m$
+found in (iii) until we have identified all the subspaces $P_m$ linked
+in this way and they will form one of our $\mathcal{P}_M$
+projectors. If $\mathcal{P}_M \ne 1$ then there will be other
+subspaces $P_m$ which we have not included so far and thus we repeat
+this procedure on the unused projectors until we identify all
+$\mathcal{P}_M$. Computationally this can be straightforwardly solved
+with some basic algorithm that can compute the connected components of
+a graph.
+
+The above procedure, whilst mathematically correct and always
+guarantees to generate the projectors $\mathcal{P}_M$, is very
+unintuitive and gives poor insight into the nature or physical meaning
+of $\mathcal{P}_M$. In order to get a better understanding of these
+subspaces we need to define a new operator $\hat{O}$, with eigenspace
+projectors $R_m$, which commutes with both $\hat{H}_0$ and
+$\c$. Physically this means that $\hat{O}$ is a compatible observable
+with $\c$ and corresponds to a quantity conserved by the
+Hamiltonian. The fact that $\hat{O}$ commutes with the Hamiltonian
+implies that the projectors can be written as a sum of Hamiltonian
+eigenstates $R_m = \sum_{h_i = h_m} | h_i \rangle \langle h_i |$ and
+thus a projector $\mathcal{P}_M = \sum_{m \in M} R_m$ is guaranteed to
+commute with the Hamiltonian and similarly since $[\hat{O}, \c] = 0$
+$\mathcal{P}_M$ will also commute with $\c$ as required. Therefore,
+$\mathcal{P}_M = \sum_{m \in M} R_m = \sum_{m \in M} P_m$ will satisfy
+all the necessary prerequisites. This is illustrated in
+Fig. \ref{fig:spaces}.
+
+\begin{figure}[hbtp!]
+	\includegraphics[width=\linewidth]{spaces}
+	\caption[A Visual Representation of the Projection Spaces of
+        the Measurement]{A visual representation of the projection
+          spaces of the measurement. The light blue areas (bottom
+          layer) are $R_m$, the eigenspaces of $\hat{O}$. The green
+          areas are measurement eigenspaces, $P_m$, and they overlap
+          non-trivially with the $R_m$ subspaces. The $\mathcal{P}_M$
+          projection space boundary (dashed line) runs through the
+          Hilbert space, $\mathcal{H}$, where there is no overlap
+          between $P_m$ and $R_m$.  \label{fig:spaces}}
+\end{figure}
+
+In the simplest case the projectors $\mathcal{P}_M$ can consist of
+only single eigenspaces of $\hat{O}$, $\mathcal{P}_M = R_m$. The
+interpretation is straightforward - measurement projects the system
+onto a eigenspace of an observable $\hat{O}$ which is a compatible
+observable with $\a$ and corresponds to a quantity conserved by the
+coherent Hamiltonian evolution. However, this may not be possible and
+we have the more general case when
+$\mathcal{P}_M = \sum_{m \in M} R_m$. In this case, one can simply
+think of all $R_{m \in M}$ as degenerate just like eigenstates of the
+measurement operator, $\a$, that are degenerate, can form a single
+eigenspace $P_m$. However, these subspaces correspond to different
+eigenvalues of $\hat{O}$ distinguishing it from conventional
+projections. Instead, the degeneracies are identified by some other
+feature.
+
+In our case, it is apparent from the form of $\Bmin$ and $\hat{H}_0$
+in Eqs. \eqref{eq:BminBeta} and \eqref{eq:H0Beta} that
+$\hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger
+\tilde{\beta_k} = \n_k + \n_{k - \pi/a}$ commutes with both operators
+for all $k$. Thus, we can easily construct
+$\hat{O} = \sum_\mathrm{RBZ} g_k \hat{O}_k$ for any arbitrary
+$g_k$. Its eigenspaces, $R_m$, can then be easily constructed and
+their relationship with $P_m$ and $\mathcal{P}_M$ is illustrated in
+Fig. \ref{fig:spaces} whilst the time evolution of
+$\langle \hat{O}_k \rangle$ for a sample trajectory is shown in
+Fig. \ref{fig:projections}(a). Note that unlike the $\c$ or $\H_0$ we
+can actually see that this observable's distribution does indeed
+freeze. These eigenspaces are composed of Fock states in momentum
+space that have the same number of atoms within each pair of $k$ and
+$k - \pi/a$ modes.
+
+Having identified and appropriate $\hat{O}$ operator we proceed to
+identifying $\mathcal{P}_M$ subspaces for the operator
+$\Bmin$. Firstly, since $\Bmin$ contains $\sin(ka)$ coefficients atoms
+in different $k$ modes that have the same $\sin(ka)$ value are
+indistinguishable to the measurement and will lie in the same $P_m$
+eigenspaces. This will happen for the pairs ($k$, $\pi/a -
+k$). Therefore, the $R_m$ spaces that have the same
+$\hat{O}_k + \hat{O}_{\pi/a - k}$ eigenvalues must belong to the same
+$\mathcal{P}_M$.
+
+Secondly, if we re-write $\hat{O}$ in terms of the $\beta_k$ and
+$\tilde{\beta}_k$ modes we get
+\begin{equation} 
+  \hat{O} = \sum_{\mathrm{RBZ}} g_k \left(
+    \beta^\dagger_k \beta_k + \tilde{\beta}_k^\dagger \tilde{\beta}_k
+  \right),
+\end{equation}
+and so it's not hard to see that
+$\B_{\mathrm{min},k} = (\beta^\dagger_k \beta_k -
+\tilde{\beta}_k^\dagger \tilde{\beta}_k)$ can have the same
+eigenvalues for different values of
+$\hat{O}_k = \beta^\dagger_k \beta_k + \tilde{\beta}_k^\dagger
+\tilde{\beta}_k$. Specifically, if a given subspace $R_m$ corresponds
+to the eigenvalue $O_k$ of $\hat{O}_k$ then the possible values of
+$B_{\mathrm{min},k}$ will be $\{-O_k, -O_k + 2, ..., O_k - 2, O_k\}$
+just like $\Delta N = N_\mathrm{even} - N_\mathrm{odd}$ can have
+multiple possible values for a fixed
+$N = N_\mathrm{even} + N_\mathrm{odd}$. Therefore, we can see that all
+$R_m$ with even values of $O_k$ will share $B_{\mathrm{min},k}$
+eigenvalues and thus they will overlap with the same $P_m$
+subspaces. The same is true for odd values of $O_k$. However, $R_m$
+with an even value of $O_k$ will never have the same value of
+$B_{\mathrm{min},k}$ as a subspace with an odd value of
+$O_k$. Therefore, a single $\mathcal{P}_M$ will contain all $R_m$ that
+have the same parities of $O_k$ for all $k$, e.g.~if it includes the
+$R_m$ with $O_k = 6$, it will also include the $R_m$ for which
+$O_k = 0, 2, 4, 6, ..., N$, but not any of $O_k = 1, 3, 5, 7, ..., N$,
+where $N$ is the total number of atoms.
+
+Finally, the $k = \pi/a$ mode is special, because $\sin(\pi) = 0$
+which means that $B_{\mathrm{min},k=\pi/a} = 0$ always. This in turn
+implies that all possible values of $O_{\pi/a}$ are degenerate to the
+measurement. Therefore, we exclude this mode when matching the
+parities of the other modes.
+
+To illustrate the above let us consider a specific example. Let us
+consider two atoms, $N=2$, on eight sites $M=8$. This particular
+configuration has eight momentum modes
+$ka = \{-\frac{3\pi}{4}, -\frac{\pi}{2}, -\frac{\pi}{4}, 0,
+\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\}$ and so the RBZ
+has only four modes,
+$\mathrm{RBZ} := \{\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4},
+\pi\}$. There are 10 different ways of splitting two atoms into these
+four modes and thus we have 10 different
+$R_m = \{O_{\pi/4a}, O_{\pi/2a} ,O_{3\pi/4a} ,O_{\pi/a}\}$ eigenspaces
+of $\hat{O}$
+\begin{table}[!htbp]
+  \centering
+  \begin{tabular}{l c c}
+    \toprule
+    $m$ & $R_m$ & Possible values of $B_\mathrm{min}$ \\ \midrule
+    0 & $\{2,0,0,0\}$ & $ -\sqrt{2}, 0, \sqrt{2}  $ \\
+    1 & $\{1,1,0,0\}$ & $  -\frac{1 + \sqrt{2}}{\sqrt{2}}, -\frac{1
+                        - \sqrt{2}}{\sqrt{2}}, \frac{1
+                        - \sqrt{2}}{\sqrt{2}}, \frac{1 +
+                        \sqrt{2}}{\sqrt{2}}  $ \\ 
+    2 & $\{1,0,1,0\}$ & $  -\sqrt{2}, 0, \sqrt{2}  $ \\
+    3 & $\{1,0,0,1\}$ & $  -\frac{1}{\sqrt{2}},
+                        \frac{1}{\sqrt{2}}  $ \\
+    4 & $\{0,2,0,0\}$ & $  -2, 0, 2  $ \\
+    5 & $\{0,1,1,0\}$ & $  -\frac{1 + \sqrt{2}}{\sqrt{2}}, -\frac{1
+                        - \sqrt{2}}{\sqrt{2}}, \frac{1
+                        - \sqrt{2}}{\sqrt{2}}, \frac{1 +
+                        \sqrt{2}}{\sqrt{2}} $ \\
+    6 & $\{0,1,0,1\}$ & $  -1, 1  $ \\
+    7 & $\{0,0,2,0\}$ & $  -\sqrt{2}, 0, \sqrt{2}  $ \\
+    8 & $\{0,0,1,1\}$ & $  -\frac{1}{\sqrt{2}},
+                        \frac{1}{\sqrt{2}}  $ \\
+    9 & $\{0,0,0,2\}$ & $0$ \\
+    \bottomrule
+  \end{tabular}
+  \caption[Eigenspace Overlaps]{A list of all $R_m$ eigenspaces for $N
+    = 2$ atoms at $M = 8$ sites. The third column displays the eigenvalues of
+    all the eigenstates of $\Bmin$ that lie in the given $R_m$.}
+  \label{tab:Rm}
+\end{table}
+In the third column we have also listed the eigenvalues of the $\Bmin$
+eigenstates that lie within the given $R_m$.
+
+We note that $ka = \pi/4$ will be degenerate with $ka = 3\pi/4$ since
+$\sin(ka)$ is the same for both. Therefore, we already know that we
+can combine $(R_0, R_2, R_7)$, $(R_1, R_5)$, and $(R_3, R_8)$, because
+those combinations have the same $O_{\pi/4a} + O_{3\pi/4a}$
+values. This is very clear in the table as these subspaces span
+exactly the same values of $B_\mathrm{min}$, i.e.~the have exactly the
+same values in the third column.
+
+Now we have to match the parities. Subspaces that have the same parity
+combination for the pair $(O_{\pi/4a} + O_{3\pi/4a}, O_{\pi/2a})$ will
+be degenerate in $\mathcal{P}_M$. Note that we excluded $O_{\pi/a}$,
+because as we discussed earlier they are all degenerate due to
+$\sin(\pi) = 0$. Therefore, the (even,even) subspace will include
+$(R_0, R_2, R_4, R_7, R_9)$, the (odd,even) will contain $(R_3, R_8)$,
+the (even, odd) will contain $(R_6)$ only, and the (odd, odd) contains
+$(R_1, R_5)$. These overlaps should be evident from the table as we
+can see that these combinations combine all $R_m$ that share any
+eigenstates of $\Bmin$ with the same eigenvalues.
+
+Therefore, we have ended up with four distinct $\mathcal{P}_M$ subspaces
+\begin{align}
+\mathcal{P}_\mathrm{even,even} = & R_0 + R_2 + R_4 + R_7 + R_9
+  \nonumber \\
+\mathcal{P}_\mathrm{odd,even} = & R_3 + R_8 \nonumber \\
+\mathcal{P}_\mathrm{even,odd} = & R_6 \nonumber \\
+\mathcal{P}_\mathrm{odd,odd} = & R_1 + R_5 \nonumber.
+\end{align}
+At this point it should be clear that these projectors satisfy all our
+requirement. The conditions $\sum_M \mathcal{P}_M = 1$ and
+$\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M$ should be
+evident from the form above. The commutator requirements are also
+easily satisfied since the subspaces $R_m$ are of an operator that
+commutes with both the Hamiltonian and the measurement operator. And
+finally, one can also verify using the table that all of these
+projectors are built from complete subspaces of $\Bmin$ (i.e.~each
+subspace $P_m$ belongs to only one $\mathcal{P}_M$) and thus
+$\mathcal{P}_M = \sum_{m \in M} P_m$.
 
 \section{Conclusions}
+
+In summary we have investigated measurement backaction resulting from
+coupling light to an ultracold gas's phase-related observables. We
+demonstrated how this can be used to prepare the Hamiltonian
+eigenstates even if significant tunnelling is occuring as the
+measurement can be engineered to not compete with the system's
+dynamics. Furthermore, we have shown that when the observable of the
+phase-related quantities does not commute with the Hamiltonian we
+still project to a specific subspace of the system that is neither an
+eigenspace of the Hamiltonian or the measurement operator. This is in
+contrast to quantum Zeno dynamics \cite{misra1977, facchi2008,
+  raimond2010, raimond2012, signoles2014} or dissipative state
+preparation \cite{diehl2008}. We showed how this projection is
+essentially an extension of the measurement postulate to weak
+measurement on dynamical systems where the competition between the two
+processes is significant.

Preamble/preamble.tex

@@ -215,3 +215,7 @@
 \newcommand{\op}{\hat{o}}
 \newcommand{\h}{\hat{h}}
 \newcommand{\N}{\hat{N}}
+\newcommand{\B}{\hat{B}}
+\newcommand{\Bmax}{\hat{B}_\mathrm{max}}
+\newcommand{\Bmin}{\hat{B}_\mathrm{min}}
+\newcommand{\D}{\hat{D}}

References/references.bib

@@ -21,6 +21,13 @@
   year={2011},
   school={LMU}
 }
+@phdthesis{StephenThesis,
+  title={Strongly correlated one-dimensional systems of cold atoms in
+                  optical lattices },
+  author={Clark, Stephen Richard James Franz},
+  year={2007},
+  school={University of Oxford}
+}
 @misc{steck,
   title={Rubidium 87 D line data},
   author={Steck, Daniel A},
@@ -58,6 +65,19 @@ year = {2010}
   year={2012},
   publisher={Springer Science \& Business Media}
 }
+@book{leggett,
+  title={Quantum liquids: Bose condensation and Cooper pairing in condensed-matter systems},
+  author={Leggett, Anthony J},
+  year={2006},
+  publisher={Oxford University Press}
+}
+@book{PitaevskiiStringari,
+  title={Bose-Einstein Condensation and Superfluidity},
+  author={Pitaevskii, Lev and Stringari, Sandro},
+  volume={164},
+  year={2016},
+  publisher={Oxford University Press}
+}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Igor's original papers
@@ -210,6 +230,18 @@ year = {2010}
   year={2015},
   publisher={APS}
 }
+@article{caballero2015njp,
+  title={Quantum properties of light scattered from structured
+                  many-body phases of ultracold atoms in quantum
+                  optical lattices},
+  author={Caballero-Benitez, Santiago F and Mekhov, Igor B},
+  journal={New Journal of Physics},
+  volume={17},
+  number={12},
+  pages={123023},
+  year={2015},
+  publisher={IOP Publishing}
+}
 @article{caballero2016,
   title = {Quantum simulators based on the global collective light-matter interaction},
   author = {Caballero-Benitez, Santiago F. and Mazzucchi, Gabriel and Mekhov, Igor B.},
@@ -224,6 +256,13 @@ year = {2010}
   doi = {10.1103/PhysRevA.93.063632},
   url = {http://link.aps.org/doi/10.1103/PhysRevA.93.063632}
 }
+@article{caballero2016a,
+  title={Bond Order via Light-Induced Synthetic Many-body Interactions
+                  of Ultracold Atoms in Optical Lattices},
+  author={Caballero-Benitez, Santiago F and Mekhov, Igor B},
+  journal={arXiv preprint arXiv:1604.02563},
+  year={2016}
+}
 @article{mazzucchi2016,
   title = {Quantum measurement-induced dynamics of many-body ultracold
                   bosonic and fermionic systems in optical lattices},
@@ -1040,3 +1079,149 @@ doi = {10.1103/PhysRevA.87.043613},
   year={2008},
   publisher={Nature Publishing Group}
 }
+@article{douglas2012,
+  title={Scattering-induced spatial superpositions in multiparticle localization},
+  author={Douglas, James S and Burnett, Keith},
+  journal={Physical Review A},
+  volume={86},
+  number={5},
+  pages={052120},
+  year={2012},
+  publisher={APS}
+}
+@article{douglas2013,
+  title={Scattering distributions in the presence of measurement backaction},
+  author={Douglas, James S and Burnett, Keith},
+  journal={Journal of Physics B: Atomic, Molecular and Optical Physics},
+  volume={46},
+  number={20},
+  pages={205301},
+  year={2013},
+  publisher={IOP Publishing}
+}
+@article{lee2014,
+  title={Classical stochastic measurement trajectories: Bosonic atomic
+                  gases in an optical cavity and quantum measurement
+                  backaction},
+  author={Lee, Mark D and Ruostekoski, Janne},
+  journal={Physical Review A},
+  volume={90},
+  number={2},
+  pages={023628},
+  year={2014},
+  publisher={APS}
+}
+@article{weinberg2016,
+  title={What happens in a measurement?},
+  author={Weinberg, Steven},
+  journal={Physical Review A},
+  volume={93},
+  number={3},
+  pages={032124},
+  year={2016},
+  publisher={APS}
+}
+@article{brune1992,
+  title={Manipulation of photons in a cavity by dispersive atom-field coupling: Quantum-nondemolition measurements and generation of ``Schr{\"o}dinger cat'' states},
+  author={Brune, M and Haroche, S and Raimond, J M and Davidovich, L and Zagury, N},
+  journal={Physical Review A},
+  volume={45},
+  number={7},
+  pages={5193},
+  year={1992},
+  publisher={APS}
+}
+@inproceedings{hubbard1963,
+  title={Electron correlations in narrow energy bands},
+  author={Hubbard, John},
+  booktitle={Proceedings of the royal society of london a:
+                  mathematical, physical and engineering sciences},
+  volume={276},
+  number={1365},
+  pages={238--257},
+  year={1963},
+  organization={The Royal Society}
+}
+@article{jaksch1998,
+  title={Cold bosonic atoms in optical lattices},
+  author={Jaksch, Dieter and Bruder, Ch and Cirac, Juan Ignacio and Gardiner, Crispin W and Zoller, Peter},
+  journal={Physical Review Letters},
+  volume={81},
+  number={15},
+  pages={3108},
+  year={1998},
+  publisher={APS}
+}
+@article{greiner2002,
+  title={Quantum phase transition from a superfluid to a Mott
+                  insulator in a gas of ultracold atoms},
+  author={Greiner, Markus and Mandel, Olaf and Esslinger, Tilman and
+                  H{\"a}nsch, Theodor W and Bloch, Immanuel},
+  journal={nature},
+  volume={415},
+  number={6867},
+  pages={39--44},
+  year={2002},
+  publisher={Nature Publishing Group}
+}
+@article{krauth1991,
+  title={Bethe ansatz for the one-dimensional boson Hubbard model},
+  author={Krauth, Werner},
+  journal={Physical Review B},
+  volume={44},
+  number={17},
+  pages={9772},
+  year={1991},
+  publisher={APS}
+}
+@article{kolovsky2004,
+  title={Quantum chaos in the Bose-Hubbard model},
+  author={Kolovsky, Andrey R and Buchleitner, Andreas},
+  journal={EPL (Europhysics Letters)},
+  volume={68},
+  number={5},
+  pages={632},
+  year={2004},
+  publisher={IOP Publishing}
+}
+@article{calzetta2006,
+  title={Bose-Einstein-condensate superfluid--Mott-insulator transition in an optical lattice},
+  author={Calzetta, Esteban and Hu, Bei-Lok and Rey, Ana Maria},
+  journal={Physical Review A},
+  volume={73},
+  number={2},
+  pages={023610},
+  year={2006},
+  publisher={APS}
+}
+@article{leggett1999,
+  title={Superfluidity},
+  author={Leggett, Anthony J},
+  journal={Reviews of Modern Physics},
+  volume={71},
+  number={2},
+  pages={S318},
+  year={1999},
+  publisher={APS}
+}
+@article{fisher1989,
+  title={Boson localization and the superfluid-insulator transition},
+  author={Fisher, Matthew PA and Weichman, Peter B and Grinstein, G and Fisher, Daniel S},
+  journal={Physical Review B},
+  volume={40},
+  number={1},
+  pages={546},
+  year={1989},
+  publisher={APS}
+}
+@article{haldane1981,
+  title={Effective harmonic-fluid approach to low-energy properties of
+                  one-dimensional quantum fluids},
+  author={Haldane, FDM},
+  journal={Physical Review Letters},
+  volume={47},
+  number={25},
+  pages={1840},
+  year={1981},
+  publisher={APS}
+}

thesis.tex

@@ -101,7 +101,7 @@
 % To use choose `chapter' option in the document class
 
 \ifdefineChapter
- \includeonly{Chapter5/chapter5}
+ \includeonly{Chapter1/chapter1}
 \fi
 
 % ******************************** Front Matter ********************************