Final version 1.0

Abstract/abstract.tex

@@ -1,5 +1,6 @@
 % ************************** Thesis Abstract *****************************
 % Use `abstract' as an option in the document class to print only the titlepage and the abstract.
+
 \begin{abstract}
   Trapping ultracold atoms in optical lattices enabled the study of
   strongly correlated phenomena in an environment that is far more

Acknowledgement/acknowledgement.tex

@@ -2,8 +2,33 @@
 
 \begin{acknowledgements}      
 
+First and foremost, I would like to thank my supervisor Dr. Igor
+Mekhov who has been an excellent mentor throughout my time at
+Oxford. It is primarily thanks to his brilliant insights and
+professionalism that I was able to reach my full potential during my
+doctoral studies. The work contained in this thesis would also not be
+possible without the help of the other members of the group, Gabriel
+Mazzucchi and Dr. Santiago Caballero-Benitez. Without our frequent
+casual discussions in the Old Library office I would have still been
+stuck on the third chapter. I would also like to acknowledge all
+members of Prof. Dieter Jaksch's and Prof. Christopher Foot's groups
+for various helpful discussions. I must also offer a special mention
+for Edward Owen who provided much needed reality checks on some of my
+wishful theoretical thinking. I would also like to express my
+gratitude to EPSRC, St. Catherine's College, the ALP sub-department,
+and the Institute of Physics for providing me with the financial means
+to live and study in Oxford as well as attend several conferences in
+the UK and abroad.
 
-\mynote{Write my own acknowledgements}
+On a personal note, I would like to thank my parents who provided me
+with all the skills necessary work towards any goals I set
+myself. Needless to say, without them I would have never been able to
+be where I am right now. I would like to thank all the new and old
+friends that have kept me company for the last four years. The time
+spent together provided a welcome respite from the sweat and toil of
+my DPhil.
 
+\vspace{2em}
+\raggedleft{Oxford, September 2016}
 
 \end{acknowledgements}

Chapter1/chapter1.tex

@@ -83,9 +83,9 @@ imprinted in the scattered light \cite{klinder2015, landig2016}.
 There are three prominent directions in which the field of quantum
 optics of quantum gases has progressed in. First, the use of quantised
 light enables direct coupling to the quantum properties of the atoms
-\cite{mekhov2007prl, mekhov2007pra, mekhov2012}. This allows us to
+\cite{mekhov2007prl, mekhov2007pra, mekhov2007NP, mekhov2012}. This
-probe the many-body system in a nondestructive manner and under
+allows us to probe the many-body system in a nondestructive manner and
-certain conditions even perform quantum non-demolition (QND)
+under certain conditions even perform quantum non-demolition (QND)
 measurements. QND measurements were originally developed in the
 context of quantum optics as a tool to measure a quantum system
 without significantly disturbing it \cite{braginsky1977, unruh1978,
@@ -200,7 +200,7 @@ measurement backaction. Whilst such interference measurements have
 been previously proposed for BECs in double-wells \cite{cirac1996,
   castin1997, ruostekoski1997}, the extension to a lattice system is
 not straightforward. However, we will show it is possible to achieve
-with our propsed setup by a careful optical arrangement. Within this
+with our proposed setup by a careful optical arrangement. Within this
 context we demonstrate a novel type of projection which occurs even
 when there is significant competition with the Hamiltonian
 dynamics. This projection is fundamentally different to the standard
@@ -215,5 +215,58 @@ Hamiltonian in a way that depends on the state of the atoms that
 scatterd the light. This can lead to new quantum phases due to new
 types of long-range interactions being mediated by the global quantum
 optical fields \cite{caballero2015, caballero2015njp, caballero2016,
-  caballero2016a}. However, this aspect of quantum optics of quantum
+  caballero2016a, elliott2016}. However, this aspect of quantum optics
-gases is beyond the scope of this thesis.
+of quantum gases is beyond the scope of this thesis.
+
+\newpage
+
+\section*{Publication List}
+
+The work contained in this thesis is based on seven publications
+\cite{kozlowski2015, elliott2015, atoms2015, mazzucchi2016,
+  kozlowski2016zeno, mazzucchi2016njp, kozlowski2016phase}:
+
+\begin{table}[hbtp!]
+  \centering
+  \begin{tabular}{r p{13cm}}
+    \toprule
+    \cite{kozlowski2015} & W. Kozlowski, S. F. Caballero-Benitez, and
+    I. B. Mekhov.  ``Probing matter-field and atom-number correlations
+    in optical lattices by global nondestructive addressing''.
+    \emph{Physical Review A}, 92(1):013613, 2015. \\ \\
+
+    \cite{elliott2015} & T. J. Elliott, W. Kozlowski,
+    S. F. Caballero-Benitez, and I. B. Mekhov. ``Multipartite
+    Entangled Spatial Modes of Ultracold Atoms Generated and
+    Controlled by Quantum Measurement''. \emph{Physical Review
+      Letters}, 114:113604, 2015. \\ \\
+
+    \cite{atoms2015} & T. J. Elliott, G. Mazzucchi, W. Kozlowski,
+    S. F. Caballero- Benitez, and I. B. Mekhov. ``Probing and
+    manipulating fermionic and bosonic quantum gases with quantum
+    light''. \emph{Atoms}, 3(3):392–406, 2015. \\ \\
+
+    \cite{mazzucchi2016} & G. Mazzucchi$^*$, W. Kozlowski$^*$,
+    S. F. Caballero-Benitez, T. J.  Elliott, and
+    I. B. Mekhov. ``Quantum measurement-induced dynamics of many- body
+    ultracold bosonic and fermionic systems in optical
+    lattices''. \emph{Physical Review A}, 93:023632,
+    2016. $^*$\emph{Equally contributing authors}. \\ \\
+
+    \cite{kozlowski2016zeno} & W. Kozlowski, S. F. Caballero-Benitez,
+    and I. B. Mekhov. ``Non- hermitian dynamics in the quantum zeno
+    limit''. \emph{Physical Review A}, 94:012123, 2016. \\ \\ 
+
+    \cite{mazzucchi2016njp} & G. Mazzucchi, W. Kozlowski,
+    S. F. Caballero-Benitez, and I. B Mekhov. ``Collective dynamics of
+    multimode bosonic systems induced by weak quan- tum
+    measurement''. \emph{New Journal of Physics}, 18(7):073017, 2016. \\ \\
+
+    \cite{kozlowski2016phase} & W. Kozlowski, S. F. Caballero-Benitez,
+    and I. B. Mekhov. ``Quantum state reduction by
+    matter-phase-related measurements in optical
+    lattices''. \emph{arXiv preprint arXiv:1605.06000}, 2016. \\
+
+    \bottomrule
+  \end{tabular}
+\end{table}

Chapter2/chapter2.tex

@@ -9,20 +9,24 @@
     \graphicspath{{Chapter2/Figs/Raster/}{Chapter2/Figs/PDF/}{Chapter2/Figs/}}
 \else
     \graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}}
-\fi
+\fi    
 
 \section{Introduction}
 
-In this chapter, we derive a general Hamiltonian that describes the
+In this chapter, we present the derivation of a general Hamiltonian
-coupling of atoms with far-detuned optical beams
+that describes the coupling of atoms with far-detuned optical beams
-\cite{mekhov2012}. This will serve as the basis from which we explore
+originally presented in Ref. \cite{mekhov2012}. This will serve as the
-the system in different parameter regimes, such as nondestructive
+basis from which we explore the system in different parameter regimes,
-measurement in free space or quantum measurement backaction in a
+such as nondestructive measurement in free space or quantum
-cavity. Another interesting direction for this field of research are
+measurement backaction in a cavity. As this model extends the
-quantum optical lattices where the trapping potential is treated
+Bose-Hubbard Hamiltonian to include the effects of interactions with
-quantum mechanically \cite{caballero2015, caballero2015njp,
+quantised light we also present a brief overview of the properties of
-  caballero2016, caballero2016a}. However this is beyond the scope of
+the Bose-Hubbard model itself and its quantum phase transiton. This is
-this work.
+followed by a description of the behaviour of the scattered light with
+a particular focus on how to couple the optical fields to phase
+observables as opposed to density observables as is typically the
+case. Finally, we conclude with an overview of possible experimental
+realisability.
 
 We consider $N$ two-level atoms in an optical lattice with $M$
 sites. For simplicity we will restrict our attention to spinless
@@ -37,7 +41,7 @@ from free particles to strongly correlated systems, to the inherent
 tunability of such lattices. Furthermore, this model is 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
+in a double-well potential) to an extended multi-site lattice with a
 low filling factor (e.g.~a system with one atom per site which will
 exhibit the Mott insulator to superfluid quantum phase transition).
 
@@ -262,7 +266,7 @@ hopping rate given by
 \begin{equation}
   J^\mathrm{cl}_{i,j} = \int \mathrm{d}^3 \b{r} w (\b{r} - \b{r}_i ) 
   \left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r}
-  - \b{r}_i),
+  - \b{r}_j),
 \end{equation}
 and $U_{ijkl}$ is the atomic interaction term given by
 \begin{equation}
@@ -731,7 +735,9 @@ neglected. This is likely to be the case since the interactions will
 be dominated by photons scattering from the much larger coherent
 probe.
 
-\section{Density and Phase Observables}
+\section[Density and Phase Observables]
+        {Density and Phase Observables\footnote{The results of this
+            section were first published in Ref. \cite{kozlowski2015}}}
 \label{sec:B}
 
 Light scatters due to its interactions with the dipole moment of the
@@ -802,6 +808,18 @@ 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. 
 
+\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}
+
 In order to calculate the $J_{i,j}$ coefficients we perform numerical
 calculations using realistic Wannier functions
 \cite{walters2013}. However, it is possible to gain some analytic
@@ -835,35 +853,6 @@ $\hat{D}$ operator since it depends on the amplitude of light in
 between the lattice sites and not at the positions of the atoms
 allowing to decouple them at specific angles.
 
-\begin{figure}
-  \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}
-
-\begin{figure}
-  \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
 $J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
 constant. This results in a diffraction maximum where each bond
@@ -889,6 +878,23 @@ 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.
 
+\begin{figure}
+  \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}
+
 Another possibility is to obtain an alternating pattern similar
 corresponding to a diffraction minimum where each bond scatters light
 in anti-phase with its neighbours giving
@@ -927,10 +933,13 @@ $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ and by
 varying the local oscillator phase, one can choose which conjugate
 operator to measure.
 
-\section{Electric Field Stength}
+\section[Electric Field Strength]
+        {Electric Field Strength\footnote{The derivation has not been
+            published before, but the final numerical results were
+            included in Ref. \cite{kozlowski2015}}}
 \label{sec:Efield}
 
-The Electric field operator at position $\b{r}$ and at time $t$ is
+The electric field operator at position $\b{r}$ and at time $t$ is
 usually written in terms of its positive and negative components:
 \begin{equation}
   \b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t),
@@ -1095,6 +1104,25 @@ Therefore, we can now express the quantity $n_{\Phi}$ as
   n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
 \end{equation}
 
+\begin{table}
+  \centering
+  \begin{tabular}{l c c}
+    \toprule
+    Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
+    $\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
+    $\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
+    $\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
+    $I$ & $4250$ Wm$^{-2}$ & N/A \\ 
+    $\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
+    $N_K$ & 10$^5$ & 147 \\ \midrule
+    $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
+    \bottomrule
+  \end{tabular}
+  \caption[Photon Scattering Rates]{Experimental parameters used in
+    estimating the photon scattering rates.}
+  \label{tab:photons}
+\end{table}
+
 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}. The two experiments were chosen as state
@@ -1131,21 +1159,38 @@ $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
 (1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz
 per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
 
-\begin{table}
+\section{Possible Experimental Issues}
-  \centering
+
-  \begin{tabular}{l c c}
+There are many possible experimental issues that we have neglected so
-    \toprule
+far in our theoretical treatment which need to be answered in order to
-    Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
+consider the experimental feasability of our proposal. The two main
-    $\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
+concerns are photodetector efficiency and heating losses.
-    $\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
+
-    $\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
+In our theoretical models we treat the detectors as if they were
-    $I$ & $4250$ Wm$^{-2}$ & N/A \\ 
+capable of detecting every scattered photon, but real photodetectors
-    $\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
+can have efficiencies as low as 5\%. For the case of nondestructive
-    $N_K$ & 10$^5$ & 147 \\ \midrule
+measurement as covered in Chapter \ref{chap:qnd} this is not an issue
-    $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
+provided a sufficient number of photons can be collected to calculate
-    \bottomrule
+reliable expectation values. The case of scattering into a cavity and
-  \end{tabular}
+the effect of efficiency on the conditioned state was addressed in
-  \caption[Photon Scattering Rates]{Experimental parameters used in
+Ref. \cite{mazzucchi2016njp} where it was shown that detector
-    estimating the photon scattering rates.}
+efficiencies as low as 1\% are still capable of resolving the dynamics
-  \label{tab:photons}
+to a good degree of accuracy and 10\% was sufficient for near unit
-\end{table}
+fidelity. However, this incredible result requires that the photon
+scattering pattern is periodic in some way, e.g.~oscillatory as was
+the case in Ref. \cite{mazzucchi2016njp} or constant. This way it is
+only necessary to detect a sufficient number of photons to deduce the
+correct phase of the oscillations or the rate for the case of a
+constant scattering rate. In this thesis we deal predominantly with
+these two cases so photodetector efficiency is not an issue.
+
+The other issue is the heating of the trapped gas which will limit the
+lifetime of the experiment. For free space scattering imaging times of
+several hundred milliseconds have been achieved by for example using
+molasses beams that simultaneously cool and trap the atoms
+\cite{weitenberg2011, weitenbergThesis}. Similar feats have been
+achieved with atoms coupled to a leaky cavity where interogation times
+of 0.8s have been achieved in Ref. \cite{brennecke2013}. Crucially,
+the cavity in said experiment has a decay rate of the order of MHz
+which is necessary to observe measurement backaction which we will
+consider in the subsequent chapters.

Chapter3/chapter3.tex

@@ -2,8 +2,10 @@
 %*********************************** Third Chapter *****************************
 %*******************************************************************************
 
-\chapter{Probing Correlations by Global 
+\chapter[Probing Correlations by Global 
-Nondestructive Addressing} %Title of the Third Chapter
+  Nondestructive Addressing] {Probing Correlations by Global
+  Nondestructive Addressing\footnote{The results of this chapter were
+    first published in Ref. \cite{kozlowski2015}}}
 \label{chap:qnd}
 
 \ifpdf
@@ -230,7 +232,7 @@ fluctuations entirely leading to absolutely no ``quantum addition''.
 
 \begin{figure}
   \centering
-  \includegraphics[width=\linewidth]{Ep1}
+  \includegraphics[width=0.8\linewidth]{Ep1}
   \caption[Light Scattering Angular Distribution]{Light intensity
     scattered into a standing wave mode from a superfluid in a 3D
     lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming

Chapter4/chapter4.tex

@@ -421,7 +421,10 @@ generalise the above result to larger Hilbert spaces with multiple
 degenerate subspaces which are of much greater interest as they reveal
 nontrivial dynamics in the system.
 
-\section{Global Measurement and ``Which-Way'' Information}
+\section[Global Measurement and ``Which-Way'' Information]
+        {Global Measurement and ``Which-Way'' Information\footnote{The
+            results of this section were first published in
+            Ref. \cite{elliott2015}}}
 \label{sec:modes}
 
 We have already mentioned that one of the key features of our model is

Chapter5/chapter5.tex

@@ -2,7 +2,11 @@
 %*********************************** Fifth Chapter *****************************
 %*******************************************************************************
 
-\chapter{Density Measurement Induced Dynamics}
+\chapter[Density Measurement Induced Dynamics]
+        {Density Measurement Induced Dynamics\footnote{The results of
+            this chapter were first published in
+            Refs. \cite{mazzucchi2016, kozlowski2016zeno,
+              mazzucchi2016njp}}}
 % Title of the Fifth Chapter
 
 \ifpdf
@@ -290,6 +294,8 @@ jump. Expanding around the peak of the Gaussian ansatz we get
 Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump
 operator above yields a Gaussian wavefunction as well, but the
 parameters change discontinuously according to
+\begingroup
+\allowdisplaybreaks
 \begin{align}
   \label{eq:jumpb2}
   b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
@@ -299,6 +305,7 @@ parameters change discontinuously according to
   z_\phi & \rightarrow z_\phi, \\
   \epsilon & \rightarrow \epsilon.
 \end{align}
+\endgroup
 The fact that the wavefunction remains Gaussian after a photodetection
 is a huge advantage, because it means that the combined time evolution
 of the system can be described with a single Gaussian ansatz in
@@ -527,15 +534,15 @@ yields
 \end{equation}
 where the approximation on the right-hand side follows from the fact
 that $\omega \approx 2$ since we are considering the $N \gg 1$ limit,
-and because we are considering the weak measurement limit
+and because we are considering the weak measurement limit $\Gamma^2 /
-$\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width
+\omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width of the
-of the initial superfluid state. This result is interesting, because
+initial superfluid state. This result is interesting, because it shows
-it shows that the width of the Gaussian distribution is squeezed as
+that the width of the Gaussian distribution is squeezed as compared
-compared with its initial state which is exactly what we see in
+with its initial state which is exactly what we see in
 Fig. \ref{fig:oscillations}(a). However, if we substitute the
 parameter values used in that trajectory we only get a reduction in
-width by about $3\%$, but the maximum amplitude oscillations in look
+width by about $3\%$, but the maximum amplitude oscillations look like
-like they have a significantly smaller width than the initial
+they have a significantly smaller width than the initial
 distribution. This discrepancy is due to the fact that the continuous
 variable approximation is only valid for $z \ll 1$ and thus it cannot
 explain the final behaviour of the system. Furthermore, it has been
@@ -1648,14 +1655,12 @@ eigenstate of $\c$ and we combine this with the requirement for it to
 be in the dark state of the tunnelling operator (eigenstate of $\H_0$
 for $U = 0$) to derive the steady state. These two conditions in
 momentum space are
-\begin{equation}
+\begin{align}
   \hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k -
-    \bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0,
+    \bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \\
-\end{equation}
-\begin{equation}
   \Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} +
     \bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle,
-\end{equation}
+\end{align}
 where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$,
 $\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice
 spacing, $N$ the total atom number, and we perform summations over the
@@ -1738,10 +1743,11 @@ discussed.
 To obtain a state with a specific value of $\Delta N$ postselection
 may be necessary, but otherwise it is not needed.  The process can be
 optimised by feedback control since the state is monitored at all
-times \cite{ivanov2014}. Furthermore, the form of the measurement
+times \cite{ivanov2014, mazzucchi2016feedback}. Furthermore, the form
-operator is very flexible and it can easily be engineered by the
+of the measurement operator is very flexible and it can easily be
-geometry of the optical setup \cite{elliott2015, mazzucchi2016} which
+engineered by the geometry of the optical setup \cite{elliott2015,
-can be used to design a state with desired properties.
+  mazzucchi2016} which can be used to design a state with desired
+properties.
 
 \section{Conclusions}
 

Chapter6/chapter6.tex

@@ -2,7 +2,10 @@
 %*********************************** Sixth Chapter *****************************
 %*******************************************************************************
 
-\chapter{Phase Measurement Induced Dynamics}
+\chapter[Phase Measurement Induced Dynamics]
+        {Phase Measurement Induced Dynamics\footnote{The results of
+            this chapter were first published in
+            Ref. \cite{kozlowski2016phase}}}
 % Title of the Sixth Chapter
 
 \ifpdf
@@ -25,7 +28,7 @@ operators just like most of the existing work \cite{LP2009, rogers2014,
 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
+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

Dedication/dedication.tex

@@ -2,7 +2,11 @@
 
 \begin{dedication} 
 
-\mynote{Write my own dedication.}
+\emph{Moim rodzicom, bez których nie byłbym w stanie osiągnąć tego
+  wszystkiego.}
 
+\emph{To my parents without whom I would not have been able to achieve
+  any of this.}
+  
 \end{dedication}
 

Preamble/preamble.tex

@@ -221,3 +221,5 @@
 \newcommand{\Bmax}{\hat{B}_\mathrm{max}}
 \newcommand{\Bmin}{\hat{B}_\mathrm{min}}
 \newcommand{\D}{\hat{D}}
+
+\usepackage[utf8]{inputenc}

thesis-info.tex

@@ -29,7 +29,7 @@ Dynamics in Ultracold Bosonic Gases}
 
 
 %% Supervisor (optional)
-%\supervisor{Prof. Kenichi Soga}
+\supervisor{Dr. Igor B. Mekhov}
 %% Supervisor Role (optional) - Supervisor (default) or advisor
 %\supervisorrole{Advisor: }
 

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,draft]{Classes/PhDThesisPSnPDF}
+\documentclass[a4paper,12pt,times,numbered,print]{Classes/PhDThesisPSnPDF}
 
 % ******************************************************************************
 % ******************************* Class Options ********************************