Final version 1.1

Abstract/abstract.tex

@@ -11,9 +11,9 @@
   what is possible in ultracold atomic systems.
 
   We show that light can serve as a nondestructive probe of the
-  quantum state of the matter. By condering a global
-  measurement scheme we show that it is possible to distinguish a
-  highly delocalised phase like a superfluid from insulators. We also
+  quantum state of matter. By condering a global measurement we show
+  that it is possible to distinguish a highly delocalised phase like a
+  superfluid from the Bose glass and Mott insulator. We also
   demonstrate that light scattering reveals not only density
   correlations, but also matter-field interference.
 

Acknowledgement/acknowledgement.tex

@@ -6,19 +6,19 @@ 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.
+doctoral studies. The work contained in this thesis would also not
+have been 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.
 
 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

Chapter1/chapter1.tex

@@ -83,8 +83,8 @@ 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, mekhov2007NP, LP2009,
-  mekhov2012}. This allows us to probe the many-body system in a
+\cite{mekhov2012, mekhov2007prl, mekhov2007pra, mekhov2007NP,
+  LP2009}. This allows us to probe the many-body system in a
 nondestructive manner and 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
@@ -145,9 +145,8 @@ the first condenste was obtained that theoretical work on the effects
 of measurement on BECs appeared \cite{cirac1996, castin1997,
   ruostekoski1997}. Recently, work has also begun on combining weak
 measurement with the strongly correlated dynamics of ultracold gases
-in optical lattices \cite{mekhov2009prl, mekhov2009pra, LP2010,
-  mekhov2012, douglas2012, LP2013, douglas2013, ashida2015,
-  ashida2015a}.
+in optical lattices \cite{mekhov2012, mekhov2009prl, mekhov2009pra,
+  LP2010, douglas2012, LP2013, douglas2013, ashida2015, ashida2015a}.
 
 In this thesis we focus on the latter by considering a quantum gas in
 an optical lattice coupled to a cavity \cite{mekhov2012}. This
@@ -244,8 +243,8 @@ The work contained in this thesis is based on seven publications
 
     \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. \\ \\
+    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
@@ -265,9 +264,9 @@ The work contained in this thesis is based on seven publications
     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. \\
+    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}

Chapter2/chapter2.tex

@@ -338,7 +338,7 @@ 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 backaction
 such as the generation of multipartite entangled spatial modes in an
-optical lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the
+optical lattice \cite{mekhov2009pra, elliott2015, atoms2015}, 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,
@@ -767,8 +767,8 @@ our global scattering scheme.
 In our model light couples to the operator $\hat{F}$ which consists of
 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 $\hat{F} \approx \hat{D}$
+b_j$. In general, the density component dominates, $\hat{D} \gg
+\hat{B}$, and thus $\hat{F} \approx \hat{D}$
 \cite{mekhov2012}. Physically, this is a consequence of the fact that
 there are more atoms to scatter light at the lattice sites than in
 between them. However, it is possible to engineer an optical geometry
@@ -780,10 +780,10 @@ light scattered from different sources. Furthermore, it is not limited
 to a double-well setup and naturally extends to a lattice structure
 which is a key advantage. 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}.
+\cite{mekhov2012, caballero2015, mekhov2008, larson2008, chen2009,
+  habibian2013, ivanov2014} and quantum measurement-induced
+preparation of many-body atomic states \cite{mekhov2009prl,
+  elliott2015, mazzucchi2016, pedersen2014}.
 
 For clarity we will consider a 1D lattice as shown in
 Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the
@@ -1187,7 +1187,8 @@ lifetime of the experiment. For free space scattering appropriate
 conditions 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 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.
+coupled to a leaky cavity in Ref. \cite{brennecke2013} where
+self-organisation effects were well observable. 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

@@ -39,7 +39,7 @@ the quantum state of the ultracold gas we can have access to not only
 density correlations, but also matter-field interference at its
 shortest possible distance in an optical lattice, i.e.~the lattice
 period. Previous work on quantum non-demolition (QND) schemes
-\cite{rogers2014, mekhov2007prl, eckert2008} probe only the density
+\cite{mekhov2007prl, rogers2014, eckert2008} probe only the density
 component as it is generally challenging to couple to the matter-field
 observables directly. Here, we will consider nondestructive probing of
 both density and interference operators.
@@ -121,21 +121,21 @@ density correlations to matter-field interference.
 \subsection{Diffraction Patterns and Bragg Conditions}
 
 We have seen in section \ref{sec:B} that typically the dominant term
-in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009,
-  mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is
-simply due to the fact that atoms are localised with lattice sites
-leading to an effective coupling with atom number operators instead of
-inter-site interference terms. Therefore, we will first consider
-nondestructive probing of the density related observables of the
-quantum gas. However, we will focus on the novel nontrivial aspects
-that go beyond the work in Ref. \cite{mekhov2012, mekhov2007prl,
+in $\hat{F}$ is the density term $\hat{D}$ \cite{mekhov2007pra,
+  LP2009, rist2010, lakomy2009, ruostekoski2009}. This is simply due
+to the fact that atoms are localised with lattice sites leading to an
+effective coupling with atom number operators instead of inter-site
+interference terms. Therefore, we will first consider nondestructive
+probing of the density related observables of the quantum
+gas. However, we will focus on the novel nontrivial aspects that go
+beyond the work in Ref. \cite{mekhov2012, mekhov2007prl,
   mekhov2007pra} which only considered a few extremal cases.
 
 As we are only interested in the quantum information imprinted in the
 state of the optical field we will simplify our analysis by
 considering the light scattering to be much faster than the atomic
 tunnelling. Therefore, our scheme is actually a QND scheme
-\cite{rogers2014, mekhov2007prl, mekhov2007pra, eckert2008} as
+\cite{mekhov2007prl, mekhov2007pra, rogers2014, eckert2008} as
 normally density-related measurements destroy the matter-phase
 coherence since it is its conjugate variable, but here we neglect the
 $\bd_i b_j$ terms. Furthermore, we will consider a deep
@@ -837,7 +837,7 @@ probing a quantum gas trapped in an optical lattice using quantised
 light. 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 two
-relevant experiments \cite{weitenberg2011, miyake2011}. Secondly, we
+relevant experiments \cite{miyake2011, weitenberg2011}. Secondly, 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}. Finally, we looked

Chapter5/chapter5.tex

@@ -44,7 +44,7 @@ correlations which enable nonlocal dynamical processes. Furthermore,
 global light scattering from multiple lattice sites creates nontrivial
 spatially nonlocal coupling to the environment, as seen in section
 \ref{sec:modes}, which is impossible to obtain with local interactions
-\cite{daley2014, diehl2008, syassen2008}. These spatial modes of
+\cite{diehl2008, syassen2008, daley2014}. These spatial modes of
 matter fields can be considered as designed systems and reservoirs
 opening the possibility of controlling dissipations in ultracold
 atomic systems without resorting to atom losses and collisions which
@@ -748,7 +748,7 @@ from the effects of interactions as they would prevent this dynamics
 by dephasing different components of the coherent excitations. Strong
 measurement, on the other hand, squeezes the quantum state by trying
 to project it onto an eigenstate of the observable
-\cite{mekhov2009prl, mekhov2009prl}. For weak interactions where the
+\cite{mekhov2009prl, mekhov2009pra}. For weak interactions where the
 ground state is a highly delocalised superfluid it is obvious that
 projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will supress
 fluctuations significantly. However, the strongly interacting regime
@@ -1744,7 +1744,7 @@ 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, mazzucchi2016feedback,
-  ivanonv2016}. Furthermore, the form of the measurement operator is
+  ivanov2016}. Furthermore, the form of the measurement operator is
 very flexible and it can easily be engineered by the geometry of the
 optical setup \cite{elliott2015, mazzucchi2016} which can be used to
 design a state with desired properties.

Chapter6/chapter6.tex

@@ -23,16 +23,16 @@ 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 like 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,
+operators just like most of the existing work \cite{mekhov2012,
+  LP2009, rogers2014, 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
@@ -137,25 +137,24 @@ is given by
   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}.
+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{mekhov2012, mekhov2009pra, atoms2015}.
 
 Unusually, we do not have to worry about the timing of the quantum
 jumps, because the measurement operator commutes with the