Fixed post-viva typos

Abstract/abstract.tex

@@ -7,11 +7,11 @@
   controllable and tunable than what was possible in condensed
   matter. Here, we consider coupling these systems to quantised light
   where the quantum nature of both the optical and matter fields play
-  equally important roles in order to push the boundaries of
-  what is possible in ultracold atomic systems.
+  equally important roles in order to push the boundaries of what is
+  possible in ultracold atomic systems.
 
   We show that light can serve as a nondestructive probe of the
-  quantum state of matter. By condering a global measurement we show
+  quantum state of matter. By considering 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
@@ -21,11 +21,10 @@
   that the measurement can efficiently compete with the local atomic
   dynamics of the quantum gas. This can generate long-range
   correlations and entanglement which in turn leads to macroscopic
-  multimode oscillations accross the whole lattice when the
-  measurement is weak and correlated tunneling, as well as selective
-  suppression and enhancement of dynamical processes beyond the
-  projective limit of the quantum Zeno effect in the strong
-  measurement regime.
+  multimode oscillations across the whole lattice when the measurement
+  is weak and correlated tunnelling, as well as selective suppression
+  and enhancement of dynamical processes beyond the projective limit
+  of the quantum Zeno effect in the strong measurement regime.
 
   We also consider quantum measurement backaction due to the
   measurement of matter-phase-related variables such as global phase

Acknowledgement/acknowledgement.tex

@@ -21,7 +21,7 @@ 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
+with all the skills necessary to 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

Chapter1/chapter1.tex

@@ -50,7 +50,7 @@ free space which are described by weakly interacting theories
 \cite{dalfovo1999}, the behaviour of ultracold gases trapped in an
 optical lattice is dominated by atomic interactions opening the
 possibility of studying strongly correlated behaviour with
-unprecendented control.
+unprecedented control.
 
 The modern field of strongly correlated ultracold gases is successful
 due to its interdisciplinarity \cite{lewenstein2007,
@@ -60,11 +60,11 @@ quantum information processing. A really new challenge is to identify
 novel phenomena which were unreasonable to consider in condensed
 matter, but will become feasible in new systems. One such direction is
 merging quantum optics and many-body physics \cite{mekhov2012,
-  ritsch2013}. Quantum optics has been developping as a branch of
+  ritsch2013}. Quantum optics has been developing as a branch of
 quantum physics independently of the progress in the many-body
 community. It describes delicate effects such as quantum measurement,
 state engineering, and systems that can generally be easily isolated
-from their environnment due to the non-interacting nature of photons
+from their environment due to the non-interacting nature of photons
 \cite{Scully}. However, they are also the perfect candidate for
 studying open systems due the advanced state of cavity technologies
 \cite{carmichael, MeasurementControl}. On the other hand ultracold
@@ -115,7 +115,7 @@ these measurements \cite{miyake2011}. Within a mean-field treatment
 this enables probing of the order parameter as well as matter-field
 quadratures and their squeezing. This can have an impact on atom-wave
 metrology and information processing in areas where quantum optics has
-already made progress, e.g.,~quantum imaging with pixellized sources
+already made progress, e.g.,~quantum imaging with pixellised sources
 of non-classical light \cite{golubev2010, kolobov1999}, as an optical
 lattice is a natural source of multimode nonclassical matter waves.
 
@@ -123,7 +123,7 @@ Second, coupling a quantum gas to a cavity also enables us to study
 open system many-body dynamics either via dissipation where we have no
 control over the coupling to the environment or via controlled state
 reduction using the measurement backaction due to
-photodetections. Initially, a lot of effort was exended in an attempt
+photodetections. Initially, a lot of effort was expanded in an attempt
 to minimise the influence of the environment in order to extend
 decoherence times. However, theoretical progress in the field has
 shown that instead being an obstacle, dissipation can actually be used
@@ -140,9 +140,9 @@ stochastic Schr\"{o}dinger equation which will be a pure state, which
 in contrast to dissipative systems where this is generally not the
 case. The question of measurement and its effect on the quantum state
 has been around since the inception of quantum theory and still
-remains a largely open question \cite{zurek2002}. It wasn't long after
-the first condenste was obtained that theoretical work on the effects
-of measurement on BECs appeared \cite{cirac1996, castin1997,
+remains a largely open question \cite{zurek2002}. It was not long
+after the first condensate 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{mekhov2012, mekhov2009prl, mekhov2009pra,
@@ -160,7 +160,7 @@ the optical fields leads to new phenomena driven by long-range
 correlations that arise from the measurement. The flexibility of the
 optical setup lets us not only consider coupling to different
 observables, but by carefully choosing the optical geometry we can
-suppress or enhace specific dynamical processes, realising spatially
+suppress or enhance specific dynamical processes, realising spatially
 nonlocal quantum Zeno dynamics.
 
 The quantum Zeno effect happens when frequent measurements slow the
@@ -186,8 +186,8 @@ extending the notion of quantum Zeno dynamics into the realm of
 non-Hermitian quantum mechanics joining the two
 paradigms. Non-Hermitian systems themselves exhibit a range of
 interesting phenomena ranging from localisation \cite{hatano1996,
-  refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$
-  symmetry} \cite{bender1998, giorgi2010, zhang2013} to spatial order
+  refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$ }
+  symmetry \cite{bender1998, giorgi2010, zhang2013} to spatial order
 \cite{otterbach2014} and novel phase transitions \cite{lee2014prx,
   lee2014prl}.
 
@@ -212,7 +212,7 @@ measurement.
 Finally, the cavity field that builds up from the scattered photons
 can also create a quantum optical potential which will modify the
 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
+scattered 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, elliott2016}. However, this aspect of quantum optics

Chapter2/chapter2.tex

@@ -21,7 +21,7 @@ such as nondestructive measurement in free space or quantum
 measurement backaction in a cavity. As this model extends the
 Bose-Hubbard Hamiltonian to include the effects of interactions with
 quantised light we also present a brief overview of the properties of
-the Bose-Hubbard model itself and its quantum phase transiton. This is
+the Bose-Hubbard model itself and its quantum phase transition. This is
 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
@@ -36,8 +36,8 @@ which yields its own set of interesting quantum phenomena
 particles. The theory can be also be generalised to continuous
 systems, but the restriction to optical lattices is convenient for a
 variety of reasons. Firstly, it allows us to precisely describe a
-many-body atomic state over a broad range of parameter values due,
-from free particles to strongly correlated systems, to the inherent
+many-body atomic state over a broad range of parameter values, from
+free particles to strongly correlated systems, due 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
@@ -108,11 +108,11 @@ The term $\H_f$ represents the optical part of the Hamiltonian,
   \H_f = \sum_l \hbar \omega_l \ad_l \a_l -
   i \hbar \sum_l \left( \eta_l^* \a_l - \eta_l \ad \right).
 \end{equation}
-The operators $\a_l$ ($\ad$) are the annihilation (creation) operators
-of light modes with frequencies $\omega_l$, wave vectors $\b{k}_l$,
-and mode functions $u_l(\b{r})$, which can be pumped by coherent
-fields with amplitudes $\eta_l$. The second part of the Hamiltonian,
-$\H_a$, is the matter-field component given by
+The operators $\a_l$ ($\ad_l$) are the annihilation (creation)
+operators of light modes with frequencies $\omega_l$, wave vectors
+$\b{k}_l$, and mode functions $u_l(\b{r})$, which can be pumped by
+coherent fields with amplitudes $\eta_l$. The second part of the
+Hamiltonian, $\H_a$, is the matter-field component given by
 \begin{equation}
 \label{eq:Ha}
   \H_a = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,a}
@@ -280,7 +280,7 @@ cases we can simply neglect all terms but the one that corresponds to
 the most significant overlap. Thus, for $J_{i,j}^\mathrm{cl}$ we will
 only consider $i$ and $j$ that correspond to nearest neighbours.
 Furthermore, since we will only be looking at lattices that have the
-same separtion between all its nearest neighbours (e.g. cubic or
+same separation between all its nearest neighbours (e.g. cubic or
 square lattice) we can define $J_{i,j}^\mathrm{cl} = - J$ (negative
 sign, because this way $J > 0$). For the inter-atomic interactions
 this simplifies to simply considering on-site collisions where
@@ -350,7 +350,7 @@ It is important to note that we are considering a situation where the
 contribution of quantised light is much weaker than that of the
 classical trapping potential. If that was not the case, it would be
 necessary to determine the Wannier functions in a self-consistent way
-which takes into account the depth of the quantum poterntial generated
+which takes into account the depth of the quantum potential generated
 by the quantised light modes. This significantly complicates the
 treatment, but can lead to interesting physics. Amongst other things,
 the atomic tunnelling and interaction coefficients will now depend on
@@ -365,7 +365,7 @@ our quantum light-matter Hamiltonian
   \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} -
   i \sum_l \kappa_l \ad_l \a_l,
 \end{equation}
-where we have phenomologically included the cavity decay rates
+where we have phenomenologically included the cavity decay rates
 $\kappa_l$ of the modes $a_l$. A crucial observation about the
 structure of this Hamiltonian is that in the interaction term, the
 light modes $a_l$ couple to the matter in a global way. Instead of
@@ -391,7 +391,7 @@ 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
+successful \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 and applied to liquid helium
 \cite{fisher1989}, but in 1998 it was shown by Jaksch \emph{et.~al.}
@@ -433,7 +433,7 @@ diagram.
 
 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
+simplicity we will consider only homogeneous 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
@@ -488,7 +488,7 @@ 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
+thermodynamic limit. In fact, the existence 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
@@ -514,7 +514,7 @@ energy term is decoupled as
 \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
+and homogeneous. $\Phi$ 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
@@ -540,15 +540,15 @@ to $\Phi$.
     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
+    filling labelled 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
+    lines of constant density in the superfluid
     phase. \label{fig:BHPhase}}
 \end{figure}
 
 The main advantage of the mean-field treatment is that it lets us
-study the quantum phase transition between the sueprfluid and Mott
+study the quantum phase transition between the superfluid and Mott
 insulator phases discussed in the previous sections. The phase
 boundaries can be obtained from second-order perturbation theory as 
 \begin{equation}
@@ -557,11 +557,11 @@ boundaries can be obtained from second-order perturbation theory as
         \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
+seen in Fig. \ref{fig:BHPhase} where each lobe represents a Mott
+insulator 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}
@@ -633,7 +633,7 @@ 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 $\mu$.
+reentrant phase transition for a fixed $\mu$.
 
 \begin{figure}
   \centering
@@ -643,7 +643,7 @@ reentrance phase transition for a fixed $\mu$.
     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
+    reentrant 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}
@@ -677,7 +677,7 @@ adiabatically follows the quantum state of matter.
 The above equation is quite general as it includes an arbitrary number
 of light modes which can be pumped directly into the cavity or
 produced via scattering from other modes. To simplify the equation
-slightly we will neglect the cavity resonancy shift,
+slightly we will neglect the cavity resonance shift,
 $U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
 rate and/or probe detuning are large enough. We will also only
 consider probing with an external coherent beam, $a_0$ with mode
@@ -745,7 +745,7 @@ atoms which for off-resonant light, like the type that we consider,
 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. Most of the exisiting work on measurement couples
+optical fields. Most of the existing work on measurement couples
 directly to atomic density operators \cite{mekhov2012, LP2009,
   rogers2014, ashida2015, ashida2015a}. However, it has been shown
 that one can couple to the interference term between two condensates
@@ -816,7 +816,7 @@ sites rather than at the positions of the atoms.
     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 $u_1$
+    $J_{i,i+1}=(-1)^i J_2$; 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}
 
@@ -996,7 +996,7 @@ probe. Therefore, we drop the sum and consider only a single wave
 vector, $\b{k}_0$, of the dominant contribution from the external
 beam corresponding to the mode $\a_0$.
 
-We now use the definititon of the atom-light coupling constant
+We now use the definition of the atom-light coupling constant
 \cite{Scully} to make the substitution
 $g_0 a_0 = -i \Omega_0 e^{-i \omega_0 t} / 2$, where
 $\Omega_0$ is the Rabi frequency for the probe-atom system. We now
@@ -1047,7 +1047,7 @@ $\b{r}_i$. Thus, the relevant many-body operator is
 \end{subequations}
 where $K$ is the number of illuminated sites. We now assume that the
 lattice is deep and that the light scattering occurs on time scales
-much shorter than atom tunneling. Therefore, we ignore all terms for
+much shorter than atom tunnelling. Therefore, we ignore all terms for
 which $i \ne j$ and the remaining integrals become $\int \mathrm{d}^3
 \b{r}_0 w^*(\b{r}_0 - \b{r}_i) f (\b{r})
 w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of
@@ -1131,10 +1131,10 @@ in free space. Miyake \emph{et al.} experimental parameters are from
 Ref. \cite{miyake2011}. The $5^2S_{1/2}$,
 $F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb
 is considered. For this transition the Rabi frequency is actually
-larger than the detuning and and effects of saturation should be taken
+larger than the detuning and effects of saturation should be taken
 into account in a more complete analysis. However, it is included for
 discussion. The detuning was specified as ``a few linewidths'' and
-note that it is much smaller than $\Omega_0$. The Rabi fequency is
+note that it is much smaller than $\Omega_0$. The Rabi frequency is
 $\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is
 obtained from definition of Rabi frequency,
 $\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$, assuming the electric
@@ -1163,7 +1163,7 @@ per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
 
 There are many possible experimental issues that we have neglected so
 far in our theoretical treatment which need to be answered in order to
-consider the experimental feasability of our proposal. The two main
+consider the experimental feasibility of our proposal. The two main
 concerns are photodetector efficiency and heating losses.
 
 In our theoretical models we treat the detectors as if they were

Chapter3/chapter3.tex

@@ -76,12 +76,12 @@ possible distance in an optical lattice, i.e. the lattice period,
 which defines key processes such as tunnelling, currents, phase
 gradients, etc. This is in contrast to standard destructive
 time-of-flight measurements which deal with far-field interference
-although a relatively near-field scheme was use in
+although a relatively near-field scheme was used in
 Ref. \cite{miyake2011}. We show how within the mean-field treatment,
 this enables measurements of the order parameter, matter-field
 quadratures and squeezing. This can have an impact on atom-wave
 metrology and information processing in areas where quantum optics
-already made progress, e.g., quantum imaging with pixellized sources
+already made progress, e.g., quantum imaging with pixellised sources
 of non-classical light \cite{golubev2010, kolobov1999}, as an optical
 lattice is a natural source of multimode nonclassical matter waves.
 
@@ -112,7 +112,7 @@ stricter requirements for a QND measurement \cite{mekhov2012,
   mekhov2007pra}) and the measurement only weakly perturbs the system,
 many consecutive measurements can be carried out with the same atoms
 without preparing a new sample. We will show how the extreme
-flexibility of the the measurement operator $\hat{F}$ allows us to
+flexibility of the measurement operator $\hat{F}$ allows us to
 probe a variety of different atomic properties in-situ ranging from
 density correlations to matter-field interference.
 
@@ -160,7 +160,7 @@ We will now define a new auxiliary quantity to aid our analysis,
 which we will call the ``quantum addition'' to light scattering. By
 construction $R$ is simply the full light intensity minus the
 classical field diffraction. In order to justify its name we will show
-that this quantity depends purely quantum mechanical properties of the
+that this quantity depends on purely quantum mechanical properties of the
 ultracold gas. We substitute $\a_1 = C \hat{D}$ using
 Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
 we make use of the shorthand notation
@@ -224,7 +224,7 @@ $R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
 )/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
 $R = |C|^2 N_K/2$, because it shows scattering from an ideal
 superfluid which has significant density fluctuations with
-correlations of infinte range. However, as the parameters of the
+correlations of infinite range. However, as the parameters of the
 lattice are tuned across the phase transition into a Mott insulator
 the signal goes to zero. This is because the Mott insulating phase has
 well localised atoms at each site which suppresses density
@@ -252,7 +252,7 @@ angles than predicted by the classical Bragg condition. Moreover, the
 classical Bragg condition is actually not satisfied which means there
 actually is no classical diffraction on top of the ``quantum
 addition'' shown here. Therefore, these features would be easy to see
-in an experiment as they wouldn't be masked by a stronger classical
+in an experiment as they would not be masked by a stronger classical
 signal.  This difference in behaviour is due to the fact that
 classical diffraction is ignorant of any quantum correlations as it is
 given by the square of the light field amplitude squared
@@ -260,7 +260,7 @@ given by the square of the light field amplitude squared
   |\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^*
   A_j \langle \n_i \rangle \langle \n_j \rangle,
 \end{equation}
-which is idependent of any two-point correlations unlike $R$. On the
+which is independent of any two-point correlations unlike $R$. On the
 other hand the full light intensity (classical signal plus ``quantum
 addition'') of the quantum light does include higher-order
 correlations
@@ -294,8 +294,8 @@ can also be negative.
 
 We will consider scattering from a superfluid, because the Mott
 insulator has no ``quantum addition'' due to a lack of density
-fluctuations. The wavefunction of a superfluid on a lattice is given
-by Eq. \eqref{eq:GSSF}. This state has infinte range correlations and
+fluctuations. The wave function of a superfluid on a lattice is given
+by Eq. \eqref{eq:GSSF}. This state has infinite range correlations and
 thus has the convenient property that all two-point density
 fluctuation correlations are equal regardless of their separation,
 i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
@@ -349,7 +349,7 @@ wave scattered mode is
 Therefore, it is straightforward to see that unless
 $2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector,
 there will be no coherent signal and we end up with the mean uniform
-signal of strength $|C|^2 N_k/2$. When this condition is satisifed all
+signal of strength $|C|^2 N_k/2$. When this condition is satisfied all
 the cosine terms will be equal and they will add up constructively
 instead of cancelling each other out. Note that this new Bragg
 condition is different from the classical one
@@ -371,7 +371,7 @@ the condition we had for $R$ and is still different from the classical
 condition $\b{k}_0 - \b{k}_1 = \b{G}$. Furthermore, just like in
 Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$.
 
-A quantum signal that isn't masked by classical diffraction is very
+A quantum signal that is not masked by classical diffraction is very
 useful for future experimental realisability. However, it is still
 unclear whether this signal would be strong enough to be
 visible. After all, a classical signal scales as $N_K^2$ whereas here
@@ -423,7 +423,7 @@ quantum phase transition is in a different universality class than its
 higher dimensional counterparts. The energy gap, which is the order
 parameter, decays exponentially slowly across the phase transition
 making it difficult to identify the phase transition even in numerical
-simulations. Here, we will show the avaialable tools provided by the
+simulations. Here, we will show the available tools provided by the
 ``quantum addition'' that allows one to nondestructively map this
 phase transition and distinguish the superfluid and Mott insulator
 phases.
@@ -463,7 +463,7 @@ difficult to directly access the excitation energy gap which defines
 this phase transition. However, a valid description of the relevant 1D
 low energy physics is provided by Luttinger liquid theory
 \cite{giamarchi} as seen in section \ref{sec:BHM1D}. In this model
-correlations in the supefluid phase as well as the superfluid density
+correlations in the superfluid phase as well as the superfluid density
 itself are characterised by the Tomonaga-Luttinger parameter,
 $K$. This parameter also identifies the critical point in the
 thermodynamic limit at $K_c = 1/2$. This quantity can be extracted
@@ -489,7 +489,7 @@ Bose-Hubbard model as it can be easily applied to many other systems
 such as fermions, photonic circuits, optical lattices with quantum
 potentials, etc. Therefore, by providing a better physical picture of
 what information is carried by the ``quantum addition'' it should be
-easier to see its usefuleness in a broader context.
+easier to see its usefulness in a broader context.
 
 We calculate the phase diagram of the Bose-Hubbard Hamiltonian given
 by
@@ -595,7 +595,7 @@ in the Mott insulating phase and they change drastically across the
 phase transition into the superfluid phase. Next, we present a case
 where it is very different. We will again consider ultracold bosons in
 an optical lattice, but this time we introduce some disorder. We do
-this by adding an additional periodic potential on top of the exisitng
+this by adding an additional periodic potential on top of the existing
 setup that is incommensurate with the original lattice. The resulting
 Hamiltonian can be shown to be
 \begin{equation}
@@ -638,7 +638,7 @@ results in a large $R_\text{max}$ and a small $W_R$.
 \begin{figure}  
   \centering
   \includegraphics[width=\linewidth]{oph22_3}
-  \caption[Mapping the Disoredered Phase Diagram]{The
+  \caption[Mapping the Disordered Phase Diagram]{The
     Mott-superfluid-glass phase diagrams for light scattering maximum
     $R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both
     quantities distinguish all three phases. Transition lines are
@@ -705,7 +705,7 @@ allows one to probe the mean-field order parameter, $\Phi$,
 directly. Normally, this is achieved by releasing the trapped gas and
 performing a time-of-flight measurement. Here, this can be achieved
 in-situ. In section \ref{sec:B} we showed that one of the possible
-optical arrangement leads to a diffraction maximum with the matter
+optical arrangements leads to a diffraction maximum with the matter
 operator
 \begin{equation}
   \label{eq:Bmax}
@@ -733,7 +733,7 @@ measurements provide us with new opportunities to study the quantum
 matter state which was previously unavailable. We will take $\Phi$ to
 be real which in the standard Bose-Hubbard Hamiltonian can be selected
 via an inherent gauge degree of freedom in the order parameter. Thus,
-the quadratures themself straightforwardly become
+the quadratures themselves straightforwardly become
 \begin{equation}
   \hat{X}^b_\alpha = \frac{\Phi}{2} (e^{-i\alpha} + e^{i\alpha}) =
   \Phi \cos(\alpha).
@@ -746,17 +746,16 @@ the quadrature is a more complicated quantity given by
   (\Delta X^b_{0,\pi/2})^2 = \frac{1}{4} + [(n - \Phi^2) \pm
   \frac{1}{2}(\langle b^2 \rangle - \Phi^2)],
 \end{equation} 
-where we have arbitrarily selected two orthogonal quadratures
-$\alpha =0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated
-from light quadrature measurements alone as we do not know the value
-of $\langle b^2 \rangle$. To obtain the value of
-$\langle b^2 \rangle$this quantity we need to consider a second-order
-light observable such as light intensity. However, in the diffraction
-maximum this signal will be dominated by a contribution proportional
-to $K^2 \Phi^4$ whereas the terms containing information on
-$\langle b^2 \rangle$ will only scale as $K$. Therefore, it would be
-difficult to extract the quantity that we need by measuring in the
-difraction maximum.
+where we have arbitrarily selected two orthogonal quadratures $\alpha
+=0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated from
+light quadrature measurements alone as we do not know the value of
+$\langle b^2 \rangle$. To obtain the value of this quantity we need to
+consider a second-order light observable such as light
+intensity. However, in the diffraction maximum this signal will be
+dominated by a contribution proportional to $K^2 \Phi^4$ whereas the
+terms containing information on $\langle b^2 \rangle$ will only scale
+as $K$. Therefore, it would be difficult to extract the quantity that
+we need by measuring in the diffraction maximum.
 
 \begin{figure}
   \centering

Chapter4/chapter4.tex

@@ -20,14 +20,14 @@
 This thesis is entirely concerned with the question of measuring a
 quantum many-body system using quantised light. However, so far we
 have only looked at expectation values in a nondestructive context
-where we neglect the effect of the quantum wavefunction collapse. We
+where we neglect the effect of the quantum wave function collapse. We
 have shown that light provides information about various statistical
 quantities of the quantum states of the atoms such as their
 correlation functions. In general, any quantum measurement affects the
-system even if it doesn't physically destroy it. In our model both
+system even if it does not physically destroy it. In our model both
 optical and matter fields are quantised and their interaction leads to
 entanglement between the two subsystems. When a photon is detected and
-the electromagnetic wavefunction of the optical field collapses, the
+the electromagnetic wave function of the optical field collapses, the
 matter state is also affected due to this entanglement resulting in
 quantum measurement backaction. Therefore, in order to determine these
 quantities multiple measurements have to be performed to establish a
@@ -75,8 +75,8 @@ realisation. In particular, we consider states conditioned upon
 measurement results such as the photodetection times. Such a
 trajectory is generally stochastic in nature as light scattering is
 not a deterministic process. Furthermore, they are in general
-discotinuous as each detection event brings about a drastic change in
-the quantum state due to the wavefunction collapse of the light field.
+discontinuous as each detection event brings about a drastic change in
+the quantum state due to the wave function collapse of the light field.
 
 Before we discuss specifics relevant to our model of quantised light
 interacting with a quantum gas we present a more general overview
@@ -105,10 +105,10 @@ there are much fewer atoms being illuminated than previously.
 
 Basic quantum mechanics tells us that such measurements will in
 general affect the quantum state in some way. Each event will cause a
-discontinuous quantum jump in the wavefunction of the system and it
+discontinuous quantum jump in the wave function of the system and it
 will have a jump operator, $\c$, associated with it. The effect of a
 detection event on the quantum state is simply the result of applying
-this jump operator to the wavefunction, $| \psi (t) \rangle$,
+this jump operator to the wave function, $| \psi (t) \rangle$,
 \begin{equation}
   \label{eq:jump}
   | \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle}
@@ -152,7 +152,7 @@ quantum jump occurs and $0$ otherwise \cite{MeasurementControl}. Note
 that this equation has a straightforward generalisation to multiple
 jump operators, but we do not consider this possibility here at all.
 
-All trajectories that we calculate in the follwing chapters are
+All trajectories that we calculate in the following chapters are
 described by the stochastic Schr\"{o}dinger equation in
 Eq. \eqref{eq:SSE}. The most straightforward way to solve it is to
 replace the differentials by small time-steps $\delta t$. Then we
@@ -165,7 +165,7 @@ applied, i.e.~$\mathrm{d}N(t) = 1$, if
 In practice, this is not the most efficient method for simulation
 \cite{MeasurementControl}. Instead we will use the following
 method. At an initial time $t = t_0$ a random number $R$ is
-generated. We then propagate the unnormalised wavefunction
+generated. We then propagate the unnormalised wave function
 $| \tilde{\psi} (t) \rangle$ using the non-Hermitian evolution given
 by
 \begin{equation}
@@ -174,9 +174,9 @@ by
 \end{equation}
 up to a time $T$ such that
 $\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$. This
-problem can be solved efficienlty using standard numerical
+problem can be solved efficiently using standard numerical
 techniques. At time $T$ a quantum jump is applied according to
-Eq. \eqref{eq:jump} which renormalises the wavefunction as well and
+Eq. \eqref{eq:jump} which renormalises the wave function as well and
 the process is repeated as long as desired. This formulation also has
 the advantage that it provides a more intuitive picture of what
 happens during a single trajectory. The quantum jumps are
@@ -194,7 +194,7 @@ processes is balanced and without any further external influence the
 distribution of outcomes over many trajectories will be entirely
 determined by the initial state even though each individual trajectory
 will be unique and conditioned on the exact detection times that
-occured during the given experimental run. However, individual
+occurred during the given experimental run. However, individual
 trajectories can have features that are not present after averaging
 and this is why we focus our attention on single experimental runs
 rather than average behaviour.
@@ -224,7 +224,7 @@ could scatter and thus include multiple jump operators reducing our
 ability to control the system.
 
 The model we derived in Eq. \eqref{eq:fullH} is in fact already in a
-form ready for quantum trajectory simulations. The phenomologically
+form ready for quantum trajectory simulations. The phenomenologically
 included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the
 non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$
 which is the jump operator we want for measurements of photons leaking
@@ -239,7 +239,7 @@ can be achieved when the cavity-probe detuning is smaller than the
 cavity decay rate, $\Delta_p \ll \kappa$
 \cite{caballero2015}. However, even though the cavity field has a
 negligible effect on the atoms, measurement backaction will not. This
-effect is of a different nature. It is due to the wavefunction
+effect is of a different nature. It is due to the wave function
 collapse due to the destruction of photons rather than an interaction
 between fields. Therefore, the final form of the Hamiltonian
 Eq. \eqref{eq:fullH} that we will be using in the following chapters
@@ -265,7 +265,7 @@ Importantly, we see that measurement introduces a new energy and time
 scale $\gamma$ which competes with the two other standard scales
 responsible for the unitary dynamics of the closed system, tunnelling,
 $J$, and on-site interaction, $U$.  If each atom scattered light
-inependently a different jump operator $\c_i$ would be required for
+independently a different jump operator $\c_i$ would be required for
 each site projecting the atomic system into a state where long-range
 coherence is degraded. This is a typical scenario for spontaneous
 emission \cite{pichler2010, sarkar2014}, or for local
@@ -343,7 +343,7 @@ is not accessible and thus must be represented as an average over all
 possible trajectories. However, there is a crucial difference between
 measurement and dissipation. When we perform a measurement we use the
 master equation to describe system evolution if we ignore the
-measuremt outcomes, but at any time we can look at the detection times
+measurement outcomes, but at any time we can look at the detection times
 and obtain a conditioned pure state for this current experimental
 run. On the other hand, for a dissipative system we simply have no
 such record of results and thus the density matrix predicted by the
@@ -398,7 +398,7 @@ know with certainty that the state must be either $| 0 \rangle$ or
 $| 1 \rangle$. We just don't know which one until we check the result
 of the measurement.
 
-We have assumed that it was a discrete wavefunction collapse that lead
+We have assumed that it was a discrete wave function collapse that lead
 to the state in Eq. \eqref{eq:rho1} in which case the conclusion we
 reached was obvious. However, the nature of the process that takes us
 from the initial state to the final state with classical uncertainty
@@ -410,8 +410,8 @@ two basis states. Therefore, if we obtained this density matrix as a
 result of applying the time evolution given by the master equation we
 would be able to identify the final states of individual trajectories
 even though we have no information about the individual trajectories
-themself. This is analogous to an approach in which decoherence due to
-coupling to the environment is used to model the wavefunction collapse
+themselves. This is analogous to an approach in which decoherence due to
+coupling to the environment is used to model the wave function collapse
 \cite{zurek2002}, but here we will be looking at projective effects
 due to weak measurement.
 
@@ -444,7 +444,7 @@ was irrelevant. Now, on the other hand, we are interested in the
 effect of these measurements on the dynamics of the system. The effect
 of measurement backaction will depend on the information that is
 encoded in the detected photon. If a scattered mode cannot
-distinuguish between two different lattice sites then we have no
+distinguish between two different lattice sites then we have no
 information about the distribution of atoms between those two sites.
 Therefore, all quantum correlations between the atoms in these sites
 are unaffected by the backaction whilst their correlations with the
@@ -494,7 +494,7 @@ that these partitions in general are not neighbours of each other,
 they are not localised, they overlap in a nontrivial way, and the
 patterns can be made more complex in higher dimensions as shown in
 Fig. \ref{fig:2dmodes} or with a more sophisticated optical setup
-\cite{caballero2016}. This has profound consquences as it can lead to
+\cite{caballero2016}. This has profound consequences as it can lead to
 the creation of long-range nonlocal correlations between lattice sites
 \cite{mazzucchi2016, kozlowski2016zeno}. The measurement does not know
 which atom within a certain mode scattered the light, there is no
@@ -538,7 +538,7 @@ Fig. \ref{fig:twomodes}.
   \includegraphics[width=0.7\textwidth]{TwoModes2}
   \caption[Two Mode Partitioning]{Top: only the odd sites scatter
     light leading to a measurement of $\hat{N}_\mathrm{odd}$ and an
-    effectove partitiong into even and odd sites. Bottom: this also
+    effective partitioning into even and odd sites. Bottom: this also
     partitions the lattice into odd and even sites, but this time
     atoms at all sites scatter light, but in anti-phase with their
     neighbours.}
@@ -546,7 +546,7 @@ Fig. \ref{fig:twomodes}.
 \end{figure}
 
 This approach can be generalised to an arbitrary number of modes,
-$Z$. For this we will conisder a deep lattice such that
+$Z$. For this we will consider a deep lattice such that
 $J_{i,i} = u_1^* (\b{r}) u_0 (\b{r})$. We will take the probe beam to
 be incident normally at a 1D lattice so that $u_0 (\b{r}) =
 1$. Therefore, the final form of the scattered light field is given by

Chapter5/chapter5.tex

@@ -32,7 +32,7 @@ which competes with the intrinsic dynamics of the bosons.
 In this chapter, we investigate the effect of quantum measurement
 backaction on the many-body state and dynamics of atoms. In
 particular, we will focus on the competition between the backaction
-and the the two standard short-range processes, tunnelling and on-site
+and the two standard short-range processes, tunnelling and on-site
 interactions, in optical lattices. We show that the possibility to
 spatially structure the measurement at a microscopic scale comparable
 to the lattice period without the need for single site resolution
@@ -46,7 +46,7 @@ spatially nonlocal coupling to the environment, as seen in section
 \ref{sec:modes}, which is impossible to obtain with local interactions
 \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
+opening the possibility of controlling dissipation in ultracold
 atomic systems without resorting to atom losses and collisions which
 are difficult to manipulate. Thus the continuous measurement of the
 light field introduces a controllable decoherence channel into the
@@ -115,7 +115,7 @@ atoms being entirely within a single mode. Finally, we note that these
 oscillating distributions are squeezed by the measurement and the
 individual components have a width smaller than the initial state. By
 contrast, in the absence of the external influence of measurement
-these distributions would spread out significantly and the center of
+these distributions would spread out significantly and the centre of
 the broad distribution would oscillate with an amplitude comparable to
 the initial imbalance, i.e.~small oscillations for a small initial
 imbalance.
@@ -128,7 +128,7 @@ imbalance.
     resulting from the competition between tunnelling and weak
     measurement induced backaction. The plots show the atom number
     distributions $p(N_l)$ in one of the modes in individual quantum
-    trajectories. These dstributions show various numbers of
+    trajectories. These distributions show various numbers of
     well-squeezed components reflecting the creation of macroscopic
     superposition states depending on the measurement
     configuration. $U/J = 0$, $\gamma/J = 0.01$, $M=N$, initial
@@ -218,7 +218,7 @@ lattice. $b_e$ ($\bd_e$) is defined similarly, but for the right site
 and the superfluid at even sites of the physical lattice.
 $\n_o = \bd_o b_o$ is the atom number operator in the left site.
 
-Even though Eq. \eqref{eq:doublewell} is relatively simple as it it is
+Even though Eq. \eqref{eq:doublewell} is relatively simple as it is
 only a non-interacting two-site model, the non-Hermitian term
 complicates the situation making the system difficult to
 solve. However, a semiclassical approach to boson dynamics in a
@@ -226,7 +226,7 @@ double-well in the limit of many atoms $N \gg 1$ has been developed in
 Ref. \cite{juliadiaz2012}. It was originally formulated to treat
 squeezing in a weakly interacting bosonic gas, but it can easily be
 applied to our system as well. In the limit of large atom number, the
-wavefunction in Eq. \eqref{eq:discretepsi} can be described using
+wave function in Eq. \eqref{eq:discretepsi} can be described using
 continuous variables by defining $\psi (x = l / N) = \sqrt{N}
 q_l$. Note that this requires the coefficients $q_l$ to vary smoothly
 which is the case for a superfluid state. We now rescale the
@@ -264,10 +264,10 @@ to obtain a valid analytic description of the oscillations and their
 growth.
 
 A superfluid state in our continuous variable approximation
-corresponds to a Gaussian wavefunction $\psi$. Furthermore, since the
+corresponds to a Gaussian wave function $\psi$. Furthermore, since the
 potential is parabolic, even with the inclusion of the non-Hermitian
 term, it will remain Gaussian during subsequent time
-evolution. Therefore, we will use a very general Gaussian wavefunction
+evolution. Therefore, we will use a very general Gaussian wave function
 of the form
 \begin{equation}
   \label{eq:ansatz}
@@ -277,7 +277,7 @@ of the form
 as our ansatz to Eq. \eqref{eq:semicl}. The parameters $b$, $\phi$,
 $z_0$, and $z_\phi$ are real-valued functions of time whereas
 $\epsilon$ is a complex-valued function of time. Physically, the value
-$b^2$ denotes the width, $z_0$ the position of the center, $\phi$ and
+$b^2$ denotes the width, $z_0$ the position of the centre, $\phi$ and
 $z_\phi$ contain the local phase information, and $\epsilon$ only
 affects the global phase and norm of the Gaussian wave packet.
 
@@ -285,14 +285,14 @@ The non-Hermitian Hamiltonian and an ansatz are not enough to describe
 the full dynamics due to measurement. We also need to know the effect
 of each quantum jump. Within the continuous variable approximation,
 our quantum jump become $\c \propto 1 + z$. We neglect the constant
-prefactors, because the wavefunction is normalised after a quantum
+prefactors, because the wave function is normalised after a quantum
 jump. Expanding around the peak of the Gaussian ansatz we get
 \begin{equation}
   1 + z \approx \exp \left[ \ln (1 + z_0) + \frac{z - z_0}{1 + z_0} -
     \frac{(z - z_0)^2}{2 (1 + z_0)^2} \right].
 \end{equation}
-Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump
-operator above yields a Gaussian wavefunction as well, but the
+Multiplying the wave function in Eq. \eqref{eq:ansatz} with the jump
+operator above yields a Gaussian wave function as well, but the
 parameters change discontinuously according to
 \begingroup
 \allowdisplaybreaks
@@ -306,16 +306,16 @@ parameters change discontinuously according to
   \epsilon & \rightarrow \epsilon.
 \end{align}
 \endgroup
-The fact that the wavefunction remains Gaussian after a photodetection
+The fact that the wave function 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
 Eq. \eqref{eq:ansatz} subject to non-Hermitian time evolution
-according to Eq. \eqref{eq:semicl} with discontinous changes to the
+according to Eq. \eqref{eq:semicl} with discontinuous changes to the
 parameter values at each quantum jump.
 
 Having identified an appropriate ansatz and the effect of quantum
-jumps we proceed with solving the dynamics of wavefunction in between
-the photodetecions. The initial values of the parameters for a
+jumps we proceed with solving the dynamics of wave function in between
+the photodetections. The initial values of the parameters for a
 superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$,
 $\phi =0$, $a_0 = 0$, $a_\phi = 0$, $\epsilon = 0$. However, we use
 the most general initial conditions at time $t = t_0$ which we denote
@@ -324,7 +324,7 @@ $z_\phi(t_0) = a_\phi$, and $\epsilon(t_0) = \epsilon_0$. The reason
 for keeping them as general as possible is that after every quantum
 jump the system changes discontinuously. The subsequent time evolution
 is obtained by solving the Schr\"{o}dinger equation with the post-jump
-paramater values as the new initial conditions.
+parameter values as the new initial conditions.
 
 By plugging the ansatz in Eq. \eqref{eq:ansatz} into the
 Schr\"{o}dinger equation in Eq. \eqref{eq:semicl} we obtain three
@@ -363,7 +363,7 @@ explicitly in the equations above.
 First, it is worth noting that all parameters of interest can be
 extracted from $p(t)$ and $q(t)$ alone. We are not interested in
 $\epsilon(t)$ as it is only related to the global phase and the norm
-of the wavefunction and it contains little physical
+of the wave function and it contains little physical
 information. Furthermore, an interesting and incredibly convenient
 feature of these equations is that the Eq. \eqref{eq:p} is a function
 of $p(t)$ alone and Eq. \eqref{eq:pq} is a function of $p(t)$ and
@@ -473,7 +473,7 @@ b^2(t) = \frac{b_0^2}{2} \left[ \left(1 + \frac{16 h^2 (1 + \phi_0^2)}
 First, these equations show that all quantities oscillate with a
 frequency $\omega$ or $2 \omega$. We are in particular interested in
 the quantity $z_0(t)$ as it represents the position of the peak of the
-wavefunction and we see that it oscillates with an amplitude
+wave function and we see that it oscillates with an amplitude
 $\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4
   \omega^2)}$. Thus we have obtained a solution that clearly shows
 oscillations of a single Gaussian wave packet. The fact that this
@@ -482,7 +482,7 @@ property of the Bose-Hubbard model itself. However, they also depend
 on the initial conditions and for these oscillations to occur, $a_0$
 and $a_\phi$ cannot be zero, but this is exactly the case for an
 initial superfluid state. We have seen in Eq. \eqref{eq:jumpz0} that
-the effect of a photodetection is to displace the wavepacket by
+the effect of a photodetection is to displace the wave packet by
 approximately $b^2$, i.e.~the width of the Gaussian, in the direction
 of the positive $z$-axis. Therefore, even though the atoms can
 oscillate in the absence of measurement it is the quantum jumps that
@@ -492,7 +492,7 @@ instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
 itself is proportional to $(1+z)^2$. This means they are most likely
 to occur at the point of maximum displacement in the positive $z$
 direction at which point a quantum jump provides positive feedback and
-further increases the amplitude of the wavefunction leading to the
+further increases the amplitude of the wave function leading to the
 growth seen in Fig. \ref{fig:oscillations}(a). The oscillations
 themselves are essentially due to the natural dynamics of coherently
 displaced atoms in a lattice , but it is the measurement that causes
@@ -517,7 +517,7 @@ visible on short time scales. Instead, we look at the long time
 limit. Unfortunately, since all the quantities are oscillatory a
 stationary long time limit does not exist especially since the quantum
 jumps provide a driving force. However, the width of the Gaussian,
-$b^2$, is unique in that it doesn't oscillate around $b^2 =
+$b^2$, is unique in that it does not oscillate around $b^2 =
 0$. Furthermore, from Eq. \eqref{eq:jumpb2} we see that even though it
 will decrease discontinuously at every jump, this effect is fairly
 small since $b^2 \ll 1$ generally. Therefore, we expect $b^2$ to
@@ -603,7 +603,7 @@ this quantity for $\hat{D} = \hat{N}_\mathrm{odd}$ averaged over
 multiple trajectories, $\langle \sigma^2_D \rangle_\mathrm{traj}$, as
 a function of $\gamma/J$ and $U/J$ for a lattice of six atoms on six
 sites (we cannot use the effective double-well model, because
-$U \ne 0$). We use a ground state of for the corresponding $U$ and $J$
+$U \ne 0$). We use a ground state for the corresponding $U$ and $J$
 values as this provides a realistic starting point and a reference for
 comparing the measurement induced dynamics. We will also consider only
 $\hat{D} = \hat{N}_\mathrm{odd}$ unless stated otherwise.
@@ -612,26 +612,25 @@ $\hat{D} = \hat{N}_\mathrm{odd}$ unless stated otherwise.
   \centering
   \includegraphics[width=\textwidth]{Squeezing}
   \caption[Squeezing in the presence of Interactions]{Atom number
-    fluctuations at odd sites for for $N = 6$ atoms at $M = 6$ sites
+    fluctuations at odd sites for $N = 6$ atoms at $M = 6$ sites
     subject to a $\hat{D} = \hat{N}_\mathrm{odd}$ measurement
     demonstrating the competition of global measurement with local
     interactions and tunnelling. Number variances are averaged over
-    100 trajectories. Error bars are too small to be shown
-    ($\sim 1\%$) which emphasizes the universal nature of the
-    squeezing. The initial state used was the ground state for the
-    corresponding $U$ and $J$ value. The fluctuations in the ground
-    state without measurement decrease as $U / J$ increases,
-    reflecting the transition between the supefluid and Mott insulator
-    phases. For weak measurement values
-    $\langle \sigma^2_D \rangle_\mathrm{traj}$ is squeezed below the
-    ground state value for $U = 0$, but it subsequently increases and
-    reaches its maximum as the atom repulsion prevents the
-    accumulation of atoms prohibiting coherent oscillations thus
-    making the squeezing less effective. In the strongly interacting
-    limit, the Mott insulator state is destroyed and the fluctuations
-    are larger than in the ground state as weak measurement isn't
-    strong enough to project into a state with smaller fluctuations
-    than the ground state.}
+    100 trajectories. Error bars are too small to be shown ($\sim
+    1\%$) which emphasizes the universal nature of the squeezing. The
+    initial state used was the ground state for the corresponding $U$
+    and $J$ value. The fluctuations in the ground state without
+    measurement decrease as $U / J$ increases, reflecting the
+    transition between the superfluid and Mott insulator phases. For
+    weak measurement values $\langle \sigma^2_D \rangle_\mathrm{traj}$
+    is squeezed below the ground state value for $U = 0$, but it
+    subsequently increases and reaches its maximum as the atom
+    repulsion prevents the accumulation of atoms prohibiting coherent
+    oscillations thus making the squeezing less effective. In the
+    strongly interacting limit, the Mott insulator state is destroyed
+    and the fluctuations are larger than in the ground state as weak
+    measurement is not strong enough to project into a state with
+    smaller fluctuations than the ground state.}
   \label{fig:squeezing}
 \end{figure}
 
@@ -654,7 +653,7 @@ be lost if we studied an ensemble average.
     dynamics of the atom-number distributions at odd sites
     illustrating competition of the global measurement with local
     interactions and tunnelling. The plots are for single quantum
-    trajectores starting from the ground state for $N = 6$ atoms on
+    trajectories starting from the ground state for $N = 6$ atoms on
     $M = 6$ sites with $\hat{D} = \hat{N}_\mathrm{odd}$,
     $\gamma/J = 0.1$. (a) Weakly interacting bosons $U/J = 1$: the
     on-site repulsion prevents the formation of well-defined
@@ -662,7 +661,7 @@ be lost if we studied an ensemble average.
     different imbalance evolve with different frequencies, the
     squeezing is not as efficient for the non-interacting case. (b)
     Strongly interacting bosons $U/J = 10$: oscillations are
-    completely supressed and the number of atoms in the mode is rather
+    completely suppressed and the number of atoms in the mode is rather
     well-defined although clearly worse than in a Mott insulator.}
   \label{fig:Utraj}
 \end{figure}
@@ -679,7 +678,7 @@ squeezed below those of the ground state followed by a rapid increase
 as $U$ is increased before peaking and eventually decreasing. We have
 already seen in the previous section and in particular
 Fig. \ref{fig:oscillations} that the macroscopic oscillations at
-$U = 0$ are well squeezed when compared to the inital state and this
+$U = 0$ are well squeezed when compared to the initial state and this
 is the case over here as well. However, as $U$ is increased the
 interactions prevent the atoms from accumulating in one place thus
 preventing oscillations with a large amplitude which effectively makes
@@ -724,7 +723,7 @@ $\c^m | \Psi_{J/U} \rangle \propto | \Psi_{J/U} \rangle + | \Phi_m
 \end{equation}
 In the weak measurement regime the effect of non-Hermitian decay is
 negligible compared to the local atomic dynamics combined with the
-quantum jumps so there is minimal dissipation occuring. Therefore,
+quantum jumps so there is minimal dissipation occurring. Therefore,
 because of the exponential growth of the excitations, even a small
 number of photons arriving in succession can destroy the ground
 state. We have neglected all dynamics in between the jumps which would
@@ -742,15 +741,15 @@ case, the squeezing will always be better than in the ground state,
 because measurement and on-site interaction cooperate in suppressing
 fluctuations. This cooperation did not exist for weak measurement,
 because it tried to induce dynamics which produced squeezed states
-(either succesfully as seen with the macroscopic oscillations or
-unsuccesfully as seen with the Mott insulator). This suffered heavily
+(either successfully as seen with the macroscopic oscillations or
+unsuccessfully as seen with the Mott insulator). This suffered heavily
 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, 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
+projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will suppress
 fluctuations significantly. However, the strongly interacting regime
 is much less evident, especially since we have just demonstrated how
 sensitive the Mott insulating phase is to the quantum jumps when the
@@ -805,13 +804,13 @@ the $\gamma/U \gg 1$ regime is reached.
 When $\gamma \rightarrow \infty$ the measurement becomes
 projective. This means that as soon as the probing begins, the system
 collapses into one of the observable's eigenstates. Furthermore, since
-this measurement is continuous and doesn't stop after the projection
+this measurement is continuous and does not stop after the projection
 the system will be frozen in this state. This effect is called the
 quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
 paradox in which a ``watched arrow never moves'' that stated that
 since an arrow in flight is not seen to move during any single
 instant, it cannot possibly be moving at all. Classically the paradox
-was resolved with a better understanding of infinity and infintesimal
+was resolved with a better understanding of infinity and infinitesimal
 changes, but in the quantum world a watched quantum arrow will in fact
 never move. The system is being continuously projected into its
 initial state before it has any chance to evolve. If degenerate
@@ -822,7 +821,7 @@ called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
 
 These effects can be easily seen in our model when
 $\gamma \rightarrow \infty$. The system will be projected into one or
-more degenerate eignstates of $\cd \c$, $| \psi_i \rangle$, for which
+more degenerate eigenstates of $\cd \c$, $| \psi_i \rangle$, for which
 we define the projector
 $P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
 denotes a single degenerate subspace. The Zeno subspace is determined
@@ -834,12 +833,12 @@ Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
 quantum jumps as their combined effect is now described by the
 projectors. Physically, in our model of ultracold bosons trapped in a
 lattice this means that tunnelling between different spatial modes is
-completely supressed since this process couples eigenstates belonging
+completely suppressed since this process couples eigenstates belonging
 to different Zeno subspaces. If a small connected part of the lattice
 was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
 tunnelling would only be prohibited between the illuminated and
 unilluminated areas, but dynamics proceeds normally within each zone
-separately. However, the goemetric patterns we have in which the modes
+separately. However, the geometric patterns we have in which the modes
 are spatially delocalised in such a way that neighbouring sites never
 belong to the same mode, e.g.  $\hat{D} = \hat{N}_\mathrm{odd}$, would
 lead to a complete suppression of tunnelling across the whole lattice
@@ -862,11 +861,11 @@ Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
 \begin{figure}[hbtp!]
   \includegraphics[width=\textwidth]{Zeno.pdf}
   \caption[Emergent Long-Range Correlated Tunnelling]{ Long-range
-    correlated tunneling and entanglement, dynamically induced by
+    correlated tunnelling and entanglement, dynamically induced by
     strong global measurement in a single quantum
     trajectory. (a),(b),(c) show different measurement geometries,
     implying different constraints. Panels (1): schematic
-    representation of the long-range tunneling processes. Panels (2):
+    representation of the long-range tunnelling processes. Panels (2):
     evolution of on-site densities. Panels (3): entanglement entropy
     growth between illuminated and non-illuminated regions. Panels
     (4): correlations between different modes (orange) and within the
@@ -879,9 +878,9 @@ Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
     correlations between the $N_I$ and $N_{NI}$ modes
     (orange). Initial state: $|1,1,1,1,1,1,1 \rangle$, $\gamma/J=100$,
     $J_{jj}=[0,0,1,1,1,0,0]$.  (b) (b.1) Even sites are illuminated,
-    freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunneling
+    freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunnelling
     is represented by any pair of one blue and one red arrow. (b.2)
-    Correlated tunneling occurs between non-neighbouring sites without
+    Correlated tunnelling occurs between non-neighbouring sites without
     changing mode populations. (b.3) Entanglement build up. (b.4)
     Negative correlations between edge sites (green) and zero
     correlations between the modes defined by $N_\text{even}$ and
@@ -949,19 +948,19 @@ neighbouring sites that belong to the same mode. Otherwise, if $i$ and
 $j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
 $\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
 where this would be the only remaining term. It is the second term
-that shows the second-order corelated tunnelling terms. This is
+that shows the second-order correlated tunnelling terms. This is
 evident from the inner sum which requires that pairs of sites ($i$,
 $j$) and ($k$, $l$) between which atoms tunnel must be nearest
 neighbours, but these pairs can be anywhere on the lattice within the
 constraints of the mode structure. This is in particular explicitly
 shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
-that the tinelling behaves like an exponential decay (overdamped
+that the tunnelling behaves like an exponential decay (overdamped
 oscillations). This also implies that the norm will decay, but this
 does not mean that there are physical losses in the system. Instead,
 the norm itself represents the probability of the system remaining in
 the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
 is now a finite probability that the stochastic nature of the
-measurement will lead to a discontinous change in the system where the
+measurement will lead to a discontinuous change in the system where the
 Zeno subspace rapidly changes which can be seen in
 Fig. \ref{fig:zeno}(a). However, later in this chapter we will see
 that steady states of this Hamiltonian exist which will no longer
@@ -973,7 +972,7 @@ selectively suppressed by the global conservation of the measured
 observable and not by the prohibitive energy costs of doubly-occupied
 sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
 profound consequences as this is the physical origin of the long-range
-correlated tunneling events represented in Eq. \eqref{eq:hz} by the
+correlated tunnelling events represented in Eq. \eqref{eq:hz} by the
 fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
 projection $P_0$ is not sensitive to individual site
 occupancies, but instead enforces a fixed value of the observable,
@@ -1002,7 +1001,7 @@ typical dynamics occurs within each region but the standard tunnelling
 between different modes is suppressed. Importantly, second-order
 processes that do not change $N_\text{K}$ are still possible since an
 atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
-from $2$ to $3$. Therefore, effective long-range tunneling between two
+from $2$ to $3$. Therefore, effective long-range tunnelling between two
 spatially disconnected zones $1$ and $3$ happens due to the two-step
 processes $1 \rightarrow 2 \rightarrow 3$ or
 $3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
@@ -1018,7 +1017,7 @@ correlated tunnelling process builds long-range entanglement between
 illuminated and non-illuminated regions as shown in
 Fig.~\ref{fig:zeno}(a.3).
 
-To make correlated tunneling visible even in the mean atom number, we
+To make correlated tunnelling visible even in the mean atom number, we
 suppress the standard Bose-Hubbard dynamics by illuminating only the
 even sites of the lattice in Fig.~\ref{fig:zeno}(b). Even though this
 measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$,
@@ -1026,10 +1025,10 @@ atoms can slowly tunnel between the odd sites of the lattice, despite
 them being spatially disconnected. This atom exchange spreads
 correlations between non-neighbouring lattice sites on a time scale
 $\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
-explanation of long-range correlated tunneling is presented in
+explanation of long-range correlated tunnelling is presented in
 Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
 the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
-such that one correlated tunneling event is represented by a pair of
+such that one correlated tunnelling event is represented by a pair of
 one red and one blue arrow. Importantly, this scheme is fully
 applicable for a lattice with large atom and site numbers, well beyond
 the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
@@ -1038,14 +1037,14 @@ assures this mode structure (in this example, two modes at odd and
 even sites) and thus the underlying pairwise global tunnelling.
 
 The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
-reservoir for the tunneling (or ``decay'') of atoms from one region to
+reservoir for the tunnelling (or ``decay'') of atoms from one region to
 another. For example, if the atoms are placed only at odd sites,
 according to Eq. \eqref{eq:hz} their tunnelling is suppressed since the
-multi-tunneling event must be successive, i.e.~an atom tunnelling into
+multi-tunnelling event must be successive, i.e.~an atom tunnelling into
 a different mode, $\varphi^\prime$, must then also tunnel back into
 its original mode, $\varphi$. If, however, one adds some atoms to even
 sites (even if they are far from the initial atoms), the correlated
-tunneling events become allowed and their rate can be tuned by the
+tunnelling events become allowed and their rate can be tuned by the
 number of added atoms. This resembles the repulsively bound pairs
 created by local interactions \cite{winkler2006, folling2007}. In
 contrast, here the atom pairs are nonlocally correlated due to the
@@ -1054,7 +1053,7 @@ consequence of the dynamics being constrained to a Zeno subspace: the
 virtual processes allowed by the measurement entangle the spatial
 modes nonlocally. Since the measurement only reveals the total number
 of atoms in the illuminated sites, but not their exact distribution,
-these multi-tunelling events cause the build-up of long-range
+these multi-tunnelling events cause the build-up of long-range
 entanglement. This is in striking contrast to the entanglement caused
 by local processes which can be very confined, especially in 1D where
 it is typically short range. This makes numerical calculations of our
@@ -1072,15 +1071,15 @@ generation of positive number correlations shown in
 Fig.~\ref{fig:zeno}(c.4) by freezing the atom number difference
 between the sites ($N_\text{odd}-N_\text{even}$). Thus, atoms can only
 enter or leave this region in pairs, which again is possible due to
-correlated tunneling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and
+correlated tunnelling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and
 manifests positive correlations. As in the previous example, two edge
 modes in Fig.~\ref{fig:zeno}(c) can be considered as a nonlocal
 reservoir for two central sites, where a constraint is applied. Note
-that, using more modes, the design of higher-order multi-tunneling
+that, using more modes, the design of higher-order multi-tunnelling
 events is possible.
 
-This global pair tunneling may play a role of a building block for
-more complicated many-body effects. For example, a pair tunneling
+This global pair tunnelling may play a role of a building block for
+more complicated many-body effects. For example, a pair tunnelling
 between the neighbouring sites has been recently shown to play
 important role in the formation of new quantum phases, e.g., pair
 superfluid \cite{sowinski2012} and lead to formulation of extended
@@ -1088,7 +1087,7 @@ Bose-Hubbard models \cite{omjyoti2015}. The search for novel
 mechanisms providing long-range interactions is crucial in many-body
 physics. One of the standard candidates is the dipole-dipole
 interaction in, e.g., dipolar molecules, where the mentioned pair
-tunneling between even neighboring sites is already considered to be
+tunnelling between even neighbouring sites is already considered to be
 long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
 suggests a fundamentally different mechanism originating from quantum
 optics: the backaction of global and spatially structured measurement,
@@ -1103,7 +1102,7 @@ the strong measurement limit in our quantum gas model. We showed that
 global measurement in the strong, but not projective, limit leads to
 correlated tunnelling events which can be highly delocalised. Multiple
 examples for different optical geometries and measurement operators
-demonstrated the incredible felixbility and potential in engineering
+demonstrated the incredible flexibility and potential in engineering
 dynamics for ultracold gases in an optical lattice. We also claimed
 that the behaviour of the system is described by the Hamiltonian given
 in Eq. \eqref{eq:hz}. Having developed a physical and intuitive
@@ -1157,7 +1156,7 @@ general are not single matrix elements. Therefore, we can write the
 master equation that describes this open system as a set of equations
 \begin{equation}
 \label{eq:master}
-  \dot{\hat{\rho}}_{mn} =  -i K P_m \left[ \h \sum_r \hat{\rho}_{rn}
+  \dot{\hat{\rho}}_{mn} =  -i \nu P_m \left[ \h \sum_r \hat{\rho}_{rn}
     - \sum_r \hat{\rho}_{mr} \h \right] P_n + \lambda^2 \left[ o_m
     o_n^*  - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right) \right] \hat{\rho}_{mn},
 \end{equation}
@@ -1204,7 +1203,7 @@ third, virtual, state that remains empty throughout the process. By
 avoiding the infinitely projective Zeno limit we open the option for
 such processes to happen in our system where transitions within a
 single Zeno subspace occur via a second, different, Zeno subspace even
-though the occupation of the intermediate states will remian
+though the occupation of the intermediate states will remain
 negligible at all times.
 
 A single quantum trajectory results in a pure state as opposed to the
@@ -1216,7 +1215,7 @@ what the pure states that make up these density matrices are. However,
 we note that for a single pure state the density matrix can consist of
 only a single diagonal submatrix $\hat{\rho}_{mm}$. To understand
 this, consider the state $| \Phi \rangle$ and take it to span exactly
-two distinct subspaces $P_a$ and $P_b$ ($a \ne b$). This wavefunction
+two distinct subspaces $P_a$ and $P_b$ ($a \ne b$). This wave function
 can thus be written as
 $| \Phi \rangle = P_a | \Phi \rangle + P_b | \Phi \rangle$. The
 corresponding density matrix is given by
@@ -1225,7 +1224,7 @@ corresponding density matrix is given by
   \rangle \langle \Phi | P_b + P_b | \Phi \rangle \langle \Phi | P_a +
   P_b | \Phi \rangle \langle \Phi | P_b.
 \end{equation}
-If the wavefunction has significant components in both subspaces then
+If the wave function has significant components in both subspaces then
 in general the density matrix will not have negligible coherences,
 $\hat{\rho}_{ab} = P_a | \Phi \rangle \langle \Phi | P_b$. A density
 matrix with just diagonal components must be in either subspace $a$,
@@ -1234,10 +1233,10 @@ $| \Phi \rangle = P_b | \Phi \rangle$. Therefore, a density matrix of
 the form $\hat{\rho} = \sum_m \hat{\rho}_{mm}$ without any cross-terms
 between different Zeno subspaces can only be composed of pure states
 that each lie predominantly within a single subspace. However, because
-we will not be dealing with the projective limit, the wavefunction
+we will not be dealing with the projective limit, the wave function
 will in general not be entirely confined to a single Zeno subspace. We
 have seen that the coherences are of order $\nu/\lambda^2$. This would
-require the wavefunction components to satisfy
+require the wave function components to satisfy
 $P_a | \Phi \rangle \approx O(1)$ and
 $P_b | \Phi \rangle \approx O(\nu/\lambda^2)$ (or vice-versa). This in
 turn implies that the population of the states outside of the dominant
@@ -1248,7 +1247,7 @@ span multiple Zeno subspaces, cannot exist in a meaningful coherent
 superposition in this limit. This means that a density matrix that
 spans multiple Zeno subspaces has only classical uncertainty about
 which subspace is currently occupied as opposed to the uncertainty due
-to a quantum superposition. This is anlogous to the simple qubit
+to a quantum superposition. This is analogous to the simple qubit
 example we considered in section \ref{sec:master}.
  
 \subsection{Quantum Measurement vs. Dissipation}
@@ -1257,10 +1256,10 @@ This is where quantum measurement deviates from dissipation. If we
 have access to a measurement record we can infer which Zeno subspace
 is occupied, because we know that only one of them can be occupied at
 any time. We have seen that since the density matrix cross-terms are
-small we know \emph{a priori} that the individual wavefunctions
+small we know \emph{a priori} that the individual wave functions
 comprising the density matrix mixture will not be coherent
 superpositions of different Zeno subspaces and thus we only have
-classical uncertainty which means we can resort to clasical
+classical uncertainty which means we can resort to classical
 probability methods. Each individual experiment will at any time be
 predominantly in a single Zeno subspace with small cross-terms and
 negligible occupations in the other subspaces. With no measurement
@@ -1402,7 +1401,7 @@ This is a crucial step as all $\hat{\rho}_{mm}$ matrices are
 decoherence free subspaces and thus they can all coexist in a mixed
 state decreasing the purity of the system without
 measurement. Physically, this means we exclude trajectories in which
-the Zeno subspace has changed (measurement isn't fully projective). By
+the Zeno subspace has changed (measurement is not fully projective). By
 substituting Eq. \eqref{eq:intermediate} into Eq. \eqref{eq:master} we
 see that this happens at a rate of $\nu^2 / \lambda^2$. However, since
 the two measurement outcomes cannot coexist any transition between
@@ -1641,7 +1640,7 @@ defined by $\hat{T} |\Psi \rangle = 0$, where
 $\hat{T} = \sum_{\langle i, j \rangle} \bd_i b_j$. This is simply the
 physical interpretation of the steady states we predicted for
 Eq. \eqref{eq:master}. Crucially, this state can only be reached if
-the dynamics aren't fully suppressed by measurement and thus,
+the dynamics are not fully suppressed by measurement and thus,
 counter-intuitively, the atomic dynamics cooperate with measurement to
 suppress itself by destructive interference. Therefore, this effect is
 beyond the scope of traditional quantum Zeno dynamics and presents a
@@ -1650,7 +1649,7 @@ dynamics and global measurement backaction.
 
 We now consider a one-dimensional lattice with $M$ sites so we extend
 the measurement to $\hat{D} = \N_\text{even}$ where every even site is
-illuminated.  The wavefunction in a Zeno subspace must be an
+illuminated.  The wave function in a Zeno subspace must be an
 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
@@ -1754,7 +1753,7 @@ design a state with desired properties.
 In this chapter we have demonstrated that global quantum measurement
 backaction can efficiently compete with standard local processes in
 many-body systems. This introduces a completely new energy and time
-scale into quantum many-body research. This is made possbile by the
+scale into quantum many-body research. This is made possible by the
 ability to structure the spatial profile of the measurement on a
 microscopic scale comparable to the lattice period without the need
 for single site addressing. The extreme flexibility of the setup

Chapter6/chapter6.tex

@@ -25,18 +25,18 @@ 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{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,
+in section \ref{sec:B} that it is possible to couple to 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
+fundamentally different as it involves direct coupling to the
 interference terms caused by atoms tunnelling rather than combining
 light scattered from different sources.
 
@@ -78,18 +78,17 @@ 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
+  \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 in the Brillouin zone. 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,
+denoted by index $k$, via $b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma}
+b_k$ and $b_k$ annihilates an atom in the Brillouin zone. Note that
+this operator is diagonal in momentum space which means that its
+eigenstates are simply 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
@@ -103,7 +102,7 @@ measurement of the from $\a_1 = C \B$ is thus
   \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
+$\c = \sqrt{2 \kappa} C \Bmax$ is applied to the wave function. We can
 rewrite the above equation as
 \begin{equation}
   \hat{H} = -\frac{J}{J^B_\mathrm{max}} \Bmax - i \gamma \Bmax^\dagger \Bmax.
@@ -179,14 +178,13 @@ 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
+lifetime. 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.
+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}
 
@@ -222,7 +220,7 @@ and the Hamiltonian:
   \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
+where the summations are performed over the reduced Brillouin 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
@@ -289,10 +287,10 @@ open systems described by the density matrix, $\hat{\rho}$,
   \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 following results will not depend on the nature nor 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
+as before, but the results are more general and can be applied to their
 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
@@ -372,7 +370,7 @@ 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
+we achieved with $P_m$. Previously we only identified 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 $P_m [\hat{H}_0, \hat{\rho}] P_n$.
@@ -495,12 +493,12 @@ 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
+observable with $\a_1$ 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
+measurement operator, $\a_1$, 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
@@ -527,14 +525,14 @@ 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
+Having identified an 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
+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
@@ -621,8 +619,8 @@ $\sin(ka)$ 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.
+exactly the same values of $B_\mathrm{min}$, i.e.~they 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
@@ -659,7 +657,7 @@ $\mathcal{P}_M = \sum_{m \in M} P_m$.
 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
+eigenstates even if significant tunnelling is occurring 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

Chapter7/chapter7.tex

@@ -58,9 +58,9 @@ In the limit of strong measurement when quantum Zeno dynamics occurs
 we showed that these nonlocal spatial modes created by the global
 measurement lead to long-range correlated tunnelling events whilst
 suppressing any other dynamics between different spatial modes of the
-measurement. Such globally paired tunneling due to a fundamentally
+measurement. Such globally paired tunnelling due to a fundamentally
 novel phenomenon can enrich physics of long-range correlated systems
-beyond relatively shortrange interactions expected from standard
+beyond relatively short range interactions expected from standard
 dipole-dipole interactions \cite{sowinski2012, omjyoti2015}. These
 nonlocal high-order processes entangle regions of the optical lattice
 that are disconnected by the measurement. Using different detection
@@ -85,8 +85,8 @@ on-site density. This defines most processes in optical lattices.  For
 example, matter-field phase changes may happen not only due to
 external gradients, but also due to intriguing effects such quantum
 jumps leading to phase flips at neighbouring sites and sudden
-cancellation of tunneling \cite{vukics2007}, which should be
-accessible by this method. Furthremore, in mean-field one can measure
+cancellation of tunnelling \cite{vukics2007}, which should be
+accessible by this method. Furthermore, in mean-field one can measure
 the matter-field amplitude (which is also the order parameter),
 quadratures and their squeezing. This can link atom optics to areas
 where quantum optics has already made progress, e.g., quantum imaging

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