Finished section on competition with interactions

Chapter4/chapter4.tex

@@ -401,6 +401,7 @@ degenerate subspaces which are of much greater interest as they reveal
 nontrivial dynamics in the system.
 
 \section{Global Measurement and ``Which-Way'' Information}
+\label{sec:modes}
 
 We have already mentioned that one of the key features of our model is
 the global nature of the measurement operators. A single light mode

Chapter5/chapter5.tex

@@ -17,34 +17,47 @@
 In the previous chapter we have introduced a theoretical framework
 which will allow us to study measurement backaction using
 discontinuous quantum jumps and non-Hermitian evolution due to null
-outcomes. We have also wrapped our quantum gas model in this quantum
+outcomesquantum trajectories. We have also wrapped our quantum gas
-trajectory formalism by considering ultracold bosons in an optical
+model in this formalism by considering ultracold bosons in an optical
 lattice coupled to a cavity which collects and enhances light
 scattered in one particular direction. One of the most important
 conclusions of the previous chapter was that the introduction of
-measurement introduces a new energy and time scale into the picture.
+measurement introduces a new energy and time scale into the picture
+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 of atoms. In particular, we will
+backaction on the many-body state and dynamics of atoms. In
-focus on the competition between the backaction and the the two
+particular, we will focus on the competition between the backaction
-standard short-range processes, tunnelling and on-site interactions,
+and the the two standard short-range processes, tunnelling and on-site
-in optical lattices. We show that the possibility to spatially
+interactions, in optical lattices. We show that the possibility to
-structure the measurement at a micrscopic scalecomparable to the
+spatially structure the measurement at a microscopic scale comparable
-lattice period without the need for single site resolution enebales us
+to the lattice period without the need for single site resolution
-to engineer efficient competition between the three processes in order
+enables us to engineer efficient competition between the three
-to generate new nontrivial dynamics. Furthermore, the global nature of
+processes in order to generate new nontrivial dynamics. However,
-the measurement creates long-range correlations which enable nonlocal
+unlike tunnelling and on-site interactions our measurement scheme is
-dynamical processes distinguishing it from the local processes.
+global in nature which makes it capable of creating long-range
+correlations which enable nonlocal dynamical processes. Furthermore,
+global light scattering from multiple lattice sites creates nontrivial
+spatially nonlocal coupling to the environment which is impossible to
+obtain with local interactions \cite{daley2014, diehl2008,
+  syassen2008}. These spatial modes of matter fields can be considered
+as designed systems and reservoirs opening the possibility of
+controlling dissipations in ultracold atomic systems without resorting
+to atom losses and collisions which are difficult to manipulate. Thus
+the continuous measurement of the light field introduces a
+controllable decoherence channel into the many-body dynamics. Such a
+quantum optical approach can broaden the field even further allowing
+quantum simulation models unobtainable using classical light and the
+design of novel systems beyond condensed matter analogues.
 
 In the weak measurement limit, where the quantum jumps do not occur
 frequently compared to the tunnelling rate, this can lead to global
 macroscopic oscillations of bosons between odd and even sites. These
 oscillations occur coherently across the whole lattice enabled by the
-fact that measurement is capable of generating nonlocal spatial modes.
+fact that measurement is capable of generating nonlocal spatial
-
+modes. When on-site interactions are included we obtain a system with
-When on-site interactions are included in the picture we obtain a
+three competing energy scales of which two correspond to local
-system with three competing energy scales of which two correspond to
+processes and one is global. This complicates the picture
-local processes and one is global. This complicates the picture
 immensely. We show how under certain circumstances interactions
 prevent measurement from generating globally coherent dynamics, but on
 the other hand when the measurement is strong both processes
@@ -53,58 +66,52 @@ collaborate in squeezing the atomic distribution.
 On the other end of the spectrum, when measurement is strong we enter
 the regime of quantum Zeno dynamics. Frequent measurements can slow
 the evolution of a quantum system leading to the quantum Zeno effect
-where a quantum state is frozen in its initial configuration. One can
+where a quantum state is frozen in its initial configuration
-also devise measurements with multi-dimensional projections which lead
+\cite{misra1977, facchi2008}. One can also devise measurements with
-to quantum Zeno dynamics where unitary evolution is uninhibited within
+multi-dimensional projections which lead to quantum Zeno dynamics
-this degenrate subspace, i.e.~the Zeno subspace. The flexible setup
+where unitary evolution is uninhibited within this degenerate
-where global light scattering can be engineered allows us to suppress
+subspace, usually called the Zeno subspace \cite{facchi2008,
-or enhance specific dynamical processes thus realising spatially
+  raimond2010, raimond2012, signoles2014}. Our flexible setup where global light
-nonlocal quantum Zeno dynamics. This unconventional variation of
+scattering can be engineered allows us to suppress or enhance specific
-quantum Zeno dynamics occurs when measurement is near, but not in, its
+dynamical processes thus realising spatially nonlocal quantum Zeno
-projective limit. The system is still confined to Zeno subspaces, but
+dynamics. This unconventional variation occurs when measurement is
-intermediate transitions are allowed via virtual Raman-like
+near, but not in, its projective limit. The system is still confined
-processes. We show that this result can, in general (i.e.~beyond the
+to Zeno subspaces, but intermediate transitions are allowed via
-ultracold gas model considered here), be approimated by a
+virtual Raman-like processes. We show that this result can, in general
+(i.e.~beyond the ultracold gas model), be approximated by a
 non-Hermitian Hamiltonian thus extending the notion of quantum Zeno
 dynamics into the realm of non-Hermitian quantum mechanics joining the
 two paradigms.
 
-The measurement process generates spatial modes of matter fields that
-can be considered as designed systems and reservoirs opening the
-possibility of controlling dissipations in ultracold atomic systems
-without resorting to atom losses and collisions which are difficult to
-manipulate. The continuous measurement of the light field introduces a
-controllable decoherence channel into the many-body
-dynamics. Furthermore, global light scattering from multiple lattice
-sites creates nontrivial spatially nonlocal coupling to the
-environment which is impossible to obtain with local
-interactions. Such a quantum optical approach can broaden the field
-even further allowing quantum simulation models unobtainable using
-classical light and the design of novel systems beyond condensed
-matter analogues.
-
 \section{Large-Scale Dynamics due to Weak Measurement}
 
 We start by considering the weak measurement limit when photon
 scattering does not occur frequently compared to the tunnelling rate
 of the atoms, i.e.~$\gamma \ll J$. When the system is probed in this
 way, the measurement is unable to project the quantum state of the
-bosons to an eigenspace thus making it impossible to establish quantum
+bosons to an eigenspace as postulated by the Copenhagen interpretation
-Zeno Dynamics. However, instead of confining the evolution of the
+of quantum mechanics. The backaction of the photodetections is simply
-quantum state, it has been shown in Refs. \cite{mazzucchi2016,
+not strong or frequent enough to confine the atoms. However, instead
-  mazzucchi2016njp} that the measurement leads to coherent global
+of confining the evolution of the quantum state, it has been shown in
-oscillations between the modes generated by the spatial profile of the
+Refs. \cite{mazzucchi2016, mazzucchi2016njp} that the measurement
-light field. Fig. \ref{fig:oscillations} illustrates the atom number
+leads to coherent global oscillations between the modes generated by
-distributions in one of the modes for $Z = 2$ ($N_\mathrm{odd}$) and
+the spatial profile of the light field which we have seen in section
-$Z = 3$ ($N_1$) \cite{mazzucchi2016}. In the absence of the external
+\ref{sec:modes}. Fig. \ref{fig:oscillations} illustrates the atom
-influence of measurement these distributions would spread out
+number distributions in the odd sites for $Z = 2$ and one of the three
-significantly and oscillate with an amplitude that is less than or
+modes for $Z = 3$. These oscillations correspond to atoms flowing from
-equal to the initial imbalance, i.e.~small oscillations for a small
+one mode to another. We only observe a small number of well defined
-initial imbalance. By contrast, here we observe a macroscopic exchange
+components which means that this flow happens in phase, all the atoms
-of atoms between the modes even in the absence of an initial
+are tunnelling between the modes together in unison. Furthermore, this
-imbalance, that the distributions consist of a small number of well
+exchange of population is macroscopic in scale. The trajectories reach
-defined components, and these components are squeezed by the weak
+a state where the maximum displacement point corresponds to all the
-measurement.
+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
+the broad distribution would oscillate with an amplitude comparable to
+the initial imbalance, i.e.~small oscillations for a small initial
+imbalance.
 
 \begin{figure}[htbp!]
   \centering
@@ -127,57 +134,62 @@ measurement.
     negative atom number differences in $p(N_\mathrm{odd})$ ($N = 100$
     bosons, $J_{j,j} = (-1)^{j+1}$). (c) Measurement for $Z = 3$ modes
     preserves three components in $p(N_1)$ ($N = 108$ bosons,
-    $J_{j,j} = e^{i 2 \pi j / 3}$.}
+    $J_{j,j} = e^{i 2 \pi j / 3}$).}
   \label{fig:oscillations}
 \end{figure}
 
-Furthermore, depending on the quantity being addressed by the
+In Figs. \ref{fig:oscillations}(b,c) we also see that the system is
-measurement, the state of the system has multiple components as seen
+composed of multiple components. This depends on the quantity that is
-in Figs. \ref{fig:oscillations}b and \ref{fig:oscillations}c. This is a
+being measured and it is a consequence of the fact that the detected
-consequence of the fact that the measured light intensity $\ad_1 \a_1$
+light intensity $\ad_1 \a_1$ is not sensitive to the light phase. The
-is not sensitive to the light phase. The measurement will not
+measurement will not distinguish between permutations of mode
-distinguish between all permutations of mode occupations that scatter
+occupations that scatter light with the same intensity, but with a
-light with the same intensity, but different phase. For example, when
+different phase. For example, when measuring
-measuring $\hat{D} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$,
+$\hat{D} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$, the light
-the light intensity will be proportional to
+intensity will be proportional to
 $\hat{D}^\dagger \hat{D} = (\hat{N}_\mathrm{odd} -
 \hat{N}_\mathrm{even})^2$ and thus it cannot distinguish between a
 positive and negative imbalance leading to the two components seen in
 Fig. \ref{fig:oscillations}. More generally, the number of components
 of the atomic state, i.e.~the degeneracy of $\ad_1 \a_1$, can be
-computed from the eigenvalues of Eq. \eqref{eq:Zmodes}, 
+computed from the eigenvalues of Eq. \eqref{eq:Zmodes},
 \begin{equation}
-  \hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l,
+  \hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l.
 \end{equation}
-noting that they can be represented as the sum of vectors on the
+Each eigenvalue can be represented as the sum of the individual terms
-complex plane with phases that are integer multiples of $2 \pi / Z$:
+in teh above sum which are vectors on the complex plane with phases
-$N_1 e^{-i 2 \pi R / Z}$, $N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since
+that are integer multiples of $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$,
-the set of possible sums of these vectors is invariant under rotations
+$N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums
-by $2 \pi l R / Z$, $l \in \mathbb{Z}$, and reflection in the real axis, the
+of these vectors is invariant under rotations by $2 \pi l R / Z$,
-state of the system is 2-fold degenerate for $Z = 2$ and $2Z$-fold
+$l \in \mathbb{Z}$, and reflection in the real axis, the state of the
-degenerate for $Z > 2$. Fig. \ref{fig:oscillations} shows the three
+system is 2-fold degenerate for $Z = 2$ (reflections leave $Z = 2$
-mode case, where there are in fact $6$ components ($2Z = 6$), but in
+unchanged) and $2Z$-fold degenerate for $Z >
-this case they all occur in pairs resulting in three visible
+2$. Fig. \ref{fig:oscillations} shows the three mode case, where there
-components.
+are in fact $6$ components ($2Z = 6$), but in this case they all occur
+in pairs resulting in only three visible components.
 
-It has also been shown in Ref. \cite{mazzucchi2016njp} that the
+We will now limit ourselves to a specific illumination pattern with
-non-interacting dynamics with quantum measurement backaction for
+$\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest
-$R$-modes reduce to an effective Bose-Hubbard Hamiltonian with
+multimode dynamics with $Z = 2$ and only a single component as seen in
-$R$-sites provided the initial state is a superfluid. In this
+Fig. \ref{fig:oscillations}a, i.e.~no multiple peaks like in
-simplified model the $N_j$ atoms in the $j$th site corresponds to a
+Figs. \ref{fig:oscillations}(b,c). This pattern can be obtained by
-superfluid of $N_j$ atoms within a single spatial mode as defined by
-Eq. \eqref{eq:Zmodes}. Furthermore, the tunnelling term in the
-Bose-Hubbard model and the quantum jumps do not affect this
-correspondence.
-
-Therefore, we will now consider an illumination pattern with
-$\hat{D} = \hat{N}_\mathrm{odd}$. This pattern can be obtained by
 crossing two beams such that their projections on the lattice are
-identical and the even sites are positioned at their
+identical and the even sites are positioned at their nodes. However,
-nodes. Fig. \ref{fig:oscillations}a shows that this leads to
+even though this is the simplest possible case and we are only dealing
-macroscopic oscillations with a single peak. We will now attempt to
+with non-interacting atoms solving the full dynamics of the
-get some physical insight into the process by using the reduced
+Bose-Hubbard Hamiltonian combined with measurement is nontrivial. The
-effective double-well model. The atomic state can be written as 
+backaction introduces a highly nonlinear global term. However, it has
+been shown in Ref. \cite{mazzucchi2016njp} that the non-interacting
+dynamics with quantum measurement backaction for $Z$-modes reduce to
+an effective Bose-Hubbard Hamiltonian with $Z$-sites provided the
+initial state is a superfluid. In this simplified model the $N_j$
+atoms in the $j$-th site correspond to a superfluid of $N_j$ atoms
+within a single spatial mode as defined in section
+\ref{sec:modes}. Therefore, we now proceed to study the dynamics for
+$\hat{D} = \hat{N}_\mathrm{odd}$ using this reduced effective
+double-well model.
+
+The atomic state can be written as
 \begin{equation}
   \label{eq:discretepsi}
   | \psi \rangle = \sum_l^N q_l |l, N - l \rangle,
@@ -192,13 +204,12 @@ jumps is given by
   \gamma \n_o^2
 \end{equation}
 and the quantum jump operator which is applied at each photodetection
-is $\c = \sqrt{2 \kappa} C \n_o$. $b_o$ ($\bd_o$) is the
+is $\c = \sqrt{2 \kappa} C \n_o$. $b_o$ ($\bd_o$) is the annihilation
-annihilation (creation) operator in the left-hand site in the
+(creation) operator in the left site of the effective double-well
-effective double-well corresponding to the superfluid at odd sites of
+corresponding to the superfluid at odd sites of the physical
-the physical lattice. $b_e$ ($\bd_e$) is defined similarly, but for
+lattice. $b_e$ ($\bd_e$) is defined similarly, but for the right site
-the right-hand site and the superfluid at even sites of the physical
+and the superfluid at even sites of the physical lattice.
-lattice. $\n_o = \bd_o b_o$ is the atom number operator in the
+$\n_o = \bd_o b_o$ is the atom number operator in the left site.
-left-hand site.
 
 Even though Eq. \eqref{eq:doublewell} is relatively simple as it it is
 only a non-interacting two-site model, the non-Hermitian term
@@ -206,16 +217,17 @@ complicates the situation making the system difficult to
 solve. However, a semiclassical approach to boson dynamics in a
 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 be easily
+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
 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
 Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by
-dividing by $NJ$ and define the relative population imbalance between
+dividing by $NJ^\mathrm{cl}$ and define the relative population
-the two wells $z = 2x - 1$. Finally, by taking the expectation value
+imbalance between the two wells $z = 2x - 1$. Finally, by taking the
-of the Hamiltonian and looking for the stationary points of
+expectation value of the Hamiltonian and looking for the stationary
+points of
 $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
 \rangle$ we obtain the semiclassical Schr\"{o}dinger equation
 \begin{equation}
@@ -230,21 +242,24 @@ $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
 \end{equation}
 where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$,
 $\omega = 2 \sqrt{1 + \Lambda - h}$, and
-$\Lambda = NU / (2J^\mathrm{cl})$. We will be considering $U = 0$ as
+$\Lambda = NU / (2J^\mathrm{cl})$. The full derivation is not
-the effective model is only valid in this limit, thus $\Lambda =
+straightforward, but the introduction of the non-Hermitian term
-0$. However, this model is valid for an actual physical double-well
+requires only a minor modification to the original formalism presented
-setup in which case interacting bosons can also be considered. The
+in detail in Ref. \cite{juliadiaz2012} so we have omitted it here. We
-equation is defined on the interval $z \in [-1, 1]$, but we have
+will also be considering $U = 0$ as the effective model is only valid
-assumed that $z \ll 1$ in order to simplify the kinetic term and
+in this limit, thus $\Lambda = 0$. However, this model is valid for an
-approximate the potential as parabolic. This does mean that this
+actual physical double-well setup in which case interacting bosons can
-approximation is not valid for the maximum amplitude oscillations seen
+also be considered. The equation is defined on the interval
-in Fig. \ref{fig:oscillations}a, but since they already appear early
+$z \in [-1, 1]$, but $z \ll 1$ has been assumed in order to simplify
-on in the trajectory we are able to obtain a valid analytic
+the kinetic term and approximate the potential as parabolic. This does
-description of the oscillations and their growth.
+mean that this approximation is not valid for the maximum amplitude
+oscillations seen in Fig. \ref{fig:oscillations}a, but since they
+already appear early on in the trajectory we are able 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
-potential is parabolic even with the inclusion of the non-Hermitian
+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
 of the form
@@ -256,17 +271,16 @@ 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, and $\phi$
+$b^2$ denotes the width, $z_0$ the position of the center, $\phi$ and
-and $z_\phi$ contain the phase information of the Gaussian wave
+$z_\phi$ contain the local phase information, and $\epsilon$ only
-packet.
+affects the global phase and norm of the Gaussian wave packet.
 
 The non-Hermitian Hamiltonian and an ansatz are not enough to describe
-the full dynamics due to measurement. We also need to derive the
+the full dynamics due to measurement. We also need to know the effect
-effect of a single quantum jump. Within the continuous variable
+of each quantum jump. Within the continuous variable approximation,
-approximation, our quantum jump become $\c \propto 1 + z$. We neglect
+our quantum jump become $\c \propto 1 + z$. We neglect the constant
-the constant prefactors, because the wavefunction is normalised after
+prefactors, because the wavefunction is normalised after a quantum
-a quantum jump. Expanding around the peak of the Gaussian ansatz we
+jump. Expanding around the peak of the Gaussian ansatz we get
-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].
@@ -280,7 +294,8 @@ parameters change discontinuously according to
   \phi & \rightarrow \frac{ \phi (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
   \label{eq:jumpz0}
   z_0 & \rightarrow z_0 + \frac{ b^2 (1 + z_0) } { (1 + z_0)^2 + b^2}, \\
-  z_\phi & \rightarrow z_\phi.
+  z_\phi & \rightarrow z_\phi, \\
+  \epsilon & \rightarrow \epsilon.
 \end{align}
 The fact that the wavefunction remains Gaussian after a photodetection
 is a huge advantage, because it means that the combined time evolution
@@ -293,17 +308,18 @@ 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
 superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$,
-$\phi =0$, $a_0 = 0$, and $a_\phi = 0$. Howver, we use the most
+$\phi =0$, $a_0 = 0$, $a_\phi = 0$, $\epsilon = 0$. However, we use
-general initial conditions at time $t = t_0$ which we denote by
+the most general initial conditions at time $t = t_0$ which we denote
-$b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$, and
+by $b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$,
-$z_\phi(t_0) = a_\phi$. The reason for keeping them as general as
+$z_\phi(t_0) = a_\phi$, and $\epsilon(t_0) = \epsilon_0$. The reason
-possible is that after every quantum jump the system changes
+for keeping them as general as possible is that after every quantum
-discontinuously. The subsequent time evolution is obtained by solving
+jump the system changes discontinuously. The subsequent time evolution
-the Schr\"{o}dinger equation with the post-jump paramater values as
+is obtained by solving the Schr\"{o}dinger equation with the post-jump
-the new initial conditions.
+paramater values as the new initial conditions.
 
 By plugging the ansatz in Eq. \eqref{eq:ansatz} into the
-Eq. \eqref{eq:semicl} we obtain three differential equations
+Schr\"{o}dinger equation in Eq. \eqref{eq:semicl} we obtain three
+differential equations
 \begin{equation}
   \label{eq:p}
   -2 h^2 p^2 + \left( \frac{ \omega^2 } { 8 } - \frac{ i \Gamma } { 4
@@ -325,14 +341,14 @@ Eq. \eqref{eq:semicl} we obtain three differential equations
 where $x = 1/b^2$, $p = (1 - i \phi)/b^2$,
 $q = (z_0 - i \phi z_\phi)/b^2$, and
 $r = (z_0^2 - \phi z_\phi^2)/b^2$. The corresponding initial
-conditions are $x(0) = x_0 = 1/b_0^2$,
+conditions are $x(t_0) = x_0 = 1/b_0^2$,
-$p(0) = p_0 = (1 - i \phi_0)/b_0^2$,
+$p(t_0) = p_0 = (1 - i \phi_0)/b_0^2$,
-$q(0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and
+$q(t_0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and
-$r(0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. The original
+$r(t_0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. The original
 parameters can be extracted from these auxiliary variables by
 $b^2 = 1 / \Re \{ p \}$, $\phi = - \Im \{ p \} / \Re \{ p \}$,
 $z_0 = \Re \{ q \} / \Re \{ p \}$,
-$z_\phi = \Im \{ q \} / \Im \{ p \}$, and $\epsilon$ is appears
+$z_\phi = \Im \{ q \} / \Im \{ p \}$, and $\epsilon$ appears
 explicitly in the equations above.
 
 First, it is worth noting that all parameters of interest can be
@@ -343,9 +359,13 @@ 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
 $q(t)$ only. Therefore, we only need to solve first two equations and
-we can neglect Eq. \eqref{eq:pqr}.
+we can neglect Eq. \eqref{eq:pqr}. However, in order to actually
+perform Monte-Carlo simulations of quantum trajectories
+Eq. \eqref{eq:pqr} would need to be solved in order to obtain correct
+jump statistics.
 
-Eq. \eqref{eq:p} can be rearranged into the form
+We start with Eq. \eqref{eq:p} and we note it can be rearranged into
+the form
 \begin{equation}
   \frac{ \mathrm{d} p } { (\zeta \omega / 4 h)^2 - p^2 } = i 4 h
   \mathrm{d} t,
@@ -378,23 +398,22 @@ Having found an expression for $p(t)$ we can now solve
 Eq. \eqref{eq:pq} for $q(t)$. To do that we first define the
 integrating factor
 \begin{equation}
-  I(t) = \exp \left[ i 4 h \int p \mathrm{d} t \right],
+  I(t) = \exp \left[ i 4 h \int p \mathrm{d} t \right] = ( \zeta
+  \omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega - 4 h p_0 )
+  e^{-i \zeta \omega t},
 \end{equation}
 which lets us rewrite Eq. \eqref{eq:pq} as
 \begin{equation}
+  \label{eq:Iq}
   \frac{\mathrm{d}} {\mathrm{d} t}(Iq) = - \frac{\Gamma}{2 h} I.
 \end{equation}
-Upon integrating the equation above we obtain
+%Upon integrating the equation above we obtain
-\begin{equation}
+%\begin{equation}
-  \label{eq:Iq}
+%  \label{eq:Iq}
-  Iq = - \frac{ \Gamma } {2 h} \int I \mathrm{d} t.
+%  Iq = - \frac{ \Gamma } {2 h} \int I \mathrm{d} t.
-\end{equation}
+%\end{equation}
-The integrating factor can be evaluated and shown to be
+Upon integrating and the substitution of the explicit form of the
-\begin{equation}
+integration factor into this equation we obtain the solution
-  I(t) = ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} + 
-  ( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega t},
-\end{equation}
-which upon substitution into Eq. \eqref{eq:Iq} yields the solution
 \begin{equation}
   \label{eq:qsol}
   q(t) = \frac{1}{2 h \zeta \omega} 
@@ -410,17 +429,19 @@ $q(t)$ in Eq. \eqref{eq:qsol} are sufficient to completely describe
 the physics of the system. Unfortunately, these expressions are fairly
 complex and it is difficult to extract the physically meaningful
 parameters in a form that is easy to analyse. Therefore, we instead
-consider the case when $\Gamma = 0$. It may seem counter-intuitive to
+consider the case when $\Gamma = 0$, but we do not neglect the effect
-neglect the term that appears due to measurement, but we are
+of quantum jumps. It may seem counter-intuitive to neglect the term
-considering the weak measurement regime where
+that appears due to measurement, but we are considering the weak
-$\gamma \ll J^\mathrm{cl}$ and thus the dynamics between the quantum
+measurement regime where $\gamma \ll J^\mathrm{cl}$ and thus the
-jumps are actually dominated by the tunnelling of atoms rather than
+dynamics between the quantum jumps are actually dominated by the
-the null outcomes. However, this is only true at times shorter than
+tunnelling of atoms rather than the null outcomes. Furthermore, the
-the average time between two consecutive quantum jumps. Therefore,
+effect of the quantum jump is independent of the value of $\Gamma$
-this approach will not yield valid answers on the time scale of a
+($\Gamma$ only determined their frequency). However, this is only true
-whole quantum trajectory, but it will give good insight into the
+at times shorter than the average time between two consecutive quantum
-dynamics immediately after a quantum jump. The solutions for $\Gamma =
+jumps. Therefore, this approach will not yield valid answers on the
-0$ are
+time scale of a whole quantum trajectory, but it will give good
+insight into the dynamics immediately after a quantum jump. The
+solutions for $\Gamma = 0$ are
 \begin{equation}
 b^2(t) = \frac{b_0^2}{2} \left[ \left(1 + \frac{16 h^2 (1 + \phi_0^2)}
     {b_0^4 \omega^2} \right) + \left(1 - \frac{16 h^2 (1 + \phi_0^2)}
@@ -445,86 +466,336 @@ 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
 $\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4
-  \omega^2)}$. For these oscillations to occur, $a_0$ and $a_\phi$
+  \omega^2)}$. Thus we have obtained a solution that clearly shows
-cannot be zero, but this is exactly the case for an initial superfluid
+oscillations of a single Gaussian wave packet. The fact that this
-state. However, we have seen in Eq. \eqref{eq:jumpz0} that the effect
+appears even when $\Gamma = 0$ shows that the oscillations are a
-of a photodetection is to displace the wavepacket by approximately
+property of the Bose-Hubbard model itself. However, they also depend
-$b^2$, i.e.~the width of the Gaussian, in the direction of the
+on the initial conditions and for these oscillations to occur, $a_0$
-positive $z$-axis. Therefore, it is the quantum jumps that are the
+and $a_\phi$ cannot be zero, but this is exactly the case for an
-driving force behind this phenomenon. The oscillations themselves are
+initial superfluid state. We have seen in Eq. \eqref{eq:jumpz0} that
-essentially due to the natural dynamics of the atoms in a lattice, but
+the effect of a photodetection is to displace the wavepacket by
-it is the measurement that causes the initial
+approximately $b^2$, i.e.~the width of the Gaussian, in the direction
-displacement. Furthermore, since the quantum jumps occur at an average
+of the positive $z$-axis. Therefore, even though the can oscillate in
-instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
+the absence of measurement it is the quantum jumps that are the
-itself is proportional to $(1+z)^2$ they are most likely to occur at
+driving force behind this phenomenon. Furthermore, these oscillations
+grow because the quantum jumps occur at an average 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 further increases the amplitude of the
+which point a quantum jump provides positive feedback and further
-wavefunction leading to the growth seen in
+increases the amplitude of the wavefunction leading to the growth seen
-Fig. \ref{fig:oscillations}a.
+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 the initial and
+more importantly coherent displacement and the positive feedback drive
+which causes the oscillations to continuously grow.
 
 We have now seen the effect of the quantum jumps and how that leads to
 oscillations between odd and even sites in a lattice. However, we have
 neglected the effect of null outcomes on the dynamics. Even though it
 is small, it will not be negligible on the time scale of a quantum
-trajectory with multiple jumps. Due to the complexity of the equations
+trajectory with multiple jumps. First, we note that all the
-in the case of $\Gamma \ne 0$ our analysis will be less rigoruous and
+oscillatory terms $p(t)$ and $q(t)$ actually appear as
-we will focus on the qualitative aspects of the dynamics.
+$\zeta \omega = (\alpha - i \beta) \omega$. Therefore, we can see that
-
+the null outcomes lead to two effects: an increase in the oscillation
-We note that all the oscillatory terms $p(t)$ and $q(t)$ actually
+frequency by a factor of $\alpha$ to $\alpha \omega$ and a damping
-appear as $\zeta \omega = (\alpha - i \beta) \omega$. Therefore, we
+term with a time scale $1/(\beta \omega)$. For weak measurement, both
-can see that the null outcomes lead to two effects: an increase in the
+$\alpha$ and $\beta$ will be close to $1$ so the effects are not
-oscillation frequency by a factor of $\alpha$ to $\alpha \omega$ and a
+visible on short time scales. Instead, we look at the long time
-damping term with a time scale $1/(\beta \omega)$. For weak
+limit. Unfortunately, since all the quantities are oscillatory a
-measurement, both $\alpha$ and $\beta$ will be close to $1$ so the
+stationary long time limit does not exist especially since the quantum
-effects are not visible on short time scales. Therefore, it would be
+jumps provide a driving force. However, the width of the Gaussian,
-worthwhile to look at the long time limit. Unfortunately, since all
-the quantities are oscillatory a long time limit is fairly meaningless
-especially since the quantum jumps provide a driving force leading to
-larger and larger oscillations. However, the width of the Gaussian,
 $b^2$, is unique in that it doesn't 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
-oscillate, but with an amplitude that decreases monotonically with
+oscillate, but with an amplitude that decreases approximately
-time, because unlike for $z_0$ the quantum jumps do not cause further
+monotonically with time due to quantum jumps and the
-displacement in this quantity. Thus, neglecting the effect of quantum
+$1/(\beta \omega)$ decay terms, because unlike for $z_0$ the quantum
-jumps and taking the long time limit yields
+jumps do not cause further displacement in this quantity. Thus,
+neglecting the effect of quantum jumps and taking the long time limit
+yields
 \begin{equation}
   \label{eq:b2}
   b^2(t \rightarrow \infty) = \frac{4 h} {\gamma \omega} \approx
   b^2_\mathrm{SF} \left( 1 - \frac{\Gamma^2}{32} \right),
 \end{equation}
 where the approximation on the right-hand side follows from the fact
-that $\omega \approx 2$ since we are considering the $N \gg 1$ limit
+that $\omega \approx 2$ since we are considering the $N \gg 1$ limit,
-and, because we are considering the weak measurement limit and so
+and because we are considering the weak measurement limit
 $\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width
 of the initial superfluid state. This result is interesting, because
 it shows that the width of the Gaussian distribution is squeezed as
-compared with its initial state. However, if we substitute the
+compared with its initial state which is exactly what we see in
-parameter values from Fig. \ref{fig:oscillations}a we only get a
+Fig. \ref{fig:oscillations}a. However, if we substitute the parameter
-reduction in width by about $3\%$. The maximum amplitude oscillations
+values used in that trajectory we only get a reduction in width by
-in Fig. \ref{fig:oscillations}a look like they have a significantly
+about $3\%$, but the maximum amplitude oscillations in look like they
-smaller width than the initial distribution. This discrepancy is due
+have a significantly smaller width than the initial distribution. This
-to the fact that the continuous variable approximation is only valid
+discrepancy is due to the fact that the continuous variable
-for $z \ll 1$ and thus it cannot explain the final behaviour of the
+approximation is only valid for $z \ll 1$ and thus it cannot explain
-system. Furthermore, it has been shown that the width of the
+the final behaviour of the system. Furthermore, it has been shown that
-distribution $b^2$ does not actually shrink to a constant value, but
+the width of the distribution $b^2$ does not actually shrink to a
-rather it keeps oscillating around the value given in
+constant value, but rather it keeps oscillating around the value given
-Eq. \eqref{eq:b2}. However, what we do see is that during the early
+in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However, what we do see
-stages of the trajectory, which should be well described by this
+is that during the early stages of the trajectory, which are well
-model, is that the width does not in fact shrink by much. It is only
+described by this model, is that the width does in fact stay roughly
-in the later stages when the oscillations reach maximal amplitude that
+constant. It is only in the later stages when the oscillations reach
-the width becomes visibly reduced.
+maximal amplitude that the width becomes visibly reduced.
 
 \section{Three-Way Competition}
 
-\section{Emergent Long-Range Correlated Tunnelling}
+Now it is time to turn on the inter-atomic interactions,
+$U/J^\mathrm{cl} \ne 0$. As a result the atomic dynamics will change
+as the measurement now competes with both the tunnelling and the
+on-site interactions. A common approach to study such open systems is
+to map a dissipative phase diagram by finding the steady state of the
+master equation for a range of parameter values
+\cite{kessler2012}. However, here we adopt a quantum optical approach
+in which we focus on the conditional dynamics of a single quantum
+trajectory as this corresponds to a single realisation of an
+experiment. The resulting evolution does not necessarily reach a
+steady state and usually occurs far from the ground state of the
+system.
 
-\section{Non-Hermitian Dynamics in the Quantum Zeno Limit}
+A key feature of the quantum trajectory approach is that each
+trajectory evolves differently as it is conditioned on the
+photodetection times which are determined stochastically. Furthermore,
+even states in the same measurement subspace, i.e.~indistinguishable
+to the measurement , can have minimal overlap. This is in contrast to
+the unconditioned solutions obtained with the master equation which
+only yields a single outcome that is an average taken over all
+possible outcomes. However, this makes it difficult to study the
+three-way competition in some meaningful way across the whole
+parameter range.
+
+Ultimately, regardless of its strength measurement always tries to
+project the quantum state onto one of its eigenstates (or eigenspaces
+if there are degeneracies). If the probe is strong enough this will
+succeed, but we have seen in the previous section that when this is
+not the case, measurement leads to new dynamical phenomena. However,
+despite this vast difference in behaviour, there is a single quantity
+that lets us determine the degree of success of the projection, namely
+the fluctuations, $\sigma_D^2$ (or equivalently the standard
+deviation, $\sigma_D$), of the observable that is being measured,
+$\hat{D}$. For a perfect projection this value is exactly zero,
+because the system at that point is in the corresponding
+eigenstate. When the system is unable to project the state into such a
+state, the variance will be non-zero. However, the smaller its value
+is the closer it is to being in such an eigenstate and on the other
+hand a large variance means that the internal processes dominate the
+competition. Finally, this quantity is perfect to study quantum
+trajectories, because its value in the long-time limit it is only a
+function of $\gamma$, $J$, and $U$. It does not depend on the explicit
+history of photodetections. Fig. \ref{fig:squeezing} shows a plot of
+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 $J$
+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.
+
+\begin{figure}[htbp!]
+  \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
+    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.}
+  \label{fig:squeezing}
+\end{figure}
+
+First, it is important to note that even though we are dealing with an
+average over many trajectories this information cannot be extracted
+from a master equation solution. This is because the variance of
+$\hat{D}$ as calculated from the density matrix would be dominated by
+the uncertainty of the final state. In other words, the fact that the
+final value of $\hat{D}$ is undetermined is included in this average
+and thus the fluctuations obtained this way are representative of the
+variance in the final outcome rather than the squeezing of an
+individual conditioned trajectory. This highlights the fact that
+interesting physics happens on a single trajectory level which would
+be lost if we studied an ensemble average.
+
+\begin{figure}[htbp!]
+  \centering
+  \includegraphics[width=\textwidth]{panel_U}
+  \caption[Trajectories in the presence of Interactions]{Conditional
+    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
+    $M = 6$ 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
+    oscillation in the population of the mode. As states with
+    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
+    well-defined although clearly worse than in a Mott insulator.}
+  \label{fig:Utraj}
+\end{figure}
+
+Looking at Fig. \ref{fig:squeezing} we see many interesting things
+happening suggesting different regimes of behaviour. For the ground
+state (i.e.~no measurement) we see that the fluctuations decrease
+monotonically as $U$ increases reflecting the superfluid to Mott
+insulator quantum phase transition. The measured state on the other
+hand behaves very differently and
+$\langle \sigma^2_D \rangle_\mathrm{traj}$ varies
+non-monotonically. For weak interactions the fluctuations are strongly
+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
+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
+the squeezing less effective as seen in Fig. \ref{fig:Utraj}a. In
+fact, we have seen towards the end of the last section how for small
+amplitude oscillations that can be described by the effective
+double-well model the width of the number distribution does not change
+by much. Even though that model is not valid for $U \ne 0$ we should
+not be surprised that without macroscopic oscillations the
+fluctuations cannot be significantly reduced.
+
+On the other end of the spectrum, for weak measurement, but strong
+on-site interactions we note that the backaction leads to a
+significant increase in fluctuations compared to the ground
+state. This is simply due to the fact that the measurement destroys
+the Mott insulating state, which has small fluctuations due to strong
+local interactions, but then subsequently is not strong enough to
+squeeze the resulting dynamics as shown in Fig. \ref{fig:Utraj}b. To
+see why this is so easy for the quantum jumps to do we look at the
+ground state in first-order perturbation theory given by
+\begin{equation}
+  | \Psi_{J/U} \rangle = \left[ 1 + \frac{J}{U} \sum_{\langle i, j
+      \rangle} \bd_i b_j \right] | \Psi_0 \rangle,
+\end{equation}
+where we have neglected the non-Hermitian term as we're in the weak
+measurement regime and $| \Psi_0 \rangle$ is the Mott insulator state and the second
+term in the brackets represents a uniform distribution of
+particle-hole excitation pairs across the lattice. In the
+$U \rightarrow \infty$ limit a quantum jump has no effect as
+$| \Psi_0 \rangle$ is already an eigenstate of $\hat{D}$. However, for
+finite $U$, each photocount will amplify the present excitations
+increasing the fluctuations in the system. In fact, consecutive
+detections lead to an exponential growth of these excitations. For
+$K \gg 1$ illuminated sites and unit filling of the lattice, the
+atomic state after $m$ consecutive quantum jumps becomes
+$\c^m | \Psi_{J/U} \rangle \propto | \Psi_{J/U} \rangle + | \Phi_m
+\rangle$ where
+\begin{equation}
+  | \Phi_m \rangle = \frac{2^m J} {K U} \sum_{i \in
+    \mathrm{odd}} \left( \bd_i b_{i-1} - \bd_{i-1} b_i - \bd_{i+1} b_i
+    + \bd_i b_{i+1} \right) | \Psi_0 \rangle.
+\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,
+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
+distribute the new excitations in a way which will affect and possibly
+reduce the effects of the subsequent quantum jumps. However, due to
+the lack of any decay channels they will remain in the system and
+subsequent jumps will still amplify them further destroying the ground
+state and thus quickly leading to a state with large fluctuations.
+
+In the strong measurement regime ($\gamma \gg J$) the measurement
+becomes more significant than the local dynamics and the system will
+freeze the state in the measurement operator eigenstates. In this
+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
+from the effects of interactions as they would prevent this dynamics
+by dephasing different components of the coherent excitations. Strong
+measurement, on the other hand, squeezes the quantum state by trying
+to project it onto an eigenstate of the observable
+\cite{mekhov2009prl, mekhov2009prl}. For weak interactions where the
+ground state is a highly delocalised superfluid it is obvious that
+projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will supress
+fluctuations significantly. However, the strongly interacting regime
+is much less evident, especially since we have just demonstrated how
+sensitive the Mott insulating phase is to the quantum jumps when the
+measurement is weak.
+
+To understand the strongly interacting case we will again use
+first-order perturbation theory and consider a postselected
+$\langle \hat{D}^\dagger \hat{D} \rangle = 0$ trajectory. This
+corresponds to a state that scatters no photons and thus is fully
+described by the non-Hermitian Hamiltonian alone. Squeezing depends on
+the measurement and interaction strengths and is common to all the
+possible trajectories so we can gain insight into the general
+behaviour by considering a specific special case. However, we will
+instead consider
+$\hat{D} = \Delta \hat{N} = \hat{N}_\mathrm{odd} -
+\hat{N}_\mathrm{even}$, because this measurement also has only $Z = 2$
+modes, but its $\langle \hat{D}^\dagger \hat{D} \rangle = 0$
+trajectory would be very close to the Mott insulating ground state,
+because $\hat{D}^\dagger \hat{D} | \Psi_0 \rangle = 0$ and we can
+expand around the Mott insulating state. Applying perturbation theory
+to obtain the modified ground state we get
+\begin{equation}
+  | \Psi_{J,U, \gamma} \rangle = \left[ 1 + \frac{J}{U - i 4 \gamma} \sum_{\langle i, j
+      \rangle} \bd_i b_j \right] | \Psi_0 \rangle.
+\end{equation}
+The variance of the measurement operator for this state is given by
+\begin{equation}
+  \sigma^2_{\Delta N} = \frac{16 J^2 M} {U^2 + 16 \gamma^2}.
+\end{equation}
+From the form of the denominator we immediately see that both
+interaction and measurement squeeze with the same quadratic dependence
+and that the squeezing is always better than in the ground state
+($\gamma = 0$) regardless of the value of $U$. Also, depending on the
+ratio of $\gamma/U$ the squeezing can be dominated by measurement
+($\gamma/U \gg 1$) or by interactions ($\gamma/U \ll 1$) or both
+processes can contribute equally ($\gamma/U \approx 1$). The
+$\hat{D} = \hat{N}_\mathrm{odd}$ measurement will behave similarly
+since the geometry is exactly the same. Furthermore, the Mott
+insulator state is also an eigenstate of this operator, just not the
+zero eigenvalue vector and thus the final state would need to be
+described using a balance of quantum jumps and non-Hermitian evolution
+complicating the picture. However, the particle-hole excitation term
+would be proportional to $(U^2 + \gamma^2)^{-1}$ instead since the
+$\gamma$ coefficient in the perturbative expansion depends on
+$(J_{i,i} - J_{i\pm1,i\pm1})^2$. We can see the system transitioning
+into the strong measurement regime in Fig. \ref{fig:squeezing} as the
+$U$-dependence flattens out with increasing measurement strength as
+the $\gamma/U \gg 1$ regime is reached.
+
+\section{Quantum Zeno Dynamics}
+
+\subsection{Emergent Long-Range Correlated Tunnelling}
+
+\subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit}
 
 % Contrast with t-J model here how U localises events, but measurement
 % does the opposite
 
-\section{Steady-State of the Non-Hermitian Hamiltonian}
+\subsection{Steady-State of the Non-Hermitian Hamiltonian}
 
 \section{Conclusions}

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