Finished first iteration of the double well section

Chapter4/chapter4.tex

@@ -528,7 +528,7 @@ be incident normally at a 1D lattice so that $u_0 (\b{r}) =
 \begin{equation}
   \label{eq:Dmodes}
   \a_1 = C \hat{D} = C \sum_m^K \exp\left[-i k_1 m d \sin \theta_1
-  \right] \hat{n}_j.
+  \right] \hat{n}_m.
 \end{equation}
 From this equation we see that it can be made periodic with a period
 $Z$ when
@@ -536,8 +536,17 @@ $Z$ when
   k_1 d \sin \theta_1 = 2\pi R / Z,
 \end{equation}
 where $R$ is just some integer and $R/Z$ are is a fraction in its
-simplest form. This partitions the 1D lattice in exactly $Z > 1$ modes
+simplest form. Therefore, we can rewrite the Eq. \eqref{eq:Dmodes} as
-by making every $Z$th lattice site scatter light with exactly the same
+a sum of the indistinguishable contributions from the $Z$ modes
-phase. It is interesting to note that these angles correspond to the
+\begin{equation}
-$K-1$ classical diffraction minima.
+  \label{eq:Zmodes}
+  \a_1 = C \hat{D} = C \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l,
+\end{equation}
+where $\hat{N}_l = \sum_{m \in l} \n_m$ is the sum of single site atom
+number operators that belong to the same mode. $\hat{N}_K$ and
+$\hat{N}_\mathrm{odd}$ are the simplest examples of these modes. This
+partitions the 1D lattice in exactly $Z > 1$ modes by making every
+$Z$th lattice site scatter light with exactly the same phase. It is
+interesting to note that these angles correspond to the $K-1$
+classical diffraction minima.
 

Chapter5/chapter5.tex

@@ -25,16 +25,506 @@ conclusions of the previous chapter was that the introduction of
 measurement introduces a new energy and time scale into the picture.
 
 In this chapter, we investigate the effect of quantum measurement
-backaction on the many-body state of atoms. 
+backaction on the many-body state 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 interactions,
+in optical lattices. We show that the possibility to spatially
+structure the measurement at a micrscopic scalecomparable to the
+lattice period without the need for single site resolution enebales us
+to engineer efficient competition between the three processes in order
+to generate new nontrivial dynamics. Furthermore, the global nature of
+the measurement creates long-range correlations which enable nonlocal
+dynamical processes distinguishing it from the local processes.
+
+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.
+
+When on-site interactions are included in the picture we obtain a
+system with three competing energy scales of which two correspond to
+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
+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
+also devise measurements with multi-dimensional projections which lead
+to quantum Zeno dynamics where unitary evolution is uninhibited within
+this degenrate subspace, i.e.~the Zeno subspace. The flexible setup
+where global light scattering can be engineered allows us to suppress
+or enhance specific dynamical processes thus realising spatially
+nonlocal quantum Zeno dynamics. This unconventional variation of
+quantum Zeno dynamics occurs when measurement is near, but not in, its
+projective limit. The system is still confined to Zeno subspaces, but
+intermediate transitions are allowed via virtual Raman-like
+processes. We show that this result can, in general (i.e.~beyond the
+ultracold gas model considered here), be approimated 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
+Zeno Dynamics. However, instead of confining the evolution of the
+quantum state, it has been shown in Refs. \cite{mazzucchi2016,
+  mazzucchi2016njp} that the measurement leads to coherent global
+oscillations between the modes generated by the spatial profile of the
+light field. Fig. \ref{fig:oscillations} illustrates the atom number
+distributions in one of the modes for $Z = 2$ ($N_\mathrm{odd}$) and
+$Z = 3$ ($N_1$) \cite{mazzucchi2016}. In the absence of the external
+influence of measurement these distributions would spread out
+significantly and oscillate with an amplitude that is less than or
+equal to the initial imbalance, i.e.~small oscillations for a small
+initial imbalance. By contrast, here we observe a macroscopic exchange
+of atoms between the modes even in the absence of an initial
+imbalance, that the distributions consist of a small number of well
+defined components, and these components are squeezed by the weak
+measurement.
+
+\begin{figure}[htbp!]
+  \centering
+  \includegraphics[width=\textwidth]{Oscillations}
+  \caption[Macroscopic Oscillations due to Weak Measurement]{Large
+    oscillations between the measurement-induced spatial modes
+    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
+    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
+    states: bosonic superfluid. (a) Measurement of the atom number at
+    odd sites $\hat{N}_\mathrm{odd}$ creates one strongly oscillating
+    component in $p(N_\mathrm{odd})$ ($N = 100$ bosons, $J_{j,j} = 1$
+    if $j$ is odd and 0 otherwise). (b) Measurement of
+    $(\hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even})^2$ introduces
+    $Z = 2$ modes and preserves the superposition of positive and
+    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,
+    $J_{j,j} = e^{i 2 \pi j / 3}$.}
+  \label{fig:oscillations}
+\end{figure}
+
+Furthermore, depending on the quantity being addressed by the
+measurement, the state of the system has multiple components as seen
+in Figs. \ref{fig:oscillations}b and \ref{fig:oscillations}c. This is a
+consequence of the fact that the measured light intensity $\ad_1 \a_1$
+is not sensitive to the light phase. The measurement will not
+distinguish between all permutations of mode occupations that scatter
+light with the same intensity, but different phase. For example, when
+measuring $\hat{D} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$,
+the light 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}, 
+\begin{equation}
+  \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
+complex plane with phases that are integer multiples of $2 \pi / Z$:
+$N_1 e^{-i 2 \pi R / Z}$, $N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since
+the set of possible sums of these vectors is invariant under rotations
+by $2 \pi l R / Z$, $l \in \mathbb{Z}$, and reflection in the real axis, the
+state of the system is 2-fold degenerate for $Z = 2$ and $2Z$-fold
+degenerate for $Z > 2$. Fig. \ref{fig:oscillations} shows the three
+mode case, where there are in fact $6$ components ($2Z = 6$), but in
+this case they all occur in pairs resulting in three visible
+components.
+
+It has also been shown in Ref. \cite{mazzucchi2016njp} that the
+non-interacting dynamics with quantum measurement backaction for
+$R$-modes reduce to an effective Bose-Hubbard Hamiltonian with
+$R$-sites provided the initial state is a superfluid. In this
+simplified model the $N_j$ atoms in the $j$th site corresponds to a
+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
+nodes. Fig. \ref{fig:oscillations}a shows that this leads to
+macroscopic oscillations with a single peak. We will now attempt to
+get some physical insight into the process by using the 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,
+\end{equation}
+where the ket $| l, N - l \rangle$, represents a superfluid with $l$
+atoms in the odd sites and $N-l$ atoms in the even sites. The
+non-Hermitian Hamiltonian describing the time evolution in between the
+jumps is given by
+\begin{equation}
+  \label{eq:doublewell}
+  \hat{H} = -J^\mathrm{cl} \left( \bd_o b_e + b_o \bd_e \right) - i
+  \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
+annihilation (creation) operator in the left-hand site in the
+effective double-well corresponding to the superfluid at odd sites of
+the physical lattice. $b_e$ ($\bd_e$) is defined similarly, but for
+the right-hand 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-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
+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
+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
+the two wells $z = 2x - 1$. Finally, by taking the 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}
+  \label{eq:semicl}
+  i h \partial_t \psi(z, t) = \mathcal{H} \psi(z, t),
+\end{equation}
+\begin{equation}
+  \label{eq:semiH}
+  \mathcal{H} \approx -2 h^2 \partial^2_z \psi(z, t) + \left[
+    \frac{\omega^2 z^2} {8} - \frac{i \Gamma} {4} \left( z + 1
+    \right)^2 \right] \psi(z, t),
+\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
+the effective model is only valid in this limit, thus $\Lambda =
+0$. However, this model is valid for an actual physical double-well
+setup in which case interacting bosons can also be considered. The
+equation is defined on the interval $z \in [-1, 1]$, but we have
+assumed that $z \ll 1$ in order to simplify the kinetic term and
+approximate the potential as parabolic. This does 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
+term, it will remain Gaussian during subsequent time
+evolution. Therefore, we will use a very general Gaussian wavefunction
+of the form
+\begin{equation}
+  \label{eq:ansatz}
+  \psi(z, t) = \frac{1}{\pi b^2}\exp\left[ i \epsilon 
+  - \frac{(z - z_0)^2} {2 b^2} + \frac{i \phi (z - z_\phi)^2} {2 b^2} \right]
+\end{equation}
+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 $\phi$
+and $z_\phi$ contain the phase information 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
+effect of a single 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 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
+parameters change discontinuously according to
+\begin{align}
+  \label{eq:jumpb2}
+  b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
+  \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.
+\end{align}
+The fact that the wavefunction remains Gaussian after a photodetection
+is a huge advantage, because it means that the combined time evolution
+of the system can be described with a single Gaussian ansatz in
+Eq. \eqref{eq:ansatz} subject to non-Hermitian time evolution
+according to Eq. \eqref{eq:semicl} with discontinous 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
+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
+general initial conditions at time $t = t_0$ which we denote by
+$b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$, and
+$z_\phi(t_0) = a_\phi$. 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.
+
+By plugging the ansatz in Eq. \eqref{eq:ansatz} into the
+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
+    } \right) + \frac{ i h } { 2 } \frac{ \mathrm{d} p } { \mathrm{d}
+    t } = 0,
+\end{equation}
+\begin{equation}
+  \label{eq:pq}
+  4 h^2 p q - \frac{ i \Gamma } { 2 } - i h \frac{ \mathrm{d} q } {
+    \mathrm{d} t } = 0
+\end{equation}
+\begin{equation}
+  \label{eq:pqr}
+  -2 h^2 (q^2 - p) - \frac{ i \Gamma } { 4 } - i h \left( \frac{ 1 } {
+      4 x } \frac{ \mathrm{d} x } {\mathrm{d} t } + i \frac{
+      \mathrm{d} \epsilon } { \mathrm{d} t } - \frac{1}{2} \frac{
+      \mathrm{d} r } { \mathrm{d} t } \right) = 0,
+\end{equation}
+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 $x(0) = x_0 = 1/b_0^2$,
+$p(0) = p_0 = (1 - i \phi_0)/b_0^2$,
+$q(0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and
+$r(0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. The original
+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
+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
+$\epsilon$ as it is only related to the global phase and the norm of
+the wavefunction 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
+$q(t)$ only. Therefore, we only need to solve first two equations and
+we can neglect Eq. \eqref{eq:pqr}.
+
+Eq. \eqref{eq:p} 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,
+\end{equation}
+where $\zeta^2 = (\alpha - i \beta)^2 = 1 - i 2 \Gamma / \omega^2$, and
+\begin{equation}
+  \alpha = \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2
+      }{ \omega^4 }}},
+\end{equation}
+\begin{equation}
+  \beta = -\sqrt{ -\frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2
+      }{ \omega^4 }}}.
+\end{equation}
+This is a standard integral\footnotemark and thus yields
+\begin{equation}
+  \label{eq:psol}
+  p(t) = \frac{ \zeta \omega } { 4 h } 
+  \frac{ ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} - ( \zeta
+    \omega - 4 h p_0 ) e^{-i \zeta \omega 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}
+
+\footnotetext{ \[ \int \frac{\mathrm{d} x}{a^2 - x^2} = \frac{1}{2a}
+    \ln \left( \frac{a+x}{a-x} \right) + \mathrm{const.} 
+    \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
+    \quad\quad\quad\quad\quad\] }
+
+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],
+\end{equation}
+which lets us rewrite Eq. \eqref{eq:pq} as
+\begin{equation}
+  \frac{\mathrm{d}} {\mathrm{d} t}(Iq) = - \frac{\Gamma}{2 h} I.
+\end{equation}
+Upon integrating the equation above we obtain
+\begin{equation}
+  \label{eq:Iq}
+  Iq = - \frac{ \Gamma } {2 h} \int I \mathrm{d} t.
+\end{equation}
+The integrating factor can be evaluated and shown to be
+\begin{equation}
+  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} 
+  \frac{4 h \zeta^2 \omega^2 q_0 - i 8 h \Gamma p_0
+    + i \Gamma [( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} - 
+    ( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega 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}
+
+The solutions we have obtained to $p(t)$ in Eq. \eqref{eq:psol} and
+$q(t)$ 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
+neglect the term that appears due to measurement, but we are
+considering the weak measurement regime where
+$\gamma \ll J^\mathrm{cl}$ and thus the dynamics between the quantum
+jumps are actually dominated by the tunnelling of atoms rather than
+the null outcomes. However, this is only true at times shorter than
+the average time between two consecutive quantum jumps. Therefore,
+this approach will not yield valid answers on the 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)}
+    {b_0^4 \omega^2} \right) \cos (2 \omega t) + \frac{8 h \phi_0}{b_0^2
+    \omega} \sin(2 \omega t) \right],
+\end{equation}
+\begin{equation}
+  \phi(t) = \frac{b_0^2 \omega} {8 h} \left[ \left( \frac{16 h^2 (1 + \phi_0^2)}
+      {b_0^4 \omega^2} - 1 \right) \sin (2 \omega t) + \frac{8 h
+      \phi_0} {b_0^2 \omega} \cos (2 \omega t) \right],
+\end{equation}
+\begin{equation}
+  z_0(t) = a_0 \cos(\omega t) + \frac{4 h \phi_0} {b_0^2 \omega} (a_0 -
+  a_\phi) \sin (\omega t),
+\end{equation}
+\begin{equation}
+  \phi(t) z_\phi(t) = \phi_0 a_\phi \cos (\omega t)  + \frac{4 h}
+  {b_0^2 \omega} (a_0 - \phi_0^2 a_\phi) \sin( \omega t).
+\end{equation}
+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
+$\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4
+  \omega^2)}$. For these oscillations to occur, $a_0$ and $a_\phi$
+cannot be zero, but this is exactly the case for an initial superfluid
+state. However, we have seen in Eq. \eqref{eq:jumpz0} that the effect
+of a photodetection is to displace the wavepacket by approximately
+$b^2$, i.e.~the width of the Gaussian, in the direction of the
+positive $z$-axis. Therefore, it is the quantum jumps that are the
+driving force behind this phenomenon. The oscillations themselves are
+essentially due to the natural dynamics of the atoms in a lattice, but
+it is the measurement that causes the initial
+displacement. Furthermore, since the quantum jumps occur at an average
+instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
+itself is proportional to $(1+z)^2$ 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
+wavefunction leading to the growth seen in
+Fig. \ref{fig:oscillations}a.
+
+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
+in the case of $\Gamma \ne 0$ our analysis will be less rigoruous and
+we will focus on the qualitative aspects of the dynamics.
+
+We note that all the oscillatory terms $p(t)$ and $q(t)$ actually
+appear as $\zeta \omega = (\alpha - i \beta) \omega$. Therefore, we
+can see that the null outcomes lead to two effects: an increase in the
+oscillation frequency by a factor of $\alpha$ to $\alpha \omega$ and a
+damping term with a time scale $1/(\beta \omega)$. For weak
+measurement, both $\alpha$ and $\beta$ will be close to $1$ so the
+effects are not visible on short time scales. Therefore, it would be
+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
+time, because unlike for $z_0$ the quantum 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
+and, because we are considering the weak measurement limit and so
+$\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ 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
+parameter values from Fig. \ref{fig:oscillations}a we only get a
+reduction in width by about $3\%$. The maximum amplitude oscillations
+in Fig. \ref{fig:oscillations}a look like they have a significantly
+smaller width than the initial distribution. This discrepancy is due
+to the fact that the continuous variable approximation is only valid
+for $z \ll 1$ and thus it cannot explain the final behaviour of the
+system. Furthermore, it has been shown that the width of the
+distribution $b^2$ does not actually shrink to a constant value, but
+rather it keeps oscillating around the value given in
+Eq. \eqref{eq:b2}. However, what we do see is that during the early
+stages of the trajectory, which should be well described by this
+model, is that the width does not in fact shrink by much. It is only
+in the later stages when the oscillations reach maximal amplitude that
+the width becomes visibly reduced.
+
 \section{Three-Way Competition}
 
 \section{Emergent Long-Range Correlated Tunnelling}
 
 \section{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}
 
 \section{Conclusions}

References/references.bib

@@ -248,6 +248,18 @@ year = {2010}
   journal={arXiv preprint arXiv:1510.04883},
   year={2015}
 }
+@article{mazzucchi2016njp,
+  author={Gabriel Mazzucchi and Wojciech Kozlowski and Santiago F
+                  Caballero-Benitez and Igor B Mekhov},
+  title={Collective dynamics of multimode bosonic systems induced by
+                  weak quantum measurement},
+  journal={New Journal of Physics},
+  volume={18},
+  number={7},
+  pages={073017},
+  url={http://stacks.iop.org/1367-2630/18/i=7/a=073017},
+  year={2016}
+}
 	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %% Other papers
@@ -803,3 +815,20 @@ doi = {10.1103/PhysRevA.87.043613},
   year={2014},
   publisher={APS}
 }
+@article{juliadiaz2012,
+  title = {Dynamic generation of spin-squeezed states in bosonic
+                  Josephson junctions},
+  author = {Juli\'a-D\'{\i}az, B. and Zibold, T. and Oberthaler,
+                  M. K. and Mel\'e-Messeguer, M. and Martorell, J. and
+                  Polls, A.},
+  journal = {Phys. Rev. A},
+  volume = {86},
+  issue = {2},
+  pages = {023615},
+  numpages = {11},
+  year = {2012},
+  month = {Aug},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevA.86.023615},
+  url = {http://link.aps.org/doi/10.1103/PhysRevA.86.023615}
+}

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