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Finished chapter 4; Added skeleton outlines of chapters 5,6, and 7; Rearranged some sections in chapter 2 and 3

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Chapter2/chapter2.tex
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@ -11,26 +11,17 @@
\graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}}
\fi
\section{Introduction}
%********************************** % First Section ****************************
In this chapter, we derive a general Hamiltonian that describes the
coupling of atoms with far-detuned optical beams
\cite{mekhov2012}. This will serve as the basis from which we explore
the system in different parameter regimes, such as nondestructive
measurement in free space or quantum measurement backaction in a
cavity. Another interesting direction for this field of research are
quantum optical lattices where the trapping potential is treated
quantum mechanically. However this is beyond the scope of this work.
\section{Ultracold Atoms in Optical Lattices}
%********************************** % Second Section ***************************
\section{Quantum Optics of Quantum Gases}
Having introduced and described the behaviour of ultracold bosons
trapped and manipulated using classical light, it is time to extend
the discussion to quantized optical fields. We will first derive a
general Hamiltonian that describes the coupling of atoms with
far-detuned optical beams \cite{mekhov2012}. This will serve as the
basis from which we explore the system in different parameter regimes,
such as nondestructive measurement in free space or quantum
measurement backaction in a cavity. Another interesting direction for
this field of research are quantum optical lattices where the trapping
potential is treated quantum mechanically. However this is beyond the
scope of this work.
\mynote{insert our paper citations here}
We consider $N$ two-level atoms in an optical lattice with $M$
@ -48,8 +39,9 @@ from a small number of sites with a large filling factor (e.g.~BECs
trapped in a double-well potential) to a an extended multi-site
lattice with a low filling factor (e.g.~a system with one atom per
site which will exhibit the Mott insulator to superfluid quantum phase
transition). \mynote{extra fermion citations, Piazza? Look up Gabi's
AF paper.}
transition).
\mynote{extra fermion citations, Piazza? Look up Gabi's AF paper.}
As we have seen in the previous section, an optical lattice can be
formed with classical light beams that form standing waves. Depending
@ -69,7 +61,7 @@ quantum potential in contrast to the classical lattice trap.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{LatticeDiagram}
\includegraphics[width=\linewidth]{LatticeDiagram}
\caption[Experimental Setup]{Atoms (green) trapped in an optical
lattice are illuminated by a coherent probe beam (red), $a_0$,
with a mode function $u_0(\b{r})$ which is at an angle $\theta_0$
@ -98,7 +90,7 @@ the light modes. However, it is much more intuitive to consider
variable angles in our model as this lends itself to a simpler
geometrical representation.
\subsection{Derivation of the Hamiltonian}
\section{Derivation of the Hamiltonian}
\label{sec:derivation}
A general many-body Hamiltonian coupled to a quantized light field in
@ -235,12 +227,12 @@ inclusion of this scattered light would not be difficult due to the
linearity of the dipoles we assumed.
Normally we will consider scattering of modes $a_l$ much weaker than
the field forming the lattice potential $V_\mathrm{cl}(\b{r})$. We now
proceed in the same way as when deriving the conventional Bose-Hubbard
Hamiltonian in the zero temperature limit. The field operators
the field forming the lattice potential
$V_\mathrm{cl}(\b{r})$. Therefore, we assume that the trapping is
entirely due to the classical potential and the field operators
$\Psi(\b{r})$ can be expanded using localised Wannier functions of
$V_\mathrm{cl}(\b{r})$ and by keeping only the lowest vibrational
state we get
$V_\mathrm{cl}(\b{r})$. By keeping only the lowest vibrational state
we get
\begin{equation}
\Psi(\b{r}) = \sum_i^M b_i w(\b{r} - \b{r}_i),
\end{equation}
@ -386,7 +378,9 @@ the quantum state of matter. This is a key element of our treatment of
the ultimate quantum regime of light-matter interaction that goes
beyond previous treatments.
\subsection{Scattered light behaviour}
\section{The Bose-Hubbard Hamiltonian}
\section{Scattered light behaviour}
\label{sec:a}
Having derived the full quantum light-matter Hamiltonian we will now
@ -472,7 +466,7 @@ neglected. This is likely to be the case since the interactions will
be dominated by photons scattering from the much larger coherent
probe.
\subsection{Density and Phase Observables}
\section{Density and Phase Observables}
\label{sec:B}
Light scatters due to its interactions with the dipole moment of the
@ -671,7 +665,7 @@ $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ and by
varying the local oscillator phase, one can choose which conjugate
operator to measure.
\subsection{Electric Field Stength}
\section{Electric Field Stength}
\label{sec:Efield}
The Electric field operator at position $\b{r}$ and at time $t$ is
@ -839,50 +833,54 @@ Therefore, we can now express the quantity $n_{\Phi}$ as
n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
\end{equation}
Estimates of the scattering rate using real experimental parameters
are given in Table \ref{tab:photons}.
are given in Table \ref{tab:photons}. Rubidium atom data has been
taken from Ref. \cite{steck}. 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
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
$\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is
obtained from definition of Rabi frequency,
$\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$, assuming the electric
field is parallel to the dipole, and the relation
$I = c \epsilon_0 |E|^2 /2$. We used
$d_\mathrm{eff} = 1.73 \times 10^{-29}$ Cm \cite{steck}. Weitenberg
\emph{et al.} experimental parameters are based on
Ref. \cite{weitenberg2011, weitenbergThesis}. The
$5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is
used. Ref. \cite{weitenberg2011} gives the free space detuning to be
$\Delta_\mathrm{free} = - 2 \pi \cdot 45$ MHz, but
Ref. \cite{weitenbergThesis} clarifies that the relevant detuning is
$\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$, where
$\Delta_\mathrm{lat} = - 2 \pi \cdot 40$
MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
$s_\mathrm{tot}$ to quantify the intensity of the beams which is
related to the Rabi frequency,
$s_\mathrm{tot} = 2 \Omega^2 / \Gamma^2$ \cite{steck,foot}. We can
extract $s_\mathrm{tot}$ for the experiment from the total scattering
rate by
$\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz
per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
\begin{table}[!htbp]
\centering
\begin{tabular}{|c|c|c|}
\hline
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline
$\omega_a$ & \multicolumn{2}{|c|}{$2 \pi \cdot 384$ THz}\\ \hline
$\Gamma$ & \multicolumn{2}{|c|}{$2 \pi \cdot 6.07$ MHz} \\ \hline
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ \hline
$I$ & $4250$ Wm$^{-2}$ & N/A \\ \hline
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ \hline
$N_K$ & 10$^5$ & 147 \\ \hline \hline
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline
\begin{tabular}{l c c}
\toprule
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
$\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
$\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
$I$ & $4250$ Wm$^{-2}$ & N/A \\
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
$N_K$ & 10$^5$ & 147 \\ \midrule
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
\bottomrule
\end{tabular}
\caption[Photon Scattering Rates]{ Rubidium data taken from
Ref. \cite{steck}; Miyake \emph{et al.}: based on
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 into account in a more complete analysis. However, it is
included for discussion. The detuning is said to be a few natural
linewidths. Note that it is much smaller than $\Omega_0$. The Rabi
fequency 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 field is parallel to the dipole, and the
relation $I = c \epsilon_0 |E|^2 /2$. We used $d_\mathrm{eff} =
1.73 \times 10^{-29}$ Cm \cite{steck}; Weitenberg \emph{et al.}:
based on Ref. \cite{weitenberg2011, weitenbergThesis}. The
$5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is
used. Ref. \cite{weitenberg2011} gives the free space detuning to
be $\Delta_\mathrm{free} = - 2 \pi \cdot 45$
MHz. Ref. \cite{weitenbergThesis} clarifies that the relevant
detuning is $\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$,
where $\Delta_\mathrm{lat} = - 2 \pi \cdot 40$
MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
$s_\mathrm{tot}$ to quantify the intensity of the beams which is
related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 /
\Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for
the experiment from the total scattering rate by
$\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60
kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.}
\caption[Photon Scattering Rates]{Experimental parameters used in
estimating the photon scattering rates.}
\label{tab:photons}
\end{table}

6
Chapter3/chapter3.tex
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@ -112,7 +112,7 @@ flexibility of the 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.
\subsection{On-site Density Measurements}
\section{On-site Density Measurements}
We have seen in section \ref{sec:B} that typically the dominant term
in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009,
@ -387,7 +387,7 @@ should be visible using currently available technology, especially
since the most prominent features, such as Bragg diffraction peaks, do
not coincide at all with the classical diffraction pattern.
\subsection{Mapping the quantum phase diagram}
\section{Mapping the quantum phase diagram}
We have shown that scattering from atom number operators leads to a
purely quantum diffraction pattern which depends on the density
@ -668,7 +668,7 @@ extracting the Tomonaga-Luttinger parameter from the scattered light,
$R$, in the same way it was done for an unperturbed Bose-Hubbard model
\cite{ejima2011}.
\subsection{Matter-field interference measurements}
\section{Matter-field interference measurements}
We have shown in section \ref{sec:B} that certain optical arrangements
lead to a different type of light-matter interaction where coupling is

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Chapter4/chapter4.tex
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@ -36,10 +36,10 @@ repeated preparations of the initial state.
In the following chapters, we consider a different approach to quantum
measurement in open systems and instead of considering expectation
values we look at a single experimental run and the resulting dynamics
due to measurement backaction. Previously we were mostly interested in
extracting information about the quantum state of the atoms from the
scattered light. The flexibility in the measurement model was used to
enable probing of as many different quantum properties of the
due to measurement backaction. Previously, we were mostly interested
in extracting information about the quantum state of the atoms from
the scattered light. The flexibility in the measurement model was used
to enable probing of as many different quantum properties of the
ultracold gas as possible. By focusing on measurement backaction we
instead investigate the effect of photodetections on the dynamics of
the many-body gas as well as the possible quantum states that we can
@ -51,21 +51,20 @@ foundation for the material that follows in which we apply these
concepts to a bosonic quantum gas. We first introduce the concept of
quantum trajectories which represent a single continuous series of
photon detections. We also present an alternative approach to open
systems in which the measurement outcomes are discarded as this will
be useful when trying to learn about dynamical features common to
every trajectory. In this case we use the density matrix formalism
which obeys the master equation. This approach is more common in
dissipative systems and we will highlight the differences between
these two different types of open systems. We conclude this chapter
with a new concept that will be central to all subsequent
discussions. In our model measurement is global, it couples to
operators that correspond to global properties of the quantum gas
rather than single-site quantities. This enables the possibility of
performing measurements that cannot distinguish certain sites from
each other. Due to a lack of ``which-way'' information this leads to
the creation of spatially nontrivial virtual lattices on top of the
physical lattice. This turns out to have significant consequences on
the dynamics of the system.
systems in which the measurement outcomes are discarded. This will be
useful when trying to learn about dynamical features common to every
trajectory. In this case we use the density matrix formalism which
obeys the master equation. This approach is more common in dissipative
systems and we will highlight the differences between these two
different types of open systems. We conclude this chapter with a new
concept that will be central to all subsequent discussions. In our
model measurement is global, it couples to operators that correspond
to global properties of the quantum gas rather than single-site
quantities. This enables the possibility of performing measurements
that cannot distinguish certain sites from each other. Due to a lack
of ``which-way'' information this leads to the creation of spatially
nontrivial virtual lattices on top of the physical lattice. This turns
out to have significant consequences on the dynamics of the system.
\section{Quantum Trajectories}
@ -83,11 +82,11 @@ interacting with a quantum gas we present a more general overview
which will be useful as some of the results in the following chapters
are more general. Measurement always consists of at least two
competing processes, two possible outcomes. If there is no competition
and only one outcome is possible then our measurement is meaningless
as it does not reveal any information about the system. In its
simplest form measurement consists of a series of events, such as the
and only one outcome is possible then our probe is meaningless as it
does not reveal any information about the system. In its simplest form
measurement consists of a series of detection events, such as the
detections of photons. Even though, on an intuitive level it seems
that we have defined only a single outcome, the event, this
that we have defined only a single outcome, the detection event, this
arrangement actually consists of two mutually exclusive outcomes. At
any point in time an event either happens or it does not, a photon is
either detected or the detector remains silent, also known as a null
@ -96,40 +95,42 @@ investigating. For example, let us consider measuring the number of
atoms by measuring the number of photons they scatter. Each atom will
on average scatter a certain number of photons contributing to the
detection rate we observe. Therefore, if we record multiple photons at
a high rate we learn that the illuminated region must contain many
atoms. On the other hand, if there are few atoms to scatter the light
we will observe few detection events which we interpret as a
continuous series of non-detection events interspersed with the
occasional detector click. This trajectory informs us that there are
much fewer atoms being illuminated than previously.
a high rate of arrival we learn that the illuminated region must
contain many atoms. On the other hand, if there are few atoms to
scatter the light we will observe few detection events which we
interpret as a continuous series of non-detection events interspersed
with the occasional detector click. This trajectory informs us that
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
will have a jump operator, $\c$, associated with it. The effect of an
event on the quantum state is simply the result of applying this jump
operator to the wavefunction, $| \psi (t) \rangle$,
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, $| \psi (t) \rangle$,
\begin{equation}
\label{eq:jump}
| \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle}
{\sqrt{\langle \cd \c \rangle (t)}},
\end{equation}
where the denominator is simply a normalising factor. The exact form
of the jump operator $\c$ will depend on the nature of the measurement
we are considering. For example, if we consider measuring the photons
escaping from a leaky cavity then $\c = \sqrt{2 \kappa} \hat{a}$,
where $\kappa$ is the cavity decay rate and $\hat{a}$ is the
annihilation operator of a photon in the cavity field. It is
interesting to note that due to renormalisation the effect of a single
quantum jump is independent of the magnitude of the operator $\c$
itself. However, larger operators lead to more frequent events and
thus more frequent applications of the jump operator.
where the denominator is simply a normalising factor
\cite{MeasurementControl}. The exact form of the jump operator $\c$
will depend on the nature of the measurement we are considering. For
example, if we consider measuring the photons escaping from a leaky
cavity then $\c = \sqrt{2 \kappa} \hat{a}$, where $\kappa$ is the
cavity decay rate and $\hat{a}$ is the annihilation operator of a
photon in the cavity field. It is interesting to note that due to
renormalisation the effect of a single quantum jump is independent of
the magnitude of the operator $\c$ itself. However, larger operators
lead to more frequent events and thus more frequent applications of
the jump operator.
The null measurement outcome will have an opposing effect to the
quantum jump, but it has to be treated differently as it does not
occur at discrete time points like the detection events
themselves. Its effect is accounted for by a modification to the
isolated Hamiltonian, $\hat{H}_0$, time evolution in the form
occur at discrete time points like the detection events themselves
\cite{MeasurementControl}. Its effect is accounted for by a
modification to the isolated Hamiltonian, $\hat{H}_0$, time evolution
in the form
\begin{equation}
| \psi (t + \mathrm{d}t) \rangle = \left\{ \hat{1} - \mathrm{d}t
\left[ i \hat{H}_0 + \frac{\cd \c}{2} - \frac{\langle \cd \c
@ -146,9 +147,9 @@ Schr\"{o}dinger equation given by
\end{equation}
where $\mathrm{d}N(t)$ is the stochastic increment to the number of
photodetections up to time $t$ which is equal to $1$ whenever a
quantum jump occurs and $0$ otherwise. Note that this equation has a
straightforward generalisation to multiple jump operators, but we do
not consider this possibility here at all.
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
described by the stochastic Schr\"{o}dinger equation in
@ -160,11 +161,12 @@ applied, i.e.~$\mathrm{d}N(t) = 1$, if
R(t) < \langle \cd \c \rangle (t) \delta t.
\end{equation}
In practice, this is not the most efficient method for
simulation. 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 $| \tilde{\psi} (t) \rangle$ using the
non-Hermitian evolution given by
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
$| \tilde{\psi} (t) \rangle$ using the non-Hermitian evolution given
by
\begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} | \tilde{\psi} (t) \rangle = -i
\left( \hat{H}_0 - i \frac{\cd \c}{2} \right) | \tilde{\psi} (t) \rangle
@ -180,7 +182,7 @@ happens during a single trajectory. The quantum jumps are
self-explanatory, but now we have a clearer picture of the effect of
null outcomes on the quantum state. Its effect is entirely encoded in
the non-Hermitian modification to the original Hamiltonian given by
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. It is now easy to see that fro a
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. It is now easy to see that for a
jump operator, $\c$, with a large magnitude the no-event outcomes will
have a more significant effect on the quantum state. At the same time
they will lead to more frequent quantum jumps which have an opposing
@ -191,7 +193,7 @@ 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. In fact, individual
occured 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.
@ -217,29 +219,30 @@ of the quantum jump operator, because light scattering in different
directions corresponds to different measurements as we have seen in
Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have
to simultaneously consider all the possible directions in which light
could scatter and thus include multiple jump operators reducing the
our ability to control the system.
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
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
from the cavity. However, we will simplify the system by considering
the regime where we can neglect the effect of the quantum potential
that builds up in the cavity. Physically, this means that whilst light
scatters due to its interaction with matter, the field that builds up
due to the scattered photons collecting in the cavity has a negligible
effect on the atomic evolution compared to its own dynamics such as
inter-site tunnelling or on-site interactions. This 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 as this effect is of a different nature. It is due
to the wavefunction 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 is
from the cavity. However, we will first simplify the system by
considering the regime where we can neglect the effect of the quantum
potential that builds up in the cavity. Physically, this means that
whilst light scatters due to its interaction with matter, the field
that builds up due to the scattered photons collecting in the cavity
has a negligible effect on the atomic evolution compared to its own
dynamics such as inter-site tunnelling or on-site interactions. This
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
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
is
\begin{equation}
\hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F}
\end{equation}
@ -253,19 +256,22 @@ the measurement and we have substituted $\a_1 = C \hat{F}$. The
quantum jumps are applied at times determined by the algorithm
described above and the jump operator is given by
\begin{equation}
\label{eq:jump}
\label{eq:jumpop}
\c = \sqrt{2 \kappa} C \hat{F}.
\end{equation}
Importantly, we see that measurement introduces a new energy and time
scale $\gamma$ which competes with the two other standard scales
responsible for the uitary dynamics of the closed system, tunnelling,
$J$, and in-site interaction, $U$. If each atom scattered light
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
each site projecting the atomic system into a state where long-range
coherence is degraded. This is a typical scenario for spontaneous
emission, or for local and fixed-range addressing. In contrast to such
emission \cite{pichler2010, sarkar2014}, or for local
\cite{syassen2008, kepesidis2012, vidanovic2014, bernier2014,
daley2014} and fixed-range addressing \cite{ates2012, everest2014}
which are typically considered in open systems. In contrast to such
situations, we consider global coherent scattering with an operator
$\c$ that is global. Therefore, the effect of measurement backaction
$\c$ that is nonlocal. Therefore, the effect of measurement backaction
is global as well and each jump affects the quantum state in a highly
nonlocal way and most importantly not only will it not degrade
long-range coherence, it will in fact lead to such long-range
@ -281,8 +287,9 @@ it difficult to obtain conclusive deterministic answers about the
behaviour of single trajectories. One possible approach that is very
common when dealing with open systems is to look at the unconditioned
state which is obtained by averaging over the random measurement
results which condition the system. The unconditioned state is no
longer a pure state and thus must be described by a density matrix,
results which condition the system \cite{MeasurementControl}. The
unconditioned state is no longer a pure state and thus must be
described by a density matrix,
\begin{equation}
\label{eq:rho}
\hat{\rho} = \sum_i p_i | \psi_i \rangle \langle \psi_i |,
@ -305,13 +312,13 @@ detection events.
We will be using the master equation and the density operator
formalism in the context of measurement. However, the exact same
methods are also applied to a different class of open systems, namely
dissipative systems. A dissipative system is an open system that
couples to an external bath in an uncontrolled way. The behaviour of
such a system is similar to a system subject measurement in which we
ignore all measurement results. One can even think of this external
coupling as a measurement who's outcome record is not accessible and
thus must be represented as an average over all possible
trajectories. However, there is a crucial difference between
dissipative systems \cite{QuantumNoise}. A dissipative system is an
open system that couples to an external bath in an uncontrolled
way. The behaviour of such a system is similar to a system subject
under observation in which we ignore all the results. One can even
think of this external coupling as a measurement who's outcome record
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
@ -326,24 +333,25 @@ A definite advantage of using the master equation for measurement is
that it includes the effect of any possible measurement
outcome. Therefore, it is useful when extracting features that are
common to many trajectories, regardless of the exact timing of the
events. In this case, we do not want to impose any specific trajectory
on the system as we are not interested in a specific experimental run,
but we would still like to identify the set of possible outcomes and
their properties. Unfortunately, calculating the inverse of
Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of
a density matrix into pure states might not even be unique. However,
if a measurement leads to a projection, i.e.~the final state becomes
confined to some subspace of the Hilbert space, then this will be
visible in the final state of the density matrix. We will show this on
an example of a qubit in the quantum state
events. However, in this case we do not want to impose any specific
trajectory on the system as we are not interested in a specific
experimental run, but we would still like to identify the set of
possible outcomes and their common properties. Unfortunately,
calculating the inverse of Eq. \eqref{eq:rho} is not an easy task. In
fact, the decomposition of a density matrix into pure states might not
even be unique. However, if a measurement leads to a projection,
i.e.~the final state becomes confined to some subspace of the Hilbert
space, then this will be visible in the final state of the density
matrix. We will show this on an example of a qubit in the quantum
state
\begin{equation}
\label{eq:qubit0}
| \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle,
\end{equation}
where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states
of the qubit and we consider performing a measurement on it in the
basis $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome.
The quantum state will have collapsed now to the $ | 0 \rangle$ with
of the qubit and we consider performing a measurement in the basis
$\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome. The
quantum state will have collapsed now to the state $ | 0 \rangle$ with
probability $| \alpha |^2$ and $| 1 \rangle$ with probability
$| \beta |^2$. The corresponding density matrix is given by
\begin{equation}
@ -353,43 +361,50 @@ $| \beta |^2$. The corresponding density matrix is given by
0 & |\beta|^2
\end{array} \right),
\end{equation}
which is now a mixed state. We note that there are no off-diagonal
terms as the system is not in a superposition between the two basis
states. Therefore, the diagonal terms represent classical
probabilities of the system being in either of the basis states. This
is in contrast to their original interpretation when the state was
given by Eq. \eqref{eq:qubit0} when they represented the quantum
uncertainty in our knowledge of the state which would have manifested
itself in the density matrix as non-zero off-diagonal terms. The
significance of these values being classical probabilities is that now
we know that the measurement has already happened and we know 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.
which is a mixed state as opposed to the initial state. We note that
there are no off-diagonal terms as the system is not in a
superposition between the two basis states. Therefore, the diagonal
terms represent classical probabilities of the system being in either
of the basis states. This is in contrast to their original
interpretation when the state was given by Eq. \eqref{eq:qubit0} when
they could not be interpreted as in such a way. The initial state was
in a quantum superposition and thus the state was indeterminate due to
the quantum uncertainty in our knowledge of the state which would have
manifested itself in the density matrix as non-zero off-diagonal
terms. The significance of these values being classical probabilities
is that now we know that the measurement has already happened and we
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
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
does not matter. The key observation is that regardless of the
trajectory taken if the final state is given by Eq. \eqref{eq:rho1} we
will definitely know that our state is either in the state
trajectory taken, if the final state is given by Eq. \eqref{eq:rho1}
we will definitely know that our state is either in the state
$| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the
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.
themself. This is analogous to an approach in which decoherence due to
coupling to the environment is used to model the wavefunction collapse
\cite{zurek2002}, but here we will be looking at projective effects
due to weak measurement.
Here we have considered a very simple case of a Hilbert space with two
non-degenerate basis states. In the following chapters we will
generalise the above result to larger Hilbert spaces with multiple
degenerate subspaces.
degenerate subspaces which are of much greater interest as they reveal
nontrivial dynamics in the system.
\section{Global Measurement and ``Which-Way'' Information}
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
couples to multiple lattice sites which then scatter the light
couples to multiple lattice sites from where atoms scatter the light
coherently into a single mode which we enhance and collect with a
cavity. If atoms at different lattice sites scatter light with a
different phase or magnitude we will be able to identify which atoms
@ -399,7 +414,8 @@ knowing which atom emitted the photon, we have no ``which-way''
information. When we were considering nondestructive measurements and
looking at expectation values, this had no consequence on our results
as we were simply interested in probing the quantum correlations of a
given ground state. Now, on the other hand, we are interested in the
given ground state and whether two sites were distinguishable or not
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
@ -409,32 +425,119 @@ Therefore, all quantum correlations between the atoms in these sites
are unaffected by the backaction whilst their correlations with the
rest of the system will change as the result of the quantum jumps.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{1DModes}
\caption[1D Modes due to Measurement Backaction]{The coefficients,
and thus the operator $\hat{D}$ is made periodic with a period of
three lattice sites. Therefore, the coefficients $J_{i,i}$ will
repeat every third site making atoms in those sites
indistinguishable to the measurement. Physically, this is due to
the fact that this periodic arrangement causes the atoms within a
single mode to scatter light with the same phase and amplitude.
The scattered light contains no information which can be used to
determine the atom distribution.}
\label{fig:1dmodes}
\end{figure}
The quantum jump operator for our model is given by
$\c = \sqrt{2 \kappa} C \hat{F}$ and we know from Eq. \eqref{eq:F}
that we have a large amount of flexibility in tuning $\hat{F}$ via the
geometry of the optical setup. We will consider the case when
$\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the argument
depends on geometry rather than the exact nature of the operator it
straightforwardly generalises to other measurement operators,
including the case when $\hat{F} = \hat{B}$. The operator $\hat{D}$ is
given by
geometry of the optical setup. A cavity aligned at a different angle
will correspond to a different measurement. We will consider the case
when $\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the
argument depends on geometry rather than the exact nature of the
operator it straightforwardly generalises to other measurement
operators, including the case when $\hat{F} = \hat{B}$ where the bonds
(inter-site operators) play the role of lattice sites. The operator
$\hat{D}$ is given by
\begin{equation}
\hat{D} = \sum_i J_{i,i} \n_i,
\end{equation}
where the coefficients $J_{i,i}$ are determined from
Eq. \eqref{eq:Jcoeff}. It is very simple to make the $J_{i,i}$
periodic as it
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{1DModes}
\caption[1D Modes due to Measurement Backaction]{}
\label{fig:cavity}
\end{figure}
Eq. \eqref{eq:Jcoeff}. These coefficients represent the coupling
strength between the atoms and the light modes and thus their spatial
variation can be easily tuned by the geometry of the optical
fields. We are in particular interested in making these coefficients
degenerate across a number of lattice sites as shown in
Figs. \ref{fig:1dmodes} and \ref{fig:2dmodes}. Note that they do not
have to be periodic, but it is much easier to make them so. This makes
all lattice sites with the same value of $J_{i,i}$ indistinguishable
to the measurement thus partitioning the lattice into a number of
distinct zones which we will refer to as modes. It is crucial to note
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
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
``which-way'' information. Therefore, from the point of view of the
observer's knowledge all atoms within a mode are identical regardless
of their spatial separation. This effect can be used to create virtual
lattices on top of the physical lattice.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{2DModes}
\caption[2D Modes due to Measurement Backaction]{}
\label{fig:cavity}
\caption[2D Modes due to Measurement Backaction]{ More complex
patterns of virtual lattice can be created in higher
dimensions. With a more complicated setup even more complicated
geometries are possible.}
\label{fig:2dmodes}
\end{figure}
We will now look at a few practical examples. The simplest case is to
measure in the diffraction maximum such that $\a_1 = C \hat{N}_K$,
where $\hat{N}_K = \sum_j^K \hat{n}_j$ is the number of atoms in the
illuminated area. If the whole lattice is illuminated we effectively
have a single mode as $J_{i,i} = 1$ for all sites. If only a subset of
lattice sites is illuminated $K < M$ then we have two modes
corresponding to the illuminated and unilluminated sites. It is
actually possible to perform such a measurement in a nonlocal way by
arranging every other site (e.g.~all the even sites) to be at a node
of both the cavity and the probe. The resulting field measures the
number of atoms in the remaining sites (all the odd sites) in a global
manner. It does not know how these atoms are distributed among these
sites as we do not have access to the individual sites. This two mode
arrangement is shown in the top panel of Fig. \ref{fig:twomodes}. A
different two-mode arrangement is possible by measuring in the
diffraction minimum such that each site scatters in anti-phase with
its neighbours as shown in the bottom panel of
Fig. \ref{fig:twomodes}.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.7\textwidth]{TwoModes1}
\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
partitions the lattice into odd and even sites, but this time
atoms at all sites scatter light, but in anti-phase with their
neighbours.}
\label{fig:twomodes}
\end{figure}
This can approach can be generalised to an arbitrary number of modes,
$Z$. For this we will conisder 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
\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.
\end{equation}
From this equation we see that it can be made periodic with a period
$Z$ when
\begin{equation}
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
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.

27
Chapter5/chapter5.tex Normal file
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@ -0,0 +1,27 @@
%*******************************************************************************
%*********************************** Fifth Chapter *****************************
%*******************************************************************************
\chapter{Density Measurement Induced Dynamics}
% Title of the Fifth Chapter
\ifpdf
\graphicspath{{Chapter5/Figs/Raster/}{Chapter5/Figs/PDF/}{Chapter5/Figs/}}
\else
\graphicspath{{Chapter5/Figs/Vector/}{Chapter5/Figs/}}
\fi
\section{Introduction}
\section{Large-Scale Dynamics due to Weak Measurement}
\section{Three-Way Competition}
\section{Emergent Long-Range Correlated Tunnelling}
\section{Non-Hermitian Dynamics in the Quantum Zeno Limit}
\section{Steady-State of the Non-Hermitian Hamiltonian}
\section{Conclusions}

23
Chapter6/chapter6.tex Normal file
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@ -0,0 +1,23 @@
%*******************************************************************************
%*********************************** Sixth Chapter *****************************
%*******************************************************************************
\chapter{Phase Measurement Induced Dynamics}
% Title of the Sixth Chapter
\ifpdf
\graphicspath{{Chapter6/Figs/Raster/}{Chapter6/Figs/PDF/}{Chapter6/Figs/}}
\else
\graphicspath{{Chapter6/Figs/Vector/}{Chapter6/Figs/}}
\fi
\section{Introduction}
\section{Diffraction Maximum and Energy Eigenstates}
\section{General Model for Weak Measurement Projection}
\section{Determining the Projection Subspace}
\section{Conclusions}

11
Chapter7/chapter7.tex Normal file
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@ -0,0 +1,11 @@
%*******************************************************************************
%*********************************** Seventh Chapter *****************************
%*******************************************************************************
\chapter{Summary and Conclusions} %Title of the Seventh Chapter
\ifpdf
\graphicspath{{Chapter7/Figs/Raster/}{Chapter7/Figs/PDF/}{Chapter7/Figs/}}
\else
\graphicspath{{Chapter7/Figs/Vector/}{Chapter7/Figs/}}
\fi

175
References/references.bib
View File

@ -15,16 +15,16 @@
year = {2003}
}
@phdthesis{weitenbergThesis,
author = {Weitenberg, Christof},
number = {April},
title = {{Single-Atom Resolved Imaging and Manipulation in an Atomic
Mott Insulator}},
year = {2011}
title={Single-atom resolved imaging and manipulation in an atomic
Mott insulator},
author={Weitenberg, Christof},
year={2011},
school={LMU}
}
@article{steck,
author = {Steck, Daniel Adam},
title = {{Rubidium 87 D Line Data Author contact information :}},
url = {http://steck.us/alkalidata}
@misc{steck,
title={Rubidium 87 D line data},
author={Steck, Daniel A},
year={2001}
}
@inbook{Scully,
author = {Scully, M. and Zubairy, S.},
@ -37,7 +37,21 @@
@misc{tnt,
howpublished="\url{http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/}"
}
@book{MeasurementControl,
author = {Wiseman, Howard M. and Milburn, Gerard J.},
title = {{Quantum Measurement and Control}},
publisher = {Cambridge University Press},
year = {2010}
}
@book{QuantumNoise,
title={Quantum noise: a handbook of Markovian and non-Markovian
quantum stochastic methods with applications to
quantum optics},
author={Gardiner, Crispin and Zoller, Peter},
volume={56},
year={2004},
publisher={Springer Science \& Business Media}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Igor's original papers
@ -190,6 +204,20 @@
year={2015},
publisher={APS}
}
@article{caballero2016,
title = {Quantum simulators based on the global collective light-matter interaction},
author = {Caballero-Benitez, Santiago F. and Mazzucchi, Gabriel and Mekhov, Igor B.},
journal = {Phys. Rev. A},
volume = {93},
issue = {6},
pages = {063632},
numpages = {22},
year = {2016},
month = {Jun},
publisher = {American Physical Society},
doi = {10.1103/PhysRevA.93.063632},
url = {http://link.aps.org/doi/10.1103/PhysRevA.93.063632}
}
@article{mazzucchi2016,
title = {Quantum measurement-induced dynamics of many-body ultracold
bosonic and fermionic systems in optical lattices},
@ -225,6 +253,15 @@
%% Other papers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@article{zurek2002,
title={Decoherence and the transition from quantum to classical-revisited},
author={Zurek, Wojciech H},
journal={Los Alamos Science},
volume={27},
pages={86--109},
year={2002},
publisher={LOS ALAMOS NATIONAL LABORATORY}
}
@article{walters2013,
title = {Ab initio derivation of Hubbard models for cold atoms in
optical lattices},
@ -652,9 +689,117 @@ doi = {10.1103/PhysRevA.87.043613},
year = {2008}
}
@article{vukics2007,
author = {Vukics, A. and Maschler, C. and Ritsch, H.},
journal = {New J. Phys},
volume = {9},
pages = {255},
year = {2007},
title={Microscopic physics of quantum self-organization of optical
lattices in cavities},
author={Vukics, Andr{\'a}s and Maschler, Christoph and Ritsch,
Helmut},
journal={New Journal of Physics},
volume={9},
number={8},
pages={255},
year={2007},
publisher={IOP Publishing}
}
@article{pichler2010,
title={Nonequilibrium dynamics of bosonic atoms in optical lattices:
Decoherence of many-body states due to spontaneous
emission},
author={Pichler, H and Daley, A J and Zoller, P},
journal={Phys. Rev. A},
volume={82},
number={6},
pages={063605},
year={2010},
publisher={APS}
}
@article{ates2012,
title={Dissipative binding of lattice bosons through distance-selective pair loss},
author={Ates, C and Olmos, B and Li, W and Lesanovsky, I},
journal={Phys. Rev. Lett.},
volume={109},
number={23},
pages={233003},
year={2012},
publisher={APS}
}
@article{everest2014,
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author={Everest, B and Hush, M R and Lesanovsky, I},
journal={Phys. Rev. B},
volume={90},
number={13},
pages={134306},
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}
@article{daley2014,
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journal={Adv. Phys.},
volume={63},
number={2},
pages={77--149},
year={2014},
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}
@article{syassen2008,
title={Strong dissipation inhibits losses and induces correlations
in cold molecular gases},
author={Syassen, Niels and Bauer, Dominik M and Lettner, Matthias
and Volz, Thomas and Dietze, Daniel and
Garcia-Ripoll, Juan J and Cirac, J Ignacio and
Rempe, Gerhard and D{\"u}rr, Stephan},
journal={Science},
volume={320},
number={5881},
pages={1329--1331},
year={2008},
publisher={American Association for the Advancement of Science}
}
@article{bernier2014,
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A slave-spin approach},
author={Bernier, Jean-S{\'e}bastien and Poletti, Dario and Kollath,
Corinna},
journal={Phys. Rev. B},
volume={90},
number={20},
pages={205125},
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}
@article{vidanovic2014,
title={Dissipation through localized loss in bosonic systems with
long-range interactions},
author={Vidanovi{\'c}, Ivana and Cocks, Daniel and Hofstetter,
Walter},
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volume={89},
number={5},
pages={053614},
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@article{kepesidis2012,
title={Bose-Hubbard model with localized particle losses},
author={Kepesidis, Kosmas V and Hartmann, Michael J},
journal={Phys. Rev. A},
volume={85},
number={6},
pages={063620},
year={2012},
publisher={APS}
}
@article{sarkar2014,
title={Light scattering and dissipative dynamics of many fermionic
atoms in an optical lattice},
author={Sarkar, Saubhik and Langer, Stephan and Schachenmayer,
Johannes and Daley, Andrew J},
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volume={90},
number={2},
pages={023618},
year={2014},
publisher={APS}
}

8
thesis.tex
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@ -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,index,chapter]{Classes/PhDThesisPSnPDF}
\documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF}
% ******************************************************************************
% ******************************* Class Options ********************************
@ -139,9 +139,9 @@
\include{Chapter2/chapter2}
\include{Chapter3/chapter3}
\include{Chapter4/chapter4}
%\include{Chapter5/chapter5}
%\include{Chapter6/chapter6}
%\include{Chapter7/chapter7}
\include{Chapter5/chapter5}
\include{Chapter6/chapter6}
\include{Chapter7/chapter7}