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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/}} \graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}}
\fi \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} \mynote{insert our paper citations here}
We consider $N$ two-level atoms in an optical lattice with $M$ 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 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 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 site which will exhibit the Mott insulator to superfluid quantum phase
transition). \mynote{extra fermion citations, Piazza? Look up Gabi's transition).
AF paper.}
\mynote{extra fermion citations, Piazza? Look up Gabi's AF paper.}
As we have seen in the previous section, an optical lattice can be As we have seen in the previous section, an optical lattice can be
formed with classical light beams that form standing waves. Depending 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!] \begin{figure}[htbp!]
\centering \centering
\includegraphics[width=1.0\textwidth]{LatticeDiagram} \includegraphics[width=\linewidth]{LatticeDiagram}
\caption[Experimental Setup]{Atoms (green) trapped in an optical \caption[Experimental Setup]{Atoms (green) trapped in an optical
lattice are illuminated by a coherent probe beam (red), $a_0$, lattice are illuminated by a coherent probe beam (red), $a_0$,
with a mode function $u_0(\b{r})$ which is at an angle $\theta_0$ 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 variable angles in our model as this lends itself to a simpler
geometrical representation. geometrical representation.
\subsection{Derivation of the Hamiltonian} \section{Derivation of the Hamiltonian}
\label{sec:derivation} \label{sec:derivation}
A general many-body Hamiltonian coupled to a quantized light field in 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. linearity of the dipoles we assumed.
Normally we will consider scattering of modes $a_l$ much weaker than 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 the field forming the lattice potential
proceed in the same way as when deriving the conventional Bose-Hubbard $V_\mathrm{cl}(\b{r})$. Therefore, we assume that the trapping is
Hamiltonian in the zero temperature limit. The field operators entirely due to the classical potential and the field operators
$\Psi(\b{r})$ can be expanded using localised Wannier functions of $\Psi(\b{r})$ can be expanded using localised Wannier functions of
$V_\mathrm{cl}(\b{r})$ and by keeping only the lowest vibrational $V_\mathrm{cl}(\b{r})$. By keeping only the lowest vibrational state
state we get we get
\begin{equation} \begin{equation}
\Psi(\b{r}) = \sum_i^M b_i w(\b{r} - \b{r}_i), \Psi(\b{r}) = \sum_i^M b_i w(\b{r} - \b{r}_i),
\end{equation} \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 the ultimate quantum regime of light-matter interaction that goes
beyond previous treatments. beyond previous treatments.
\subsection{Scattered light behaviour} \section{The Bose-Hubbard Hamiltonian}
\section{Scattered light behaviour}
\label{sec:a} \label{sec:a}
Having derived the full quantum light-matter Hamiltonian we will now 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 be dominated by photons scattering from the much larger coherent
probe. probe.
\subsection{Density and Phase Observables} \section{Density and Phase Observables}
\label{sec:B} \label{sec:B}
Light scatters due to its interactions with the dipole moment of the 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 varying the local oscillator phase, one can choose which conjugate
operator to measure. operator to measure.
\subsection{Electric Field Stength} \section{Electric Field Stength}
\label{sec:Efield} \label{sec:Efield}
The Electric field operator at position $\b{r}$ and at time $t$ is 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. n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
\end{equation} \end{equation}
Estimates of the scattering rate using real experimental parameters 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] \begin{table}[!htbp]
\centering \centering
\begin{tabular}{|c|c|c|} \begin{tabular}{l c c}
\hline \toprule
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
$\omega_a$ & \multicolumn{2}{|c|}{$2 \pi \cdot 384$ THz}\\ \hline $\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
$\Gamma$ & \multicolumn{2}{|c|}{$2 \pi \cdot 6.07$ MHz} \\ \hline $\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ \hline $\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
$I$ & $4250$ Wm$^{-2}$ & N/A \\ \hline $I$ & $4250$ Wm$^{-2}$ & N/A \\
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ \hline $\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
$N_K$ & 10$^5$ & 147 \\ \hline \hline $N_K$ & 10$^5$ & 147 \\ \midrule
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
\bottomrule
\end{tabular} \end{tabular}
\caption[Photon Scattering Rates]{ Rubidium data taken from \caption[Photon Scattering Rates]{Experimental parameters used in
Ref. \cite{steck}; Miyake \emph{et al.}: based on estimating the photon scattering rates.}
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 $s_\mathrm{tot} = 2.5$.}
\label{tab:photons} \label{tab:photons}
\end{table} \end{table}

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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 probe a variety of different atomic properties in-situ ranging from
density correlations to matter-field interference. 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 We have seen in section \ref{sec:B} that typically the dominant term
in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009, 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 since the most prominent features, such as Bragg diffraction peaks, do
not coincide at all with the classical diffraction pattern. 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 We have shown that scattering from atom number operators leads to a
purely quantum diffraction pattern which depends on the density 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 $R$, in the same way it was done for an unperturbed Bose-Hubbard model
\cite{ejima2011}. \cite{ejima2011}.
\subsection{Matter-field interference measurements} \section{Matter-field interference measurements}
We have shown in section \ref{sec:B} that certain optical arrangements We have shown in section \ref{sec:B} that certain optical arrangements
lead to a different type of light-matter interaction where coupling is 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 In the following chapters, we consider a different approach to quantum
measurement in open systems and instead of considering expectation measurement in open systems and instead of considering expectation
values we look at a single experimental run and the resulting dynamics values we look at a single experimental run and the resulting dynamics
due to measurement backaction. Previously we were mostly interested in due to measurement backaction. Previously, we were mostly interested
extracting information about the quantum state of the atoms from the in extracting information about the quantum state of the atoms from
scattered light. The flexibility in the measurement model was used to the scattered light. The flexibility in the measurement model was used
enable probing of as many different quantum properties of the to enable probing of as many different quantum properties of the
ultracold gas as possible. By focusing on measurement backaction we ultracold gas as possible. By focusing on measurement backaction we
instead investigate the effect of photodetections on the dynamics of instead investigate the effect of photodetections on the dynamics of
the many-body gas as well as the possible quantum states that we can 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 concepts to a bosonic quantum gas. We first introduce the concept of
quantum trajectories which represent a single continuous series of quantum trajectories which represent a single continuous series of
photon detections. We also present an alternative approach to open photon detections. We also present an alternative approach to open
systems in which the measurement outcomes are discarded as this will systems in which the measurement outcomes are discarded. This will be
be useful when trying to learn about dynamical features common to useful when trying to learn about dynamical features common to every
every trajectory. In this case we use the density matrix formalism trajectory. In this case we use the density matrix formalism which
which obeys the master equation. This approach is more common in obeys the master equation. This approach is more common in dissipative
dissipative systems and we will highlight the differences between systems and we will highlight the differences between these two
these two different types of open systems. We conclude this chapter different types of open systems. We conclude this chapter with a new
with a new concept that will be central to all subsequent concept that will be central to all subsequent discussions. In our
discussions. In our model measurement is global, it couples to model measurement is global, it couples to operators that correspond
operators that correspond to global properties of the quantum gas to global properties of the quantum gas rather than single-site
rather than single-site quantities. This enables the possibility of quantities. This enables the possibility of performing measurements
performing measurements that cannot distinguish certain sites from that cannot distinguish certain sites from each other. Due to a lack
each other. Due to a lack of ``which-way'' information this leads to of ``which-way'' information this leads to the creation of spatially
the creation of spatially nontrivial virtual lattices on top of the nontrivial virtual lattices on top of the physical lattice. This turns
physical lattice. This turns out to have significant consequences on out to have significant consequences on the dynamics of the system.
the dynamics of the system.
\section{Quantum Trajectories} \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 which will be useful as some of the results in the following chapters
are more general. Measurement always consists of at least two are more general. Measurement always consists of at least two
competing processes, two possible outcomes. If there is no competition competing processes, two possible outcomes. If there is no competition
and only one outcome is possible then our measurement is meaningless and only one outcome is possible then our probe is meaningless as it
as it does not reveal any information about the system. In its does not reveal any information about the system. In its simplest form
simplest form measurement consists of a series of events, such as the measurement consists of a series of detection events, such as the
detections of photons. Even though, on an intuitive level it seems 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 arrangement actually consists of two mutually exclusive outcomes. At
any point in time an event either happens or it does not, a photon is 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 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 atoms by measuring the number of photons they scatter. Each atom will
on average scatter a certain number of photons contributing to the on average scatter a certain number of photons contributing to the
detection rate we observe. Therefore, if we record multiple photons at detection rate we observe. Therefore, if we record multiple photons at
a high rate we learn that the illuminated region must contain many a high rate of arrival we learn that the illuminated region must
atoms. On the other hand, if there are few atoms to scatter the light contain many atoms. On the other hand, if there are few atoms to
we will observe few detection events which we interpret as a scatter the light we will observe few detection events which we
continuous series of non-detection events interspersed with the interpret as a continuous series of non-detection events interspersed
occasional detector click. This trajectory informs us that there are with the occasional detector click. This trajectory informs us that
much fewer atoms being illuminated than previously. there are much fewer atoms being illuminated than previously.
Basic quantum mechanics tells us that such measurements will in Basic quantum mechanics tells us that such measurements will in
general affect the quantum state in some way. Each event will cause a general affect the quantum state in some way. Each event will cause a
discontinuous quantum jump in the wavefunction of the system and it discontinuous quantum jump in the wavefunction of the system and it
will have a jump operator, $\c$, associated with it. The effect of an will have a jump operator, $\c$, associated with it. The effect of a
event on the quantum state is simply the result of applying this jump detection event on the quantum state is simply the result of applying
operator to the wavefunction, $| \psi (t) \rangle$, this jump operator to the wavefunction, $| \psi (t) \rangle$,
\begin{equation} \begin{equation}
\label{eq:jump} \label{eq:jump}
| \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle} | \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle}
{\sqrt{\langle \cd \c \rangle (t)}}, {\sqrt{\langle \cd \c \rangle (t)}},
\end{equation} \end{equation}
where the denominator is simply a normalising factor. The exact form where the denominator is simply a normalising factor
of the jump operator $\c$ will depend on the nature of the measurement \cite{MeasurementControl}. The exact form of the jump operator $\c$
we are considering. For example, if we consider measuring the photons will depend on the nature of the measurement we are considering. For
escaping from a leaky cavity then $\c = \sqrt{2 \kappa} \hat{a}$, example, if we consider measuring the photons escaping from a leaky
where $\kappa$ is the cavity decay rate and $\hat{a}$ is the cavity then $\c = \sqrt{2 \kappa} \hat{a}$, where $\kappa$ is the
annihilation operator of a photon in the cavity field. It is cavity decay rate and $\hat{a}$ is the annihilation operator of a
interesting to note that due to renormalisation the effect of a single photon in the cavity field. It is interesting to note that due to
quantum jump is independent of the magnitude of the operator $\c$ renormalisation the effect of a single quantum jump is independent of
itself. However, larger operators lead to more frequent events and the magnitude of the operator $\c$ itself. However, larger operators
thus more frequent applications of the jump operator. 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 The null measurement outcome will have an opposing effect to the
quantum jump, but it has to be treated differently as it does not quantum jump, but it has to be treated differently as it does not
occur at discrete time points like the detection events occur at discrete time points like the detection events themselves
themselves. Its effect is accounted for by a modification to the \cite{MeasurementControl}. Its effect is accounted for by a
isolated Hamiltonian, $\hat{H}_0$, time evolution in the form modification to the isolated Hamiltonian, $\hat{H}_0$, time evolution
in the form
\begin{equation} \begin{equation}
| \psi (t + \mathrm{d}t) \rangle = \left\{ \hat{1} - \mathrm{d}t | \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 \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} \end{equation}
where $\mathrm{d}N(t)$ is the stochastic increment to the number of 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 photodetections up to time $t$ which is equal to $1$ whenever a
quantum jump occurs and $0$ otherwise. Note that this equation has a quantum jump occurs and $0$ otherwise \cite{MeasurementControl}. Note
straightforward generalisation to multiple jump operators, but we do that this equation has a straightforward generalisation to multiple
not consider this possibility here at all. jump operators, but we do not consider this possibility here at all.
All trajectories that we calculate in the follwing chapters are All trajectories that we calculate in the follwing chapters are
described by the stochastic Schr\"{o}dinger equation in 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. R(t) < \langle \cd \c \rangle (t) \delta t.
\end{equation} \end{equation}
In practice, this is not the most efficient method for In practice, this is not the most efficient method for simulation
simulation. Instead we will use the following method. At an initial \cite{MeasurementControl}. Instead we will use the following
time $t = t_0$ a random number $R$ is generated. We then propagate the method. At an initial time $t = t_0$ a random number $R$ is
unnormalised wavefunction $| \tilde{\psi} (t) \rangle$ using the generated. We then propagate the unnormalised wavefunction
non-Hermitian evolution given by $| \tilde{\psi} (t) \rangle$ using the non-Hermitian evolution given
by
\begin{equation} \begin{equation}
\frac{\mathrm{d}}{\mathrm{d}t} | \tilde{\psi} (t) \rangle = -i \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 \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 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 null outcomes on the quantum state. Its effect is entirely encoded in
the non-Hermitian modification to the original Hamiltonian given by 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 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 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 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 distribution of outcomes over many trajectories will be entirely
determined by the initial state even though each individual trajectory determined by the initial state even though each individual trajectory
will be unique and conditioned on the exact detection times that 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 trajectories can have features that are not present after averaging
and this is why we focus our attention on single experimental runs and this is why we focus our attention on single experimental runs
rather than average behaviour. 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 directions corresponds to different measurements as we have seen in
Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have
to simultaneously consider all the possible directions in which light to simultaneously consider all the possible directions in which light
could scatter and thus include multiple jump operators reducing the could scatter and thus include multiple jump operators reducing our
our ability to control the system. ability to control the system.
The model we derived in Eq. \eqref{eq:fullH} is in fact already in a The model we derived in Eq. \eqref{eq:fullH} is in fact already in a
form ready for quantum trajectory simulations. The phenomologically form ready for quantum trajectory simulations. The phenomologically
included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the
non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$ 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 which is the jump operator we want for measurements of photons leaking
from the cavity. However, we will simplify the system by considering from the cavity. However, we will first simplify the system by
the regime where we can neglect the effect of the quantum potential considering the regime where we can neglect the effect of the quantum
that builds up in the cavity. Physically, this means that whilst light potential that builds up in the cavity. Physically, this means that
scatters due to its interaction with matter, the field that builds up whilst light scatters due to its interaction with matter, the field
due to the scattered photons collecting in the cavity has a negligible that builds up due to the scattered photons collecting in the cavity
effect on the atomic evolution compared to its own dynamics such as has a negligible effect on the atomic evolution compared to its own
inter-site tunnelling or on-site interactions. This can be achieved dynamics such as inter-site tunnelling or on-site interactions. This
when the cavity-probe detuning is smaller than the cavity decay rate, can be achieved when the cavity-probe detuning is smaller than the
$\Delta_p \ll \kappa$ \cite{caballero2015}. However, even though the cavity decay rate, $\Delta_p \ll \kappa$
cavity field has a negligible effect on the atoms, measurement \cite{caballero2015}. However, even though the cavity field has a
backaction will not as this effect is of a different nature. It is due negligible effect on the atoms, measurement backaction will not. This
to the wavefunction collapse due to the destruction of photons rather effect is of a different nature. It is due to the wavefunction
than an interaction between fields. Therefore, the final form of the collapse due to the destruction of photons rather than an interaction
Hamiltonian Eq. \eqref{eq:fullH} that we will be using in the between fields. Therefore, the final form of the Hamiltonian
following chapters is Eq. \eqref{eq:fullH} that we will be using in the following chapters
is
\begin{equation} \begin{equation}
\hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F} \hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F}
\end{equation} \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 quantum jumps are applied at times determined by the algorithm
described above and the jump operator is given by described above and the jump operator is given by
\begin{equation} \begin{equation}
\label{eq:jump} \label{eq:jumpop}
\c = \sqrt{2 \kappa} C \hat{F}. \c = \sqrt{2 \kappa} C \hat{F}.
\end{equation} \end{equation}
Importantly, we see that measurement introduces a new energy and time Importantly, we see that measurement introduces a new energy and time
scale $\gamma$ which competes with the two other standard scales scale $\gamma$ which competes with the two other standard scales
responsible for the uitary dynamics of the closed system, tunnelling, responsible for the unitary dynamics of the closed system, tunnelling,
$J$, and in-site interaction, $U$. If each atom scattered light $J$, and on-site interaction, $U$. If each atom scattered light
inependently a different jump operator $\c_i$ would be required for inependently a different jump operator $\c_i$ would be required for
each site projecting the atomic system into a state where long-range each site projecting the atomic system into a state where long-range
coherence is degraded. This is a typical scenario for spontaneous 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 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 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 nonlocal way and most importantly not only will it not degrade
long-range coherence, it will in fact lead to such long-range 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 behaviour of single trajectories. One possible approach that is very
common when dealing with open systems is to look at the unconditioned common when dealing with open systems is to look at the unconditioned
state which is obtained by averaging over the random measurement state which is obtained by averaging over the random measurement
results which condition the system. The unconditioned state is no results which condition the system \cite{MeasurementControl}. The
longer a pure state and thus must be described by a density matrix, unconditioned state is no longer a pure state and thus must be
described by a density matrix,
\begin{equation} \begin{equation}
\label{eq:rho} \label{eq:rho}
\hat{\rho} = \sum_i p_i | \psi_i \rangle \langle \psi_i |, \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 We will be using the master equation and the density operator
formalism in the context of measurement. However, the exact same formalism in the context of measurement. However, the exact same
methods are also applied to a different class of open systems, namely methods are also applied to a different class of open systems, namely
dissipative systems. A dissipative system is an open system that dissipative systems \cite{QuantumNoise}. A dissipative system is an
couples to an external bath in an uncontrolled way. The behaviour of open system that couples to an external bath in an uncontrolled
such a system is similar to a system subject measurement in which we way. The behaviour of such a system is similar to a system subject
ignore all measurement results. One can even think of this external under observation in which we ignore all the results. One can even
coupling as a measurement who's outcome record is not accessible and think of this external coupling as a measurement who's outcome record
thus must be represented as an average over all possible is not accessible and thus must be represented as an average over all
trajectories. However, there is a crucial difference between possible trajectories. However, there is a crucial difference between
measurement and dissipation. When we perform a measurement we use the measurement and dissipation. When we perform a measurement we use the
master equation to describe system evolution if we ignore the master equation to describe system evolution if we ignore the
measuremt outcomes, but at any time we can look at the detection times 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 that it includes the effect of any possible measurement
outcome. Therefore, it is useful when extracting features that are outcome. Therefore, it is useful when extracting features that are
common to many trajectories, regardless of the exact timing of the common to many trajectories, regardless of the exact timing of the
events. In this case, we do not want to impose any specific trajectory events. However, in this case we do not want to impose any specific
on the system as we are not interested in a specific experimental run, trajectory on the system as we are not interested in a specific
but we would still like to identify the set of possible outcomes and experimental run, but we would still like to identify the set of
their properties. Unfortunately, calculating the inverse of possible outcomes and their common properties. Unfortunately,
Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of calculating the inverse of Eq. \eqref{eq:rho} is not an easy task. In
a density matrix into pure states might not even be unique. However, fact, the decomposition of a density matrix into pure states might not
if a measurement leads to a projection, i.e.~the final state becomes even be unique. However, if a measurement leads to a projection,
confined to some subspace of the Hilbert space, then this will be i.e.~the final state becomes confined to some subspace of the Hilbert
visible in the final state of the density matrix. We will show this on space, then this will be visible in the final state of the density
an example of a qubit in the quantum state matrix. We will show this on an example of a qubit in the quantum
state
\begin{equation} \begin{equation}
\label{eq:qubit0} \label{eq:qubit0}
| \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle, | \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle,
\end{equation} \end{equation}
where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states
of the qubit and we consider performing a measurement on it in the of the qubit and we consider performing a measurement in the basis
basis $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome. $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome. The
The quantum state will have collapsed now to the $ | 0 \rangle$ with quantum state will have collapsed now to the state $ | 0 \rangle$ with
probability $| \alpha |^2$ and $| 1 \rangle$ with probability probability $| \alpha |^2$ and $| 1 \rangle$ with probability
$| \beta |^2$. The corresponding density matrix is given by $| \beta |^2$. The corresponding density matrix is given by
\begin{equation} \begin{equation}
@ -353,43 +361,50 @@ $| \beta |^2$. The corresponding density matrix is given by
0 & |\beta|^2 0 & |\beta|^2
\end{array} \right), \end{array} \right),
\end{equation} \end{equation}
which is now a mixed state. We note that there are no off-diagonal which is a mixed state as opposed to the initial state. We note that
terms as the system is not in a superposition between the two basis there are no off-diagonal terms as the system is not in a
states. Therefore, the diagonal terms represent classical superposition between the two basis states. Therefore, the diagonal
probabilities of the system being in either of the basis states. This terms represent classical probabilities of the system being in either
is in contrast to their original interpretation when the state was of the basis states. This is in contrast to their original
given by Eq. \eqref{eq:qubit0} when they represented the quantum interpretation when the state was given by Eq. \eqref{eq:qubit0} when
uncertainty in our knowledge of the state which would have manifested they could not be interpreted as in such a way. The initial state was
itself in the density matrix as non-zero off-diagonal terms. The in a quantum superposition and thus the state was indeterminate due to
significance of these values being classical probabilities is that now the quantum uncertainty in our knowledge of the state which would have
we know that the measurement has already happened and we know that the manifested itself in the density matrix as non-zero off-diagonal
state must be either $| 0 \rangle$ or $| 1 \rangle$. We just don't terms. The significance of these values being classical probabilities
know which one until we check the result of the measurement. 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 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 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 reached was obvious. However, the nature of the process that takes us
from the initial state to the final state with classical uncertainty from the initial state to the final state with classical uncertainty
does not matter. The key observation is that regardless of the 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 trajectory taken, if the final state is given by Eq. \eqref{eq:rho1}
will definitely know that our state is either in the state we will definitely know that our state is either in the state
$| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the $| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the
two basis states. Therefore, if we obtained this density matrix as a two basis states. Therefore, if we obtained this density matrix as a
result of applying the time evolution given by the master equation we result of applying the time evolution given by the master equation we
would be able to identify the final states of individual trajectories would be able to identify the final states of individual trajectories
even though we have no information about the 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 Here we have considered a very simple case of a Hilbert space with two
non-degenerate basis states. In the following chapters we will non-degenerate basis states. In the following chapters we will
generalise the above result to larger Hilbert spaces with multiple 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} \section{Global Measurement and ``Which-Way'' Information}
We have already mentioned that one of the key features of our model is 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 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 coherently into a single mode which we enhance and collect with a
cavity. If atoms at different lattice sites scatter light 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 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 information. When we were considering nondestructive measurements and
looking at expectation values, this had no consequence on our results looking at expectation values, this had no consequence on our results
as we were simply interested in probing the quantum correlations of a 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 effect of these measurements on the dynamics of the system. The effect
of measurement backaction will depend on the information that is of measurement backaction will depend on the information that is
encoded in the detected photon. If a scattered mode cannot 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 are unaffected by the backaction whilst their correlations with the
rest of the system will change as the result of the quantum jumps. 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 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} $\c = \sqrt{2 \kappa} C \hat{F}$ and we know from Eq. \eqref{eq:F}
that we have a large amount of flexibility in tuning $\hat{F}$ via the 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 geometry of the optical setup. A cavity aligned at a different angle
$\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the argument will correspond to a different measurement. We will consider the case
depends on geometry rather than the exact nature of the operator it when $\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the
straightforwardly generalises to other measurement operators, argument depends on geometry rather than the exact nature of the
including the case when $\hat{F} = \hat{B}$. The operator $\hat{D}$ is operator it straightforwardly generalises to other measurement
given by 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} \begin{equation}
\hat{D} = \sum_i J_{i,i} \n_i, \hat{D} = \sum_i J_{i,i} \n_i,
\end{equation} \end{equation}
where the coefficients $J_{i,i}$ are determined from where the coefficients $J_{i,i}$ are determined from
Eq. \eqref{eq:Jcoeff}. It is very simple to make the $J_{i,i}$ Eq. \eqref{eq:Jcoeff}. These coefficients represent the coupling
periodic as it strength between the atoms and the light modes and thus their spatial
variation can be easily tuned by the geometry of the optical
\begin{figure}[htbp!] fields. We are in particular interested in making these coefficients
\centering degenerate across a number of lattice sites as shown in
\includegraphics[width=1.0\textwidth]{1DModes} Figs. \ref{fig:1dmodes} and \ref{fig:2dmodes}. Note that they do not
\caption[1D Modes due to Measurement Backaction]{} have to be periodic, but it is much easier to make them so. This makes
\label{fig:cavity} all lattice sites with the same value of $J_{i,i}$ indistinguishable
\end{figure} 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!] \begin{figure}[htbp!]
\centering \centering
\includegraphics[width=1.0\textwidth]{2DModes} \includegraphics[width=1.0\textwidth]{2DModes}
\caption[2D Modes due to Measurement Backaction]{} \caption[2D Modes due to Measurement Backaction]{ More complex
\label{fig:cavity} 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} \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 @@
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@ -190,6 +204,20 @@
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publisher={APS} publisher={APS}
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@ -225,6 +253,15 @@
%% Other papers %% Other papers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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8
thesis.tex
View File

@ -1,7 +1,7 @@
% ******************************* PhD Thesis Template ************************** % ******************************* PhD Thesis Template **************************
% Please have a look at the README.md file for info on how to use the 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 ******************************** % ******************************* Class Options ********************************
@ -139,9 +139,9 @@
\include{Chapter2/chapter2} \include{Chapter2/chapter2}
\include{Chapter3/chapter3} \include{Chapter3/chapter3}
\include{Chapter4/chapter4} \include{Chapter4/chapter4}
%\include{Chapter5/chapter5} \include{Chapter5/chapter5}
%\include{Chapter6/chapter6} \include{Chapter6/chapter6}
%\include{Chapter7/chapter7} \include{Chapter7/chapter7}