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% ************************** Thesis Abstract ***************************** % ************************** Thesis Abstract *****************************
% Use `abstract' as an option in the document class to print only the titlepage and the abstract. % Use `abstract' as an option in the document class to print only the titlepage and the abstract.
\begin{abstract} \begin{abstract}
Trapping ultracold atoms in optical lattices enabled the study of Trapping ultracold atoms in optical lattices enabled the study of
strongly correlated phenomena in an environment that is far more strongly correlated phenomena in an environment that is far more

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@ -2,8 +2,33 @@
\begin{acknowledgements} \begin{acknowledgements}
First and foremost, I would like to thank my supervisor Dr. Igor
Mekhov who has been an excellent mentor throughout my time at
Oxford. It is primarily thanks to his brilliant insights and
professionalism that I was able to reach my full potential during my
doctoral studies. The work contained in this thesis would also not be
possible without the help of the other members of the group, Gabriel
Mazzucchi and Dr. Santiago Caballero-Benitez. Without our frequent
casual discussions in the Old Library office I would have still been
stuck on the third chapter. I would also like to acknowledge all
members of Prof. Dieter Jaksch's and Prof. Christopher Foot's groups
for various helpful discussions. I must also offer a special mention
for Edward Owen who provided much needed reality checks on some of my
wishful theoretical thinking. I would also like to express my
gratitude to EPSRC, St. Catherine's College, the ALP sub-department,
and the Institute of Physics for providing me with the financial means
to live and study in Oxford as well as attend several conferences in
the UK and abroad.
\mynote{Write my own acknowledgements} On a personal note, I would like to thank my parents who provided me
with all the skills necessary work towards any goals I set
myself. Needless to say, without them I would have never been able to
be where I am right now. I would like to thank all the new and old
friends that have kept me company for the last four years. The time
spent together provided a welcome respite from the sweat and toil of
my DPhil.
\vspace{2em}
\raggedleft{Oxford, September 2016}
\end{acknowledgements} \end{acknowledgements}

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@ -83,9 +83,9 @@ imprinted in the scattered light \cite{klinder2015, landig2016}.
There are three prominent directions in which the field of quantum There are three prominent directions in which the field of quantum
optics of quantum gases has progressed in. First, the use of quantised optics of quantum gases has progressed in. First, the use of quantised
light enables direct coupling to the quantum properties of the atoms light enables direct coupling to the quantum properties of the atoms
\cite{mekhov2007prl, mekhov2007pra, mekhov2012}. This allows us to \cite{mekhov2007prl, mekhov2007pra, mekhov2007NP, mekhov2012}. This
probe the many-body system in a nondestructive manner and under allows us to probe the many-body system in a nondestructive manner and
certain conditions even perform quantum non-demolition (QND) under certain conditions even perform quantum non-demolition (QND)
measurements. QND measurements were originally developed in the measurements. QND measurements were originally developed in the
context of quantum optics as a tool to measure a quantum system context of quantum optics as a tool to measure a quantum system
without significantly disturbing it \cite{braginsky1977, unruh1978, without significantly disturbing it \cite{braginsky1977, unruh1978,
@ -200,7 +200,7 @@ measurement backaction. Whilst such interference measurements have
been previously proposed for BECs in double-wells \cite{cirac1996, been previously proposed for BECs in double-wells \cite{cirac1996,
castin1997, ruostekoski1997}, the extension to a lattice system is castin1997, ruostekoski1997}, the extension to a lattice system is
not straightforward. However, we will show it is possible to achieve not straightforward. However, we will show it is possible to achieve
with our propsed setup by a careful optical arrangement. Within this with our proposed setup by a careful optical arrangement. Within this
context we demonstrate a novel type of projection which occurs even context we demonstrate a novel type of projection which occurs even
when there is significant competition with the Hamiltonian when there is significant competition with the Hamiltonian
dynamics. This projection is fundamentally different to the standard dynamics. This projection is fundamentally different to the standard
@ -215,5 +215,58 @@ Hamiltonian in a way that depends on the state of the atoms that
scatterd the light. This can lead to new quantum phases due to new scatterd the light. This can lead to new quantum phases due to new
types of long-range interactions being mediated by the global quantum types of long-range interactions being mediated by the global quantum
optical fields \cite{caballero2015, caballero2015njp, caballero2016, optical fields \cite{caballero2015, caballero2015njp, caballero2016,
caballero2016a}. However, this aspect of quantum optics of quantum caballero2016a, elliott2016}. However, this aspect of quantum optics
gases is beyond the scope of this thesis. of quantum gases is beyond the scope of this thesis.
\newpage
\section*{Publication List}
The work contained in this thesis is based on seven publications
\cite{kozlowski2015, elliott2015, atoms2015, mazzucchi2016,
kozlowski2016zeno, mazzucchi2016njp, kozlowski2016phase}:
\begin{table}[hbtp!]
\centering
\begin{tabular}{r p{13cm}}
\toprule
\cite{kozlowski2015} & W. Kozlowski, S. F. Caballero-Benitez, and
I. B. Mekhov. ``Probing matter-field and atom-number correlations
in optical lattices by global nondestructive addressing''.
\emph{Physical Review A}, 92(1):013613, 2015. \\ \\
\cite{elliott2015} & T. J. Elliott, W. Kozlowski,
S. F. Caballero-Benitez, and I. B. Mekhov. ``Multipartite
Entangled Spatial Modes of Ultracold Atoms Generated and
Controlled by Quantum Measurement''. \emph{Physical Review
Letters}, 114:113604, 2015. \\ \\
\cite{atoms2015} & T. J. Elliott, G. Mazzucchi, W. Kozlowski,
S. F. Caballero- Benitez, and I. B. Mekhov. ``Probing and
manipulating fermionic and bosonic quantum gases with quantum
light''. \emph{Atoms}, 3(3):392406, 2015. \\ \\
\cite{mazzucchi2016} & G. Mazzucchi$^*$, W. Kozlowski$^*$,
S. F. Caballero-Benitez, T. J. Elliott, and
I. B. Mekhov. ``Quantum measurement-induced dynamics of many- body
ultracold bosonic and fermionic systems in optical
lattices''. \emph{Physical Review A}, 93:023632,
2016. $^*$\emph{Equally contributing authors}. \\ \\
\cite{kozlowski2016zeno} & W. Kozlowski, S. F. Caballero-Benitez,
and I. B. Mekhov. ``Non- hermitian dynamics in the quantum zeno
limit''. \emph{Physical Review A}, 94:012123, 2016. \\ \\
\cite{mazzucchi2016njp} & G. Mazzucchi, W. Kozlowski,
S. F. Caballero-Benitez, and I. B Mekhov. ``Collective dynamics of
multimode bosonic systems induced by weak quan- tum
measurement''. \emph{New Journal of Physics}, 18(7):073017, 2016. \\ \\
\cite{kozlowski2016phase} & W. Kozlowski, S. F. Caballero-Benitez,
and I. B. Mekhov. ``Quantum state reduction by
matter-phase-related measurements in optical
lattices''. \emph{arXiv preprint arXiv:1605.06000}, 2016. \\
\bottomrule
\end{tabular}
\end{table}

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@ -9,20 +9,24 @@
\graphicspath{{Chapter2/Figs/Raster/}{Chapter2/Figs/PDF/}{Chapter2/Figs/}} \graphicspath{{Chapter2/Figs/Raster/}{Chapter2/Figs/PDF/}{Chapter2/Figs/}}
\else \else
\graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}} \graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}}
\fi \fi
\section{Introduction} \section{Introduction}
In this chapter, we derive a general Hamiltonian that describes the In this chapter, we present the derivation of a general Hamiltonian
coupling of atoms with far-detuned optical beams that describes the coupling of atoms with far-detuned optical beams
\cite{mekhov2012}. This will serve as the basis from which we explore originally presented in Ref. \cite{mekhov2012}. This will serve as the
the system in different parameter regimes, such as nondestructive basis from which we explore the system in different parameter regimes,
measurement in free space or quantum measurement backaction in a such as nondestructive measurement in free space or quantum
cavity. Another interesting direction for this field of research are measurement backaction in a cavity. As this model extends the
quantum optical lattices where the trapping potential is treated Bose-Hubbard Hamiltonian to include the effects of interactions with
quantum mechanically \cite{caballero2015, caballero2015njp, quantised light we also present a brief overview of the properties of
caballero2016, caballero2016a}. However this is beyond the scope of the Bose-Hubbard model itself and its quantum phase transiton. This is
this work. followed by a description of the behaviour of the scattered light with
a particular focus on how to couple the optical fields to phase
observables as opposed to density observables as is typically the
case. Finally, we conclude with an overview of possible experimental
realisability.
We consider $N$ two-level atoms in an optical lattice with $M$ We consider $N$ two-level atoms in an optical lattice with $M$
sites. For simplicity we will restrict our attention to spinless sites. For simplicity we will restrict our attention to spinless
@ -37,7 +41,7 @@ from free particles to strongly correlated systems, to the inherent
tunability of such lattices. Furthermore, this model is capable of tunability of such lattices. Furthermore, this model is capable of
describing a range of different experimental setups ranging from a describing a range of different experimental setups ranging from a
small number of sites with a large filling factor (e.g.~BECs trapped 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 in a double-well potential) to an extended multi-site lattice with a
low filling factor (e.g.~a system with one atom per site which will low filling factor (e.g.~a system with one atom per site which will
exhibit the Mott insulator to superfluid quantum phase transition). exhibit the Mott insulator to superfluid quantum phase transition).
@ -262,7 +266,7 @@ hopping rate given by
\begin{equation} \begin{equation}
J^\mathrm{cl}_{i,j} = \int \mathrm{d}^3 \b{r} w (\b{r} - \b{r}_i ) J^\mathrm{cl}_{i,j} = \int \mathrm{d}^3 \b{r} w (\b{r} - \b{r}_i )
\left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r} \left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r}
- \b{r}_i), - \b{r}_j),
\end{equation} \end{equation}
and $U_{ijkl}$ is the atomic interaction term given by and $U_{ijkl}$ is the atomic interaction term given by
\begin{equation} \begin{equation}
@ -731,7 +735,9 @@ 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.
\section{Density and Phase Observables} \section[Density and Phase Observables]
{Density and Phase Observables\footnote{The results of this
section were first published in Ref. \cite{kozlowski2015}}}
\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
@ -802,6 +808,18 @@ between the light modes and the nearest neighbour Wannier overlap,
$W_1(x)$. This can be achieved by concentrating the light between the $W_1(x)$. This can be achieved by concentrating the light between the
sites rather than at the positions of the atoms. sites rather than at the positions of the atoms.
\begin{figure}[hbtp!]
\centering
\includegraphics[width=0.8\linewidth]{BDiagram}
\caption[Maximising Light-Matter Coupling between Lattice
Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
between lattice sites. The thin black line indicates the trapping
potential (not to scale). (a) Arrangement for the uniform pattern
$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
$J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure}
In order to calculate the $J_{i,j}$ coefficients we perform numerical In order to calculate the $J_{i,j}$ coefficients we perform numerical
calculations using realistic Wannier functions calculations using realistic Wannier functions
\cite{walters2013}. However, it is possible to gain some analytic \cite{walters2013}. However, it is possible to gain some analytic
@ -835,35 +853,6 @@ $\hat{D}$ operator since it depends on the amplitude of light in
between the lattice sites and not at the positions of the atoms between the lattice sites and not at the positions of the atoms
allowing to decouple them at specific angles. allowing to decouple them at specific angles.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{BDiagram}
\caption[Maximising Light-Matter Coupling between Lattice
Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
between lattice sites. The thin black line indicates the trapping
potential (not to scale). (a) Arrangement for the uniform pattern
$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
$J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{WF_S}
\caption[Wannier Function Products]{The Wannier function products:
(a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
in diffraction maximum (c) and minimum (d) as are shown as
functions of standing wave shifts $\varphi$ or, if one were to
measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
oscillator phase $\beta$. The black points indicate the positions,
where light measures matter interference $\hat{B} \ne 0$, and the
density-term is suppressed, $\hat{D} = 0$. The trapping potential
depth is approximately 5 recoil energies.}
\label{fig:WannierProducts}
\end{figure}
The simplest case is to find a diffraction maximum where The simplest case is to find a diffraction maximum where
$J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a $J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
constant. This results in a diffraction maximum where each bond constant. This results in a diffraction maximum where each bond
@ -889,6 +878,23 @@ arrangement of light modes maximizes the interference signal,
$\hat{B}$, by suppressing the density signal, $\hat{D}$, via $\hat{B}$, by suppressing the density signal, $\hat{D}$, via
interference compensating for the spreading of the Wannier functions. interference compensating for the spreading of the Wannier functions.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{WF_S}
\caption[Wannier Function Products]{The Wannier function products:
(a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
in diffraction maximum (c) and minimum (d) as are shown as
functions of standing wave shifts $\varphi$ or, if one were to
measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
oscillator phase $\beta$. The black points indicate the positions,
where light measures matter interference $\hat{B} \ne 0$, and the
density-term is suppressed, $\hat{D} = 0$. The trapping potential
depth is approximately 5 recoil energies.}
\label{fig:WannierProducts}
\end{figure}
Another possibility is to obtain an alternating pattern similar Another possibility is to obtain an alternating pattern similar
corresponding to a diffraction minimum where each bond scatters light corresponding to a diffraction minimum where each bond scatters light
in anti-phase with its neighbours giving in anti-phase with its neighbours giving
@ -927,10 +933,13 @@ $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ 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.
\section{Electric Field Stength} \section[Electric Field Strength]
{Electric Field Strength\footnote{The derivation has not been
published before, but the final numerical results were
included in Ref. \cite{kozlowski2015}}}
\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
usually written in terms of its positive and negative components: usually written in terms of its positive and negative components:
\begin{equation} \begin{equation}
\b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t), \b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t),
@ -1095,6 +1104,25 @@ 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}
\begin{table}
\centering
\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]{Experimental parameters used in
estimating the photon scattering rates.}
\label{tab:photons}
\end{table}
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}. Rubidium atom data has been are given in Table \ref{tab:photons}. Rubidium atom data has been
taken from Ref. \cite{steck}. The two experiments were chosen as state taken from Ref. \cite{steck}. The two experiments were chosen as state
@ -1131,21 +1159,38 @@ $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz (1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz
per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$. per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
\begin{table} \section{Possible Experimental Issues}
\centering
\begin{tabular}{l c c} There are many possible experimental issues that we have neglected so
\toprule far in our theoretical treatment which need to be answered in order to
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule consider the experimental feasability of our proposal. The two main
$\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\ concerns are photodetector efficiency and heating losses.
$\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ In our theoretical models we treat the detectors as if they were
$I$ & $4250$ Wm$^{-2}$ & N/A \\ capable of detecting every scattered photon, but real photodetectors
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ can have efficiencies as low as 5\%. For the case of nondestructive
$N_K$ & 10$^5$ & 147 \\ \midrule measurement as covered in Chapter \ref{chap:qnd} this is not an issue
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ provided a sufficient number of photons can be collected to calculate
\bottomrule reliable expectation values. The case of scattering into a cavity and
\end{tabular} the effect of efficiency on the conditioned state was addressed in
\caption[Photon Scattering Rates]{Experimental parameters used in Ref. \cite{mazzucchi2016njp} where it was shown that detector
estimating the photon scattering rates.} efficiencies as low as 1\% are still capable of resolving the dynamics
\label{tab:photons} to a good degree of accuracy and 10\% was sufficient for near unit
\end{table} fidelity. However, this incredible result requires that the photon
scattering pattern is periodic in some way, e.g.~oscillatory as was
the case in Ref. \cite{mazzucchi2016njp} or constant. This way it is
only necessary to detect a sufficient number of photons to deduce the
correct phase of the oscillations or the rate for the case of a
constant scattering rate. In this thesis we deal predominantly with
these two cases so photodetector efficiency is not an issue.
The other issue is the heating of the trapped gas which will limit the
lifetime of the experiment. For free space scattering imaging times of
several hundred milliseconds have been achieved by for example using
molasses beams that simultaneously cool and trap the atoms
\cite{weitenberg2011, weitenbergThesis}. Similar feats have been
achieved with atoms coupled to a leaky cavity where interogation times
of 0.8s have been achieved in Ref. \cite{brennecke2013}. Crucially,
the cavity in said experiment has a decay rate of the order of MHz
which is necessary to observe measurement backaction which we will
consider in the subsequent chapters.

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@ -2,8 +2,10 @@
%*********************************** Third Chapter ***************************** %*********************************** Third Chapter *****************************
%******************************************************************************* %*******************************************************************************
\chapter{Probing Correlations by Global \chapter[Probing Correlations by Global
Nondestructive Addressing} %Title of the Third Chapter Nondestructive Addressing] {Probing Correlations by Global
Nondestructive Addressing\footnote{The results of this chapter were
first published in Ref. \cite{kozlowski2015}}}
\label{chap:qnd} \label{chap:qnd}
\ifpdf \ifpdf
@ -230,7 +232,7 @@ fluctuations entirely leading to absolutely no ``quantum addition''.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{Ep1} \includegraphics[width=0.8\linewidth]{Ep1}
\caption[Light Scattering Angular Distribution]{Light intensity \caption[Light Scattering Angular Distribution]{Light intensity
scattered into a standing wave mode from a superfluid in a 3D scattered into a standing wave mode from a superfluid in a 3D
lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming

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@ -421,7 +421,10 @@ generalise the above result to larger Hilbert spaces with multiple
degenerate subspaces which are of much greater interest as they reveal degenerate subspaces which are of much greater interest as they reveal
nontrivial dynamics in the system. nontrivial dynamics in the system.
\section{Global Measurement and ``Which-Way'' Information} \section[Global Measurement and ``Which-Way'' Information]
{Global Measurement and ``Which-Way'' Information\footnote{The
results of this section were first published in
Ref. \cite{elliott2015}}}
\label{sec:modes} \label{sec:modes}
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

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@ -2,7 +2,11 @@
%*********************************** Fifth Chapter ***************************** %*********************************** Fifth Chapter *****************************
%******************************************************************************* %*******************************************************************************
\chapter{Density Measurement Induced Dynamics} \chapter[Density Measurement Induced Dynamics]
{Density Measurement Induced Dynamics\footnote{The results of
this chapter were first published in
Refs. \cite{mazzucchi2016, kozlowski2016zeno,
mazzucchi2016njp}}}
% Title of the Fifth Chapter % Title of the Fifth Chapter
\ifpdf \ifpdf
@ -290,6 +294,8 @@ jump. Expanding around the peak of the Gaussian ansatz we get
Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump
operator above yields a Gaussian wavefunction as well, but the operator above yields a Gaussian wavefunction as well, but the
parameters change discontinuously according to parameters change discontinuously according to
\begingroup
\allowdisplaybreaks
\begin{align} \begin{align}
\label{eq:jumpb2} \label{eq:jumpb2}
b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\ b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
@ -299,6 +305,7 @@ parameters change discontinuously according to
z_\phi & \rightarrow z_\phi, \\ z_\phi & \rightarrow z_\phi, \\
\epsilon & \rightarrow \epsilon. \epsilon & \rightarrow \epsilon.
\end{align} \end{align}
\endgroup
The fact that the wavefunction remains Gaussian after a photodetection The fact that the wavefunction remains Gaussian after a photodetection
is a huge advantage, because it means that the combined time evolution is a huge advantage, because it means that the combined time evolution
of the system can be described with a single Gaussian ansatz in of the system can be described with a single Gaussian ansatz in
@ -527,15 +534,15 @@ yields
\end{equation} \end{equation}
where the approximation on the right-hand side follows from the fact where the approximation on the right-hand side follows from the fact
that $\omega \approx 2$ since we are considering the $N \gg 1$ limit, that $\omega \approx 2$ since we are considering the $N \gg 1$ limit,
and because we are considering the weak measurement limit and because we are considering the weak measurement limit $\Gamma^2 /
$\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width of the
of the initial superfluid state. This result is interesting, because initial superfluid state. This result is interesting, because it shows
it shows that the width of the Gaussian distribution is squeezed as that the width of the Gaussian distribution is squeezed as compared
compared with its initial state which is exactly what we see in with its initial state which is exactly what we see in
Fig. \ref{fig:oscillations}(a). However, if we substitute the Fig. \ref{fig:oscillations}(a). However, if we substitute the
parameter values used in that trajectory we only get a reduction in parameter values used in that trajectory we only get a reduction in
width by about $3\%$, but the maximum amplitude oscillations in look width by about $3\%$, but the maximum amplitude oscillations look like
like they have a significantly smaller width than the initial they have a significantly smaller width than the initial
distribution. This discrepancy is due to the fact that the continuous distribution. This discrepancy is due to the fact that the continuous
variable approximation is only valid for $z \ll 1$ and thus it cannot variable approximation is only valid for $z \ll 1$ and thus it cannot
explain the final behaviour of the system. Furthermore, it has been explain the final behaviour of the system. Furthermore, it has been
@ -1648,14 +1655,12 @@ eigenstate of $\c$ and we combine this with the requirement for it to
be in the dark state of the tunnelling operator (eigenstate of $\H_0$ be in the dark state of the tunnelling operator (eigenstate of $\H_0$
for $U = 0$) to derive the steady state. These two conditions in for $U = 0$) to derive the steady state. These two conditions in
momentum space are momentum space are
\begin{equation} \begin{align}
\hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k - \hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k -
\bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \\
\end{equation}
\begin{equation}
\Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} + \Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} +
\bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle, \bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle,
\end{equation} \end{align}
where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$, where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$,
$\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice $\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice
spacing, $N$ the total atom number, and we perform summations over the spacing, $N$ the total atom number, and we perform summations over the
@ -1738,10 +1743,11 @@ discussed.
To obtain a state with a specific value of $\Delta N$ postselection To obtain a state with a specific value of $\Delta N$ postselection
may be necessary, but otherwise it is not needed. The process can be may be necessary, but otherwise it is not needed. The process can be
optimised by feedback control since the state is monitored at all optimised by feedback control since the state is monitored at all
times \cite{ivanov2014}. Furthermore, the form of the measurement times \cite{ivanov2014, mazzucchi2016feedback}. Furthermore, the form
operator is very flexible and it can easily be engineered by the of the measurement operator is very flexible and it can easily be
geometry of the optical setup \cite{elliott2015, mazzucchi2016} which engineered by the geometry of the optical setup \cite{elliott2015,
can be used to design a state with desired properties. mazzucchi2016} which can be used to design a state with desired
properties.
\section{Conclusions} \section{Conclusions}

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@ -2,7 +2,10 @@
%*********************************** Sixth Chapter ***************************** %*********************************** Sixth Chapter *****************************
%******************************************************************************* %*******************************************************************************
\chapter{Phase Measurement Induced Dynamics} \chapter[Phase Measurement Induced Dynamics]
{Phase Measurement Induced Dynamics\footnote{The results of
this chapter were first published in
Ref. \cite{kozlowski2016phase}}}
% Title of the Sixth Chapter % Title of the Sixth Chapter
\ifpdf \ifpdf
@ -25,7 +28,7 @@ operators just like most of the existing work \cite{LP2009, rogers2014,
section \ref{sec:B} that it is possible to couple to the the relative section \ref{sec:B} that it is possible to couple to the the relative
phase differences between sites in an optical lattice by illuminating phase differences between sites in an optical lattice by illuminating
the bonds between them. Furthermore, we have also shown how it can be the bonds between them. Furthermore, we have also shown how it can be
applied to probe the Bose Hubbard order parameter or even matter-field applied to probe the Bose-Hubbard order parameter or even matter-field
quadratures in Chapter \ref{chap:qnd}. This concept has also been quadratures in Chapter \ref{chap:qnd}. This concept has also been
applied to the study of quantum optical potentials formed in a cavity applied to the study of quantum optical potentials formed in a cavity
and shown to lead to a host of interesting quantum phase diagrams and shown to lead to a host of interesting quantum phase diagrams

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@ -2,7 +2,11 @@
\begin{dedication} \begin{dedication}
\mynote{Write my own dedication.} \emph{Moim rodzicom, bez których nie byłbym w stanie osiągnąć tego
wszystkiego.}
\emph{To my parents without whom I would not have been able to achieve
any of this.}
\end{dedication} \end{dedication}

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@ -221,3 +221,5 @@
\newcommand{\Bmax}{\hat{B}_\mathrm{max}} \newcommand{\Bmax}{\hat{B}_\mathrm{max}}
\newcommand{\Bmin}{\hat{B}_\mathrm{min}} \newcommand{\Bmin}{\hat{B}_\mathrm{min}}
\newcommand{\D}{\hat{D}} \newcommand{\D}{\hat{D}}
\usepackage[utf8]{inputenc}

File diff suppressed because it is too large Load Diff

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@ -29,7 +29,7 @@ Dynamics in Ultracold Bosonic Gases}
%% Supervisor (optional) %% Supervisor (optional)
%\supervisor{Prof. Kenichi Soga} \supervisor{Dr. Igor B. Mekhov}
%% Supervisor Role (optional) - Supervisor (default) or advisor %% Supervisor Role (optional) - Supervisor (default) or advisor
%\supervisorrole{Advisor: } %\supervisorrole{Advisor: }

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