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6
Abstract/abstract.tex
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@ -11,9 +11,9 @@
what is possible in ultracold atomic systems. what is possible in ultracold atomic systems.
We show that light can serve as a nondestructive probe of the We show that light can serve as a nondestructive probe of the
quantum state of the matter. By condering a global quantum state of matter. By condering a global measurement we show
measurement scheme we show that it is possible to distinguish a that it is possible to distinguish a highly delocalised phase like a
highly delocalised phase like a superfluid from insulators. We also superfluid from the Bose glass and Mott insulator. We also
demonstrate that light scattering reveals not only density demonstrate that light scattering reveals not only density
correlations, but also matter-field interference. correlations, but also matter-field interference.

26
Acknowledgement/acknowledgement.tex
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@ -6,19 +6,19 @@ First and foremost, I would like to thank my supervisor Dr. Igor
Mekhov who has been an excellent mentor throughout my time at Mekhov who has been an excellent mentor throughout my time at
Oxford. It is primarily thanks to his brilliant insights and Oxford. It is primarily thanks to his brilliant insights and
professionalism that I was able to reach my full potential during my professionalism that I was able to reach my full potential during my
doctoral studies. The work contained in this thesis would also not be doctoral studies. The work contained in this thesis would also not
possible without the help of the other members of the group, Gabriel have been possible without the help of the other members of the group,
Mazzucchi and Dr. Santiago Caballero-Benitez. Without our frequent Gabriel Mazzucchi and Dr. Santiago Caballero-Benitez. Without our
casual discussions in the Old Library office I would have still been frequent casual discussions in the Old Library office I would have
stuck on the third chapter. I would also like to acknowledge all still been stuck on the third chapter. I would also like to
members of Prof. Dieter Jaksch's and Prof. Christopher Foot's groups acknowledge all members of Prof. Dieter Jaksch's and Prof. Christopher
for various helpful discussions. I must also offer a special mention Foot's groups for various helpful discussions. I must also offer a
for Edward Owen who provided much needed reality checks on some of my special mention for Edward Owen who provided much needed reality
wishful theoretical thinking. I would also like to express my checks on some of my wishful theoretical thinking. I would also like
gratitude to EPSRC, St. Catherine's College, the ALP sub-department, to express my gratitude to EPSRC, St. Catherine's College, the ALP
and the Institute of Physics for providing me with the financial means sub-department, and the Institute of Physics for providing me with the
to live and study in Oxford as well as attend several conferences in financial means to live and study in Oxford as well as attend several
the UK and abroad. conferences in the UK and abroad.
On a personal note, I would like to thank my parents who provided me 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 with all the skills necessary work towards any goals I set

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Chapter1/chapter1.tex
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@ -83,8 +83,8 @@ 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, mekhov2007NP, LP2009, \cite{mekhov2012, mekhov2007prl, mekhov2007pra, mekhov2007NP,
mekhov2012}. This allows us to probe the many-body system in a LP2009}. This allows us to probe the many-body system in a
nondestructive manner and under certain conditions even perform nondestructive manner and under certain conditions even perform
quantum non-demolition (QND) measurements. QND measurements were quantum non-demolition (QND) measurements. QND measurements were
originally developed in the context of quantum optics as a tool to originally developed in the context of quantum optics as a tool to
@ -145,9 +145,8 @@ the first condenste was obtained that theoretical work on the effects
of measurement on BECs appeared \cite{cirac1996, castin1997, of measurement on BECs appeared \cite{cirac1996, castin1997,
ruostekoski1997}. Recently, work has also begun on combining weak ruostekoski1997}. Recently, work has also begun on combining weak
measurement with the strongly correlated dynamics of ultracold gases measurement with the strongly correlated dynamics of ultracold gases
in optical lattices \cite{mekhov2009prl, mekhov2009pra, LP2010, in optical lattices \cite{mekhov2012, mekhov2009prl, mekhov2009pra,
mekhov2012, douglas2012, LP2013, douglas2013, ashida2015, LP2010, douglas2012, LP2013, douglas2013, ashida2015, ashida2015a}.
ashida2015a}.
In this thesis we focus on the latter by considering a quantum gas in In this thesis we focus on the latter by considering a quantum gas in
an optical lattice coupled to a cavity \cite{mekhov2012}. This an optical lattice coupled to a cavity \cite{mekhov2012}. This
@ -244,8 +243,8 @@ The work contained in this thesis is based on seven publications
\cite{atoms2015} & T. J. Elliott, G. Mazzucchi, W. Kozlowski, \cite{atoms2015} & T. J. Elliott, G. Mazzucchi, W. Kozlowski,
S. F. Caballero- Benitez, and I. B. Mekhov. ``Probing and S. F. Caballero- Benitez, and I. B. Mekhov. ``Probing and
manipulating fermionic and bosonic quantum gases with quantum Manipulating Fermionic and Bosonic Quantum Gases with Quantum
light''. \emph{Atoms}, 3(3):392–406, 2015. \\ \\ Light''. \emph{Atoms}, 3(3):392–406, 2015. \\ \\
\cite{mazzucchi2016} & G. Mazzucchi$^*$, W. Kozlowski$^*$, \cite{mazzucchi2016} & G. Mazzucchi$^*$, W. Kozlowski$^*$,
S. F. Caballero-Benitez, T. J. Elliott, and S. F. Caballero-Benitez, T. J. Elliott, and
@ -265,9 +264,9 @@ The work contained in this thesis is based on seven publications
2016. \\ \\ 2016. \\ \\
\cite{kozlowski2016phase} & W. Kozlowski, S. F. Caballero-Benitez, \cite{kozlowski2016phase} & W. Kozlowski, S. F. Caballero-Benitez,
and I. B. Mekhov. ``Quantum state reduction by and I. B. Mekhov. ``Quantum State Reduction by
matter-phase-related measurements in optical Matter-Phase-Related Measurements in Optical
lattices''. \emph{arXiv preprint arXiv:1605.06000}, 2016. \\ Lattices''. \emph{arXiv preprint arXiv:1605.06000}, 2016. \\
\bottomrule \bottomrule
\end{tabular} \end{tabular}

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Chapter2/chapter2.tex
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@ -338,7 +338,7 @@ are entirely determined by the values of the $J^{l,m}_{i,j}$
coefficients and despite its simplicity, this is sufficient to give coefficients and despite its simplicity, this is sufficient to give
rise to a host of interesting phenomena via measurement backaction rise to a host of interesting phenomena via measurement backaction
such as the generation of multipartite entangled spatial modes in an such as the generation of multipartite entangled spatial modes in an
optical lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the optical lattice \cite{mekhov2009pra, elliott2015, atoms2015}, the
appearance of long-range correlated tunnelling capable of entangling appearance of long-range correlated tunnelling capable of entangling
distant lattice sites, and in the case of fermions, the break-up and distant lattice sites, and in the case of fermions, the break-up and
protection of strongly interacting pairs \cite{mazzucchi2016, protection of strongly interacting pairs \cite{mazzucchi2016,
@ -767,8 +767,8 @@ our global scattering scheme.
In our model light couples to the operator $\hat{F}$ which consists of In our model light couples to the operator $\hat{F}$ which consists of
a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase
component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i
b_j$. In general, the density component dominates, b_j$. In general, the density component dominates, $\hat{D} \gg
$\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$ \hat{B}$, and thus $\hat{F} \approx \hat{D}$
\cite{mekhov2012}. Physically, this is a consequence of the fact that \cite{mekhov2012}. Physically, this is a consequence of the fact that
there are more atoms to scatter light at the lattice sites than in there are more atoms to scatter light at the lattice sites than in
between them. However, it is possible to engineer an optical geometry between them. However, it is possible to engineer an optical geometry
@ -780,10 +780,10 @@ light scattered from different sources. Furthermore, it is not limited
to a double-well setup and naturally extends to a lattice structure to a double-well setup and naturally extends to a lattice structure
which is a key advantage. Such a counter-intuitive configuration may which is a key advantage. Such a counter-intuitive configuration may
affect works on quantum gases trapped in quantum potentials affect works on quantum gases trapped in quantum potentials
\cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013, \cite{mekhov2012, caballero2015, mekhov2008, larson2008, chen2009,
ivanov2014, caballero2015} and quantum measurement-induced habibian2013, ivanov2014} and quantum measurement-induced
preparation of many-body atomic states \cite{mazzucchi2016, preparation of many-body atomic states \cite{mekhov2009prl,
mekhov2009prl, pedersen2014, elliott2015}. elliott2015, mazzucchi2016, pedersen2014}.
For clarity we will consider a 1D lattice as shown in For clarity we will consider a 1D lattice as shown in
Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the
@ -1187,7 +1187,8 @@ lifetime of the experiment. For free space scattering appropriate
conditions have been achieved by for example using molasses beams that conditions have been achieved by for example using molasses beams that
simultaneously cool and trap the atoms \cite{weitenberg2011, simultaneously cool and trap the atoms \cite{weitenberg2011,
weitenbergThesis}. Similar feats have been achieved with atoms weitenbergThesis}. Similar feats have been achieved with atoms
coupled to a leaky cavity in Ref. \cite{brennecke2013}. Crucially, the coupled to a leaky cavity in Ref. \cite{brennecke2013} where
cavity in said experiment has a decay rate of the order of MHz which self-organisation effects were well observable. Crucially, the cavity
is necessary to observe measurement backaction which we will consider in said experiment has a decay rate of the order of MHz which is
in the subsequent chapters. necessary to observe measurement backaction which we will consider in
the subsequent chapters.

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Chapter3/chapter3.tex
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@ -39,7 +39,7 @@ the quantum state of the ultracold gas we can have access to not only
density correlations, but also matter-field interference at its density correlations, but also matter-field interference at its
shortest possible distance in an optical lattice, i.e.~the lattice shortest possible distance in an optical lattice, i.e.~the lattice
period. Previous work on quantum non-demolition (QND) schemes period. Previous work on quantum non-demolition (QND) schemes
\cite{rogers2014, mekhov2007prl, eckert2008} probe only the density \cite{mekhov2007prl, rogers2014, eckert2008} probe only the density
component as it is generally challenging to couple to the matter-field component as it is generally challenging to couple to the matter-field
observables directly. Here, we will consider nondestructive probing of observables directly. Here, we will consider nondestructive probing of
both density and interference operators. both density and interference operators.
@ -121,21 +121,21 @@ density correlations to matter-field interference.
\subsection{Diffraction Patterns and Bragg Conditions} \subsection{Diffraction Patterns and Bragg Conditions}
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{mekhov2007pra,
mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is LP2009, rist2010, lakomy2009, ruostekoski2009}. This is simply due
simply due to the fact that atoms are localised with lattice sites to the fact that atoms are localised with lattice sites leading to an
leading to an effective coupling with atom number operators instead of effective coupling with atom number operators instead of inter-site
inter-site interference terms. Therefore, we will first consider interference terms. Therefore, we will first consider nondestructive
nondestructive probing of the density related observables of the probing of the density related observables of the quantum
quantum gas. However, we will focus on the novel nontrivial aspects gas. However, we will focus on the novel nontrivial aspects that go
that go beyond the work in Ref. \cite{mekhov2012, mekhov2007prl, beyond the work in Ref. \cite{mekhov2012, mekhov2007prl,
mekhov2007pra} which only considered a few extremal cases. mekhov2007pra} which only considered a few extremal cases.
As we are only interested in the quantum information imprinted in the As we are only interested in the quantum information imprinted in the
state of the optical field we will simplify our analysis by state of the optical field we will simplify our analysis by
considering the light scattering to be much faster than the atomic considering the light scattering to be much faster than the atomic
tunnelling. Therefore, our scheme is actually a QND scheme tunnelling. Therefore, our scheme is actually a QND scheme
\cite{rogers2014, mekhov2007prl, mekhov2007pra, eckert2008} as \cite{mekhov2007prl, mekhov2007pra, rogers2014, eckert2008} as
normally density-related measurements destroy the matter-phase normally density-related measurements destroy the matter-phase
coherence since it is its conjugate variable, but here we neglect the coherence since it is its conjugate variable, but here we neglect the
$\bd_i b_j$ terms. Furthermore, we will consider a deep $\bd_i b_j$ terms. Furthermore, we will consider a deep
@ -837,7 +837,7 @@ probing a quantum gas trapped in an optical lattice using quantised
light. Firstly, we showed that the density term in scattering has an light. Firstly, we showed that the density term in scattering has an
angular distribution richer than classical diffraction, derived angular distribution richer than classical diffraction, derived
generalized Bragg conditions, and estimated parameters for two generalized Bragg conditions, and estimated parameters for two
relevant experiments \cite{weitenberg2011, miyake2011}. Secondly, we relevant experiments \cite{miyake2011, weitenberg2011}. Secondly, we
demonstrated how the method accesses effects beyond mean-field and demonstrated how the method accesses effects beyond mean-field and
distinguishes all the phases in the Mott-superfluid-glass transition, distinguishes all the phases in the Mott-superfluid-glass transition,
which is currently a challenge \cite{derrico2014}. Finally, we looked which is currently a challenge \cite{derrico2014}. Finally, we looked

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Chapter5/chapter5.tex
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@ -44,7 +44,7 @@ correlations which enable nonlocal dynamical processes. Furthermore,
global light scattering from multiple lattice sites creates nontrivial global light scattering from multiple lattice sites creates nontrivial
spatially nonlocal coupling to the environment, as seen in section spatially nonlocal coupling to the environment, as seen in section
\ref{sec:modes}, which is impossible to obtain with local interactions \ref{sec:modes}, which is impossible to obtain with local interactions
\cite{daley2014, diehl2008, syassen2008}. These spatial modes of \cite{diehl2008, syassen2008, daley2014}. These spatial modes of
matter fields can be considered as designed systems and reservoirs matter fields can be considered as designed systems and reservoirs
opening the possibility of controlling dissipations in ultracold opening the possibility of controlling dissipations in ultracold
atomic systems without resorting to atom losses and collisions which atomic systems without resorting to atom losses and collisions which
@ -748,7 +748,7 @@ from the effects of interactions as they would prevent this dynamics
by dephasing different components of the coherent excitations. Strong by dephasing different components of the coherent excitations. Strong
measurement, on the other hand, squeezes the quantum state by trying measurement, on the other hand, squeezes the quantum state by trying
to project it onto an eigenstate of the observable to project it onto an eigenstate of the observable
\cite{mekhov2009prl, mekhov2009prl}. For weak interactions where the \cite{mekhov2009prl, mekhov2009pra}. For weak interactions where the
ground state is a highly delocalised superfluid it is obvious that ground state is a highly delocalised superfluid it is obvious that
projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will supress projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will supress
fluctuations significantly. However, the strongly interacting regime fluctuations significantly. However, the strongly interacting regime
@ -1744,7 +1744,7 @@ 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, mazzucchi2016feedback, times \cite{ivanov2014, mazzucchi2016feedback,
ivanonv2016}. Furthermore, the form of the measurement operator is ivanov2016}. Furthermore, the form of the measurement operator is
very flexible and it can easily be engineered by the geometry of the very flexible and it can easily be engineered by the geometry of the
optical setup \cite{elliott2015, mazzucchi2016} which can be used to optical setup \cite{elliott2015, mazzucchi2016} which can be used to
design a state with desired properties. design a state with desired properties.

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Chapter6/chapter6.tex
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@ -23,16 +23,16 @@ with atomic density, not the matter-wave amplitude. Therefore, it is
challenging to couple light to the phase of the matter-field, as is challenging to couple light to the phase of the matter-field, as is
typical in quantum optics for optical fields. In the previous chapter typical in quantum optics for optical fields. In the previous chapter
we only considered measurement that couples directly to atomic density we only considered measurement that couples directly to atomic density
operators just like most of the existing work \cite{LP2009, rogers2014, operators just like most of the existing work \cite{mekhov2012,
mekhov2012, ashida2015, ashida2015a}. However, we have shown in LP2009, rogers2014, ashida2015, ashida2015a}. However, we have shown
section \ref{sec:B} that it is possible to couple to the the relative in section \ref{sec:B} that it is possible to couple to the the
phase differences between sites in an optical lattice by illuminating relative phase differences between sites in an optical lattice by
the bonds between them. Furthermore, we have also shown how it can be illuminating the bonds between them. Furthermore, we have also shown
applied to probe the Bose-Hubbard order parameter or even matter-field how it can be applied to probe the Bose-Hubbard order parameter or
quadratures in Chapter \ref{chap:qnd}. This concept has also been even matter-field quadratures in Chapter \ref{chap:qnd}. This concept
applied to the study of quantum optical potentials formed in a cavity has also been applied to the study of quantum optical potentials
and shown to lead to a host of interesting quantum phase diagrams formed in a cavity and shown to lead to a host of interesting quantum
\cite{caballero2015, caballero2015njp, caballero2016, phase diagrams \cite{caballero2015, caballero2015njp, caballero2016,
caballero2016a}. This is a multi-site generalisation of previous caballero2016a}. This is a multi-site generalisation of previous
double-well schemes \cite{cirac1996, castin1997, ruostekoski1997, double-well schemes \cite{cirac1996, castin1997, ruostekoski1997,
ruostekoski1998, rist2012}, although the physical mechanism is ruostekoski1998, rist2012}, although the physical mechanism is
@ -137,25 +137,24 @@ is given by
p(B_\mathrm{max}, m, t) = \frac{B_\mathrm{max}^{2m}} {F(t)} \exp\left[ - 2 p(B_\mathrm{max}, m, t) = \frac{B_\mathrm{max}^{2m}} {F(t)} \exp\left[ - 2
\gamma B_\mathrm{max}^2 t \right] p_0 (B_\mathrm{max}), \gamma B_\mathrm{max}^2 t \right] p_0 (B_\mathrm{max}),
\end{equation} \end{equation}
where where $p_0(B_\mathrm{max}) = \sum_{J_\mathrm{max} h_l = J
$p_0(B_\mathrm{max}) = \sum_{J_\mathrm{max} h_l = J B_\mathrm{max}} B_\mathrm{max}} |z_l^0|^2$. This distribution will have two distinct
|z_l^0|^2$. This distribution will have two distinct peaks at peaks at $B_\mathrm{max} = \pm \sqrt{m/2\kappa |C|^2 t}$ and an
$B_\mathrm{max} = \pm \sqrt{m/2\kappa |C|^2 t}$ and an initially broad initially broad distribution will narrow down around these two peaks
distribution will narrow down around these two peaks with successive with successive photocounts. The final state is in a superposition,
photocounts. The final state is in a superposition, because we measure because we measure the photon number, $\ad_1 \a_1$ and not field
the photon number, $\ad_1 \a_1$ and not field amplitude. Therefore, amplitude. Therefore, the measurement is insensitive to the phase of
the measurement is insensitive to the phase of $\a_1 = C \B$ and we $\a_1 = C \B$ and we get a superposition of $\pm B_\mathrm{max}$. This
get a superposition of $\pm B_\mathrm{max}$. This is exactly the same is exactly the same situation that we saw for the macroscopic
situation that we saw for the macroscopic oscillations of two distinct oscillations of two distinct components when the atom number
components when the atom number difference between two modes is difference between two modes is measured as seen in
measured as seen in Fig. \ref{fig:oscillations}(b). However, this Fig. \ref{fig:oscillations}(b). However, this means that the matter is
means that the matter is still entangled with the light as the two still entangled with the light as the two states scatter light with
states scatter light with different phase which the photocount different phase which the photocount detector cannot
detector cannot distinguish. Fortunately, this is easily mitigated at distinguish. Fortunately, this is easily mitigated at the end of the
the end of the experiment by switching off the probe beam and allowing experiment by switching off the probe beam and allowing the cavity to
the cavity to empty out or by measuring the light phase (quadrature) empty out or by measuring the light phase (quadrature) to isolate one
to isolate one of the components \cite{mekhov2009pra, mekhov2012, of the components \cite{mekhov2012, mekhov2009pra, atoms2015}.
atoms2015}.
Unusually, we do not have to worry about the timing of the quantum Unusually, we do not have to worry about the timing of the quantum
jumps, because the measurement operator commutes with the jumps, because the measurement operator commutes with the

418
References/references.bib
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