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Abstract/abstract.tex
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@ -4,7 +4,7 @@
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
controllable and tunable than what was possible in condensed controllable and tunable than what was possible in condensed
matter. Here, we consider coupling these systems to quantized light matter. Here, we consider coupling these systems to quantised light
where the quantum nature of both the optical and matter fields play where the quantum nature of both the optical and matter fields play
equally important roles in order to push the boundaries of equally important roles in order to push the boundaries of
what is possible in ultracold atomic systems. what is possible in ultracold atomic systems.
@ -30,7 +30,7 @@
measurement of matter-phase-related variables such as global phase measurement of matter-phase-related variables such as global phase
coherence. We show how this unconventional approach opens up new coherence. We show how this unconventional approach opens up new
opportunities to affect system evolution and demonstrate how this opportunities to affect system evolution and demonstrate how this
can lead to a new class of measurement projections, thus extending can lead to a new class of measurement projections thus extending
the measurement postulate for the case of strong competition with the measurement postulate for the case of strong competition with
the system’s own evolution. the system’s own evolution.
\end{abstract} \end{abstract}

158
Chapter1/chapter1.tex
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@ -11,8 +11,8 @@
\fi \fi
The field of ultracold gases has been a rapidly growing field ever The field of ultracold gases has been rapidly growing ever since the
since the first Bose-Einstein condensate (BEC) was obtained in 1995 first Bose-Einstein condensate (BEC) was obtained in 1995
\cite{anderson1995, bradley1995, davis1995}. This new quantum state of \cite{anderson1995, bradley1995, davis1995}. This new quantum state of
matter is characterised by a macroscopic occupancy of the single matter is characterised by a macroscopic occupancy of the single
particle ground state at which point the whole system behaves like a particle ground state at which point the whole system behaves like a
@ -24,7 +24,7 @@ meant that the degree of control and isolation from the environment
was far greater than was possible in condensed matter systems was far greater than was possible in condensed matter systems
\cite{lewenstein2007, bloch2008}. Initially, the main focus of the \cite{lewenstein2007, bloch2008}. Initially, the main focus of the
research was on the properties of coherent matter waves, such as research was on the properties of coherent matter waves, such as
interference properties \cite{andrews1997}, long range phase coherence interference properties \cite{andrews1997}, long-range phase coherence
\cite{bloch2000}, or quantised vortices \cite{matthews1999, \cite{bloch2000}, or quantised vortices \cite{matthews1999,
madison2000, abo2001}. Fermi degeneracy in ultracold gases was madison2000, abo2001}. Fermi degeneracy in ultracold gases was
obtained shortly afterwards opening a similar field for fermions obtained shortly afterwards opening a similar field for fermions
@ -52,38 +52,38 @@ optical lattice is dominated by atomic interactions opening the
possibility of studying strongly correlated behaviour with possibility of studying strongly correlated behaviour with
unprecendented control. unprecendented control.
The modern field of ultracold gases is successful due to its The modern field of strongly correlated ultracold gases is successful
interdisciplinarity \cite{lewenstein2007, bloch2008}. Originally due to its interdisciplinarity \cite{lewenstein2007,
condensed matter effects are now mimicked in controlled atomic systems bloch2008}. Originally condensed matter effects are now mimicked in
finding applications in areas such as quantum information controlled atomic systems finding applications in areas such as
processing. A really new challenge is to identify novel phenomena quantum information processing. A really new challenge is to identify
which were unreasonable to consider in condensed matter, but will novel phenomena which were unreasonable to consider in condensed
become feasible in new systems. One such direction is merging quantum matter, but will become feasible in new systems. One such direction is
optics and many-body physics \cite{mekhov2012, ritsch2013}. Quantum merging quantum optics and many-body physics \cite{mekhov2012,
optics has been developping as a branch of quantum physics ritsch2013}. Quantum optics has been developping as a branch of
independently of the progress in the many-body community. It describes quantum physics independently of the progress in the many-body
delicate effects such as quantum measurement, state engineering, and community. It describes delicate effects such as quantum measurement,
systems that can generally be easily isolated from their environnment state engineering, and systems that can generally be easily isolated
due to the non-interacting nature of photons \cite{Scully}. However, from their environnment due to the non-interacting nature of photons
they are also the perfect candidate for studying open systems due the \cite{Scully}. However, they are also the perfect candidate for
advanced state of cavity technologies \cite{carmichael, studying open systems due the advanced state of cavity technologies
MeasurementControl}. On the other hand ultracold gases are now used \cite{carmichael, MeasurementControl}. On the other hand ultracold
to study strongly correlated behaviour of complex macroscopic gases are now used to study strongly correlated behaviour of complex
ensembles where decoherence is not so easy to avoid or control. Recent macroscopic ensembles where decoherence is not so easy to avoid or
experimental progress in combining the two fields offered a very control. Recent experimental progress in combining the two fields
promising candidate for taking many-body physics in a direction that offered a very promising candidate for taking many-body physics in a
would not be possible for condensed matter \cite{baumann2010, direction that would not be possible for condensed matter
wolke2012, schmidt2014}. Two very recent breakthrough experiments \cite{baumann2010, wolke2012, schmidt2014}. Furthermore, two very
have even managed to couple an ultracold gas trapped in an optical recent breakthrough experiments have even managed to couple an
lattice to an optical cavity enabling the study of strongly correlated ultracold gas trapped in an optical lattice to an optical cavity
systems coupled to quantized light fields where the quantum properties enabling the study of strongly correlated systems coupled to quantised
of the atoms become imprinted in the scattered light light fields where the quantum properties of the atoms become
\cite{klinder2015, landig2016}. 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, mekhov2007prl, mekhov2012}. This allows us to \cite{mekhov2007prl, mekhov2007pra, mekhov2012}. This allows us to
probe the many-body system in a nondestructive manner and under probe the many-body system in a nondestructive manner and under
certain conditions even perform quantum non-demolition (QND) certain conditions even perform quantum non-demolition (QND)
measurements. QND measurements were originally developed in the measurements. QND measurements were originally developed in the
@ -92,56 +92,57 @@ without significantly disturbing it \cite{braginsky1977, unruh1978,
brune1990, brune1992}. This has naturally been extended into the brune1990, brune1992}. This has naturally been extended into the
realm of ultracold gases where such non-demolition schemes have been realm of ultracold gases where such non-demolition schemes have been
applied to both fermionic \cite{eckert2008qnd, roscilde2009} and applied to both fermionic \cite{eckert2008qnd, roscilde2009} and
bosonic \cite{hauke2013, rogers2014}. In this thesis, we consider bosonic systems \cite{hauke2013, rogers2014}. In this thesis, we
light scattering in free space from a bosonic ultracold gas and show consider light scattering in free space from a bosonic ultracold gas
that there are many prominent features that go beyond classical and show that there are many prominent features that go beyond
optics. Even the scattering angular distribution is nontrivial with classical optics. Even the scattering angular distribution is
Bragg conditions that are significantly different from the classical nontrivial with Bragg conditions that are significantly different from
case. Furthermore, we show that the direct coupling of quantised light the classical case. Furthermore, we show that the direct coupling of
to the atomic systems enables the nondestructive probing beyond a quantised light to the atomic systems enables the nondestructive
standard mean-field description. We demonstrate this by showing that probing beyond a standard mean-field description. We demonstrate this
the whole phase diagram of a disordered one-dimensional Bose-Hubbard by showing that the whole phase diagram of a disordered
Hamiltonian, which consists of the superfluid, Mott insulating, and one-dimensional Bose-Hubbard Hamiltonian, which consists of the
Bose glass phases, can be mapped from the properties of the scattered superfluid, Mott insulating, and Bose glass phases, can be mapped from
light. Additionally, we go beyond standard QND approaches, which only the properties of the scattered light. Additionally, we go beyond
consider coupling to density observables, by also considering the standard QND approaches, which only consider coupling to density
direct coupling of the quantised light to the interference between observables, by also considering the direct coupling of the quantised
neighbouring lattice sites. We show that not only is this possible to light to the interference between neighbouring lattice sites. We show
achieve in a nondestructive manner, it is also achieved without the that not only is this possible to achieve in a nondestructive manner,
need for single-site resolution. This is in contrast to the standard it is also achieved without the need for single-site resolution. This
destructive time-of-flight measurements currently used to perform is in contrast to the standard destructive time-of-flight measurements
these measurements \cite{miyake2011}. Within a mean-field treatment currently used to perform these measurements \cite{miyake2011}. Within
this enables probing of the order parameter as well as matter-field a mean-field treatment this enables probing of the order parameter as
quadratures and their squeezing. This can have an impact on atom-wave well as matter-field quadratures and their squeezing. This can have an
metrology and information processing in areas where quantum optics has impact on atom-wave metrology and information processing in areas
already made progress, e.g.,~quantum imaging with pixellized sources where quantum optics has already made progress, e.g.,~quantum imaging
of non-classical light \cite{golubev2010, kolobov1999}, as an optical with pixellized sources of non-classical light \cite{golubev2010,
lattice is a natural source of multimode nonclassical matter waves. kolobov1999}, as an optical lattice is a natural source of multimode
nonclassical matter waves.
Second, coupling a quantum gas to a cavity also enables us to study Second, coupling a quantum gas to a cavity also enables us to study
open system many-body dynamics either via dissipation where we have no open system many-body dynamics either via dissipation where we have no
control over the coupling to the environment or via controlled state control over the coupling to the environment or via controlled state
reduction using the measurement backaction due to photodetections. A reduction using the measurement backaction due to
lot of effort was expanded in an attempt to minimise the influence of photodetections. Initially, a lot of effort was exended in an attempt
the environment in order to extend decoherence times. However, to minimise the influence of the environment in order to extend
theoretical progress in the field has shown that instead being an decoherence times. However, theoretical progress in the field has
obstacle, dissipation can actually be used as a tool in engineering shown that instead being an obstacle, dissipation can actually be used
quantum states \cite{diehl2008}. Furthermore, as the environment as a tool in engineering quantum states \cite{diehl2008}. Furthermore,
coupling is varied the system may exhibit sudden changes in the as the environment coupling is varied the system may exhibit sudden
properties of its steady state giving rise to dissipative phase changes in the properties of its steady state giving rise to
transitions \cite{carmichael1980, werner2005, capriotti2005, dissipative phase transitions \cite{carmichael1980, werner2005,
morrison2008, eisert2010, bhaseen2012, diehl2010, capriotti2005, morrison2008, eisert2010, bhaseen2012, diehl2010,
vznidarivc2011}. An alternative approach to open systems is to look vznidarivc2011}. An alternative approach to open systems is to look
at quantum measurement where we consider a quantum state conditioned at quantum measurement where we consider a quantum state conditioned
on the outcome of a single experimental run \cite{carmichael, on the outcome of a single experimental run \cite{carmichael,
MeasurementControl}. In this approach we consider the solutions to a MeasurementControl}. In this approach we consider the solutions to a
stochastic Schr\"{o}dinger equation which will be a pure state, which stochastic Schr\"{o}dinger equation which will be a pure state, which
in contrast to dissipative systems is generally not the case. The in contrast to dissipative systems where this is generally not the
question of measurement and its effect on the quantum state has been case. The question of measurement and its effect on the quantum state
around since the inception of quantum theory and still remains a has been around since the inception of quantum theory and still
largely open question \cite{zurek2002}. It wasn't long after the first remains a largely open question \cite{zurek2002}. It wasn't long after
condenste was obtained that theoretical work on the effects of the first condenste was obtained that theoretical work on the effects
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, mekhov2012, in optical lattices \cite{mekhov2009prl, mekhov2009pra, mekhov2012,
@ -183,11 +184,12 @@ correlations and entanglement. Furthermore, we show that this
behaviour can be approximated by a non-Hermitian Hamiltonian thus behaviour can be approximated by a non-Hermitian Hamiltonian thus
extending the notion of quantum Zeno dynamics into the realm of extending the notion of quantum Zeno dynamics into the realm of
non-Hermitian quantum mechanics joining the two non-Hermitian quantum mechanics joining the two
paradigms. Non-Hermitian systems themself exhibit a range of paradigms. Non-Hermitian systems themselves exhibit a range of
interesting phenomena ranging from localisation \cite{hatano1996, interesting phenomena ranging from localisation \cite{hatano1996,
refael2006} and $\mathcal{PT}$ symmetry \cite{bender1998, refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$
giorgi2010, zhang2013} to spatial order \cite{otterbach2014} and symmetry} \cite{bender1998, giorgi2010, zhang2013} to spatial order
novel phase transitions \cite{lee2014prx, lee2014prl}. \cite{otterbach2014} and novel phase transitions \cite{lee2014prx,
lee2014prl}.
Just like for the nondestructive measurements we also consider Just like for the nondestructive measurements we also consider
measurement backaction due to coupling to the interference terms measurement backaction due to coupling to the interference terms
@ -197,8 +199,8 @@ density, this allows to enter a new regime of quantum control using
measurement backaction. Whilst such interference measurements have 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, but we will show it is possible to achieve with not straightforward. However, we will show it is possible to achieve
our propsed setup by a careful optical arrangement. Within this with our propsed 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

383
Chapter2/chapter2.tex
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@ -32,14 +32,30 @@ which yields its own set of interesting quantum phenomena
particles. The theory can be also be generalised to continuous particles. The theory can be also be generalised to continuous
systems, but the restriction to optical lattices is convenient for a systems, but the restriction to optical lattices is convenient for a
variety of reasons. Firstly, it allows us to precisely describe a variety of reasons. Firstly, it allows us to precisely describe a
many-body atomic state over a broad range of parameter values due to many-body atomic state over a broad range of parameter values due,
the inherent tunability of such lattices. Furthermore, this model is from free particles to strongly correlated systems, to the inherent
capable of describing a range of different experimental setups ranging tunability of such lattices. Furthermore, this model is capable of
from a small number of sites with a large filling factor (e.g.~BECs describing a range of different experimental setups ranging from a
trapped in a double-well potential) to a an extended multi-site small number of sites with a large filling factor (e.g.~BECs trapped
lattice with a low filling factor (e.g.~a system with one atom per in a double-well potential) to a an extended multi-site lattice with a
site which will exhibit the Mott insulator to superfluid quantum phase low filling factor (e.g.~a system with one atom per site which will
transition). exhibit the Mott insulator to superfluid quantum phase transition).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{LatticeDiagram}
\caption[Experimental Setup in Free Space]{Atoms (green) trapped in
an optical lattice are illuminated by a coherent probe beam (red),
$a_0$, with a mode function $u_0(\b{r})$ which is at an angle
$\theta_0$ to the normal to the lattice. The light scatters (blue)
into the mode $\a_1$ in free space or into a cavity and is
measured by a detector. Its mode function is given by $u_1(\b{r})$
and it is at an angle $\theta_1$ relative to the normal to the
lattice. If the experiment is in free space light can scatter in
any direction. A cavity on the other hand enhances scattering in
one particular direction.}
\label{fig:LatticeDiagram}
\end{figure}
An optical lattice can be formed with classical light beams that form An optical lattice can be formed with classical light beams that form
standing waves. Depending on the detuning with respect to the atomic standing waves. Depending on the detuning with respect to the atomic
@ -50,49 +66,34 @@ trapped bosons (green) are illuminated with a coherent probe beam
measured with a detector. The most straightforward measurement is to measured with a detector. The most straightforward measurement is to
simply count the number of photons with a photodetector, but it is simply count the number of photons with a photodetector, but it is
also possible to perform a quadrature measurement by using a homodyne also possible to perform a quadrature measurement by using a homodyne
detection scheme. The experiment can be performed in free space where detection scheme \cite{carmichael, atoms2015}. The experiment can be
light can scatter in any direction. The atoms can also be placed performed in free space where light can scatter in any direction. The
inside a cavity which has the advantage of being able to enhance light atoms can also be placed inside a cavity which has the advantage of
scattering in a particular direction. Furthermore, cavities allow for being able to enhance light scattering in a particular direction
the formation of a fully quantum potential in contrast to the \cite{bux2013, kessler2014, landig2015}. Furthermore, cavities allow
for the formation of a fully quantum potential in contrast to the
classical lattice trap. classical lattice trap.
\begin{figure}[htbp!] For simplicity we will be considering one-dimensional lattices most of
\centering the time. However, the model itself is derived for any number of
\includegraphics[width=\linewidth]{LatticeDiagram}
\caption[Experimental Setup]{Atoms (green) trapped in an optical
lattice are illuminated by a coherent probe beam (red), $a_0$,
with a mode function $u_0(\b{r})$ which is at an angle $\theta_0$
to the normal to the lattice. The light scatters (blue) into the
mode $\a_1$ in free space or into a cavity and is measured by a
detector. Its mode function is given by $u_1(\b{r})$ and it is at
an angle $\theta_1$ relative to the normal to the lattice. If the
experiment is in free space light can scatter in any direction. A
cavity on the other hand enhances scattering in one particular
direction.}
\label{fig:LatticeDiagram}
\end{figure}
For simplicity, we will be considering one-dimensional lattices most
of the time. However, the model itself is derived for any number of
dimensions and since none of our arguments will ever rely on dimensions and since none of our arguments will ever rely on
dimensionality our results straightforwardly generalise to 2- and 3-D dimensionality our results straightforwardly generalise to two- and
systems. This simplification allows us to present a much more three-dimensional systems. This simplification allows us to present a
intuitive picture of the physical setup where we only need to concern much more intuitive picture of the physical setup where we only need
ourselves with a single angle for each optical mode. As shown in to concern ourselves with a single angle for each optical mode. As
Fig. \ref{fig:LatticeDiagram} the angle between the normal to the shown in Fig. \ref{fig:LatticeDiagram} the angle between the normal to
lattice and the probe and detected beam are denoted by $\theta_0$ and the lattice and the probe and detected beam are denoted by $\theta_0$
$\theta_1$ respectively. We will consider these angles to be tunable and $\theta_1$ respectively. We will consider these angles to be
although the same effect can be achieved by varying the wavelength of tunable although the same effect can be achieved by varying the
the light modes. However, it is much more intuitive to consider wavelength of the light modes. However, it is much more intuitive to
variable angles in our model as this lends itself to a simpler consider variable angles in our model as this lends itself to a
geometrical representation. simpler geometrical representation.
\section{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 quantised light field in
second quantized can be separated into three parts, second quantised can be separated into three parts,
\begin{equation} \begin{equation}
\label{eq:FullH} \label{eq:FullH}
\H = \H_f + \H_a + \H_{fa}. \H = \H_f + \H_a + \H_{fa}.
@ -149,7 +150,7 @@ atomic raising, lowering and population difference operators, where
$|g \rangle$ and $| e \rangle$ denote the ground and excited states of $|g \rangle$ and $| e \rangle$ denote the ground and excited states of
the two-level atom respectively. $g_l$ are the atom-light coupling the two-level atom respectively. $g_l$ are the atom-light coupling
constants for each mode. It is the inclusion of the interaction of the constants for each mode. It is the inclusion of the interaction of the
boson with quantized light that distinguishes our work from the boson with quantised light that distinguishes our work from the
typical approach to ultracold atoms where all the optical fields, typical approach to ultracold atoms where all the optical fields,
including the trapping potentials, are treated classically. including the trapping potentials, are treated classically.
@ -243,7 +244,7 @@ $\H_{1,fa} = \H_{1,fa}^\mathrm{eff}$ given by Eq. \eqref{eq:aeff} and
\eqref{eq:faeff} respectively yields the following generalised \eqref{eq:faeff} respectively yields the following generalised
Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$, Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$,
\begin{equation} \begin{equation}
\H = \H_f + \sum_{i,j}^M J^\mathrm{cl}_{i,j} \bd_i b_j + \H = \H_f + \sum_{m,n}^M J^\mathrm{cl}_{m,n} \bd_m b_n +
\sum_{i,j,k,l}^M \frac{U_{ijkl}}{2} \bd_i \bd_j b_k b_l + \sum_{i,j,k,l}^M \frac{U_{ijkl}}{2} \bd_i \bd_j b_k b_l +
\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m
\left( \sum_{i,j}^K J^{l,m}_{i,j} \bd_i b_j \right). \left( \sum_{i,j}^K J^{l,m}_{i,j} \bd_i b_j \right).
@ -276,14 +277,13 @@ the most significant overlap. Thus, for $J_{i,j}^\mathrm{cl}$ we will
only consider $i$ and $j$ that correspond to nearest neighbours. only consider $i$ and $j$ that correspond to nearest neighbours.
Furthermore, since we will only be looking at lattices that have the Furthermore, since we will only be looking at lattices that have the
same separtion between all its nearest neighbours (e.g. cubic or same separtion between all its nearest neighbours (e.g. cubic or
square lattice) we can define $J_{i,j}^\mathrm{cl} = - J^\mathrm{cl}$ square lattice) we can define $J_{i,j}^\mathrm{cl} = - J$ (negative
(negative sign, because this way $J^\mathrm{cl} > 0$). For the sign, because this way $J > 0$). For the inter-atomic interactions
inter-atomic interactions this simplifies to simply considering this simplifies to simply considering on-site collisions where
on-site collisions where $i=j=k=l$ and we define $U_{iiii} = $i=j=k=l$ and we define $U_{iiii} = U$. Finally, we end up with the
U$. Finally, we end up with the canonical form for the Bose-Hubbard canonical form for the Bose-Hubbard Hamiltonian
Hamiltonian
\begin{equation} \begin{equation}
\H_a = -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + \H_a = -J \sum_{\langle i,j \rangle}^M \bd_i b_j +
\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1), \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1),
\end{equation} \end{equation}
where $\langle i,j \rangle$ denotes a summation over nearest where $\langle i,j \rangle$ denotes a summation over nearest
@ -322,39 +322,41 @@ nearest-neighbour tunnelling operators
\end{equation} \end{equation}
where $K$ denotes a sum over the illuminated sites and we neglect where $K$ denotes a sum over the illuminated sites and we neglect
couplings beyond nearest neighbours for the same reason as before when couplings beyond nearest neighbours for the same reason as before when
deriving the matter Hamiltonian. deriving the matter Hamiltonian. Thus the interaction part of the
Hamiltonian is given by
\begin{equation}
\H_{fa} = \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m}
\end{equation}
\mynote{make sure all group papers are cited here} These equations These equations encapsulate the simplicity and flexibility of the
encapsulate the simplicity and flexibility of the measurement scheme measurement scheme that we are proposing. The operators given above
that we are proposing. The operators given above are entirely are entirely determined by the values of the $J^{l,m}_{i,j}$
determined by the values of the $J^{l,m}_{i,j}$ coefficients and coefficients and despite its simplicity, this is sufficient to give
despite its simplicity, this is sufficient to give rise to a host of rise to a host of interesting phenomena via measurement backaction
interesting phenomena via measurement back-action such as the such as the generation of multipartite entangled spatial modes in an
generation of multipartite entangled spatial modes in an optical optical lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the
lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the appearance appearance of long-range correlated tunnelling capable of entangling
of long-range correlated tunnelling capable of entangling distant distant lattice sites, and in the case of fermions, the break-up and
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,
kozlowski2016zeno}. Additionally, these coefficients are easy to kozlowski2016zeno}. Additionally, these coefficients are easy to
manipulate experimentally by adjusting the optical geometry via the manipulate experimentally by adjusting the optical geometry via the
light mode functions $u_l(\b{r})$. light mode functions $u_l(\b{r})$.
It is important to note that we are considering a situation where the It is important to note that we are considering a situation where the
contribution of quantized light is much weaker than that of the contribution of quantised light is much weaker than that of the
classical trapping potential. If that was not the case, it would be classical trapping potential. If that was not the case, it would be
necessary to determine the Wannier functions in a self-consistent way necessary to determine the Wannier functions in a self-consistent way
which takes into account the depth of the quantum poterntial generated which takes into account the depth of the quantum poterntial generated
by the quantized light modes. This significantly complicates the by the quantised light modes. This significantly complicates the
treatment, but can lead to interesting physics. Amongst other things, treatment, but can lead to interesting physics. Amongst other things,
the atomic tunnelling and interaction coefficients will now depend on the atomic tunnelling and interaction coefficients will now depend on
the quantum state of light. \mynote{cite Santiago's papers and the quantum state of light \cite{mekhov2008}.
Maschler/Igor EPJD}
Therefore, combining these final simplifications we finally arrive at Therefore, combining these final simplifications we finally arrive at
our quantum light-matter Hamiltonian our quantum light-matter Hamiltonian
\begin{equation} \begin{equation}
\label{eq:fullH} \label{eq:fullH}
\H = \H_f -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + \H = \H_f -J \sum_{\langle i,j \rangle}^M \bd_i b_j +
\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) + \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) +
\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} - \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} -
i \sum_l \kappa_l \ad_l \a_l, i \sum_l \kappa_l \ad_l \a_l,
@ -387,20 +389,21 @@ approximation of the Coulomb screening interaction as a simple on-site
interaction. Despite these enormous simplifications the model was very interaction. Despite these enormous simplifications the model was very
succesful \cite{leggett}. The Bose-Hubbard model is an even simpler succesful \cite{leggett}. The Bose-Hubbard model is an even simpler
variation where instead of fermions we consider spinless bosons. It variation where instead of fermions we consider spinless bosons. It
was originally devised as a toy model, but in 1998 it was shown by was originally devised as a toy model and applied to liquid helium
Jaksch \emph{et.~al.} that it can be realised with ultracold atoms in \cite{fisher1989}, but in 1998 it was shown by Jaksch \emph{et.~al.}
an optical lattice \cite{jaksch1998}. Shortly afterwards it was that it can be realised with ultracold atoms in an optical lattice
obtained in a ground-breaking experiment \cite{greiner2002}. The model \cite{jaksch1998}. Shortly afterwards it was obtained in a
has been the subject of intense research since then, because despite ground-breaking experiment \cite{greiner2002}. The model has been the
its simplicity it possesses highly nontrivial properties such as the subject of intense research since then, because despite its simplicity
superfluid to Mott insulator quantum phase transition. Furthermore, it it possesses highly nontrivial properties such as the superfluid to
is one of the most controllable quantum many-body systems thus Mott insulator quantum phase transition. Furthermore, it is one of the
providing a solid basis for new experiments and technologies. most controllable quantum many-body systems thus providing a solid
basis for new experiments and technologies.
The model we have derived is essentially an extension of the The model we have derived is essentially an extension of the
well-known Bose-Hubbard model that also includes interactions with well-known Bose-Hubbard model that also includes interactions with
quantised light. Therefore, it should come as no surprise that if we quantised light. Therefore, it should come as no surprise that if we
eliminate all the quantized fields from the Hamiltonian we obtain eliminate all the quantised fields from the Hamiltonian we obtain
exactly the Bose-Hubbard model exactly the Bose-Hubbard model
\begin{equation} \begin{equation}
\H_a = -J \sum_{\langle i,j \rangle}^M \bd_i b_j + \H_a = -J \sum_{\langle i,j \rangle}^M \bd_i b_j +
@ -433,7 +436,7 @@ momentum space defined by the annihilation operator
\begin{equation} \begin{equation}
b_\b{k} = \frac{1} {\sqrt{M}} \sum_m b_m e^{i \b{k} \cdot \b{r}_m}, b_\b{k} = \frac{1} {\sqrt{M}} \sum_m b_m e^{i \b{k} \cdot \b{r}_m},
\end{equation} \end{equation}
where $\b{k}$ denotes the wavevector running over the first Brillouin where $\b{k}$ denotes the wave vector running over the first Brillouin
zone. The Hamiltonian then is given by zone. The Hamiltonian then is given by
\begin{equation} \begin{equation}
\H_a = \sum_\b{k} \epsilon_\b{k} \bd_\b{k} b_\b{k}, \H_a = \sum_\b{k} \epsilon_\b{k} \bd_\b{k} b_\b{k},
@ -496,7 +499,7 @@ it will not exhibit a quantum phase transition.
Now that we have a basic understanding of the two limiting cases we Now that we have a basic understanding of the two limiting cases we
can now consider the model in between these two extremes. We have can now consider the model in between these two extremes. We have
already mentioned that an exact solution is not known, but fotunately already mentioned that an exact solution is not known, but fortunately
a very good mean-field approximation exists \cite{fisher1989}. In this a very good mean-field approximation exists \cite{fisher1989}. In this
approach the interaction term is treated exactly, but the kinetic approach the interaction term is treated exactly, but the kinetic
energy term is decoupled as energy term is decoupled as
@ -512,7 +515,7 @@ neighbouring sites and is the mean -field order parameter. This
decoupling effect means that we can write the Hamiltonian as $\hat{H}_a decoupling effect means that we can write the Hamiltonian as $\hat{H}_a
= \sum_m^M \hat{h}_a$, where = \sum_m^M \hat{h}_a$, where
\begin{equation} \begin{equation}
\hat{h}_a = -z J ( \Phi \bd \Phi^* b) + z J | \Phi |^2 + \frac{U}{2} \hat{h}_a = -z J ( \Phi \bd + \Phi^* b) + z J | \Phi |^2 + \frac{U}{2}
\n (\n - 1) - \mu \n, \n (\n - 1) - \mu \n,
\end{equation} \end{equation}
and $z$ is the coordination number, i.e. the number of nearest and $z$ is the coordination number, i.e. the number of nearest
@ -522,10 +525,24 @@ the Hamiltonian no longer conserves the total atom number. The ground
state of the $| \Psi_0 \rangle$ of the overall system will be a state of the $| \Psi_0 \rangle$ of the overall system will be a
site-wise product of the individual ground states of site-wise product of the individual ground states of
$\hat{h}_a$. These can be found very easily using standard $\hat{h}_a$. These can be found very easily using standard
diagonalisation techniques. $\Phi$ is then self-consistently diagonalisation techniques where $\Phi$ is self-consistently
determined by minimising the energy of the ground state with respect determined by minimising the energy of the ground state with respect
to $\Phi$. to $\Phi$.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{BHPhase}
\caption[Mean-Field Bose-Hubbard Phase Diagram]{Mean-field phase
diagram of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
Ref. \cite{StephenThesis}. The shaded regions are the Mott
insulator lobes and each lobe corresponds to a different on-site
filling labeled by $n$. The rest of the space corresponds to the
superfluid phase. The dashed lines are the phase boundaries
obtained from first-order perturbation theory. The solid lines are
are lines of constant density in the superfluid
phase. \label{fig:BHPhase}}
\end{figure}
The main advantage of the mean-field treatment is that it lets us The main advantage of the mean-field treatment is that it lets us
study the quantum phase transition between the sueprfluid and Mott study the quantum phase transition between the sueprfluid and Mott
insulator phases discussed in the previous sections. The phase insulator phases discussed in the previous sections. The phase
@ -551,30 +568,69 @@ highlights the fact that the interactions, which are treated exactly
here, are the dominant driver leading to this phase transition and here, are the dominant driver leading to this phase transition and
other strongly correlated effects in the Bose-Hubbard model. other strongly correlated effects in the Bose-Hubbard model.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{BHPhase}
\caption[Mean-Field Bose-Hubbard Phase Diagram]{Mean-field phase
diagram of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
Ref. \cite{StephenThesis}. The shaded regions are the Mott
insulator lobes and each lobe corresponds to a different on-site
filling labeled by $n$. The rest of the space corresponds to the
superfluid phase. The dashed lines are the phase boundaries
obtained from first-order perturbation theory. The solid lines are
are lines of constant density in the superfluid
phase. \label{fig:BHPhase}}
\end{figure}
\subsection{The Bose-Hubbard Model in One Dimension} \subsection{The Bose-Hubbard Model in One Dimension}
\label{sec:BHM1D}
The mean-field theory in the previous section is very useful tool for The mean-field theory in the previous section is very useful tool for
studying the quantum phase transition in the Bose-Hubbard studying the quantum phase transition in the Bose-Hubbard
model. However, it is effectively an infinite-dimensional theory and model. However, it is effectively an infinite-dimensional theory and
in practice it only works in two dimensions or more. The phase in practice it only works in two dimensions or more. The phase
transition in 1D is poorly described, because it actually belongs to a transition in 1D is poorly described, because it actually belongs to a
different universality class. This is clearly seen from the one different universality class \cite{cazalilla2011, ejima2011,
kuhner2000, pino2012, pino2013}. This is clearly seen from the one
dimensional phase diagram shown in Fig. \ref{fig:1DPhase}. dimensional phase diagram shown in Fig. \ref{fig:1DPhase}.
Some general conclusions can be obtained by looking at Haldane's
prescription for Luttinger liquids \cite{haldane1981,
giamarchi}. Without a periodic potential the low-energy physics of
the system is described by the Hamiltonian
\begin{equation}
\hat{H}_a = \frac{1}{2 \pi} \int \mathrm{d} x \left\{ v K [ \hat{\Pi}(x)
]^2 + \frac{v} {K} [\partial_x \hat{\Phi}(x) ]^2 \right\},
\end{equation}
where we have expressed the bosonic field operators in terms of a
density operator $\hat{\rho}(x)$ and a phase operator $\hat{\Phi}(x)$
as $\hat{\Psi}(x) = \sqrt{\hat{\rho}(x)} e^{i \hat{\Phi}(x)}$
and $\hat{\Pi}(x)$ is the density fluctuation operator. Provided the
parameters $v$ and $K$ can be correctly determined this Hamiltonian
gives the correct description of the gapless superfluid phase of the
Bose-Hubbard model. Most importantly it gives an expression for the
spatial correlation functions such as
\begin{equation}
\langle \bd_i b_j \rangle = A \left( \frac{\alpha} {|i - j|} \right)^{K/2},
\end{equation}
where $A$ is some amplitude and $\alpha$ is a necessary cutoff to
regularise the theory at short distances. Unlike the superfluid ground
state in Eq. \eqref{eq:GSSF} this state does not have infinite range
correlations. They decay according to a power-law. However, for
non-interacting systems $K = 0$ and long-range order is re-established
as before though it is important to note that in higher dimensions
this long-range order persists in the whole superfluid phase even with
interactions present.
In order to describe the phase transition and the Mott insulating
phase it is necessary to introduce a periodic lattice potential. It
can be shown that this system exhibits at $T = 0$ a
Berezinskii-Kosterlitz-Thouless phase transition as the parameter $K$
is varied with a critical point at $K_c = \frac{1}{2}$ where
$K < \frac{1}{2}$ is a superfluid. Above $K = \frac{1}{2}$ the value
of $K$ jumps discontinuously to $K \rightarrow \infty$ producing the
Mott insulator phase. Unlike the gapless superfluid phase the spatial
correlations decay exponentially as
\begin{equation}
\langle \bd_i b_j \rangle = B e^{ - |i - j|/\xi},
\end{equation}
where $B$ is some constant and the correlation length is given by
$\xi = v / \Delta$ where $\Delta$ is the energy gap.
Using advanced numerical methods such as density matrix
renormalisation group (DMRG) calculations it is possible to identify
the critical point by fitting the power-law decay correlations in
order to obtain $K$. The resulting phase transition is shown in
Fig. \ref{fig:1DPhase} and the critical point was shown to be
$(U/zJ) = 1.68$. Note that unusually the phase diagram exhibits a
reentrance phase transition for a fixed $\mu$.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=0.8\linewidth]{1DPhase} \includegraphics[width=0.8\linewidth]{1DPhase}
@ -588,55 +644,6 @@ dimensional phase diagram shown in Fig. \ref{fig:1DPhase}.
by an 'x'. \label{fig:1DPhase}} by an 'x'. \label{fig:1DPhase}}
\end{figure} \end{figure}
Some general conclusions can be obtained by looking at Haldane's
prescription for Luttinger liquids \cite{haldane1981,
giamarchi}. Without a periodic potential the low-energy physics of
the system is described by the Hamiltonian
\begin{equation}
\hat{H}_a = \frac{1}{2 \pi} \int \mathrm{d} x \left\{ v K [ \hat{\Pi}(x)
]^2 + \frac{v} {K} [\partial_x \hat{\Phi}(x) ]^2 \right\},
\end{equation}
where we have expressed the bosonic field operators in terms of a
density operator $\hat{\rho}(x)$ and a phase operator $\hat{\Phi}(x)$
as $\hat{\Psi}(x) = \sqrt{\hat{\rho}(x)} e^{i \hat{\hat{\Phi}(x)}}$
and $\hat{\Pi}(x)$ is the density fluctuation operator. Provided the
parameters $v$ and $K$ can be correctly determined this Hamiltonian
gives the correct description of the gapless superfluid phase of the
Bose-Hubbard model. Most importantly it gives an expression for the
spatial correlation functions such as
\begin{equation}
\langle \bd_m b_n \rangle = A \left( \frac{\alpha} {|m - n|} \right)^{K/2},
\end{equation}
where $A$ is some amplitude and $\alpha$ is a necessary cutoff to
regularise the theory at short distances. Unlike the superfluid ground
state in Eq. \eqref{eq:GSSF} this state does not have infinite range
correlations. They decay according to a power law. However, for
non-interacting systems $K = 0$ and long-range order is re-established
as before though it is important to note that in higher dimensions
this long-range order persists in the whole superfluid phase even with
interactions present. In order to describe the phase transition and
the Mott insulating phase it is necessary to introduce a periodic
lattice potential. It can be shown that this system exhibits at
$T = 0$ a Berezinskii-Kosterlitz-Thouless phase transition as the
parameter $K$ is varied with a critical point at $K = \frac{1}{2}$
where $K < \frac{1}{2}$ is a superfluid. Above $K = \frac{1}{2}$ the
value of $K$ jumps discontinuously to $K \rightarrow \infty$ producing
the Mott insulator phase. Unlike the gapless superfluid phase the
spatial correlations decay exponentially as
\begin{equation}
\langle \bd_m b_n \rangle = B e^{ - |m - n|/\xi},
\end{equation}
where $B$ is some constant and the correlation length is given by
$\xi = v / \Delta$ where $\Delta$ is the energy gap.
Using advanced numerical methods such as density matrix
renormalisation group (DMRG) calculations it is possible to identify
the critical point by fitting the power-law decay correlations in
order to obtain $K$. The resulting phase transition is shown in
Fig. \ref{fig:1DPhase} and the critical point was shown to be $(U/zJ) = 1.68$. Note
that unusually the phase diagram exhibits a reentrance phase
transition for a fixed $\mu$.
\section{Scattered light behaviour} \section{Scattered light behaviour}
\label{sec:a} \label{sec:a}
@ -683,11 +690,12 @@ $\hat{F}_{1,0}$
where we have defined $C = U_{1,0} a_0 / (\Delta_{p} + i \kappa)$ where we have defined $C = U_{1,0} a_0 / (\Delta_{p} + i \kappa)$
which is essentially the Rayleigh scattering coefficient into the which is essentially the Rayleigh scattering coefficient into the
cavity. Furthermore, since there is no longer any ambiguity in the cavity. Furthermore, since there is no longer any ambiguity in the
indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv indices $l$ and $m$, we have dropped indices on
\Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv $\Delta_{1p} \equiv \Delta_p$, $\kappa_1 \equiv \kappa$, and
\hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv $\hat{F}_{1,0} \equiv \hat{F}$. We also do the same for the operators
\hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and the coefficients $\hat{D}_{1,0} \equiv \hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and
$J^{1,0}_{i,j} \equiv J_{i,j}$. the coefficients $J^{1,0}_{i,j} \equiv J_{i,j}$. We will adhere to
this convention from now on.
The operator $\a_1$ itself is not an observable. However, it is The operator $\a_1$ itself is not an observable. However, it is
possible to combine the outgoing light field with a stronger local possible to combine the outgoing light field with a stronger local
@ -754,22 +762,22 @@ 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,
$\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However, $\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$
it is possible to engineer an optical geometry in which $\hat{D} = 0$ \cite{mekhov2012}. Physically, this is a consequence of the fact that
leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is there are more atoms to scatter light at the lattice sites than in
fundamentally different from the aforementioned double-well proposals between them. However, it is possible to engineer an optical geometry
as it directly couples to the interference terms caused by atoms in which $\hat{D} = 0$ leaving $\hat{B}$ as the dominant term in
tunnelling rather than combining light scattered from different $\hat{F}$. This approach is fundamentally different from the
sources. Furthermore, it is not limited to a double-wellsetup and aforementioned double-well proposals as it directly couples to the
naturally extends to a lattice structure which is a key interference terms caused by atoms tunnelling rather than combining
advantage. Such a counter-intuitive configuration may affect works on light scattered from different sources. Furthermore, it is not limited
quantum gases trapped in quantum potentials \cite{mekhov2012, to a double-well setup and naturally extends to a lattice structure
mekhov2008, larson2008, chen2009, habibian2013, ivanov2014, which is a key advantage. Such a counter-intuitive configuration may
caballero2015} and quantum measurement-induced preparation of affect works on quantum gases trapped in quantum potentials
many-body atomic states \cite{mazzucchi2016, mekhov2009prl, \cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013,
pedersen2014, elliott2015}. ivanov2014, caballero2015} and quantum measurement-induced
preparation of many-body atomic states \cite{mazzucchi2016,
\mynote{add citiations above if necessary} mekhov2009prl, pedersen2014, elliott2015}.
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
@ -794,9 +802,6 @@ 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.
\mynote{Potentially expand details of the derivation of these
equations}
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
@ -812,25 +817,25 @@ probe beam to be standing waves which gives the following expressions
for the $\hat{D}$ and $\hat{B}$ operators for the $\hat{D}$ and $\hat{B}$ operators
\begin{align} \begin{align}
\label{eq:FTs} \label{eq:FTs}
\hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_- \hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_i\hat{n}_i\cos(k_-
x_m +\varphi_-) \nonumber\\ x_i +\varphi_-) \nonumber\\
& + \mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+ x_m +\varphi_+)], & + \mathcal{F}[W_0](k_+)\sum_i\hat{n}_i\cos(k_+ x_i +\varphi_+)],
\nonumber\\ \nonumber\\
\hat{B} = & \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m \hat{B} = & \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_i\hat{B}_i\cos(k_- x_i
+\frac{k_-d}{2}+\varphi_-) \nonumber\\ +\frac{k_-d}{2}+\varphi_-) \nonumber\\
& +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+ & +\mathcal{F}[W_1](k_+)\sum_i\hat{B}_i\cos(k_+
x_m +\frac{k_+d}{2}+\varphi_+)], x_i +\frac{k_+d}{2}+\varphi_+)],
\end{align} \end{align}
where $k_\pm = k_{0x} \pm k_{1x}$, where $k_\pm = k_{0x} \pm k_{1x}$,
$k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$), $k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$),
$\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and $\hat{B}_i=\bd_ib_{i+1}+b_i\bd_{i+1}$, and
$\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the $\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the
$\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the $\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the
$\hat{D}$ operator since it depends on the amplitude of light in $\hat{D}$ 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}[hbtp!] \begin{figure}
\centering \centering
\includegraphics[width=0.8\linewidth]{BDiagram} \includegraphics[width=0.8\linewidth]{BDiagram}
\caption[Maximising Light-Matter Coupling between Lattice \caption[Maximising Light-Matter Coupling between Lattice
@ -842,7 +847,7 @@ allowing to decouple them at specific angles.
is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}} is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure} \end{figure}
\begin{figure}[hbtp!] \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{WF_S} \includegraphics[width=\linewidth]{WF_S}
\caption[Wannier Function Products]{The Wannier function products: \caption[Wannier Function Products]{The Wannier function products:
@ -864,7 +869,7 @@ $J_{i,i+1} = J^B_\mathrm{max}$, 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
(inter-site term) scatters light in phase and the operator is given by (inter-site term) scatters light in phase and the operator is given by
\begin{equation} \begin{equation}
\hat{B} = J^B_\mathrm{max} \sum_m^K \hat{B}_m . \hat{B}_\mathrm{max} = J^B_\mathrm{max} \sum_i^K \hat{B}_i .
\end{equation} \end{equation}
This can be achieved by crossing the light modes such that This can be achieved by crossing the light modes such that
$\theta_0 = -\theta_1$ and $k_{0x} = k_{1x} = \pi/d$ and choosing the $\theta_0 = -\theta_1$ and $k_{0x} = k_{1x} = \pi/d$ and choosing the
@ -888,7 +893,7 @@ 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
\begin{equation} \begin{equation}
\hat{B} = J^B_\mathrm{min} \sum_m^K (-1)^m \hat{B}_m, \hat{B}_\mathrm{min} = J^B_\mathrm{min} \sum_i^K (-1)^i \hat{B}_i,
\end{equation} \end{equation}
where $J^B_\mathrm{min}$ is a constant. We consider an arrangement where $J^B_\mathrm{min}$ is a constant. We consider an arrangement
where the beams are arranged such that $k_{0x} = 0$ and where the beams are arranged such that $k_{0x} = 0$ and
@ -896,9 +901,9 @@ $k_{1x} = \pi/d$ which gives the following expressions for the density
and interference terms and interference terms
\begin{align} \begin{align}
\label{eq:DMin} \label{eq:DMin}
\hat{D} = & \mathcal{F}[W_0]\left(\frac{\pi}{d}\right) \sum_m (-1)^m \hat{n}_m \hat{D} = & \mathcal{F}[W_0]\left(\frac{\pi}{d}\right) \sum_i (-1)^i \hat{n}_i
\cos(\varphi_0) \cos(\varphi_1) \nonumber \\ \cos(\varphi_0) \cos(\varphi_1) \nonumber \\
\hat{B} = & -\mathcal{F}[W_1]\left(\frac{\pi}{d}\right) \sum_m (-1)^m \hat{B}_m \hat{B} = & -\mathcal{F}[W_1]\left(\frac{\pi}{d}\right) \sum_i (-1)^i \hat{B}_i
\cos(\varphi_0) \sin(\varphi_1). \cos(\varphi_0) \sin(\varphi_1).
\end{align} \end{align}
For $\varphi_0 = 0$ the corresponding $J_{i,j}$ coefficients are given For $\varphi_0 = 0$ the corresponding $J_{i,j}$ coefficients are given
@ -944,7 +949,7 @@ where
$\hat{\epsilon}_{\b{k}}$ is a unit polarization vector, $\b{k}$ is the $\hat{\epsilon}_{\b{k}}$ is a unit polarization vector, $\b{k}$ is the
wave vector, wave vector,
$\mathcal{E}_{\b{k}} = \sqrt{\hbar \omega_{\b{k}} / 2 \epsilon_0 V}$, $\mathcal{E}_{\b{k}} = \sqrt{\hbar \omega_{\b{k}} / 2 \epsilon_0 V}$,
$\epsilon_0$ is the free space permittivity, $V$ is the quantization $\epsilon_0$ is the free space permittivity, $V$ is the quantisation
volume and $\a_\b{k}$ and $\a_\b{k}^\dagger$ are the annihilation and volume and $\a_\b{k}$ and $\a_\b{k}^\dagger$ are the annihilation and
creation operators respectively of a photon in mode $\b{k}$, and creation operators respectively of a photon in mode $\b{k}$, and
$\omega_\b{k}$ is the angular frequency of mode $\b{k}$. $\omega_\b{k}$ is the angular frequency of mode $\b{k}$.
@ -1039,7 +1044,7 @@ which $i \ne j$ and the remaining integrals become $\int \mathrm{d}^3
w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of
the many body operator is then the many body operator is then
\begin{equation} \begin{equation}
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E \b{\hat{E}}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E
\sum_{j = 1}^K \hat{n}_j \frac{u_0 (\b{r}_j)}{|\b{r} - \sum_{j = 1}^K \hat{n}_j \frac{u_0 (\b{r}_j)}{|\b{r} -
\b{r}_j|} e^ {i \b{k}_1 \cdot (\b{r} - \b{r}_j \b{r}_j|} e^ {i \b{k}_1 \cdot (\b{r} - \b{r}_j
) - i \omega_0 t }, ) - i \omega_0 t },
@ -1092,8 +1097,10 @@ Therefore, we can now express the quantity $n_{\Phi}$ as
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}. Miyake \emph{et al.} experimental taken from Ref. \cite{steck}. The two experiments were chosen as state
parameters are from Ref. \cite{miyake2011}. The $5^2S_{1/2}$, of the art setups that collected light scattered from ultracold atoms
in free space. 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 $F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb
is considered. For this transition the Rabi frequency is actually is considered. For this transition the Rabi frequency is actually
larger than the detuning and and effects of saturation should be taken larger than the detuning and and effects of saturation should be taken

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@ -19,10 +19,9 @@ Nondestructive Addressing} %Title of the Third Chapter
Having developed the basic theoretical framework within which we can Having developed the basic theoretical framework within which we can
treat the fully quantum regime of light-matter interactions we now treat the fully quantum regime of light-matter interactions we now
consider possible applications. There are three prominent directions consider possible applications. We will first look at nondestructive
in which we can proceed: nondestructive probing of the quantum state measurement where measurement backaction can be neglected and we focus
of matter, quantum measurement backaction induced dynamics and quantum on what expectation values can be extracted via optical methods.
optical lattices. Here, we deal with the first of the three options.
In this chapter we develop a method to measure properties of ultracold In this chapter we develop a method to measure properties of ultracold
gases in optical lattices by light scattering. In the previous chapter gases in optical lattices by light scattering. In the previous chapter
@ -57,13 +56,15 @@ beyond mean-field prediction. We demonstrate this by showing that this
scheme is capable of distinguishing all three phases in the Mott scheme is capable of distinguishing all three phases in the Mott
insulator - superfluid - Bose glass phase transition in a 1D insulator - superfluid - Bose glass phase transition in a 1D
disordered optical lattice which is not very well described by a disordered optical lattice which is not very well described by a
mean-field treatment. We underline that transitions in 1D are much mean-field treatment \cite{cazalilla2011, ejima2011, kuhner2000,
pino2012, pino2013}. We underline that transitions in 1D are much
more visible when changing an atomic density rather than for more visible when changing an atomic density rather than for
fixed-density scattering. It was only recently that an experiment fixed-density scattering. It was only recently that an experiment
distinguished a Mott insulator from a Bose glass \cite{derrico2014} distinguished a Mott insulator from a Bose glass via a series of
via a series of destructive measurements. Our proposal, on the other destructive measurements \cite{derrico2014}. Our proposal, on the
hand, is nondestructive and is capable of extracting all the relevant other hand, is nondestructive and is capable of extracting all the
information in a single experiment making our proposal timely. relevant information in a single experiment making our proposal
timely.
Having shown the possibilities created by this nondestructive Having shown the possibilities created by this nondestructive
measurement scheme we move on to considering light scattering from the measurement scheme we move on to considering light scattering from the
@ -102,7 +103,7 @@ the scattered light.
Here, we will use this fact that the light is sensitive to the atomic Here, we will use this fact that the light is sensitive to the atomic
quantum state due to the coupling of the optical and matter fields via quantum state due to the coupling of the optical and matter fields via
operators in order to develop a method to probe the properties of an operators in order to develop a method to probe the properties of an
ultracold gas. Therefore, we neglect the measurement back-action and ultracold gas. Therefore, we neglect the measurement backaction and
we will only consider expectation values of light observables. Since we will only consider expectation values of light observables. Since
the scheme is nondestructive (in some cases, it even satisfies the the scheme is nondestructive (in some cases, it even satisfies the
stricter requirements for a QND measurement \cite{mekhov2012, stricter requirements for a QND measurement \cite{mekhov2012,
@ -115,6 +116,8 @@ density correlations to matter-field interference.
\section{On-site Density Measurements} \section{On-site Density Measurements}
\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{LP2009,
mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is
@ -210,11 +213,11 @@ diffraction. Furthermore, these peaks can be tuned very easily with
$\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
dependence of $R$ for the case when the probe is a travelling wave dependence of $R$ for the case when the probe is a travelling wave
scattering from an ideal superfluid in a 3D optical lattice into a scattering from an ideal superfluid in a 3D optical lattice into a
standing wave scattered mode. The first noticeable feature is the standing wave mode. The first noticeable feature is the isotropic
isotropic background which does not exist in classical background which does not exist in classical diffraction. This
diffraction. This background yields information about density background yields information about density fluctuations which,
fluctuations which, according to mean-field estimates (i.e.~inter-site according to mean-field estimates (i.e.~inter-site correlations are
correlations are ignored), are related by ignored), are related by
$R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 $R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
)/2$. In Fig. \ref{fig:scattering} we can see a significant signal of )/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, because it shows scattering from an ideal $R = |C|^2 N_K/2$, because it shows scattering from an ideal
@ -225,7 +228,7 @@ the signal goes to zero. This is because the Mott insulating phase has
well localised atoms at each site which suppresses density well localised atoms at each site which suppresses density
fluctuations entirely leading to absolutely no ``quantum addition''. fluctuations entirely leading to absolutely no ``quantum addition''.
\begin{figure}[htbp!] \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{Ep1} \includegraphics[width=\linewidth]{Ep1}
\caption[Light Scattering Angular Distribution]{Light intensity \caption[Light Scattering Angular Distribution]{Light intensity
@ -290,7 +293,7 @@ can also be negative.
We will consider scattering from a superfluid, because the Mott We will consider scattering from a superfluid, because the Mott
insulator has no ``quantum addition'' due to a lack of density insulator has no ``quantum addition'' due to a lack of density
fluctuations. The wavefunction of a superfluid on a lattice is given fluctuations. The wavefunction of a superfluid on a lattice is given
by \textbf{Eq. (??)}. This state has infinte range correlations and by Eq. \eqref{eq:GSSF}. This state has infinte range correlations and
thus has the convenient property that all two-point density thus has the convenient property that all two-point density
fluctuation correlations are equal regardless of their separation, fluctuation correlations are equal regardless of their separation,
i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$ i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
@ -332,10 +335,10 @@ second term as it is always negative and it has the same angular
distribution as the classical diffraction pattern and thus it is distribution as the classical diffraction pattern and thus it is
mostly zero except when the classical Bragg condition is mostly zero except when the classical Bragg condition is
satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle
such that the Bragg is not satisfied this term is essentially such that the Bragg condition is not satisfied this term is
zero. Therefore, we are left with the first term $\sum_i^K |A_i|^2$ essentially zero. Therefore, we are left with the first term
which for a travelling wave probe and a standing wave scattered mode $\sum_i^K |A_i|^2$ which for a travelling wave probe and a standing
is wave scattered mode is
\begin{equation} \begin{equation}
\sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) = \sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) =
\frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2 \frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2
@ -388,7 +391,7 @@ 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.
\section{Mapping the quantum phase diagram} \subsection{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
@ -397,8 +400,8 @@ signal should be strong enough to be visible using currently available
technology. However, so far we have not looked at what this can tell technology. However, so far we have not looked at what this can tell
us about the quantum state of matter. We have briefly mentioned that a us about the quantum state of matter. We have briefly mentioned that a
deep superfluid will scatter a lot of light due to its infinite range deep superfluid will scatter a lot of light due to its infinite range
correlations and a Mott insulator will not contriute any ``quantum correlations and a Mott insulator will not contribute any ``quantum
addition'' at all, but we have not look at the quantum phase addition'' at all, but we have not looked at the quantum phase
transition between these two phases. In two or higher dimensions this transition between these two phases. In two or higher dimensions this
has a rather simple answer as the Bose-Hubbard phase transition is has a rather simple answer as the Bose-Hubbard phase transition is
described well by mean-field theories and it has a sharp transition at described well by mean-field theories and it has a sharp transition at
@ -410,8 +413,9 @@ much more information.
There are many situations where the mean-field approximation is not a There are many situations where the mean-field approximation is not a
valid description of the physics. A prominent example is the valid description of the physics. A prominent example is the
Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000,
pino2012, pino2013}. Observing the transition in 1D by light at pino2012, pino2013} as we have seen in section
fixed density was considered to be difficult \cite{rogers2014} or even \ref{sec:BHM1D}. Observing the transition in 1D by light at fixed
density was considered to be difficult \cite{rogers2014} or even
impossible \cite{roth2003}. This is because the one-dimensional impossible \cite{roth2003}. This is because the one-dimensional
quantum phase transition is in a different universality class than its quantum phase transition is in a different universality class than its
higher dimensional counterparts. The energy gap, which is the order higher dimensional counterparts. The energy gap, which is the order
@ -437,7 +441,7 @@ decay algebraically \cite{giamarchi}.
The method we propose gives us direct access to the structure factor, The method we propose gives us direct access to the structure factor,
which is a function of the two-point correlation $\langle \delta which is a function of the two-point correlation $\langle \delta
\hat{n}_i \delta \hat{n}_j \rangle$. This quantity can be extracted \hat{n}_i \delta \hat{n}_j \rangle$. This quantity can be extracted
from the measured light intensity bu considering the ``quantum from the measured light intensity by considering the ``quantum
addition''. We will consider the case when both the probe and addition''. We will consider the case when both the probe and
scattered modes are plane waves which can be easily achieved in free scattered modes are plane waves which can be easily achieved in free
space. We will again consider the case of light being maximally space. We will again consider the case of light being maximally
@ -449,8 +453,6 @@ addition is given by
\hat{n}_j \rangle. \hat{n}_j \rangle.
\end{equation} \end{equation}
\mynote{can put in more detail here with equations}
This alone allows us to analyse the phase transition quantitatively This alone allows us to analyse the phase transition quantitatively
using our method. Unlike in higher dimensions where an order parameter using our method. Unlike in higher dimensions where an order parameter
can be easily defined within the mean-field approximation as a simple can be easily defined within the mean-field approximation as a simple
@ -458,21 +460,22 @@ expectation value, the situation in 1D is more complex as it is
difficult to directly access the excitation energy gap which defines difficult to directly access the excitation energy gap which defines
this phase transition. However, a valid description of the relevant 1D this phase transition. However, a valid description of the relevant 1D
low energy physics is provided by Luttinger liquid theory low energy physics is provided by Luttinger liquid theory
\cite{giamarchi}. In this model correlations in the supefluid phase as \cite{giamarchi} as seen in section \ref{sec:BHM1D}. In this model
well as the superfluid density itself are characterised by the correlations in the supefluid phase as well as the superfluid density
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies itself are characterised by the Tomonaga-Luttinger parameter,
the critical point in the thermodynamic limit at $K_b = 1/2$. This $K$. This parameter also identifies the critical point in the
quantity can be extracted from various correlation functions and in thermodynamic limit at $K_c = 1/2$. This quantity can be extracted
our case it can be extracted directly from $R$ \cite{ejima2011}. This from various correlation functions and in our case it can be extracted
quantity was used in numerical calculations that used highly efficient directly from $R$ \cite{ejima2011}. This quantity was used in
density matrix renormalisation group (DMRG) methods to calculate the numerical calculations that used highly efficient density matrix
ground state to subsequently fit the Luttinger theory to extract this renormalisation group (DMRG) methods to calculate the ground state to
parameter $K_b$. These calculations yield a theoretical estimate of subsequently fit the Luttinger theory to extract this parameter
the critical point in the thermodynamic limit for commensurate filling $K$. These calculations yield a theoretical estimate of the critical
in 1D to be at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our point in the thermodynamic limit for commensurate filling in 1D to be
proposal provides a method to directly measure $R$ nondestructively in at $U/2J \approx 1.64$ \cite{ejima2011}. Our proposal provides a
a lab which can then be used to experimentally determine the location method to directly measure $R$ nondestructively in a lab which can
of the critical point in 1D. then be used to experimentally determine the location of the critical
point in 1D.
However, whilst such an approach will yield valuable quantitative However, whilst such an approach will yield valuable quantitative
results we will instead focus on its qualitative features which give a results we will instead focus on its qualitative features which give a
@ -489,7 +492,7 @@ easier to see its usefuleness in a broader context.
We calculate the phase diagram of the Bose-Hubbard Hamiltonian given We calculate the phase diagram of the Bose-Hubbard Hamiltonian given
by by
\begin{equation} \begin{equation}
\hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle} \hat{H}_\mathrm{dis} = -J \sum_{\langle i, j \rangle}
\bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) - \mu \bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) - \mu
\sum_i \hat{n}_i, \sum_i \hat{n}_i,
\end{equation} \end{equation}
@ -501,35 +504,34 @@ DMRG methods \cite{tnt} from which we can compute all the necessary
atomic observables. Experiments typically use an additional harmonic atomic observables. Experiments typically use an additional harmonic
confining potential on top of the optical lattice to keep the atoms in confining potential on top of the optical lattice to keep the atoms in
place which means that the chemical potential will vary in place which means that the chemical potential will vary in
space. However, with careful consideration of the full space. However, with careful consideration of the full ($\mu/2J$,
($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in $U/2J$) phase diagrams in Fig. \ref{fig:SFMI}(d,e) our analysis can
Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the still be applied to the system \cite{batrouni2002}.
system \cite{batrouni2002}.
\begin{figure}[htbp!] \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{oph11} \includegraphics[width=\linewidth]{oph11_3}
\caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular \caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular
dependence of scattered light $R$ for a superfluid (thin black, dependence of scattered light $R$ for a superfluid (thin purple,
left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green, left scale, $U/2J = 0$) and Mott insulator (thick blue, right
right scale, $U/2J^\text{cl} =10$). The two phases differ in both scale, $U/2J =10$). The two phases differ in both their value of
their value of $R_\text{max}$ as well as $W_R$ showing that $R_\text{max}$ as well as $W_R$ showing that density correlations
density correlations in the two phases differ in magnitude as well in the two phases differ in magnitude as well as extent. Light
as extent. Light scattering maximum $R_\text{max}$ is shown in (b, scattering maximum $R_\text{max}$ is shown in (b, d) and the width
d) and the width $W_R$ in (c, e). It is very clear that varying $W_R$ in (c, e). It is very clear that varying chemical potential
chemical potential $\mu$ or density $\langle n\rangle$ sharply $\mu$ or density $\langle n\rangle$ sharply identifies the
identifies the superfluid-Mott insulator transition in both superfluid-Mott insulator transition in both quantities. (b) and
quantities. (b) and (c) are cross-sections of the phase diagrams (c) are cross-sections of the phase diagrams (d) and (e) at
(d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple), $U/2J=2$ (thick blue), 3 (thin purple), and 4 (dashed
and 4 (dashed blue). Insets show density dependencies for the blue). Insets show density dependencies for the $U/(2 J) = 3$
$U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.} line. $K=M=N=25$.}
\label{fig:SFMI} \label{fig:SFMI}
\end{figure} \end{figure}
We then consider probing these ground states using our optical scheme We then consider probing these ground states using our optical scheme
and we calculate the ``quantum addition'', $R$, based on these ground and we calculate the ``quantum addition'', $R$, based on these ground
states. The angular dependence of $R$ for a Mott insulator and a states. The angular dependence of $R$ for a Mott insulator and a
superfluid is shown in Fig. \ref{fig:SFMI}a, and we note that there superfluid is shown in Fig. \ref{fig:SFMI}(a), and we note that there
are two variables distinguishing the states. Firstly, maximal $R$, are two variables distinguishing the states. Firstly, maximal $R$,
$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$, $R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
probes the fluctuations and compressibility $\kappa'$ probes the fluctuations and compressibility $\kappa'$
@ -537,26 +539,25 @@ probes the fluctuations and compressibility $\kappa'$
\rangle$). The Mott insulator is incompressible and thus will have \rangle$). The Mott insulator is incompressible and thus will have
very small on-site fluctuations and it will scatter little light very small on-site fluctuations and it will scatter little light
leading to a small $R_\text{max}$. The deeper the system is in the leading to a small $R_\text{max}$. The deeper the system is in the
insulating phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the insulating phase (i.e. the larger the $U/2J$ ratio is), the smaller
smaller these values will be until ultimately it will scatter no light these values will be until ultimately it will scatter no light at all
at all in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}(a) this
this can be seen in the value of the peak in $R$. The value can be seen in the value of the peak in $R$. The value $R_\text{max}$
$R_\text{max}$ in the superfluid phase ($U/2J^\text{cl} = 0$) is in the superfluid phase ($U/2J = 0$) is larger than its value in the
larger than its value in the Mott insulating phase Mott insulating phase ($U/2J = 10$) by a factor of
($U/2J^\text{cl} = 10$) by a factor of
$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of $\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
$R_\text{max}$ changes across the phase transition. There are a few $R_\text{max}$ changes across the phase transition. There are a few
things to note at this point. Firstly, if we follow the transition things to note at this point. Firstly, if we follow the transition
along the line corresponding to commensurate filling (i.e.~any line along the line corresponding to commensurate filling (i.e.~any line
that is in between the two white lines in Fig. \ref{fig:SFMI}d) we see that is in between the two white lines in Fig. \ref{fig:SFMI}(d)) we
that the transition is very smooth and it is hard to see a definite see that the transition is very smooth and it is hard to see a
critical point. This is due to the energy gap closing exponentially definite critical point. This is due to the energy gap closing
slowly which makes precise identification of the critical point exponentially slowly which makes precise identification of the
extremely difficult. The best option at this point would be to fit critical point extremely difficult. The best option at this point
Tomonaga-Luttinger theory to the results in order to find this would be to fit Tomonaga-Luttinger theory to the results in order to
critical point. However, we note that there is a drastic change in find this critical point. However, we note that there is a drastic
signal as the chemical potential (and thus the density) is change in signal as the chemical potential (and thus the density) is
varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how varied. This is highlighted in Fig. \ref{fig:SFMI}(b) which shows how
the Mott insulator can be easily identified by a dip in the quantity the Mott insulator can be easily identified by a dip in the quantity
$R_\text{max}$. $R_\text{max}$.
@ -568,10 +569,10 @@ large $W_R$. On the other hand, the superfluid in 1D exhibits pseudo
long-range order which manifests itself in algebraically decaying long-range order which manifests itself in algebraically decaying
two-point correlations \cite{giamarchi} which significantly reduces two-point correlations \cite{giamarchi} which significantly reduces
the dip in the $R$. This can be seen in the dip in the $R$. This can be seen in
Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see Fig. \ref{fig:SFMI}(a). Furthermore, just like for $R_\text{max}$ we
that the transition is much sharper as $\mu$ is varied. This is shown see that the transition is much sharper as $\mu$ is varied. This is
in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between shown in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle
a superfluid and an insulating state is fairly significant between a superfluid and an insulating state is fairly significant
$\sim 20^\circ$ which should make the two phases easy to identify $\sim 20^\circ$ which should make the two phases easy to identify
using this measure. In this particular case, measuring $W_R$ in the using this measure. In this particular case, measuring $W_R$ in the
Mott phase is not very practical as the insulating phase does not Mott phase is not very practical as the insulating phase does not
@ -596,7 +597,7 @@ this by adding an additional periodic potential on top of the exisitng
setup that is incommensurate with the original lattice. The resulting setup that is incommensurate with the original lattice. The resulting
Hamiltonian can be shown to be Hamiltonian can be shown to be
\begin{equation} \begin{equation}
\hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle} \hat{H}_\mathrm{dis} = -J \sum_{\langle i, j \rangle}
\bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) + \bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) +
\frac{V}{2} \sum_i \left[ 1 + \cos (2 r \pi m + 2 \phi) \right] \frac{V}{2} \sum_i \left[ 1 + \cos (2 r \pi m + 2 \phi) \right]
\hat{n}_i, \hat{n}_i,
@ -608,7 +609,7 @@ first two terms are the standard Bose-Hubbard Hamiltonian and the only
modification is an additional spatially varying potential shift. We modification is an additional spatially varying potential shift. We
will only consider the phase diagram at fixed density as the will only consider the phase diagram at fixed density as the
introduction of disorder makes the usual interpretation of the phase introduction of disorder makes the usual interpretation of the phase
diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a diagram in the ($\mu/2J$, $U/2J$) plane for a
fixed ratio $V/U$ complicated due to the presence of multiple fixed ratio $V/U$ complicated due to the presence of multiple
compressible and incompressible phases between successive Mott compressible and incompressible phases between successive Mott
insulator lobes \cite{roux2008}. Therefore, the chemical potential no insulator lobes \cite{roux2008}. Therefore, the chemical potential no
@ -629,12 +630,12 @@ decaying correlations. This gives a large $R_\text{max}$ and a large
$W_R$. A Mott insulator also has exponentially decaying correlations $W_R$. A Mott insulator also has exponentially decaying correlations
since it is an insulator, but it is incompressible. Thus, it will since it is an insulator, but it is incompressible. Thus, it will
scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a
superfluid has long range correlations and large compressibility which superfluid has long-range correlations and large compressibility which
results in a large $R_\text{max}$ and a small $W_R$. results in a large $R_\text{max}$ and a small $W_R$.
\begin{figure}[htbp!] \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{oph22} \includegraphics[width=\linewidth]{oph22_3}
\caption[Mapping the Disoredered Phase Diagram]{The \caption[Mapping the Disoredered Phase Diagram]{The
Mott-superfluid-glass phase diagrams for light scattering maximum Mott-superfluid-glass phase diagrams for light scattering maximum
$R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both $R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both
@ -682,9 +683,6 @@ an optical lattice as this gives an in-situ method for probing the
inter-site interference terms at its shortest possible distance, inter-site interference terms at its shortest possible distance,
i.e.~the lattice period. i.e.~the lattice period.
% I mention mean-field here, but do not explain it. That should be
% done in Chapter 2.1}
Unlike in the previous sections, here we will use the mean-field Unlike in the previous sections, here we will use the mean-field
description of the Bose-Hubbard model in order to obtain a simple description of the Bose-Hubbard model in order to obtain a simple
physical picture of what information is contained in the quantum physical picture of what information is contained in the quantum
@ -709,7 +707,8 @@ optical arrangement leads to a diffraction maximum with the matter
operator operator
\begin{equation} \begin{equation}
\label{eq:Bmax} \label{eq:Bmax}
\hat{B} = J^B_\mathrm{max} \sum_i \left( \bd_i b_{i+1} + b_i \bd_{i+1} \right), \hat{B}_\mathrm{max} = J^B_\mathrm{max} \sum_i \left( \bd_i b_{i+1}
+ b_i \bd_{i+1} \right),
\end{equation} \end{equation}
where $J^B_\mathrm{max} = \mathcal{F}[W_1](2\pi/d)$. Therefore, by measuring the where $J^B_\mathrm{max} = \mathcal{F}[W_1](2\pi/d)$. Therefore, by measuring the
expectation value of the quadrature we obtain the following quantity expectation value of the quadrature we obtain the following quantity
@ -757,21 +756,22 @@ $\langle b^2 \rangle$ will only scale as $K$. Therefore, it would be
difficult to extract the quantity that we need by measuring in the difficult to extract the quantity that we need by measuring in the
difraction maximum. difraction maximum.
\begin{figure}[htbp!] \begin{figure}
\centering \centering
\includegraphics[width=\linewidth]{Quads} \includegraphics[width=\linewidth]{QuadsC}
\captionsetup{justification=centerlast,font=small} \caption[Mean-Field Matter Quadratures]{Mean-field quadratures and
\caption[Mean-Field Matter Quadratures]{Photon number scattered in a resulting photon scattering rates. (a) The variances of
diffraction minimum, given by Eq. (\ref{intensity}), where
$\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More
light is scattered from a MI than a SF due to the large
uncertainty in phase in the insulator. (a) The variances of
quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$ quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$
(dashed) of the matter field across the phase transition. Level (dashed) of the matter field across the phase transition. Level
1/4 is the minimal (Heisenberg) uncertainty. There are three 1/4 is the minimal (Heisenberg) uncertainty. There are three
important points along the phase transition: the coherent state important points along the phase transition: the coherent state
(SF) at A, the amplitude-squeezed state at B, and the Fock state (SF) at A, the amplitude-squeezed state at B, and the Fock state
(MI) at C. (b) The uncertainties plotted in phase space.} (MI) at C. (b) The uncertainties plotted in phase space. (c)
Photon number scattered in a diffraction minimum, given by
Eq. (\ref{intensity}), where
$\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More
light is scattered from a MI than a SF due to the large
uncertainty in phase in the insulator.}
\label{Quads} \label{Quads}
\end{figure} \end{figure}
@ -793,7 +793,7 @@ proportional to $K^2$ and thus we obtain the following quantity
(\frac{\pi}{d}) [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ], (\frac{\pi}{d}) [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ],
\end{equation} \end{equation}
This is plotted in Fig. \ref{Quads} as a function of This is plotted in Fig. \ref{Quads} as a function of
$U/(zJ^\text{cl})$. Now, we can easily deduce the value of $U/(zJ)$. Now, we can easily deduce the value of
$\langle b^2 \rangle$ since we will already know the mean density, $\langle b^2 \rangle$ since we will already know the mean density,
$n$, from our experimental setup and we have seen that we can obtain $n$, from our experimental setup and we have seen that we can obtain
$\Phi^2$ from the diffraction maximum. Thus, we now have access to the $\Phi^2$ from the diffraction maximum. Thus, we now have access to the
@ -831,8 +831,8 @@ themselves only in high-order correlations \cite{kaszlikowski2008}.
\section{Conclusions} \section{Conclusions}
In this chapter we explored the possibility of nondestructively In this chapter we explored the possibility of nondestructively
probing a quantum gas trapped in an optical lattice using quantized 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{weitenberg2011, miyake2011}. Secondly, we
@ -847,7 +847,7 @@ measurements which deal with far-field interference. This quantity
defines most processes in optical lattices. E.g. matter-field phase defines most processes in optical lattices. E.g. matter-field phase
changes may happen not only due to external gradients, but also due to changes may happen not only due to external gradients, but also due to
intriguing effects such quantum jumps leading to phase flips at intriguing effects such quantum jumps leading to phase flips at
neighbouring sites and sudden cancellation of tunneling neighbouring sites and sudden cancellation of tunnelling
\cite{vukics2007}, which should be accessible by our method. We showed \cite{vukics2007}, which should be accessible by our method. We showed
how in mean-field, one can measure the matter-field amplitude (order how in mean-field, one can measure the matter-field amplitude (order
parameter), quadratures and squeezing. This can link atom optics to parameter), quadratures and squeezing. This can link atom optics to

45
Chapter4/chapter4.tex
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@ -18,14 +18,14 @@
\section{Introduction} \section{Introduction}
This thesis is entirely concerned with the question of measuring a This thesis is entirely concerned with the question of measuring a
quantum many-body system using quantized light. However, so far we quantum many-body system using quantised light. However, so far we
have only looked at expectation values in a nondestructive context have only looked at expectation values in a nondestructive context
where we neglect the effect of the quantum wavefunction collapse. We where we neglect the effect of the quantum wavefunction collapse. We
have shown that light provides information about various statistical have shown that light provides information about various statistical
quantities of the quantum states of the atoms such as their quantities of the quantum states of the atoms such as their
correlation functions. In general, any quantum measurement affects the correlation functions. In general, any quantum measurement affects the
system even if it doesn't physically destroy it. In our model both system even if it doesn't physically destroy it. In our model both
optical and matter fields are quantized and their interaction leads to optical and matter fields are quantised and their interaction leads to
entanglement between the two subsystems. When a photon is detected and entanglement between the two subsystems. When a photon is detected and
the electromagnetic wavefunction of the optical field collapses, the the electromagnetic wavefunction of the optical field collapses, the
matter state is also affected due to this entanglement resulting in matter state is also affected due to this entanglement resulting in
@ -78,7 +78,7 @@ not a deterministic process. Furthermore, they are in general
discotinuous as each detection event brings about a drastic change in discotinuous as each detection event brings about a drastic change in
the quantum state due to the wavefunction collapse of the light field. the quantum state due to the wavefunction collapse of the light field.
Before we discuss specifics relevant to our model of quantized light Before we discuss specifics relevant to our model of quantised light
interacting with a quantum gas we present a more general overview 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
@ -118,8 +118,8 @@ where the denominator is simply a normalising factor
\cite{MeasurementControl}. The exact form of the jump operator $\c$ \cite{MeasurementControl}. The exact form of the jump operator $\c$
will depend on the nature of the measurement we are considering. For will depend on the nature of the measurement we are considering. For
example, if we consider measuring the photons escaping from a leaky example, if we consider measuring the photons escaping from a leaky
cavity then $\c = \sqrt{2 \kappa} \hat{a}$, where $\kappa$ is the cavity then $\c = \sqrt{2 \kappa} \a$, where $\kappa$ is the
cavity decay rate and $\hat{a}$ is the annihilation operator of a cavity decay rate and $\a$ is the annihilation operator of a
photon in the cavity field. It is interesting to note that due to photon in the cavity field. It is interesting to note that due to
renormalisation the effect of a single quantum jump is independent of renormalisation the effect of a single quantum jump is independent of
the magnitude of the operator $\c$ itself. However, larger operators the magnitude of the operator $\c$ itself. However, larger operators
@ -283,7 +283,7 @@ In Chapter \ref{chap:qnd} we used highly efficient DMRG methods
\cite{tnt} to calculate the ground state of the Bose-Hubbard \cite{tnt} to calculate the ground state of the Bose-Hubbard
Hamiltonian. Related techniques such as Time-Evolving Block Decimation Hamiltonian. Related techniques such as Time-Evolving Block Decimation
(TEBD) or t-DMRG are often used for numerical calculations of time (TEBD) or t-DMRG are often used for numerical calculations of time
evolution. However, despite the fact our Hamiltonian in evolution. However, despite the fact that our Hamiltonian in
Eq. \eqref{eq:backaction} is simply the Bose-Hubbard model with a Eq. \eqref{eq:backaction} is simply the Bose-Hubbard model with a
non-Hermitian term added due to measurement it is actually difficult non-Hermitian term added due to measurement it is actually difficult
to apply these methods to our system. The problem lies in the fact to apply these methods to our system. The problem lies in the fact
@ -294,11 +294,11 @@ correlations. Unfortunately, the global nature of the measurement we
consider violates the assumptions made in deriving the area law and, consider violates the assumptions made in deriving the area law and,
as we shall see in the following chapters, leads to long-range as we shall see in the following chapters, leads to long-range
correlations regardless of coupling strength. Therefore, we resort to correlations regardless of coupling strength. Therefore, we resort to
using exact methods such as exact diagonalisation which we solve with using alternative methods such as exact diagonalisation which we solve
well-known ordinary differential equation solvers. This means that we with well-known ordinary differential equation solvers. This means
can at most simulate a few atoms, but as we shall see it is the that we can at most simulate a few atoms, but as we shall see it is
geometry of the measurement that matters the most and these effects the geometry of the measurement that matters the most and these
are already visible in smaller systems. effects are already visible in smaller systems.
\section{The Master Equation} \section{The Master Equation}
\label{sec:master} \label{sec:master}
@ -355,17 +355,16 @@ 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. However, in this case we do not want to impose any specific events. In this case we do not want to impose any specific trajectory
trajectory on the system as we are not interested in a specific on the system as we are not interested in a specific experimental run,
experimental run, but we would still like to identify the set of but we would still like to identify the set of possible outcomes and
possible outcomes and their common properties. Unfortunately, their common properties. Unfortunately, calculating the inverse of
calculating the inverse of Eq. \eqref{eq:rho} is not an easy task. In Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of
fact, the decomposition of a density matrix into pure states might not a density matrix into pure states might not even be unique. However,
even be unique. However, if a measurement leads to a projection, if a measurement leads to a projection, i.e.~the final state becomes
i.e.~the final state becomes confined to some subspace of the Hilbert confined to some subspace of the Hilbert space, then this will be
space, then this will be visible in the final state of the density visible in the final state of the density matrix. We will show this on
matrix. We will show this on an example of a qubit in the quantum an example of a qubit in the quantum state
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,
@ -543,7 +542,7 @@ Fig. \ref{fig:twomodes}.
\label{fig:twomodes} \label{fig:twomodes}
\end{figure} \end{figure}
This can approach can be generalised to an arbitrary number of modes, This approach can be generalised to an arbitrary number of modes,
$Z$. For this we will conisder a deep lattice such that $Z$. 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 $J_{i,i} = u_1^* (\b{r}) u_0 (\b{r})$. We will take the probe beam to
be incident normally at a 1D lattice so that $u_0 (\b{r}) = be incident normally at a 1D lattice so that $u_0 (\b{r}) =

190
Chapter5/chapter5.tex
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@ -17,9 +17,9 @@
In the previous chapter we have introduced a theoretical framework In the previous chapter we have introduced a theoretical framework
which will allow us to study measurement backaction using which will allow us to study measurement backaction using
discontinuous quantum jumps and non-Hermitian evolution due to null discontinuous quantum jumps and non-Hermitian evolution due to null
outcomesquantum trajectories. We have also wrapped our quantum gas outcomes using quantum trajectories. We have also wrapped our quantum
model in this formalism by considering ultracold bosons in an optical gas model in this formalism by considering ultracold bosons in an
lattice coupled to a cavity which collects and enhances light optical lattice coupled to a cavity which collects and enhances light
scattered in one particular direction. One of the most important scattered in one particular direction. One of the most important
conclusions of the previous chapter was that the introduction of conclusions of the previous chapter was that the introduction of
measurement introduces a new energy and time scale into the picture measurement introduces a new energy and time scale into the picture
@ -38,17 +38,18 @@ unlike tunnelling and on-site interactions our measurement scheme is
global in nature which makes it capable of creating long-range global in nature which makes it capable of creating long-range
correlations which enable nonlocal dynamical processes. Furthermore, 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 which is impossible to spatially nonlocal coupling to the environment, as seen in section
obtain with local interactions \cite{daley2014, diehl2008, \ref{sec:modes}, which is impossible to obtain with local interactions
syassen2008}. These spatial modes of matter fields can be considered \cite{daley2014, diehl2008, syassen2008}. These spatial modes of
as designed systems and reservoirs opening the possibility of matter fields can be considered as designed systems and reservoirs
controlling dissipations in ultracold atomic systems without resorting opening the possibility of controlling dissipations in ultracold
to atom losses and collisions which are difficult to manipulate. Thus atomic systems without resorting to atom losses and collisions which
the continuous measurement of the light field introduces a are difficult to manipulate. Thus the continuous measurement of the
controllable decoherence channel into the many-body dynamics. Such a light field introduces a controllable decoherence channel into the
quantum optical approach can broaden the field even further allowing many-body dynamics. Such a quantum optical approach can broaden the
quantum simulation models unobtainable using classical light and the field even further allowing quantum simulation models unobtainable
design of novel systems beyond condensed matter analogues. using classical light and the design of novel systems beyond condensed
matter analogues.
In the weak measurement limit, where the quantum jumps do not occur In the weak measurement limit, where the quantum jumps do not occur
frequently compared to the tunnelling rate, this can lead to global frequently compared to the tunnelling rate, this can lead to global
@ -159,7 +160,7 @@ computed from the eigenvalues of Eq. \eqref{eq:Zmodes},
\hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l. \hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l.
\end{equation} \end{equation}
Each eigenvalue can be represented as the sum of the individual terms Each eigenvalue can be represented as the sum of the individual terms
in teh above sum which are vectors on the complex plane with phases in the above sum which are vectors on the complex plane with phases
that are integer multiples of $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$, that are integer multiples of $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$,
$N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums $N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums
of these vectors is invariant under rotations by $2 \pi l R / Z$, of these vectors is invariant under rotations by $2 \pi l R / Z$,
@ -173,7 +174,7 @@ in pairs resulting in only three visible components.
We will now limit ourselves to a specific illumination pattern with We will now limit ourselves to a specific illumination pattern with
$\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest $\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest
multimode dynamics with $Z = 2$ and only a single component as seen in multimode dynamics with $Z = 2$ and only a single component as seen in
Fig. \ref{fig:oscillations}a, i.e.~no multiple peaks like in Fig. \ref{fig:oscillations}(a), i.e.~no multiple peaks like in
Figs. \ref{fig:oscillations}(b,c). This pattern can be obtained by Figs. \ref{fig:oscillations}(b,c). This pattern can be obtained by
crossing two beams such that their projections on the lattice are crossing two beams such that their projections on the lattice are
identical and the even sites are positioned at their nodes. However, identical and the even sites are positioned at their nodes. However,
@ -202,7 +203,7 @@ non-Hermitian Hamiltonian describing the time evolution in between the
jumps is given by jumps is given by
\begin{equation} \begin{equation}
\label{eq:doublewell} \label{eq:doublewell}
\hat{H} = -J^\mathrm{cl} \left( \bd_o b_e + b_o \bd_e \right) - i \hat{H} = -J \left( \bd_o b_e + b_o \bd_e \right) - i
\gamma \n_o^2 \gamma \n_o^2
\end{equation} \end{equation}
and the quantum jump operator which is applied at each photodetection and the quantum jump operator which is applied at each photodetection
@ -226,10 +227,9 @@ continuous variables by defining $\psi (x = l / N) = \sqrt{N}
q_l$. Note that this requires the coefficients $q_l$ to vary smoothly q_l$. Note that this requires the coefficients $q_l$ to vary smoothly
which is the case for a superfluid state. We now rescale the which is the case for a superfluid state. We now rescale the
Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by
dividing by $NJ^\mathrm{cl}$ and define the relative population dividing by $NJ$ and define the relative population imbalance between
imbalance between the two wells $z = 2x - 1$. Finally, by taking the the two wells $z = 2x - 1$. Finally, by taking the expectation value
expectation value of the Hamiltonian and looking for the stationary of the Hamiltonian and looking for the stationary points of
points of
$\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
\rangle$ we obtain the semiclassical Schr\"{o}dinger equation \rangle$ we obtain the semiclassical Schr\"{o}dinger equation
\begin{equation} \begin{equation}
@ -243,21 +243,21 @@ $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
\right)^2 \right] \psi(z, t), \right)^2 \right] \psi(z, t),
\end{equation} \end{equation}
where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$, where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$,
$\omega = 2 \sqrt{1 + \Lambda - h}$, and $\omega = 2 \sqrt{1 + \Lambda - h}$, and $\Lambda = NU / (2J)$. The
$\Lambda = NU / (2J^\mathrm{cl})$. The full derivation is not full derivation is not straightforward, but the introduction of the
straightforward, but the introduction of the non-Hermitian term non-Hermitian term requires only a minor modification to the original
requires only a minor modification to the original formalism presented formalism presented in detail in Ref. \cite{juliadiaz2012} so we have
in detail in Ref. \cite{juliadiaz2012} so we have omitted it here. We omitted it here. We will also be considering $U = 0$ as the effective
will also be considering $U = 0$ as the effective model is only valid model is only valid in this limit, thus $\Lambda = 0$. However, this
in this limit, thus $\Lambda = 0$. However, this model is valid for an model is valid for an actual physical double-well setup in which case
actual physical double-well setup in which case interacting bosons can interacting bosons can also be considered. The equation is defined on
also be considered. The equation is defined on the interval the interval $z \in [-1, 1]$, but $z \ll 1$ has been assumed in order
$z \in [-1, 1]$, but $z \ll 1$ has been assumed in order to simplify to simplify the kinetic term and approximate the potential as
the kinetic term and approximate the potential as parabolic. This does parabolic. This does mean that this approximation is not valid for the
mean that this approximation is not valid for the maximum amplitude maximum amplitude oscillations seen in Fig. \ref{fig:oscillations}(a),
oscillations seen in Fig. \ref{fig:oscillations}a, but since they but since they already appear early on in the trajectory we are able
already appear early on in the trajectory we are able to obtain a to obtain a valid analytic description of the oscillations and their
valid analytic description of the oscillations and their growth. growth.
A superfluid state in our continuous variable approximation A superfluid state in our continuous variable approximation
corresponds to a Gaussian wavefunction $\psi$. Furthermore, since the corresponds to a Gaussian wavefunction $\psi$. Furthermore, since the
@ -355,8 +355,8 @@ explicitly in the equations above.
First, it is worth noting that all parameters of interest can be First, it is worth noting that all parameters of interest can be
extracted from $p(t)$ and $q(t)$ alone. We are not interested in extracted from $p(t)$ and $q(t)$ alone. We are not interested in
$\epsilon$ as it is only related to the global phase and the norm of $\epsilon(t)$ as it is only related to the global phase and the norm
the wavefunction and it contains little physical of the wavefunction and it contains little physical
information. Furthermore, an interesting and incredibly convenient information. Furthermore, an interesting and incredibly convenient
feature of these equations is that the Eq. \eqref{eq:p} is a function feature of these equations is that the Eq. \eqref{eq:p} is a function
of $p(t)$ alone and Eq. \eqref{eq:pq} is a function of $p(t)$ and of $p(t)$ alone and Eq. \eqref{eq:pq} is a function of $p(t)$ and
@ -434,7 +434,7 @@ parameters in a form that is easy to analyse. Therefore, we instead
consider the case when $\Gamma = 0$, but we do not neglect the effect consider the case when $\Gamma = 0$, but we do not neglect the effect
of quantum jumps. It may seem counter-intuitive to neglect the term of quantum jumps. It may seem counter-intuitive to neglect the term
that appears due to measurement, but we are considering the weak that appears due to measurement, but we are considering the weak
measurement regime where $\gamma \ll J^\mathrm{cl}$ and thus the measurement regime where $\gamma \ll J$ and thus the
dynamics between the quantum jumps are actually dominated by the dynamics between the quantum jumps are actually dominated by the
tunnelling of atoms rather than the null outcomes. Furthermore, the tunnelling of atoms rather than the null outcomes. Furthermore, the
effect of the quantum jump is independent of the value of $\Gamma$ effect of the quantum jump is independent of the value of $\Gamma$
@ -477,20 +477,23 @@ and $a_\phi$ cannot be zero, but this is exactly the case for an
initial superfluid state. We have seen in Eq. \eqref{eq:jumpz0} that initial superfluid state. We have seen in Eq. \eqref{eq:jumpz0} that
the effect of a photodetection is to displace the wavepacket by the effect of a photodetection is to displace the wavepacket by
approximately $b^2$, i.e.~the width of the Gaussian, in the direction approximately $b^2$, i.e.~the width of the Gaussian, in the direction
of the positive $z$-axis. Therefore, even though the can oscillate in of the positive $z$-axis. Therefore, even though the atoms can
the absence of measurement it is the quantum jumps that are the oscillate in the absence of measurement it is the quantum jumps that
driving force behind this phenomenon. Furthermore, these oscillations are the driving force behind this phenomenon. Furthermore, these
grow because the quantum jumps occur at an average instantaneous rate oscillations grow because the quantum jumps occur at an average
proportional to $\langle \cd \c \rangle (t)$ which itself is instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
proportional to $(1+z)^2$. This means they are most likely to occur at itself is proportional to $(1+z)^2$. This means they are most likely
the point of maximum displacement in the positive $z$ direction at to occur at the point of maximum displacement in the positive $z$
which point a quantum jump provides positive feedback and further direction at which point a quantum jump provides positive feedback and
increases the amplitude of the wavefunction leading to the growth seen further increases the amplitude of the wavefunction leading to the
in Fig. \ref{fig:oscillations}a. The oscillations themselves are growth seen in Fig. \ref{fig:oscillations}(a). The oscillations
essentially due to the natural dynamics of coherently displaced atoms themselves are essentially due to the natural dynamics of coherently
in a lattice , but it is the measurement that causes the initial and displaced atoms in a lattice , but it is the measurement that causes
more importantly coherent displacement and the positive feedback drive the initial and more importantly coherent displacement and the
which causes the oscillations to continuously grow. positive feedback drive which causes the oscillations to continuously
grow. Furthermore, it is by engineering the measurement, and through
it the geometry of the modes, that we have control over the nature of
the correlated dynamics of the oscillations.
We have now seen the effect of the quantum jumps and how that leads to We have now seen the effect of the quantum jumps and how that leads to
oscillations between odd and even sites in a lattice. However, we have oscillations between odd and even sites in a lattice. However, we have
@ -529,35 +532,35 @@ $\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width
of the initial superfluid state. This result is interesting, because of the initial superfluid state. This result is interesting, because
it shows that the width of the Gaussian distribution is squeezed as it shows that the width of the Gaussian distribution is squeezed as
compared with its initial state which is exactly what we see in compared with its initial state which is exactly what we see in
Fig. \ref{fig:oscillations}a. However, if we substitute the parameter Fig. \ref{fig:oscillations}(a). However, if we substitute the
values used in that trajectory we only get a reduction in width by parameter values used in that trajectory we only get a reduction in
about $3\%$, but the maximum amplitude oscillations in look like they width by about $3\%$, but the maximum amplitude oscillations in look
have a significantly smaller width than the initial distribution. This like they have a significantly smaller width than the initial
discrepancy is due to the fact that the continuous variable distribution. This discrepancy is due to the fact that the continuous
approximation is only valid for $z \ll 1$ and thus it cannot explain variable approximation is only valid for $z \ll 1$ and thus it cannot
the final behaviour of the system. Furthermore, it has been shown that explain the final behaviour of the system. Furthermore, it has been
the width of the distribution $b^2$ does not actually shrink to a shown that the width of the distribution $b^2$ does not actually
constant value, but rather it keeps oscillating around the value given shrink to a constant value, but rather it keeps oscillating around the
in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However, what we do see value given in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However,
is that during the early stages of the trajectory, which are well what we do see is that during the early stages of the trajectory,
described by this model, is that the width does in fact stay roughly which are well described by this model, is that the width does in fact
constant. It is only in the later stages when the oscillations reach stay roughly constant. It is only in the later stages when the
maximal amplitude that the width becomes visibly reduced. oscillations reach maximal amplitude that the width becomes visibly
reduced.
\subsection{Three-Way Competition} \subsection{Three-Way Competition}
Now it is time to turn on the inter-atomic interactions, Now it is time to turn on the inter-atomic interactions, $U/J \ne
$U/J^\mathrm{cl} \ne 0$. As a result the atomic dynamics will change 0$. As a result the atomic dynamics will change as the measurement now
as the measurement now competes with both the tunnelling and the competes with both the tunnelling and the on-site interactions. A
on-site interactions. A common approach to study such open systems is common approach to study such open systems is to map a dissipative
to map a dissipative phase diagram by finding the steady state of the phase diagram by finding the steady state of the master equation for a
master equation for a range of parameter values range of parameter values \cite{kessler2012}. However, here we adopt a
\cite{kessler2012}. However, here we adopt a quantum optical approach quantum optical approach in which we focus on the conditional dynamics
in which we focus on the conditional dynamics of a single quantum of a single quantum trajectory as this corresponds to a single
trajectory as this corresponds to a single realisation of an realisation of an experiment. The resulting evolution does not
experiment. The resulting evolution does not necessarily reach a necessarily reach a steady state and usually occurs far from the
steady state and usually occurs far from the ground state of the ground state of the system.
system.
A key feature of the quantum trajectory approach is that each A key feature of the quantum trajectory approach is that each
trajectory evolves differently as it is conditioned on the trajectory evolves differently as it is conditioned on the
@ -673,7 +676,7 @@ $U = 0$ are well squeezed when compared to the inital state and this
is the case over here as well. However, as $U$ is increased the is the case over here as well. However, as $U$ is increased the
interactions prevent the atoms from accumulating in one place thus interactions prevent the atoms from accumulating in one place thus
preventing oscillations with a large amplitude which effectively makes preventing oscillations with a large amplitude which effectively makes
the squeezing less effective as seen in Fig. \ref{fig:Utraj}a. In the squeezing less effective as seen in Fig. \ref{fig:Utraj}(a). In
fact, we have seen towards the end of the last section how for small fact, we have seen towards the end of the last section how for small
amplitude oscillations that can be described by the effective amplitude oscillations that can be described by the effective
double-well model the width of the number distribution does not change double-well model the width of the number distribution does not change
@ -687,7 +690,7 @@ significant increase in fluctuations compared to the ground
state. This is simply due to the fact that the measurement destroys state. This is simply due to the fact that the measurement destroys
the Mott insulating state, which has small fluctuations due to strong the Mott insulating state, which has small fluctuations due to strong
local interactions, but then subsequently is not strong enough to local interactions, but then subsequently is not strong enough to
squeeze the resulting dynamics as shown in Fig. \ref{fig:Utraj}b. To squeeze the resulting dynamics as shown in Fig. \ref{fig:Utraj}(b). To
see why this is so easy for the quantum jumps to do we look at the see why this is so easy for the quantum jumps to do we look at the
ground state in first-order perturbation theory given by ground state in first-order perturbation theory given by
\begin{equation} \begin{equation}
@ -720,9 +723,10 @@ number of photons arriving in succession can destroy the ground
state. We have neglected all dynamics in between the jumps which would state. We have neglected all dynamics in between the jumps which would
distribute the new excitations in a way which will affect and possibly distribute the new excitations in a way which will affect and possibly
reduce the effects of the subsequent quantum jumps. However, due to reduce the effects of the subsequent quantum jumps. However, due to
the lack of any decay channels they will remain in the system and the lack of any serious decay channels they will remain in the system
subsequent jumps will still amplify them further destroying the ground and subsequent jumps will still amplify them further destroying the
state and thus quickly leading to a state with large fluctuations. ground state and thus quickly leading to a state with large
fluctuations.
In the strong measurement regime ($\gamma \gg J$) the measurement In the strong measurement regime ($\gamma \gg J$) the measurement
becomes more significant than the local dynamics and the system will becomes more significant than the local dynamics and the system will
@ -964,7 +968,7 @@ sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
profound consequences as this is the physical origin of the long-range profound consequences as this is the physical origin of the long-range
correlated tunneling events represented in Eq. \eqref{eq:hz} by the correlated tunneling events represented in Eq. \eqref{eq:hz} by the
fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
projection $\hat{P}_0$ is not sensitive to individual site projection $P_0$ is not sensitive to individual site
occupancies, but instead enforces a fixed value of the observable, occupancies, but instead enforces a fixed value of the observable,
i.e.~a single Zeno subspace. This is a striking difference with the i.e.~a single Zeno subspace. This is a striking difference with the
$t$-$J$ and other strongly interacting models. The strong interaction $t$-$J$ and other strongly interacting models. The strong interaction
@ -980,7 +984,7 @@ consist of many sites the stable configuration can be restored by a
tunnelling event from a completely different lattice site that belongs tunnelling event from a completely different lattice site that belongs
to the same mode. to the same mode.
In Fig.~\ref{fig:zeno}a we consider illuminating only the central In Fig.~\ref{fig:zeno}(a) we consider illuminating only the central
region of the optical lattice and detecting light in the diffraction region of the optical lattice and detecting light in the diffraction
maximum, thus we freeze the atom number in the $K$ illuminated sites maximum, thus we freeze the atom number in the $K$ illuminated sites
$\hat{N}_\text{K}$~\cite{mekhov2009prl,mekhov2009pra}. The measurement $\hat{N}_\text{K}$~\cite{mekhov2009prl,mekhov2009pra}. The measurement
@ -1043,7 +1047,7 @@ consequence of the dynamics being constrained to a Zeno subspace: the
virtual processes allowed by the measurement entangle the spatial virtual processes allowed by the measurement entangle the spatial
modes nonlocally. Since the measurement only reveals the total number modes nonlocally. Since the measurement only reveals the total number
of atoms in the illuminated sites, but not their exact distribution, of atoms in the illuminated sites, but not their exact distribution,
these multi-tunelling events cause the build-up of long range these multi-tunelling events cause the build-up of long-range
entanglement. This is in striking contrast to the entanglement caused entanglement. This is in striking contrast to the entanglement caused
by local processes which can be very confined, especially in 1D where by local processes which can be very confined, especially in 1D where
it is typically short range. This makes numerical calculations of our it is typically short range. This makes numerical calculations of our
@ -1184,7 +1188,7 @@ the individual Zeno subspace density matrices. One can easily recover
the projective Zeno limit by considering $\lambda \rightarrow \infty$ the projective Zeno limit by considering $\lambda \rightarrow \infty$
when all the subspaces completely decouple. This is exactly the when all the subspaces completely decouple. This is exactly the
$\gamma \rightarrow \infty$ limit discussed in the previous $\gamma \rightarrow \infty$ limit discussed in the previous
section. However, we have seen that it is crucial we only consider, section. However, we have seen that it is crucial we only consider
$\lambda^2 \gg \nu$, but not infinite. If the subspaces do not $\lambda^2 \gg \nu$, but not infinite. If the subspaces do not
decouple completely, then transitions within a single subspace can decouple completely, then transitions within a single subspace can
occur via other subspaces in a manner similar to Raman transitions. In occur via other subspaces in a manner similar to Raman transitions. In
@ -1443,7 +1447,7 @@ have the projector $P_0$ applied from one side,
e.g.~$\hat{\rho}_{0m}$. The term $\delta \c \hat{\rho} \delta \cd$ e.g.~$\hat{\rho}_{0m}$. The term $\delta \c \hat{\rho} \delta \cd$
applies the fluctuation operator from both sides so it does not matter applies the fluctuation operator from both sides so it does not matter
in this case, but it becomes relevant for terms such as in this case, but it becomes relevant for terms such as
$\delta \cd \delta \c \hat{\rho}$. It is important to note that this $ \hat{\rho} \delta \cd \delta \c$. It is important to note that this
term does not automatically vanish, but when the explicit form of our term does not automatically vanish, but when the explicit form of our
approximate density matrix is inserted, it is in fact zero. Therefore, approximate density matrix is inserted, it is in fact zero. Therefore,
we can omit this term using the information we gained from we can omit this term using the information we gained from
@ -1585,7 +1589,7 @@ multiplying each eigenvector with its corresponding time evolution
z_1 - \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t / z_1 - \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t /
\gamma} \\ \gamma} \\
\end{array} \end{array}
\right), \nonumber \right),
\end{equation} \end{equation}
where $z_i$ denote the overlap between the eigenvectors and the where $z_i$ denote the overlap between the eigenvectors and the
initial state, $z_i = \langle v_i | \Psi (0) \rangle$, with initial state, $z_i = \langle v_i | \Psi (0) \rangle$, with
@ -1646,11 +1650,11 @@ for $U = 0$) to derive the steady state. These two conditions in
momentum space are momentum space are
\begin{equation} \begin{equation}
\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, \nonumber \bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0,
\end{equation} \end{equation}
\begin{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, \nonumber \bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle,
\end{equation} \end{equation}
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
@ -1679,7 +1683,7 @@ we can now write the equation for the $N$-particle steady state
| \Psi \rangle \propto \left[ \prod_{i=1}^{(N - |\Delta N|)/2} | \Psi \rangle \propto \left[ \prod_{i=1}^{(N - |\Delta N|)/2}
\left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger \left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger
\right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{| \right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{|
\Delta N |} | 0 \rangle, \nonumber \Delta N |} | 0 \rangle,
\end{equation} \end{equation}
where $\phi_{i,k}$ are coefficients that depend on the trajectory where $\phi_{i,k}$ are coefficients that depend on the trajectory
taken to reach this state and $|0 \rangle$ is the vacuum state defined taken to reach this state and $|0 \rangle$ is the vacuum state defined

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@ -20,7 +20,7 @@ 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 most of the existing work \cite{LP2009, rogers2014, operators just like most of the existing work \cite{LP2009, rogers2014,
mekhov2012, ashida2015, ashida2015a}. However, we have shown in mekhov2012, ashida2015, ashida2015a}. However, we have shown in
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
@ -44,7 +44,7 @@ quantum optical potentials. All three have been covered in the context
of density-based measurement either here or in other works. However, of density-based measurement either here or in other works. However,
coupling to phase observables in lattices has only been proposed and coupling to phase observables in lattices has only been proposed and
considered in the context of nondestructive measurements (see Chapter considered in the context of nondestructive measurements (see Chapter
\ref{chap:qnd}) and quantum optical potentials. \cite{caballero2015, \ref{chap:qnd}) and quantum optical potentials \cite{caballero2015,
caballero2015njp, caballero2016, caballero2016a}. In this chapter, caballero2015njp, caballero2016, caballero2016a}. In this chapter,
we go in a new direction by considering the effect of measurement we go in a new direction by considering the effect of measurement
backaction on the atomic gas that results from such coupling. We backaction on the atomic gas that results from such coupling. We
@ -82,12 +82,11 @@ maximum of scattered light, when our measurement operator is given by
where the second equality follows from converting to momentum space, where the second equality follows from converting to momentum space,
denoted by index $k$, via denoted by index $k$, via
$b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates $b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates
an atom with momentum $k$ in the range an atom in the Brillouin zone. Note that this operator is diagonal in
$\{ \frac{(M-1) \pi} {M}, \frac{(M-2) \pi} {M}, ..., \pi \}$. Note momentum space which means that its eigenstates are simply Fock
that this operator is diagonal in momentum space which means that its momentum Fock states. We have also seen in Chapter
eigenstates are simply Fock momentum Fock states. We have also seen in \ref{chap:backaction} how the global nature of the jump operators
Chapter \ref{chap:backaction} how the global nature of the jump introduces a nonlocal quadratic term to the Hamiltonian,
operators introduces a nonlocal quadratic term to the Hamiltonian,
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the $\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the
competition between tunnelling and measurement backaction we again competition between tunnelling and measurement backaction we again
consider non-interacting atoms, $U = 0$. Therefore, $\B$ is consider non-interacting atoms, $U = 0$. Therefore, $\B$ is
@ -159,7 +158,7 @@ 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
Hamiltonian. This highlights an important feature of this measurement Hamiltonian. This highlights an important feature of this measurement
- it does not compete with atomic tunnelling, and represents a quantum - it does not compete with atomic tunnelling, and represents a quantum
nondemolition (QND) measurement of the phase-related observable non-demolition (QND) measurement of the phase-related observable
\cite{brune1992}. Eq. \eqref{eq:bmax} shows that regardless of the \cite{brune1992}. Eq. \eqref{eq:bmax} shows that regardless of the
initial state or the photocount trajectory the system will project initial state or the photocount trajectory the system will project
onto a superposition of eigenstates of the $\Bmax$ operator. In fact, onto a superposition of eigenstates of the $\Bmax$ operator. In fact,
@ -231,7 +230,7 @@ modes whilst a uniform pattern had only one mode, $b_k$. Furthermore,
note the similarities to note the similarities to
$\D = \Delta \hat{N} = \hat{N}_\mathrm{even} - \hat{N}_\mathrm{odd}$ $\D = \Delta \hat{N} = \hat{N}_\mathrm{even} - \hat{N}_\mathrm{odd}$
which is the density measurement operator obtained by illuminated the which is the density measurement operator obtained by illuminated the
alttice such that neighbouring sites scatter light in anti-phase. This lattice such that neighbouring sites scatter light in anti-phase. This
further highlights the importance of geometry for global measurement. further highlights the importance of geometry for global measurement.
Trajectory simulations confirm that there is no steady state. However, Trajectory simulations confirm that there is no steady state. However,
@ -320,15 +319,30 @@ impossible unless the measurement is strong enough for the quantum
Zeno effect to occur. Zeno effect to occur.
We now go beyond what we previously did and define a new type of We now go beyond what we previously did and define a new type of
projector $\mathcal{P}_M = \sum_{m \in M} P_m$, such that projector
$\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M$ and \begin{equation}
$\sum_M \mathcal{P}_M = \hat{1}$ where $M$ denotes some arbitrary \mathcal{P}_M = \sum_{m \in M} P_m,
subspace. The first equation implies that the subspaces can be built \end{equation}
from $P_m$ whilst the second and third equation are properties of such that
\begin{equation}
\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M
\end{equation}
\begin{equation}
\sum_M \mathcal{P}_M = \hat{1}
\end{equation}
where $M$ denotes some arbitrary subspace. The first equation implies that
the subspaces can be built from
$P_m$ whilst the second and third equation are properties of
projectors and specify that these projectors do not overlap and that projectors and specify that these projectors do not overlap and that
they cover the whole Hilbert space. Furthermore, we will also require they cover the whole Hilbert space. Furthermore, we will also require
that $[\mathcal{P}_M, \hat{H}_0 ] = 0$ and $[\mathcal{P}_M, \c] = that
0$. The second commutator simply follows from the definition \begin{equation}
[\mathcal{P}_M, \hat{H}_0 ] = 0,
\end{equation}
\begin{equation}
[\mathcal{P}_M, \c] = 0.
\end{equation}
The second commutator simply follows from the definition
$\mathcal{P}_M = \sum_{m \in M} P_m$, but the first one is $\mathcal{P}_M = \sum_{m \in M} P_m$, but the first one is
non-trivial. However, if we can show that non-trivial. However, if we can show that
$\mathcal{P}_M = \sum_{m \in M} | h_m \rangle \langle h_m |$, where $\mathcal{P}_M = \sum_{m \in M} | h_m \rangle \langle h_m |$, where
@ -407,19 +421,27 @@ eigenstates of the two operators overlap.
To find $\mathcal{P}_M$ we need to identify the subspaces $M$ which To find $\mathcal{P}_M$ we need to identify the subspaces $M$ which
satisfy the following relation satisfy the following relation
$\sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle \langle h_m \begin{equation}
|$. This can be done iteratively by (i) selecting some $P_m$, (ii) \mathcal{P}_M = \sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle
identifying the $| h_m \rangle$ which overlap with this subspace, \langle h_m |.
(iii) identifying any other $P_m$ which also overlap with these \end{equation}
$| h_m \rangle$ from step (ii). We repeat (ii)-(iii) for all the $P_m$ This can be done iteratively by
found in (iii) until we have identified all the subspaces $P_m$ linked \begin{enumerate}
in this way and they will form one of our $\mathcal{P}_M$ \item selecting some $P_m$,
projectors. If $\mathcal{P}_M \ne 1$ then there will be other \item identifying the $| h_m \rangle$ which overlap with this
subspaces $P_m$ which we have not included so far and thus we repeat subspace,
this procedure on the unused projectors until we identify all \item identifying any other $P_m$ which also overlap with these
$\mathcal{P}_M$. Computationally this can be straightforwardly solved $| h_m \rangle$ from step (ii).
with some basic algorithm that can compute the connected components of \item Repeat 2-3 for all the $P_m$ found in 3 until we
a graph. have identified all the subspaces $P_m$ linked in this way and
they will form one of our $\mathcal{P}_M$ projectors. If
$\mathcal{P}_M \ne 1$ then there will be other subspaces $P_m$
which we have not included so far and thus we repeat this
procedure on the unused projectors until we identify all
$\mathcal{P}_M$.
\end{enumerate}
Computationally this can be straightforwardly solved with some basic
algorithm that can compute the connected components of a graph.
The above procedure, whilst mathematically correct and always The above procedure, whilst mathematically correct and always
guarantees to generate the projectors $\mathcal{P}_M$, is very guarantees to generate the projectors $\mathcal{P}_M$, is very
@ -427,16 +449,31 @@ unintuitive and gives poor insight into the nature or physical meaning
of $\mathcal{P}_M$. In order to get a better understanding of these of $\mathcal{P}_M$. In order to get a better understanding of these
subspaces we need to define a new operator $\hat{O}$, with eigenspace subspaces we need to define a new operator $\hat{O}$, with eigenspace
projectors $R_m$, which commutes with both $\hat{H}_0$ and projectors $R_m$, which commutes with both $\hat{H}_0$ and
$\c$. Physically this means that $\hat{O}$ is a compatible observable $\c$,
with $\c$ and corresponds to a quantity conserved by the \begin{equation}
Hamiltonian. The fact that $\hat{O}$ commutes with the Hamiltonian [\hat{O}, \hat{H}_0 ] = 0,
implies that the projectors can be written as a sum of Hamiltonian \end{equation}
eigenstates $R_m = \sum_{h_i = h_m} | h_i \rangle \langle h_i |$ and \begin{equation}
thus a projector $\mathcal{P}_M = \sum_{m \in M} R_m$ is guaranteed to [\hat{O}, \c] = 0.
commute with the Hamiltonian and similarly since $[\hat{O}, \c] = 0$ \end{equation}
$\mathcal{P}_M$ will also commute with $\c$ as required. Therefore, Physically this means that $\hat{O}$ is a compatible observable with
$\mathcal{P}_M = \sum_{m \in M} R_m = \sum_{m \in M} P_m$ will satisfy $\c$ and corresponds to a quantity conserved by the Hamiltonian. The
all the necessary prerequisites. This is illustrated in fact that $\hat{O}$ commutes with the Hamiltonian implies that the
projectors can be written as a sum of Hamiltonian eigenstates
\begin{equation}
R_m = \sum_{h_i = h_m} | h_i \rangle \langle h_i |
\end{equation}
and thus a projector
\begin{equation}
\mathcal{P}_M = \sum_{m \in M} R_m
\end{equation}
is guaranteed to commute with the Hamiltonian and similarly since
$[\hat{O}, \c] = 0$ $\mathcal{P}_M$ will also commute with $\c$ as
required. Therefore,
\begin{equation}
\mathcal{P}_M = \sum_{m \in M} R_m = \sum_{m \in M} P_m
\end{equation}
will satisfy all the necessary prerequisites. This is illustrated in
Fig. \ref{fig:spaces}. Fig. \ref{fig:spaces}.
\begin{figure}[hbtp!] \begin{figure}[hbtp!]
@ -469,13 +506,18 @@ feature.
In our case, it is apparent from the form of $\Bmin$ and $\hat{H}_0$ In our case, it is apparent from the form of $\Bmin$ and $\hat{H}_0$
in Eqs. \eqref{eq:BminBeta} and \eqref{eq:H0Beta} that in Eqs. \eqref{eq:BminBeta} and \eqref{eq:H0Beta} that
$\hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger \begin{equation}
\tilde{\beta_k} = \n_k + \n_{k - \pi/a}$ commutes with both operators \hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger
for all $k$. Thus, we can easily construct \tilde{\beta_k} = \n_k + \n_{k - \pi/a}
$\hat{O} = \sum_\mathrm{RBZ} g_k \hat{O}_k$ for any arbitrary \end{equation}
$g_k$. Its eigenspaces, $R_m$, can then be easily constructed and commutes with both operators for all $k$. Thus, we can easily
their relationship with $P_m$ and $\mathcal{P}_M$ is illustrated in construct
Fig. \ref{fig:spaces} whilst the time evolution of \begin{equation}
\hat{O} = \sum_\mathrm{RBZ} g_k \hat{O}_k
\end{equation}
for any arbitrary $g_k$. Its eigenspaces, $R_m$, can then be easily
constructed and their relationship with $P_m$ and $\mathcal{P}_M$ is
illustrated in Fig. \ref{fig:spaces} whilst the time evolution of
$\langle \hat{O}_k \rangle$ for a sample trajectory is shown in $\langle \hat{O}_k \rangle$ for a sample trajectory is shown in
Fig. \ref{fig:projections}(a). Note that unlike the $\c$ or $\H_0$ we Fig. \ref{fig:projections}(a). Note that unlike the $\c$ or $\H_0$ we
can actually see that this observable's distribution does indeed can actually see that this observable's distribution does indeed

9
Chapter7/chapter7.tex
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@ -107,13 +107,14 @@ In this thesis we have covered significant areas of the broad field
that is quantum optics of quantum gases, but there is much more that that is quantum optics of quantum gases, but there is much more that
has been left untouched. Here, we have only considered spinless has been left untouched. Here, we have only considered spinless
bosons, but the theory can also been extended to fermions bosons, but the theory can also been extended to fermions
\cite{atoms2015, mazzucchi2016, mazzucchi2016af} and \cite{atoms2015, mazzucchi2016, mazzucchi2016af} and molecules
molecules \cite{LP2013} and potentially even photonic circuits \cite{LP2013} and potentially even photonic circuits
\cite{mazzucchi2016njp}. Furthermore, the question of quantum \cite{mazzucchi2016njp}. Furthermore, the question of quantum
measurement and its properties has been a subject of heated debate measurement and its properties has been a subject of heated debate
since the very origins of quantum theory yet it is still as mysterious since the very origins of quantum theory yet it is still as mysterious
as it was at the beginning of the $20^\mathrm{th}$ century. However, as it was at the beginning of the $20^\mathrm{th}$ century. However,
this work has hopefully demonstrated that coupling quantised light this work has hopefully demonstrated that coupling quantised light
fields to many-body systems provides a very rich playground for fields to many-body systems provides a very rich playground for
exploring new quantum mechanical phenomena beyond what would otherwise exploring new quantum mechanical phenomena especially the competition
be possible in other fields. between weak quantum measurement and many-body dynamics in ultracold
bosonic gases.

2
Preamble/preamble.tex
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@ -203,6 +203,8 @@
% ***************************** Shorthand operator notation ******************** % ***************************** Shorthand operator notation ********************
\DeclareMathAlphabet{\mathcal}{OMS}{cmsy}{m}{n}
\renewcommand{\H}{\hat{H}} \renewcommand{\H}{\hat{H}}
\newcommand{\n}{\hat{n}} \newcommand{\n}{\hat{n}}
\newcommand{\dn}{\delta \hat{n}} \newcommand{\dn}{\delta \hat{n}}

32
References/references.bib
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@ -1746,3 +1746,35 @@ doi = {10.1103/PhysRevA.87.043613},
month = {Jan}, month = {Jan},
publisher = {American Physical Society}, publisher = {American Physical Society},
} }
@article{bux2013,
title={Control of matter-wave superradiance with a high-finesse ring cavity},
author={Bux, Simone and Tomczyk, Hannah and Schmidt, D and
Courteille, Ph W and Piovella, N and Zimmermann, C},
journal={Phys. Rev. A},
volume={87},
number={2},
pages={023607},
year={2013},
publisher={APS}
}
@article{kessler2014,
title={Steering matter wave superradiance with an ultranarrow-band
optical cavity},
author={Ke{\ss}ler, H and Klinder, J and Wolke, M and Hemmerich, A},
journal={Phys. Rev. Lett.},
volume={113},
number={7},
pages={070404},
year={2014},
publisher={APS}
}
@article{landig2015,
title={Measuring the dynamic structure factor of a quantum gas
undergoing a structural phase transition},
author={Landig, Renate and Brennecke, Ferdinand and Mottl, Rafael
and Donner, Tobias and Esslinger, Tilman},
journal={Nat. Comms.},
volume={6},
year={2015},
publisher={Nature Publishing Group}
}

2
thesis.tex
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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]{Classes/PhDThesisPSnPDF} \documentclass[a4paper,12pt,times,numbered,print,draft]{Classes/PhDThesisPSnPDF}
% ****************************************************************************** % ******************************************************************************
% ******************************* Class Options ******************************** % ******************************* Class Options ********************************