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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
strongly correlated phenomena in an environment that is far more
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
equally important roles in order to push the boundaries of
what is possible in ultracold atomic systems.
@ -30,7 +30,7 @@
measurement of matter-phase-related variables such as global phase
coherence. We show how this unconventional approach opens up new
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 system’s own evolution.
\end{abstract}

158
Chapter1/chapter1.tex
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@ -11,8 +11,8 @@
\fi
The field of ultracold gases has been a rapidly growing field ever
since the first Bose-Einstein condensate (BEC) was obtained in 1995
The field of ultracold gases has been rapidly growing ever since the
first Bose-Einstein condensate (BEC) was obtained in 1995
\cite{anderson1995, bradley1995, davis1995}. This new quantum state of
matter is characterised by a macroscopic occupancy of the single
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
\cite{lewenstein2007, bloch2008}. Initially, the main focus of the
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,
madison2000, abo2001}. Fermi degeneracy in ultracold gases was
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
unprecendented control.
The modern field of ultracold gases is successful due to its
interdisciplinarity \cite{lewenstein2007, bloch2008}. Originally
condensed matter effects are now mimicked in controlled atomic systems
finding applications in areas such as quantum information
processing. A really new challenge is to identify novel phenomena
which were unreasonable to consider in condensed matter, but will
become feasible in new systems. One such direction is merging quantum
optics and many-body physics \cite{mekhov2012, ritsch2013}. Quantum
optics has been developping as a branch of quantum physics
independently of the progress in the many-body community. It describes
delicate effects such as quantum measurement, state engineering, and
systems that can generally be easily isolated from their environnment
due to the non-interacting nature of photons \cite{Scully}. However,
they are also the perfect candidate for studying open systems due the
advanced state of cavity technologies \cite{carmichael,
MeasurementControl}. On the other hand ultracold gases are now used
to study strongly correlated behaviour of complex macroscopic
ensembles where decoherence is not so easy to avoid or control. Recent
experimental progress in combining the two fields offered a very
promising candidate for taking many-body physics in a direction that
would not be possible for condensed matter \cite{baumann2010,
wolke2012, schmidt2014}. Two very recent breakthrough experiments
have even managed to couple an ultracold gas trapped in an optical
lattice to an optical cavity enabling the study of strongly correlated
systems coupled to quantized light fields where the quantum properties
of the atoms become imprinted in the scattered light
\cite{klinder2015, landig2016}.
The modern field of strongly correlated ultracold gases is successful
due to its interdisciplinarity \cite{lewenstein2007,
bloch2008}. Originally condensed matter effects are now mimicked in
controlled atomic systems finding applications in areas such as
quantum information processing. A really new challenge is to identify
novel phenomena which were unreasonable to consider in condensed
matter, but will become feasible in new systems. One such direction is
merging quantum optics and many-body physics \cite{mekhov2012,
ritsch2013}. Quantum optics has been developping as a branch of
quantum physics independently of the progress in the many-body
community. It describes delicate effects such as quantum measurement,
state engineering, and systems that can generally be easily isolated
from their environnment due to the non-interacting nature of photons
\cite{Scully}. However, they are also the perfect candidate for
studying open systems due the advanced state of cavity technologies
\cite{carmichael, MeasurementControl}. On the other hand ultracold
gases are now used to study strongly correlated behaviour of complex
macroscopic ensembles where decoherence is not so easy to avoid or
control. Recent experimental progress in combining the two fields
offered a very promising candidate for taking many-body physics in a
direction that would not be possible for condensed matter
\cite{baumann2010, wolke2012, schmidt2014}. Furthermore, two very
recent breakthrough experiments have even managed to couple an
ultracold gas trapped in an optical lattice to an optical cavity
enabling the study of strongly correlated systems coupled to quantised
light fields where the quantum properties of the atoms become
imprinted in the scattered light \cite{klinder2015, landig2016}.
There are three prominent directions in which the field of quantum
optics of quantum gases has progressed in. First, the use of quantised
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
certain conditions even perform quantum non-demolition (QND)
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
realm of ultracold gases where such non-demolition schemes have been
applied to both fermionic \cite{eckert2008qnd, roscilde2009} and
bosonic \cite{hauke2013, rogers2014}. In this thesis, we consider
light scattering in free space from a bosonic ultracold gas and show
that there are many prominent features that go beyond classical
optics. Even the scattering angular distribution is nontrivial with
Bragg conditions that are significantly different from the classical
case. Furthermore, we show that the direct coupling of quantised light
to the atomic systems enables the nondestructive probing beyond a
standard mean-field description. We demonstrate this by showing that
the whole phase diagram of a disordered one-dimensional Bose-Hubbard
Hamiltonian, which consists of the superfluid, Mott insulating, and
Bose glass phases, can be mapped from the properties of the scattered
light. Additionally, we go beyond standard QND approaches, which only
consider coupling to density observables, by also considering the
direct coupling of the quantised light to the interference between
neighbouring lattice sites. We show that not only is this possible to
achieve in a nondestructive manner, it is also achieved without the
need for single-site resolution. This is in contrast to the standard
destructive time-of-flight measurements currently used to perform
these measurements \cite{miyake2011}. Within a mean-field treatment
this enables probing of the order parameter as well as matter-field
quadratures and their squeezing. This can have an impact on atom-wave
metrology and information processing in areas where quantum optics has
already made progress, e.g.,~quantum imaging with pixellized sources
of non-classical light \cite{golubev2010, kolobov1999}, as an optical
lattice is a natural source of multimode nonclassical matter waves.
bosonic systems \cite{hauke2013, rogers2014}. In this thesis, we
consider light scattering in free space from a bosonic ultracold gas
and show that there are many prominent features that go beyond
classical optics. Even the scattering angular distribution is
nontrivial with Bragg conditions that are significantly different from
the classical case. Furthermore, we show that the direct coupling of
quantised light to the atomic systems enables the nondestructive
probing beyond a standard mean-field description. We demonstrate this
by showing that the whole phase diagram of a disordered
one-dimensional Bose-Hubbard Hamiltonian, which consists of the
superfluid, Mott insulating, and Bose glass phases, can be mapped from
the properties of the scattered light. Additionally, we go beyond
standard QND approaches, which only consider coupling to density
observables, by also considering the direct coupling of the quantised
light to the interference between neighbouring lattice sites. We show
that not only is this possible to achieve in a nondestructive manner,
it is also achieved without the need for single-site resolution. This
is in contrast to the standard destructive time-of-flight measurements
currently used to perform these measurements \cite{miyake2011}. Within
a mean-field treatment this enables probing of the order parameter as
well as matter-field quadratures and their squeezing. This can have an
impact on atom-wave metrology and information processing in areas
where quantum optics has already made progress, e.g.,~quantum imaging
with pixellized sources of non-classical light \cite{golubev2010,
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
open system many-body dynamics either via dissipation where we have no
control over the coupling to the environment or via controlled state
reduction using the measurement backaction due to photodetections. A
lot of effort was expanded in an attempt to minimise the influence of
the environment in order to extend decoherence times. However,
theoretical progress in the field has shown that instead being an
obstacle, dissipation can actually be used as a tool in engineering
quantum states \cite{diehl2008}. Furthermore, as the environment
coupling is varied the system may exhibit sudden changes in the
properties of its steady state giving rise to dissipative phase
transitions \cite{carmichael1980, werner2005, capriotti2005,
morrison2008, eisert2010, bhaseen2012, diehl2010,
reduction using the measurement backaction due to
photodetections. Initially, a lot of effort was exended in an attempt
to minimise the influence of the environment in order to extend
decoherence times. However, theoretical progress in the field has
shown that instead being an obstacle, dissipation can actually be used
as a tool in engineering quantum states \cite{diehl2008}. Furthermore,
as the environment coupling is varied the system may exhibit sudden
changes in the properties of its steady state giving rise to
dissipative phase transitions \cite{carmichael1980, werner2005,
capriotti2005, morrison2008, eisert2010, bhaseen2012, diehl2010,
vznidarivc2011}. An alternative approach to open systems is to look
at quantum measurement where we consider a quantum state conditioned
on the outcome of a single experimental run \cite{carmichael,
MeasurementControl}. In this approach we consider the solutions to a
stochastic Schr\"{o}dinger equation which will be a pure state, which
in contrast to dissipative systems is generally not the case. The
question of measurement and its effect on the quantum state has been
around since the inception of quantum theory and still remains a
largely open question \cite{zurek2002}. It wasn't long after the first
condenste was obtained that theoretical work on the effects of
measurement on BECs appeared \cite{cirac1996, castin1997,
in contrast to dissipative systems where this is generally not the
case. The question of measurement and its effect on the quantum state
has been around since the inception of quantum theory and still
remains a largely open question \cite{zurek2002}. It wasn't long after
the first condenste was obtained that theoretical work on the effects
of measurement on BECs appeared \cite{cirac1996, castin1997,
ruostekoski1997}. Recently, work has also begun on combining weak
measurement with the strongly correlated dynamics of ultracold gases
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
extending the notion of quantum Zeno dynamics into the realm of
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,
refael2006} and $\mathcal{PT}$ symmetry \cite{bender1998,
giorgi2010, zhang2013} to spatial order \cite{otterbach2014} and
novel phase transitions \cite{lee2014prx, lee2014prl}.
refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$
symmetry} \cite{bender1998, giorgi2010, zhang2013} to spatial order
\cite{otterbach2014} and novel phase transitions \cite{lee2014prx,
lee2014prl}.
Just like for the nondestructive measurements we also consider
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
been previously proposed for BECs in double-wells \cite{cirac1996,
castin1997, ruostekoski1997}, the extension to a lattice system is
not straightforward, but we will show it is possible to achieve with
our propsed setup by a careful optical arrangement. Within this
not straightforward. However, we will show it is possible to achieve
with our propsed setup by a careful optical arrangement. Within this
context we demonstrate a novel type of projection which occurs even
when there is significant competition with the Hamiltonian
dynamics. This projection is fundamentally different to the standard

385
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
systems, but the restriction to optical lattices is convenient for 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
the inherent tunability of such lattices. Furthermore, this model is
capable of describing a range of different experimental setups ranging
from a small number of sites with a large filling factor (e.g.~BECs
trapped in a double-well potential) to a an extended multi-site
lattice with a low filling factor (e.g.~a system with one atom per
site which will exhibit the Mott insulator to superfluid quantum phase
transition).
many-body atomic state over a broad range of parameter values due,
from free particles to strongly correlated systems, to the inherent
tunability of such lattices. Furthermore, this model is capable of
describing a range of different experimental setups ranging from a
small number of sites with a large filling factor (e.g.~BECs trapped
in a double-well potential) to a an extended multi-site lattice with a
low filling factor (e.g.~a system with one atom per site which will
exhibit the Mott insulator to superfluid quantum phase transition).
\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
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
simply count the number of photons with a photodetector, but it is
also possible to perform a quadrature measurement by using a homodyne
detection scheme. The experiment can be performed in free space where
light can scatter in any direction. The atoms can also be placed
inside a cavity which has the advantage of being able to enhance light
scattering in a particular direction. Furthermore, cavities allow for
the formation of a fully quantum potential in contrast to the
detection scheme \cite{carmichael, atoms2015}. The experiment can be
performed in free space where light can scatter in any direction. The
atoms can also be placed inside a cavity which has the advantage of
being able to enhance light scattering in a particular direction
\cite{bux2013, kessler2014, landig2015}. Furthermore, cavities allow
for the formation of a fully quantum potential in contrast to the
classical lattice trap.
\begin{figure}[htbp!]
\centering
\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
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
dimensionality our results straightforwardly generalise to 2- and 3-D
systems. This simplification allows us to present a much more
intuitive picture of the physical setup where we only need to concern
ourselves with a single angle for each optical mode. As shown in
Fig. \ref{fig:LatticeDiagram} the angle between the normal to the
lattice and the probe and detected beam are denoted by $\theta_0$ and
$\theta_1$ respectively. We will consider these angles to be tunable
although the same effect can be achieved by varying the wavelength of
the light modes. However, it is much more intuitive to consider
variable angles in our model as this lends itself to a simpler
geometrical representation.
dimensionality our results straightforwardly generalise to two- and
three-dimensional systems. This simplification allows us to present a
much more intuitive picture of the physical setup where we only need
to concern ourselves with a single angle for each optical mode. As
shown in Fig. \ref{fig:LatticeDiagram} the angle between the normal to
the lattice and the probe and detected beam are denoted by $\theta_0$
and $\theta_1$ respectively. We will consider these angles to be
tunable although the same effect can be achieved by varying the
wavelength of the light modes. However, it is much more intuitive to
consider variable angles in our model as this lends itself to a
simpler geometrical representation.
\section{Derivation of the Hamiltonian}
\label{sec:derivation}
A general many-body Hamiltonian coupled to a quantized light field in
second quantized can be separated into three parts,
A general many-body Hamiltonian coupled to a quantised light field in
second quantised can be separated into three parts,
\begin{equation}
\label{eq:FullH}
\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
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
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,
including the trapping potentials, are treated classically.
@ -205,7 +206,7 @@ An effective Hamiltonian which results in the same optical equations
of motion can be written as
$\H^\mathrm{eff}_1 = \H_f + \H^\mathrm{eff}_{1,a} +
\H^\mathrm{eff}_{1,fa}$. The effective atomic and interaction
Hamiltonians are
Hamiltonians are
\begin{equation}
\label{eq:aeff}
\H^\mathrm{eff}_{1,a} = \frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}),
@ -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
Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$,
\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 +
\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).
@ -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.
Furthermore, since we will only be looking at lattices that have the
same separtion between all its nearest neighbours (e.g. cubic or
square lattice) we can define $J_{i,j}^\mathrm{cl} = - J^\mathrm{cl}$
(negative sign, because this way $J^\mathrm{cl} > 0$). For the
inter-atomic interactions this simplifies to simply considering
on-site collisions where $i=j=k=l$ and we define $U_{iiii} =
U$. Finally, we end up with the canonical form for the Bose-Hubbard
Hamiltonian
square lattice) we can define $J_{i,j}^\mathrm{cl} = - J$ (negative
sign, because this way $J > 0$). For the inter-atomic interactions
this simplifies to simply considering on-site collisions where
$i=j=k=l$ and we define $U_{iiii} = U$. Finally, we end up with the
canonical form for the Bose-Hubbard Hamiltonian
\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),
\end{equation}
where $\langle i,j \rangle$ denotes a summation over nearest
@ -322,39 +322,41 @@ nearest-neighbour tunnelling operators
\end{equation}
where $K$ denotes a sum over the illuminated sites and we neglect
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
encapsulate the simplicity and flexibility of the measurement scheme
that we are proposing. The operators given above are entirely
determined by the values of the $J^{l,m}_{i,j}$ coefficients and
despite its simplicity, this is sufficient to give rise to a host of
interesting phenomena via measurement back-action such as the
generation of multipartite entangled spatial modes in an optical
lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the appearance
of long-range correlated tunnelling capable of entangling distant
lattice sites, and in the case of fermions, the break-up and
These equations encapsulate the simplicity and flexibility of the
measurement scheme that we are proposing. The operators given above
are entirely determined by the values of the $J^{l,m}_{i,j}$
coefficients and despite its simplicity, this is sufficient to give
rise to a host of interesting phenomena via measurement backaction
such as the generation of multipartite entangled spatial modes in an
optical lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the
appearance of long-range correlated tunnelling capable of entangling
distant lattice sites, and in the case of fermions, the break-up and
protection of strongly interacting pairs \cite{mazzucchi2016,
kozlowski2016zeno}. Additionally, these coefficients are easy to
manipulate experimentally by adjusting the optical geometry via the
light mode functions $u_l(\b{r})$.
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
necessary to determine the Wannier functions in a self-consistent way
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,
the atomic tunnelling and interaction coefficients will now depend on
the quantum state of light. \mynote{cite Santiago's papers and
Maschler/Igor EPJD}
the quantum state of light \cite{mekhov2008}.
Therefore, combining these final simplifications we finally arrive at
our quantum light-matter Hamiltonian
\begin{equation}
\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{\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,
@ -387,20 +389,21 @@ approximation of the Coulomb screening interaction as a simple on-site
interaction. Despite these enormous simplifications the model was very
succesful \cite{leggett}. The Bose-Hubbard model is an even simpler
variation where instead of fermions we consider spinless bosons. It
was originally devised as a toy model, but in 1998 it was shown by
Jaksch \emph{et.~al.} that it can be realised with ultracold atoms in
an optical lattice \cite{jaksch1998}. Shortly afterwards it was
obtained in a ground-breaking experiment \cite{greiner2002}. The model
has been the subject of intense research since then, because despite
its simplicity it possesses highly nontrivial properties such as the
superfluid to Mott insulator quantum phase transition. Furthermore, it
is one of the most controllable quantum many-body systems thus
providing a solid basis for new experiments and technologies.
was originally devised as a toy model and applied to liquid helium
\cite{fisher1989}, but in 1998 it was shown by Jaksch \emph{et.~al.}
that it can be realised with ultracold atoms in an optical lattice
\cite{jaksch1998}. Shortly afterwards it was obtained in a
ground-breaking experiment \cite{greiner2002}. The model has been the
subject of intense research since then, because despite its simplicity
it possesses highly nontrivial properties such as the superfluid to
Mott insulator quantum phase transition. Furthermore, it is one of the
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
well-known Bose-Hubbard model that also includes interactions with
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
\begin{equation}
\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}
b_\b{k} = \frac{1} {\sqrt{M}} \sum_m b_m e^{i \b{k} \cdot \b{r}_m},
\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
\begin{equation}
\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
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
approach the interaction term is treated exactly, but the kinetic
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 $\hat{H}_a
= \sum_m^M \hat{h}_a$, where
\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,
\end{equation}
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
site-wise product of the individual ground states of
$\hat{h}_a$. 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
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
study the quantum phase transition between the sueprfluid and Mott
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
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}
\label{sec:BHM1D}
The mean-field theory in the previous section is very useful tool for
studying the quantum phase transition in the Bose-Hubbard
model. However, it is effectively an infinite-dimensional theory and
in practice it only works in two dimensions or more. The phase
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}.
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}
\centering
\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}}
\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}
\label{sec:a}
@ -683,11 +690,12 @@ $\hat{F}_{1,0}$
where we have defined $C = U_{1,0} a_0 / (\Delta_{p} + i \kappa)$
which is essentially the Rayleigh scattering coefficient into the
cavity. Furthermore, since there is no longer any ambiguity in the
indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv
\Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv
\hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv
\hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and the coefficients
$J^{1,0}_{i,j} \equiv J_{i,j}$.
indices $l$ and $m$, we have dropped indices on
$\Delta_{1p} \equiv \Delta_p$, $\kappa_1 \equiv \kappa$, and
$\hat{F}_{1,0} \equiv \hat{F}$. We also do the same for the operators
$\hat{D}_{1,0} \equiv \hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and
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
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
component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i
b_j$. In general, the density component dominates,
$\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However,
it is possible to engineer an optical geometry in which $\hat{D} = 0$
leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is
fundamentally different from the aforementioned double-well proposals
as it directly couples to the interference terms caused by atoms
tunnelling rather than combining light scattered from different
sources. Furthermore, it is not limited to a double-wellsetup and
naturally extends to a lattice structure which is a key
advantage. Such a counter-intuitive configuration may affect works on
quantum gases trapped in quantum potentials \cite{mekhov2012,
mekhov2008, larson2008, chen2009, habibian2013, ivanov2014,
caballero2015} and quantum measurement-induced preparation of
many-body atomic states \cite{mazzucchi2016, mekhov2009prl,
pedersen2014, elliott2015}.
\mynote{add citiations above if necessary}
$\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$
\cite{mekhov2012}. Physically, this is a consequence of the fact that
there are more atoms to scatter light at the lattice sites than in
between them. However, it is possible to engineer an optical geometry
in which $\hat{D} = 0$ leaving $\hat{B}$ as the dominant term in
$\hat{F}$. This approach is fundamentally different from the
aforementioned double-well proposals as it directly couples to the
interference terms caused by atoms tunnelling rather than combining
light scattered from different sources. Furthermore, it is not limited
to a double-well setup and naturally extends to a lattice structure
which is a key advantage. Such a counter-intuitive configuration may
affect works on quantum gases trapped in quantum potentials
\cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013,
ivanov2014, caballero2015} and quantum measurement-induced
preparation of many-body atomic states \cite{mazzucchi2016,
mekhov2009prl, pedersen2014, elliott2015}.
For clarity we will consider a 1D lattice as shown in
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
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
calculations using realistic Wannier functions
\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
\begin{align}
\label{eq:FTs}
\hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_-
x_m +\varphi_-) \nonumber\\
& + \mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+ x_m +\varphi_+)],
\hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_i\hat{n}_i\cos(k_-
x_i +\varphi_-) \nonumber\\
& + \mathcal{F}[W_0](k_+)\sum_i\hat{n}_i\cos(k_+ x_i +\varphi_+)],
\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\\
& +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
x_m +\frac{k_+d}{2}+\varphi_+)],
& +\mathcal{F}[W_1](k_+)\sum_i\hat{B}_i\cos(k_+
x_i +\frac{k_+d}{2}+\varphi_+)],
\end{align}
where $k_\pm = k_{0x} \pm k_{1x}$,
$k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$),
$\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, 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
$\hat{B}$ 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
between the lattice sites and not at the positions of the atoms
allowing to decouple them at specific angles.
\begin{figure}[hbtp!]
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{BDiagram}
\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}}
\end{figure}
\begin{figure}[hbtp!]
\begin{figure}
\centering
\includegraphics[width=\linewidth]{WF_S}
\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
(inter-site term) scatters light in phase and the operator is given by
\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}
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
@ -888,7 +893,7 @@ Another possibility is to obtain an alternating pattern similar
corresponding to a diffraction minimum where each bond scatters light
in anti-phase with its neighbours giving
\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}
where $J^B_\mathrm{min}$ is a constant. We consider an arrangement
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
\begin{align}
\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 \\
\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).
\end{align}
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
wave vector,
$\mathcal{E}_{\b{k}} = \sqrt{\hbar \omega_{\b{k}} / 2 \epsilon_0 V}$,
$\epsilon_0$ 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
creation operators respectively of a photon in mode $\b{k}$, and
$\omega_\b{k}$ 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
the many body operator is then
\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} -
\b{r}_j|} e^ {i \b{k}_1 \cdot (\b{r} - \b{r}_j
) - 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
are given in Table \ref{tab:photons}. Rubidium atom data has been
taken from Ref. \cite{steck}. Miyake \emph{et al.} experimental
parameters are from Ref. \cite{miyake2011}. The $5^2S_{1/2}$,
taken from Ref. \cite{steck}. The two experiments were chosen as state
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
is considered. For this transition the Rabi frequency is actually
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
treat the fully quantum regime of light-matter interactions we now
consider possible applications. There are three prominent directions
in which we can proceed: nondestructive probing of the quantum state
of matter, quantum measurement backaction induced dynamics and quantum
optical lattices. Here, we deal with the first of the three options.
consider possible applications. We will first look at nondestructive
measurement where measurement backaction can be neglected and we focus
on what expectation values can be extracted via optical methods.
In this chapter we develop a method to measure properties of ultracold
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
insulator - superfluid - Bose glass phase transition in a 1D
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
fixed-density scattering. It was only recently that an experiment
distinguished a Mott insulator from a Bose glass \cite{derrico2014}
via a series of destructive measurements. Our proposal, on the other
hand, is nondestructive and is capable of extracting all the relevant
information in a single experiment making our proposal timely.
distinguished a Mott insulator from a Bose glass via a series of
destructive measurements \cite{derrico2014}. Our proposal, on the
other hand, is nondestructive and is capable of extracting all the
relevant information in a single experiment making our proposal
timely.
Having shown the possibilities created by this nondestructive
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
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
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
the scheme is nondestructive (in some cases, it even satisfies the
stricter requirements for a QND measurement \cite{mekhov2012,
@ -115,6 +116,8 @@ density correlations to matter-field interference.
\section{On-site Density Measurements}
\subsection{Diffraction Patterns and Bragg Conditions}
We have seen in section \ref{sec:B} that typically the dominant term
in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009,
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
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
standing wave scattered mode. The first noticeable feature is the
isotropic background which does not exist in classical
diffraction. This background yields information about density
fluctuations which, according to mean-field estimates (i.e.~inter-site
correlations are ignored), are related by
standing wave mode. The first noticeable feature is the isotropic
background which does not exist in classical diffraction. This
background yields information about density fluctuations which,
according to mean-field estimates (i.e.~inter-site correlations are
ignored), are related by
$R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
)/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, 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
fluctuations entirely leading to absolutely no ``quantum addition''.
\begin{figure}[htbp!]
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Ep1}
\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
insulator has no ``quantum addition'' due to a lack of density
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
fluctuation correlations are equal regardless of their separation,
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
mostly zero except when the classical Bragg condition is
satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle
such that the Bragg is not satisfied this term is essentially
zero. Therefore, we are left with the first term $\sum_i^K |A_i|^2$
which for a travelling wave probe and a standing wave scattered mode
is
such that the Bragg condition is not satisfied this term is
essentially zero. Therefore, we are left with the first term
$\sum_i^K |A_i|^2$ which for a travelling wave probe and a standing
wave scattered mode is
\begin{equation}
\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
@ -388,7 +391,7 @@ should be visible using currently available technology, especially
since the most prominent features, such as Bragg diffraction peaks, do
not coincide at all with the classical diffraction pattern.
\section{Mapping the quantum phase diagram}
\subsection{Mapping the Quantum Phase Diagram}
We have shown that scattering from atom number operators leads to a
purely quantum diffraction pattern which depends on the density
@ -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
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
correlations and a Mott insulator will not contriute any ``quantum
addition'' at all, but we have not look at the quantum phase
correlations and a Mott insulator will not contribute any ``quantum
addition'' at all, but we have not looked at the quantum phase
transition between these two phases. In two or higher dimensions this
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
@ -410,8 +413,9 @@ much more information.
There are many situations where the mean-field approximation is not a
valid description of the physics. A prominent example is the
Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000,
pino2012, pino2013}. Observing the transition in 1D by light at
fixed density was considered to be difficult \cite{rogers2014} or even
pino2012, pino2013} as we have seen in section
\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
quantum phase transition is in a different universality class than its
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,
which is a function of the two-point correlation $\langle \delta
\hat{n}_i \delta \hat{n}_j \rangle$. 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
scattered modes are plane waves which can be easily achieved in free
space. We will again consider the case of light being maximally
@ -449,8 +453,6 @@ addition is given by
\hat{n}_j \rangle.
\end{equation}
\mynote{can put in more detail here with equations}
This alone allows us to analyse the phase transition quantitatively
using our method. Unlike in higher dimensions where an order parameter
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
this phase transition. However, a valid description of the relevant 1D
low energy physics is provided by Luttinger liquid theory
\cite{giamarchi}. In this model correlations in the supefluid phase as
well as the superfluid density itself are characterised by the
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
the critical point in the thermodynamic limit at $K_b = 1/2$. This
quantity can be extracted from various correlation functions and in
our case it can be extracted directly from $R$ \cite{ejima2011}. This
quantity was used in numerical calculations that used highly efficient
density matrix renormalisation group (DMRG) methods to calculate the
ground state to subsequently fit the Luttinger theory to extract this
parameter $K_b$. These calculations yield a theoretical estimate of
the critical point in the thermodynamic limit for commensurate filling
in 1D to be at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our
proposal provides a method to directly measure $R$ nondestructively in
a lab which can then be used to experimentally determine the location
of the critical point in 1D.
\cite{giamarchi} as seen in section \ref{sec:BHM1D}. In this model
correlations in the supefluid phase as well as the superfluid density
itself are characterised by the Tomonaga-Luttinger parameter,
$K$. This parameter also identifies the critical point in the
thermodynamic limit at $K_c = 1/2$. This quantity can be extracted
from various correlation functions and in our case it can be extracted
directly from $R$ \cite{ejima2011}. This quantity was used in
numerical calculations that used highly efficient density matrix
renormalisation group (DMRG) methods to calculate the ground state to
subsequently fit the Luttinger theory to extract this parameter
$K$. These calculations yield a theoretical estimate of the critical
point in the thermodynamic limit for commensurate filling in 1D to be
at $U/2J \approx 1.64$ \cite{ejima2011}. Our proposal provides a
method to directly measure $R$ nondestructively in a lab which can
then be used to experimentally determine the location of the critical
point in 1D.
However, whilst such an approach will yield valuable quantitative
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
by
\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
\sum_i \hat{n}_i,
\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
confining potential on top of the optical lattice to keep the atoms in
place which means that the chemical potential will vary in
space. However, with careful consideration of the full
($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
system \cite{batrouni2002}.
space. However, with careful consideration of the full ($\mu/2J$,
$U/2J$) phase diagrams in Fig. \ref{fig:SFMI}(d,e) our analysis can
still be applied to the system \cite{batrouni2002}.
\begin{figure}[htbp!]
\begin{figure}
\centering
\includegraphics[width=\linewidth]{oph11}
\includegraphics[width=\linewidth]{oph11_3}
\caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular
dependence of scattered light $R$ for a superfluid (thin black,
left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green,
right scale, $U/2J^\text{cl} =10$). The two phases differ in both
their value of $R_\text{max}$ as well as $W_R$ showing that
density correlations in the two phases differ in magnitude as well
as extent. Light scattering maximum $R_\text{max}$ is shown in (b,
d) and the width $W_R$ in (c, e). It is very clear that varying
chemical potential $\mu$ or density $\langle n\rangle$ sharply
identifies the superfluid-Mott insulator transition in both
quantities. (b) and (c) are cross-sections of the phase diagrams
(d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple),
and 4 (dashed blue). Insets show density dependencies for the
$U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.}
dependence of scattered light $R$ for a superfluid (thin purple,
left scale, $U/2J = 0$) and Mott insulator (thick blue, right
scale, $U/2J =10$). The two phases differ in both their value of
$R_\text{max}$ as well as $W_R$ showing that density correlations
in the two phases differ in magnitude as well as extent. Light
scattering maximum $R_\text{max}$ is shown in (b, d) and the width
$W_R$ in (c, e). It is very clear that varying chemical potential
$\mu$ or density $\langle n\rangle$ sharply identifies the
superfluid-Mott insulator transition in both quantities. (b) and
(c) are cross-sections of the phase diagrams (d) and (e) at
$U/2J=2$ (thick blue), 3 (thin purple), and 4 (dashed
blue). Insets show density dependencies for the $U/(2 J) = 3$
line. $K=M=N=25$.}
\label{fig:SFMI}
\end{figure}
We then consider probing these ground states using our optical scheme
and we calculate the ``quantum addition'', $R$, based on these ground
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 $R$,
$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
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
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
insulating phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the
smaller these values will be until ultimately it will scatter no light
at all in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a
this can be seen in the value of the peak in $R$. The value
$R_\text{max}$ in the superfluid phase ($U/2J^\text{cl} = 0$) is
larger than its value in the Mott insulating phase
($U/2J^\text{cl} = 10$) by a factor of
insulating phase (i.e. the larger the $U/2J$ ratio is), the smaller
these values will be until ultimately it will scatter no light at all
in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}(a) this
can be seen in the value of the peak in $R$. The value $R_\text{max}$
in the superfluid phase ($U/2J = 0$) is larger than its value in the
Mott insulating phase ($U/2J = 10$) by a factor 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
things to note at this point. Firstly, if we follow the transition
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 the transition is very smooth and it is hard to see a definite
critical point. This is due to the energy gap closing exponentially
slowly which makes precise identification of the critical point
extremely difficult. The best option at this point would be to fit
Tomonaga-Luttinger theory to the results in order to find this
critical point. However, we note that there is a drastic 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
that is in between the two white lines in Fig. \ref{fig:SFMI}(d)) we
see that the transition is very smooth and it is hard to see a
definite critical point. This is due to the energy gap closing
exponentially slowly which makes precise identification of the
critical point extremely difficult. The best option at this point
would be to fit Tomonaga-Luttinger theory to the results in order to
find this critical point. However, we note that there is a drastic
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
the Mott insulator can be easily identified by a dip in the quantity
$R_\text{max}$.
@ -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
two-point correlations \cite{giamarchi} which significantly reduces
the dip in the $R$. This can be seen in
Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see
that the transition is much sharper as $\mu$ is varied. This is shown
in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between
a superfluid and an insulating state is fairly significant
Fig. \ref{fig:SFMI}(a). Furthermore, just like for $R_\text{max}$ we
see that the transition is much sharper as $\mu$ is varied. This is
shown in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle
between a superfluid and an insulating state is fairly significant
$\sim 20^\circ$ which should make the two phases easy to identify
using this measure. In this particular case, measuring $W_R$ in the
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
Hamiltonian can be shown to be
\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) +
\frac{V}{2} \sum_i \left[ 1 + \cos (2 r \pi m + 2 \phi) \right]
\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
will only consider the phase diagram at fixed density as the
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
compressible and incompressible phases between successive Mott
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
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
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 $W_R$.
\begin{figure}[htbp!]
\begin{figure}
\centering
\includegraphics[width=\linewidth]{oph22}
\includegraphics[width=\linewidth]{oph22_3}
\caption[Mapping the Disoredered Phase Diagram]{The
Mott-superfluid-glass phase diagrams for light scattering maximum
$R_\text{max}/N_K$ (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,
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
description of the Bose-Hubbard model in order to obtain a simple
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
\begin{equation}
\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}
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
@ -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
difraction maximum.
\begin{figure}[htbp!]
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Quads}
\captionsetup{justification=centerlast,font=small}
\caption[Mean-Field Matter Quadratures]{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. (a) The variances of
\includegraphics[width=\linewidth]{QuadsC}
\caption[Mean-Field Matter Quadratures]{Mean-field quadratures and
resulting photon scattering rates. (a) The variances of
quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$
(dashed) of the matter field across the phase transition. Level
1/4 is the minimal (Heisenberg) uncertainty. There are three
important points along the phase transition: the coherent 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}
\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 ) ],
\end{equation}
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,
$n$, 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
@ -831,8 +831,8 @@ themselves only in high-order correlations \cite{kaszlikowski2008}.
\section{Conclusions}
In this chapter we explored the possibility of nondestructively
probing a quantum gas trapped in an optical lattice using quantized
light. Firstly, we showed that the density-term in scattering has an
probing a quantum gas trapped in an optical lattice using quantised
light. Firstly, we showed that the density term in scattering has an
angular distribution richer than classical diffraction, derived
generalized Bragg conditions, and estimated parameters for two
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
changes may happen not only due to external gradients, but also due to
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
how in mean-field, one can measure the matter-field amplitude (order
parameter), quadratures and squeezing. This can link atom optics to

45
Chapter4/chapter4.tex
View File

@ -18,14 +18,14 @@
\section{Introduction}
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
where we neglect the effect of the quantum wavefunction collapse. We
have shown that light provides information about various statistical
quantities of the quantum states of the atoms such as their
correlation functions. In general, any quantum measurement affects the
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
the electromagnetic wavefunction of the optical field collapses, the
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
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
which will be useful as some of the results in the following chapters
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 $\c$
will depend on the nature of the measurement we are considering. For
example, if we consider measuring the photons escaping from a leaky
cavity then $\c = \sqrt{2 \kappa} \hat{a}$, where $\kappa$ is the
cavity decay rate and $\hat{a}$ is the annihilation operator of a
cavity then $\c = \sqrt{2 \kappa} \a$, where $\kappa$ is the
cavity decay rate and $\a$ is the annihilation operator of a
photon in the cavity field. It is interesting to note that due to
renormalisation the effect of a single quantum jump is independent of
the magnitude of the operator $\c$ itself. However, larger operators
@ -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
Hamiltonian. Related techniques such as Time-Evolving Block Decimation
(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
non-Hermitian term added due to measurement it is actually difficult
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,
as we shall see in the following chapters, leads to long-range
correlations regardless of coupling strength. Therefore, we resort to
using exact methods such as exact diagonalisation which we solve with
well-known ordinary differential equation solvers. This means that we
can at most simulate a few atoms, but as we shall see it is the
geometry of the measurement that matters the most and these effects
are already visible in smaller systems.
using alternative methods such as exact diagonalisation which we solve
with well-known ordinary differential equation solvers. This means
that we can at most simulate a few atoms, but as we shall see it is
the geometry of the measurement that matters the most and these
effects are already visible in smaller systems.
\section{The Master Equation}
\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
outcome. Therefore, it is useful when extracting features that are
common to many trajectories, regardless of the exact timing of the
events. However, in this case we do not want to impose any specific
trajectory on the system as we are not interested in a specific
experimental run, but we would still like to identify the set of
possible outcomes and their common properties. Unfortunately,
calculating the inverse of Eq. \eqref{eq:rho} is not an easy task. In
fact, the decomposition of a density matrix into pure states might not
even be unique. However, if a measurement leads to a projection,
i.e.~the final state becomes confined to some subspace of the Hilbert
space, then this will be visible in the final state of the density
matrix. We will show this on an example of a qubit in the quantum
state
events. In this case we do not want to impose any specific trajectory
on the system as we are not interested in a specific experimental run,
but we would still like to identify the set of possible outcomes and
their common properties. Unfortunately, calculating the inverse of
Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of
a density matrix into pure states might not even be unique. However,
if a measurement leads to a projection, i.e.~the final state becomes
confined to some subspace of the Hilbert space, then this will be
visible in the final state of the density matrix. We will show this on
an example of a qubit in the quantum state
\begin{equation}
\label{eq:qubit0}
| \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle,
@ -543,7 +542,7 @@ Fig. \ref{fig:twomodes}.
\label{fig:twomodes}
\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
$J_{i,i} = u_1^* (\b{r}) u_0 (\b{r})$. We will take the probe beam to
be incident normally at a 1D lattice so that $u_0 (\b{r}) =

190
Chapter5/chapter5.tex
View File

@ -17,9 +17,9 @@
In the previous chapter we have introduced a theoretical framework
which will allow us to study measurement backaction using
discontinuous quantum jumps and non-Hermitian evolution due to null
outcomesquantum trajectories. We have also wrapped our quantum gas
model in this formalism by considering ultracold bosons in an optical
lattice coupled to a cavity which collects and enhances light
outcomes using quantum trajectories. We have also wrapped our quantum
gas model in this formalism by considering ultracold bosons in an
optical lattice coupled to a cavity which collects and enhances light
scattered in one particular direction. One of the most important
conclusions of the previous chapter was that the introduction of
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
correlations which enable nonlocal dynamical processes. Furthermore,
global light scattering from multiple lattice sites creates nontrivial
spatially nonlocal coupling to the environment which is impossible to
obtain with local interactions \cite{daley2014, diehl2008,
syassen2008}. These spatial modes of matter fields can be considered
as designed systems and reservoirs opening the possibility of
controlling dissipations in ultracold atomic systems without resorting
to atom losses and collisions which are difficult to manipulate. Thus
the continuous measurement of the light field introduces a
controllable decoherence channel into the many-body dynamics. Such a
quantum optical approach can broaden the field even further allowing
quantum simulation models unobtainable using classical light and the
design of novel systems beyond condensed matter analogues.
spatially nonlocal coupling to the environment, as seen in section
\ref{sec:modes}, which is impossible to obtain with local interactions
\cite{daley2014, diehl2008, syassen2008}. These spatial modes of
matter fields can be considered as designed systems and reservoirs
opening the possibility of controlling dissipations in ultracold
atomic systems without resorting to atom losses and collisions which
are difficult to manipulate. Thus the continuous measurement of the
light field introduces a controllable decoherence channel into the
many-body dynamics. Such a quantum optical approach can broaden the
field even further allowing quantum simulation models unobtainable
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
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.
\end{equation}
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 $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$,
$N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums
of these vectors is invariant under rotations by $2 \pi l R / Z$,
@ -173,7 +174,7 @@ in pairs resulting in only three visible components.
We will now limit ourselves to a specific illumination pattern with
$\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest
multimode dynamics with $Z = 2$ and only a single component as seen in
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
crossing two beams such that their projections on the lattice are
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
\begin{equation}
\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
\end{equation}
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
which is the case for a superfluid state. We now rescale the
Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by
dividing by $NJ^\mathrm{cl}$ and define the relative population
imbalance between the two wells $z = 2x - 1$. Finally, by taking the
expectation value of the Hamiltonian and looking for the stationary
points of
dividing by $NJ$ and define the relative population imbalance between
the two wells $z = 2x - 1$. Finally, by taking the expectation value
of the Hamiltonian and looking for the stationary points of
$\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
\rangle$ we obtain the semiclassical Schr\"{o}dinger equation
\begin{equation}
@ -243,21 +243,21 @@ $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
\right)^2 \right] \psi(z, t),
\end{equation}
where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$,
$\omega = 2 \sqrt{1 + \Lambda - h}$, and
$\Lambda = NU / (2J^\mathrm{cl})$. The full derivation is not
straightforward, but the introduction of the non-Hermitian term
requires only a minor modification to the original formalism presented
in detail in Ref. \cite{juliadiaz2012} so we have omitted it here. We
will also be considering $U = 0$ as the effective model is only valid
in this limit, thus $\Lambda = 0$. However, this model is valid for an
actual physical double-well setup in which case interacting bosons can
also be considered. The equation is defined on the interval
$z \in [-1, 1]$, but $z \ll 1$ has been assumed in order to simplify
the kinetic term and approximate the potential as parabolic. This does
mean that this approximation is not valid for the maximum amplitude
oscillations seen in Fig. \ref{fig:oscillations}a, but since they
already appear early on in the trajectory we are able to obtain a
valid analytic description of the oscillations and their growth.
$\omega = 2 \sqrt{1 + \Lambda - h}$, and $\Lambda = NU / (2J)$. The
full derivation is not straightforward, but the introduction of the
non-Hermitian term requires only a minor modification to the original
formalism presented in detail in Ref. \cite{juliadiaz2012} so we have
omitted it here. We will also be considering $U = 0$ as the effective
model is only valid in this limit, thus $\Lambda = 0$. However, this
model is valid for an actual physical double-well setup in which case
interacting bosons can also be considered. The equation is defined on
the interval $z \in [-1, 1]$, but $z \ll 1$ has been assumed in order
to simplify the kinetic term and approximate the potential as
parabolic. This does mean that this approximation is not valid for the
maximum amplitude oscillations seen in Fig. \ref{fig:oscillations}(a),
but since they already appear early on in the trajectory we are able
to obtain a valid analytic description of the oscillations and their
growth.
A superfluid state in our continuous variable approximation
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
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
the wavefunction and it contains little physical
$\epsilon(t)$ as it is only related to the global phase and the norm
of the wavefunction and it contains little physical
information. Furthermore, an interesting and incredibly convenient
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
@ -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
of quantum jumps. It may seem counter-intuitive to neglect the term
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
tunnelling of atoms rather than the null outcomes. Furthermore, the
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
the effect of a photodetection is to displace the wavepacket by
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
the absence of measurement it is the quantum jumps that are the
driving force behind this phenomenon. Furthermore, these oscillations
grow because the quantum jumps occur at an average instantaneous rate
proportional to $\langle \cd \c \rangle (t)$ which itself is
proportional to $(1+z)^2$. This means they are most likely to occur at
the point of maximum displacement in the positive $z$ direction at
which point a quantum jump provides positive feedback and further
increases the amplitude of the wavefunction leading to the growth seen
in Fig. \ref{fig:oscillations}a. The oscillations themselves are
essentially due to the natural dynamics of coherently displaced atoms
in a lattice , but it is the measurement that causes the initial and
more importantly coherent displacement and the positive feedback drive
which causes the oscillations to continuously grow.
of the positive $z$-axis. Therefore, even though the atoms can
oscillate in the absence of measurement it is the quantum jumps that
are the driving force behind this phenomenon. Furthermore, these
oscillations grow because the quantum jumps occur at an average
instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
itself is proportional to $(1+z)^2$. This means they are most likely
to occur at the point of maximum displacement in the positive $z$
direction at which point a quantum jump provides positive feedback and
further increases the amplitude of the wavefunction leading to the
growth seen in Fig. \ref{fig:oscillations}(a). The oscillations
themselves are essentially due to the natural dynamics of coherently
displaced atoms in a lattice , but it is the measurement that causes
the initial and more importantly coherent displacement and the
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
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
it shows that the width of the Gaussian distribution is squeezed as
compared with its initial state which is exactly what we see in
Fig. \ref{fig:oscillations}a. However, if we substitute the parameter
values used in that trajectory we only get a reduction in width by
about $3\%$, but the maximum amplitude oscillations in look like they
have a significantly smaller width than the initial distribution. This
discrepancy is due to the fact that the continuous variable
approximation is only valid for $z \ll 1$ and thus it cannot explain
the final behaviour of the system. Furthermore, it has been shown that
the width of the distribution $b^2$ does not actually shrink to a
constant value, but rather it keeps oscillating around the value given
in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However, what we do see
is that during the early stages of the trajectory, which are well
described by this model, is that the width does in fact stay roughly
constant. It is only in the later stages when the oscillations reach
maximal amplitude that the width becomes visibly reduced.
Fig. \ref{fig:oscillations}(a). However, if we substitute the
parameter values used in that trajectory we only get a reduction in
width by about $3\%$, but the maximum amplitude oscillations in look
like they have a significantly smaller width than the initial
distribution. This discrepancy is due to the fact that the continuous
variable approximation is only valid for $z \ll 1$ and thus it cannot
explain the final behaviour of the system. Furthermore, it has been
shown that the width of the distribution $b^2$ does not actually
shrink to a constant value, but rather it keeps oscillating around the
value given in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However,
what we do see is that during the early stages of the trajectory,
which are well described by this model, is that the width does in fact
stay roughly constant. It is only in the later stages when the
oscillations reach maximal amplitude that the width becomes visibly
reduced.
\subsection{Three-Way Competition}
Now it is time to turn on the inter-atomic interactions,
$U/J^\mathrm{cl} \ne 0$. As a result the atomic dynamics will change
as the measurement now competes with both the tunnelling and the
on-site interactions. A common approach to study such open systems is
to map a dissipative phase diagram by finding the steady state of the
master equation for a range of parameter values
\cite{kessler2012}. However, here we adopt a quantum optical approach
in which we focus on the conditional dynamics of a single quantum
trajectory as this corresponds to a single realisation of an
experiment. The resulting evolution does not necessarily reach a
steady state and usually occurs far from the ground state of the
system.
Now it is time to turn on the inter-atomic interactions, $U/J \ne
0$. As a result the atomic dynamics will change as the measurement now
competes with both the tunnelling and the on-site interactions. A
common approach to study such open systems is to map a dissipative
phase diagram by finding the steady state of the master equation for a
range of parameter values \cite{kessler2012}. However, here we adopt a
quantum optical approach in which we focus on the conditional dynamics
of a single quantum trajectory as this corresponds to a single
realisation of an experiment. The resulting evolution does not
necessarily reach a steady state and usually occurs far from the
ground state of the system.
A key feature of the quantum trajectory approach is that each
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
interactions prevent the atoms from accumulating in one place thus
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
amplitude oscillations that can be described by the effective
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
the Mott insulating state, which has small fluctuations due to strong
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
ground state in first-order perturbation theory given by
\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
distribute the new excitations in a way which will affect and possibly
reduce the effects of the subsequent quantum jumps. However, due to
the lack of any decay channels they will remain in the system and
subsequent jumps will still amplify them further destroying the ground
state and thus quickly leading to a state with large fluctuations.
the lack of any serious decay channels they will remain in the system
and subsequent jumps will still amplify them further destroying the
ground state and thus quickly leading to a state with large
fluctuations.
In the strong measurement regime ($\gamma \gg J$) the measurement
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
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
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,
i.e.~a single Zeno subspace. This is a striking difference with the
$t$-$J$ 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
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
maximum, thus we freeze the atom number in the $K$ illuminated sites
$\hat{N}_\text{K}$~\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
modes nonlocally. Since the measurement only reveals the total number
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
by local processes which can be very confined, especially in 1D where
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 $\lambda \rightarrow \infty$
when all the subspaces completely decouple. This is exactly the
$\gamma \rightarrow \infty$ 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
decouple completely, then transitions within a single subspace can
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 $\delta \c \hat{\rho} \delta \cd$
applies the fluctuation operator from both sides so it does not matter
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
approximate density matrix is inserted, it is in fact zero. Therefore,
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 /
\gamma} \\
\end{array}
\right), \nonumber
\right),
\end{equation}
where $z_i$ denote the overlap between the eigenvectors and the
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
\begin{equation}
\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}
\begin{equation}
\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}
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
@ -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}
\left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger
\right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{|
\Delta N |} | 0 \rangle, \nonumber
\Delta N |} | 0 \rangle,
\end{equation}
where $\phi_{i,k}$ are coefficients that depend on the trajectory
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
typical in quantum optics for optical fields. In the previous chapter
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
section \ref{sec:B} that it is possible to couple to the the relative
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,
coupling to phase observables in lattices has only been proposed and
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,
we go in a new direction by considering the effect of measurement
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,
denoted by index $k$, via
$b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates
an atom with momentum $k$ in the range
$\{ \frac{(M-1) \pi} {M}, \frac{(M-2) \pi} {M}, ..., \pi \}$. Note
that this operator is diagonal in momentum space which means that its
eigenstates are simply Fock momentum Fock states. We have also seen in
Chapter \ref{chap:backaction} how the global nature of the jump
operators introduces a nonlocal quadratic term to the Hamiltonian,
an atom in the Brillouin zone. Note that this operator is diagonal in
momentum space which means that its eigenstates are simply Fock
momentum Fock states. We have also seen in Chapter
\ref{chap:backaction} how the global nature of the jump operators
introduces a nonlocal quadratic term to the Hamiltonian,
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the
competition between tunnelling and measurement backaction we again
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
Hamiltonian. This highlights an important feature of this measurement
- 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
initial state or the photocount trajectory the system will project
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
$\D = \Delta \hat{N} = \hat{N}_\mathrm{even} - \hat{N}_\mathrm{odd}$
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.
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.
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
$\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M$ and
$\sum_M \mathcal{P}_M = \hat{1}$ 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
projector
\begin{equation}
\mathcal{P}_M = \sum_{m \in M} P_m,
\end{equation}
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
they cover the whole Hilbert space. Furthermore, we will also require
that $[\mathcal{P}_M, \hat{H}_0 ] = 0$ and $[\mathcal{P}_M, \c] =
0$. The second commutator simply follows from the definition
that
\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
non-trivial. However, if we can show that
$\mathcal{P}_M = \sum_{m \in M} | h_m \rangle \langle h_m |$, where
@ -407,19 +421,27 @@ eigenstates of the two operators overlap.
To find $\mathcal{P}_M$ we need to identify the subspaces $M$ which
satisfy the following relation
$\sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle \langle h_m
|$. This can be done iteratively by (i) selecting some $P_m$, (ii)
identifying the $| h_m \rangle$ which overlap with this subspace,
(iii) identifying any other $P_m$ which also overlap with these
$| h_m \rangle$ from step (ii). We repeat (ii)-(iii) for all the $P_m$
found in (iii) until we 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
$\mathcal{P}_M$. Computationally this can be straightforwardly solved
with some basic algorithm that can compute the connected components of
a graph.
\begin{equation}
\mathcal{P}_M = \sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle
\langle h_m |.
\end{equation}
This can be done iteratively by
\begin{enumerate}
\item selecting some $P_m$,
\item identifying the $| h_m \rangle$ which overlap with this
subspace,
\item identifying any other $P_m$ which also overlap with these
$| h_m \rangle$ from step (ii).
\item Repeat 2-3 for all the $P_m$ found in 3 until we
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
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
subspaces we need to define a new operator $\hat{O}$, with eigenspace
projectors $R_m$, which commutes with both $\hat{H}_0$ and
$\c$. Physically this means that $\hat{O}$ is a compatible observable
with $\c$ and corresponds to a quantity conserved by the
Hamiltonian. The fact that $\hat{O}$ commutes with the Hamiltonian
implies that the projectors can be written as a sum of Hamiltonian
eigenstates $R_m = \sum_{h_i = h_m} | h_i \rangle \langle h_i |$ and
thus a projector $\mathcal{P}_M = \sum_{m \in M} R_m$ 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,
$\mathcal{P}_M = \sum_{m \in M} R_m = \sum_{m \in M} P_m$ will satisfy
all the necessary prerequisites. This is illustrated in
$\c$,
\begin{equation}
[\hat{O}, \hat{H}_0 ] = 0,
\end{equation}
\begin{equation}
[\hat{O}, \c] = 0.
\end{equation}
Physically this means that $\hat{O}$ is a compatible observable with
$\c$ and corresponds to a quantity conserved by the Hamiltonian. The
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}.
\begin{figure}[hbtp!]
@ -469,13 +506,18 @@ feature.
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
$\hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger
\tilde{\beta_k} = \n_k + \n_{k - \pi/a}$ commutes with both operators
for all $k$. Thus, we can easily construct
$\hat{O} = \sum_\mathrm{RBZ} g_k \hat{O}_k$ 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
\begin{equation}
\hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger
\tilde{\beta_k} = \n_k + \n_{k - \pi/a}
\end{equation}
commutes with both operators for all $k$. Thus, we can easily
construct
\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
Fig. \ref{fig:projections}(a). Note that unlike the $\c$ or $\H_0$ we
can actually see that this observable's distribution does indeed

9
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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
has been left untouched. Here, we have only considered spinless
bosons, but the theory can also been extended to fermions
\cite{atoms2015, mazzucchi2016, mazzucchi2016af} and
molecules \cite{LP2013} and potentially even photonic circuits
\cite{atoms2015, mazzucchi2016, mazzucchi2016af} and molecules
\cite{LP2013} and potentially even photonic circuits
\cite{mazzucchi2016njp}. Furthermore, the question of quantum
measurement and its properties has been a subject of heated debate
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,
this work has hopefully demonstrated that coupling quantised light
fields to many-body systems provides a very rich playground for
exploring new quantum mechanical phenomena beyond what would otherwise
be possible in other fields.
exploring new quantum mechanical phenomena especially the competition
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 ********************
\DeclareMathAlphabet{\mathcal}{OMS}{cmsy}{m}{n}
\renewcommand{\H}{\hat{H}}
\newcommand{\n}{\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},
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 **************************
% 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 ********************************