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Abstract/abstract.tex
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% ************************** Thesis Abstract *****************************
% Use `abstract' as an option in the document class to print only the titlepage and the abstract.
\begin{abstract}
This is where you write your abstract ...
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
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.
We show that light can serve as a nondestructive probe of the
quantum state of the matter. By condering a global
measurement scheme we show that it is possible to distinguish a
highly delocalised phase like a superfluid from insulators. We also
demonstrate that light scattering reveals not only density
correlations, but also matter-field interference.
By taking into account the effect of measurement backaction we show
that the measurement can efficiently compete with the local atomic
dynamics of the quantum gas. This can generate long-range
correlations and entanglement which in turn leads to macroscopic
multimode oscillations accross the whole lattice when the
measurement is weak and correlated tunneling, as well as selective
suppression and enhancement of dynamical processes beyond the
projective limit of the quantum Zeno effect in the strong
measurement regime.
We also consider quantum measurement backaction due to the
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
the measurement postulate for the case of strong competition with
the system’s own evolution.
\end{abstract}

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Chapter1/chapter1.tex
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The field of ultracold gases has been a rapidly growing field ever
since the first Bose-Einstein condensate was obtained in 1995. 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 single quantum object. This was revolutionary as
it enabled the study of coherent properties of macroscopic systems
rather than single atoms or photons. Furthermore, the advanced state
of laser cooling and manipulation technologies meant that the degree
of control and isolation from the environment was far greater than was
possible in condensed matter systems. Initially, the main focus of the
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
single many-body quantum object \cite{PitaevskiiStringari}. This was
revolutionary as it enabled the study of coherent properties of
macroscopic systems rather than single atoms or photons. Furthermore,
the advanced state of laser cooling and manipulation technologies
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, long range phase coherence, or quantised
vortices. Fermi degeneracy in ultracold gases was obtained shortly
afterwards opening a similar field for fermions.
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
\cite{demarco1999, schreck2001, truscott2001}.
In 1998 it was shown that a degenerate ultracold gas trapped in an
optical lattice is a near-perfect realisation of the Bose-Hubbard
model and in 2002 it was already demonstrated in a ground-breaking
experiment. The Bose-Hubbard Hamiltonian was already known in the
field of condensed matter where it was considered a simple toy
model. Despite its simplicity the model exhibits a variety of
different interesting phenomena such as the quantum phase transition
from a delocalised superfluid state to a Mott insulator as the on-site
interaction is increased above a critical point. In contrast to a
thermodynamic phase transition, a quantum phase transition is driven
by quantum fluctuations and can occur at zero temperature. The ability
model \cite{jaksch1998} and in 2002 it was already demonstrated in a
ground-breaking experiment \cite{greiner2002}. The Bose-Hubbard
Hamiltonian was previously known in the field of condensed matter
where it was considered a simple toy model. Despite its simplicity the
model exhibits a variety of different interesting phenomena such as
the quantum phase transition from a delocalised superfluid state to a
Mott insulator as the on-site interaction is increased above a
critical point which was originally studied in the context of liquid
helium \cite{fisher1989}. In contrast to a thermodynamic phase
transition, a quantum phase transition is driven by quantum
fluctuations and can occur at zero temperature. The ability to achieve
a Bose-Hubbard Hamiltonian where the model parameters can be easily
tuned by varying the lattice potential opened up a new regime in the
many-body physics of atomic gases. Unlike Bose-Einstein condensates in
free space which are described by weakly interacting theories
\cite{dalfovo1999}, the behaviour of ultracold gases trapped in an
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 [1, 2]. Originally condensed matter effects are
now mimicked in controlled atomic systems finding applications in
areas such as quantum information processing (QIP). 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 [3,
4]. The former describes delicate effects such as quantum
measurement and state engineering, but for systems without strong
many-body correlations (e.g. atomic ensembles). In the latter,
decoherence destroys these effects in conventional condensed
matter. Due to recent experimental progress, e.g. Bose-Einstein
condensates (BEC) in cavities [5–7], quantum optics of quantum gases
can close this gap.
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}.
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
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
context of quantum optics as a tool to measure a quantum system
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.
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,
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,
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,
douglas2012, douglas2013, ashida2015, ashida2015a}.
In this thesis we focus on the latter by considering a quantum gas in
an optical lattice coupled to a cavity \cite{mekhov2012}. This
provides us with a flexible setup where the global light scattering
can be engineered. We show that this introduces a new competing energy
scale into the system and by considering continuous measurement, as
opposed to discrete projective measurements, we demonstrate the
quantum backaction can effectively compete with the standard
short-range processes of the Bose-Hubbard model. The global nature of
the optical fields leads to new phenomena driven by long-range
correlations that arise from the measurement. The flexibility of the
optical setup lets us not only consider coupling to different
observables, but by carefully choosing the optical geometry we can
suppress or enhace specific dynamical processes, realising spatially
nonlocal quantum Zeno dynamics.
The quantum Zeno effect happens when frequent measurements slow the
evolution of a quantum system \cite{misra1977, facchi2008}. This
effect was already considered by von Neumann and it has been
successfully observed in a variety of systems \cite{itano1990,
nagels1997, kwiat1999, balzer2000, streed2006, hosten2006,
bernu2008}. The generalisation of this effect to measurements with
multidimensional projections leads to quantum Zeno dynamics where
unitary evolution is uninhibited within this degenerate subspace,
i.e. the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
signoles2014}. Here, by combining quantum optical measurements with
the complex Hilbert space of a many-body quantum gas we go beyond
conventional quantum Zeno dynamics. By considering the case of
measurement near, but not in, the projective limit the system is still
confined to a Zeno subspace, but intermediate transitions are allowed
via virtual Raman-like processes. In a lattice system, like the
Bose-Hubbard model we can use global measurement to engineer these
dynamics to be highly nonlocal leading to the generation of long-range
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
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}.
Just like for the nondestructive measurements we also consider
measurement backaction due to coupling to the interference terms
between the lattice sites. This effectively amounts to coupling to the
phase observables of the system. As this is the conjugate variable of
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
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
formulation of the Copenhagen postulate projection or the quantum Zeno
effect \cite{misra1977, facchi2008} thus providing an extension of the
measurement postulate to dynamical systems subject to weak
measurement.
Finally, the cavity field that builds up from the scattered photons
can also create a quantum optical potential which will modify the
Hamiltonian in a way that depends on the state of the atoms that
scatterd the light. This can lead to new quantum phases due to new
types of long-range interactions being mediated by the global quantum
optical fields \cite{caballero2015, caballero2015njp, caballero2016,
caballero2016a}. However, this aspect of quantum optics of quantum
gases is beyond the scope of this thesis.

40
Chapter2/chapter2.tex
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@ -20,9 +20,9 @@ the system in different parameter regimes, such as nondestructive
measurement in free space or quantum measurement backaction in a
cavity. Another interesting direction for this field of research are
quantum optical lattices where the trapping potential is treated
quantum mechanically. However this is beyond the scope of this work.
\mynote{insert our paper citations here}
quantum mechanically \cite{caballero2015, caballero2015njp,
caballero2016, caballero2016a}. However this is beyond the scope of
this work.
We consider $N$ two-level atoms in an optical lattice with $M$
sites. For simplicity we will restrict our attention to spinless
@ -39,25 +39,23 @@ 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).
transition).
\mynote{extra fermion citations, Piazza? Look up Gabi's AF paper.}
As we have seen in the previous section, an optical lattice can be
formed with classical light beams that form standing waves. Depending
on the detuning with respect to the atomic resonance, the nodes or
antinodes form the lattice sites in which atoms accumulate. As shown
in Fig. \ref{fig:LatticeDiagram} the trapped bosons (green) are
illuminated with a coherent probe beam (red) and scatter light into a
different mode (blue) which is then 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 classical lattice trap.
An optical lattice can be formed with classical light beams that form
standing waves. Depending on the detuning with respect to the atomic
resonance, the nodes or antinodes form the lattice sites in which
atoms accumulate. As shown in Fig. \ref{fig:LatticeDiagram} the
trapped bosons (green) are illuminated with a coherent probe beam
(red) and scatter light into a different mode (blue) which is then
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
classical lattice trap.
\begin{figure}[htbp!]
\centering

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Chapter7/chapter7.tex
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@ -9,3 +9,111 @@
\else
\graphicspath{{Chapter7/Figs/Vector/}{Chapter7/Figs/}}
\fi
Quantum optics of quantum gases explores the ultimate quantum regime
of light-matter interactions where both the optical and matter fields
are fully quantised. It provides a very rich system in which new
phenomena can be observed, engineered, and controlled beyond what
would be possible in condensed matter. Combined with rapid and
promising experimental progress in this field the theoretical
proposals have the potential of directing the research in the
foreseeable future \cite{baumann2010, wolke2012, schmidt2014,
klinder2015, landig2016}.
In this thesis we focused on the coupling between global quantised
optical fields and an ultracold bosonic quantum gas. By considering
global fields as opposed to localised light-matter interactions we
were able to introduce several nonlocal properties to the Hamiltonian
in a controllable manner which would otherwise be impossible to
implement. We showed how this can be useful in the context of
nondestructive probing by showing that it can easily distinguish
between a highly delocalised quantum state such as a superfluid and
insulating states such as the Mott insulator and the Bose glass phases
which is currently a challenge \cite{derrico2014}. Furthermore, we
have seen how the correlation length, which would be inaccessible in
localised measurements, was immediately visible in our scheme and lead
to an angular scattering pattern that was far richer than it was for
the classical case. This is best highlighted by the fact that it would
be visible even when classically no light would scatter coherently at
all.
More interestingly, the global nature of the measurement was also
capable of creating such long-range correlations itself when we
considered measurement backaction. This was most visible when we saw
how weak measurement was capable of driving global macroscopic
multimode oscillations between different spatial modes, such as odd
and even sites, across the whole lattice which could be used for
quantum information and metrology. Such dynamical states show spatial
density-density correlations with nontrivial periods and long-range
coherence, thus having supersolid properties, but as an essentially
dynamical version. Furthermore, the tunability of the optical
arrangement meant that we had extreme flexibility in choosing our
observables, effectively tailoring the long-range entanglement and
correlations in the system. We have also shown how global measurement
when combined with both atomic tunnelling and interactions leads to
highly nontrivial dynamics in which backaction can either compete or
cooperate with on-site repulsion in squeezing the atomic variables.
In the limit of strong measurement when quantum Zeno dynamics occurs
we showed that these nonlocal spatial modes created by the global
measurement lead to long-range correlated tunnelling events whilst
suppressing any other dynamics between different spatial modes of the
measurement. Such globally paired tunneling due to a fundamentally
novel phenomenon can enrich physics of long-range correlated systems
beyond relatively shortrange interactions expected from standard
dipole-dipole interactions \cite{sowinski2012, omjyoti2015}. These
nonlocal high-order processes entangle regions of the optical lattice
that are disconnected by the measurement. Using different detection
schemes, we showed how to tailor density-density correlations between
distant lattice sites. Quantum optical engineering of nonlocal
coupling to environment, combined with quantum measurement, can allow
the design of nontrivial system-bath interactions, enabling new links
to quantum simulations \cite{stannigel2013} and thermodynamics
\cite{erez2008}. Interestingly, these dynamics also provide a link to
non-Hermitian quantum mechanics as this regime of measurement can be
accurately described with a non-Hermitian Hamiltonian. Furthermore, we
show that this allows for a rather novel type of competition between
measurement and tunnelling where both processes actually cooperate to
produce a steady state in which tunnelling is suppressed by
destructive matter-wave interference.
A unique feature of our global measurement scheme meant that we could
couple directly to the phase observables of the system by coupling to
the interference between the lattice sites, which represents the
shortest meaningful distance in an optical lattice, rather than their
on-site density. This defines most processes in optical lattices. For
example, 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 \cite{vukics2007}, which should be
accessible by this method. Furthremore, in mean-field one can measure
the matter-field amplitude (which is also the order parameter),
quadratures and their squeezing. This can link atom optics to areas
where quantum optics has already made progress, e.g., quantum imaging
\cite{golubev2010, kolobov1999}, using an optical lattice as an array
of multimode nonclassical matter- field sources with a high degree of
entanglement for quantum information processing. We have also shown
how this scheme of coupling to phase observables can be used in the
context of quantum measurement backaction to achieve a new degree of
control. We used this result to show a generalisation of weak
measurement on dynamical systems by showing that there is now a new
class of projections available even when the measurement is not a
compatible observable of the Hamiltonian. This an interesting result
as the projections themselves are unlike those postulated by the
Copenhagen interpretation, those present in quantum Zeno dynamic, or
even those possible to engineer using dissipative methods.
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{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.

525
References/references.bib
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@ -2,6 +2,15 @@
%% Books, theses, reference material
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@book{carmichael,
title={An open systems approach to quantum optics: lectures
presented at the Universit{\'e} Libre de Bruxelles,
October 28 to November 4, 1991},
author={Carmichael, Howard},
volume={18},
year={2009},
publisher={Springer Science \& Business Media}
}
@book{foot,
author = {Foot, C. J.},
title = {{Atomic Physics}},
@ -243,8 +252,10 @@ year = {2010}
publisher={IOP Publishing}
}
@article{caballero2016,
title = {Quantum simulators based on the global collective light-matter interaction},
author = {Caballero-Benitez, Santiago F. and Mazzucchi, Gabriel and Mekhov, Igor B.},
title = {Quantum simulators based on the global collective
light-matter interaction},
author = {Caballero-Benitez, Santiago F. and Mazzucchi, Gabriel and
Mekhov, Igor B.},
journal = {Phys. Rev. A},
volume = {93},
issue = {6},
@ -1225,3 +1236,513 @@ doi = {10.1103/PhysRevA.87.043613},
year={1981},
publisher={APS}
}
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title={Observation of Bose-Einstein Condensation in a Dilute Atomic
Vapor},
author={Anderson, MH and Ensher, JR and Matthews, MR and Wieman, CE
and Cornell, EA},
journal={science},
volume={269},
pages={14},
year={1995}
}
@article{bradley1995,
title={Evidence of Bose-Einstein condensation in an atomic gas with
attractive interactions},
author={Bradley, Cl C and Sackett, CA and Tollett, JJ and Hulet,
Randall G},
journal={Physical Review Letters},
volume={75},
number={9},
pages={1687},
year={1995},
publisher={APS}
}
@article{davis1995,
title={Bose-Einstein condensation in a gas of sodium atoms},
author={Davis, Kendall B and Mewes, M-O and Andrews, Michael R and
Van Druten, NJ and Durfee, DS and Kurn, DM and
Ketterle, Wolfgang},
journal={Physical review letters},
volume={75},
number={22},
pages={3969},
year={1995},
publisher={APS}
}
@article{bloch2008,
title={Many-body physics with ultracold gases},
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volume={80},
number={3},
pages={885},
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publisher={APS}
}
@article{lewenstein2007,
title={Ultracold atomic gases in optical lattices: mimicking
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author={Lewenstein, Maciej and Sanpera, Anna and Ahufinger, Veronica
and Damski, Bogdan and Sen, Aditi and Sen, Ujjwal},
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volume={56},
number={2},
pages={243--379},
year={2007},
publisher={Taylor \& Francis}
}
@article{andrews1997,
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and Kurn, DM and Ketterle, W},
journal={Science},
volume={275},
number={5300},
pages={637--641},
year={1997},
publisher={American Association for the Advancement of Science}
}
@article{bloch2000,
title={Measurement of the spatial coherence of a trapped Bose gas at
the phase transition},
author={Bloch, I and H{\"a}nsch, Th W and Esslinger, T},
journal={Nature},
volume={403},
number={6766},
pages={166--170},
year={2000},
publisher={Nature Publishing Group}
}
@article{matthews1999,
title={Vortices in a Bose-Einstein condensate},
author={Matthews, Michael Robin and Anderson, Brian P and Haljan, PC
and Hall, DS and Wieman, CE and Cornell, EA},
journal={Physical Review Letters},
volume={83},
number={13},
pages={2498},
year={1999},
publisher={APS}
}
@article{madison2000,
title={Vortex formation in a stirred Bose-Einstein condensate},
author={Madison, KW and Chevy, F and Wohlleben, W and Dalibard, J},
journal={Physical Review Letters},
volume={84},
number={5},
pages={806},
year={2000},
publisher={APS}
}
@article{abo2001,
title={Observation of vortex lattices in Bose-Einstein condensates},
author={Abo-Shaeer, JR and Raman, C and Vogels, JM and Ketterle, Wolfgang},
journal={Science},
volume={292},
number={5516},
pages={476--479},
year={2001},
publisher={American Association for the Advancement of Science}
}
@article{demarco1999,
title={Onset of Fermi degeneracy in a trapped atomic gas},
author={DeMarco, Brian and Jin, Deborah S},
journal={Science},
volume={285},
number={5434},
pages={1703--1706},
year={1999},
publisher={American Association for the Advancement of Science}
}
@article{schreck2001,
title={Quasipure Bose-Einstein condensate immersed in a Fermi sea},
author={Schreck, F and Khaykovich, Lev and Corwin, KL and Ferrari, G
and Bourdel, Thomas and Cubizolles, Julien and
Salomon, Christophe},
journal={Physical Review Letters},
volume={87},
number={8},
pages={080403},
year={2001},
publisher={APS}
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4
thesis.tex
View File

@ -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,chapter]{Classes/PhDThesisPSnPDF}
\documentclass[a4paper,12pt,times,numbered,print]{Classes/PhDThesisPSnPDF}
% ******************************************************************************
% ******************************* Class Options ********************************
@ -101,7 +101,7 @@
% To use choose `chapter' option in the document class
\ifdefineChapter
\includeonly{Chapter1/chapter1}
\includeonly{Abstract/Abstract}
\fi
% ******************************** Front Matter ********************************