Final version 1.0
This commit is contained in:
Wojciech Kozlowski
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% ************************** Thesis Abstract *****************************
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% Use `abstract' as an option in the document class to print only the titlepage and the abstract.
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\begin{abstract}
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Trapping ultracold atoms in optical lattices enabled the study of
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strongly correlated phenomena in an environment that is far more
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@ -2,8 +2,33 @@
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\begin{acknowledgements}
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First and foremost, I would like to thank my supervisor Dr. Igor
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Mekhov who has been an excellent mentor throughout my time at
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Oxford. It is primarily thanks to his brilliant insights and
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professionalism that I was able to reach my full potential during my
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doctoral studies. The work contained in this thesis would also not be
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possible without the help of the other members of the group, Gabriel
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Mazzucchi and Dr. Santiago Caballero-Benitez. Without our frequent
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casual discussions in the Old Library office I would have still been
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stuck on the third chapter. I would also like to acknowledge all
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members of Prof. Dieter Jaksch's and Prof. Christopher Foot's groups
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for various helpful discussions. I must also offer a special mention
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for Edward Owen who provided much needed reality checks on some of my
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wishful theoretical thinking. I would also like to express my
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gratitude to EPSRC, St. Catherine's College, the ALP sub-department,
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and the Institute of Physics for providing me with the financial means
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to live and study in Oxford as well as attend several conferences in
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the UK and abroad.
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\mynote{Write my own acknowledgements}
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On a personal note, I would like to thank my parents who provided me
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with all the skills necessary work towards any goals I set
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myself. Needless to say, without them I would have never been able to
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be where I am right now. I would like to thank all the new and old
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friends that have kept me company for the last four years. The time
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spent together provided a welcome respite from the sweat and toil of
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my DPhil.
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\vspace{2em}
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\raggedleft{Oxford, September 2016}
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\end{acknowledgements}
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@ -83,9 +83,9 @@ imprinted in the scattered light \cite{klinder2015, landig2016}.
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There are three prominent directions in which the field of quantum
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optics of quantum gases has progressed in. First, the use of quantised
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light enables direct coupling to the quantum properties of the atoms
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\cite{mekhov2007prl, mekhov2007pra, mekhov2012}. This allows us to
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probe the many-body system in a nondestructive manner and under
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certain conditions even perform quantum non-demolition (QND)
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\cite{mekhov2007prl, mekhov2007pra, mekhov2007NP, mekhov2012}. This
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allows us to probe the many-body system in a nondestructive manner and
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under certain conditions even perform quantum non-demolition (QND)
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measurements. QND measurements were originally developed in the
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context of quantum optics as a tool to measure a quantum system
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without significantly disturbing it \cite{braginsky1977, unruh1978,
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@ -200,7 +200,7 @@ measurement backaction. Whilst such interference measurements have
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been previously proposed for BECs in double-wells \cite{cirac1996,
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castin1997, ruostekoski1997}, the extension to a lattice system is
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not straightforward. However, we will show it is possible to achieve
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with our propsed setup by a careful optical arrangement. Within this
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with our proposed setup by a careful optical arrangement. Within this
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context we demonstrate a novel type of projection which occurs even
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when there is significant competition with the Hamiltonian
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dynamics. This projection is fundamentally different to the standard
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@ -215,5 +215,58 @@ Hamiltonian in a way that depends on the state of the atoms that
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scatterd the light. This can lead to new quantum phases due to new
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types of long-range interactions being mediated by the global quantum
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optical fields \cite{caballero2015, caballero2015njp, caballero2016,
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caballero2016a}. However, this aspect of quantum optics of quantum
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gases is beyond the scope of this thesis.
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caballero2016a, elliott2016}. However, this aspect of quantum optics
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of quantum gases is beyond the scope of this thesis.
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\newpage
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\section*{Publication List}
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The work contained in this thesis is based on seven publications
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\cite{kozlowski2015, elliott2015, atoms2015, mazzucchi2016,
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kozlowski2016zeno, mazzucchi2016njp, kozlowski2016phase}:
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\begin{table}[hbtp!]
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\centering
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\begin{tabular}{r p{13cm}}
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\toprule
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\cite{kozlowski2015} & W. Kozlowski, S. F. Caballero-Benitez, and
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I. B. Mekhov. ``Probing matter-field and atom-number correlations
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in optical lattices by global nondestructive addressing''.
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\emph{Physical Review A}, 92(1):013613, 2015. \\ \\
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\cite{elliott2015} & T. J. Elliott, W. Kozlowski,
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S. F. Caballero-Benitez, and I. B. Mekhov. ``Multipartite
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Entangled Spatial Modes of Ultracold Atoms Generated and
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Controlled by Quantum Measurement''. \emph{Physical Review
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Letters}, 114:113604, 2015. \\ \\
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\cite{atoms2015} & T. J. Elliott, G. Mazzucchi, W. Kozlowski,
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S. F. Caballero- Benitez, and I. B. Mekhov. ``Probing and
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manipulating fermionic and bosonic quantum gases with quantum
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light''. \emph{Atoms}, 3(3):392–406, 2015. \\ \\
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\cite{mazzucchi2016} & G. Mazzucchi$^*$, W. Kozlowski$^*$,
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S. F. Caballero-Benitez, T. J. Elliott, and
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I. B. Mekhov. ``Quantum measurement-induced dynamics of many- body
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ultracold bosonic and fermionic systems in optical
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lattices''. \emph{Physical Review A}, 93:023632,
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2016. $^*$\emph{Equally contributing authors}. \\ \\
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\cite{kozlowski2016zeno} & W. Kozlowski, S. F. Caballero-Benitez,
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and I. B. Mekhov. ``Non- hermitian dynamics in the quantum zeno
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limit''. \emph{Physical Review A}, 94:012123, 2016. \\ \\
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\cite{mazzucchi2016njp} & G. Mazzucchi, W. Kozlowski,
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S. F. Caballero-Benitez, and I. B Mekhov. ``Collective dynamics of
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multimode bosonic systems induced by weak quan- tum
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measurement''. \emph{New Journal of Physics}, 18(7):073017, 2016. \\ \\
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\cite{kozlowski2016phase} & W. Kozlowski, S. F. Caballero-Benitez,
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and I. B. Mekhov. ``Quantum state reduction by
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matter-phase-related measurements in optical
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lattices''. \emph{arXiv preprint arXiv:1605.06000}, 2016. \\
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\bottomrule
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\end{tabular}
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\end{table}
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@ -9,20 +9,24 @@
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\graphicspath{{Chapter2/Figs/Raster/}{Chapter2/Figs/PDF/}{Chapter2/Figs/}}
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\else
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\graphicspath{{Chapter2/Figs/Vector/}{Chapter2/Figs/}}
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\fi
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\fi
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\section{Introduction}
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In this chapter, we derive a general Hamiltonian that describes the
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coupling of atoms with far-detuned optical beams
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\cite{mekhov2012}. This will serve as the basis from which we explore
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the system in different parameter regimes, such as nondestructive
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measurement in free space or quantum measurement backaction in a
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cavity. Another interesting direction for this field of research are
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quantum optical lattices where the trapping potential is treated
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quantum mechanically \cite{caballero2015, caballero2015njp,
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caballero2016, caballero2016a}. However this is beyond the scope of
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this work.
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In this chapter, we present the derivation of a general Hamiltonian
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that describes the coupling of atoms with far-detuned optical beams
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originally presented in Ref. \cite{mekhov2012}. This will serve as the
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basis from which we explore the system in different parameter regimes,
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such as nondestructive measurement in free space or quantum
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measurement backaction in a cavity. As this model extends the
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Bose-Hubbard Hamiltonian to include the effects of interactions with
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quantised light we also present a brief overview of the properties of
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the Bose-Hubbard model itself and its quantum phase transiton. This is
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followed by a description of the behaviour of the scattered light with
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a particular focus on how to couple the optical fields to phase
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observables as opposed to density observables as is typically the
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case. Finally, we conclude with an overview of possible experimental
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realisability.
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We consider $N$ two-level atoms in an optical lattice with $M$
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sites. For simplicity we will restrict our attention to spinless
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@ -37,7 +41,7 @@ from free particles to strongly correlated systems, to the inherent
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tunability of such lattices. Furthermore, this model is capable of
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describing a range of different experimental setups ranging from a
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small number of sites with a large filling factor (e.g.~BECs trapped
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in a double-well potential) to a an extended multi-site lattice with a
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in a double-well potential) to an extended multi-site lattice with a
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low filling factor (e.g.~a system with one atom per site which will
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exhibit the Mott insulator to superfluid quantum phase transition).
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@ -262,7 +266,7 @@ hopping rate given by
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\begin{equation}
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J^\mathrm{cl}_{i,j} = \int \mathrm{d}^3 \b{r} w (\b{r} - \b{r}_i )
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\left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r}
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- \b{r}_i),
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- \b{r}_j),
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\end{equation}
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and $U_{ijkl}$ is the atomic interaction term given by
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\begin{equation}
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@ -731,7 +735,9 @@ neglected. This is likely to be the case since the interactions will
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be dominated by photons scattering from the much larger coherent
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probe.
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\section{Density and Phase Observables}
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\section[Density and Phase Observables]
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{Density and Phase Observables\footnote{The results of this
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section were first published in Ref. \cite{kozlowski2015}}}
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\label{sec:B}
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Light scatters due to its interactions with the dipole moment of the
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@ -802,6 +808,18 @@ between the light modes and the nearest neighbour Wannier overlap,
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$W_1(x)$. This can be achieved by concentrating the light between the
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sites rather than at the positions of the atoms.
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\begin{figure}[hbtp!]
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\centering
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\includegraphics[width=0.8\linewidth]{BDiagram}
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\caption[Maximising Light-Matter Coupling between Lattice
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Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
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between lattice sites. The thin black line indicates the trapping
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potential (not to scale). (a) Arrangement for the uniform pattern
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$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
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$J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
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is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
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\end{figure}
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In order to calculate the $J_{i,j}$ coefficients we perform numerical
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calculations using realistic Wannier functions
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\cite{walters2013}. However, it is possible to gain some analytic
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@ -835,35 +853,6 @@ $\hat{D}$ operator since it depends on the amplitude of light in
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between the lattice sites and not at the positions of the atoms
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allowing to decouple them at specific angles.
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\begin{figure}
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\centering
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\includegraphics[width=0.8\linewidth]{BDiagram}
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\caption[Maximising Light-Matter Coupling between Lattice
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Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
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between lattice sites. The thin black line indicates the trapping
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potential (not to scale). (a) Arrangement for the uniform pattern
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$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
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$J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
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is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{WF_S}
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\caption[Wannier Function Products]{The Wannier function products:
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(a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
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axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
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Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
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in diffraction maximum (c) and minimum (d) as are shown as
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functions of standing wave shifts $\varphi$ or, if one were to
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measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
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oscillator phase $\beta$. The black points indicate the positions,
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where light measures matter interference $\hat{B} \ne 0$, and the
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density-term is suppressed, $\hat{D} = 0$. The trapping potential
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depth is approximately 5 recoil energies.}
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\label{fig:WannierProducts}
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\end{figure}
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The simplest case is to find a diffraction maximum where
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$J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
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constant. This results in a diffraction maximum where each bond
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@ -889,6 +878,23 @@ arrangement of light modes maximizes the interference signal,
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$\hat{B}$, by suppressing the density signal, $\hat{D}$, via
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interference compensating for the spreading of the Wannier functions.
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{WF_S}
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\caption[Wannier Function Products]{The Wannier function products:
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(a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
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axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
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Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
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in diffraction maximum (c) and minimum (d) as are shown as
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functions of standing wave shifts $\varphi$ or, if one were to
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measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
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oscillator phase $\beta$. The black points indicate the positions,
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where light measures matter interference $\hat{B} \ne 0$, and the
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density-term is suppressed, $\hat{D} = 0$. The trapping potential
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depth is approximately 5 recoil energies.}
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\label{fig:WannierProducts}
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\end{figure}
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Another possibility is to obtain an alternating pattern similar
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corresponding to a diffraction minimum where each bond scatters light
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in anti-phase with its neighbours giving
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@ -927,10 +933,13 @@ $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ and by
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varying the local oscillator phase, one can choose which conjugate
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operator to measure.
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\section{Electric Field Stength}
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\section[Electric Field Strength]
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{Electric Field Strength\footnote{The derivation has not been
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published before, but the final numerical results were
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included in Ref. \cite{kozlowski2015}}}
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\label{sec:Efield}
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The Electric field operator at position $\b{r}$ and at time $t$ is
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The electric field operator at position $\b{r}$ and at time $t$ is
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usually written in terms of its positive and negative components:
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\begin{equation}
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\b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t),
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@ -1095,6 +1104,25 @@ Therefore, we can now express the quantity $n_{\Phi}$ as
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n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
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\end{equation}
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\begin{table}
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\centering
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\begin{tabular}{l c c}
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\toprule
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Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
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$\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
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$\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
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$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
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$I$ & $4250$ Wm$^{-2}$ & N/A \\
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$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
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$N_K$ & 10$^5$ & 147 \\ \midrule
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$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
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\bottomrule
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\end{tabular}
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\caption[Photon Scattering Rates]{Experimental parameters used in
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estimating the photon scattering rates.}
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\label{tab:photons}
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\end{table}
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Estimates of the scattering rate using real experimental parameters
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are given in Table \ref{tab:photons}. Rubidium atom data has been
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taken from Ref. \cite{steck}. The two experiments were chosen as state
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@ -1131,21 +1159,38 @@ $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
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(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 kHz
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per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
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\begin{table}
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\centering
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\begin{tabular}{l c c}
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\toprule
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Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
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$\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
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$\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
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$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
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$I$ & $4250$ Wm$^{-2}$ & N/A \\
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$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
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$N_K$ & 10$^5$ & 147 \\ \midrule
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$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
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\bottomrule
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\end{tabular}
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\caption[Photon Scattering Rates]{Experimental parameters used in
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estimating the photon scattering rates.}
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\label{tab:photons}
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\end{table}
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\section{Possible Experimental Issues}
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There are many possible experimental issues that we have neglected so
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far in our theoretical treatment which need to be answered in order to
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consider the experimental feasability of our proposal. The two main
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concerns are photodetector efficiency and heating losses.
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In our theoretical models we treat the detectors as if they were
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capable of detecting every scattered photon, but real photodetectors
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can have efficiencies as low as 5\%. For the case of nondestructive
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measurement as covered in Chapter \ref{chap:qnd} this is not an issue
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provided a sufficient number of photons can be collected to calculate
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reliable expectation values. The case of scattering into a cavity and
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the effect of efficiency on the conditioned state was addressed in
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Ref. \cite{mazzucchi2016njp} where it was shown that detector
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efficiencies as low as 1\% are still capable of resolving the dynamics
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to a good degree of accuracy and 10\% was sufficient for near unit
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fidelity. However, this incredible result requires that the photon
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scattering pattern is periodic in some way, e.g.~oscillatory as was
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the case in Ref. \cite{mazzucchi2016njp} or constant. This way it is
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only necessary to detect a sufficient number of photons to deduce the
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correct phase of the oscillations or the rate for the case of a
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constant scattering rate. In this thesis we deal predominantly with
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these two cases so photodetector efficiency is not an issue.
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The other issue is the heating of the trapped gas which will limit the
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lifetime of the experiment. For free space scattering imaging times of
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several hundred milliseconds have been achieved by for example using
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molasses beams that simultaneously cool and trap the atoms
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\cite{weitenberg2011, weitenbergThesis}. Similar feats have been
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achieved with atoms coupled to a leaky cavity where interogation times
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of 0.8s have been achieved in Ref. \cite{brennecke2013}. Crucially,
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the cavity in said experiment has a decay rate of the order of MHz
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which is necessary to observe measurement backaction which we will
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consider in the subsequent chapters.
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@ -2,8 +2,10 @@
|
||||
%*********************************** Third Chapter *****************************
|
||||
%*******************************************************************************
|
||||
|
||||
\chapter{Probing Correlations by Global
|
||||
Nondestructive Addressing} %Title of the Third Chapter
|
||||
\chapter[Probing Correlations by Global
|
||||
Nondestructive Addressing] {Probing Correlations by Global
|
||||
Nondestructive Addressing\footnote{The results of this chapter were
|
||||
first published in Ref. \cite{kozlowski2015}}}
|
||||
\label{chap:qnd}
|
||||
|
||||
\ifpdf
|
||||
@ -230,7 +232,7 @@ fluctuations entirely leading to absolutely no ``quantum addition''.
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{Ep1}
|
||||
\includegraphics[width=0.8\linewidth]{Ep1}
|
||||
\caption[Light Scattering Angular Distribution]{Light intensity
|
||||
scattered into a standing wave mode from a superfluid in a 3D
|
||||
lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming
|
||||
|
||||
@ -421,7 +421,10 @@ generalise the above result to larger Hilbert spaces with multiple
|
||||
degenerate subspaces which are of much greater interest as they reveal
|
||||
nontrivial dynamics in the system.
|
||||
|
||||
\section{Global Measurement and ``Which-Way'' Information}
|
||||
\section[Global Measurement and ``Which-Way'' Information]
|
||||
{Global Measurement and ``Which-Way'' Information\footnote{The
|
||||
results of this section were first published in
|
||||
Ref. \cite{elliott2015}}}
|
||||
\label{sec:modes}
|
||||
|
||||
We have already mentioned that one of the key features of our model is
|
||||
|
||||
@ -2,7 +2,11 @@
|
||||
%*********************************** Fifth Chapter *****************************
|
||||
%*******************************************************************************
|
||||
|
||||
\chapter{Density Measurement Induced Dynamics}
|
||||
\chapter[Density Measurement Induced Dynamics]
|
||||
{Density Measurement Induced Dynamics\footnote{The results of
|
||||
this chapter were first published in
|
||||
Refs. \cite{mazzucchi2016, kozlowski2016zeno,
|
||||
mazzucchi2016njp}}}
|
||||
% Title of the Fifth Chapter
|
||||
|
||||
\ifpdf
|
||||
@ -290,6 +294,8 @@ jump. Expanding around the peak of the Gaussian ansatz we get
|
||||
Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump
|
||||
operator above yields a Gaussian wavefunction as well, but the
|
||||
parameters change discontinuously according to
|
||||
\begingroup
|
||||
\allowdisplaybreaks
|
||||
\begin{align}
|
||||
\label{eq:jumpb2}
|
||||
b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
|
||||
@ -299,6 +305,7 @@ parameters change discontinuously according to
|
||||
z_\phi & \rightarrow z_\phi, \\
|
||||
\epsilon & \rightarrow \epsilon.
|
||||
\end{align}
|
||||
\endgroup
|
||||
The fact that the wavefunction remains Gaussian after a photodetection
|
||||
is a huge advantage, because it means that the combined time evolution
|
||||
of the system can be described with a single Gaussian ansatz in
|
||||
@ -527,15 +534,15 @@ yields
|
||||
\end{equation}
|
||||
where the approximation on the right-hand side follows from the fact
|
||||
that $\omega \approx 2$ since we are considering the $N \gg 1$ limit,
|
||||
and because we are considering the weak measurement limit
|
||||
$\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ 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
|
||||
and because we are considering the weak measurement limit $\Gamma^2 /
|
||||
\omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ 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
|
||||
width by about $3\%$, but the maximum amplitude oscillations 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
|
||||
@ -1648,14 +1655,12 @@ eigenstate of $\c$ and we combine this with the requirement for it to
|
||||
be in the dark state of the tunnelling operator (eigenstate of $\H_0$
|
||||
for $U = 0$) to derive the steady state. These two conditions in
|
||||
momentum space are
|
||||
\begin{equation}
|
||||
\begin{align}
|
||||
\hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k -
|
||||
\bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0,
|
||||
\end{equation}
|
||||
\begin{equation}
|
||||
\bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \\
|
||||
\Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} +
|
||||
\bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle,
|
||||
\end{equation}
|
||||
\end{align}
|
||||
where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$,
|
||||
$\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice
|
||||
spacing, $N$ the total atom number, and we perform summations over the
|
||||
@ -1738,10 +1743,11 @@ discussed.
|
||||
To obtain a state with a specific value of $\Delta N$ postselection
|
||||
may be necessary, but otherwise it is not needed. The process can be
|
||||
optimised by feedback control since the state is monitored at all
|
||||
times \cite{ivanov2014}. Furthermore, the form of the measurement
|
||||
operator is very flexible and it can easily be engineered by the
|
||||
geometry of the optical setup \cite{elliott2015, mazzucchi2016} which
|
||||
can be used to design a state with desired properties.
|
||||
times \cite{ivanov2014, mazzucchi2016feedback}. Furthermore, the form
|
||||
of the measurement operator is very flexible and it can easily be
|
||||
engineered by the geometry of the optical setup \cite{elliott2015,
|
||||
mazzucchi2016} which can be used to design a state with desired
|
||||
properties.
|
||||
|
||||
\section{Conclusions}
|
||||
|
||||
|
||||
@ -2,7 +2,10 @@
|
||||
%*********************************** Sixth Chapter *****************************
|
||||
%*******************************************************************************
|
||||
|
||||
\chapter{Phase Measurement Induced Dynamics}
|
||||
\chapter[Phase Measurement Induced Dynamics]
|
||||
{Phase Measurement Induced Dynamics\footnote{The results of
|
||||
this chapter were first published in
|
||||
Ref. \cite{kozlowski2016phase}}}
|
||||
% Title of the Sixth Chapter
|
||||
|
||||
\ifpdf
|
||||
@ -25,7 +28,7 @@ operators just like most of the existing work \cite{LP2009, rogers2014,
|
||||
section \ref{sec:B} that it is possible to couple to the the relative
|
||||
phase differences between sites in an optical lattice by illuminating
|
||||
the bonds between them. Furthermore, we have also shown how it can be
|
||||
applied to probe the Bose Hubbard order parameter or even matter-field
|
||||
applied to probe the Bose-Hubbard order parameter or even matter-field
|
||||
quadratures in Chapter \ref{chap:qnd}. This concept has also been
|
||||
applied to the study of quantum optical potentials formed in a cavity
|
||||
and shown to lead to a host of interesting quantum phase diagrams
|
||||
|
||||
@ -2,7 +2,11 @@
|
||||
|
||||
\begin{dedication}
|
||||
|
||||
\mynote{Write my own dedication.}
|
||||
\emph{Moim rodzicom, bez których nie byłbym w stanie osiągnąć tego
|
||||
wszystkiego.}
|
||||
|
||||
\emph{To my parents without whom I would not have been able to achieve
|
||||
any of this.}
|
||||
|
||||
\end{dedication}
|
||||
|
||||
|
||||
@ -221,3 +221,5 @@
|
||||
\newcommand{\Bmax}{\hat{B}_\mathrm{max}}
|
||||
\newcommand{\Bmin}{\hat{B}_\mathrm{min}}
|
||||
\newcommand{\D}{\hat{D}}
|
||||
|
||||
\usepackage[utf8]{inputenc}
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@ -29,7 +29,7 @@ Dynamics in Ultracold Bosonic Gases}
|
||||
|
||||
|
||||
%% Supervisor (optional)
|
||||
%\supervisor{Prof. Kenichi Soga}
|
||||
\supervisor{Dr. Igor B. Mekhov}
|
||||
%% Supervisor Role (optional) - Supervisor (default) or advisor
|
||||
%\supervisorrole{Advisor: }
|
||||
|
||||
|
||||
@ -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,draft]{Classes/PhDThesisPSnPDF}
|
||||
\documentclass[a4paper,12pt,times,numbered,print]{Classes/PhDThesisPSnPDF}
|
||||
|
||||
% ******************************************************************************
|
||||
% ******************************* Class Options ********************************
|
||||
|
||||
Reference in New Issue
Block a user