diff --git a/Abstract/abstract.tex b/Abstract/abstract.tex index 942ec36..6f84146 100644 --- a/Abstract/abstract.tex +++ b/Abstract/abstract.tex @@ -1,5 +1,36 @@ % ************************** 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} diff --git a/Chapter1/chapter1.tex b/Chapter1/chapter1.tex index dfff3ac..175b3cf 100644 --- a/Chapter1/chapter1.tex +++ b/Chapter1/chapter1.tex @@ -12,46 +12,206 @@ 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. diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index a6375d1..56beb5c 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -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 diff --git a/Chapter6/Figs/Projections.pdf b/Chapter6/Figs/Projections.pdf index a703c2c..03e65e7 100644 Binary files a/Chapter6/Figs/Projections.pdf and b/Chapter6/Figs/Projections.pdf differ diff --git a/Chapter7/chapter7.tex b/Chapter7/chapter7.tex index a1d642b..f7edf37 100644 --- a/Chapter7/chapter7.tex +++ b/Chapter7/chapter7.tex @@ -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. diff --git a/References/references.bib b/References/references.bib index dbb2295..9a5ffd4 100644 --- a/References/references.bib +++ b/References/references.bib @@ -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. 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