218 lines
13 KiB
TeX
218 lines
13 KiB
TeX
%*******************************************************************************
|
|
%*********************************** First Chapter *****************************
|
|
%*******************************************************************************
|
|
|
|
\chapter{Introduction} %Title of the First Chapter
|
|
|
|
\ifpdf
|
|
\graphicspath{{Chapter1/Figs/Raster/}{Chapter1/Figs/PDF/}{Chapter1/Figs/}}
|
|
\else
|
|
\graphicspath{{Chapter1/Figs/Vector/}{Chapter1/Figs/}}
|
|
\fi
|
|
|
|
|
|
The field of ultracold gases has been a rapidly growing field ever
|
|
since the first Bose-Einstein condensate (BEC) was obtained in 1995
|
|
\cite{anderson1995, bradley1995, davis1995}. This new quantum state of
|
|
matter is characterised by a macroscopic occupancy of the single
|
|
particle ground state at which point the whole system behaves like a
|
|
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 \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 \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 \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.
|