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15
Abstract/abstract.tex
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@ -7,11 +7,11 @@
controllable and tunable than what was possible in condensed
matter. Here, we consider coupling these systems to quantised light
where the quantum nature of both the optical and matter fields play
equally important roles in order to push the boundaries of
what is possible in ultracold atomic systems.
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 matter. By condering a global measurement we show
quantum state of matter. By considering a global measurement we show
that it is possible to distinguish a highly delocalised phase like a
superfluid from the Bose glass and Mott insulator. We also
demonstrate that light scattering reveals not only density
@ -21,11 +21,10 @@
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.
multimode oscillations across the whole lattice when the measurement
is weak and correlated tunnelling, 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

2
Acknowledgement/acknowledgement.tex
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@ -21,7 +21,7 @@ financial means to live and study in Oxford as well as attend several
conferences in the UK and abroad.
On a personal note, I would like to thank my parents who provided me
with all the skills necessary work towards any goals I set
with all the skills necessary to work towards any goals I set
myself. Needless to say, without them I would have never been able to
be where I am right now. I would like to thank all the new and old
friends that have kept me company for the last four years. The time

24
Chapter1/chapter1.tex
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@ -50,7 +50,7 @@ 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.
unprecedented control.
The modern field of strongly correlated ultracold gases is successful
due to its interdisciplinarity \cite{lewenstein2007,
@ -60,11 +60,11 @@ 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
ritsch2013}. Quantum optics has been developing 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
from their environment 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
@ -115,7 +115,7 @@ 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
already made progress, e.g.,~quantum imaging with pixellised sources
of non-classical light \cite{golubev2010, kolobov1999}, as an optical
lattice is a natural source of multimode nonclassical matter waves.
@ -123,7 +123,7 @@ 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. Initially, a lot of effort was exended in an attempt
photodetections. Initially, 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
@ -140,9 +140,9 @@ stochastic Schr\"{o}dinger equation which will be a pure state, which
in contrast to dissipative systems where this is generally not the
case. The question of measurement and its effect on the quantum state
has been around since the inception of quantum theory and still
remains a largely open question \cite{zurek2002}. It wasn't long after
the first condenste was obtained that theoretical work on the effects
of measurement on BECs appeared \cite{cirac1996, castin1997,
remains a largely open question \cite{zurek2002}. It was not long
after the first condensate 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{mekhov2012, mekhov2009prl, mekhov2009pra,
@ -160,7 +160,7 @@ 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
suppress or enhance specific dynamical processes, realising spatially
nonlocal quantum Zeno dynamics.
The quantum Zeno effect happens when frequent measurements slow the
@ -186,8 +186,8 @@ extending the notion of quantum Zeno dynamics into the realm of
non-Hermitian quantum mechanics joining the two
paradigms. Non-Hermitian systems themselves exhibit a range of
interesting phenomena ranging from localisation \cite{hatano1996,
refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$
symmetry} \cite{bender1998, giorgi2010, zhang2013} to spatial order
refael2006} and {\fontfamily{cmr}\selectfont $\mathcal{PT}$ }
symmetry \cite{bender1998, giorgi2010, zhang2013} to spatial order
\cite{otterbach2014} and novel phase transitions \cite{lee2014prx,
lee2014prl}.
@ -212,7 +212,7 @@ 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
scattered 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, elliott2016}. However, this aspect of quantum optics

66
Chapter2/chapter2.tex
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@ -21,7 +21,7 @@ such as nondestructive measurement in free space or quantum
measurement backaction in a cavity. As this model extends the
Bose-Hubbard Hamiltonian to include the effects of interactions with
quantised light we also present a brief overview of the properties of
the Bose-Hubbard model itself and its quantum phase transiton. This is
the Bose-Hubbard model itself and its quantum phase transition. This is
followed by a description of the behaviour of the scattered light with
a particular focus on how to couple the optical fields to phase
observables as opposed to density observables as is typically the
@ -36,8 +36,8 @@ which yields its own set of interesting quantum phenomena
particles. The theory can be also be generalised to continuous
systems, but the restriction to optical lattices is convenient for a
variety of reasons. Firstly, it allows us to precisely describe a
many-body atomic state over a broad range of parameter values due,
from free particles to strongly correlated systems, to the inherent
many-body atomic state over a broad range of parameter values, from
free particles to strongly correlated systems, due to the inherent
tunability of such lattices. Furthermore, this model is capable of
describing a range of different experimental setups ranging from a
small number of sites with a large filling factor (e.g.~BECs trapped
@ -108,11 +108,11 @@ The term $\H_f$ represents the optical part of the Hamiltonian,
\H_f = \sum_l \hbar \omega_l \ad_l \a_l -
i \hbar \sum_l \left( \eta_l^* \a_l - \eta_l \ad \right).
\end{equation}
The operators $\a_l$ ($\ad$) are the annihilation (creation) operators
of light modes with frequencies $\omega_l$, wave vectors $\b{k}_l$,
and mode functions $u_l(\b{r})$, which can be pumped by coherent
fields with amplitudes $\eta_l$. The second part of the Hamiltonian,
$\H_a$, is the matter-field component given by
The operators $\a_l$ ($\ad_l$) are the annihilation (creation)
operators of light modes with frequencies $\omega_l$, wave vectors
$\b{k}_l$, and mode functions $u_l(\b{r})$, which can be pumped by
coherent fields with amplitudes $\eta_l$. The second part of the
Hamiltonian, $\H_a$, is the matter-field component given by
\begin{equation}
\label{eq:Ha}
\H_a = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,a}
@ -280,7 +280,7 @@ cases we can simply neglect all terms but the one that corresponds to
the most significant overlap. Thus, for $J_{i,j}^\mathrm{cl}$ we will
only consider $i$ and $j$ that correspond to nearest neighbours.
Furthermore, since we will only be looking at lattices that have the
same separtion between all its nearest neighbours (e.g. cubic or
same separation between all its nearest neighbours (e.g. cubic or
square lattice) we can define $J_{i,j}^\mathrm{cl} = - J$ (negative
sign, because this way $J > 0$). For the inter-atomic interactions
this simplifies to simply considering on-site collisions where
@ -350,7 +350,7 @@ It is important to note that we are considering a situation where the
contribution of quantised light is much weaker than that of the
classical trapping potential. If that was not the case, it would be
necessary to determine the Wannier functions in a self-consistent way
which takes into account the depth of the quantum poterntial generated
which takes into account the depth of the quantum potential generated
by the quantised light modes. This significantly complicates the
treatment, but can lead to interesting physics. Amongst other things,
the atomic tunnelling and interaction coefficients will now depend on
@ -365,7 +365,7 @@ our quantum light-matter Hamiltonian
\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} -
i \sum_l \kappa_l \ad_l \a_l,
\end{equation}
where we have phenomologically included the cavity decay rates
where we have phenomenologically included the cavity decay rates
$\kappa_l$ of the modes $a_l$. A crucial observation about the
structure of this Hamiltonian is that in the interaction term, the
light modes $a_l$ couple to the matter in a global way. Instead of
@ -391,7 +391,7 @@ electrons within transition metals \cite{hubbard1963}. The key
elements were the confinement of the motion to a lattice and the
approximation of the Coulomb screening interaction as a simple on-site
interaction. Despite these enormous simplifications the model was very
succesful \cite{leggett}. The Bose-Hubbard model is an even simpler
successful \cite{leggett}. The Bose-Hubbard model is an even simpler
variation where instead of fermions we consider spinless bosons. It
was originally devised as a toy model and applied to liquid helium
\cite{fisher1989}, but in 1998 it was shown by Jaksch \emph{et.~al.}
@ -433,7 +433,7 @@ diagram.
Before trying to understand the quantum phase transition itself we
will look at the ground states of the two extremal cases first and for
simplicity we will consider only homogenous systems with periodic
simplicity we will consider only homogeneous systems with periodic
boundary conditions here. We will begin with the $U = 0$ limit where
the Hamiltonian's kinetic term can be diagonalised by transforming to
momentum space defined by the annihilation operator
@ -488,7 +488,7 @@ site and thus not only are there no long-range correlations, there are
actually no two-site correlations in this state at all. Additionally,
this state is gapped, i.e.~it costs a finite amount of energy to
excite the system into its lowest excited state even in the
thermodynamic limit. In fact, the existance of the gap is the defining
thermodynamic limit. In fact, the existence of the gap is the defining
property that distinguishes the Mott insulator from the superfluid in
the Bose-Hubbard model. Note that we have only considered a
commensurate case (number of atoms an integer multiple of the number
@ -514,7 +514,7 @@ energy term is decoupled as
\end{equation}
where the expectation values are for the $T = 0$ ground state. Note
that we have assumed that $\Phi = \langle b_m \rangle $ is non-zero
and homogenous. $\Phi$ describes the influence of hopping between
and homogeneous. $\Phi$ describes the influence of hopping between
neighbouring sites and is the mean -field order parameter. This
decoupling effect means that we can write the Hamiltonian as $\hat{H}_a
= \sum_m^M \hat{h}_a$, where
@ -540,15 +540,15 @@ to $\Phi$.
diagram of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
Ref. \cite{StephenThesis}. The shaded regions are the Mott
insulator lobes and each lobe corresponds to a different on-site
filling labeled by $n$. The rest of the space corresponds to the
filling labelled by $n$. The rest of the space corresponds to the
superfluid phase. The dashed lines are the phase boundaries
obtained from first-order perturbation theory. The solid lines are
are lines of constant density in the superfluid
lines of constant density in the superfluid
phase. \label{fig:BHPhase}}
\end{figure}
The main advantage of the mean-field treatment is that it lets us
study the quantum phase transition between the sueprfluid and Mott
study the quantum phase transition between the superfluid and Mott
insulator phases discussed in the previous sections. The phase
boundaries can be obtained from second-order perturbation theory as
\begin{equation}
@ -557,11 +557,11 @@ boundaries can be obtained from second-order perturbation theory as
\right) + \left( \frac{zJ}{U} \right)^2} \right].
\end{equation}
This yields a phase diagram with multiple Mott insulating lobes as
seen in Fig. \ref{fig:BHPhase} where each lobe represents a Mott insualtor with a
different number of atoms per site. The tip of the lobe which
corresponds to the quantum phase transition along a line of fixed
density is obtained by equating the two branches of the above equation
to get
seen in Fig. \ref{fig:BHPhase} where each lobe represents a Mott
insulator with a different number of atoms per site. The tip of the
lobe which corresponds to the quantum phase transition along a line of
fixed density is obtained by equating the two branches of the above
equation to get
\begin{equation}
\left( \frac{J} {U} \right) = \frac{1} {z (2g + 1 + \sqrt{4g^2 + 4g} )}.
\end{equation}
@ -633,7 +633,7 @@ the critical point by fitting the power-law decay correlations in
order to obtain $K$. The resulting phase transition is shown in
Fig. \ref{fig:1DPhase} and the critical point was shown to be
$(U/zJ) = 1.68$. Note that unusually the phase diagram exhibits a
reentrance phase transition for a fixed $\mu$.
reentrant phase transition for a fixed $\mu$.
\begin{figure}
\centering
@ -643,7 +643,7 @@ reentrance phase transition for a fixed $\mu$.
Ref. \cite{StephenThesis}. The shaded region corresponds to the
$n = 1$ Mott insulator lobe. The sharp tip distinguishes it from
the mean-field phase diagram. The red dotted line denotes the
reentrance phase transition which occurs for a fixed $\mu$.The
reentrant phase transition which occurs for a fixed $\mu$.The
critical point for the fixed density $n = 1$ transition is denoted
by an 'x'. \label{fig:1DPhase}}
\end{figure}
@ -677,7 +677,7 @@ adiabatically follows the quantum state of matter.
The above equation is quite general as it includes an arbitrary number
of light modes which can be pumped directly into the cavity or
produced via scattering from other modes. To simplify the equation
slightly we will neglect the cavity resonancy shift,
slightly we will neglect the cavity resonance shift,
$U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
rate and/or probe detuning are large enough. We will also only
consider probing with an external coherent beam, $a_0$ with mode
@ -745,7 +745,7 @@ atoms which for off-resonant light, like the type that we consider,
results in an effective coupling with atomic density, not the
matter-wave amplitude. Therefore, it is challenging to couple light to
the phase of the matter-field as is typical in quantum optics for
optical fields. Most of the exisiting work on measurement couples
optical fields. Most of the existing work on measurement couples
directly to atomic density operators \cite{mekhov2012, LP2009,
rogers2014, ashida2015, ashida2015a}. However, it has been shown
that one can couple to the interference term between two condensates
@ -816,7 +816,7 @@ sites rather than at the positions of the atoms.
between lattice sites. The thin black line indicates the trapping
potential (not to scale). (a) Arrangement for the uniform pattern
$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
$J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
$J_{i,i+1}=(-1)^i J_2$; here $u_0=1$ so it is not shown and $u_1$
is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure}
@ -996,7 +996,7 @@ probe. Therefore, we drop the sum and consider only a single wave
vector, $\b{k}_0$, of the dominant contribution from the external
beam corresponding to the mode $\a_0$.
We now use the definititon of the atom-light coupling constant
We now use the definition of the atom-light coupling constant
\cite{Scully} to make the substitution
$g_0 a_0 = -i \Omega_0 e^{-i \omega_0 t} / 2$, where
$\Omega_0$ is the Rabi frequency for the probe-atom system. We now
@ -1047,7 +1047,7 @@ $\b{r}_i$. Thus, the relevant many-body operator is
\end{subequations}
where $K$ is the number of illuminated sites. We now assume that the
lattice is deep and that the light scattering occurs on time scales
much shorter than atom tunneling. Therefore, we ignore all terms for
much shorter than atom tunnelling. Therefore, we ignore all terms for
which $i \ne j$ and the remaining integrals become $\int \mathrm{d}^3
\b{r}_0 w^*(\b{r}_0 - \b{r}_i) f (\b{r})
w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of
@ -1131,10 +1131,10 @@ in free space. Miyake \emph{et al.} experimental parameters are from
Ref. \cite{miyake2011}. The $5^2S_{1/2}$,
$F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb
is considered. For this transition the Rabi frequency is actually
larger than the detuning and and effects of saturation should be taken
larger than the detuning and effects of saturation should be taken
into account in a more complete analysis. However, it is included for
discussion. The detuning was specified as ``a few linewidths'' and
note that it is much smaller than $\Omega_0$. The Rabi fequency is
note that it is much smaller than $\Omega_0$. The Rabi frequency is
$\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is
obtained from definition of Rabi frequency,
$\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$, assuming the electric
@ -1163,7 +1163,7 @@ per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.
There are many possible experimental issues that we have neglected so
far in our theoretical treatment which need to be answered in order to
consider the experimental feasability of our proposal. The two main
consider the experimental feasibility of our proposal. The two main
concerns are photodetector efficiency and heating losses.
In our theoretical models we treat the detectors as if they were

55
Chapter3/chapter3.tex
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@ -76,12 +76,12 @@ possible distance in an optical lattice, i.e. the lattice period,
which defines key processes such as tunnelling, currents, phase
gradients, etc. This is in contrast to standard destructive
time-of-flight measurements which deal with far-field interference
although a relatively near-field scheme was use in
although a relatively near-field scheme was used in
Ref. \cite{miyake2011}. We show how within the mean-field treatment,
this enables measurements of the order parameter, matter-field
quadratures and squeezing. This can have an impact on atom-wave
metrology and information processing in areas where quantum optics
already made progress, e.g., quantum imaging with pixellized sources
already made progress, e.g., quantum imaging with pixellised sources
of non-classical light \cite{golubev2010, kolobov1999}, as an optical
lattice is a natural source of multimode nonclassical matter waves.
@ -112,7 +112,7 @@ stricter requirements for a QND measurement \cite{mekhov2012,
mekhov2007pra}) and the measurement only weakly perturbs the system,
many consecutive measurements can be carried out with the same atoms
without preparing a new sample. We will show how the extreme
flexibility of the the measurement operator $\hat{F}$ allows us to
flexibility of the measurement operator $\hat{F}$ allows us to
probe a variety of different atomic properties in-situ ranging from
density correlations to matter-field interference.
@ -160,7 +160,7 @@ We will now define a new auxiliary quantity to aid our analysis,
which we will call the ``quantum addition'' to light scattering. By
construction $R$ is simply the full light intensity minus the
classical field diffraction. In order to justify its name we will show
that this quantity depends purely quantum mechanical properties of the
that this quantity depends on purely quantum mechanical properties of the
ultracold gas. We substitute $\a_1 = C \hat{D}$ using
Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
we make use of the shorthand notation
@ -224,7 +224,7 @@ $R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
)/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, because it shows scattering from an ideal
superfluid which has significant density fluctuations with
correlations of infinte range. However, as the parameters of the
correlations of infinite range. However, as the parameters of the
lattice are tuned across the phase transition into a Mott insulator
the signal goes to zero. This is because the Mott insulating phase has
well localised atoms at each site which suppresses density
@ -252,7 +252,7 @@ angles than predicted by the classical Bragg condition. Moreover, the
classical Bragg condition is actually not satisfied which means there
actually is no classical diffraction on top of the ``quantum
addition'' shown here. Therefore, these features would be easy to see
in an experiment as they wouldn't be masked by a stronger classical
in an experiment as they would not be masked by a stronger classical
signal. This difference in behaviour is due to the fact that
classical diffraction is ignorant of any quantum correlations as it is
given by the square of the light field amplitude squared
@ -260,7 +260,7 @@ given by the square of the light field amplitude squared
|\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^*
A_j \langle \n_i \rangle \langle \n_j \rangle,
\end{equation}
which is idependent of any two-point correlations unlike $R$. On the
which is independent of any two-point correlations unlike $R$. On the
other hand the full light intensity (classical signal plus ``quantum
addition'') of the quantum light does include higher-order
correlations
@ -295,7 +295,7 @@ can also be negative.
We will consider scattering from a superfluid, because the Mott
insulator has no ``quantum addition'' due to a lack of density
fluctuations. The wave function of a superfluid on a lattice is given
by Eq. \eqref{eq:GSSF}. This state has infinte range correlations and
by Eq. \eqref{eq:GSSF}. This state has infinite range correlations and
thus has the convenient property that all two-point density
fluctuation correlations are equal regardless of their separation,
i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
@ -349,7 +349,7 @@ wave scattered mode is
Therefore, it is straightforward to see that unless
$2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector,
there will be no coherent signal and we end up with the mean uniform
signal of strength $|C|^2 N_k/2$. When this condition is satisifed all
signal of strength $|C|^2 N_k/2$. When this condition is satisfied all
the cosine terms will be equal and they will add up constructively
instead of cancelling each other out. Note that this new Bragg
condition is different from the classical one
@ -371,7 +371,7 @@ the condition we had for $R$ and is still different from the classical
condition $\b{k}_0 - \b{k}_1 = \b{G}$. Furthermore, just like in
Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$.
A quantum signal that isn't masked by classical diffraction is very
A quantum signal that is not masked by classical diffraction is very
useful for future experimental realisability. However, it is still
unclear whether this signal would be strong enough to be
visible. After all, a classical signal scales as $N_K^2$ whereas here
@ -423,7 +423,7 @@ quantum phase transition is in a different universality class than its
higher dimensional counterparts. The energy gap, which is the order
parameter, decays exponentially slowly across the phase transition
making it difficult to identify the phase transition even in numerical
simulations. Here, we will show the avaialable tools provided by the
simulations. Here, we will show the available tools provided by the
``quantum addition'' that allows one to nondestructively map this
phase transition and distinguish the superfluid and Mott insulator
phases.
@ -463,7 +463,7 @@ difficult to directly access the excitation energy gap which defines
this phase transition. However, a valid description of the relevant 1D
low energy physics is provided by Luttinger liquid theory
\cite{giamarchi} as seen in section \ref{sec:BHM1D}. In this model
correlations in the supefluid phase as well as the superfluid density
correlations in the superfluid phase as well as the superfluid density
itself are characterised by the Tomonaga-Luttinger parameter,
$K$. This parameter also identifies the critical point in the
thermodynamic limit at $K_c = 1/2$. This quantity can be extracted
@ -489,7 +489,7 @@ Bose-Hubbard model as it can be easily applied to many other systems
such as fermions, photonic circuits, optical lattices with quantum
potentials, etc. Therefore, by providing a better physical picture of
what information is carried by the ``quantum addition'' it should be
easier to see its usefuleness in a broader context.
easier to see its usefulness in a broader context.
We calculate the phase diagram of the Bose-Hubbard Hamiltonian given
by
@ -595,7 +595,7 @@ in the Mott insulating phase and they change drastically across the
phase transition into the superfluid phase. Next, we present a case
where it is very different. We will again consider ultracold bosons in
an optical lattice, but this time we introduce some disorder. We do
this by adding an additional periodic potential on top of the exisitng
this by adding an additional periodic potential on top of the existing
setup that is incommensurate with the original lattice. The resulting
Hamiltonian can be shown to be
\begin{equation}
@ -638,7 +638,7 @@ results in a large $R_\text{max}$ and a small $W_R$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{oph22_3}
\caption[Mapping the Disoredered Phase Diagram]{The
\caption[Mapping the Disordered Phase Diagram]{The
Mott-superfluid-glass phase diagrams for light scattering maximum
$R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both
quantities distinguish all three phases. Transition lines are
@ -705,7 +705,7 @@ allows one to probe the mean-field order parameter, $\Phi$,
directly. Normally, this is achieved by releasing the trapped gas and
performing a time-of-flight measurement. Here, this can be achieved
in-situ. In section \ref{sec:B} we showed that one of the possible
optical arrangement leads to a diffraction maximum with the matter
optical arrangements leads to a diffraction maximum with the matter
operator
\begin{equation}
\label{eq:Bmax}
@ -733,7 +733,7 @@ measurements provide us with new opportunities to study the quantum
matter state which was previously unavailable. We will take $\Phi$ to
be real which in the standard Bose-Hubbard Hamiltonian can be selected
via an inherent gauge degree of freedom in the order parameter. Thus,
the quadratures themself straightforwardly become
the quadratures themselves straightforwardly become
\begin{equation}
\hat{X}^b_\alpha = \frac{\Phi}{2} (e^{-i\alpha} + e^{i\alpha}) =
\Phi \cos(\alpha).
@ -746,17 +746,16 @@ the quadrature is a more complicated quantity given by
(\Delta X^b_{0,\pi/2})^2 = \frac{1}{4} + [(n - \Phi^2) \pm
\frac{1}{2}(\langle b^2 \rangle - \Phi^2)],
\end{equation}
where we have arbitrarily selected two orthogonal quadratures
$\alpha =0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated
from light quadrature measurements alone as we do not know the value
of $\langle b^2 \rangle$. To obtain the value of
$\langle b^2 \rangle$this quantity we need to consider a second-order
light observable such as light intensity. However, in the diffraction
maximum this signal will be dominated by a contribution proportional
to $K^2 \Phi^4$ whereas the terms containing information on
$\langle b^2 \rangle$ will only scale as $K$. Therefore, it would be
difficult to extract the quantity that we need by measuring in the
difraction maximum.
where we have arbitrarily selected two orthogonal quadratures $\alpha
=0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated from
light quadrature measurements alone as we do not know the value of
$\langle b^2 \rangle$. To obtain the value of this quantity we need to
consider a second-order light observable such as light
intensity. However, in the diffraction maximum this signal will be
dominated by a contribution proportional to $K^2 \Phi^4$ whereas the
terms containing information on $\langle b^2 \rangle$ will only scale
as $K$. Therefore, it would be difficult to extract the quantity that
we need by measuring in the diffraction maximum.
\begin{figure}
\centering

26
Chapter4/chapter4.tex
View File

@ -24,7 +24,7 @@ where we neglect the effect of the quantum wavefunction collapse. We
have shown that light provides information about various statistical
quantities of the quantum states of the atoms such as their
correlation functions. In general, any quantum measurement affects the
system even if it doesn't physically destroy it. In our model both
system even if it does not physically destroy it. In our model both
optical and matter fields are quantised and their interaction leads to
entanglement between the two subsystems. When a photon is detected and
the electromagnetic wave function of the optical field collapses, the
@ -75,7 +75,7 @@ realisation. In particular, we consider states conditioned upon
measurement results such as the photodetection times. Such a
trajectory is generally stochastic in nature as light scattering is
not a deterministic process. Furthermore, they are in general
discotinuous as each detection event brings about a drastic change in
discontinuous as each detection event brings about a drastic change in
the quantum state due to the wave function collapse of the light field.
Before we discuss specifics relevant to our model of quantised light
@ -152,7 +152,7 @@ quantum jump occurs and $0$ otherwise \cite{MeasurementControl}. Note
that this equation has a straightforward generalisation to multiple
jump operators, but we do not consider this possibility here at all.
All trajectories that we calculate in the follwing chapters are
All trajectories that we calculate in the following chapters are
described by the stochastic Schr\"{o}dinger equation in
Eq. \eqref{eq:SSE}. The most straightforward way to solve it is to
replace the differentials by small time-steps $\delta t$. Then we
@ -174,7 +174,7 @@ by
\end{equation}
up to a time $T$ such that
$\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$. This
problem can be solved efficienlty using standard numerical
problem can be solved efficiently using standard numerical
techniques. At time $T$ a quantum jump is applied according to
Eq. \eqref{eq:jump} which renormalises the wave function as well and
the process is repeated as long as desired. This formulation also has
@ -194,7 +194,7 @@ processes is balanced and without any further external influence the
distribution of outcomes over many trajectories will be entirely
determined by the initial state even though each individual trajectory
will be unique and conditioned on the exact detection times that
occured during the given experimental run. However, individual
occurred during the given experimental run. However, individual
trajectories can have features that are not present after averaging
and this is why we focus our attention on single experimental runs
rather than average behaviour.
@ -224,7 +224,7 @@ could scatter and thus include multiple jump operators reducing our
ability to control the system.
The model we derived in Eq. \eqref{eq:fullH} is in fact already in a
form ready for quantum trajectory simulations. The phenomologically
form ready for quantum trajectory simulations. The phenomenologically
included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the
non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$
which is the jump operator we want for measurements of photons leaking
@ -265,7 +265,7 @@ Importantly, we see that measurement introduces a new energy and time
scale $\gamma$ which competes with the two other standard scales
responsible for the unitary dynamics of the closed system, tunnelling,
$J$, and on-site interaction, $U$. If each atom scattered light
inependently a different jump operator $\c_i$ would be required for
independently a different jump operator $\c_i$ would be required for
each site projecting the atomic system into a state where long-range
coherence is degraded. This is a typical scenario for spontaneous
emission \cite{pichler2010, sarkar2014}, or for local
@ -343,7 +343,7 @@ is not accessible and thus must be represented as an average over all
possible trajectories. However, there is a crucial difference between
measurement and dissipation. When we perform a measurement we use the
master equation to describe system evolution if we ignore the
measuremt outcomes, but at any time we can look at the detection times
measurement outcomes, but at any time we can look at the detection times
and obtain a conditioned pure state for this current experimental
run. On the other hand, for a dissipative system we simply have no
such record of results and thus the density matrix predicted by the
@ -410,7 +410,7 @@ two basis states. Therefore, if we obtained this density matrix as a
result of applying the time evolution given by the master equation we
would be able to identify the final states of individual trajectories
even though we have no information about the individual trajectories
themself. This is analogous to an approach in which decoherence due to
themselves. This is analogous to an approach in which decoherence due to
coupling to the environment is used to model the wave function collapse
\cite{zurek2002}, but here we will be looking at projective effects
due to weak measurement.
@ -444,7 +444,7 @@ was irrelevant. Now, on the other hand, we are interested in the
effect of these measurements on the dynamics of the system. The effect
of measurement backaction will depend on the information that is
encoded in the detected photon. If a scattered mode cannot
distinuguish between two different lattice sites then we have no
distinguish between two different lattice sites then we have no
information about the distribution of atoms between those two sites.
Therefore, all quantum correlations between the atoms in these sites
are unaffected by the backaction whilst their correlations with the
@ -494,7 +494,7 @@ that these partitions in general are not neighbours of each other,
they are not localised, they overlap in a nontrivial way, and the
patterns can be made more complex in higher dimensions as shown in
Fig. \ref{fig:2dmodes} or with a more sophisticated optical setup
\cite{caballero2016}. This has profound consquences as it can lead to
\cite{caballero2016}. This has profound consequences as it can lead to
the creation of long-range nonlocal correlations between lattice sites
\cite{mazzucchi2016, kozlowski2016zeno}. The measurement does not know
which atom within a certain mode scattered the light, there is no
@ -538,7 +538,7 @@ Fig. \ref{fig:twomodes}.
\includegraphics[width=0.7\textwidth]{TwoModes2}
\caption[Two Mode Partitioning]{Top: only the odd sites scatter
light leading to a measurement of $\hat{N}_\mathrm{odd}$ and an
effectove partitiong into even and odd sites. Bottom: this also
effective partitioning into even and odd sites. Bottom: this also
partitions the lattice into odd and even sites, but this time
atoms at all sites scatter light, but in anti-phase with their
neighbours.}
@ -546,7 +546,7 @@ Fig. \ref{fig:twomodes}.
\end{figure}
This approach can be generalised to an arbitrary number of modes,
$Z$. For this we will conisder a deep lattice such that
$Z$. For this we will consider a deep lattice such that
$J_{i,i} = u_1^* (\b{r}) u_0 (\b{r})$. We will take the probe beam to
be incident normally at a 1D lattice so that $u_0 (\b{r}) =
1$. Therefore, the final form of the scattered light field is given by

137
Chapter5/chapter5.tex
View File

@ -32,7 +32,7 @@ which competes with the intrinsic dynamics of the bosons.
In this chapter, we investigate the effect of quantum measurement
backaction on the many-body state and dynamics of atoms. In
particular, we will focus on the competition between the backaction
and the the two standard short-range processes, tunnelling and on-site
and the two standard short-range processes, tunnelling and on-site
interactions, in optical lattices. We show that the possibility to
spatially structure the measurement at a microscopic scale comparable
to the lattice period without the need for single site resolution
@ -46,7 +46,7 @@ spatially nonlocal coupling to the environment, as seen in section
\ref{sec:modes}, which is impossible to obtain with local interactions
\cite{diehl2008, syassen2008, daley2014}. These spatial modes of
matter fields can be considered as designed systems and reservoirs
opening the possibility of controlling dissipations in ultracold
opening the possibility of controlling dissipation in ultracold
atomic systems without resorting to atom losses and collisions which
are difficult to manipulate. Thus the continuous measurement of the
light field introduces a controllable decoherence channel into the
@ -115,7 +115,7 @@ atoms being entirely within a single mode. Finally, we note that these
oscillating distributions are squeezed by the measurement and the
individual components have a width smaller than the initial state. By
contrast, in the absence of the external influence of measurement
these distributions would spread out significantly and the center of
these distributions would spread out significantly and the centre of
the broad distribution would oscillate with an amplitude comparable to
the initial imbalance, i.e.~small oscillations for a small initial
imbalance.
@ -128,7 +128,7 @@ imbalance.
resulting from the competition between tunnelling and weak
measurement induced backaction. The plots show the atom number
distributions $p(N_l)$ in one of the modes in individual quantum
trajectories. These dstributions show various numbers of
trajectories. These distributions show various numbers of
well-squeezed components reflecting the creation of macroscopic
superposition states depending on the measurement
configuration. $U/J = 0$, $\gamma/J = 0.01$, $M=N$, initial
@ -218,7 +218,7 @@ lattice. $b_e$ ($\bd_e$) is defined similarly, but for the right site
and the superfluid at even sites of the physical lattice.
$\n_o = \bd_o b_o$ is the atom number operator in the left site.
Even though Eq. \eqref{eq:doublewell} is relatively simple as it it is
Even though Eq. \eqref{eq:doublewell} is relatively simple as it is
only a non-interacting two-site model, the non-Hermitian term
complicates the situation making the system difficult to
solve. However, a semiclassical approach to boson dynamics in a
@ -277,7 +277,7 @@ of the form
as our ansatz to Eq. \eqref{eq:semicl}. The parameters $b$, $\phi$,
$z_0$, and $z_\phi$ are real-valued functions of time whereas
$\epsilon$ is a complex-valued function of time. Physically, the value
$b^2$ denotes the width, $z_0$ the position of the center, $\phi$ and
$b^2$ denotes the width, $z_0$ the position of the centre, $\phi$ and
$z_\phi$ contain the local phase information, and $\epsilon$ only
affects the global phase and norm of the Gaussian wave packet.
@ -310,12 +310,12 @@ 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
Eq. \eqref{eq:ansatz} subject to non-Hermitian time evolution
according to Eq. \eqref{eq:semicl} with discontinous changes to the
according to Eq. \eqref{eq:semicl} with discontinuous changes to the
parameter values at each quantum jump.
Having identified an appropriate ansatz and the effect of quantum
jumps we proceed with solving the dynamics of wave function in between
the photodetecions. The initial values of the parameters for a
the photodetections. The initial values of the parameters for a
superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$,
$\phi =0$, $a_0 = 0$, $a_\phi = 0$, $\epsilon = 0$. However, we use
the most general initial conditions at time $t = t_0$ which we denote
@ -324,7 +324,7 @@ $z_\phi(t_0) = a_\phi$, and $\epsilon(t_0) = \epsilon_0$. The reason
for keeping them as general as possible is that after every quantum
jump the system changes discontinuously. The subsequent time evolution
is obtained by solving the Schr\"{o}dinger equation with the post-jump
paramater values as the new initial conditions.
parameter values as the new initial conditions.
By plugging the ansatz in Eq. \eqref{eq:ansatz} into the
Schr\"{o}dinger equation in Eq. \eqref{eq:semicl} we obtain three
@ -517,7 +517,7 @@ visible on short time scales. Instead, we look at the long time
limit. Unfortunately, since all the quantities are oscillatory a
stationary long time limit does not exist especially since the quantum
jumps provide a driving force. However, the width of the Gaussian,
$b^2$, is unique in that it doesn't oscillate around $b^2 =
$b^2$, is unique in that it does not oscillate around $b^2 =
0$. Furthermore, from Eq. \eqref{eq:jumpb2} we see that even though it
will decrease discontinuously at every jump, this effect is fairly
small since $b^2 \ll 1$ generally. Therefore, we expect $b^2$ to
@ -603,7 +603,7 @@ this quantity for $\hat{D} = \hat{N}_\mathrm{odd}$ averaged over
multiple trajectories, $\langle \sigma^2_D \rangle_\mathrm{traj}$, as
a function of $\gamma/J$ and $U/J$ for a lattice of six atoms on six
sites (we cannot use the effective double-well model, because
$U \ne 0$). We use a ground state of for the corresponding $U$ and $J$
$U \ne 0$). We use a ground state for the corresponding $U$ and $J$
values as this provides a realistic starting point and a reference for
comparing the measurement induced dynamics. We will also consider only
$\hat{D} = \hat{N}_\mathrm{odd}$ unless stated otherwise.
@ -612,26 +612,25 @@ $\hat{D} = \hat{N}_\mathrm{odd}$ unless stated otherwise.
\centering
\includegraphics[width=\textwidth]{Squeezing}
\caption[Squeezing in the presence of Interactions]{Atom number
fluctuations at odd sites for for $N = 6$ atoms at $M = 6$ sites
fluctuations at odd sites for $N = 6$ atoms at $M = 6$ sites
subject to a $\hat{D} = \hat{N}_\mathrm{odd}$ measurement
demonstrating the competition of global measurement with local
interactions and tunnelling. Number variances are averaged over
100 trajectories. Error bars are too small to be shown
($\sim 1\%$) which emphasizes the universal nature of the
squeezing. The initial state used was the ground state for the
corresponding $U$ and $J$ value. The fluctuations in the ground
state without measurement decrease as $U / J$ increases,
reflecting the transition between the supefluid and Mott insulator
phases. For weak measurement values
$\langle \sigma^2_D \rangle_\mathrm{traj}$ is squeezed below the
ground state value for $U = 0$, but it subsequently increases and
reaches its maximum as the atom repulsion prevents the
accumulation of atoms prohibiting coherent oscillations thus
making the squeezing less effective. In the strongly interacting
limit, the Mott insulator state is destroyed and the fluctuations
are larger than in the ground state as weak measurement isn't
strong enough to project into a state with smaller fluctuations
than the ground state.}
100 trajectories. Error bars are too small to be shown ($\sim
1\%$) which emphasizes the universal nature of the squeezing. The
initial state used was the ground state for the corresponding $U$
and $J$ value. The fluctuations in the ground state without
measurement decrease as $U / J$ increases, reflecting the
transition between the superfluid and Mott insulator phases. For
weak measurement values $\langle \sigma^2_D \rangle_\mathrm{traj}$
is squeezed below the ground state value for $U = 0$, but it
subsequently increases and reaches its maximum as the atom
repulsion prevents the accumulation of atoms prohibiting coherent
oscillations thus making the squeezing less effective. In the
strongly interacting limit, the Mott insulator state is destroyed
and the fluctuations are larger than in the ground state as weak
measurement is not strong enough to project into a state with
smaller fluctuations than the ground state.}
\label{fig:squeezing}
\end{figure}
@ -654,7 +653,7 @@ be lost if we studied an ensemble average.
dynamics of the atom-number distributions at odd sites
illustrating competition of the global measurement with local
interactions and tunnelling. The plots are for single quantum
trajectores starting from the ground state for $N = 6$ atoms on
trajectories starting from the ground state for $N = 6$ atoms on
$M = 6$ sites with $\hat{D} = \hat{N}_\mathrm{odd}$,
$\gamma/J = 0.1$. (a) Weakly interacting bosons $U/J = 1$: the
on-site repulsion prevents the formation of well-defined
@ -662,7 +661,7 @@ be lost if we studied an ensemble average.
different imbalance evolve with different frequencies, the
squeezing is not as efficient for the non-interacting case. (b)
Strongly interacting bosons $U/J = 10$: oscillations are
completely supressed and the number of atoms in the mode is rather
completely suppressed and the number of atoms in the mode is rather
well-defined although clearly worse than in a Mott insulator.}
\label{fig:Utraj}
\end{figure}
@ -679,7 +678,7 @@ squeezed below those of the ground state followed by a rapid increase
as $U$ is increased before peaking and eventually decreasing. We have
already seen in the previous section and in particular
Fig. \ref{fig:oscillations} that the macroscopic oscillations at
$U = 0$ are well squeezed when compared to the inital state and this
$U = 0$ are well squeezed when compared to the initial state and this
is the case over here as well. However, as $U$ is increased the
interactions prevent the atoms from accumulating in one place thus
preventing oscillations with a large amplitude which effectively makes
@ -724,7 +723,7 @@ $\c^m | \Psi_{J/U} \rangle \propto | \Psi_{J/U} \rangle + | \Phi_m
\end{equation}
In the weak measurement regime the effect of non-Hermitian decay is
negligible compared to the local atomic dynamics combined with the
quantum jumps so there is minimal dissipation occuring. Therefore,
quantum jumps so there is minimal dissipation occurring. Therefore,
because of the exponential growth of the excitations, even a small
number of photons arriving in succession can destroy the ground
state. We have neglected all dynamics in between the jumps which would
@ -742,15 +741,15 @@ case, the squeezing will always be better than in the ground state,
because measurement and on-site interaction cooperate in suppressing
fluctuations. This cooperation did not exist for weak measurement,
because it tried to induce dynamics which produced squeezed states
(either succesfully as seen with the macroscopic oscillations or
unsuccesfully as seen with the Mott insulator). This suffered heavily
(either successfully as seen with the macroscopic oscillations or
unsuccessfully as seen with the Mott insulator). This suffered heavily
from the effects of interactions as they would prevent this dynamics
by dephasing different components of the coherent excitations. Strong
measurement, on the other hand, squeezes the quantum state by trying
to project it onto an eigenstate of the observable
\cite{mekhov2009prl, mekhov2009pra}. For weak interactions where the
ground state is a highly delocalised superfluid it is obvious that
projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will supress
projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will suppress
fluctuations significantly. However, the strongly interacting regime
is much less evident, especially since we have just demonstrated how
sensitive the Mott insulating phase is to the quantum jumps when the
@ -805,13 +804,13 @@ the $\gamma/U \gg 1$ regime is reached.
When $\gamma \rightarrow \infty$ the measurement becomes
projective. This means that as soon as the probing begins, the system
collapses into one of the observable's eigenstates. Furthermore, since
this measurement is continuous and doesn't stop after the projection
this measurement is continuous and does not stop after the projection
the system will be frozen in this state. This effect is called the
quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
paradox in which a ``watched arrow never moves'' that stated that
since an arrow in flight is not seen to move during any single
instant, it cannot possibly be moving at all. Classically the paradox
was resolved with a better understanding of infinity and infintesimal
was resolved with a better understanding of infinity and infinitesimal
changes, but in the quantum world a watched quantum arrow will in fact
never move. The system is being continuously projected into its
initial state before it has any chance to evolve. If degenerate
@ -822,7 +821,7 @@ called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
These effects can be easily seen in our model when
$\gamma \rightarrow \infty$. The system will be projected into one or
more degenerate eignstates of $\cd \c$, $| \psi_i \rangle$, for which
more degenerate eigenstates of $\cd \c$, $| \psi_i \rangle$, for which
we define the projector
$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
denotes a single degenerate subspace. The Zeno subspace is determined
@ -834,12 +833,12 @@ Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
quantum jumps as their combined effect is now described by the
projectors. Physically, in our model of ultracold bosons trapped in a
lattice this means that tunnelling between different spatial modes is
completely supressed since this process couples eigenstates belonging
completely suppressed since this process couples eigenstates belonging
to different Zeno subspaces. If a small connected part of the lattice
was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
tunnelling would only be prohibited between the illuminated and
unilluminated areas, but dynamics proceeds normally within each zone
separately. However, the goemetric patterns we have in which the modes
separately. However, the geometric patterns we have in which the modes
are spatially delocalised in such a way that neighbouring sites never
belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, would
lead to a complete suppression of tunnelling across the whole lattice
@ -862,11 +861,11 @@ Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
\begin{figure}[hbtp!]
\includegraphics[width=\textwidth]{Zeno.pdf}
\caption[Emergent Long-Range Correlated Tunnelling]{ Long-range
correlated tunneling and entanglement, dynamically induced by
correlated tunnelling and entanglement, dynamically induced by
strong global measurement in a single quantum
trajectory. (a),(b),(c) show different measurement geometries,
implying different constraints. Panels (1): schematic
representation of the long-range tunneling processes. Panels (2):
representation of the long-range tunnelling processes. Panels (2):
evolution of on-site densities. Panels (3): entanglement entropy
growth between illuminated and non-illuminated regions. Panels
(4): correlations between different modes (orange) and within the
@ -879,9 +878,9 @@ Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
correlations between the $N_I$ and $N_{NI}$ modes
(orange). Initial state: $|1,1,1,1,1,1,1 \rangle$, $\gamma/J=100$,
$J_{jj}=[0,0,1,1,1,0,0]$. (b) (b.1) Even sites are illuminated,
freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunneling
freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunnelling
is represented by any pair of one blue and one red arrow. (b.2)
Correlated tunneling occurs between non-neighbouring sites without
Correlated tunnelling occurs between non-neighbouring sites without
changing mode populations. (b.3) Entanglement build up. (b.4)
Negative correlations between edge sites (green) and zero
correlations between the modes defined by $N_\text{even}$ and
@ -949,19 +948,19 @@ neighbouring sites that belong to the same mode. Otherwise, if $i$ and
$j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
$\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
where this would be the only remaining term. It is the second term
that shows the second-order corelated tunnelling terms. This is
that shows the second-order correlated tunnelling terms. This is
evident from the inner sum which requires that pairs of sites ($i$,
$j$) and ($k$, $l$) between which atoms tunnel must be nearest
neighbours, but these pairs can be anywhere on the lattice within the
constraints of the mode structure. This is in particular explicitly
shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
that the tinelling behaves like an exponential decay (overdamped
that the tunnelling behaves like an exponential decay (overdamped
oscillations). This also implies that the norm will decay, but this
does not mean that there are physical losses in the system. Instead,
the norm itself represents the probability of the system remaining in
the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
is now a finite probability that the stochastic nature of the
measurement will lead to a discontinous change in the system where the
measurement will lead to a discontinuous change in the system where the
Zeno subspace rapidly changes which can be seen in
Fig. \ref{fig:zeno}(a). However, later in this chapter we will see
that steady states of this Hamiltonian exist which will no longer
@ -973,7 +972,7 @@ selectively suppressed by the global conservation of the measured
observable and not by the prohibitive energy costs of doubly-occupied
sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
profound consequences as this is the physical origin of the long-range
correlated tunneling events represented in Eq. \eqref{eq:hz} by the
correlated tunnelling events represented in Eq. \eqref{eq:hz} by the
fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
projection $P_0$ is not sensitive to individual site
occupancies, but instead enforces a fixed value of the observable,
@ -1002,7 +1001,7 @@ typical dynamics occurs within each region but the standard tunnelling
between different modes is suppressed. Importantly, second-order
processes that do not change $N_\text{K}$ are still possible since an
atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
from $2$ to $3$. Therefore, effective long-range tunneling between two
from $2$ to $3$. Therefore, effective long-range tunnelling between two
spatially disconnected zones $1$ and $3$ happens due to the two-step
processes $1 \rightarrow 2 \rightarrow 3$ or
$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
@ -1018,7 +1017,7 @@ correlated tunnelling process builds long-range entanglement between
illuminated and non-illuminated regions as shown in
Fig.~\ref{fig:zeno}(a.3).
To make correlated tunneling visible even in the mean atom number, we
To make correlated tunnelling visible even in the mean atom number, we
suppress the standard Bose-Hubbard dynamics by illuminating only the
even sites of the lattice in Fig.~\ref{fig:zeno}(b). Even though this
measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$,
@ -1026,10 +1025,10 @@ atoms can slowly tunnel between the odd sites of the lattice, despite
them being spatially disconnected. This atom exchange spreads
correlations between non-neighbouring lattice sites on a time scale
$\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
explanation of long-range correlated tunneling is presented in
explanation of long-range correlated tunnelling is presented in
Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
such that one correlated tunneling event is represented by a pair of
such that one correlated tunnelling event is represented by a pair of
one red and one blue arrow. Importantly, this scheme is fully
applicable for a lattice with large atom and site numbers, well beyond
the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
@ -1038,14 +1037,14 @@ assures this mode structure (in this example, two modes at odd and
even sites) and thus the underlying pairwise global tunnelling.
The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
reservoir for the tunneling (or ``decay'') of atoms from one region to
reservoir for the tunnelling (or ``decay'') of atoms from one region to
another. For example, if the atoms are placed only at odd sites,
according to Eq. \eqref{eq:hz} their tunnelling is suppressed since the
multi-tunneling event must be successive, i.e.~an atom tunnelling into
multi-tunnelling event must be successive, i.e.~an atom tunnelling into
a different mode, $\varphi^\prime$, must then also tunnel back into
its original mode, $\varphi$. If, however, one adds some atoms to even
sites (even if they are far from the initial atoms), the correlated
tunneling events become allowed and their rate can be tuned by the
tunnelling events become allowed and their rate can be tuned by the
number of added atoms. This resembles the repulsively bound pairs
created by local interactions \cite{winkler2006, folling2007}. In
contrast, here the atom pairs are nonlocally correlated due to the
@ -1054,7 +1053,7 @@ consequence of the dynamics being constrained to a Zeno subspace: the
virtual processes allowed by the measurement entangle the spatial
modes nonlocally. Since the measurement only reveals the total number
of atoms in the illuminated sites, but not their exact distribution,
these multi-tunelling events cause the build-up of long-range
these multi-tunnelling events cause the build-up of long-range
entanglement. This is in striking contrast to the entanglement caused
by local processes which can be very confined, especially in 1D where
it is typically short range. This makes numerical calculations of our
@ -1072,15 +1071,15 @@ generation of positive number correlations shown in
Fig.~\ref{fig:zeno}(c.4) by freezing the atom number difference
between the sites ($N_\text{odd}-N_\text{even}$). Thus, atoms can only
enter or leave this region in pairs, which again is possible due to
correlated tunneling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and
correlated tunnelling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and
manifests positive correlations. As in the previous example, two edge
modes in Fig.~\ref{fig:zeno}(c) can be considered as a nonlocal
reservoir for two central sites, where a constraint is applied. Note
that, using more modes, the design of higher-order multi-tunneling
that, using more modes, the design of higher-order multi-tunnelling
events is possible.
This global pair tunneling may play a role of a building block for
more complicated many-body effects. For example, a pair tunneling
This global pair tunnelling may play a role of a building block for
more complicated many-body effects. For example, a pair tunnelling
between the neighbouring sites has been recently shown to play
important role in the formation of new quantum phases, e.g., pair
superfluid \cite{sowinski2012} and lead to formulation of extended
@ -1088,7 +1087,7 @@ Bose-Hubbard models \cite{omjyoti2015}. The search for novel
mechanisms providing long-range interactions is crucial in many-body
physics. One of the standard candidates is the dipole-dipole
interaction in, e.g., dipolar molecules, where the mentioned pair
tunneling between even neighboring sites is already considered to be
tunnelling between even neighbouring sites is already considered to be
long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
suggests a fundamentally different mechanism originating from quantum
optics: the backaction of global and spatially structured measurement,
@ -1103,7 +1102,7 @@ the strong measurement limit in our quantum gas model. We showed that
global measurement in the strong, but not projective, limit leads to
correlated tunnelling events which can be highly delocalised. Multiple
examples for different optical geometries and measurement operators
demonstrated the incredible felixbility and potential in engineering
demonstrated the incredible flexibility and potential in engineering
dynamics for ultracold gases in an optical lattice. We also claimed
that the behaviour of the system is described by the Hamiltonian given
in Eq. \eqref{eq:hz}. Having developed a physical and intuitive
@ -1157,7 +1156,7 @@ general are not single matrix elements. Therefore, we can write the
master equation that describes this open system as a set of equations
\begin{equation}
\label{eq:master}
\dot{\hat{\rho}}_{mn} = -i K P_m \left[ \h \sum_r \hat{\rho}_{rn}
\dot{\hat{\rho}}_{mn} = -i \nu P_m \left[ \h \sum_r \hat{\rho}_{rn}
- \sum_r \hat{\rho}_{mr} \h \right] P_n + \lambda^2 \left[ o_m
o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right) \right] \hat{\rho}_{mn},
\end{equation}
@ -1204,7 +1203,7 @@ third, virtual, state that remains empty throughout the process. By
avoiding the infinitely projective Zeno limit we open the option for
such processes to happen in our system where transitions within a
single Zeno subspace occur via a second, different, Zeno subspace even
though the occupation of the intermediate states will remian
though the occupation of the intermediate states will remain
negligible at all times.
A single quantum trajectory results in a pure state as opposed to the
@ -1248,7 +1247,7 @@ span multiple Zeno subspaces, cannot exist in a meaningful coherent
superposition in this limit. This means that a density matrix that
spans multiple Zeno subspaces has only classical uncertainty about
which subspace is currently occupied as opposed to the uncertainty due
to a quantum superposition. This is anlogous to the simple qubit
to a quantum superposition. This is analogous to the simple qubit
example we considered in section \ref{sec:master}.
\subsection{Quantum Measurement vs. Dissipation}
@ -1260,7 +1259,7 @@ any time. We have seen that since the density matrix cross-terms are
small we know \emph{a priori} that the individual wave functions
comprising the density matrix mixture will not be coherent
superpositions of different Zeno subspaces and thus we only have
classical uncertainty which means we can resort to clasical
classical uncertainty which means we can resort to classical
probability methods. Each individual experiment will at any time be
predominantly in a single Zeno subspace with small cross-terms and
negligible occupations in the other subspaces. With no measurement
@ -1402,7 +1401,7 @@ This is a crucial step as all $\hat{\rho}_{mm}$ matrices are
decoherence free subspaces and thus they can all coexist in a mixed
state decreasing the purity of the system without
measurement. Physically, this means we exclude trajectories in which
the Zeno subspace has changed (measurement isn't fully projective). By
the Zeno subspace has changed (measurement is not fully projective). By
substituting Eq. \eqref{eq:intermediate} into Eq. \eqref{eq:master} we
see that this happens at a rate of $\nu^2 / \lambda^2$. However, since
the two measurement outcomes cannot coexist any transition between
@ -1641,7 +1640,7 @@ defined by $\hat{T} |\Psi \rangle = 0$, where
$\hat{T} = \sum_{\langle i, j \rangle} \bd_i b_j$. This is simply the
physical interpretation of the steady states we predicted for
Eq. \eqref{eq:master}. Crucially, this state can only be reached if
the dynamics aren't fully suppressed by measurement and thus,
the dynamics are not fully suppressed by measurement and thus,
counter-intuitively, the atomic dynamics cooperate with measurement to
suppress itself by destructive interference. Therefore, this effect is
beyond the scope of traditional quantum Zeno dynamics and presents a
@ -1754,7 +1753,7 @@ design a state with desired properties.
In this chapter we have demonstrated that global quantum measurement
backaction can efficiently compete with standard local processes in
many-body systems. This introduces a completely new energy and time
scale into quantum many-body research. This is made possbile by the
scale into quantum many-body research. This is made possible by the
ability to structure the spatial profile of the measurement on a
microscopic scale comparable to the lattice period without the need
for single site addressing. The extreme flexibility of the setup

68
Chapter6/chapter6.tex
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@ -25,18 +25,18 @@ typical in quantum optics for optical fields. In the previous chapter
we only considered measurement that couples directly to atomic density
operators just like most of the existing work \cite{mekhov2012,
LP2009, rogers2014, ashida2015, ashida2015a}. However, we have shown
in section \ref{sec:B} that it is possible to couple to the the
relative phase differences between sites in an optical lattice by
illuminating 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 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 \cite{caballero2015, caballero2015njp, caballero2016,
in section \ref{sec:B} that it is possible to couple to 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
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
\cite{caballero2015, caballero2015njp, caballero2016,
caballero2016a}. This is a multi-site generalisation of previous
double-well schemes \cite{cirac1996, castin1997, ruostekoski1997,
ruostekoski1998, rist2012}, although the physical mechanism is
fundametally different as it involves direct coupling to the
fundamentally different as it involves direct coupling to the
interference terms caused by atoms tunnelling rather than combining
light scattered from different sources.
@ -78,18 +78,17 @@ density contribution from $\hat{D}$ is suppressed. We will first
consider the case when the profile is uniform, i.e.~the diffraction
maximum of scattered light, when our measurement operator is given by
\begin{equation}
\B = \Bmax = J^B_\mathrm{max} \sum_j^K \left( \bd_j b_{j+1} b_j
\B = \Bmax = J^B_\mathrm{max} \sum_j^K \left( \bd_j b_{j+1} + b_j
\bd_{j+1} \right)
= 2 J^B_\mathrm{max} \sum_k \bd_k b_k \cos(ka),
\end{equation}
where the second equality follows from converting to momentum space,
denoted by index $k$, via
$b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates
an atom in the Brillouin zone. Note that this operator is diagonal in
momentum space which means that its eigenstates are simply Fock
momentum Fock states. We have also seen in Chapter
\ref{chap:backaction} how the global nature of the jump operators
introduces a nonlocal quadratic term to the Hamiltonian,
denoted by index $k$, via $b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma}
b_k$ and $b_k$ annihilates an atom in the Brillouin zone. Note that
this operator is diagonal in momentum space which means that its
eigenstates are simply momentum Fock states. We have also seen in
Chapter \ref{chap:backaction} how the global nature of the jump
operators introduces a nonlocal quadratic term to the Hamiltonian,
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the
competition between tunnelling and measurement backaction we again
consider non-interacting atoms, $U = 0$. Therefore, $\B$ is
@ -179,14 +178,13 @@ spreads the atoms), effectively requiring strong couplings for a
projection as seen in the previous chapter. Here a projection is
achieved at any measurement strength which allows for a weaker probe
and thus effectively less heating and a longer experimental
lifetime. Furthermore. This is in contrast to the quantum Zeno effect
which requires a very strong probe to compete effectively with the
lifetime. This is in contrast to the quantum Zeno effect which
requires a very strong probe to compete effectively with the
Hamiltonian \cite{misra1977, facchi2008, raimond2010, raimond2012,
signoles2014}. Interestingly, the $\Bmax$ measurement will even
establish phase coherence across the lattice,
$\langle \bd_j b_j \rangle \ne 0$, in contrast to density based
measurements where the opposite is true: Fock states with no
coherences are favoured.
establish phase coherence across the lattice, $\langle \bd_j b_j
\rangle \ne 0$, in contrast to density based measurements where the
opposite is true: Fock states with no coherences are favoured.
\section{General Model for Weak Measurement Projection}
@ -222,7 +220,7 @@ and the Hamiltonian:
\hat{H}_0 = - 2 J \sum_{\mathrm{RBZ}} \cos(ka) \left( \beta_k^\dagger
\tilde{\beta}_k + \tilde{\beta}^\dagger_k \beta_k \right),
\end{equation}
where the summations are performed over the reduced Brilluoin Zone
where the summations are performed over the reduced Brillouin Zone
(RBZ), $0 < k \le \pi/a$, to ensure the transformation is
canonical. We see that the measurement operator now consists of two
types of modes, $\beta_k$ and $\tilde{\beta_k}$, which are
@ -289,10 +287,10 @@ open systems described by the density matrix, $\hat{\rho}$,
\left[ \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho}
\cd \c \right) \right].
\end{equation}
The following results will not depend on the nature or exact form of
The following results will not depend on the nature nor exact form of
the jump operator $\c$. However, whenever we refer to our simulations
or our model we will be considering $\c = \sqrt{2 \kappa} C(\D + \B)$
as before, but the results are more general and can be applied to ther
as before, but the results are more general and can be applied to their
setups. This equation describes the state of the system if we discard
all knowledge of the outcome. The commutator describes coherent
dynamics due to the isolated Hamiltonian and the remaining terms are
@ -372,7 +370,7 @@ depends only on itself. Every submatrix
$\mathcal{P}_M \dot{\hat{\rho}} \mathcal{P}_N$ depends only on the
current state of itself and its evolution is ignorant of anything else
in the total density matrix. This is in contrast to the partitioning
we achieved with $P_m$. Previously we only identitified subspaces that
we achieved with $P_m$. Previously we only identified subspaces that
were decoherence free, i.e.~unaffected by measurement. However, those
submatrices could still couple with the rest of the density matrix via
the coherent term $P_m [\hat{H}_0, \hat{\rho}] P_n$.
@ -495,12 +493,12 @@ In the simplest case the projectors $\mathcal{P}_M$ can consist of
only single eigenspaces of $\hat{O}$, $\mathcal{P}_M = R_m$. The
interpretation is straightforward - measurement projects the system
onto a eigenspace of an observable $\hat{O}$ which is a compatible
observable with $\a$ and corresponds to a quantity conserved by the
observable with $\a_1$ and corresponds to a quantity conserved by the
coherent Hamiltonian evolution. However, this may not be possible and
we have the more general case when
$\mathcal{P}_M = \sum_{m \in M} R_m$. In this case, one can simply
think of all $R_{m \in M}$ as degenerate just like eigenstates of the
measurement operator, $\a$, that are degenerate, can form a single
measurement operator, $\a_1$, that are degenerate, can form a single
eigenspace $P_m$. However, these subspaces correspond to different
eigenvalues of $\hat{O}$ distinguishing it from conventional
projections. Instead, the degeneracies are identified by some other
@ -527,14 +525,14 @@ freeze. These eigenspaces are composed of Fock states in momentum
space that have the same number of atoms within each pair of $k$ and
$k - \pi/a$ modes.
Having identified and appropriate $\hat{O}$ operator we proceed to
Having identified an appropriate $\hat{O}$ operator we proceed to
identifying $\mathcal{P}_M$ subspaces for the operator
$\Bmin$. Firstly, since $\Bmin$ contains $\sin(ka)$ coefficients atoms
in different $k$ modes that have the same $\sin(ka)$ value are
indistinguishable to the measurement and will lie in the same $P_m$
eigenspaces. This will happen for the pairs ($k$, $\pi/a -
k$). Therefore, the $R_m$ spaces that have the same
$\hat{O}_k + \hat{O}_{\pi/a - k}$ eigenvalues must belong to the same
k$). Therefore, the $R_m$ spaces that have the same $\hat{O}_k +
\hat{O}_{\pi/a - k}$ eigenvalues must belong to the same
$\mathcal{P}_M$.
Secondly, if we re-write $\hat{O}$ in terms of the $\beta_k$ and
@ -621,8 +619,8 @@ $\sin(ka)$ is the same for both. Therefore, we already know that we
can combine $(R_0, R_2, R_7)$, $(R_1, R_5)$, and $(R_3, R_8)$, because
those combinations have the same $O_{\pi/4a} + O_{3\pi/4a}$
values. This is very clear in the table as these subspaces span
exactly the same values of $B_\mathrm{min}$, i.e.~the have exactly the
same values in the third column.
exactly the same values of $B_\mathrm{min}$, i.e.~they have exactly
the same values in the third column.
Now we have to match the parities. Subspaces that have the same parity
combination for the pair $(O_{\pi/4a} + O_{3\pi/4a}, O_{\pi/2a})$ will
@ -659,7 +657,7 @@ $\mathcal{P}_M = \sum_{m \in M} P_m$.
In summary we have investigated measurement backaction resulting from
coupling light to an ultracold gas's phase-related observables. We
demonstrated how this can be used to prepare the Hamiltonian
eigenstates even if significant tunnelling is occuring as the
eigenstates even if significant tunnelling is occurring as the
measurement can be engineered to not compete with the system's
dynamics. Furthermore, we have shown that when the observable of the
phase-related quantities does not commute with the Hamiltonian we

6
Chapter7/chapter7.tex
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@ -58,7 +58,7 @@ 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
measurement. Such globally paired tunnelling due to a fundamentally
novel phenomenon can enrich physics of long-range correlated systems
beyond relatively short range interactions expected from standard
dipole-dipole interactions \cite{sowinski2012, omjyoti2015}. These
@ -85,8 +85,8 @@ 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
cancellation of tunnelling \cite{vukics2007}, which should be
accessible by this method. Furthermore, 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

2
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@ -1,7 +1,7 @@
% ******************************* PhD Thesis Template **************************
% Please have a look at the README.md file for info on how to use the template
\documentclass[a4paper,12pt,times,numbered,print]{Classes/PhDThesisPSnPDF}
\documentclass[a4paper,12pt,times,numbered,print,oneside]{Classes/PhDThesisPSnPDF}
% ******************************************************************************
% ******************************* Class Options ********************************