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Finished first subsection on correlated tunnelling

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@ -791,6 +791,287 @@ the $\gamma/U \gg 1$ regime is reached.
\subsection{Emergent Long-Range Correlated Tunnelling}
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. Since this
measurement is continuous and doesn't stop after the projection the
system will be frozen in this state. This effect is called the quantum
Zeno effect from Zeno's classical paradox in which a ``watched arrow
never moves'' that stated 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 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 away. If degenerate eigenspaces exist then we can observe
quantum Zeno dynamics where unitary evolution is uninhibited within
such a degenerate subspace, called the Zeno subspace.
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
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
randomly as per the Copenhagen postulates and thus it depends on the
initial state. If the projection is into the subspace
$\varphi^\prime$, the subsequent evolution is described by the
projected Hamiltonian
$P_{\varphi^\prime} \hat{H}_0 P_{\varphi^\prime}$. We have used the
original 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
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. Therefore, the goemetric patterns we have in which the
modes are sptailly 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 as there is no way for an atom to tunnel within this Zeno
subspace.
This is an interesting example of the quantum Zeno effect and dynamics
and it can be used to prohibit parts of the dynamics of the
Bose-Hubbard Hamiltonian in order to engineer desired Hamiltonians for
quantum simulations or other applications. However, the infinite
projective limit is uninteresting in the context of a global
measurement scheme. The same effects and Hamiltonians can be achieved
using multiple independent measurements which address a few sites
each. The only advantage of the global scheme is that it might be
simpler to achieve as it requires a less complicated optical setup. In
order to take advantage of the nonlocal nature of the measurement it
turns out that we need to consider a finite limit for $\gamma \gg
J$. By considering a non-infinite $\gamma$ we observe additional
dynamics while the usual atomic tunnelling is still heavily
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
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):
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
same mode (green); $N_I$ ($N_{NI}$) is the atom number in the
illuminated (non-illuminated) mode. (a) (a.1) Atom number in the
central region is frozen: system is divided into three
regions. (a.2) Standard dynamics happens within each region, but
not between them. (a.3) Entanglement build up. (a.4) Negative
correlations between non-illuminated regions (green) and zero
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
is represented by any pair of one blue and one red arrow. (b.2)
Correlated tunneling 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
$N_\text{odd}$ (orange). Initial state: $|0,1,2,1,0 \rangle$,
$\gamma/J=100$, $J_{jj}=[0,1,0,1,0]$. (c) (c.1,2) Atom number
difference between two central sites is frozen. (c.3) Entanglement
build up. (c.4) In contrast to previous examples, sites in the
same zones are positively correlated (green), while atoms in
different zones are negatively correlated (orange). Initial state:
$|0,2,2,0 \rangle$, $\gamma/J=100$, $J_{jj}=[0,-1,1,0]$. 1D
lattice, $U/J=0$.}
\label{fig:zeno}
\end{figure}
There are two crucial features of the resulting dynamics that are of
note. First, just like in the infinite quantum Zeno limit the
evolution between nearest neighbours within the same mode is
unperturbed whilst tunnelling between different modes is heavily
suppressed by the measurement. Therefore, we see the usual quantum
Zeno dynamics within a single Zeno subspace and just like before, it
is also possible to use the global probing scheme to engineer these
eigenspaces and select which tunnelling processes should be
uninhibited and which should be suppressed. However, there is a second
effect that was not present before. In Fig. \ref{fig:zeno} we can
observe tunnelling that violates the boundaries established by the
spatial modes. When $\gamma$ is finite, second-order processes can now
occur, i.e.~two correlated tunnelling events, via an intermediate
(virtual) state outside of the Zeno subspace as long as the Zeno
subspace of the final state remains the same. Crucially, these
tunnelling events are only correlated in time, but not in space. This
means that the two events do not have to occur for the same atom or
even at the same site in the lattice. As long as the Zeno subspace is
preserved, these processes can occur anywhere in the system, that is a
pair of atoms separated by many sites is able to tunnel in a
correlated manner. This is only possible due to the possibility of
creating extensive and spatially nonlocal modes as described in
section \ref{sec:modes} which in turn is enabled by the global nature
of the measurement. This would not be possible to achieve with local
measurements as this would lead to Zeno subspaces described entirely
by local variables which cannot be preserved by such delocalised
tunnelling events.
In the subsequent sections we will rigorously derive the following
Hamiltonian for the non-interacting dynamics within a single Zeno
subspace, $\varphi = 0$, for a lattice where the measurement defines
$Z = 2$ distinct modes, e.g. $\hat{D} = \hat{N}_K$ or
$\hat{D} = \hat{N}_\mathrm{odd}$
\begin{equation}
\label{eq:hz}
\hat{H}_\varphi = P_0 \left[ -J \sum_{\langle i, j \rangle}
b^\dagger_i b_j - i \frac{J^2} {A \gamma} \sum_{\varphi}
\sum_{\substack{\langle i \in \varphi, j \in \varphi^\prime
\rangle \\ \langle k \in \varphi^\prime, l \in \varphi
\rangle}} b^\dagger_i b_j b^\dagger_k b_l \right] P_0,
\end{equation}
where
$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
constant that depends on the measurement scheme, $\varphi$ denotes a
set of site belonging to a single mode and $\varphi^\prime$ is the
set's complement (e.g.~odd and even or illuminated and non-illuminated
sites). We see that this Hamiltonian consists of two parts. The first
term corresponds to the standard quantum Zeno first-order dynamics
that occurs within a Zeno subspace, i.e.~tunnelling between
neighbouring sites that belong to the same mode. Otherwise,
$P_0 \bd_i b_j P_0 = 0$. If $\gamma \rightarrow \infty$ we would
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 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 as far
apart from each other as possible within 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 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 Zeno subspace rapidly
changes. However, later in this chapter we will see that steady states
of this Hamiltonian exist which will no longer change Zeno subspaces.
Crucially, what sets this effect apart from usual many-body dynamics
with short-range interactions is that first order processes are
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 \eqref{eq:hz} by the fact
that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. This is
because the projection $\hat{P}_0$ is not sensitive to individual site
occupancies, but instead enforces a fixed value of the observable,
i.e.~a single Zeno subspace.
In Fig.~\ref{fig:zeno}a we consider illuminating only the central
region of the optical lattice and detecting light in the diffraction
maximum, thus we freeze the atom number in the $K$ illuminated sites
$\hat{N}_\text{K}$~\cite{mekhov2009prl,mekhov2009pra}. The measurement
scheme defines two different spatial modes: the non-illuminated zones
$1$ and $3$ and the illuminated one $2$. Figure~\ref{fig:zeno}(a.2)
illustrates the evolution of the mean density at each lattice site:
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 $3$. Therfore, effective long-range tunneling 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
the negative (anti-)correlations
$\langle \delta N_1 \delta N_3 \rangle = \langle N_1 N_3 \rangle -
\langle N_1 \rangle \langle N_3 \rangle$ showing that an atom
disappearing from zone $1$ appears in zone $3$, while there are no
number correlations between illuminated and non-illuminated regions,
$\langle( \delta N_1 + \delta N_3 ) \delta N_2 \rangle = 0$ as shown
in Fig.~\ref{fig:zeno}(a.4). In contrast to fully-projective
measurement, the existence of an intermediate (virtual) step in the
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
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 $N_\text{odd}$,
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
$\sim \gamma/J^2$. The schematic explanation of long-range correlated
tunneling 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 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 see in \eqref{eq:hz}
it is the geometry of quantum measurement that assures this mode
structure (in this example, two modes at odd and even sites) and thus
the underlying pairwise global tunnelling.
This global pair tunneling may play a role of a building block for
more complicated many-body effects. For example, a pair tunneling
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
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
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,
which as we prove can successfully compete with other short-range
processes in many-body systems. This opens promising opportunities for
future research.
The scheme in Fig.~\ref{fig:zeno}(b.1) can help to design a nonlocal
reservoir for the tunneling (or ``decay'') of atoms from one region to
another. For example, if the atoms are placed only at odd sites,
according to \eqref{eq:hz} their tunnelling is suppressed since the
multi-tunneling 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
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
global measurement. Additionally, these long-range correlations are a
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
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
system for large atom numbers really difficult, since well-known
methods such as DMRG and MPS \cite{schollwock2005} (which are successful
for short-range interactions) rely on the limited extent of
entanglement.
The negative number correlations are typical for systems with
constraints (superselection rules) such as fixed atom number. The
effective dynamics due to our global, but spatially structured,
measurement introduces more general constraints to the evolution of
the system. For example, in Fig.~\ref{fig:zeno}(c) we show the
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
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
events is possible.
\subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit}
% Contrast with t-J model here how U localises events, but measurement

66
References/references.bib
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@ -52,6 +52,12 @@ year = {2010}
year={2004},
publisher={Springer Science \& Business Media}
}
@book{auerbach,
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publisher={Springer Science \& Business Media}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Igor's original papers
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publisher={APS}
}
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}

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thesis-info.tex
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@ -1,6 +1,7 @@
% ************************ Thesis Information & Meta-data **********************
%% The title of the thesis
\title{Interaction of Quantized Light with Ultracold Bosons}
\title{Competition between Weak Quantum Measurement and Many-Body
Dynamics in Ultracold Bosonic Gases}
%\texorpdfstring is used for PDF metadata. Usage:
%\texorpdfstring{LaTeX_Version}{PDF Version (non-latex)} eg.,
%\texorpdfstring{$sigma$}{sigma}