Proof read first section on strong measurement
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@ -793,19 +793,21 @@ the $\gamma/U \gg 1$ regime is reached.
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When $\gamma \rightarrow \infty$ the measurement becomes
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When $\gamma \rightarrow \infty$ the measurement becomes
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projective. This means that as soon as the probing begins, the system
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projective. This means that as soon as the probing begins, the system
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collapses into one of the observable's eigenstates. Since this
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collapses into one of the observable's eigenstates. Furthermore, since
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measurement is continuous and doesn't stop after the projection the
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this measurement is continuous and doesn't stop after the projection
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system will be frozen in this state. This effect is called the quantum
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the system will be frozen in this state. This effect is called the
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Zeno effect from Zeno's classical paradox in which a ``watched arrow
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quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
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never moves'' that stated since an arrow in flight is not seen to move
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paradox in which a ``watched arrow never moves'' that stated that
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during any single instant, it cannot possibly be moving at
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since an arrow in flight is not seen to move during any single
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all. Classically the paradox was resolved with a better understanding
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instant, it cannot possibly be moving at all. Classically the paradox
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of infinity and infintesimal changes, but in the quantum world a
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was resolved with a better understanding of infinity and infintesimal
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watched quantum arrow will in fact never move. The system is being
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changes, but in the quantum world a watched quantum arrow will in fact
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continuously projected into its initial state before it has any chance
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never move. The system is being continuously projected into its
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to evolve away. If degenerate eigenspaces exist then we can observe
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initial state before it has any chance to evolve. If degenerate
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quantum Zeno dynamics where unitary evolution is uninhibited within
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eigenspaces exist then we can observe quantum Zeno dynamics where
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such a degenerate subspace, called the Zeno subspace.
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unitary evolution is uninhibited within such a degenerate subspace,
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called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
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signoles2014}.
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These effects can be easily seen in our model when
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These effects can be easily seen in our model when
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$\gamma \rightarrow \infty$. The system will be projected into one or
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$\gamma \rightarrow \infty$. The system will be projected into one or
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@ -814,12 +816,11 @@ we define the projector
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$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
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$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
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denotes a single degenerate subspace. The Zeno subspace is determined
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denotes a single degenerate subspace. The Zeno subspace is determined
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randomly as per the Copenhagen postulates and thus it depends on the
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randomly as per the Copenhagen postulates and thus it depends on the
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initial state. If the projection is into the subspace
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initial state. If the projection is into the subspace $\varphi$, the
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$\varphi^\prime$, the subsequent evolution is described by the
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subsequent evolution is described by the projected Hamiltonian
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projected Hamiltonian
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$P_{\varphi} \hat{H}_0 P_{\varphi}$. We have used the original
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$P_{\varphi^\prime} \hat{H}_0 P_{\varphi^\prime}$. We have used the
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Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
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original Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or
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quantum jumps as their combined effect is now described by the
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the quantum jumps as their combined effect is now described by the
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projectors. Physically, in our model of ultracold bosons trapped in a
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projectors. Physically, in our model of ultracold bosons trapped in a
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lattice this means that tunnelling between different spatial modes is
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lattice this means that tunnelling between different spatial modes is
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completely supressed since this process couples eigenstates belonging
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completely supressed since this process couples eigenstates belonging
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@ -827,26 +828,24 @@ to different Zeno subspaces. If a small connected part of the lattice
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was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
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was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
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tunnelling would only be prohibited between the illuminated and
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tunnelling would only be prohibited between the illuminated and
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unilluminated areas, but dynamics proceeds normally within each zone
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unilluminated areas, but dynamics proceeds normally within each zone
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separately. Therefore, the goemetric patterns we have in which the
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separately. However, the goemetric patterns we have in which the modes
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modes are sptailly delocalised in such a way that neighbouring sites
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are spatially delocalised in such a way that neighbouring sites never
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never belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$,
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belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, would
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would lead to a complete suppression of tunnelling across the whole
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lead to a complete suppression of tunnelling across the whole lattice
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lattice as there is no way for an atom to tunnel within this Zeno
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as there is no way for an atom to tunnel within this Zeno subspace
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subspace.
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without first having to leave it.
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This is an interesting example of the quantum Zeno effect and dynamics
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This is an interesting example of the quantum Zeno effect and dynamics
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and it can be used to prohibit parts of the dynamics of the
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and it can be used to prohibit parts of the dynamics of the
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Bose-Hubbard Hamiltonian in order to engineer desired Hamiltonians for
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Bose-Hubbard Model in order to engineer desired Hamiltonians for
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quantum simulations or other applications. However, the infinite
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quantum simulations or other applications. However, the infinite
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projective limit is uninteresting in the context of a global
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projective limit is uninteresting in the context of a global
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measurement scheme. The same effects and Hamiltonians can be achieved
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measurement scheme. The same effects and Hamiltonians can be achieved
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using multiple independent measurements which address a few sites
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using multiple independent measurements which address a few sites
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each. The only advantage of the global scheme is that it might be
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each. In order to take advantage of the nonlocal nature of the
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simpler to achieve as it requires a less complicated optical setup. In
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measurement it turns out that we need to consider a finite limit for
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order to take advantage of the nonlocal nature of the measurement it
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$\gamma \gg J$. By considering a non-infinite $\gamma$ we observe
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turns out that we need to consider a finite limit for $\gamma \gg
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additional dynamics while the usual atomic tunnelling is still heavily
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J$. By considering a non-infinite $\gamma$ we observe additional
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dynamics while the usual atomic tunnelling is still heavily
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Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
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Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
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\begin{figure}[hbtp!]
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\begin{figure}[hbtp!]
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@ -897,22 +896,22 @@ eigenspaces and select which tunnelling processes should be
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uninhibited and which should be suppressed. However, there is a second
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uninhibited and which should be suppressed. However, there is a second
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effect that was not present before. In Fig. \ref{fig:zeno} we can
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effect that was not present before. In Fig. \ref{fig:zeno} we can
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observe tunnelling that violates the boundaries established by the
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observe tunnelling that violates the boundaries established by the
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spatial modes. When $\gamma$ is finite, second-order processes can now
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spatial modes. When $\gamma$ is finite, second-order processes,
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occur, i.e.~two correlated tunnelling events, via an intermediate
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i.e.~two correlated tunnelling events, can now occur via an
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(virtual) state outside of the Zeno subspace as long as the Zeno
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intermediate (virtual) state outside of the Zeno subspace as long as
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subspace of the final state remains the same. Crucially, these
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the Zeno subspace of the final state remains the same. Crucially,
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tunnelling events are only correlated in time, but not in space. This
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these tunnelling events are only correlated in time, but not in
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means that the two events do not have to occur for the same atom or
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space. This means that the two events do not have to occur for the
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even at the same site in the lattice. As long as the Zeno subspace is
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same atom or even at the same site in the lattice. As long as the Zeno
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preserved, these processes can occur anywhere in the system, that is a
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subspace is preserved, these processes can occur anywhere in the
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pair of atoms separated by many sites is able to tunnel in a
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system. That is, a pair of atoms separated by many sites is able to
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correlated manner. This is only possible due to the possibility of
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tunnel in a correlated manner. This is only possible due to the
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creating extensive and spatially nonlocal modes as described in
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ability of creating extensive and spatially nonlocal modes as
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section \ref{sec:modes} which in turn is enabled by the global nature
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described in section \ref{sec:modes} which in turn is enabled by the
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of the measurement. This would not be possible to achieve with local
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global nature of the measurement. This would not be possible to
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measurements as this would lead to Zeno subspaces described entirely
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achieve with local measurements as the Zeno subspaces would be
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by local variables which cannot be preserved by such delocalised
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described entirely by local variables which cannot be preserved by
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tunnelling events.
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such delocalised tunnelling events.
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In the subsequent sections we will rigorously derive the following
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In the subsequent sections we will rigorously derive the following
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Hamiltonian for the non-interacting dynamics within a single Zeno
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Hamiltonian for the non-interacting dynamics within a single Zeno
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@ -930,30 +929,32 @@ $\hat{D} = \hat{N}_\mathrm{odd}$
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where
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where
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$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
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$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
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constant that depends on the measurement scheme, $\varphi$ denotes a
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constant that depends on the measurement scheme, $\varphi$ denotes a
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set of site belonging to a single mode and $\varphi^\prime$ is the
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set of sites belonging to a single mode and $\varphi^\prime$ is the
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set's complement (e.g.~odd and even or illuminated and non-illuminated
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set's complement (e.g.~odd and even or illuminated and non-illuminated
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sites). We see that this Hamiltonian consists of two parts. The first
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sites). We see that this Hamiltonian consists of two parts. The first
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term corresponds to the standard quantum Zeno first-order dynamics
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term corresponds to the standard quantum Zeno first-order dynamics
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that occurs within a Zeno subspace, i.e.~tunnelling between
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that occurs within a Zeno subspace, i.e.~tunnelling between
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neighbouring sites that belong to the same mode. Otherwise,
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neighbouring sites that belong to the same mode. Otherwise, if $i$ and
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$P_0 \bd_i b_j P_0 = 0$. If $\gamma \rightarrow \infty$ we would
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$j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
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recover the quantum Zeno Hamiltonian where this would be the only
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$\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
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remaining term. It is the second term that shows the second-order
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where this would be the only remaining term. It is the second term
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corelated tunnelling terms. This is evident from the inner sum which
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that shows the second-order corelated tunnelling terms. This is
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requires that pairs of sites ($i$, $j$) and ($k$, $l$) between which
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evident from the inner sum which requires that pairs of sites ($i$,
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atoms tunnel must be nearest neighbours, but these pairs can be as far
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$j$) and ($k$, $l$) between which atoms tunnel must be nearest
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apart from each other as possible within the mode structure. This is
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neighbours, but these pairs can be anywhere on the lattice within the
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in particular explicitly shown in Figs. \ref{fig:zeno}(a,b). The
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constraints of the mode structure. This is in particular explicitly
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imaginary coefficient means that the tinelling behaves like an
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shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
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exponential decay (overdamped oscillations). This also implies that
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that the tinelling behaves like an exponential decay (overdamped
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the norm will decay, but this does not mean that there are physical
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oscillations). This also implies that the norm will decay, but this
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losses in the system. Instead, the norm itself represents the
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does not mean that there are physical losses in the system. Instead,
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probability of the system remaining in the $\varphi = 0$ Zeno
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the norm itself represents the probability of the system remaining in
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subspace. Since $\gamma$ is not infinite there is now a finite
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the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
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probability that the stochastic nature of the measurement will lead to
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is now a finite probability that the stochastic nature of the
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a discontinous change in the system where the Zeno subspace rapidly
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measurement will lead to a discontinous change in the system where the
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changes. However, later in this chapter we will see that steady states
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Zeno subspace rapidly changes which can be seen in
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of this Hamiltonian exist which will no longer change Zeno subspaces.
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Fig. \ref{fig:zeno}(a). However, later in this chapter we will see
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that steady states of this Hamiltonian exist which will no longer
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change Zeno subspaces.
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Crucially, what sets this effect apart from usual many-body dynamics
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Crucially, what sets this effect apart from usual many-body dynamics
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with short-range interactions is that first order processes are
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with short-range interactions is that first order processes are
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@ -961,11 +962,23 @@ selectively suppressed by the global conservation of the measured
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observable and not by the prohibitive energy costs of doubly-occupied
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observable and not by the prohibitive energy costs of doubly-occupied
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sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
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sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
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profound consequences as this is the physical origin of the long-range
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profound consequences as this is the physical origin of the long-range
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correlated tunneling events represented in \eqref{eq:hz} by the fact
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correlated tunneling events represented in Eq. \eqref{eq:hz} by the
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that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. This is
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fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
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because the projection $\hat{P}_0$ is not sensitive to individual site
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projection $\hat{P}_0$ is not sensitive to individual site
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occupancies, but instead enforces a fixed value of the observable,
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occupancies, but instead enforces a fixed value of the observable,
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i.e.~a single Zeno subspace.
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i.e.~a single Zeno subspace. This is a striking difference with the
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$t$-$J$ and other strongly interacting models. The strong interaction
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also leads to correlated events in which atoms can tunnel over each
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other by creating an unstable doubly occupied site during the
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intermediate step. However, these correlated events are by their
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nature localised. Due to interactions the doubly-occupied site cannot
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be present in the final state which means that any tunnelling event
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that created this unstable configuration must be followed by another
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tunnelling event which takes an atom away. In the case of global
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measurement this process is delocalised, because since the modes
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consist of many sites the stable configuration can be restored by a
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tunnelling event from a completely different lattice site that belongs
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to the same mode.
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In Fig.~\ref{fig:zeno}a we consider illuminating only the central
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In Fig.~\ref{fig:zeno}a we consider illuminating only the central
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region of the optical lattice and detecting light in the diffraction
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region of the optical lattice and detecting light in the diffraction
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@ -978,7 +991,7 @@ typical dynamics occurs within each region but the standard tunnelling
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between different modes is suppressed. Importantly, second-order
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between different modes is suppressed. Importantly, second-order
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processes that do not change $N_\text{K}$ are still possible since an
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processes that do not change $N_\text{K}$ are still possible since an
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atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
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atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
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from $2$ to $3$. Therfore, effective long-range tunneling between two
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from $2$ to $3$. Therefore, effective long-range tunneling between two
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spatially disconnected zones $1$ and $3$ happens due to the two-step
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spatially disconnected zones $1$ and $3$ happens due to the two-step
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processes $1 \rightarrow 2 \rightarrow 3$ or
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processes $1 \rightarrow 2 \rightarrow 3$ or
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$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
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$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
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@ -1001,39 +1014,22 @@ measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$,
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atoms can slowly tunnel between the odd sites of the lattice, despite
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atoms can slowly tunnel between the odd sites of the lattice, despite
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them being spatially disconnected. This atom exchange spreads
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them being spatially disconnected. This atom exchange spreads
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correlations between non-neighbouring lattice sites on a time scale
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correlations between non-neighbouring lattice sites on a time scale
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$\sim \gamma/J^2$. The schematic explanation of long-range correlated
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$\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
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tunneling is presented in Fig.~\ref{fig:zeno}(b.1): the atoms can
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explanation of long-range correlated tunneling is presented in
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tunnel only in pairs to assure the globally conserved values of
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Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
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$N_\text{even}$ and $N_\text{odd}$, such that one correlated tunneling
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the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
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event is represented by a pair of one red and one blue
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such that one correlated tunneling event is represented by a pair of
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arrow. Importantly, this scheme is fully applicable for a lattice with
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one red and one blue arrow. Importantly, this scheme is fully
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large atom and site numbers, well beyond the numerical example in
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applicable for a lattice with large atom and site numbers, well beyond
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Fig.~\ref{fig:zeno}(b.1), because as we can see in \eqref{eq:hz}
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the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
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it is the geometry of quantum measurement that assures this mode
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see in Eq. \eqref{eq:hz} it is the geometry of quantum measurement that
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structure (in this example, two modes at odd and even sites) and thus
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assures this mode structure (in this example, two modes at odd and
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the underlying pairwise global tunnelling.
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even sites) and thus the underlying pairwise global tunnelling.
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This global pair tunneling may play a role of a building block for
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The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
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more complicated many-body effects. For example, a pair tunneling
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between the neighbouring sites has been recently shown to play
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important role in the formation of new quantum phases, e.g., pair
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superfluid \cite{sowinski2012} and lead to formulation of extended
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Bose-Hubbard models \cite{omjyoti2015}. The search for novel
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mechanisms providing long-range interactions is crucial in many-body
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physics. One of the standard candidates is the dipole-dipole
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interaction in, e.g., dipolar molecules, where the mentioned pair
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tunneling between even neighboring sites is already considered to be
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long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
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suggests a fundamentally different mechanism originating from quantum
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optics: the backaction of global and spatially structured measurement,
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which as we prove can successfully compete with other short-range
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processes in many-body systems. This opens promising opportunities for
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future research.
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The scheme in Fig.~\ref{fig:zeno}(b.1) can help to design a nonlocal
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reservoir for the tunneling (or ``decay'') of atoms from one region to
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reservoir for the tunneling (or ``decay'') of atoms from one region to
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another. For example, if the atoms are placed only at odd sites,
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another. For example, if the atoms are placed only at odd sites,
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according to \eqref{eq:hz} their tunnelling is suppressed since the
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according to Eq. \eqref{eq:hz} their tunnelling is suppressed since the
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multi-tunneling event must be successive, i.e.~an atom tunnelling into
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multi-tunneling event must be successive, i.e.~an atom tunnelling into
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a different mode, $\varphi^\prime$, must then also tunnel back into
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a different mode, $\varphi^\prime$, must then also tunnel back into
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its original mode, $\varphi$. If, however, one adds some atoms to even
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its original mode, $\varphi$. If, however, one adds some atoms to even
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@ -1052,8 +1048,8 @@ entanglement. This is in striking contrast to the entanglement caused
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by local processes which can be very confined, especially in 1D where
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by local processes which can be very confined, especially in 1D where
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it is typically short range. This makes numerical calculations of our
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it is typically short range. This makes numerical calculations of our
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system for large atom numbers really difficult, since well-known
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system for large atom numbers really difficult, since well-known
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methods such as DMRG and MPS \cite{schollwock2005} (which are successful
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methods such as DMRG and MPS \cite{schollwock2005} (which are
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for short-range interactions) rely on the limited extent of
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successful for short-range interactions) rely on the limited extent of
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entanglement.
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entanglement.
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The negative number correlations are typical for systems with
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The negative number correlations are typical for systems with
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@ -1072,10 +1068,26 @@ reservoir for two central sites, where a constraint is applied. Note
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that, using more modes, the design of higher-order multi-tunneling
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that, using more modes, the design of higher-order multi-tunneling
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events is possible.
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events is possible.
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This global pair tunneling may play a role of a building block for
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more complicated many-body effects. For example, a pair tunneling
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between the neighbouring sites has been recently shown to play
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important role in the formation of new quantum phases, e.g., pair
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superfluid \cite{sowinski2012} and lead to formulation of extended
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Bose-Hubbard models \cite{omjyoti2015}. The search for novel
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mechanisms providing long-range interactions is crucial in many-body
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physics. One of the standard candidates is the dipole-dipole
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interaction in, e.g., dipolar molecules, where the mentioned pair
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tunneling between even neighboring sites is already considered to be
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long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
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|
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.
|
||||||
|
|
||||||
\subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit}
|
\subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit}
|
||||||
|
|
||||||
% Contrast with t-J model here how U localises events, but measurement
|
|
||||||
% does the opposite
|
|
||||||
|
|
||||||
\subsection{Steady-State of the Non-Hermitian Hamiltonian}
|
\subsection{Steady-State of the Non-Hermitian Hamiltonian}
|
||||||
|
|
||||||
|
@ -1,7 +1,7 @@
|
|||||||
% ******************************* PhD Thesis Template **************************
|
% ******************************* PhD Thesis Template **************************
|
||||||
% Please have a look at the README.md file for info on how to use the template
|
% Please have a look at the README.md file for info on how to use the template
|
||||||
|
|
||||||
\documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF}
|
\documentclass[a4paper,12pt,times,numbered,print,chapter]{Classes/PhDThesisPSnPDF}
|
||||||
|
|
||||||
% ******************************************************************************
|
% ******************************************************************************
|
||||||
% ******************************* Class Options ********************************
|
% ******************************* Class Options ********************************
|
||||||
|
Reference in New Issue
Block a user