Proof read first section on strong measurement

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Wojciech Kozlowski 2016-08-03 14:51:21 +01:00
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@ -793,19 +793,21 @@ the $\gamma/U \gg 1$ regime is reached.
When $\gamma \rightarrow \infty$ the measurement becomes When $\gamma \rightarrow \infty$ the measurement becomes
projective. This means that as soon as the probing begins, the system projective. This means that as soon as the probing begins, the system
collapses into one of the observable's eigenstates. Since this collapses into one of the observable's eigenstates. Furthermore, since
measurement is continuous and doesn't stop after the projection the this measurement is continuous and doesn't stop after the projection
system will be frozen in this state. This effect is called the quantum the system will be frozen in this state. This effect is called the
Zeno effect from Zeno's classical paradox in which a ``watched arrow quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
never moves'' that stated since an arrow in flight is not seen to move paradox in which a ``watched arrow never moves'' that stated that
during any single instant, it cannot possibly be moving at since an arrow in flight is not seen to move during any single
all. Classically the paradox was resolved with a better understanding instant, it cannot possibly be moving at all. Classically the paradox
of infinity and infintesimal changes, but in the quantum world a was resolved with a better understanding of infinity and infintesimal
watched quantum arrow will in fact never move. The system is being changes, but in the quantum world a watched quantum arrow will in fact
continuously projected into its initial state before it has any chance never move. The system is being continuously projected into its
to evolve away. If degenerate eigenspaces exist then we can observe initial state before it has any chance to evolve. If degenerate
quantum Zeno dynamics where unitary evolution is uninhibited within eigenspaces exist then we can observe quantum Zeno dynamics where
such a degenerate subspace, called the Zeno subspace. unitary evolution is uninhibited within such a degenerate subspace,
called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
signoles2014}.
These effects can be easily seen in our model when These effects can be easily seen in our model when
$\gamma \rightarrow \infty$. The system will be projected into one or $\gamma \rightarrow \infty$. The system will be projected into one or
@ -814,12 +816,11 @@ we define the projector
$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$ $P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
denotes a single degenerate subspace. The Zeno subspace is determined denotes a single degenerate subspace. The Zeno subspace is determined
randomly as per the Copenhagen postulates and thus it depends on the randomly as per the Copenhagen postulates and thus it depends on the
initial state. If the projection is into the subspace initial state. If the projection is into the subspace $\varphi$, the
$\varphi^\prime$, the subsequent evolution is described by the subsequent evolution is described by the projected Hamiltonian
projected Hamiltonian $P_{\varphi} \hat{H}_0 P_{\varphi}$. We have used the original
$P_{\varphi^\prime} \hat{H}_0 P_{\varphi^\prime}$. We have used the Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
original Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or quantum jumps as their combined effect is now described by the
the quantum jumps as their combined effect is now described by the
projectors. Physically, in our model of ultracold bosons trapped in a projectors. Physically, in our model of ultracold bosons trapped in a
lattice this means that tunnelling between different spatial modes is lattice this means that tunnelling between different spatial modes is
completely supressed since this process couples eigenstates belonging completely supressed since this process couples eigenstates belonging
@ -827,26 +828,24 @@ to different Zeno subspaces. If a small connected part of the lattice
was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
tunnelling would only be prohibited between the illuminated and tunnelling would only be prohibited between the illuminated and
unilluminated areas, but dynamics proceeds normally within each zone unilluminated areas, but dynamics proceeds normally within each zone
separately. Therefore, the goemetric patterns we have in which the separately. However, the goemetric patterns we have in which the modes
modes are sptailly delocalised in such a way that neighbouring sites are spatially delocalised in such a way that neighbouring sites never
never belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, would
would lead to a complete suppression of tunnelling across the whole lead to a complete suppression of tunnelling across the whole lattice
lattice as there is no way for an atom to tunnel within this Zeno as there is no way for an atom to tunnel within this Zeno subspace
subspace. without first having to leave it.
This is an interesting example of the quantum Zeno effect and dynamics 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 and it can be used to prohibit parts of the dynamics of the
Bose-Hubbard Hamiltonian in order to engineer desired Hamiltonians for Bose-Hubbard Model in order to engineer desired Hamiltonians for
quantum simulations or other applications. However, the infinite quantum simulations or other applications. However, the infinite
projective limit is uninteresting in the context of a global projective limit is uninteresting in the context of a global
measurement scheme. The same effects and Hamiltonians can be achieved measurement scheme. The same effects and Hamiltonians can be achieved
using multiple independent measurements which address a few sites using multiple independent measurements which address a few sites
each. The only advantage of the global scheme is that it might be each. In order to take advantage of the nonlocal nature of the
simpler to achieve as it requires a less complicated optical setup. In measurement it turns out that we need to consider a finite limit for
order to take advantage of the nonlocal nature of the measurement it $\gamma \gg J$. By considering a non-infinite $\gamma$ we observe
turns out that we need to consider a finite limit for $\gamma \gg additional dynamics while the usual atomic tunnelling is still heavily
J$. 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}. Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
\begin{figure}[hbtp!] \begin{figure}[hbtp!]
@ -897,22 +896,22 @@ eigenspaces and select which tunnelling processes should be
uninhibited and which should be suppressed. However, there is a second uninhibited and which should be suppressed. However, there is a second
effect that was not present before. In Fig. \ref{fig:zeno} we can effect that was not present before. In Fig. \ref{fig:zeno} we can
observe tunnelling that violates the boundaries established by the observe tunnelling that violates the boundaries established by the
spatial modes. When $\gamma$ is finite, second-order processes can now spatial modes. When $\gamma$ is finite, second-order processes,
occur, i.e.~two correlated tunnelling events, via an intermediate i.e.~two correlated tunnelling events, can now occur via an
(virtual) state outside of the Zeno subspace as long as the Zeno intermediate (virtual) state outside of the Zeno subspace as long as
subspace of the final state remains the same. Crucially, these the Zeno subspace of the final state remains the same. Crucially,
tunnelling events are only correlated in time, but not in space. This these tunnelling events are only correlated in time, but not in
means that the two events do not have to occur for the same atom or space. This means that the two events do not have to occur for the
even at the same site in the lattice. As long as the Zeno subspace is same atom or even at the same site in the lattice. As long as the Zeno
preserved, these processes can occur anywhere in the system, that is a subspace is preserved, these processes can occur anywhere in the
pair of atoms separated by many sites is able to tunnel in a system. That is, a pair of atoms separated by many sites is able to
correlated manner. This is only possible due to the possibility of tunnel in a correlated manner. This is only possible due to the
creating extensive and spatially nonlocal modes as described in ability of creating extensive and spatially nonlocal modes as
section \ref{sec:modes} which in turn is enabled by the global nature described in section \ref{sec:modes} which in turn is enabled by the
of the measurement. This would not be possible to achieve with local global nature of the measurement. This would not be possible to
measurements as this would lead to Zeno subspaces described entirely achieve with local measurements as the Zeno subspaces would be
by local variables which cannot be preserved by such delocalised described entirely by local variables which cannot be preserved by
tunnelling events. such delocalised tunnelling events.
In the subsequent sections we will rigorously derive the following In the subsequent sections we will rigorously derive the following
Hamiltonian for the non-interacting dynamics within a single Zeno Hamiltonian for the non-interacting dynamics within a single Zeno
@ -930,30 +929,32 @@ $\hat{D} = \hat{N}_\mathrm{odd}$
where where
$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a $A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
constant that depends on the measurement scheme, $\varphi$ denotes 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 of sites belonging to a single mode and $\varphi^\prime$ is the
set's complement (e.g.~odd and even or illuminated and non-illuminated 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 sites). We see that this Hamiltonian consists of two parts. The first
term corresponds to the standard quantum Zeno first-order dynamics term corresponds to the standard quantum Zeno first-order dynamics
that occurs within a Zeno subspace, i.e.~tunnelling between that occurs within a Zeno subspace, i.e.~tunnelling between
neighbouring sites that belong to the same mode. Otherwise, neighbouring sites that belong to the same mode. Otherwise, if $i$ and
$P_0 \bd_i b_j P_0 = 0$. If $\gamma \rightarrow \infty$ we would $j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
recover the quantum Zeno Hamiltonian where this would be the only $\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
remaining term. It is the second term that shows the second-order where this would be the only remaining term. It is the second term
corelated tunnelling terms. This is evident from the inner sum which that shows the second-order corelated tunnelling terms. This is
requires that pairs of sites ($i$, $j$) and ($k$, $l$) between which evident from the inner sum which requires that pairs of sites ($i$,
atoms tunnel must be nearest neighbours, but these pairs can be as far $j$) and ($k$, $l$) between which atoms tunnel must be nearest
apart from each other as possible within the mode structure. This is neighbours, but these pairs can be anywhere on the lattice within the
in particular explicitly shown in Figs. \ref{fig:zeno}(a,b). The constraints of the mode structure. This is in particular explicitly
imaginary coefficient means that the tinelling behaves like an shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
exponential decay (overdamped oscillations). This also implies that that the tinelling behaves like an exponential decay (overdamped
the norm will decay, but this does not mean that there are physical oscillations). This also implies that the norm will decay, but this
losses in the system. Instead, the norm itself represents the does not mean that there are physical losses in the system. Instead,
probability of the system remaining in the $\varphi = 0$ Zeno the norm itself represents the probability of the system remaining in
subspace. Since $\gamma$ is not infinite there is now a finite the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
probability that the stochastic nature of the measurement will lead to is now a finite probability that the stochastic nature of the
a discontinous change in the system where the Zeno subspace rapidly measurement will lead to a discontinous change in the system where the
changes. However, later in this chapter we will see that steady states Zeno subspace rapidly changes which can be seen in
of this Hamiltonian exist which will no longer change Zeno subspaces. Fig. \ref{fig:zeno}(a). 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 Crucially, what sets this effect apart from usual many-body dynamics
with short-range interactions is that first order processes are with short-range interactions is that first order processes are
@ -961,11 +962,23 @@ selectively suppressed by the global conservation of the measured
observable and not by the prohibitive energy costs of doubly-occupied 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 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 profound consequences as this is the physical origin of the long-range
correlated tunneling events represented in \eqref{eq:hz} by the fact correlated tunneling events represented in Eq. \eqref{eq:hz} by the
that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. This is fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
because the projection $\hat{P}_0$ is not sensitive to individual site projection $\hat{P}_0$ is not sensitive to individual site
occupancies, but instead enforces a fixed value of the observable, occupancies, but instead enforces a fixed value of the observable,
i.e.~a single Zeno subspace. i.e.~a single Zeno subspace. This is a striking difference with the
$t$-$J$ and other strongly interacting models. The strong interaction
also leads to correlated events in which atoms can tunnel over each
other by creating an unstable doubly occupied site during the
intermediate step. However, these correlated events are by their
nature localised. Due to interactions the doubly-occupied site cannot
be present in the final state which means that any tunnelling event
that created this unstable configuration must be followed by another
tunnelling event which takes an atom away. In the case of global
measurement this process is delocalised, because since the modes
consist of many sites the stable configuration can be restored by a
tunnelling event from a completely different lattice site that belongs
to the same mode.
In Fig.~\ref{fig:zeno}a we consider illuminating only the central In Fig.~\ref{fig:zeno}a we consider illuminating only the central
region of the optical lattice and detecting light in the diffraction region of the optical lattice and detecting light in the diffraction
@ -978,7 +991,7 @@ typical dynamics occurs within each region but the standard tunnelling
between different modes is suppressed. Importantly, second-order between different modes is suppressed. Importantly, second-order
processes that do not change $N_\text{K}$ are still possible since an 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 atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
from $2$ to $3$. Therfore, effective long-range tunneling between two from $2$ to $3$. Therefore, effective long-range tunneling between two
spatially disconnected zones $1$ and $3$ happens due to the two-step spatially disconnected zones $1$ and $3$ happens due to the two-step
processes $1 \rightarrow 2 \rightarrow 3$ or processes $1 \rightarrow 2 \rightarrow 3$ or
$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for $3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
@ -1001,39 +1014,22 @@ measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$,
atoms can slowly tunnel between the odd sites of the lattice, despite atoms can slowly tunnel between the odd sites of the lattice, despite
them being spatially disconnected. This atom exchange spreads them being spatially disconnected. This atom exchange spreads
correlations between non-neighbouring lattice sites on a time scale correlations between non-neighbouring lattice sites on a time scale
$\sim \gamma/J^2$. The schematic explanation of long-range correlated $\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
tunneling is presented in Fig.~\ref{fig:zeno}(b.1): the atoms can explanation of long-range correlated tunneling is presented in
tunnel only in pairs to assure the globally conserved values of Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
$N_\text{even}$ and $N_\text{odd}$, such that one correlated tunneling the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
event is represented by a pair of one red and one blue such that one correlated tunneling event is represented by a pair of
arrow. Importantly, this scheme is fully applicable for a lattice with one red and one blue arrow. Importantly, this scheme is fully
large atom and site numbers, well beyond the numerical example in applicable for a lattice with large atom and site numbers, well beyond
Fig.~\ref{fig:zeno}(b.1), because as we can see in \eqref{eq:hz} the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
it is the geometry of quantum measurement that assures this mode see in Eq. \eqref{eq:hz} it is the geometry of quantum measurement that
structure (in this example, two modes at odd and even sites) and thus assures this mode structure (in this example, two modes at odd and
the underlying pairwise global tunnelling. even sites) and thus the underlying pairwise global tunnelling.
This global pair tunneling may play a role of a building block for The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
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 reservoir for the tunneling (or ``decay'') of atoms from one region to
another. For example, if the atoms are placed only at odd sites, another. For example, if the atoms are placed only at odd sites,
according to \eqref{eq:hz} their tunnelling is suppressed since the 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-tunneling event must be successive, i.e.~an atom tunnelling into
a different mode, $\varphi^\prime$, must then also tunnel back into a different mode, $\varphi^\prime$, must then also tunnel back into
its original mode, $\varphi$. If, however, one adds some atoms to even its original mode, $\varphi$. If, however, one adds some atoms to even
@ -1052,8 +1048,8 @@ entanglement. This is in striking contrast to the entanglement caused
by local processes which can be very confined, especially in 1D where by local processes which can be very confined, especially in 1D where
it is typically short range. This makes numerical calculations of our it is typically short range. This makes numerical calculations of our
system for large atom numbers really difficult, since well-known system for large atom numbers really difficult, since well-known
methods such as DMRG and MPS \cite{schollwock2005} (which are successful methods such as DMRG and MPS \cite{schollwock2005} (which are
for short-range interactions) rely on the limited extent of successful for short-range interactions) rely on the limited extent of
entanglement. entanglement.
The negative number correlations are typical for systems with The negative number correlations are typical for systems with
@ -1072,10 +1068,26 @@ 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-tunneling
events is possible. 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
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.
\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}

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@ -1,7 +1,7 @@
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