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Chapter4/chapter4.tex
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@ -280,6 +280,7 @@ correlations itself.
\mynote{introduce citations from PRX/PRA above}
\section{The Master Equation}
\label{sec:master}
A quantum trajectory is stochastic in nature, it depends on the exact
timings of the quantum jumps which are determined randomly. This makes

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Chapter5/chapter5.tex
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@ -83,7 +83,9 @@ non-Hermitian Hamiltonian thus extending the notion of quantum Zeno
dynamics into the realm of non-Hermitian quantum mechanics joining the
two paradigms.
\section{Large-Scale Dynamics due to Weak Measurement}
\section{Quantum Measurement Induced Dynamics}
\subsection{Large-Scale Dynamics due to Weak Measurement}
We start by considering the weak measurement limit when photon
scattering does not occur frequently compared to the tunnelling rate
@ -542,7 +544,7 @@ described by this model, is that the width does in fact stay roughly
constant. It is only in the later stages when the oscillations reach
maximal amplitude that the width becomes visibly reduced.
\section{Three-Way Competition}
\subsection{Three-Way Competition}
Now it is time to turn on the inter-atomic interactions,
$U/J^\mathrm{cl} \ne 0$. As a result the atomic dynamics will change
@ -787,8 +789,6 @@ into the strong measurement regime in Fig. \ref{fig:squeezing} as the
$U$-dependence flattens out with increasing measurement strength as
the $\gamma/U \gg 1$ regime is reached.
\section{Quantum Zeno Dynamics}
\subsection{Emergent Long-Range Correlated Tunnelling}
When $\gamma \rightarrow \infty$ the measurement becomes
@ -1085,10 +1085,716 @@ 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}
\section{Non-Hermitian Dynamics in the Quantum Zeno Limit}
In the previous section we provided a rather high-level analysis of
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
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
understanding of the dynamics in the quantum Zeno limit we will now
provide a more rigorous, low-level and fundamental understanding of
the process.
\subsection{Suppression of Coherences in the Density Matrix}
\subsection{Steady-State of the Non-Hermitian Hamiltonian}
At this point we deviate from the quantum trajectory approach and we
resort to a master equation as introduced in section
\ref{sec:master}. We do this, because we have seen that the emergent
long-range correlated tunnelling is a feature of all trajectories and
mostly depends on the geometry of the measurement. Therefore, a
general approach starting from an unconditioned state should be able
to reveal these features. However, we will later make use of the fact
that we are in possession of a measurement record and obtain a
conditioned state. Furthermore, we first consider the most general
case of an open system subject to a quantum measurement and only limit
ourselves to the quantum gas model later on. This demonstrates that
the dynamics we observed in the previous section are a feature of
measurement rather than our specific model.
\section{Conclusions}
As introduced in section \ref{sec:master} we consider a state
described by the density matrix $\hat{\rho}$ whose isolated behaviour
is described by the Hamiltonian $\H_0$ and when measured the jump
operator $\c$ is applied to the state at each detection
\cite{MeasurementControl}. The master equation describing its time
evolution when we ignore the measurement outcomes is given by
\begin{equation}
\dot{\hat{\rho}} = -i [ \H_0 , \hat{\rho} ] + \c \hat{\rho} \cd - \frac{1}{2}(
\cd\c \hat{\rho} + \hat{\rho} \cd\c ).
\end{equation}
We also define $\c = \lambda \op$ and $\H_0 = \nu \h$. The exact
definition of $\lambda$ and $\nu$ is not so important as long as these
coefficients can be considered to be some measure of the relative size
of these operators. They would have to be determined on a case-by-case
basis, because the operators $\c$ and $\H_0$ may be unbounded. If
these operators are bounded, one can simply define them such that
$||\op|| \sim O(1)$ and $||\h|| \sim O(1)$. If they are unbounded, one
possible approach would be to identify the relevant subspace of which
dynamics we are interested in and scale the operators such that the
eigenvalues of $\op$ and $\h$ in this subspace are $\sim O(1)$.
We will once again use projectors $P_m$ which have no effect on states
within a degenerate subspace of $\c$ ($\op$) with eigenvalue $c_m$
($o_m$), but annihilate everything else. For convenience we will also
use the following definition $\hat{\rho}_{mn} = P_m \hat{\rho} P_n$.
Note that these are submatrices of the density matrix, which in
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}
- \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}
where the first term describes coherent evolution whereas the second
term causes dissipation.
First, note that for the density submatrices for which $m = n$,
$\hat{\rho}_{mm}$, the dissipative term vanishes. This means that
these submatrices are subject to coherent evolution only and do not
experience losses and they are thus decoherence free subspaces. It is
crucial to note that these submatrices are simply the density matrices
of the individual degenerate Zeno subspaces. Interestingly, any state
that consists only of these decoherence free subspaces, i.e.~
$\hat{\rho} = \sum_m \hat{\rho}_{mm}$, and that commutes with the
Hamiltonian, $[\hat{\rho}, \hat{H}_0] = 0$, will be a steady state.
This can be seen by substituting this ansatz into
Eq. \eqref{eq:master} which yields $\dot{\hat{\rho}}_{mn} = 0$ for all
$m$ and $n$. These states can be prepared dissipatively using known
techniques \cite{diehl2008}, but it is not required that the state be
a dark state of the dissipative operator as is usually the case.
Second, we consider a large detection rate, $\lambda^2 \gg \nu$, for
which the coherences, i.e.~ the density submatrices $\hat{\rho}_{mn}$
for which $m \ne n$, will be heavily suppressed by dissipation. We can
adiabatically eliminate these cross-terms by setting
$\dot{\hat{\rho}}_{mn} = 0$, to get
\begin{equation}
\label{eq:intermediate}
\hat{\rho}_{mn} = \frac{\nu}{\lambda^2} \frac{i P_m \left[ \h \sum_r \hat{\rho}_{rn} - \sum_r \hat{\rho}_{mr} \h \right] P_n } {o_m o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right)}
\end{equation}
which tells us that they are of order $\nu/\lambda^2 \ll
1$. Therefore, the resulting density matrix will be given by
$\hat{\rho} \approx \sum_m \hat{\rho}_{mm}$ which consists solely of
the individual Zeno subspace density matrices. One can easily recover
the projective Zeno limit by considering $\lambda \rightarrow \infty$
when all the subspaces completely decouple. This is exactly the
$\gamma \rightarrow \infty$ limit discussed in the previous
section. However, we have seen that it is crucial we only consider,
$\lambda^2 \gg \nu$, but not infinite. If the subspaces do not
decouple completely, then transitions within a single subspace can
occur via other subspaces in a manner similar to Raman transitions. In
Raman transitions population is transferred between two states via a
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
negligible at all times.
A single quantum trajectory results in a pure state as opposed to the
density matrix and in general, there are many density matrices that
have non-zero and non-negligible $m = n$ submatrices,
$\hat{\rho}_{mm}$, even when the coherences are small. They correspond
to a mixed states containing many Zeno subspaces and it is not clear
what the pure states that make up these density matrices are. However,
we note that for a single pure state the density matrix can consist of
only a single diagonal submatrix $\hat{\rho}_{mm}$. To understand
this, consider the state $| \Phi \rangle$ and take it to span exactly
two distinct subspaces $P_a$ and $P_b$ ($a \ne b$). This wavefunction
can thus be written as
$| \Phi \rangle = P_a | \Phi \rangle + P_b | \Phi \rangle$. The
corresponding density matrix is given by
\begin{equation}
\hat{\rho}_\Psi = P_a | \Phi \rangle \langle \Phi | P_a + P_a | \Phi
\rangle \langle \Phi | P_b + P_b | \Phi \rangle \langle \Phi | P_a +
P_b | \Phi \rangle \langle \Phi | P_b.
\end{equation}
If the wavefunction has significant components in both subspaces then
in general the density matrix will not have negligible coherences,
$\hat{\rho}_{ab} = P_a | \Phi \rangle \langle \Phi | P_b$. A density
matrix with just diagonal components must be in either subspace $a$,
$| \Phi \rangle = P_a | \Phi \rangle$, or in subspace $b$,
$| \Phi \rangle = P_b | \Phi \rangle$. Therefore, a density matrix of
the form $\hat{\rho} = \sum_m \hat{\rho}_{mm}$ without any cross-terms
between different Zeno subspaces can only be composed of pure states
that each lie predominantly within a single subspace. However, because
we will not be dealing with the projective limit, the wavefunction
will in general not be entirely confined to a single Zeno subspace. We
have seen that the coherences are of order $\nu/\lambda^2$. This would
require the wavefunction components to satisfy
$P_a | \Phi \rangle \approx O(1)$ and
$P_b | \Phi \rangle \approx O(\nu/\lambda^2)$ (or vice-versa). This in
turn implies that the population of the states outside of the dominant
subspace (and thus the submatrix $\hat{\rho}_{bb}$) will be of order
$\langle \Phi | P_b^2 | \Phi \rangle \approx
O(\nu^2/\lambda^4)$. Therefore, these pure states, even though they
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
example we considered in section \ref{sec:master}.
\subsection{Quantum Measurement vs. Dissipation}
This is where quantum measurement deviates from dissipation. If we
have access to a measurement record we can infer which Zeno subspace
is occupied, because we know that only one of them can be occupied at
any time. We have seen that since the density matrix cross-terms are
small we know \emph{a priori} that the individual wavefunctions
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
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
record our density matrix would be a mixture of all these
possibilities. We can try and determine the Zeno subspace around which
the state evolves in a single experiment from the number of
detections, $m$, in time $t$.
The detection distribution on time-scales shorter than dissipation (so
we can approximate as if we were in a fully Zeno regime) can be
obtained by integrating over the detection times \cite{mekhov2009pra}
to get
\begin{equation}
P(m,t) = \sum_n \frac{[|c_n|^2 t]^m} {m!} e^{-|c_n|^2 t} \mathrm{Tr} (\rho_{nn}).
\end{equation}
For a state that is predominantly in one Zeno subspace, the
distribution will be approximately Poissonian (up to
$O(\nu^2 / \lambda^4)$, the population of the other
subspaces). Therefore, in a single experiment we will measure
$m = |c_0|^2t \pm \sqrt{|c_0|^2t}$ detections (note, we have assumed
$|c_0|^2 t$ is large enough to approximate the distribution as
normal. This is not necessary, we simply use it here to not have to
worry about the asymmetry in the deviation around the mean value). The
uncertainty does not come from the fact that $\lambda$ is not
infinite. The jumps are random events with a Poisson
distribution. Therefore, even in the full projective limit we will not
observe the same detection trajectory in each experiment even though
the system evolves in exactly the same way and remains in a perfectly
pure state.
If the basis of $\c$ is continuous (e.g. free particle position or
momentum) then the deviation around the mean will be our upper bound
on the deviation of the system from a pure state evolving around a
single Zeno subspace. However, continuous systems are beyond the scope
of this work and we will confine ourselves to discrete systems. Though
it is important to remember that continuous systems can be treated
this way, but the error estimate (and thus the mixedness of the state)
will be different.
For a discrete system it is easier to exclude all possibilities except
for one. The error in our estimate of $|c_0|^2$ in a single experiment
decreases as $1/\sqrt{t}$ and thus it can take a long time to
confidently determine $|c_0|^2$ to a sufficient precision this
way. However, since we know that it can only take one of the possible
values from the set $\{|c_n|^2 \}$ it is much easier to instead
exclude all the other values.
In an experiment we can use Bayes' theorem to infer the state of our
system as follows
\begin{equation}
p(c_n = c_0 | m) = \frac{ p(m | c_n = c_0) p(c_n = c_0) }{ p(m) },
\end{equation}
where $p(x)$ denotes the probability of the discrete event $x$ and
$p(x|y)$ the conditional probability of $x$ given $y$. We know that
$p(m | c_n = c_0)$ is simply given by a Poisson distribution with mean
$|c_0|^2 t$. $p(m)$ is just a normalising factor and $p(c_n = c_0)$ is
our \emph{a priori} knowledge of the state. Therefore, one can get the
probability of being in the right Zeno subspace from
\begin{align}
p(c_n & = c_0 | m) = \frac{ p_0(c_n = c_0) \frac{ \left( |c_0|^2 t
\right)^{2m} } {m!} e^{-|c_0|^2 t}} {\sum_n p_0(c_n) \frac{
\left( |c_n|^2 t \right)^{2m} } {m!} e^{-|c_n|^2 t}} \nonumber \\
& = p_0(c_n = c_0) \left[ \sum_n p_0(c_n) \left( \frac{
|c_n|^2 } { |c_0|^2 } \right)^{2m} e^{\left( |c_0|^2 -
|c_n|^2 \right) t} \right]^{-1},
\end{align}
where $p_0$ denotes probabilities at $t = 0$. In a real experiment one
could prepare the initial state to be close to the Zeno subspace of
interest and thus it would be easier to deduce the state. Furthermore,
in the middle of an experiment if we have already established the Zeno
subspace this will be reflected in these \emph{a priori} probabilities
again making it easier to infer the correct subspace. However, we will
consider the worst case scenario which might be useful if we don't
know the initial state or if the Zeno subspace changes during the
experiment, a uniform $p_0(c_n)$.
This probability is a rather complicated function as $m$ is a
stochastic quantity that also increases with $t$. We want it to be as
close to $1$ as possible. In order to devise an appropriate condition
for this we note that in the first line all terms in the denominator
are Poisson distributions of $m$. Therefore, if the mean values
$|c_n|^2 t$ are sufficiently spaced out, only one of the terms in the
sum will be significant for a given $m$ and if this happens to be the
one that corresponds to $c_0$ we get a probability close to
unity. Therefore, we set the condition such that it is highly unlikely
that our measured $m$ could be produced by two different distributions
\begin{align}
\sqrt{|c_0|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0 \\
\sqrt{|c_n|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0
\end{align}
The left-hand side is the standard deviation of $m$ if the system was
in subspace $P_0$ or $P_n$. The right-hand side is the difference in
the mean detections between the subspace $n$ and the one we are
interested in. The condition becomes more strict if the subspaces
become less distinguishable as it becomes harder to confidently
determine the correct state. Once again, using $\c = \lambda \hat{o}$
where $\hat{o} \sim O(1)$ we get
\begin{equation}
t \gg \frac{1}{\lambda^2} \frac{|o_{0,n}|^2} {(|o_0|^2 - |o_n|^2|)^2}.
\end{equation}
Since detections happen on average at an average rate of order
$\lambda^2$ we only need to wait for a few detections to satisfy this
condition. Therefore, we see that even in the worst case scenario of
complete ignorance of the state of the system we can very easily
determine the correct subspace. Once it is established for the first
time, the \emph{a priori} information can be updated and it will
become even easier to monitor the system.
However, it is important to note that physically once the quantum
jumps deviate too much from the mean value the system is more likely
to change the Zeno subspace (due to measurement backaction) and the
detection rate will visibly change. Therefore, if we observe a
consistent detection rate it is extremely unlikely that it can be
produced by two different Zeno subspaces so in fact it is even easier
to determine the correct state, but the above estimate serves as a
good lower bound on the necessary detection time.
Having derived the necessary conditions to confidently determine which
Zeno subspace is being observed in the experiment we can make another
approximation thanks to measurement which would be impossible in a
purely dissipative open system. If we observe a number of detections
consistent with the subspace $P_m = P_0$ we can set
$\hat{\rho}_{mn} \approx 0$ for all cases when both $m \ne 0$ and
$n \ne 0$ leaving our density matrix in the form
\begin{equation}
\label{eq:approxrho}
\hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} +
\hat{\rho}_{r0}).
\end{equation}
We can do this, because the other states are inconsistent with the
measurement record. We know from the previous section that the system
must lie predominantly in only one of the Zeno subspaces and when that
is the case, $\hat{\rho}_{0r} \approx O(\nu/\lambda^2)$ and for
$m \ne 0$ and $n \ne 0$ we have
$\hat{\rho}_{mn} \approx O(\nu^2/\lambda^4)$. Therefore, this amounts
to keeping first order terms in $\nu/\lambda^2$ in our approximation.
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
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
them happens in discrete transitions (which we know about from the
change in the detection rate as each Zeno subspace will correspond to
a different rate) and not as continuous coherent evolution. Therefore,
we can postselect in a manner similar to Refs. \cite{otterbach2014,
lee2014prx, lee2014prl}, but our requirements are significantly more
relaxed - we do not require a specific single trajectory, only that it
remains within a Zeno subspace. Furthermore, upon reaching a steady
state, these transitions become impossible as the coherences
vanish. This approximation is analogous to optical Raman transitions
where the population of the excited state is neglected. Here, we can
make a similar approximation and neglect all but one Zeno subspace
thanks to the additional knowledge we gain from knowing the
measurement outcomes.
\subsection{The Non-Hermitian Hamiltonian}
Rewriting the master equation using $\c = c_0 + \delta \c$, where
$c_0$ is the eigenvalue corresponding to the eigenspace defined by the
projector $P_0$ which we used to obtain the density matrix in
Eq. \eqref{eq:approxrho}, we get
\begin{equation}
\label{eq:finalrho}
\dot{\hat{\rho}} = -i \left( \H_\mathrm{eff} \hat{\rho} - \hat{\rho}
\H_\mathrm{eff}^\dagger \right) + \delta \c \hat{\rho} \delta \cd,
\end{equation}
\begin{equation}
\label{eq:Ham}
\H_\mathrm{eff} = \H_0 + i \left( c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right).
\end{equation}
The first term in Eq. \eqref{eq:finalrho} describes coherent evolution
due to the non-Hermitian Hamiltonian $\H_\mathrm{eff}$ and the second
term is decoherence due to our ignorance of measurement outcomes. When
we substitute our approximation of the density matrix
$\hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} +
\hat{\rho}_{r0})$ into Eq. \eqref{eq:finalrho}, the last term
vanishes, $\delta \c \hat{\rho} \delta \cd = 0$. This happens, because
$\delta \c P_0 \hat{\rho} = \hat{\rho} P_0 \delta \c^\dagger = 0$. The
projector annihilates all states except for those with eigenvalue
$c_0$ and so the operator $\delta \c = \c - c_0$ will always evaluate
to $c_0 - c_0 = 0$. Recall that we defined
$\hat{\rho}_{mn} = P_m \hat{\rho} P_n$ which means that every term in
our approximate density matrix contains the projector $P_0$. However,
it is important to note that this argument does not apply to other
second order terms in the master equation, because some terms only
have the projector $P_0$ applied from one side,
e.g.~$\hat{\rho}_{0m}$. The term $\delta \c \hat{\rho} \delta \cd$
applies the fluctuation operator from both sides so it does not matter
in this case, but it becomes relevant for terms such as
$\delta \cd \delta \c \hat{\rho}$. It is important to note that this
term does not automatically vanish, but when the explicit form of our
approximate density matrix is inserted, it is in fact zero. Therefore,
we can omit this term using the information we gained from
measurement, but keep other second order terms, such as
$\delta \cd \delta \c \rho$ in the Hamiltonian which are the origin of
other second-order dynamics. This could not be the case in a
dissipative system.
Ultimately we find that a system under continuous measurement for
which $\lambda^2 \gg \nu$ in the Zeno subspace $P_0$ is described by
the deterministic non-Hermitian Hamiltonian $\H_\mathrm{eff}$ in
Eq. \eqref{eq:Ham} and thus obeys the following Schr\"{o}dinger
equation
\begin{equation}
i \frac{\mathrm{d} | \Psi \rangle}{\mathrm{d}t} = \left[\H_0 + i \left(
c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right) \right] |
\Psi \rangle.
\end{equation}
Of the three terms in the parentheses the first two represent the
effects of quantum jumps due to detections (which one can think of as
`reference frame' shifts between different degenerate eigenspaces) and
the last term is the non-Hermitian decay due to information gain from
no detections. It is important to emphasize that even though we
obtained a deterministic equation, we have not neglected the
stochastic nature of the detection events. The detection trajectory
seen in an experiment will have fluctuations around the mean
determined by the Zeno subspace, but there simply are many possible
measurement records with the same outcome. This is just like the fully
projective Zeno limit where the system remains perfectly pure in one
of the possible projections, but the detections remain randomly
distributed in time.
One might then be concerned that purity is preserved even though we
might be averaging over many trajectories within this Zeno
subspace. We have neglected the small terms $\hat{\rho}_{m,n}$
($m,n \ne 0$) which are $O(\nu^2/\lambda^4)$ and thus they are not
correctly accounted for by our approximation. This means that we have
an $O(\nu^2/\lambda^4)$ error in our density matrix. The purity
given by
\begin{equation}
\mathrm{Tr}(\hat{\rho}^2) = \mathrm{Tr}(\hat{\rho}^2_{00} + \sum_{m \ne
0} \hat{\rho}_{0m}\hat{\rho}_{m0}) + \mathrm{Tr}(\sum_{m,n\ne0}
\hat{\rho}_{mn} \hat{\rho}_{nm})
\end{equation}
where the second term contains the terms not accounted for by our
approximation thus introduces an $O(\nu^4/\lambda^8)$
error. Therefore, this discrepancy is negligible in our
approximation. The pure state predicted by $\H_\mathrm{eff}$ is only
an approximation, albeit a good one, and the real state will be mixed
to a small extent. Whilst perfect purity within the Zeno subspace
$\hat{\rho}_{00}$ is expected due to the measurement's strong
decoupling effect, the nearly perfect purity when transitions outside
the Zeno subspace are included is a nontrivial result. Similarly, in
Raman transitions the population of the neglected excited state is
also non-zero, but negligible. Furthermore, this equation does not
actually require the adiabatic elimination used in
Eq. \eqref{eq:intermediate} (we only used it to convince ourselves
that the coherences are small) and such situations may be considered
provided all approximations remain valid. In a similar way the limit
of linear optics is derived from the physics of a two-level nonlinear
medium, when the population of the upper state is neglected and the
adiabatic elimination of coherences is not required.
\subsection{Non-Hermitian Dynamics in Ultracold Gases}
We finally return to our quantum gas model inside of a cavity. We
start by considering the simplest case of a global multi-site
measurement of the form $\hat{D} = \hat{N}_K = \sum_i^K \n_i$, where
the sum is over $K$ illuminated sites. The effective Hamiltonian
becomes
\begin{equation}
\label{eq:nHH2}
\hat{H}_\mathrm{eff} = \hat{H}_0 - i \gamma \left( \delta \hat{N}_K \right)^2,
\end{equation}
where $ \delta \hat{N}_K = \hat{N}_K - N^0_K$ and $N^0_K$ is the Zeno
subspace eigenvalue. It is now obvious that continuous measurement
squeezes the fluctuations in the measured quantity, as expected, and
that the only competing process is the system's own dynamics.
In this case, if we adiabatically eliminate the density matrix
cross-terms and substitute Eq. \eqref{eq:intermediate} into
Eq. \eqref{eq:master} for this system we obtain an effective
Hamiltonian within the Zeno subspace defined by $N_K$
\begin{equation}
\H_\varphi = P_0 \left[ \H_0 - i \frac{J^2}{\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 $\varphi$ denotes a 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 sites) and $P_0$ is the projector onto
the eigenspace with $N_K^0$ atoms in the illuminated area. We focus on
the case when the second term is not only significant, but also leads
to dynamics within a Zeno subspace that are not allowed by
conventional quantum Zeno dynamics accounted for by the first
term. The second term represents second-order transitions via other
subspaces which act as intermediate states much like virtual states in
optical Raman transitions. This is in contrast to the conventional
understanding of the Zeno dynamics for infinitely frequent projective
measurements (corresponding to $\gamma \rightarrow \infty$) where such
processes are forbidden \cite{facchi2008}. Thus, it is the weak
quantum measurement that effectively couples the states. Note that
this is a special case of the equation in Eq. \eqref{eq:hz} which can
be obtained by considering a more general two mode setup.
\subsection{Small System Example}
To get clear physical insight, we initially consider three atoms in
three sites and choose our measurement operator such that
$\hat{D} = \n_2$, i.e.~only the middle site is subject to measurement,
and the Zeno subspace defined by $n_2 = 1$. Such an illumination
pattern can be achieved with global addressing by crossing two beams
and placing the nodes at the odd sites and the antinodes at even
sites. This means that $P_0 \H_0 P_0 = 0$. However, the
first and third sites are connected via the second term. Diagonalising
the Hamiltonian reveals that out of its ten eigenvalues all but three
have a significant negative imaginary component of the order $\gamma$
which means that the corresponding eigenstates decay on a time scale
of a single quantum jump and thus quickly become negligible. The three
remaining eigenvectors are dominated by the linear superpositions of
the three Fock states $|2,1,0 \rangle$, $|1, 1, 1 \rangle$, and
$|0,1,2 \rangle$. Whilst it is not surprising that these components
are the only ones that remain as they are the only ones that actually
lie in the Zeno subspace $n_2 = 1$, it is impossible to solve the full
dynamics by just considering these Fock states alone as they are not
coupled to each other in $\hat{H}_0$. The components lying outside of
the Zeno subspace have to be included to allow intermediate steps to
occur via states that do not belong in this subspace, much like
virtual states in optical Raman transitions.
An approximate solution for $U=0$ can be written for the
$\{|2,1,0 \rangle, |1,1,1 \rangle, |0,1,2 \rangle\}$ subspace by
multiplying each eigenvector with its corresponding time evolution
\begin{equation}
| \Psi(t) \rangle \propto \left( \begin{array}{c}
z_1 + \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t / \gamma} \\
-\sqrt{2} \left(z_1 - z_3 e^{-12 J^2 t / \gamma} \right) \\
z_1 - \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t /
\gamma} \\
\end{array}
\right), \nonumber
\end{equation}
where $z_i$ denote the overlap between the eigenvectors and the
initial state, $z_i = \langle v_i | \Psi (0) \rangle$, with
$| v_1 \rangle = (1, -\sqrt{2}, 1)/2$,
$| v_2 \rangle = (1, 0, -1)/\sqrt{2}$, and
$| v_3 \rangle = (1, \sqrt{2}, 1)/2$. The steady state as
$t \rightarrow \infty$ is given by
$| v_1 \rangle = (1, -\sqrt{2}, 1)/2$. This solution is illustrated in
Fig. \ref{fig:comp} which clearly demonstrates dynamics beyond the
canonical understanding of quantum Zeno dynamics as tunnelling occurs
between states coupled via a different Zeno subspace.
\begin{figure}[hbtp!]
\includegraphics[width=\linewidth]{comp}
\caption[Fock State Populations in a Zeno
Subspace]{Populations of the Fock states in the Zeno subspace
for $\gamma/J = 100$ and initial state $| 2,1,0 \rangle$. It
is clear that quantum Zeno dynamics occurs via Raman-like
processes even though none of these states are connected in
$\hat{H}_0$. The dynamics occurs via virtual intermediate
states outside the Zeno subspace. The system also tends to a
steady state which minimises tunnelling effectively
suppressing fluctuations. The lines are solutions to the
non-Hermitian Hamiltonian, and the dots are points from a
stochastic trajectory calculation.\label{fig:comp}}
\end{figure}
\subsection{Steady State of non-Hermitian Dynamics}
A distinctive difference between Bose-Hubbard model ground states and
the final steady state,
$| \Psi \rangle = [|2,1,0 \rangle - \sqrt{2} |1,1,1\rangle +
|0,1,2\rangle]/2$, is that its components are not in phase. Squeezing
due to measurement naturally competes with inter-site tunnelling which
tends to spread the atoms. However, from Eq. \eqref{eq:nHH2} we see
the final state will always be the eigenvector with the smallest
fluctuations as it will have an eigenvalue with the largest imaginary
component. This naturally corresponds to the state where tunnelling
between Zeno subspaces (here between every site) is minimised by
destructive matter-wave interference, i.e.~the tunnelling dark state
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,
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
new perspective on the competition between a system's short-range
dynamics and global measurement backaction.
We now consider a one-dimensional lattice with $M$ sites so we extend
the measurement to $\hat{D} = \N_\text{even}$ where every even site is
illuminated. The wavefunction in a Zeno subspace must be an
eigenstate of $\c$ and we combine this with the requirement for it to
be in the dark state of the tunnelling operator (eigenstate of $\H_0$
for $U = 0$) to derive the steady state. These two conditions in
momentum space are
\begin{equation}
\hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k -
\bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \nonumber
\end{equation}
\begin{equation}
\Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} +
\bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle, \nonumber
\end{equation}
where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$,
$\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice
spacing, $N$ the total atom number, and we perform summations over the
reduced Brillouin zone (RBZ), $-\pi/2a < k \le \pi/2a$, as the
symmetries of the system are clearer this way. Now we define
\begin{equation}
\hat{\alpha}_k^\dagger = \bd_k \bd_q - \bd_{-k} \bd_{-q},
\end{equation}
\begin{equation}
\hat{\beta}_\varphi^\dagger = \bd_{\pi/2a} + \varphi \bd_{-\pi/2a},
\end{equation}
where $\varphi = \Delta N / | \Delta N |$, which create the smallest
possible states that satisfy the two equations for $\Delta N = 0$ and
$\Delta N \ne 0$ respectively. Therefore, by noting that
\begin{align}
\left[ \hat{T}, \hat{\alpha}_k^\dagger \right] & = 0, \\
\left[ \hat{T}, \hat{\beta}_\varphi^\dagger \right] & = 0, \\
\left[ \Delta \N, \hat{\alpha}_k^\dagger \right] & = 0, \\
\left[ \Delta \N, \hat{\beta}_\varphi^\dagger \right] & = \varphi
\hat{\beta}_\varphi^\dagger,
\end{align}
we can now write the equation for the $N$-particle steady state
\begin{equation}
\label{eq:ss}
| \Psi \rangle \propto \left[ \prod_{i=1}^{(N - |\Delta N|)/2}
\left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger
\right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{|
\Delta N |} | 0 \rangle, \nonumber
\end{equation}
where $\phi_{i,k}$ are coefficients that depend on the trajectory
taken to reach this state and $|0 \rangle$ is the vacuum state defined
by $b_k |0 \rangle = 0$. Since this a dark state (an eigenstate of
$\H_0$) of the atomic dynamics, this state will remain stationary even
with measurement switched-off. Interestingly, this state is very
different from the ground states of the Bose-Hubbard Hamiltonian, it
is even orthogonal to the superfluid state, and thus it cannot be
obtained by cooling or projecting from an initial ground state. The
combination of tunnelling with measurement is necessary.
\begin{figure}[hbtp!]
\includegraphics[width=\linewidth]{figure3}
\caption[Non-Hermitian Steady State]{A trajectory simulation
for eight atoms in eight sites, initially in
$|1,1,1,1,1,1,1,1 \rangle$, with periodic boundary
conditions and $\gamma/J = 100$. (a), The fluctuations in
$\c$ where the stochastic nature of the process is clearly
visible on a single trajectory level. However, the general
trend is captured by the non-Hermitian Hamiltonian. (b), The
local density variance. Whilst the fluctuations in the
global measurement operator decrease, the fluctuations in
local density increase due to tunnelling via states outside
the Zeno subspace. (c), The momentum distribution. The
initial Fock state has a flat distribution which with time
approaches the steady state distribution of two identical
and symmetric distributions centred at $k = \pi/2a$ and
$k = -\pi/2a$.\label{fig:steady}}
\end{figure}
In order to prepare the steady state one has to run the experiment and
wait until the photocount rate remains constant for a sufficiently
long time. Such a trajectory is illustrated in Fig. \ref{fig:steady}
and compared to a deterministic trajectory calculated using the
non-Hermitian Hamiltonian. It is easy to see from
Fig. \ref{fig:steady}(a) how the stochastic fluctuations around the
mean value of the observable have no effect on the general behaviour
of the system in the strong measurement regime. By discarding these
fluctuations we no longer describe a pure state, but we showed how
this only leads to a negligible error. Fig. \ref{fig:steady}(b) shows
the local density variance in the lattice. Not only does it grow
showing evidence of tunnelling between illuminated and non-illuminated
sites, but it grows to significant values. This is in contrast to
conventional quantum Zeno dynamics where no tunnelling would be
allowed at all. Finally, Fig. \ref{fig:steady}(c) shows the momentum
distribution of the trajectory. We can clearly see that it deviates
significantly from the initial flat distribution of the Fock
state. Furthermore, the steady state does not have any atoms in the
$k=0$ state and thus is orthogonal to the superfluid state as
discussed.
To obtain a state with a specific value of $\Delta N$ postselection
may be necessary, but otherwise it is not needed. The process can be
optimised by feedback control since the state is monitored at all
times \cite{ivanov2014}. Furthermore, the form of the measurement
operator is very flexible and it can easily be engineered by the
geometry of the optical setup \cite{elliott2015, mazzucchi2016} which
can be used to design a state with desired properties.
\section{Conclusions}
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
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
considered allowed us to effectively tailor long-range entanglement
and correlations present in the system. We showed that the competition
between the global backaction and usual atomic dynamics leads to the
production of spatially multimode macroscopic superpositions which
exhibit large-scale oscillatory dynamics which could be used for
quantum information and metrology. We subsequently demonstrated that
when on-site atomic interactions are introduced the dynamics become
much more complicated with different regimes of behaviour where
measurement and interactions can either compete or cooperate. In the
strong measurement regime we showed that conventional quantum Zeno
dynamics can be realised, but more interestingly, by considering a
strong, but not projective, limit of measurement we observe a new type
of nonlocal dynamics. It turns out that a global measurement scheme
leads to correlations between spatially separated tunnelling events
which conserve the Zeno subspace via Raman-like processes which would
be forbidden in the canonical fully projective limit. We subsequently
presented a rigorous analysis of the underlying process of this new
type of quantum Zeno dynamics in which we showed that in this limit
quantum trajectories can be described by a deterministic non-Hermitian
Hamiltonian. In contrast to previous works, it is independent of the
underlying system and there is no need to postselect a particular
exotic trajectory \cite{lee2014prx, lee2014prl}. Finally, we have
shown that the system will always tend towards the eigenstate of the
Hamiltonian with the best squeezing of the observable and the atomic
dynamics, which normally tend to spread the distribution, cooperates
with measurement to produce a state in which tunnelling is suppressed
by destructive matter-wave interference. A dark state of the
tunnelling operator will have zero fluctuations and we provided an
expression for the steady state which is significantly different from
the ground state of the Hamiltonian. This is in contrast to previous
works on dissipative state preparation where the steady state had to
be a dark state of the measurement operator \cite{diehl2008}.
Such globally paired tunnelling due to a fundamentally new phenomenon,
global quantum measurement backaction, can enrich the physics of
long-range correlated systems beyond relatively short-range
interactions expected from standard dipole-dipole interactions
\cite{sowinski2012, omjyoti2015}. These nonlocal high-order processes
entangle regions of the optical lattice that are disconnected by the
measurement. Using different detection schemes, we showed how to
tailor density-density correlations between distant lattice
sites. Quantum optical engineering of nonlocal coupling to
environment, combined with quantum measurement, can allow the design
of nontrivial system-bath interactions, enabling new links to quantum
simulations~\cite{stannigel2013} and thermodynamics~\cite{erez2008}
and extend these directions to the field of non-Hermitian quantum
mechanics, where quantum optical setups are particularly
promising~\cite{lee2014prl}. Importantly, both systems and baths,
designed by our method, can be strongly correlated systems with
internal long-range entanglement.

5
Preamble/preamble.tex
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@ -211,4 +211,7 @@
\renewcommand{\a}{a} % in case we decide to put hats on
\renewcommand{\c}{\hat{c}}
\newcommand{\cd}{\hat{c}^\dagger}
\renewcommand{\b}[1]{\mathbf{#1}}
\renewcommand{\b}[1]{\mathbf{#1}}
\newcommand{\op}{\hat{o}}
\newcommand{\h}{\hat{h}}
\newcommand{\N}{\hat{N}}

62
References/references.bib
View File

@ -978,3 +978,65 @@ doi = {10.1103/PhysRevA.87.043613},
year = {2005},
publisher = {American Physical Society},
}
@article{otterbach2014,
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Bose-Einstein Condensates},
author = {Otterbach, Johannes and Lemeshko, Mikhail},
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volume = {113},
issue = {7},
pages = {070401},
numpages = {6},
year = {2014},
month = {Aug},
publisher = {American Physical Society},
}
@article{lee2014prx,
title = {Heralded Magnetism in Non-Hermitian Atomic Systems},
author = {Lee, Tony E. and Chan, Ching-Kit},
journal = {Phys. Rev. X},
volume = {4},
issue = {4},
pages = {041001},
numpages = {13},
year = {2014},
month = {Oct},
publisher = {American Physical Society},
}
@article{lee2014prl,
title = {Entanglement and Spin Squeezing in Non-Hermitian Phase Transitions},
author = {Lee, Tony E. and Reiter, Florentin and Moiseyev, Nimrod},
journal = {Phys. Rev. Lett.},
volume = {113},
issue = {25},
pages = {250401},
numpages = {5},
year = {2014},
month = {Dec},
publisher = {American Physical Society},
}
@article{stannigel2013,
title = {{Constrained Dynamics via the Zeno Effect in Quantum
Simulation: Implementing Non-Abelian Lattice Gauge
Theories with Cold Atoms}},
author = {Stannigel, K. and Hauke, P. and Marcos, D. and Hafezi,
M. and Diehl, S. and Dalmonte, M. and Zoller, P.},
journal = {Phys. Rev. Lett.},
volume = {112},
issue = {12},
pages = {120406},
numpages = {6},
year = {2014},
month = {Mar},
publisher = {American Physical Society},
doi = {10.1103/PhysRevLett.112.120406}
}
@article{erez2008,
title={{Repulsively bound atom pairs in an optical lattice}},
author={Erez, N. and Gordon, G. and Nest, M. and Kurizki, G.},
journal={Nature},
volume={452},
pages = {724},
year={2008},
publisher={Nature Publishing Group}
}

2
thesis.tex
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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,chapter]{Classes/PhDThesisPSnPDF}
\documentclass[a4paper,12pt,times,numbered,print]{Classes/PhDThesisPSnPDF}
% ******************************************************************************
% ******************************* Class Options ********************************