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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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Chapter5/chapter5.tex
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@ -793,19 +793,21 @@ the $\gamma/U \gg 1$ regime is reached.
When $\gamma \rightarrow \infty$ the measurement becomes
projective. This means that as soon as the probing begins, the system
collapses into one of the observable's eigenstates. Since this
measurement is continuous and doesn't stop after the projection the
system will be frozen in this state. This effect is called the quantum
Zeno effect from Zeno's classical paradox in which a ``watched arrow
never moves'' that stated since an arrow in flight is not seen to move
during any single instant, it cannot possibly be moving at
all. Classically the paradox was resolved with a better understanding
of infinity and infintesimal changes, but in the quantum world a
watched quantum arrow will in fact never move. The system is being
continuously projected into its initial state before it has any chance
to evolve away. If degenerate eigenspaces exist then we can observe
quantum Zeno dynamics where unitary evolution is uninhibited within
such a degenerate subspace, called the Zeno subspace.
collapses into one of the observable's eigenstates. Furthermore, since
this measurement is continuous and doesn't stop after the projection
the system will be frozen in this state. This effect is called the
quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
paradox in which a ``watched arrow never moves'' that stated that
since an arrow in flight is not seen to move during any single
instant, it cannot possibly be moving at all. Classically the paradox
was resolved with a better understanding of infinity and infintesimal
changes, but in the quantum world a watched quantum arrow will in fact
never move. The system is being continuously projected into its
initial state before it has any chance to evolve. If degenerate
eigenspaces exist then we can observe quantum Zeno dynamics where
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
$\gamma \rightarrow \infty$. 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 $\varphi$
denotes a single degenerate subspace. The Zeno subspace is determined
randomly as per the Copenhagen postulates and thus it depends on the
initial state. If the projection is into the subspace
$\varphi^\prime$, the subsequent evolution is described by the
projected Hamiltonian
$P_{\varphi^\prime} \hat{H}_0 P_{\varphi^\prime}$. We have used the
original Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or
the quantum jumps as their combined effect is now described by the
initial state. If the projection is into the subspace $\varphi$, the
subsequent evolution is described by the projected Hamiltonian
$P_{\varphi} \hat{H}_0 P_{\varphi}$. We have used the original
Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
quantum jumps as their combined effect is now described by the
projectors. Physically, in our model of ultracold bosons trapped in a
lattice this means that tunnelling between different spatial modes is
completely supressed since this process couples eigenstates belonging
@ -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
tunnelling would only be prohibited between the illuminated and
unilluminated areas, but dynamics proceeds normally within each zone
separately. Therefore, the goemetric patterns we have in which the
modes are sptailly delocalised in such a way that neighbouring sites
never belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$,
would lead to a complete suppression of tunnelling across the whole
lattice as there is no way for an atom to tunnel within this Zeno
subspace.
separately. However, the goemetric patterns we have in which the modes
are spatially delocalised in such a way that neighbouring sites never
belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, would
lead to a complete suppression of tunnelling across the whole lattice
as there is no way for an atom to tunnel within this Zeno subspace
without first having to leave it.
This is an interesting example of the quantum Zeno effect and dynamics
and it can be used to prohibit parts of the dynamics of the
Bose-Hubbard Hamiltonian in order to engineer desired Hamiltonians for
Bose-Hubbard Model in order to engineer desired Hamiltonians for
quantum simulations or other applications. However, the infinite
projective limit is uninteresting in the context of a global
measurement scheme. The same effects and Hamiltonians can be achieved
using multiple independent measurements which address a few sites
each. The only advantage of the global scheme is that it might be
simpler to achieve as it requires a less complicated optical setup. In
order to take advantage of the nonlocal nature of the measurement it
turns out that we need to consider a finite limit for $\gamma \gg
J$. By considering a non-infinite $\gamma$ we observe additional
dynamics while the usual atomic tunnelling is still heavily
each. In order to take advantage of the nonlocal nature of the
measurement it turns out that we need to consider a finite limit for
$\gamma \gg J$. By considering a non-infinite $\gamma$ we observe
additional dynamics while the usual atomic tunnelling is still heavily
Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
\begin{figure}[hbtp!]
@ -897,22 +896,22 @@ eigenspaces and select which tunnelling processes should be
uninhibited and which should be suppressed. However, there is a second
effect that was not present before. In Fig. \ref{fig:zeno} we can
observe tunnelling that violates the boundaries established by the
spatial modes. When $\gamma$ is finite, second-order processes can now
occur, i.e.~two correlated tunnelling events, via an intermediate
(virtual) state outside of the Zeno subspace as long as the Zeno
subspace of the final state remains the same. Crucially, these
tunnelling events are only correlated in time, but not in space. This
means that the two events do not have to occur for the same atom or
even at the same site in the lattice. As long as the Zeno subspace is
preserved, these processes can occur anywhere in the system, that is a
pair of atoms separated by many sites is able to tunnel in a
correlated manner. This is only possible due to the possibility of
creating extensive and spatially nonlocal modes as described in
section \ref{sec:modes} which in turn is enabled by the global nature
of the measurement. This would not be possible to achieve with local
measurements as this would lead to Zeno subspaces described entirely
by local variables which cannot be preserved by such delocalised
tunnelling events.
spatial modes. When $\gamma$ is finite, second-order processes,
i.e.~two correlated tunnelling events, can now occur via an
intermediate (virtual) state outside of the Zeno subspace as long as
the Zeno subspace of the final state remains the same. Crucially,
these tunnelling events are only correlated in time, but not in
space. This means that the two events do not have to occur for the
same atom or even at the same site in the lattice. As long as the Zeno
subspace is preserved, these processes can occur anywhere in the
system. That is, a pair of atoms separated by many sites is able to
tunnel in a correlated manner. This is only possible due to the
ability of creating extensive and spatially nonlocal modes as
described in section \ref{sec:modes} which in turn is enabled by the
global nature of the measurement. This would not be possible to
achieve with local measurements as the Zeno subspaces would be
described entirely by local variables which cannot be preserved by
such delocalised tunnelling events.
In the subsequent sections we will rigorously derive the following
Hamiltonian for the non-interacting dynamics within a single Zeno
@ -930,30 +929,32 @@ $\hat{D} = \hat{N}_\mathrm{odd}$
where
$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
constant that depends on the measurement scheme, $\varphi$ denotes a
set of site belonging to a single mode and $\varphi^\prime$ is the
set 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). We see that this Hamiltonian consists of two parts. The first
term corresponds to the standard quantum Zeno first-order dynamics
that occurs within a Zeno subspace, i.e.~tunnelling between
neighbouring sites that belong to the same mode. Otherwise,
$P_0 \bd_i b_j P_0 = 0$. If $\gamma \rightarrow \infty$ we would
recover the quantum Zeno Hamiltonian where this would be the only
remaining term. It is the second term that shows the second-order
corelated tunnelling terms. This is evident from the inner sum which
requires that pairs of sites ($i$, $j$) and ($k$, $l$) between which
atoms tunnel must be nearest neighbours, but these pairs can be as far
apart from each other as possible within the mode structure. This is
in particular explicitly shown in Figs. \ref{fig:zeno}(a,b). The
imaginary coefficient means that the tinelling behaves like an
exponential decay (overdamped oscillations). This also implies that
the norm will decay, but this does not mean that there are physical
losses in the system. Instead, the norm itself represents the
probability of the system remaining in the $\varphi = 0$ Zeno
subspace. Since $\gamma$ is not infinite there is now a finite
probability that the stochastic nature of the measurement will lead to
a discontinous change in the system where the Zeno subspace rapidly
changes. However, later in this chapter we will see that steady states
of this Hamiltonian exist which will no longer change Zeno subspaces.
neighbouring sites that belong to the same mode. Otherwise, if $i$ and
$j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
$\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
where this would be the only remaining term. It is the second term
that shows the second-order corelated tunnelling terms. This is
evident from the inner sum which requires that pairs of sites ($i$,
$j$) and ($k$, $l$) between which atoms tunnel must be nearest
neighbours, but these pairs can be anywhere on the lattice within the
constraints of the mode structure. This is in particular explicitly
shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
that the tinelling behaves like an exponential decay (overdamped
oscillations). This also implies that the norm will decay, but this
does not mean that there are physical losses in the system. Instead,
the norm itself represents the probability of the system remaining in
the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
is now a finite probability that the stochastic nature of the
measurement will lead to a discontinous change in the system where the
Zeno subspace rapidly changes which can be seen in
Fig. \ref{fig:zeno}(a). However, later in this chapter we will see
that steady states of this Hamiltonian exist which will no longer
change Zeno subspaces.
Crucially, what sets this effect apart from usual many-body dynamics
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
sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
profound consequences as this is the physical origin of the long-range
correlated tunneling events represented in \eqref{eq:hz} by the fact
that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. This is
because the projection $\hat{P}_0$ is not sensitive to individual site
correlated tunneling events represented in Eq. \eqref{eq:hz} by the
fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
projection $\hat{P}_0$ is not sensitive to individual site
occupancies, but instead enforces a fixed value of the observable,
i.e.~a single Zeno subspace.
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
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
processes that do not change $N_\text{K}$ are still possible since an
atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
from $2$ to $3$. Therfore, effective long-range tunneling between two
from $2$ to $3$. Therefore, effective long-range tunneling between two
spatially disconnected zones $1$ and $3$ happens due to the two-step
processes $1 \rightarrow 2 \rightarrow 3$ or
$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
@ -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
them being spatially disconnected. This atom exchange spreads
correlations between non-neighbouring lattice sites on a time scale
$\sim \gamma/J^2$. The schematic explanation of long-range correlated
tunneling is presented in Fig.~\ref{fig:zeno}(b.1): the atoms can
tunnel only in pairs to assure the globally conserved values of
$N_\text{even}$ and $N_\text{odd}$, such that one correlated tunneling
event is represented by a pair of one red and one blue
arrow. Importantly, this scheme is fully applicable for a lattice with
large atom and site numbers, well beyond the numerical example in
Fig.~\ref{fig:zeno}(b.1), because as we can see in \eqref{eq:hz}
it is the geometry of quantum measurement that assures this mode
structure (in this example, two modes at odd and even sites) and thus
the underlying pairwise global tunnelling.
$\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
explanation of long-range correlated tunneling is presented in
Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
such that one correlated tunneling event is represented by a pair of
one red and one blue arrow. Importantly, this scheme is fully
applicable for a lattice with large atom and site numbers, well beyond
the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
see in Eq. \eqref{eq:hz} it is the geometry of quantum measurement that
assures this mode structure (in this example, two modes at odd and
even sites) and thus the underlying pairwise global tunnelling.
This global pair tunneling may play a role of a building block for
more complicated many-body effects. For example, a pair tunneling
between the neighbouring sites has been recently shown to play
important role in the formation of new quantum phases, e.g., pair
superfluid \cite{sowinski2012} and lead to formulation of extended
Bose-Hubbard models \cite{omjyoti2015}. The search for novel
mechanisms providing long-range interactions is crucial in many-body
physics. One of the standard candidates is the dipole-dipole
interaction in, e.g., dipolar molecules, where the mentioned pair
tunneling between even neighboring sites is already considered to be
long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
suggests a fundamentally different mechanism originating from quantum
optics: the backaction of global and spatially structured measurement,
which as we prove can successfully compete with other short-range
processes in many-body systems. This opens promising opportunities for
future research.
The scheme in Fig.~\ref{fig:zeno}(b.1) can help to design a nonlocal
The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
reservoir for the tunneling (or ``decay'') of atoms from one region to
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
a different mode, $\varphi^\prime$, must then also tunnel back into
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
it is typically short range. This makes numerical calculations of our
system for large atom numbers really difficult, since well-known
methods such as DMRG and MPS \cite{schollwock2005} (which are successful
for short-range interactions) rely on the limited extent of
methods such as DMRG and MPS \cite{schollwock2005} (which are
successful for short-range interactions) rely on the limited extent of
entanglement.
The negative number correlations are typical for systems with
@ -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
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}
% Contrast with t-J model here how U localises events, but measurement
% does the opposite
\subsection{Steady-State of the Non-Hermitian Hamiltonian}

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