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\chapter{Quantum Measurement Backaction}  
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\section{Introduction}

This thesis is entirely concerned with the question of measuring a
quantum many-body system using quantized light. However, so far we
have only looked at expectation values in a nondestructive context
where we neglect the effect of the quantum wavefunction collapse. We
have shown that light provides information about various statistical
quantities of the quantum states of the atoms such as their
correlation functions. In general, any quantum measurement affects the
system even if it doesn't physically destroy it. In our model both
optical and matter fields are quantized and their interaction leads to
entanglement between the two subsystems. When a photon is detected and
the electromagnetic wavefunction of the optical field collapses, the
matter state is also affected due to this entanglement resulting in
quantum measurement backaction. Therefore, in order to determine these
quantities multiple measurements have to be performed to establish a
precise measurement of the expectation value which will require
repeated preparations of the initial state.

In the following chapters, we consider a different approach to quantum
measurement in open systems and instead of considering expectation
values we look at a single experimental run and the resulting dynamics
due to measurement backaction. Previously we were mostly interested in
extracting information about the quantum state of the atoms from the
scattered light. The flexibility in the measurement model was used to
enable probing of as many different quantum properties of the
ultracold gas as possible. By focusing on measurement backaction we
instead investigate the effect of photodetections on the dynamics of
the many-body gas as well as the possible quantum states that we can
prepare instead of what information can be extracted.

In this chapter, we introduce the necessary theory of quantum
measurement and backaction in open systems in order to lay a
foundation for the material that follows in which we apply these
concepts to a bosonic quantum gas. We first introduce the concept of
quantum trajectories which represent a single continuous series of
photon detections. We also present an alternative approach to open
systems in which the measurement outcomes are discarded as this will
be useful when trying to learn about dynamical features common to
every trajectory. In this case we use the density matrix formalism
which obeys the master equation. This approach is more common in
dissipative systems and we will highlight the differences between
these two different types of open systems. We conclude this chapter
with a new concept that will be central to all subsequent
discussions. In our model measurement is global, it couples to
operators that correspond to global properties of the quantum gas
rather than single-site quantities. This enables the possibility of
performing measurements that cannot distinguish certain sites from
each other. Due to a lack of ``which-way'' information this leads to
the creation of spatially nontrivial virtual lattices on top of the
physical lattice. This turns out to have significant consequences on
the dynamics of the system.

\section{Quantum Trajectories}

A simple intuitive concept of a quantum trajectory is that it is the
path taken by a quantum state over time during a single experimental
realisation. In particular, we consider states conditioned upon
measurement results such as the photodetection times. Such a
trajectory is generally stochastic in nature as light scattering is
not a deterministic process. Furthermore, they are in general
discotinuous as each detection event brings about a drastic change in
the quantum state due to the wavefunction collapse of the light field.

Before we discuss specifics relevant to our model of quantized light
interacting with a quantum gas we present a more general overview
which will be useful as some of the results in the following chapters
are more general. Measurement always consists of at least two
competing processes, two possible outcomes. If there is no competition
and only one outcome is possible then our measurement is meaningless
as it does not reveal any information about the system. In its
simplest form measurement consists of a series of events, such as the
detections of photons. Even though, on an intuitive level it seems
that we have defined only a single outcome, the event, this
arrangement actually consists of two mutually exclusive outcomes. At
any point in time an event either happens or it does not, a photon is
either detected or the detector remains silent, also known as a null
result. Both outcomes reveal some information about the system we are
investigating. For example, let us consider measuring the number of
atoms by measuring the number of photons they scatter. Each atom will
on average scatter a certain number of photons contributing to the
detection rate we observe. Therefore, if we record multiple photons at
a high rate we learn that the illuminated region must contain many
atoms. On the other hand, if there are few atoms to scatter the light
we will observe few detection events which we interpret as a
continuous series of non-detection events interspersed with the
occasional detector click. This trajectory informs us that there are
much fewer atoms being illuminated than previously.

Basic quantum mechanics tells us that such measurements will in
general affect the quantum state in some way. Each event will cause a
discontinuous quantum jump in the wavefunction of the system and it
will have a jump operator, $\c$, associated with it. The effect of an
event on the quantum state is simply the result of applying this jump
operator to the wavefunction, $| \psi (t) \rangle$,
\begin{equation}
  \label{eq:jump}
  | \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle}
  {\sqrt{\langle \cd \c \rangle (t)}},
\end{equation}
where the denominator is simply a normalising factor. The exact form
of the jump operator $\c$ will depend on the nature of the measurement
we are considering. For example, if we consider measuring the photons
escaping from a leaky cavity then $\c = \sqrt{2 \kappa} \hat{a}$,
where $\kappa$ is the cavity decay rate and $\hat{a}$ is the
annihilation operator of a photon in the cavity field. It is
interesting to note that due to renormalisation the effect of a single
quantum jump is independent of the magnitude of the operator $\c$
itself. However, larger operators lead to more frequent events and
thus more frequent applications of the jump operator.

The null measurement outcome will have an opposing effect to the
quantum jump, but it has to be treated differently as it does not
occur at discrete time points like the detection events
themselves. Its effect is accounted for by a modification to the
isolated Hamiltonian, $\hat{H}_0$, time evolution in the form
\begin{equation}
  | \psi (t + \mathrm{d}t) \rangle = \left\{ \hat{1} - \mathrm{d}t
    \left[ i \hat{H}_0 + \frac{\cd \c}{2} - \frac{\langle \cd \c
        \rangle (t)}{2} \right] \right\} | \psi (t) \rangle.
\end{equation}
The effect of both outcomes can be included in a single stochastic
Schr\"{o}dinger equation given by
\begin{equation}
  \label{eq:SSE}
  \mathrm{d} | \psi(t) \rangle = \left[ \mathrm{d} N(t) \left(
      \frac{\c} {\sqrt{ \langle \cd \c \rangle (t)}} - \hat{1} \right)
    + \mathrm{d} t \left( \frac{\langle \cd \c \rangle (t)}{2} -
      \frac{ \cd \c}{2} - i \hat{H}_0 \right) \right] | \psi(t) \rangle,
\end{equation}
where $\mathrm{d}N(t)$ is the stochastic increment to the number of
photodetections up to time $t$ which is equal to $1$ whenever a
quantum jump occurs and $0$ otherwise. Note that this equation has a
straightforward generalisation to multiple jump operators, but we do
not consider this possibility here at all.

All trajectories that we calculate in the follwing chapters are
described by the stochastic Schr\"{o}dinger equation in
Eq. \eqref{eq:SSE}. The most straightforward way to solve it is to
replace the differentials by small time-steps $\delta t$. Then we
generate a random number $R(t)$ at every time-step and a jump is
applied, i.e.~$\mathrm{d}N(t) = 1$, if
\begin{equation}
  R(t) < \langle \cd \c \rangle (t) \delta t.
\end{equation}

In practice, this is not the most efficient method for
simulation. Instead we will use the following method. At an initial
time $t = t_0$ a random number $R$ is generated. We then propagate the
unnormalised wavefunction $| \tilde{\psi} (t) \rangle$ using the
non-Hermitian evolution given by
\begin{equation}
  \frac{\mathrm{d}}{\mathrm{d}t} | \tilde{\psi} (t) \rangle = -i
  \left( \hat{H}_0 - i \frac{\cd \c}{2} \right) | \tilde{\psi} (t) \rangle
\end{equation}
up to a time $T$ such that
$\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$. This
problem can be solved efficienlty using standard numerical
techniques. At time $T$ a quantum jump is applied according to
Eq. \eqref{eq:jump} which renormalises the wavefunction as well and
the process is repeated as long as desired. This formulation also has
the advantage that it provides a more intuitive picture of what
happens during a single trajectory. The quantum jumps are
self-explanatory, but now we have a clearer picture of the effect of
null outcomes on the quantum state. Its effect is entirely encoded in
the non-Hermitian modification to the original Hamiltonian given by
$\hat{H} = \hat{H}_0 - i \cd \c / 2$. It is now easy to see that fro a
jump operator, $\c$, with a large magnitude the no-event outcomes will
have a more significant effect on the quantum state. At the same time
they will lead to more frequent quantum jumps which have an opposing
effect to the non-Hermitian evolution, because the jump condition is
satisfied $\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$
more frequently. In general, the competition between these two
processes is balanced and without any further external influence the
distribution of outcomes over many trajectories will be entirely
determined by the initial state even though each individual trajectory
will be unique and conditioned on the exact detection times that
occured during the given experimental run. In fact, individual
trajectories can have features that are not present after averaging
and this is why we focus our attention on single experimental runs
rather than average behaviour.

\begin{figure}[htbp!]
  \centering
  \includegraphics[width=1.0\textwidth]{setup}
  \caption[Experimental Setup with Cavity]{Atoms in an optical lattice
    are probed by a coherent light beam (red), and the light scattered
    (blue) at a particular angle is enhanced and collected by a leaky
    cavity. The photons escaping the cavity are detected, perturbing
    the atomic evolution via measurement backaction.}
  \label{fig:cavity}
\end{figure}

The quantum trajectory theory can now be very straightforwardly
applied to our model of ultracold bosons in an optical
lattice. However, from now on we will only consider the case when the
atomic system is coupled to a single mode cavity in order to enhance
light scattering in one particular direction as shown in
Fig. \ref{fig:cavity}. This way we have complete control over the form
of the quantum jump operator, because light scattering in different
directions corresponds to different measurements as we have seen in
Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have
to simultaneously consider all the possible directions in which light
could scatter and thus include multiple jump operators reducing the
our ability to control the system.

The model we derived in Eq. \eqref{eq:fullH} is in fact already in a
form ready for quantum trajectory simulations. The phenomologically
included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the
non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$
which is the jump operator we want for measurements of photons leaking
from the cavity. However, we will simplify the system by considering
the regime where we can neglect the effect of the quantum potential
that builds up in the cavity. Physically, this means that whilst light
scatters due to its interaction with matter, the field that builds up
due to the scattered photons collecting in the cavity has a negligible
effect on the atomic evolution compared to its own dynamics such as
inter-site tunnelling or on-site interactions. This can be achieved
when the cavity-probe detuning is smaller than the cavity decay rate,
$\Delta_p \ll \kappa$ \cite{caballero2015}. However, even though the
cavity field has a negligible effect on the atoms, measurement
backaction will not as this effect is of a different nature. It is due
to the wavefunction collapse due to the destruction of photons rather
than an interaction between fields. Therefore, the final form of the
Hamiltonian Eq. \eqref{eq:fullH} that we will be using in the
following chapters is
\begin{equation}
  \hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F}
\end{equation}
\begin{equation}
  \hat{H}_0 = -J \sum_{\langle i, j \rangle} \bd_i b_j + \frac{U}{2}
                \sum_i \n_i (\n_i - 1),
\end{equation}
where $\hat{H}_0$ is simply the Bose-Hubbard Hamiltonian,
$\gamma = \kappa |C|^2$ is a quantity that measures the strength of
the measurement and we have substituted $\a_1 = C \hat{F}$. The
quantum jumps are applied at times determined by the algorithm
described above and the jump operator is given by
\begin{equation}
  \label{eq:jump}
  \c = \sqrt{2 \kappa} C \hat{F}.
\end{equation}
Importantly, we see that measurement introduces a new energy and time
scale $\gamma$ which competes with the two other standard scales
responsible for the uitary dynamics of the closed system, tunnelling,
$J$, and in-site interaction, $U$.  If each atom scattered light
inependently a different jump operator $\c_i$ would be required for
each site projecting the atomic system into a state where long-range
coherence is degraded. This is a typical scenario for spontaneous
emission, or for local and fixed-range addressing. In contrast to such
situations, we consider global coherent scattering with an operator
$\c$ that is global. Therefore, the effect of measurement backaction
is global as well and each jump affects the quantum state in a highly
nonlocal way and most importantly not only will it not degrade
long-range coherence, it will in fact lead to such long-range
correlations itself.

\mynote{introduce citations from PRX/PRA above}

\section{The Master Equation}

A quantum trajectory is stochastic in nature, it depends on the exact
timings of the quantum jumps which are determined randomly. This makes
it difficult to obtain conclusive deterministic answers about the
behaviour of single trajectories. One possible approach that is very
common when dealing with open systems is to look at the unconditioned
state which is obtained by averaging over the random measurement
results which condition the system. The unconditioned state is no
longer a pure state and thus must be described by a density matrix,
\begin{equation}
  \label{eq:rho}
  \hat{\rho} = \sum_i p_i | \psi_i \rangle \langle \psi_i |,
\end{equation}
where $p_i$ is the probability the system is in the pure state
$| \psi_i \rangle$. If more than one $p_i$ value is non-zero then the
state is mixed, it cannot be represented by a single pure state. The
time evolution of the density operator obeys the master equation given
by
\begin{equation}
  \dot{\hat{\rho}} = -i \left[ \hat{H}_0, \hat{\rho} \right] + \c
  \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho}
    \cd \c \right).
\end{equation}
Physically, the unconditioned state, $\rho$, represents our knowledge
of the quantum system if we are ignorant of the measurement outcome
(or we choose to ignore it), i.e.~we do not know the timings of the
detection events. 

We will be using the master equation and the density operator
formalism in the context of measurement. However, the exact same
methods are also applied to a different class of open systems, namely
dissipative systems. A dissipative system is an open system that
couples to an external bath in an uncontrolled way. The behaviour of
such a system is similar to a system subject measurement in which we
ignore all measurement results. One can even think of this external
coupling as a measurement who's outcome record is not accessible and
thus must be represented as an average over all possible
trajectories. However, there is a crucial difference between
measurement and dissipation. When we perform a measurement we use the
master equation to describe system evolution if we ignore the
measuremt outcomes, but at any time we can look at the detection times
and obtain a conditioned pure state for this current experimental
run. On the other hand, for a dissipative system we simply have no
such record of results and thus the density matrix predicted by the
master equation, which in general will be a mixed state, represents
our best knowledge of the system. In order to obtain a pure state, it
would be necessary to perform an actual measurement.

A definite advantage of using the master equation for measurement is
that it includes the effect of any possible measurement
outcome. Therefore, it is useful when extracting features that are
common to many trajectories, regardless of the exact timing of the
events. In this case, we do not want to impose any specific trajectory
on the system as we are not interested in a specific experimental run,
but we would still like to identify the set of possible outcomes and
their properties. Unfortunately, calculating the inverse of
Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of
a density matrix into pure states might not even be unique. However,
if a measurement leads to a projection, i.e.~the final state becomes
confined to some subspace of the Hilbert space, then this will be
visible in the final state of the density matrix. We will show this on
an example of a qubit in the quantum state
\begin{equation}
  \label{eq:qubit0}
  | \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle,
\end{equation}
where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states
of the qubit and we consider performing a measurement on it in the
basis $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome.
The quantum state will have collapsed now to the $ | 0 \rangle$ with
probability $| \alpha |^2$ and $| 1 \rangle$ with probability
$| \beta |^2$. The corresponding density matrix is given by
\begin{equation}
  \label{eq:rho1}
  \hat{\rho} = \left( \begin{array}{cc}
                        | \alpha |^2 & 0 \\
                        0 & |\beta|^2 
                      \end{array} \right),
\end{equation}
which is now a mixed state. We note that there are no off-diagonal
terms as the system is not in a superposition between the two basis
states. Therefore, the diagonal terms represent classical
probabilities of the system being in either of the basis states. This
is in contrast to their original interpretation when the state was
given by Eq. \eqref{eq:qubit0} when they represented the quantum
uncertainty in our knowledge of the state which would have manifested
itself in the density matrix as non-zero off-diagonal terms. The
significance of these values being classical probabilities is that now
we know that the measurement has already happened and we know that the
state must be either $| 0 \rangle$ or $| 1 \rangle$. We just don't
know which one until we check the result of the measurement.

We have assumed that it was a discrete wavefunction collapse that lead
to the state in Eq. \eqref{eq:rho1} in which case the conclusion we
reached was obvious. However, the nature of the process that takes us
from the initial state to the final state with classical uncertainty
does not matter. The key observation is that regardless of the
trajectory taken if the final state is given by Eq. \eqref{eq:rho1} we
will definitely know that our state is either in the state
$| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the
two basis states. Therefore, if we obtained this density matrix as a
result of applying the time evolution given by the master equation we
would be able to identify the final states of individual trajectories
even though we have no information about the individual trajectories
themself.

Here we have considered a very simple case of a Hilbert space with two
non-degenerate basis states. In the following chapters we will
generalise the above result to larger Hilbert spaces with multiple
degenerate subspaces.

\section{Global Measurement and ``Which-Way'' Information}

We have already mentioned that one of the key features of our model is
the global nature of the measurement operators. A single light mode
couples to multiple lattice sites which then scatter the light
coherently into a single mode which we enhance and collect with a
cavity. If atoms at different lattice sites scatter light with a
different phase or magnitude we will be able to identify which atoms
contributed to the light we detected. However, if they scatter the
photons in phase and with the same amplitude then we have no way of
knowing which atom emitted the photon, we have no ``which-way''
information. When we were considering nondestructive measurements and
looking at expectation values, this had no consequence on our results
as we were simply interested in probing the quantum correlations of a
given ground state. Now, on the other hand, we are interested in the
effect of these measurements on the dynamics of the system. The effect
of measurement backaction will depend on the information that is
encoded in the detected photon. If a scattered mode cannot
distinuguish between two different lattice sites then we have no
information about the distribution of atoms between those two sites.
Therefore, all quantum correlations between the atoms in these sites
are unaffected by the backaction whilst their correlations with the
rest of the system will change as the result of the quantum jumps.

The quantum jump operator for our model is given by
$\c = \sqrt{2 \kappa} C \hat{F}$ and we know from Eq. \eqref{eq:F}
that we have a large amount of flexibility in tuning $\hat{F}$ via the
geometry of the optical setup. We will consider the case when
$\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the argument
depends on geometry rather than the exact nature of the operator it
straightforwardly generalises to other measurement operators,
including the case when $\hat{F} = \hat{B}$. The operator $\hat{D}$ is
given by 
\begin{equation}
  \hat{D} = \sum_i J_{i,i} \n_i,
\end{equation}
where the coefficients $J_{i,i}$ are determined from
Eq. \eqref{eq:Jcoeff}. It is very simple to make the $J_{i,i}$
periodic as it 

\begin{figure}[htbp!]
  \centering
  \includegraphics[width=1.0\textwidth]{1DModes}
  \caption[1D Modes due to Measurement Backaction]{}
  \label{fig:cavity}
\end{figure}

\begin{figure}[htbp!]
  \centering
  \includegraphics[width=1.0\textwidth]{2DModes}
  \caption[2D Modes due to Measurement Backaction]{}
  \label{fig:cavity}
\end{figure}