441 lines
23 KiB
TeX
441 lines
23 KiB
TeX
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%*********************************** Fourth Chapter *****************************
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\chapter{Quantum Measurement Backaction}
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% Title of the Fourth Chapter
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%********************************** %First Section **************************************
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\section{Introduction}
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This thesis is entirely concerned with the question of measuring a
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quantum many-body system using quantized light. However, so far we
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have only looked at expectation values in a nondestructive context
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where we neglect the effect of the quantum wavefunction collapse. We
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have shown that light provides information about various statistical
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quantities of the quantum states of the atoms such as their
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correlation functions. In general, any quantum measurement affects the
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system even if it doesn't physically destroy it. In our model both
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optical and matter fields are quantized and their interaction leads to
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entanglement between the two subsystems. When a photon is detected and
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the electromagnetic wavefunction of the optical field collapses, the
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matter state is also affected due to this entanglement resulting in
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quantum measurement backaction. Therefore, in order to determine these
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quantities multiple measurements have to be performed to establish a
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precise measurement of the expectation value which will require
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repeated preparations of the initial state.
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In the following chapters, we consider a different approach to quantum
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measurement in open systems and instead of considering expectation
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values we look at a single experimental run and the resulting dynamics
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due to measurement backaction. Previously we were mostly interested in
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extracting information about the quantum state of the atoms from the
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scattered light. The flexibility in the measurement model was used to
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enable probing of as many different quantum properties of the
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ultracold gas as possible. By focusing on measurement backaction we
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instead investigate the effect of photodetections on the dynamics of
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the many-body gas as well as the possible quantum states that we can
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prepare instead of what information can be extracted.
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In this chapter, we introduce the necessary theory of quantum
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measurement and backaction in open systems in order to lay a
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foundation for the material that follows in which we apply these
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concepts to a bosonic quantum gas. We first introduce the concept of
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quantum trajectories which represent a single continuous series of
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photon detections. We also present an alternative approach to open
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systems in which the measurement outcomes are discarded as this will
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be useful when trying to learn about dynamical features common to
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every trajectory. In this case we use the density matrix formalism
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which obeys the master equation. This approach is more common in
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dissipative systems and we will highlight the differences between
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these two different types of open systems. We conclude this chapter
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with a new concept that will be central to all subsequent
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discussions. In our model measurement is global, it couples to
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operators that correspond to global properties of the quantum gas
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rather than single-site quantities. This enables the possibility of
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performing measurements that cannot distinguish certain sites from
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each other. Due to a lack of ``which-way'' information this leads to
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the creation of spatially nontrivial virtual lattices on top of the
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physical lattice. This turns out to have significant consequences on
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the dynamics of the system.
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\section{Quantum Trajectories}
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A simple intuitive concept of a quantum trajectory is that it is the
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path taken by a quantum state over time during a single experimental
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realisation. In particular, we consider states conditioned upon
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measurement results such as the photodetection times. Such a
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trajectory is generally stochastic in nature as light scattering is
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not a deterministic process. Furthermore, they are in general
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discotinuous as each detection event brings about a drastic change in
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the quantum state due to the wavefunction collapse of the light field.
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Before we discuss specifics relevant to our model of quantized light
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interacting with a quantum gas we present a more general overview
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which will be useful as some of the results in the following chapters
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are more general. Measurement always consists of at least two
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competing processes, two possible outcomes. If there is no competition
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and only one outcome is possible then our measurement is meaningless
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as it does not reveal any information about the system. In its
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simplest form measurement consists of a series of events, such as the
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detections of photons. Even though, on an intuitive level it seems
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that we have defined only a single outcome, the event, this
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arrangement actually consists of two mutually exclusive outcomes. At
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any point in time an event either happens or it does not, a photon is
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either detected or the detector remains silent, also known as a null
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result. Both outcomes reveal some information about the system we are
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investigating. For example, let us consider measuring the number of
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atoms by measuring the number of photons they scatter. Each atom will
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on average scatter a certain number of photons contributing to the
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detection rate we observe. Therefore, if we record multiple photons at
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a high rate we learn that the illuminated region must contain many
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atoms. On the other hand, if there are few atoms to scatter the light
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we will observe few detection events which we interpret as a
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continuous series of non-detection events interspersed with the
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occasional detector click. This trajectory informs us that there are
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much fewer atoms being illuminated than previously.
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Basic quantum mechanics tells us that such measurements will in
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general affect the quantum state in some way. Each event will cause a
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discontinuous quantum jump in the wavefunction of the system and it
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will have a jump operator, $\c$, associated with it. The effect of an
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event on the quantum state is simply the result of applying this jump
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operator to the wavefunction, $| \psi (t) \rangle$,
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\begin{equation}
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\label{eq:jump}
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| \psi(t + \mathrm{d}t) \rangle = \frac{\c | \psi(t) \rangle}
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{\sqrt{\langle \cd \c \rangle (t)}},
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\end{equation}
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where the denominator is simply a normalising factor. The exact form
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of the jump operator $\c$ will depend on the nature of the measurement
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we are considering. For example, if we consider measuring the photons
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escaping from a leaky cavity then $\c = \sqrt{2 \kappa} \hat{a}$,
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where $\kappa$ is the cavity decay rate and $\hat{a}$ is the
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annihilation operator of a photon in the cavity field. It is
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interesting to note that due to renormalisation the effect of a single
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quantum jump is independent of the magnitude of the operator $\c$
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itself. However, larger operators lead to more frequent events and
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thus more frequent applications of the jump operator.
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The null measurement outcome will have an opposing effect to the
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quantum jump, but it has to be treated differently as it does not
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occur at discrete time points like the detection events
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themselves. Its effect is accounted for by a modification to the
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isolated Hamiltonian, $\hat{H}_0$, time evolution in the form
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\begin{equation}
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| \psi (t + \mathrm{d}t) \rangle = \left\{ \hat{1} - \mathrm{d}t
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\left[ i \hat{H}_0 + \frac{\cd \c}{2} - \frac{\langle \cd \c
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\rangle (t)}{2} \right] \right\} | \psi (t) \rangle.
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\end{equation}
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The effect of both outcomes can be included in a single stochastic
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Schr\"{o}dinger equation given by
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\begin{equation}
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\label{eq:SSE}
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\mathrm{d} | \psi(t) \rangle = \left[ \mathrm{d} N(t) \left(
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\frac{\c} {\sqrt{ \langle \cd \c \rangle (t)}} - \hat{1} \right)
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+ \mathrm{d} t \left( \frac{\langle \cd \c \rangle (t)}{2} -
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\frac{ \cd \c}{2} - i \hat{H}_0 \right) \right] | \psi(t) \rangle,
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\end{equation}
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where $\mathrm{d}N(t)$ is the stochastic increment to the number of
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photodetections up to time $t$ which is equal to $1$ whenever a
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quantum jump occurs and $0$ otherwise. Note that this equation has a
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straightforward generalisation to multiple jump operators, but we do
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not consider this possibility here at all.
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All trajectories that we calculate in the follwing chapters are
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described by the stochastic Schr\"{o}dinger equation in
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Eq. \eqref{eq:SSE}. The most straightforward way to solve it is to
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replace the differentials by small time-steps $\delta t$. Then we
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generate a random number $R(t)$ at every time-step and a jump is
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applied, i.e.~$\mathrm{d}N(t) = 1$, if
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\begin{equation}
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R(t) < \langle \cd \c \rangle (t) \delta t.
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\end{equation}
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In practice, this is not the most efficient method for
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simulation. Instead we will use the following method. At an initial
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time $t = t_0$ a random number $R$ is generated. We then propagate the
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unnormalised wavefunction $| \tilde{\psi} (t) \rangle$ using the
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non-Hermitian evolution given by
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\begin{equation}
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\frac{\mathrm{d}}{\mathrm{d}t} | \tilde{\psi} (t) \rangle = -i
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\left( \hat{H}_0 - i \frac{\cd \c}{2} \right) | \tilde{\psi} (t) \rangle
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\end{equation}
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up to a time $T$ such that
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$\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$. This
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problem can be solved efficienlty using standard numerical
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techniques. At time $T$ a quantum jump is applied according to
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Eq. \eqref{eq:jump} which renormalises the wavefunction as well and
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the process is repeated as long as desired. This formulation also has
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the advantage that it provides a more intuitive picture of what
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happens during a single trajectory. The quantum jumps are
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self-explanatory, but now we have a clearer picture of the effect of
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null outcomes on the quantum state. Its effect is entirely encoded in
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the non-Hermitian modification to the original Hamiltonian given by
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$\hat{H} = \hat{H}_0 - i \cd \c / 2$. It is now easy to see that fro a
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jump operator, $\c$, with a large magnitude the no-event outcomes will
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have a more significant effect on the quantum state. At the same time
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they will lead to more frequent quantum jumps which have an opposing
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effect to the non-Hermitian evolution, because the jump condition is
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satisfied $\langle \tilde{\psi} (T) | \tilde{\psi} (T) \rangle = R$
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more frequently. In general, the competition between these two
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processes is balanced and without any further external influence the
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distribution of outcomes over many trajectories will be entirely
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determined by the initial state even though each individual trajectory
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will be unique and conditioned on the exact detection times that
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occured during the given experimental run. In fact, individual
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trajectories can have features that are not present after averaging
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and this is why we focus our attention on single experimental runs
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rather than average behaviour.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=1.0\textwidth]{setup}
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\caption[Experimental Setup with Cavity]{Atoms in an optical lattice
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are probed by a coherent light beam (red), and the light scattered
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(blue) at a particular angle is enhanced and collected by a leaky
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cavity. The photons escaping the cavity are detected, perturbing
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the atomic evolution via measurement backaction.}
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\label{fig:cavity}
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\end{figure}
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The quantum trajectory theory can now be very straightforwardly
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applied to our model of ultracold bosons in an optical
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lattice. However, from now on we will only consider the case when the
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atomic system is coupled to a single mode cavity in order to enhance
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light scattering in one particular direction as shown in
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Fig. \ref{fig:cavity}. This way we have complete control over the form
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of the quantum jump operator, because light scattering in different
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directions corresponds to different measurements as we have seen in
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Eq. \eqref{eq:Jcoeff}. On the other hand, in free space we would have
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to simultaneously consider all the possible directions in which light
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could scatter and thus include multiple jump operators reducing the
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our ability to control the system.
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The model we derived in Eq. \eqref{eq:fullH} is in fact already in a
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form ready for quantum trajectory simulations. The phenomologically
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included cavity decay rate $-i \kappa \ad_1 \a_1$ is in fact the
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non-Hermitian term $-i \cd \c / 2$, where $\c = \sqrt{2 \kappa} \a_1$
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which is the jump operator we want for measurements of photons leaking
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from the cavity. However, we will simplify the system by considering
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the regime where we can neglect the effect of the quantum potential
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that builds up in the cavity. Physically, this means that whilst light
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scatters due to its interaction with matter, the field that builds up
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due to the scattered photons collecting in the cavity has a negligible
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effect on the atomic evolution compared to its own dynamics such as
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inter-site tunnelling or on-site interactions. This can be achieved
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when the cavity-probe detuning is smaller than the cavity decay rate,
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$\Delta_p \ll \kappa$ \cite{caballero2015}. However, even though the
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cavity field has a negligible effect on the atoms, measurement
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backaction will not as this effect is of a different nature. It is due
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to the wavefunction collapse due to the destruction of photons rather
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than an interaction between fields. Therefore, the final form of the
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Hamiltonian Eq. \eqref{eq:fullH} that we will be using in the
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following chapters is
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\begin{equation}
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\hat{H} = \hat{H}_0 - i \gamma \hat{F}^\dagger \hat{F}
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\end{equation}
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\begin{equation}
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\hat{H}_0 = -J \sum_{\langle i, j \rangle} \bd_i b_j + \frac{U}{2}
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\sum_i \n_i (\n_i - 1),
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\end{equation}
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where $\hat{H}_0$ is simply the Bose-Hubbard Hamiltonian,
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$\gamma = \kappa |C|^2$ is a quantity that measures the strength of
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the measurement and we have substituted $\a_1 = C \hat{F}$. The
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quantum jumps are applied at times determined by the algorithm
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described above and the jump operator is given by
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\begin{equation}
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\label{eq:jump}
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\c = \sqrt{2 \kappa} C \hat{F}.
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\end{equation}
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Importantly, we see that measurement introduces a new energy and time
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scale $\gamma$ which competes with the two other standard scales
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responsible for the uitary dynamics of the closed system, tunnelling,
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$J$, and in-site interaction, $U$. If each atom scattered light
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inependently a different jump operator $\c_i$ would be required for
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each site projecting the atomic system into a state where long-range
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coherence is degraded. This is a typical scenario for spontaneous
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emission, or for local and fixed-range addressing. In contrast to such
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situations, we consider global coherent scattering with an operator
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$\c$ that is global. Therefore, the effect of measurement backaction
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is global as well and each jump affects the quantum state in a highly
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nonlocal way and most importantly not only will it not degrade
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long-range coherence, it will in fact lead to such long-range
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correlations itself.
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\mynote{introduce citations from PRX/PRA above}
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\section{The Master Equation}
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A quantum trajectory is stochastic in nature, it depends on the exact
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timings of the quantum jumps which are determined randomly. This makes
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it difficult to obtain conclusive deterministic answers about the
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behaviour of single trajectories. One possible approach that is very
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common when dealing with open systems is to look at the unconditioned
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state which is obtained by averaging over the random measurement
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results which condition the system. The unconditioned state is no
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longer a pure state and thus must be described by a density matrix,
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\begin{equation}
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\label{eq:rho}
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\hat{\rho} = \sum_i p_i | \psi_i \rangle \langle \psi_i |,
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\end{equation}
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where $p_i$ is the probability the system is in the pure state
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$| \psi_i \rangle$. If more than one $p_i$ value is non-zero then the
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state is mixed, it cannot be represented by a single pure state. The
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time evolution of the density operator obeys the master equation given
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by
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\begin{equation}
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\dot{\hat{\rho}} = -i \left[ \hat{H}_0, \hat{\rho} \right] + \c
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\hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho}
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\cd \c \right).
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\end{equation}
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Physically, the unconditioned state, $\rho$, represents our knowledge
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of the quantum system if we are ignorant of the measurement outcome
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(or we choose to ignore it), i.e.~we do not know the timings of the
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detection events.
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We will be using the master equation and the density operator
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formalism in the context of measurement. However, the exact same
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methods are also applied to a different class of open systems, namely
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dissipative systems. A dissipative system is an open system that
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couples to an external bath in an uncontrolled way. The behaviour of
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such a system is similar to a system subject measurement in which we
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ignore all measurement results. One can even think of this external
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coupling as a measurement who's outcome record is not accessible and
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thus must be represented as an average over all possible
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trajectories. However, there is a crucial difference between
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measurement and dissipation. When we perform a measurement we use the
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master equation to describe system evolution if we ignore the
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measuremt outcomes, but at any time we can look at the detection times
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and obtain a conditioned pure state for this current experimental
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run. On the other hand, for a dissipative system we simply have no
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such record of results and thus the density matrix predicted by the
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master equation, which in general will be a mixed state, represents
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our best knowledge of the system. In order to obtain a pure state, it
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would be necessary to perform an actual measurement.
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A definite advantage of using the master equation for measurement is
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that it includes the effect of any possible measurement
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outcome. Therefore, it is useful when extracting features that are
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common to many trajectories, regardless of the exact timing of the
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events. In this case, we do not want to impose any specific trajectory
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on the system as we are not interested in a specific experimental run,
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but we would still like to identify the set of possible outcomes and
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their properties. Unfortunately, calculating the inverse of
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Eq. \eqref{eq:rho} is not an easy task. In fact, the decomposition of
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a density matrix into pure states might not even be unique. However,
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if a measurement leads to a projection, i.e.~the final state becomes
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confined to some subspace of the Hilbert space, then this will be
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visible in the final state of the density matrix. We will show this on
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an example of a qubit in the quantum state
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\begin{equation}
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\label{eq:qubit0}
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| \psi \rangle = \alpha |0 \rangle + \beta | 1 \rangle,
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\end{equation}
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where $| 0 \rangle$ and $| 1 \rangle$ represent the two basis states
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of the qubit and we consider performing a measurement on it in the
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basis $\{| 0 \rangle, | 1 \rangle \}$, but we don't check the outcome.
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The quantum state will have collapsed now to the $ | 0 \rangle$ with
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probability $| \alpha |^2$ and $| 1 \rangle$ with probability
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$| \beta |^2$. The corresponding density matrix is given by
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\begin{equation}
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\label{eq:rho1}
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\hat{\rho} = \left( \begin{array}{cc}
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| \alpha |^2 & 0 \\
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0 & |\beta|^2
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\end{array} \right),
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\end{equation}
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which is now a mixed state. We note that there are no off-diagonal
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terms as the system is not in a superposition between the two basis
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states. Therefore, the diagonal terms represent classical
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probabilities of the system being in either of the basis states. This
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is in contrast to their original interpretation when the state was
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given by Eq. \eqref{eq:qubit0} when they represented the quantum
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uncertainty in our knowledge of the state which would have manifested
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itself in the density matrix as non-zero off-diagonal terms. The
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significance of these values being classical probabilities is that now
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we know that the measurement has already happened and we know that the
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state must be either $| 0 \rangle$ or $| 1 \rangle$. We just don't
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know which one until we check the result of the measurement.
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We have assumed that it was a discrete wavefunction collapse that lead
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to the state in Eq. \eqref{eq:rho1} in which case the conclusion we
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reached was obvious. However, the nature of the process that takes us
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from the initial state to the final state with classical uncertainty
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does not matter. The key observation is that regardless of the
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trajectory taken if the final state is given by Eq. \eqref{eq:rho1} we
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will definitely know that our state is either in the state
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$| 0 \rangle$ or $| 1 \rangle$ and not in some superposition of the
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two basis states. Therefore, if we obtained this density matrix as a
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result of applying the time evolution given by the master equation we
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would be able to identify the final states of individual trajectories
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even though we have no information about the individual trajectories
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themself.
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Here we have considered a very simple case of a Hilbert space with two
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non-degenerate basis states. In the following chapters we will
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generalise the above result to larger Hilbert spaces with multiple
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degenerate subspaces.
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\section{Global Measurement and ``Which-Way'' Information}
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We have already mentioned that one of the key features of our model is
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the global nature of the measurement operators. A single light mode
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couples to multiple lattice sites which then scatter the light
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coherently into a single mode which we enhance and collect with a
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cavity. If atoms at different lattice sites scatter light with a
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different phase or magnitude we will be able to identify which atoms
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contributed to the light we detected. However, if they scatter the
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photons in phase and with the same amplitude then we have no way of
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knowing which atom emitted the photon, we have no ``which-way''
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information. When we were considering nondestructive measurements and
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looking at expectation values, this had no consequence on our results
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as we were simply interested in probing the quantum correlations of a
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given ground state. Now, on the other hand, we are interested in the
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effect of these measurements on the dynamics of the system. The effect
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of measurement backaction will depend on the information that is
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encoded in the detected photon. If a scattered mode cannot
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distinuguish between two different lattice sites then we have no
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information about the distribution of atoms between those two sites.
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Therefore, all quantum correlations between the atoms in these sites
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are unaffected by the backaction whilst their correlations with the
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rest of the system will change as the result of the quantum jumps.
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The quantum jump operator for our model is given by
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$\c = \sqrt{2 \kappa} C \hat{F}$ and we know from Eq. \eqref{eq:F}
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that we have a large amount of flexibility in tuning $\hat{F}$ via the
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geometry of the optical setup. We will consider the case when
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$\hat{F} = \hat{D}$ given by Eq. \eqref{eq:D}, but since the argument
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depends on geometry rather than the exact nature of the operator it
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straightforwardly generalises to other measurement operators,
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including the case when $\hat{F} = \hat{B}$. The operator $\hat{D}$ is
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given by
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\begin{equation}
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\hat{D} = \sum_i J_{i,i} \n_i,
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\end{equation}
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|
where the coefficients $J_{i,i}$ are determined from
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|
Eq. \eqref{eq:Jcoeff}. It is very simple to make the $J_{i,i}$
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|
periodic as it
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|
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|
\begin{figure}[htbp!]
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|
\centering
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|
\includegraphics[width=1.0\textwidth]{1DModes}
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|
\caption[1D Modes due to Measurement Backaction]{}
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|
\label{fig:cavity}
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|
\end{figure}
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|
|
|
\begin{figure}[htbp!]
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|
\centering
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|
\includegraphics[width=1.0\textwidth]{2DModes}
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|
\caption[2D Modes due to Measurement Backaction]{}
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|
\label{fig:cavity}
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|
\end{figure}
|