Working on chapter 4

Chapter4/chapter4.tex

@@ -110,6 +110,7 @@ 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}
@@ -118,11 +119,17 @@ 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. The null
+annihilation operator of a photon in the cavity field. It is
-measurement outcome has to be treated differently as it does not occur
+interesting to note that due to renormalisation the effect of a single
-at discrete time points like the detection events themselves. Its
+quantum jump is independent of the magnitude of the operator $\c$
-effect is accounted for by a modification to the isolated Hamiltonian,
+itself. However, larger operators lead to more frequent events and
-$\hat{H}_0$, time evolution in the form
+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
@@ -135,24 +142,299 @@ Schr\"{o}dinger equation given by
   \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} \right) \right] | \psi(t) \rangle,
+      \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 which we do
+straightforward generalisation to multiple jump operators, but we do
-not consider here at all.
+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-setps $\delta t$. Then we
+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}

References/references.bib

@@ -179,6 +179,17 @@
   year={2015},
   publisher={Multidisciplinary Digital Publishing Institute}
 }
+@article{caballero2015,
+  title={Quantum optical lattices for emergent many-body phases of
+                  ultracold atoms},
+  author={Caballero-Benitez, Santiago F and Mekhov, Igor B},
+  journal={Physical review letters},
+  volume={115},
+  number={24},
+  pages={243604},
+  year={2015},
+  publisher={APS}
+}
 @article{mazzucchi2016,
   title = {Quantum measurement-induced dynamics of many-body ultracold
                   bosonic and fermionic systems in optical lattices},
@@ -422,17 +433,6 @@ doi = {10.1103/PhysRevA.87.043613},
   year={2014},
   publisher={Springer}
 }
-@article{caballero2015,
-  title={Quantum optical lattices for emergent many-body phases of
-                  ultracold atoms},
-  author={Caballero-Benitez, Santiago F and Mekhov, Igor B},
-  journal={Physical review letters},
-  volume={115},
-  number={24},
-  pages={243604},
-  year={2015},
-  publisher={APS}
-}
 @article{pedersen2014,
   author={Mads Kock Pedersen and Jens Jakob W H Sorensen and Malte C
                   Tichy and Jacob F Sherson},

thesis.tex

@@ -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,index,chapter]{Classes/PhDThesisPSnPDF}
 
 % ******************************************************************************
 % ******************************* Class Options ********************************