From bd128c8b5dd5cb4846d30b2a89d82e1863a3e6d7 Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Tue, 19 Jul 2016 19:03:30 +0100 Subject: [PATCH] Started on chapter 4 --- Chapter4/chapter4.tex | 158 ++++++++++++++++++++++++++++++++++++++++++ Preamble/preamble.tex | 2 + thesis.tex | 6 +- 3 files changed, 163 insertions(+), 3 deletions(-) create mode 100644 Chapter4/chapter4.tex diff --git a/Chapter4/chapter4.tex b/Chapter4/chapter4.tex new file mode 100644 index 0000000..77d68ed --- /dev/null +++ b/Chapter4/chapter4.tex @@ -0,0 +1,158 @@ +%******************************************************************************* +%*********************************** Fourth Chapter ***************************** +%******************************************************************************* + +\chapter{Quantum Measurement Backaction} +% Title of the Fourth Chapter + +\ifpdf + \graphicspath{{Chapter4/Figs/Raster/}{Chapter4/Figs/PDF/}{Chapter4/Figs/}} +\else + \graphicspath{{Chapter4/Figs/Vector/}{Chapter4/Figs/}} +\fi + + +%********************************** %First Section ************************************** + +\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} + | \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. The null +measurement outcome 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} \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 +not consider 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 +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} + +\section{The Master Equation} + +\section{Global Measurement and ``Which-Way'' Information} \ No newline at end of file diff --git a/Preamble/preamble.tex b/Preamble/preamble.tex index 4da1549..5106e1d 100644 --- a/Preamble/preamble.tex +++ b/Preamble/preamble.tex @@ -209,4 +209,6 @@ \newcommand{\ad}{a^\dagger} \newcommand{\bd}{b^\dagger} \renewcommand{\a}{a} % in case we decide to put hats on +\renewcommand{\c}{\hat{c}} +\newcommand{\cd}{\hat{c}^\dagger} \renewcommand{\b}[1]{\mathbf{#1}} \ No newline at end of file diff --git a/thesis.tex b/thesis.tex index 0e6eb48..77d227b 100644 --- a/thesis.tex +++ b/thesis.tex @@ -101,7 +101,7 @@ % To use choose `chapter' option in the document class \ifdefineChapter - \includeonly{Chapter3/chapter3} + \includeonly{Chapter4/chapter4} \fi % ******************************** Front Matter ******************************** @@ -115,7 +115,7 @@ \include{Dedication/dedication} -\include{Declaration/declaration} +%\include{Declaration/declaration} \include{Acknowledgement/acknowledgement} \include{Abstract/abstract} @@ -138,7 +138,7 @@ \include{Chapter1/chapter1} \include{Chapter2/chapter2} \include{Chapter3/chapter3} -%\include{Chapter4/chapter4} +\include{Chapter4/chapter4} %\include{Chapter5/chapter5} %\include{Chapter6/chapter6} %\include{Chapter7/chapter7}