diff --git a/Chapter1/chapter1.tex b/Chapter1/chapter1.tex index cca0034..9fbc68f 100644 --- a/Chapter1/chapter1.tex +++ b/Chapter1/chapter1.tex @@ -2,7 +2,7 @@ %*********************************** First Chapter ***************************** %******************************************************************************* -\chapter{Chapter title} %Title of the First Chapter +\chapter{Introduction} %Title of the First Chapter \ifpdf \graphicspath{{Chapter1/Figs/Raster/}{Chapter1/Figs/PDF/}{Chapter1/Figs/}} diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index e5b45a4..2a533aa 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -44,7 +44,184 @@ from a small number of sites with a large filling factor (e.g.~BECs trapped in a double-well potential) to a an extended multi-site lattice with a low filling factor (e.g.~a system with one atom per site will exhibit the Mott insulator to superfluid quantum phase -transition). \mynote{extra fermion citations, Piazza?} +transition). \mynote{extra fermion citations, Piazza? Look up Gabi's + AF paper.} +\mynote{Potentially some more crap, but come to think of it the + content will strongly depend on what was included in the preceding + section on plain ultracold bosons} +As we have seen in the previous section, an optical lattice can be +formed with classical light beams that form standing waves. Depending +on the detuning with respect to the atomic resonance, the nodes or +antinodes form the lattice sites in which atoms accumulate. As shown +in Fig. \ref{fig:LatticeDiagram} the trapped bosons (green) are +illuminated with a coherent probe beam (red) and scatter light into a +different mode (blue) which is then measured with a detector. The most +straightforward measurement is to simply count the number of photons +with a photodetector, but it is also possible to perform a quadrature +measurement by using a homodyne detection scheme. The experiment can +be performed in free space where light can scatter in any +direction. The atoms can also be placed inside a cavity which has the +advantage of being able to enhance light scattering in a particular +direction. Furthermore, cavities allow for the formation of a fully +quantum potential in contrast to the classical lattice trap. +\begin{figure}[htbp!] + \centering + \includegraphics[width=1.0\textwidth]{LatticeDiagram} + \caption[LatticeDiagram]{Atoms (green) trapped in an optical lattice + are illuminated by a coherent probe beam (red). The light scatters + (blue) in free space or into a cavity and is measured by a + detector. If the experiment is in free space light can scatter in + any direction. A cavity on the other hand enhances scattering in + one particular direction.} + \label{fig:LatticeDiagram} +\end{figure} + +For simplicity, we will be considering one-dimensional lattices most +of the time. However, the model itself is derived for any number of +dimensions and since none of our arguments will ever rely on +dimensionality our results straightforwardly generalise to 2- and 3-D +systems. This simplification allows us to present a much simpler +picture of the physical setup where we only need to concern ourselves +with a single angle for each optical mode. As shown in +Fig. \ref{fig:LatticeDiagram} the angle between the normal to the +lattice and the probe and detected beam are denoted by $\theta_0$ and +$\theta_1$ respectively. We will consider these angles to be tunable +although the same effect can be achieved by varying the wavelength of +the light modes. However, it is much more intuitive to consider +variable angles in our model as this lends itself to a simpler +geometrical representation. + +\subsection{Derivation of the Hamiltonian} + +A general many-body Hamiltonian coupled to a quantized light field in +second quantized can be separated into three parts, +\begin{equation} +\label{eq:TwoH} + \H = \H_f + \H_a + \H_{fa}. +\end{equation} +The term $\H_f$ represents the optical part of the Hamiltonian, +\begin{equation} +\label{eq:Hf} + \H_f = \sum_l \hbar \omega_l \ad_l \a_l - + i \hbar \sum_l \left( \eta_l^* \a_l - \eta_l \ad \right). +\end{equation} +The operators $\a_l$ ($\ad$) are the annihilation (creation) operators +of light modes with frequencies $\omega_l$, wave vectors $\b{k}_l$, +and mode functions $u_l(\b{r})$, which can be pumped by coherent +fields with amplitudes $\eta_l$. The second part of the Hamiltonian, +$\H_a$, is the matter-field component given by +\begin{equation} +\label{eq:Ha} + \H_a = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,a} + \Psi(\b{r}) + \frac{2 \pi a_s \hbar^2}{m} \int \mathrm{d}^3 \b{r} + \Psi^\dagger(\b{r}) \Psi^\dagger(\b{r}) \Psi(\b{r}) \Psi(\b{r}). +\end{equation} +Here, $\Psi(\b{r})$ ($\Psi^\dagger(\b{r})$) are the matter-field +operators that annihilate (create) an atom at position $\b{r}$, $a_s$ +is the $s$-wave scattering length characterising the interatomic +interaction, and $\H_{1,a}$ is the atomic part of the single-particle +Hamiltonian $\H_1$. The final component of the total Hamiltonian is +the interaction given by +\begin{equation} + \label{eq:Hfa} + \H_{fa} = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,fa} + \Psi(\b{r}), +\end{equation} +where $\H_{1,fa}$ is the interaction part of the single-particle +Hamiltonian, $\H_1$. + +The single-particle Hamiltonian in the rotating-wave and dipole +approximation is given by +\begin{equation} + \H_1 = \H_f + \H_{1,a} + \H_{1,fa}, +\end{equation} +\begin{equation} + \H_{1,a} = \frac{\b{p}^2} {2 m_a} + \frac{\hbar \omega_a}{2} \sigma_z, +\end{equation} +\begin{equation} + \H_{1,fa} = - i \hbar \sum_l \left[ \sigma^+ g_l \a_l u_l(\b{r}) - \sigma^- g^*_l + \ad_l u^*_l(\b{r}) \right]. +\end{equation} +In the equations above, $\b{p}$ and $\b{r}$ are the momentum and +position operators of an atom of mass $m_a$ and resonance frequency +$\omega_a$. The operators $\sigma^+ = |g \rangle \langle e|$, +$\sigma^- = |e \rangle \langle g|$, and +$\sigma_z = |e \rangle \langle e| - |g \rangle \langle g|$ are the +atomic raising, lowering and population difference operators, where +$|g \rangle$ and $| e \rangle$ denote the ground and excited states of +the two-level atom respectively. $g_l$ are the atom-light coupling +constants for each mode. It is the inclusion of the interaction of the +boson with quantized light that distinguishes our work from the +typical approach to ultracold atoms where all the optical fields, +including the trapping potentials, are treated classically. + +We will now simplify the single-particle Hamiltonian by adiabatically +eliminating the upper excited level of the atom. The equations of +motion for the time evolution of operator $\hat{A}$ in the Heisenberg +picture are given by +\begin{equation} + \dot{\hat{A}} = \frac{i}{\hbar} \left[\H, \hat{A} \right]. +\end{equation} +Therefore, the Heisenberg equation for the lowering operator of a +single particle is +\begin{equation} + \dot{\sigma}^- = \frac{i}{\hbar} \left[\H_1, \hat{\sigma}^- \right] + = \hbar \omega_a \sigma^- + i \hbar \sum_l \sigma_z g_l \a_l u_l(\b{r}). +\end{equation} +We will consider nonresonant interactions between light and atoms +where the detunings between the light fields and the atomic resonance, +$\Delta_{la} = \omega_l - \omega_a$, are much larger than the +spontaneous emission rate and Rabi frequencies $g_l \a_l$. Therefore, +the atom will be predominantly found in the ground state and we can +set $\sigma_z = -1$ which is also known as the linear dipole +approximation as the dipoles respond linearly to the light amplitude +when the excited state has negligible population. Moreover, we can +adiabatically eliminate the polarization $\sigma^-$. Firstly we will +re-write its equation of motion in a frame rotating at $\omega_p$, the +external probe frequency, such that +$\sigma^- = \tilde{\sigma}^- \exp(i \omega_p t)$, and similarly for +$\tilde{\a}_l$. The resulting equation is given by +\begin{equation} + \dot{\tilde{\sigma}}^- = - \hbar \Delta_a \tilde{\sigma}^- - i \hbar + \sum_l g_l \tilde{\a}_l u_l(\b{r}), +\end{equation} +where $\Delta_a = \omega_p - \omega_a$ is the atom-probe +detuning. Within this rotating frame we will take +$\dot{\tilde{\sigma}}^- \approx 0$ and thus obtain the following +equation for the lowering operator +\begin{equation} + \sigma^- = - \frac{i}{\Delta_a} \sum_l g_l \a_l u_l(\b{r}). +\end{equation} +Therefore, by inserting this expression into the Heisenberg equation +for the light mode $m$ given by +\begin{equation} + \dot{\a}_m = - \sigma^- g^*_m u^*_m(\b{r}) +\end{equation} +we get the following equation of motion +\begin{equation} + \dot{\a}_m = \frac{i}{\Delta_a} \sum_l g_l g^*_m u_l(\b{r}) + u^*_m(\b{r}) \a_l. +\end{equation} +An effective Hamiltonian which results in the same optical equations +of motion can be written as +$\H^\mathrm{eff}_1 = \H_f + \H^\mathrm{eff}_{1,a} + +\H^\mathrm{eff}_{1,fa}$. The effective atomic and interaction +Hamiltonians are +\begin{equation} + \H^\mathrm{eff}_{1,a} = \frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}), +\end{equation} +\begin{equation} + \H^\mathrm{eff}_{1,fa} = \frac{\hbar}{\Delta_a} \sum_{l,m} + u_l^*(\b{r}) u_m(\b{r}) g_l g_m \ad_l \a_m, +\end{equation} +where we have explicitly extracted +$V_\mathrm{cl}(\b{r}) = \hbar g_\mathrm{cl}^2 |a_\mathrm{cl} +u_\mathrm{cl}(\b{r})|^2 / \Delta_{\mathrm{cl},a}$, the classical +trapping potential, from the interaction terms. However, we consider +the trapping beam to be sufficiently detuned from the other light +modes that we can neglect any scattering between them. However, a +later inclusion of this scattered light would not be difficult due to +the linearity of the dipoles we assumed. diff --git a/Preamble/preamble.tex b/Preamble/preamble.tex index 4b78b32..35701bb 100644 --- a/Preamble/preamble.tex +++ b/Preamble/preamble.tex @@ -200,3 +200,10 @@ \fi % Example todo: \mynote{Hey! I have a note} + +% ***************************** Shorthand operator notation ******************** + +\renewcommand{\H}{\hat{H}} +\newcommand{\ad}{a^\dagger} +\renewcommand{\a}{a} % in case we decide to put hats on +\renewcommand{\b}[1]{\mathbf{#1}} \ No newline at end of file diff --git a/thesis.tex b/thesis.tex index d34819e..2494023 100644 --- a/thesis.tex +++ b/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,draft]{Classes/PhDThesisPSnPDF} +\documentclass[a4paper,12pt,times,numbered,print,index,draft,chapter]{Classes/PhDThesisPSnPDF} % ****************************************************************************** % ******************************* Class Options ******************************** @@ -101,7 +101,7 @@ % To use choose `chapter' option in the document class \ifdefineChapter - \includeonly{Chapter3/chapter3} + \includeonly{Chapter2/chapter2} \fi % ******************************** Front Matter ********************************