diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index f305752..d3d8416 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -70,12 +70,12 @@ 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.} + \caption[Experimental Setup]{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} @@ -401,6 +401,38 @@ operators, we can easily have access to higher moments by simply simply considering higher moments of the $\a_l$ such as the photon number $\ad_l \a_l$. +\subsection{Simplifications for Two Modes and Linear Coupling} + +So far we have derived a general Hamiltonian for an arbitrary number +of light modes. However, most of the time we will be considering a +much simpler setup, namely the case with a coherent probe beam and a +single scattered mode as shown in Fig. \ref{fig:LatticeDiagram}. We +can use this fact to simplify our equations. Even though we consider +only one scattered mode, this treatment is also applicable to free +space scattering where the atoms can emit light in any direction. We +will simply neglect multiple scattering events, thus we can simply +vary the scattered mode's angle to obtain results for all possible +directions. + +We consider the two modes indexed with $l = 0$ and $1$, $\a_0$ and $\a_1$. We +will take $\a_0$ to be the external probe beam which is incident on the +quantum gas (it is not pumped into the cavity via $\eta_0$). +Furthermore, we will consider it to be in a coherent state. Therefore, +we can treat $a_0$ in our equations like a complex number instead of a +full operator. This leaves $\a_1$ as our scattered light mode. We can +now simplify the Hamiltonian to +\begin{align} + \H = & \hbar \left( \omega_1 + U_{1,1} \hat{F}_{1,1} \right) \ad_1 + \a_1 -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + + \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) \nonumber \\ & + + \hbar U_{0,0} |a_0|^2 \hat{F}_{0,0} + \hbar U_{0,1} \left[ a_0^* + \a_1 \hat{F}_{0,1} + \ad_1 a_0 \hat{F}_{1,0} \right] - i \kappa + \ad_1 \a_1, +\end{align} + +\subsection{Density and Phase Observables} + + \subsection{Electric Field Stength} The Electric field operator at position $\b{r}$ and at time $t$ is @@ -572,47 +604,47 @@ Estimates of the scattering rate using real experimental parameters are given in Table \ref{tab:photons}. \begin{table}[!htbp] - \begin{center} - \begin{tabular}{|c|c|c|} - \hline - Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline - $\omega_a$ & \multicolumn{2}{|c|}{$2 \pi \cdot 384$ THz}\\ \hline - $\Gamma$ & \multicolumn{2}{|c|}{$2 \pi \cdot 6.07$ MHz} \\ \hline - $\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ \hline - $I$ & $4250$ Wm$^{-2}$ & N/A \\ \hline - $\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ \hline - $N_K$ & 10$^5$ & 147 \\ \hline \hline - $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline - \end{tabular} - \end{center} - \caption{ Rubidium data taken from Ref. \cite{steck}; Miyake - \emph{et al.}: based on Ref. \cite{miyake2011}. The $5^2S_{1/2}$, - $F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of - $^{87}$Rb is considered. For this transition the Rabi frequency is - actually larger than the detuning and and effects of saturation - should be taken into account in a more complete analysis. However, - it is included for discussion. The detuning is said to be a few - natural linewidths. Note that it is much smaller than - $\Omega_0$. The Rabi fequency is $\Omega_0 = - (d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is obtained - from definition of Rabi frequency, $\Omega = \mathbf{d} \cdot - \mathbf{E} / \hbar$, assuming the electric field is parallel to - the dipole, and the relation $I = c \epsilon_0 |E|^2 /2$. We used - $d_\mathrm{eff} = 1.73 \times 10^{-29}$ Cm \cite{steck}; - Weitenberg \emph{et al.}: based on Ref. \cite{weitenberg2011, - weitenbergThesis}. The $5^2S_{1/2} \rightarrow 5^2P_{3/2}$ - transition of $^{87}$Rb is used. Ref. \cite{weitenberg2011} gives - the free space detuning to be $\Delta_\mathrm{free} = - 2 \pi - \cdot 45$ MHz. Ref. \cite{weitenbergThesis} clarifies that the - relevant detuning is $\Delta = \Delta_\mathrm{free} + - \Delta_\mathrm{lat}$, where $\Delta_\mathrm{lat} = - 2 \pi \cdot - 40$ MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter + \centering + \begin{tabular}{|c|c|c|} + \hline + Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline + $\omega_a$ & \multicolumn{2}{|c|}{$2 \pi \cdot 384$ THz}\\ \hline + $\Gamma$ & \multicolumn{2}{|c|}{$2 \pi \cdot 6.07$ MHz} \\ \hline + $\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ \hline + $I$ & $4250$ Wm$^{-2}$ & N/A \\ \hline + $\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ \hline + $N_K$ & 10$^5$ & 147 \\ \hline \hline + $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline + \end{tabular} + \caption[Photon Scattering Rates]{ Rubidium data taken from + Ref. \cite{steck}; Miyake \emph{et al.}: based on + Ref. \cite{miyake2011}. The $5^2S_{1/2}$, $F=2 \rightarrow + 5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb is + considered. For this transition the Rabi frequency is actually + larger than the detuning and and effects of saturation should be + taken into account in a more complete analysis. However, it is + included for discussion. The detuning is said to be a few natural + linewidths. Note that it is much smaller than $\Omega_0$. The Rabi + fequency is $\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c + \epsilon_0}$ which is obtained from definition of Rabi + frequency, $\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$, + assuming the electric field is parallel to the dipole, and the + relation $I = c \epsilon_0 |E|^2 /2$. We used $d_\mathrm{eff} = + 1.73 \times 10^{-29}$ Cm \cite{steck}; Weitenberg \emph{et al.}: + based on Ref. \cite{weitenberg2011, weitenbergThesis}. The + $5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is + used. Ref. \cite{weitenberg2011} gives the free space detuning to + be $\Delta_\mathrm{free} = - 2 \pi \cdot 45$ + MHz. Ref. \cite{weitenbergThesis} clarifies that the relevant + detuning is $\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$, + where $\Delta_\mathrm{lat} = - 2 \pi \cdot 40$ + MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter $s_\mathrm{tot}$ to quantify the intensity of the beams which is related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 / \Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for the experiment from the total scattering rate by $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) / (1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 - kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = - 2.5$.\label{tab:photons}} + kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.} + \label{tab:photons} \end{table} diff --git a/Chapter3/chapter3.tex b/Chapter3/chapter3.tex new file mode 100644 index 0000000..42b2a5e --- /dev/null +++ b/Chapter3/chapter3.tex @@ -0,0 +1,15 @@ +%******************************************************************************* +%*********************************** Third Chapter ***************************** +%******************************************************************************* + +\chapter{Probing Matter-Field and Atom-Number Correlations in Optical Lattices + by Global Nondestructive Addressing} %Title of the Third Chapter + +\ifpdf + \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}} +\else + \graphicspath{{Chapter3/Figs/Vector/}{Chapter3/Figs/}} +\fi + + +%********************************** %First Section ************************************** diff --git a/thesis.tex b/thesis.tex index 2494023..6648dc4 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,chapter]{Classes/PhDThesisPSnPDF} +\documentclass[a4paper,12pt,times,numbered,print,index,draft]{Classes/PhDThesisPSnPDF} % ****************************************************************************** % ******************************* Class Options ******************************** @@ -101,7 +101,7 @@ % To use choose `chapter' option in the document class \ifdefineChapter - \includeonly{Chapter2/chapter2} + \includeonly{Chapter3/chapter3} \fi % ******************************** Front Matter ******************************** @@ -137,7 +137,7 @@ \include{Chapter1/chapter1} \include{Chapter2/chapter2} -%\include{Chapter3/chapter3} +\include{Chapter3/chapter3} %\include{Chapter4/chapter4} %\include{Chapter5/chapter5} %\include{Chapter6/chapter6}