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Wojciech Kozlowski 2016-07-11 23:47:13 +01:00
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118
Chapter2/chapter2.tex
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@ -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}

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Chapter3/chapter3.tex Normal file
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@ -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 **************************************

6
thesis.tex
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@ -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}