Finished for the day

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Wojciech Kozlowski 2016-07-11 23:47:13 +01:00
parent 85bc2f1d0e
commit 3719e20e6a
3 changed files with 93 additions and 46 deletions

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@ -70,12 +70,12 @@ quantum potential in contrast to the classical lattice trap.
\begin{figure}[htbp!] \begin{figure}[htbp!]
\centering \centering
\includegraphics[width=1.0\textwidth]{LatticeDiagram} \includegraphics[width=1.0\textwidth]{LatticeDiagram}
\caption[LatticeDiagram]{Atoms (green) trapped in an optical lattice \caption[Experimental Setup]{Atoms (green) trapped in an optical
are illuminated by a coherent probe beam (red). The light scatters lattice are illuminated by a coherent probe beam (red). The light
(blue) in free space or into a cavity and is measured by a scatters (blue) in free space or into a cavity and is measured by
detector. If the experiment is in free space light can scatter in a detector. If the experiment is in free space light can scatter
any direction. A cavity on the other hand enhances scattering in in any direction. A cavity on the other hand enhances scattering
one particular direction.} in one particular direction.}
\label{fig:LatticeDiagram} \label{fig:LatticeDiagram}
\end{figure} \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 simply considering higher moments of the $\a_l$ such as the photon
number $\ad_l \a_l$. 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} \subsection{Electric Field Stength}
The Electric field operator at position $\b{r}$ and at time $t$ is The Electric field operator at position $\b{r}$ and at time $t$ is
@ -572,7 +604,7 @@ Estimates of the scattering rate using real experimental parameters
are given in Table \ref{tab:photons}. are given in Table \ref{tab:photons}.
\begin{table}[!htbp] \begin{table}[!htbp]
\begin{center} \centering
\begin{tabular}{|c|c|c|} \begin{tabular}{|c|c|c|}
\hline \hline
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline
@ -584,35 +616,35 @@ are given in Table \ref{tab:photons}.
$N_K$ & 10$^5$ & 147 \\ \hline \hline $N_K$ & 10$^5$ & 147 \\ \hline \hline
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline $n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline
\end{tabular} \end{tabular}
\end{center} \caption[Photon Scattering Rates]{ Rubidium data taken from
\caption{ Rubidium data taken from Ref. \cite{steck}; Miyake Ref. \cite{steck}; Miyake \emph{et al.}: based on
\emph{et al.}: based on Ref. \cite{miyake2011}. The $5^2S_{1/2}$, Ref. \cite{miyake2011}. The $5^2S_{1/2}$, $F=2 \rightarrow
$F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of 5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb is
$^{87}$Rb is considered. For this transition the Rabi frequency is considered. For this transition the Rabi frequency is actually
actually larger than the detuning and and effects of saturation larger than the detuning and and effects of saturation should be
should be taken into account in a more complete analysis. However, taken into account in a more complete analysis. However, it is
it is included for discussion. The detuning is said to be a few included for discussion. The detuning is said to be a few natural
natural linewidths. Note that it is much smaller than linewidths. Note that it is much smaller than $\Omega_0$. The Rabi
$\Omega_0$. The Rabi fequency is $\Omega_0 = fequency is $\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c
(d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is obtained \epsilon_0}$ which is obtained from definition of Rabi
from definition of Rabi frequency, $\Omega = \mathbf{d} \cdot frequency, $\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$,
\mathbf{E} / \hbar$, assuming the electric field is parallel to assuming the electric field is parallel to the dipole, and the
the dipole, and the relation $I = c \epsilon_0 |E|^2 /2$. We used relation $I = c \epsilon_0 |E|^2 /2$. We used $d_\mathrm{eff} =
$d_\mathrm{eff} = 1.73 \times 10^{-29}$ Cm \cite{steck}; 1.73 \times 10^{-29}$ Cm \cite{steck}; Weitenberg \emph{et al.}:
Weitenberg \emph{et al.}: based on Ref. \cite{weitenberg2011, based on Ref. \cite{weitenberg2011, weitenbergThesis}. The
weitenbergThesis}. The $5^2S_{1/2} \rightarrow 5^2P_{3/2}$ $5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is
transition of $^{87}$Rb is used. Ref. \cite{weitenberg2011} gives used. Ref. \cite{weitenberg2011} gives the free space detuning to
the free space detuning to be $\Delta_\mathrm{free} = - 2 \pi be $\Delta_\mathrm{free} = - 2 \pi \cdot 45$
\cdot 45$ MHz. Ref. \cite{weitenbergThesis} clarifies that the MHz. Ref. \cite{weitenbergThesis} clarifies that the relevant
relevant detuning is $\Delta = \Delta_\mathrm{free} + detuning is $\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$,
\Delta_\mathrm{lat}$, where $\Delta_\mathrm{lat} = - 2 \pi \cdot where $\Delta_\mathrm{lat} = - 2 \pi \cdot 40$
40$ MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
$s_\mathrm{tot}$ to quantify the intensity of the beams which is $s_\mathrm{tot}$ to quantify the intensity of the beams which is
related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 / related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 /
\Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for \Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for
the experiment from the total scattering rate by the experiment from the total scattering rate by
$\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) / $\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60 (1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60
kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.}
2.5$.\label{tab:photons}} \label{tab:photons}
\end{table} \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 **************************************

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@ -1,7 +1,7 @@
% ******************************* PhD Thesis Template ************************** % ******************************* PhD Thesis Template **************************
% Please have a look at the README.md file for info on how to use the 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 ******************************** % ******************************* Class Options ********************************
@ -101,7 +101,7 @@
% To use choose `chapter' option in the document class % To use choose `chapter' option in the document class
\ifdefineChapter \ifdefineChapter
\includeonly{Chapter2/chapter2} \includeonly{Chapter3/chapter3}
\fi \fi
% ******************************** Front Matter ******************************** % ******************************** Front Matter ********************************
@ -137,7 +137,7 @@
\include{Chapter1/chapter1} \include{Chapter1/chapter1}
\include{Chapter2/chapter2} \include{Chapter2/chapter2}
%\include{Chapter3/chapter3} \include{Chapter3/chapter3}
%\include{Chapter4/chapter4} %\include{Chapter4/chapter4}
%\include{Chapter5/chapter5} %\include{Chapter5/chapter5}
%\include{Chapter6/chapter6} %\include{Chapter6/chapter6}