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Wojciech Kozlowski
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@ -70,12 +70,12 @@ quantum potential in contrast to the classical lattice trap.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=1.0\textwidth]{LatticeDiagram}
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\caption[LatticeDiagram]{Atoms (green) trapped in an optical lattice
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are illuminated by a coherent probe beam (red). The light scatters
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(blue) in free space or into a cavity and is measured by a
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detector. If the experiment is in free space light can scatter in
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any direction. A cavity on the other hand enhances scattering in
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one particular direction.}
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\caption[Experimental Setup]{Atoms (green) trapped in an optical
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lattice are illuminated by a coherent probe beam (red). The light
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scatters (blue) in free space or into a cavity and is measured by
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a detector. If the experiment is in free space light can scatter
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in any direction. A cavity on the other hand enhances scattering
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in one particular direction.}
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\label{fig:LatticeDiagram}
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\end{figure}
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@ -401,6 +401,38 @@ operators, we can easily have access to higher moments by simply
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simply considering higher moments of the $\a_l$ such as the photon
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number $\ad_l \a_l$.
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\subsection{Simplifications for Two Modes and Linear Coupling}
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So far we have derived a general Hamiltonian for an arbitrary number
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of light modes. However, most of the time we will be considering a
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much simpler setup, namely the case with a coherent probe beam and a
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single scattered mode as shown in Fig. \ref{fig:LatticeDiagram}. We
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can use this fact to simplify our equations. Even though we consider
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only one scattered mode, this treatment is also applicable to free
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space scattering where the atoms can emit light in any direction. We
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will simply neglect multiple scattering events, thus we can simply
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vary the scattered mode's angle to obtain results for all possible
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directions.
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We consider the two modes indexed with $l = 0$ and $1$, $\a_0$ and $\a_1$. We
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will take $\a_0$ to be the external probe beam which is incident on the
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quantum gas (it is not pumped into the cavity via $\eta_0$).
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Furthermore, we will consider it to be in a coherent state. Therefore,
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we can treat $a_0$ in our equations like a complex number instead of a
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full operator. This leaves $\a_1$ as our scattered light mode. We can
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now simplify the Hamiltonian to
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\begin{align}
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\H = & \hbar \left( \omega_1 + U_{1,1} \hat{F}_{1,1} \right) \ad_1
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\a_1 -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j +
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\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) \nonumber \\ &
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+ \hbar U_{0,0} |a_0|^2 \hat{F}_{0,0} + \hbar U_{0,1} \left[ a_0^*
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\a_1 \hat{F}_{0,1} + \ad_1 a_0 \hat{F}_{1,0} \right] - i \kappa
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\ad_1 \a_1,
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\end{align}
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\subsection{Density and Phase Observables}
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\subsection{Electric Field Stength}
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The Electric field operator at position $\b{r}$ and at time $t$ is
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@ -572,7 +604,7 @@ Estimates of the scattering rate using real experimental parameters
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are given in Table \ref{tab:photons}.
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\begin{table}[!htbp]
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\begin{center}
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\centering
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\begin{tabular}{|c|c|c|}
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\hline
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Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline
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@ -584,35 +616,35 @@ are given in Table \ref{tab:photons}.
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$N_K$ & 10$^5$ & 147 \\ \hline \hline
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$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline
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\end{tabular}
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\end{center}
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\caption{ Rubidium data taken from Ref. \cite{steck}; Miyake
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\emph{et al.}: based on Ref. \cite{miyake2011}. The $5^2S_{1/2}$,
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$F=2 \rightarrow 5^2P_{3/2}$, $F^\prime = 3$ transition of
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$^{87}$Rb is considered. For this transition the Rabi frequency is
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actually larger than the detuning and and effects of saturation
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should be taken into account in a more complete analysis. However,
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it is included for discussion. The detuning is said to be a few
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natural linewidths. Note that it is much smaller than
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$\Omega_0$. The Rabi fequency is $\Omega_0 =
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(d_\mathrm{eff}/\hbar)\sqrt{2 I / c \epsilon_0}$ which is obtained
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from definition of Rabi frequency, $\Omega = \mathbf{d} \cdot
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\mathbf{E} / \hbar$, assuming the electric field is parallel to
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the dipole, and the relation $I = c \epsilon_0 |E|^2 /2$. We used
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$d_\mathrm{eff} = 1.73 \times 10^{-29}$ Cm \cite{steck};
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Weitenberg \emph{et al.}: based on Ref. \cite{weitenberg2011,
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weitenbergThesis}. The $5^2S_{1/2} \rightarrow 5^2P_{3/2}$
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transition of $^{87}$Rb is used. Ref. \cite{weitenberg2011} gives
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the free space detuning to be $\Delta_\mathrm{free} = - 2 \pi
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\cdot 45$ MHz. Ref. \cite{weitenbergThesis} clarifies that the
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relevant detuning is $\Delta = \Delta_\mathrm{free} +
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\Delta_\mathrm{lat}$, where $\Delta_\mathrm{lat} = - 2 \pi \cdot
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40$ MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
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\caption[Photon Scattering Rates]{ Rubidium data taken from
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Ref. \cite{steck}; Miyake \emph{et al.}: based on
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Ref. \cite{miyake2011}. The $5^2S_{1/2}$, $F=2 \rightarrow
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5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb is
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considered. For this transition the Rabi frequency is actually
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larger than the detuning and and effects of saturation should be
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taken into account in a more complete analysis. However, it is
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included for discussion. The detuning is said to be a few natural
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linewidths. Note that it is much smaller than $\Omega_0$. The Rabi
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fequency is $\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c
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\epsilon_0}$ which is obtained from definition of Rabi
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frequency, $\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$,
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assuming the electric field is parallel to the dipole, and the
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relation $I = c \epsilon_0 |E|^2 /2$. We used $d_\mathrm{eff} =
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1.73 \times 10^{-29}$ Cm \cite{steck}; Weitenberg \emph{et al.}:
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based on Ref. \cite{weitenberg2011, weitenbergThesis}. The
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$5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is
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used. Ref. \cite{weitenberg2011} gives the free space detuning to
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be $\Delta_\mathrm{free} = - 2 \pi \cdot 45$
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MHz. Ref. \cite{weitenbergThesis} clarifies that the relevant
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detuning is $\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$,
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where $\Delta_\mathrm{lat} = - 2 \pi \cdot 40$
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MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
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$s_\mathrm{tot}$ to quantify the intensity of the beams which is
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related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 /
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\Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for
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the experiment from the total scattering rate by
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$\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
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(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60
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kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} =
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2.5$.\label{tab:photons}}
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kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.}
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\label{tab:photons}
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\end{table}
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15
Chapter3/chapter3.tex
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15
Chapter3/chapter3.tex
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@ -0,0 +1,15 @@
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%*******************************************************************************
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%*********************************** Third Chapter *****************************
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%*******************************************************************************
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\chapter{Probing Matter-Field and Atom-Number Correlations in Optical Lattices
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by Global Nondestructive Addressing} %Title of the Third Chapter
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\ifpdf
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\graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}}
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\else
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\graphicspath{{Chapter3/Figs/Vector/}{Chapter3/Figs/}}
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\fi
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%********************************** %First Section **************************************
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@ -1,7 +1,7 @@
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% ******************************* PhD Thesis Template **************************
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% Please have a look at the README.md file for info on how to use the template
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\documentclass[a4paper,12pt,times,numbered,print,index,draft,chapter]{Classes/PhDThesisPSnPDF}
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\documentclass[a4paper,12pt,times,numbered,print,index,draft]{Classes/PhDThesisPSnPDF}
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% ******************************************************************************
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% ******************************* Class Options ********************************
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@ -101,7 +101,7 @@
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% To use choose `chapter' option in the document class
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\ifdefineChapter
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\includeonly{Chapter2/chapter2}
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\includeonly{Chapter3/chapter3}
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\fi
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% ******************************** Front Matter ********************************
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@ -137,7 +137,7 @@
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\include{Chapter1/chapter1}
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\include{Chapter2/chapter2}
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%\include{Chapter3/chapter3}
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\include{Chapter3/chapter3}
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%\include{Chapter4/chapter4}
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%\include{Chapter5/chapter5}
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%\include{Chapter6/chapter6}
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