From 85bc2f1d0e0810905db56629dbfc261fba24723a Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Fri, 8 Jul 2016 19:49:03 +0100 Subject: [PATCH] Finished section on electric field strength --- Chapter2/chapter2.tex | 53 +++++++++++++++++++++++++++++++++++++-- References/references.bib | 37 +++++++++++++++++++++++++++ 2 files changed, 88 insertions(+), 2 deletions(-) diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index 1f69375..f305752 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -450,7 +450,8 @@ between the two levels, and $c$ is the speed of light in vacuum. We have already derived an expression for the atomic lowering operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by \begin{equation} - \sigma^- = - \frac{i} {\Delta_a} \sum_\b{k} g_\b{k} a_\b{k} u_\b{k}(\b{r}). + \sigma^-(\b{r}, t) = - \frac{i} {\Delta_a} \sum_\b{k} g_\b{k} + a_\b{k}(t) u_\b{k}(\b{r}). \end{equation} We will want to substitute this expression into Eq. \eqref{eq:Ep}, but first we note that in the setup we consider the coherent probe will be @@ -458,7 +459,7 @@ much stronger than the scattered modes. This in turn implies that the expression for $\sigma^-$ will be dominated by this external probe. Therefore, we drop the sum and consider only a single wave vector, $\b{k}_0$, of the dominant contribution from the external -beam. +beam corresponding to the mode $\a_0$. We now use the definititon of the atom-light coupling constant \cite{Scully} to make the substitution @@ -567,3 +568,51 @@ Therefore, we can now express the quantity $n_{\Phi}$ as \begin{equation} n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K. \end{equation} +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 + $s_\mathrm{tot}$ 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}} +\end{table} diff --git a/References/references.bib b/References/references.bib index c93116f..ad6832a 100644 --- a/References/references.bib +++ b/References/references.bib @@ -1,3 +1,40 @@ +@book{foot, + author = {Foot, C. J.}, + title = {{Atomic Physics}}, + publisher = {Oxford University Press}, + year = {2005} +} +@phdthesis{weitenbergThesis, +author = {Weitenberg, Christof}, +number = {April}, +title = {{Single-Atom Resolved Imaging and Manipulation in an Atomic Mott Insulator}}, +year = {2011} +} +@article{steck, +author = {Steck, Daniel Adam}, +title = {{Rubidium 87 D Line Data Author contact information :}}, +url = {http://steck.us/alkalidata} +} +@article{weitenberg2011, + title={Coherent light scattering from a two-dimensional Mott insulator}, + author={Weitenberg, Christof and Schau{\ss}, Peter and Fukuhara, Takeshi and Cheneau, Marc and Endres, Manuel and Bloch, Immanuel and Kuhr, Stefan}, + journal={Physical review letters}, + volume={106}, + number={21}, + pages={215301}, + year={2011}, + publisher={APS} +} +@article{miyake2011, + title={Bragg scattering as a probe of atomic wave functions and quantum phase transitions in optical lattices}, + author={Miyake, Hirokazu and Siviloglou, Georgios A and Puentes, Graciana and Pritchard, David E and Ketterle, Wolfgang and Weld, David M}, + journal={Physical review letters}, + volume={107}, + number={17}, + pages={175302}, + year={2011}, + publisher={APS} +} @article{mekhov2012, title={Quantum optics with ultracold quantum gases: towards the full quantum regime of the light--matter interaction}, author={Mekhov, Igor B and Ritsch, Helmut},