%\documentclass[a0,portrait]{a0poster}

% May 2007 version 

\documentclass[a0,portrait,amstex,amssymb,aps,graphics,floats]{a0poster}
\usepackage{pstricks,multicol,pst-grad,color,graphicx}

%%% 28 March

%\usepackage{graphics}
%\usepackage{thmdefs}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{colordvi}
\usepackage{epsfig}
\usepackage{psfig}
%\usepackage{alltt}
\usepackage{color}
\usepackage{times}
\usepackage{UQ_poster_2}
%\usepackage{floatfig}
\usepackage{wrapfig}
%\newcommand{\nocolor}{
%\definecolor{PgBkG}{rgb}{1,1,1}
%\definecolor{PgBkG}{rgb}{1,1,1}
%\definecolor{Txt}{rgb}{0,0,1}
%\definecolor{Sec}{rgb}{0,0,0}
%\definecolor{Emph}{cmyk}{0,0,0,1}}

\definecolor{PgBkG}{rgb}{1,1,0.8}
\definecolor{PgBkG}{cmyk}{0,0,0.15,0}

%\definecolor{PgBkG}{rgb}{1,1,0.85}
%\definecolor{PgBkG}{rgb}{0,.35,0}

\definecolor{Eqn}{cmyk}{1,1,0,.4}

\definecolor{Txt}{cmyk}{1,1,0,.3}
%\definecolor{Txt}{cmyk}{1,1,0,.6}
%\definecolor{Txt}{rgb}{.95,.95,.75}

\definecolor{Sec}{rgb}{0.8,0,0.0}
%\definecolor{Sec}{cmyk}{0,1,.4,.1}

\definecolor{Emph}{rgb}{0.7,0.1,0}
%\definecolor{Emph}{cmyk}{.8,0,.8,.4}


\definecolor{bckgrnd}{rgb}{1,1,0.8}
%\definecolor{bckgrnd}{cmyk}{0.3,0.2,0.9,0.01}
%\definecolor{bckgrnd}{cmyk}{0,0,0,0}

\definecolor{pink}{cmyk}{0,1,0,0}
%\definecolor{red}{rgb}{1,0,0}
%\definecolor{blue}{rgb}{0,0,1}
%\definecolor{green}{rgb}{0,1,0}
\definecolor{gray}{cmyk}{0,0,0,.75}
\definecolor{orange}{cmyk}{0,.5,1,.3}

\definecolor{blue}{cmyk}{0.5,0.5,0.0,0.5}
\definecolor{red}{cmyk}{0.1,0.9,0.9,0.15}
%\definecolor{red}{rgb}{0.7,0.1,0}
\definecolor{purple}{cmyk}{0.2,0.9,0.1,0.2}
\definecolor{green}{rgb}{0.0,0.6,0.0}


\newcommand{\mnote}[1]{\marginpar{\raggedright{\Em #1}}}
\newcommand{\fg}[2]{\message{#2}\par\noindent\vbox to #1 cm
{\vfill\special{epsf=#2}}\par} \newdimen\boxsize

\newcommand{\BOX}[1]{\boxsize=\hsize\advance\boxsize by -17.8pt\vskip 5mm
\framebox{\parbox{\boxsize}{#1}}\vskip 5mm}

%\newcommand{\BOX}[1]{\boxsize=\hsize\advance\boxsize by -6.8pt\vskip 5mm
%\framebox{\parbox{\boxsize}{#1}}\vskip 5mm}

% And here are the colours that I am going to use
\newcommand{\Section}[1]{{\color{Sec}\sf\section*{#1}}}
%\newcommand{\Section}[1]{{\color{green}\sf\section*{#1}}}
\newcommand{\Subsection}[1]{{\color{Sec}\subsection*{#1}}}
\newcommand{\Subsubsection}[1]{{\color{Sec}\subsubsection*{#1}}}
\newenvironment{Figure}{\vskip 2pt\noindent}{\vskip 2pt\noindent}
\newcommand{\Caption}[1]{\par\noindent{\small #1} \par}
\newcommand{\Em}{\em\color{Emph}}




\newcommand{\chd } {\color{Emph}}

\AtBeginDocument{
  \renewcommand{\labelitemi}{{\chd $\bullet$~}}
}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}
\sf



\title{{\color{Sec}{\textbf{\hspace{9cm}FINITE-SIZE~~EFFECTS~~IN~~ATOM-ATOM\newline
CORRELATIONS~~IN~~MOLECULAR~~DISSOCIATION}}}}
\author{Magnus~\"{O}gren and Kar\'en~V.~Kheruntsyan} % \underline{Magnus~\"{O}gren}


\makeheader

\vspace{4cm}
%% Column 1
\begin{center}
\col{
\vspace{-2.3cm}
\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{\sf {\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~~~~~~~~Motivation{\color{yellow}g}~~~~~~~~~~~}}}}}}

\vspace{-1.0cm}


Dissociation of a BEC of molecular dimers can produce pair-correlated atomic ensembles, with either fermionic or bosonic atom statistics, experiment e.g. at JILA \cite{JILA}.

% Through dissociation of a bosonic molecule (dimer), two atoms, obeying
% \begin{center}
% \begin{minipage}[l]{.47\textwidth}
% 
%   %\normalsize
% 
%   %\emph{Figure 1:}
% %\vspace{-1.0cm}
% either bosonic or fermionic statistics, can be
% created \cite{JILA}.
% \end{minipage}
% %\hspace{5mm}
% \begin{minipage}[r]{.47\textwidth}
%     \begin{center}
% 
% 	\includegraphics[clip=true,angle=0,width=1.1\textwidth]{bwDiagram3noa.eps} % Diagram3.jpg}%Diagram2c.eps}%
% 
% %\vspace{1cm}
% 
% %      \includegraphics[clip=true,angle=0,width=1.0\textwidth]{Jin_circle.eps}
%     \end{center}
% \end{minipage}
% \end{center}


%Weather the dimer or the atoms correspond to the lowest
%energy can be controlled e.g. with a Feschbach resonance, resulting in
%a kinetic energy $\hbar\Delta$ for the atoms after a swep. 
\vspace{0.25cm}
\vspace{0.25cm}

\\ {\bf Questions:}

\vspace{0.05cm}

\labelitemi How does the spatial inhomogeneity of the molecular BEC affects the strength of atom-atom correlations?

\vspace{0.5cm}

\labelitemi What is the correlation width?

\vspace{0.25cm}

\v

\vspace{1.25cm}


\begin{center}
\begin{minipage}[l]{.49\textwidth}

  %\normalsize

  %\emph{Figure 1:}
Due to
momentum conservation the two atoms are of equal but opposite momenta,
$-\mathbf{k}$ and $\mathbf{k}$.
\end{minipage}
\hspace{5mm}
\begin{minipage}[r]{.45\textwidth}
    \begin{center}

%	\includegraphics[clip=true,angle=0,width=1.0\textwidth]{bwDiagram3noa.eps}

%\vspace{1cm}

      \includegraphics[clip=true,angle=0,width=1.0\textwidth]{noise07fig1.eps}
    \end{center}
\end{minipage}
\end{center}


% We study how a
% non-uniform initial molecular field, change the fermionic atom-atom
% correlation properties, compared to a uniform system.
\vspace{0.75cm}

We compare our results for non-uniform initial molecular BEC with those obtained in idealised uniform systems.





\end{pbox}

\vspace{-0.9cm}
\vspace{-1.0cm}

\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~Effective Quantum Field~\hspace{0.42cm}}}}}}}
\vspace{-3.7cm} %3.8
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~~~~~Theory Model~~~~~~~~~~~}}}}}}

\vspace{-1.0cm}

The coupled molecular ($ \hat{\Psi}_0$) and atomic ($ \hat{\Psi}_{\sigma}$) fields can be described by
\vspace{0.25cm}
\begin{equation*}
\hat{H}=\hat{H}_0-i\hbar \chi\int d\mathbf{x}
\left(\hat{\Psi}^{\dagger}_0 \hat{\Psi}_{\uparrow}
\hat{\Psi}_{\downarrow} - \hat{\Psi}^{\dagger}_{\downarrow}
\hat{\Psi}^{\dagger}_{\uparrow} \hat{\Psi}_0 \right),
\end{equation*}
\vspace{0.5cm}
where $\sigma=\uparrow,\downarrow$ denotes different spin states (in the bosonic case
one can also have dissociation into only one atomic state).

% \vspace{0.25cm}
% 
% \labelitemi Single mode approx.: $\langle
% \hat{\Psi}_0 \rangle=\beta\left(t\right)$
% 
% \labelitemi Undep. pump a. (UPA): $\langle
% \hat{\Psi}_0 \rangle=\beta_0$
% 
% \vspace{0.25cm}
% 
% \v


Within the undepleted molecule approximation, $\langle \hat{\Psi}_0\left( \mathbf{x},t \right) \rangle \rightarrow \sqrt{n_0\left( \mathbf{x} \right)}$, the Heisenberg equations for creation-/annihilation operators of plane
wave modes for the ($-/+$ fermionic-/bosonic-) atoms are
\vspace{0.25cm}
\begin{equation*}
\frac{ \partial \hat{a}_{\mathbf{k},\uparrow} }{\partial t}=-i\Delta_{\mathbf{k}} 
\hat{a}_{\mathbf{k},\uparrow} \mp \frac{1}{L^{D/3}} \sum_{\mathbf{k}'}
G_{\mathbf{k}'+\mathbf{k}} \hat{a}^{\dagger}_{\mathbf{k}',\downarrow},
\end{equation*}

\vspace{-0.5cm}

\begin{equation*}
\frac{ \partial \hat{a}^{\dagger}_{\mathbf{k},\downarrow} }{\partial t}=i\Delta_{\mathbf{k}} 
\hat{a}^{\dagger}_{\mathbf{k},\downarrow} + \frac{1}{L^{D/3}} \sum_{\mathbf{k}'}
G_{\mathbf{k}'-\mathbf{k}} \hat{a}_{-\mathbf{k}',\uparrow},
\end{equation*}
\vspace{0.25cm}
where $\Delta_{\mathbf{k}}=\Delta+\hbar\mathbf{k}^2/\left( 2m_{at}\right)$, $L$ is the size and $D$ the dimension of the system.
Here $G_{\mathbf{k}}$ is the fourier transform of $\chi\sqrt{n_0\left( \mathbf{x} \right)}$. 


\end{pbox}
\vspace{-0.9cm}
\vspace{-1.0cm}




\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{\sf {\huge {\textbf {\center \hspace{-1.5cm}
\colorbox{yellow}{~~Uniform Molecular Field~~}}}}}

\vspace{-1.0cm}

The case where $G_{\mathbf{k}}=G_0,~else~0$ can be solved analytically for bosonic/fermionic atoms, within the
undepleted molecule approximation \cite{Karen}. See the dashed black lines in the following four figures. 

%This gives limits for the maximal correlation
%properties in those systems and should be compared with any %numerical solutions.

\end{pbox}

\vspace{-3cm}

}
%-------------- end of first column
\col{
\vspace{-2.3cm}


\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{\sf {\huge {\textbf {\center \hspace{-1.5cm}
\colorbox{yellow}{{\color{yellow}g}~~~~~~~~~~Non-uniform~~~~~~~~~~~}}}}}
\vspace{-3.45cm}
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~~~Molecular Field~~~~~~~~~~}}}}}}

\vspace{-1.0cm}

For bosons, first principal calculations (+P) have been carried
out \cite{Craig}. For fermions we have done calculations (systems of ODE:s) both for the undepleted molecular approximation and for a time-dependant
single mode (uniform case).


{\par\centering\resizebox*{15.5cm}{!}{\includegraphics{noise07fig2.eps}}\par}

\end{pbox}
\vspace{-0.9cm}
\vspace{-1cm}

\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~Mode occupancies and~\hspace{0.7cm}}}}}}}
\vspace{-3.45cm}
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~atom populations~~~~~~~~}}}}}}

\vspace{-1.0cm}

We simulate occupancy dynamic in momentum space: % ($t=1.1~ms$)

\begin{center}
	\begin{minipage}[l]{.49\textwidth}
		\hspace{-1.5cm}
		\includegraphics[clip=true,angle=0,width=1.12\textwidth]{noise07fig3.eps}
	\end{minipage}
\hspace{-0.7cm}
	\begin{minipage}[r]{.49\textwidth}
    		%\begin{center}
      			\includegraphics[clip=true,angle=0,width=1.12\textwidth]{noise07fig4.eps}
    		%\end{center}
	\end{minipage}
\end{center}




%{\par\centering\resizebox*{7cm}{!}{\includegraphics{noise07fig5.eps}}\par}


The number of atoms are obtained by summing up the mode occupancies:
%$N_{\sigma} \left( t \right) = \sum_{\mathbf{k}} n_{\mathbf{k},\sigma} \left( t \right)$.

%\begin{equation*}
%N_{\sigma} \left( t \right) = \sum_{\mathbf{k}} %n_{\mathbf{k},\sigma} \left( t \right).
%\end{equation*}


{\par\centering\resizebox*{16cm}{!}{\includegraphics{noise07fig5.eps}}\par}


\end{pbox}

\vspace{-0.9cm}
\vspace{-1.0cm}

\begin{pbox}[1 1 .8]
\vspace{-1.4cm}
\Section{{\huge {\textbf{\center{\hspace{-1.5cm}
\colorbox{yellow}{~~~~Correlation Functions~~\hspace{0.73cm}}}}}}}

\vspace{-1.0cm}

To study correlations in momentum space the following definition is made:
\begin{equation*}
g_{\uparrow,\downarrow} \left( \mathbf{k},\mathbf{k}' ,t \right)
=\frac{ \langle \Delta  \hat{n}_{\mathbf{k},\uparrow}  \Delta
  \hat{n}_{\mathbf{k}',\downarrow} \rangle }{\sqrt{ \langle
    \hat{n}_{\mathbf{k},\uparrow} \rangle  \langle  \hat{n}_{\mathbf{k}',\downarrow}  \rangle } }.
\end{equation*}

Again here the uniform case can be done analytically within the undepleted molecular approximation
$g_{\uparrow,\downarrow} \left( \mathbf{k},\mathbf{k}'=-\mathbf{k} ,t
\right)=1\mp n_{\mathbf{k}} \left( t \right)$ ($-/+$ fermions/bosons), it has zero width i.e. $g_{\uparrow,\downarrow} \left( \mathbf{k},\mathbf{k}\neq -\mathbf{k}' ,t
\right)=0$.


For a non-uniform molecular field, several momentum states start to couple, resulting in a finite width for the correlation function.

\vspace{0.3cm}

\begin{center}
\begin{minipage}[r]{.5\textwidth}
  \normalsize
At JILA one have studied similar correlations from a cigar-shaped $^{40}K_2$ BEC \cite{JILA}. Experiments with $^{6}Li_2$ are in progress at Swinburne :)
\end{minipage}
\hspace{5mm}
\begin{minipage}[l]{.4\textwidth}
    \begin{center}
      \includegraphics[clip=true,angle=0,width=0.55\textwidth]{noise07fig6.eps}
    \end{center}
\end{minipage}
\end{center}

\begin{center}
\includegraphics[clip=true,angle=0,width=0.775\textwidth]{noise07fig7.eps}
\end{center}

\normalsize
%Experiments with $^{6}Li_2$ are in progress at Swinburne :-)

\end{pbox}
\vspace{-3cm}%\vspace{-0.9cm}



}
% end of second column ------------------------------------------------------
\col{
\vspace{-2.3cm}



\begin{pbox}[1 1 .8]

\vspace{-1.4cm}
\Section{\sf {\huge {\textbf {\center \hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~~~~1D Simulations~~~~~~~~~}}}}}

\vspace{-1.0cm}

Narrow molecular condensate, {\bf (c)}:
{\par\centering\resizebox*{18.4cm}{!}{\includegraphics{noise07fig8.eps}}\par}



Cut at the resonance $k=-k'=k_0$: % for {\bf (a)}, {\bf (b)}, {\bf (c)}:
{\par\centering\resizebox*{19cm}{!}{\includegraphics{noise07fig9.eps}}\par}

{\par\centering\resizebox*{19cm}{!}{\includegraphics{noise07fig10.eps}}\par}

{\par\centering\resizebox*{19cm}{!}{\includegraphics{noise07fig11.eps}}\par}

Illustration of the inequality: \\ $g_{\uparrow,\downarrow} \left( \mathbf{k},\mathbf{k}'=-\mathbf{k} ,t
\right) \leq 1\mp n_{\mathbf{k}} \left( t \right)$ (the equality holds for the uniform case):%, {\bf (a)}:
{\par\centering\resizebox*{19.4cm}{!}{\includegraphics{noise07fig12.eps}}\par}

%Maximal correlation is reach for the uniform system where $g_{\uparrow,\downarrow} \left( \mathbf{k},\mathbf{k}'=-\mathbf{k} ,t
%\right) = 1\mp n_{\mathbf{k}} \left( t \right)$.



\vspace{-0.1cm}

\end{pbox}


\vspace{-0.9cm}
\vspace{-1.0cm}

\begin{pbox}[1 1 0.8]
\vspace{-1.4cm}
\Section{\sf {\huge{\textbf{\center\hspace{-1.5cm}
\colorbox{yellow}{~~~~~~~~~~~~~~~Summary~~~~~~~~~~~~~\hspace{0.5cm}}}}}}

\vspace{-1.0cm}

The width of the correlation function becomes broader for a narrower
molecular field, more specifically the correlation function and the molecular density in momentum space are proportional and can be accuratelly described in terms of a Bessel function: \\ $g\propto J_1^2\left(R_{TF}k\right)/k^2$, of width $\simeq3.2/R_{TF}$.



\begin{thebibliography}{10}

\vspace{-1.2cm}

\bibitem{JILA} M. Greiner {\it et al.}, Phys. Rev. Lett. {\bf94}, 110401 (2005).

\bibitem{Karen} K.V. Kheruntsyan, Phys. Rev. Lett. {\bf96}, 110401
  (2006).

\bibitem{Craig} C.M. Savage {\it et al.}, Phys. Rev. A {\bf 74}, 033620 (2006).

\end{thebibliography}


\end{pbox}
\vspace{-3cm}
}

\end{center}

%\makefooter

\end{document}
