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\title{Coherence induced correlation and anti-correlations}


\author{Yanhong Xiao,$^{1,**}$, Tun Wang,$^{2,3}$, Maria Baryakhtar,$^4$,Michael Crescimanno,$^5$, Michael Hohensee$^1$, David F. Phillips$^1$, Susanne F. Yelin$^{2,6}$, Ronald L. Walsworth, $^{1,4}$}
\address{$^1$Harvard-Smithsonian Center for Astrophysics, Cambridge,
MA, 02138 USA}
\address{$^2$Department of Physics, University of
Connecticut,Storrs,CT 06269, USA}
\address{$^3$ITP, Univ. Innsbruck, Switzerland} 
\address{$^4$Department of Physics, Harvard University, Cambridge,
MA, 02138 USA}
\address{$^5$Department of Physics and Astronomy, Youngstown State
University, Youngstown, OH 44555-2001}
\address{$^6$ITAMP, Cambridge, MA,02138 USA}
\address{$^{**}$Corresponding author: yxiao@cfa.harvard.edu}

\date{\today}

\begin{abstract}

Although electromagnetically induced transparency (EIT)
is well understood from the steady state evaluation of a simple
quantum optics model, noise induced by this two-photon resonance 
has recently received much recent experimental interest 
because it is somewhat less well understood and is being evaluated
for fundamental metrology. We report on 
a simple intuitive and quantitative physical picture for the conversion 
of laser phase noise into intensity noise in EIT. This view
relates the observed transition between intensity noise correlation and 
anticorrelation in terms of the changes in the slope of the 
one-photon resonances as one detunes through the two photon resonance. 
This analytic approach quantitatively connects the steady state 
solution of the quantum optics model with experimental signatures
including the power-broadening-free resonance, correlation zeros at 
detuning equal width and contrast versus overall one-photon detuning. 



\end{abstract}

\maketitle %% null function with osajnl.sty

] %% activate for two-column option
%Intro: 
%	Diodes: intensity -vs- phase noise
%	The reason for the paper: simple explanation
%		g_2 in terms of the slopes of the one-photon response
%		In this sense this is really a theoretical paper. 	


Noise in the optical response of two- and three- level atomic systems
illuminated by lasers has been studied since the late 80's\cite{haslwanter, dalton, rzazewski}. These studies have sought to characterize
the observed changes in the light's intensity 
noise spectra in terms of fundamental non-linear optical processes
\cite{fleischhauer2, walser}. Understanding 
intensity noise from vapor cells has ramifications for CPT 
clocks \cite{dalton}, EIT based magnetometers and quantum memory. 

Diode lasers have low intensity noise and high phase noise which is
associated with a large spectral linewidth typically ranging from
tens of kHz to hundreds of MHz. Phase noise converts to intensity
noise after near-resonant interaction with an atomic medium. 
Throughout this note we
focus only on multiplicative noise processes and ignore endogenous 
noise sources from the vapor cell, as experiments indicate that
this later source of noise is typically small. 
Recently experimental studies \cite{cruz, sautenkov, 
ariunbold}
have revealed new aspects of the noise spectra from EIT prepared 
media, such as the power-broadening-free resonance in the intensity 
cross correlation \cite{sautenkov, ariunbold}, of potential 
interest in precision spectroscopy and quantum optics metrology. 

We consider EIT in a generic three level $\Lambda$ system composed of
two metastable ground states $|a\rangle$, $|b\rangle$ and an excited
state $|e\rangle$. For simplicity we assume that two 
laser fields with the same Rabi frequency
$\Omega_a=\Omega_b=\Omega$ are on resonance with transitions
$|a\rangle\rightarrow |e\rangle$ and $|b\rangle\rightarrow|e\rangle$
respectively.  To further simplify and relate to the light from 
laser diodes, we assume that there are essentially no intensity 
fluctuations of these light fields, but that that their 
frequency varies jointly over a large bandwidth. 
After the cell the intensities of each light field
are separately digitized and crosscorrelated in a spectrum analyzer, 
giving a signal proportional to $g_2(t) \equiv 
\frac{\langle\delta I_1(t)\delta I_2(0)\rangle}
{\sqrt{\langle(\delta I_1)^2\rangle\langle(\delta I_2)^2}\rangle}$, 
where $\delta I_{1,2}$ are intensity fluctuations 
for the two fields. In the optically thin limit in which 
we also ignore endogenous noise sources in the cell, the intensity 
noise cross correlator can be written entirely in terms of the 
variations of the optical susceptibilities, 

\begin{equation}
g^2(t)=\frac{\langle Im(\Omega_a\delta \rho_{ea}(t))\,Im(\Omega_b\delta
\rho_{eb}(0))\rangle}{\sqrt{\langle(Im(\Omega_a \delta
\rho_{ea}))^2\rangle\langle(Im(\Omega_b \delta \rho_{eb}))^2}\rangle}
\label{eq:g2define}
\end{equation}
with $\langle\,\rangle$ denoting the average over the frequancy 
excursions, the $\rho_{ij}$ the density matrix. 


%%Fig.1
\begin{figure}[htb]
\includegraphics[clip,width=1\linewidth]{Zeeman-setup.eps}
\caption{Schematics of experimental setup for Zeeman EIT.}
\label{setup.fig}
\end{figure}

%%Fig.2
\begin{figure}[htb]
%\includegraphics[clip,width=1\linewidth]
\includegraphics[clip,width=0.5\linewidth]{sep2.eps}
\includegraphics[clip,width=0.5\linewidth]{sep3.eps}
\includegraphics[clip,width=0.8\linewidth]{sep1.eps}
\caption{One-photon transmission spectrums of the two EIT fields for
three two-photon detunings. Laser frequency sweeps through
transitions $|F=2\rangle\rightarrow|F^{\prime}=2\rangle$and
$|F=2\rangle\rightarrow|F^{\prime}=1\rangle$, from left to right.
Large offset of two spectrums up to $120$ MHz appears for
$\Delta=550$Hz. Other parameters: $\Gamma\approx 100$ Hz, laser power
$170$ $\mu$W, beam diameter $3.7$ mm, cell with $2$ Torr Ne buffer
gas.} 
\label{separation.fig}
\end{figure}
% labbook p92, large beam, \Delta=0.55 kHz.


The offset of the two one-photon resonances is directly related to
$g^2(0)$ . To illustrate this connection, we assume the laser
frequency is locked at a fixed one-photon detuning, for instance
$\delta_0=0$ (position is indicated by the dashed lines in
Fig.~\ref{separation.fig}). Since the offset is caused by ground
state coherence which cannot follow the fast fluctuations of laser
phase (hence frequency), the slope relations will remain the same as
that shown in Fig.~\ref{separation.fig} from where we can deduce that
$g^2(0)\approx 1$ for zero (Fig.~\ref{separation.fig}(a))and large
$\Delta$ (Fig.~\ref{separation.fig}(b))because the slopes are
completely parallel, and $\approx -1$ for $\Delta=550$ Hz because the
slopes are completely opposite(Fig.~\ref{separation.fig}(c)). 

We now address at what two-photon detunings transitions between
anti-correlation and correlation happen.  To simply the analysis, we
assume that one-photon detuning is zero (then $d=0$ in
Eq.~(\ref{eq:g2define})). Using Eq.~(\ref{eq:g2define}), the
numerator of the $g^2$ in Eq.~(\ref{eq:g2define}) can be rewritten as
%
\begin{eqnarray}\label{long-equation}
G^2(0)=\langle \delta Im(\rho_{ea}(t))\delta Im(\rho_{eb}(t))\rangle \\
= 0.25\left\{\langle(\delta Im(\rho_{ea}+\rho_{eb}))^2\rangle
-\langle(\delta Im(\rho_{ea}-\rho_{eb}))^2\rangle\right\} \nonumber\\
%
= 0.25\Omega((\frac{1}{2}+Re(\rho_{ab}))^2(\delta\frac{\gamma/2}{(\gamma/2)^2+\delta_0^2})^2 \nonumber \\
%
-Im^2(\rho_{ab})(\delta \frac{\delta_0}{(\gamma/2)^2+\delta_0^2})^2
)\nonumber
\end{eqnarray}
%

Finally, we remark on the role of two-photon phase noise, which
exists if two-independent laser sources are used, or the EIT fields
are produced by phase modulating a laser with a RF source which is
not phase noise free. In our Zeeman EIT experiment, amplitude noise
in the magnetic field can induce effective two-photon phase noise. We
have used a stable current source for the magnetic field generation,
and checked that the two-photon noise influence is negligible. In
cases where it is not negligible, $g^2(0)$ value will be closer to
zero for the entire $g^2(0)$ v.s.$\Delta$ curve, due to the
unsyncronized phase noise itself (reduce correlation) and decreased
ground state coherence(reduce anticorrelation).

In conclusion, we demonstrate that EIT ground state coherence can
lead to large offset of the otherwise overlapped one-photon
transmission spectrums of the two EIT fields. This phenomena is the
essence of anticorrelation in the intensity fluctuations in the two
EIT fields. Our physics picture provides intuition to the noise
processes in the system, and is useful for applications.

We are grateful to J. Vanier for bringing our attention to laser
phase noise in EIT and to I. Novikova for useful discussions. This
work was supported by ONR, DARPA, ITAMP and the Smithsonian Institution.


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%\pagebreak

\begin{thebibliography}{99}
%\frenchspacing


\bibitem{rzazewski} K. Rzazewski, B. Stone, and M. Wilkens, 
``Laser-noise-induced intensity fluctuations in resonance flourescence,'' 
{\it Phys. Rev. A} {\bf 40}, 2788-2791 (1989).

% early noise spec paper. Intensity-Intensity correlation in flourescence
% this is  a theory paper. two level system. 


\bibitem{yabuzaki} T. Yabuzaki, T. Mitsui, U. Tanaka, 
"New type of high-resolution spectroscopy with a diode laser," 
{\it Phys. Rev. Lett.} {\bf 67}, 
2453-2457 (1991). 

% p2453_1
% first historical reference to suggestion of noise spectroscopy in EIT
% amplitude flucts in laser diode are claimed to be below the shot noise limit.

\bibitem{haslwanter} Th. Haslwanter, H. Ritsch, J. Cooper, and P. Zoller,
``Lasernoise-induced population fluctuations in two- and threelevel systems,'' {\it Phys. Rev. A} {\bf 38}, 5652-5659 (1988).

% early noise spect paper

\bibitem{dalton} B. J. Dalton and P. L. Knight,
``The effects of laser field fluctuations on coherent population 
trapping,'' {\it J. Phys. B} {\bf 15}, 3997-4015 (1982).

% early noise spect paper.

\bibitem{fleischhauer2} M. Fleischhauer and T. Richter, "Pulse matching 
and correlation of phase fluctuations in a  $\Lambda$ system," 
{\it Phys. Rev. A} {\bf 51} 2430-2442 (1995). 

% realy closely related. Theory paper. Has a fast and a slow time scale. 
% Fast is the optical pumping time, and slow being the DS relaxation time. 


\bibitem{walser} R. Walser and P. Zoller, "Laser-noise-induced polarization
fluctuations as a spectroscopic tool",
{\it Phys. Rev. A} {\bf 49}, 5067-5077 (1994).

% p5067_1

\bibitem {martinelli} M. Martinelli, P. Valente, H. Failache, D. Felinto, L.S.
Cruz, P. Nussenzveig, A. Lezama, Phys. Rev.A 69, 043809 (2004)

% interesting paper. Same setup. cell stabilized DL. But they say it is an ECL.% And claim 1 MHz width. Lots of power, like 3 mW
% PD's had a 10 MHz bandwidth. Signal sent to a 1 GHz spectrum analyzer. 
% overkill? 


\bibitem {cruz} L.S. Cruz, D. Felinto, J.G. Aguirre G´omez, M. Martinelli, P. Valente, A. Lezama, and P. Nussenzveig,
"Laser-noise-induced correlations and anti-correlations
in electromagnetically induced transparency," 
{\it Eur. Phys. J.} D {\bf 41}, 531-539 (2007). 

% This is the same as the above, but with the phase diffusion model. 
% this paper looks more comprehensive than the one above. 

\bibitem{sautenkov} V.A. Sautenkov, Y.V. Rostovtsev, M.O. Scully, 
"  ,"
{\it Phys. Rev. A} {\bf  72}, 065801 (2005), quant-ph/0508026. 

% the observation and the assertion that it is associated with the PM to AM 
% on the resonance. not much of a model. This paper is just an advertisement. 

\bibitem{ariunbold} G.O. Ariunbold, V.A. Sautenkov, Y.V. Rostovtsev, M.O.
Scully, 
" ,"
arXiv:quant-ph/0603025 (2005).

\end{thebibliography}

\end{document}
