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\begin{document}

%\title{Bidirectional illumination of a single atom}
%\title{Strong light scattering by an atom illuminated from two sides}
\title{Nonlinear photon-atom coupling with 4Pi microscopy} 
%\title{Nonlinear quantum optics with an atom 4Pi microscope} 
%\title{A single atom as a quantum nonlinear medium}
%\title{A single atom as a nonlinear medium for single photons}
\author{Yue-Sum Chin}
\affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\author{Matthias Steiner}
\affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542}
\author{Christian Kurtsiefer}
\affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\affiliation{Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542}
\email[]{}
\date{\today}

%\begin{abstract}

%\end{abstract}

\maketitle
\textbf{ 
Implementing nonlinear interactions between single photons and single atoms is at the forefront of optical physics. %a long-standing goal of optical physics. %with importance for numerous (quantum) technological applications.  
Motivated by the prospects of deterministic all-optical quantum logic, many efforts are currently underway to find suitable experimental techniques~\cite{Tiecke2014,Shomroni2014,Hacker2016}. 
Focusing the incident photons onto the atom with a lens yielded promising results~\cite{Wineland1987,Vamivakas2007,Tey2008,Wrigge2008,Piro2011,Maser2016}, but is limited by diffraction to moderate interaction strengths~\cite{Tey2009}.  
However, techniques to exceed the diffraction limit are known from high-resolution imaging. %~\cite{Hell2015}.
Here, we adapt a super-resolution imaging technique, 4Pi microscopy~\cite{Hell1992}, to efficiently couple light to a single atom. 
We observe 36.6(3)\% extinction of the incident field, and a modified photon statistics of the transmitted field -- indicating nonlinear interaction at the single-photon level. 
Our results pave the way to few-photon nonlinear optics with individual atoms in free space. 
  }

\noindent
%To couple efficiently to an atom in free space, that is in the absence of resonators or waveguides, 
%the incident light needs to be shaped to match the dipole radiation pattern of the atom~\cite{Leuchs2012}. 
%Any focusing device is characterized by a coupling efficiency~$\Lambda$ which is the spatial mode overlap of the atomic dipole mode with the input mode, where $\Lambda = 1$ implies complete mode matching.
%In theory, already the most basic optical element, a lens, can transform a Gaussian beam so that at the focal point a significant part, namely up to $35\%$, of the incident intensity matches the atomic dipole mode~\cite{Tey2009}.
%This requires a device which focuses the light to a small spot size. 
%Unfortunately, this mode overlap falls short of the 50\% needed for deterministic photon-atom coupling~\cite{Zumofen2008} which motivates the development of alternative focusing schemes~\cite{Sondermann2007}.     
%In theory, a lens can transform a Gaussian beam so that at the focal point a significant part, namely up to $35\%$, of the incident intensity matches the atomic dipole mode~\cite{Tey2009}.
%Unfortunately, this mode overlap falls short of the 50\% needed for deterministic photon-atom coupling~\cite{Zumofen2008} which motivates the development of alternative focusing schemes~\cite{Sondermann2007}.     
%Here we use the 4Pi arrangement which consists of two opposing lenses with coinciding focal points to further increase the mode overlap~(Fig.~\ref{fig:figure1}a).  
%The path of the incident field is split and the atom is coherently illuminated by the two counter-propagating parts of the field simultaneously. 
%Thus, the mode overlap is doubled provided that the two parts of the field interfere constructively at the location of the atom.

%To confirm this intuitive argument, we numerically calculate the spatial dependence of the mode matching close to the focal point. 
%We consider a Gaussian field driving a circularly polarized dipole transition and follow Ref.~\cite{Tey2009} to obtain the spatial mode overlap~$\Lambda\in [0,1]$ of the atomic dipole mode with the propagating excitation mode. 
%Figure~\ref{fig:figure1}b-c shows the spatial dependence of $\Lambda$ for focusing parameters corresponding to an objective with numerical aperture NA=0.95.
%As expected, in the 4Pi arrangement, the maximum mode overlap is doubled to 70\% and is tightly localized to a small volume, approximately a sphere of diameter $\lambda/2$. 
%Thus, deterministic photon-atom coupling is, in principle, feasible in the 4Pi arrangement. 
%We show in Fig.~\ref{fig:figure1}d-e the spatial dependence of $\Lambda$ for our experimental parameters~(input beam waist~$w_0=2.7\,$mm at lens, focal length~$f=5.95$\,mm). 
%The maximum mode overlap is again doubled in the 4Pi arrangement. 
%Notably, the spatial dependence exhibits a standing wave pattern which requires to position the atom in an anti-node of the incident field. 

To realize nonlinear interactions between a few propagating photons and a single atom in free space, the photons need to be tightly focused to a small volume~\cite{Sondermann2007,Leuchs2012}. 
From high-resolution imaging it is well-known that a small focal volume requires optical elements which cover a large fraction of the solid angle~\cite{Abbe1873}. 
%This requires optics which cover a large fraction of the solid angle~\cite{Sondermann2007}. 
%For example, in standard confocal microscopy the incident light is focused through a lens which uses only one hemisphere of the solid angle. 
%The result is a poor axial resolution due to the strongly elongated focal
%volume along the optical axis.  
While standard confocal optical microscopy accomplished already very small probe volumes, the excitation light is focused through a lens that can cover only up to half of the solid angle, limiting the axial resolution due to a focal volume elongated along the optical axis.
This limitation has been overcome by using two opposing lenses with coinciding focal points, known as 4Pi arrangement~\cite{Hell1992}:   
%For example, focusing through a lens uses only one hemisphere of the solid angle, and results in a strongly elongated focal volume.
%For example, focusing through a lens results in a strongly elongated focal volume along the axial direction because only one hemisphere of the solid angle is used. 
%Whereas focusing with the 4Pi configuration, which consists of two opposing lenses in a confocal arrangement, produces an axially compressed focal volume. 
%However, the focal volume can be shortened with the 4Pi arrangement consisting of two opposing lenses with coinciding focal points~(Fig.~\ref{fig:figure1}a). 
%Here we split a beam into two paths and coherently illuminate the atom by the two counter-propagating parts of the field simultaneously~(Fig.~\ref{fig:figure1}a). 
The path of the incident beam is split, and the object is coherently illuminated by two counter-propagating parts of the field simultaneously~(Fig.~\ref{fig:figure1}a). 
In this way the input mode covers almost the entire solid angle, limited only by the numerical aperture of the focusing lenses. 
The symmetry between imaging and excitation of quantum emitter suggests that a 4Pi arrangement can also be used to efficiently couple light to an atom.  
This intuitive argument is confirmed by numerical simulations of the electric field distribution near the focal point, from which we obtain the spatial mode overlap of the atomic dipole mode with the input mode, referred to as the light-atom coupling efficiency~$\Lambda= |E_\textrm{focus}|^2/|E_\textrm{max}|^2$, where $E_\textrm{focus(max)}$ is the (maximally possible) amplitude of the incident electric field component parallel to the atomic dipole~(Fig.~\ref{fig:figure1}b-f)~\cite{Tey2009,Golla2012}. 


In our experiment, we hold a single $^{87}$Rb atom between two lenses with a far off-resonant optical dipole trap (FORT) operating at a wavelength 851\,nm~\cite{Schlosser2001}. 
%Our implementation consists of a a single $^{87}$Rb atom held between two lenses with a far off-resonant trap (FORT)~\cite{Schlosser2001} operating at a wavelength 851\,nm. 
%In the experiment we hold a single $^{87}$Rb atom between two lenses with a far off-resonant trap (FORT)~\cite{Schlosser2001} operating at a wavelength 851\,nm and probe the interaction by driving the closed \mbox{5\hflev{S}{1}{2}{2}, $m_F$=-2} to  \mbox{5\hflev{P}{3}{2}{3}, $m_F$=-3} transition near 780\,nm~\cite{Chin2017}. 
We compare 4Pi and one-sided illumination by performing a transmission experiment with a weak coherent field driving the closed transition~\mbox{5\hflev{S}{1}{2}{2}, $m_F$=-2} to  \mbox{5\hflev{P}{3}{2}{3}, $m_F$=-3} near 780\,nm~\cite{Chin2017}. 
%We probe the interaction by driving the closed \mbox{5\hflev{S}{1}{2}{2}, $m_F$=-2} to  \mbox{5\hflev{P}{3}{2}{3}, $m_F$=-3} transition near 780\,nm~\cite{Chin2017}. 
The probe beam originates from a collimated output of a single mode fiber. 
After splitting into path~1 and path~2, the beam is focused onto the atom through lenses~$L_{1}$ and $L_{2}$~(see Fig.~\ref{fig:figure1}a). 
The opposing lens re-collimates the probe beam, which is then via an asymmetric beam splitter coupled into a single mode fiber connected to avalanche photodetector~$D_1$ or $D_2$, respectively~(see Supplementary Information for details).  
%We use asymmetric beam splitters to experimentally access the transmission~(see Supplementary Information for details).    
The electric fields at the detectors are superpositions of the probe field and the field scattered by the atom. 
To derive the total electric field, we adapt the theoretical description of Ref.~\cite{Tey2009,Slodifmmodeheckclsecika2010} to account for the contributions of the two counter-propagating probe fields. 
The optical power~$P_1$ at detector~$D_1$ depends then on the power in the individual beam paths~$P_\textrm{1(2),in}$ and the light-atom coupling efficiency~$\Lambda_{1(2)}$ of path~1(2), 
\begin{equation}\label{eq:P_i}
P_\textrm{1} =  \left( \sqrt{P_\textrm{1,in}} - 2\Lambda_1 \sqrt{P_\textrm{1,in}} - 2 \sqrt{\Lambda_1 \Lambda_2}  \sqrt{P_\textrm{2,in}}\right)^2,
\end{equation}
where we assume that the two fields interfere constructively at the focal point. 
Similarly, the power at detector~$D_2$ is obtained by exchanging subscripts $1\leftrightarrow2$.  
From equation~\ref{eq:P_i} we obtain the expected values for % our observables,
 the individual transmission $T_\textrm{1(2)} = P_{1(2)}/P_\textrm{1(2),in}$, and the total transmission $T_\textrm{total} = (P_\textrm{1}+ P_\textrm{2})/ (P_\textrm{1,in}+ P_\textrm{2,in})$. 
For example, for a one-sided illumination through lens~$L_1$, i.e. $P_\textrm{2,in}=0$, the transmission measured at detector~$D_1$ takes the well known expression $T_1  =   \left(1- 2\Lambda_1\right)^2$~\cite{Tey2009,Slodifmmodeheckclsecika2010}. 
In the 4Pi configuration, we determine the total coupling~$\Lambda_\textrm{total}$ from the total transmission $T_\textrm{total}= \left(1- 2\Lambda_\textrm{total}\right)^2$. 
From equation~\ref{eq:P_i} we find that the power splitting~$P_\textrm{2,in}=P_\textrm{1,in} \Lambda_1/\Lambda_2$ optimizes the total coupling to~$\Lambda_\textrm{total}=\Lambda_1+\Lambda_2$.

Figure~\ref{fig:histo_spec}a shows the transmission spectrum of a weak coherent field for one-sided illumination, either via path 1 (blue) or path 2 (red). %when we tune the frequency of the probe field. 
Comparing the resonant transmission $T_1= 77.9(2)\%$ and $T_2= 79.8(3)\%$ to equation~\ref{eq:P_i}, we find similar coupling efficiencies, $\Lambda_1 = 0.059(1)$ and~$\Lambda_2 = 0.053(1)$, as expected for our symmetric arrangement with two nominally identical lenses. 
Therefore, the maximum coupling expected in the 4Pi configuration is~$\Lambda_\textrm{total}=\Lambda_1+\Lambda_2= 0.112(4)$, assuming perfect phase matching of the fields and ideal positioning of the atom.  
%Aberrations and the atomic motion cause the mode overlaps~$\Lambda_i$ to be lower than predicted for an ideal lens.  

In the 4Pi configuration the atom needs to be precisely placed at an anti-node of the incident field~(Fig.~\ref{fig:figure1}e). 
To this end, we tightly confine the atom along the optical axis with an additional blue-detuned standing wave dipole trap~(761\,nm). 
As the atom is loaded probabilistically into the optical lattice, we use a simple postselection technique to check whether the atom is trapped close to an anti-node of the incident field~(see Methods).
%%type1
%Each experimental sequence consists of two transmission measurements. % referred to as `coupling' and `postselection' cycle. 
%In the first transmission measurement, we tune the probe frequency to determine the light-atom coupling~$\Lambda$. 
%The second transmission measurement, with the probe frequency fixed on resonant to the atomic transition, is used to independently determine the coupling for postselection. 
%We then discard events in which the second transmission is higher than a threshold value~(see Methods).
%%type2
%To determine the light-atom coupling~$\Lambda$ we tune the probe frequency in the `coupling' cycle while the probe frequency during the `postselection' cycle is resonant with the atomic transition. 
%We then discard `probe' cycles if the transmission in the `postselection' cycle is higher than a threshold value~(see Methods). 
Figure~\ref{fig:histo_spec}b shows the observed transmission when the atom is illuminated in the 4Pi arrangement. 
The increased light-atom coupling is evident from the strong reduction of transmission. 
%Illuminating the atom from both sides simultaneously with a near resonant probe field, we observe a strong light scattering~(Fig.~\ref{fig:histo_spec}b). 
The individual transmissions $T_\textrm{1}=62.3(5)\%$, $T_\textrm{2}=64.6(5)\%$, and the total transmission $T_\textrm{total} = 63.4(3)\%$ are significantly lower compared to the one-sided illumination.
The corresponding total coupling of $\Lambda_\textrm{total}=0.102(1)$ is close to the theoretical prediction of~$0.112(4)$. 

We next show that for a symmetric arrangement~$\Lambda_1\approx\Lambda_2$, the highest interaction is achieved with an equal power splitting $P_\textrm{2,in}\approx P_\textrm{1,in}$. 
Figure~\ref{fig:ratio} displays the resonant transmissions for different relative beam power in the two paths.  
For imbalanced beam power, the total transmission is increased, albeit with a fairly weak dependence. 
In contrast, we find a strong dependence of the individual transmissions on the relative beam power: 
For $P_\textrm{1,in}\approx 12 P_\textrm{2,in}$, the total transmission is
still low, $T_\textrm{total} =71.2(8)\%$, but the two values for the
individual transmissions are no longer equal: $T_\textrm{1,4Pi}=74.0(8)\%$,
$T_\textrm{2,4Pi}=41(2)\%$, 
%The observed transmission values are 
in qualitative agreement with equation~\ref{eq:P_i}~(solid lines in Fig.~\ref{fig:ratio}).

The nonlinear character of the photon-atom interaction can induce effective attractive or repulsive interactions between two photons~\cite{Shen2007}. % if the initial field contains multi-photon contributions. %~\cite{Birnbaum2005,Hoi2012,Peyronel2012}. %, for example, a coherent state. 
These interactions can be observed as modification of the photon statistics of the transmitted field if the initial field contains multi-photon contributions~\cite{Birnbaum2005,Dayan2008,Reinhard2012,Hoi2012,Peyronel2012}. 
For a weak coherent driving field, %of weak amplitude %, i.e. ignoring three or higher photon contributions, 
the second-order correlation function~$g^{(2)}(\tau)$ takes the specific form~\cite{Chang2007,Zheng2010} 
%the~$g^{(2)}(t)$ of the transmitted field is given by
\begin{equation}\label{eq:g2}
 g^{(2)}(\tau) = e^{-\Gamma_0 \tau} \left( \left(\frac{2\Lambda}{1-2\Lambda}\right)^2 - e^{\frac{\Gamma_0 \tau}{2}} \right)^2,
\end{equation}
where $\Gamma_0=2\pi \times 6.07\,$MHz is the excited state linewidth. %~\cite{Chang2007,Zheng2010}. 
By time-tagging the detection events at detector~$D_1$ and $D_2$ during the probe phase, we obtain $g^{(2)}(\tau)= \langle p_1(t) p_2(t+\tau)\rangle / ( \langle p_1(t)\rangle \langle p_2(t+\tau)\rangle)$, where $p_{1(2)}(t)$ is the detection probability at detector~$D_{1(2)}$ at time~$t$, and $\langle \rangle$ denotes the long time average. 
To acquire sufficient statistics, we use $50\%$ more photons in the probe pulse as compared to Fig.~\ref{fig:histo_spec}, and also atoms which are not optimally coupled to the probe field~(see Methods). 
From the resulting average transmission~$T_\textrm{total} =70.3(3)\%$, we deduce an average coupling~$\Lambda_\textrm{total}=0.0808(5)$ for this experiment. %, see equation~\ref{eq:tx}.  
As shown in Fig.~\ref{fig:g2}, we find a clear signature of nonlinear photon-atom interaction in the intensity correlations of the transmitted light. 
The observed photon anti-bunching~$g^{(2)}(0) = 0.934(7)\%$ is in good agreement with equation~\ref{eq:g2}. 
Here, for fair comparison with equation~\ref{eq:g2}, we account for a small photon bunching effect~($\approx1.7\%$, see Methods) due to the diffusive atomic motion~\cite{Gomer1998,Weber2006}. 
%Our results show a proof of principle of modification of photon statistics.  
%The possibility of significant changes of the photon statistics at higher interaction strength is shown in Fig.~\ref{fig:g2}b.
For stronger light-atom coupling the changes of the photon statistics are expected to be more significant~(Fig.~\ref{fig:g2}b). 
%When the photon-atom interaction is very weak, $\Lambda\ll1$, the atom does not significantly change the photon statistics, $g^{(2)}(0)\approx 1$~(Fig.~\ref{eq:g2}a). 
Notably, for $\Lambda =0.25$ the transmitted and the reflected light show
anti-bunching ($g^{(2)}(0) =0$), that means the atom acts as a photon turnstile and converts a coherent field completely into a single photon field. 
The transmission for this light-atom coupling is~$T_\textrm{total}=25\%$ (see equation~\ref{eq:P_i}). 
Photon bunching ($g^{(2)}(0) > 1$) for large values of $\Lambda$ signals an enhanced probability for multiple photons to be transmitted, essentially because the atom cannot scatter multiple photons simultaneously.    
%The quadratic dependence of $g^{(2)}(0)$ on the $\Lambda$ renders the observation of 
%In contrast for $\Lambda \lesssim0.3 $, photon anti-bunching,~$g^{(2)}(0) < 1$, is expected.  

Our work establishes the 4Pi arrangement as an effective technique to couple a propagating field to an atom. 
This opens exciting prospects to implement effective interactions between photons with tightly focused free space modes and single atoms. 
Strongly interacting photons could find application in imaging, metrology, quantum computing and cryptography, and constitute a novel platform to study many-body physics~\cite{Chang2008,Chang2014}. 
The presented approach forms an experimental alternative to waveguide/cavity quantum electrodynamics~\cite{Birnbaum2005,Reiserer2015} and Rydberg quantum optics~\cite{Pritchard2010,Peyronel2012,Firstenberg2016}.  
While the achieved nonlinearity of the photon-atom interaction, observed as modification of the photon statistics, does not create strongly correlated photons yet, the 4Pi arrangement eases the technical requirements to the focusing lens considerably, making the implementation of strong photon-photon interaction feasible. 
In the near future, we expect that by using higher numerical aperture lenses, the 4Pi arrangement will allow the efficient conversion of a coherent beam into single photons.  

\section*{Methods}
\subsection*{Experimental sequence and postselection of the atom position}
The experimental sequence starts with loading a single atom from a cold ensemble in a magneto-optical trap  into a far-off resonant dipole trap. 
Once trapped, the atom undergoes molasses cooling for 5\,ms. 
%A single atom, upon loading, is cooled by polarization gradient cooling for 5\,ms to 16\,$\mu$K. 
We then apply a bias magnetic field of 0.74\,mT along the optical axis and optically pump the atom into the 5\hflev{S}{1}{2}{2}, $m_F$=-2 state. 
Subsequently, we perform two transmission experiments during which we switch on the probe field for 1\,ms each. 
The first transmission measurement is used to determine the light-atom coupling~$\Lambda$, the second one to check whether the atom is trapped at an anti-node of probe field. 
To obtain the relative transmission, we also detect the instantaneous probe power for each transmission experiment by optically pumping the atom into the \mbox{5\hflev{S}{1}{2}{1}} hyperfine state, which shifts the atom out of resonance with the probe field by 6.8\,GHz, and reapply the probe field. 

The postselection of the atom position is performed as follows:
%The position of the atom is postselected based on the detected transmission during second probe cycle. 
We select the detection events in the first transmission experiment conditioned on the number of photons detected in the second one. 
The frequency of the probe field during the second transmission experiment is set to be resonant with the atomic transition. 
%We then discard events for the case of the second transmission is higher than a threshold value. 
For the data shown in Fig.~\ref{fig:histo_spec}b and Fig.~\ref{fig:ratio} we use a threshold which selects approximately 0.5\% of the total events as a trade-off between data acquisition rate and selectiveness of the atomic position.
%Note that this is done at the end of each measurement cycle to avoid additional photon scattering induced heating that reduces the interaction strength. 
%In addition, to increase the likelihood of atom loading into a desired site, we routinely position the standing wave blue-detuned FORT to a position of lowest average transmission by tuning the frequency of the laser. 
%For Fig.~\ref{fig:histo_spec}b and Fig.~\ref{fig:ratio}, we retain $\sim$0.5\% of the data. 
To measure the second-order correlation function of the transmitted light (Fig.~\ref{fig:g2}a),  we choose a higher threshold which selects $\sim$10\% of the experimental cycles. 

%We remark for the case of one side illumination, such postselection does not result in lower transmission. 

\subsection*{Normalization of second-order correlation function}
We measure the second order correlation function  of the transmitted light using detector~$D_1$ and $D_2$ as the two detectors of a Hanbury-Brown and Twiss setup.
The photodetection events are time tagged during the probe phase, and sorted into a time delay histogram. 
We obtain the normalized correlation function~$g^{(2)}(\tau)$ by dividing the number of occurrences by $r_1 r_2 \Delta t T$, where $r_{1(2)}$ is the mean count rate at detector $D_{1(2)}$, $\Delta t$ is the time bin width, and $T$ is the total measurement time. 
%For each 1\,ms-long measurement cycle, we obtain the normalized correlation function~$g^{(2)}_i(\tau)$~(index $i$ describes the measurement cycle) by dividing the number of occurrences by $r_1 r_2 \Delta t T$, where $r_{1(2)}$ is the mean count rate at detector $D_{1(2)}$, $\Delta t$ is the time bin width, and $T$ is the total measurement time. 
%We then obtain $g^{(2)}_0(\tau)$ from the  weighted mean
%\begin{equation}\label{eq:g2norm}
% g^{(2)}_0(\tau) =  \frac{\sum^{N}_{i=1} g^{(2)}_i(\tau)  (r_{1,i}+r_{2,i})}{\sum^{N}_{i=1} (r_{1,i}+r_{2,i})}.
%\end{equation}
For times $100\,$ns $< \tau < 1\,\mu$s, we find super-Poissonian intensity
correlations $g^{(2)}(\tau)>1$, which are induced by the atomic motion through the trap. 
Although the amplitude of the correlations is small, we nevertheless perform a deconvolution for a better comparison to Eq.~\ref{eq:g2}. 
The correlations are expected to decay exponentially for diffusive motion, thus we fit  $f(\tau)= 1 + a_0 \exp{(-\tau/\tau_d)}$ to $g^{(2)}(\tau)$, resulting in $\tau_d=0.71(8)\,\mu$s and $a_0=0.019(2)$. % and a reduced $\chi^2=1.6$. 
%We note that $t_0$ is much larger than the excited state lifetime~$26\,$ns. 
Figure~\ref{fig:g2} shows the second order correlation function after deconvolution of the diffusive motion, i.e., after division by $f(\tau)$~(see Supplementary Information). 



%\subsection*{Coupling efficiency}

\bibliographystyle{apsrev4-1}
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\bibliography{fourpi.bib}{}

\begin{acknowledgments}
We acknowledge the support of this work by the Ministry of Education in
Singapore (AcRF Tier 1) and the National Research Foundation, Prime Minister's
office.
M.\,Steiner acknowledges support by the Lee Kuan Yew Postdoctoral Fellowship.
\end{acknowledgments}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{figure/figure1/figure1.pdf}
    \caption{\label{fig:figure1}
    \textbf{Concept of 4Pi illumination.}
    \textbf{a}, Schematics of the optical setup. 
    The probe beam~(black arrow) is split into path 1 (blue arrows) and path 2 (red arrows). 
    The two beams then illuminate the atom from counter-propagating directions. 
    Asymmetric beamsplitters are used to sample the probe light after passing the atom. 
    The probe light in path~1(2) is coupled into a single mode fiber connected to detector~$D_{1(2)}$.  
    By blocking one path, we recover the commonly employed one-sided illumination. % which we compare to the 4Pi arrangement. 
    BS:\,beam splitter, $L_{1(2)}$:\,high numerical aperture lens, $D_{1(2)}$:\,avalanche photodetector. 
    \textbf{b-e}, Numerical results of the coupling efficiency~$\Lambda$ near the focal point considering a Gaussian field resonantly driving a circularly polarized dipole transition near 780\,nm~\cite{Tey2009}. 
    The field is assumed to constructively interfere at the focal point for the 4Pi configuration. 
    %Numerical results of the coupling efficiency~$\Lambda$ between the atomic dipole mode and the probe mode near the focal spot. 
    %We consider a Gaussian field driving resonantly a circularly polarized dipole transition near 780\,nm~\cite{Tey2009}. 
    %We find that the maximum coupling is doubled in the 4Pi arrangement compared to illumination from only one side. 
    \textbf{b/c}, Focusing parameters corresponding to an objective with numerical aperture NA$=0.95$. 
    %The highest values for $\Lambda$ are tightly localized to a small volume, approximately a sphere of diameter $\lambda$.  
    \textbf{d/e}, Focusing parameters used in this experiment~(input beam waist~$w_0=2.7\,$mm at lens, focal length~$f=5.95$\,mm). 
    %\textbf{b/c}, One side/4Pi configuration for tight focusing. 
    %\textbf{d/e}, One side/4Pi configuration for moderate focusing. 
 }
\end{figure}


\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{figure/histo_spec/hist_spec.eps}
    \caption{\label{fig:histo_spec} 
    %\textbf{Probing light-atom interaction strength.}
    \textbf{ Extinction of a weak coherent probe beam.} 
    \textbf{a}, One-sided illumination via path~1~(blue diamonds) or path~2~(red squares). 
    Solid lines are Lorentzian fits. 
    The inset shows the normalized histogram of detected photons during the probe cycle (solid line) and reference cycle (gray) for the resonant data point. 
    \textbf{b}, Same as \textbf{a} but with 4Pi illumination. 
    The total transmission (black circles) is obtained from the sum of detectors~$D_1$ and $D_2$. 
    Error bars represent one standard deviation of propagated Poissonian counting uncertainties. 
    The FORT shifts the resonance frequency by approximately $38.5\,$MHz compared to the natural transition frequency. 
}
\end{figure}


\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth/2]{figure/ratio/ratio.eps}
    \caption{\label{fig:ratio}
    \textbf{Resonant transmission for different power splittings between path~1 and path~2.}
    Transmission at detector $D_1$ (top), $D_2$ (center) and the total transmission $D_1+D_2$ (bottom). 
    The total number of incident photons is kept constant. 
    Solid lines are  $T_\textrm{1(2)}$ and $T_\textrm{total}$ derived from equation~\ref{eq:P_i}. 
    %theoretical predictions using the interaction strength measured in one side illumination. 
    Error bars represent one standard deviation of propagated Poissonian counting uncertainties. 
 }
\end{figure}

\begin{figure}[t]
  \centering
  \includegraphics[width=\columnwidth]{figure/g2/g2.eps}
    \caption{\label{fig:g2}
    \textbf{Modified photon statistics due to nonlinear interaction. }
    \textbf{a}, Intensity correlation of transmitted light with a time bin width of 5\,ns.  
    Solid line is the theoretical prediction without free parameter~(see equation~\ref{eq:g2}). % independently determined from transmission. 
    \textbf{b}, Dependence on the coupling efficiency~$\Lambda$. 
    The inset is a zoom into the region of our data point for clarity, and the 
    solid line is $g^{(2)}(0)$ from equation~\ref{eq:g2}. 
 }
\end{figure}

\end{document}
