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\begin{document}


%DIF < \preprint{}
\title{Transmission spectroscopy of a single atom in the presence of tensor light shifts}
%DIF < \title{Transmission spectroscopy of a single atom in the presence of transverse Zeeman and light shifts}
\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{Yue-Sum Chin}
\affiliation{Centre for Quantum Technologies, 3 Science Drive 2, Singapore 117543}
\author{Christian Kurtsiefer}
\DIFaddbegin \email{christian.kurtsiefer@gmail.com}
\DIFaddend \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}
%DIF < \thanks{}
%DIF <  \email[]{christian.kurtsiefer@gmail.com}
\date{\today}



\begin{abstract}
We investigate the interplay between Zeeman and light shifts in the transmission spectrum of an optically trapped, spin-polarized Rubidium atom. 
%DIF < We find that the spectral shape of the transmission is strongly modified by the tensor light shift, which is caused by the linearly polarized dipole trap. 
%DIF < Varying the magnetic field strength and the depth of the dipole trap, we
%DIF < identify operating conditions favorable for quantum information applications
%DIF < in that efficient light-atom coupling and a long coherence time between ground
%DIF < state hyperfine levels concur. 
The spectral shape of the transmission changes from  multiple, broad resonances to a single, narrow Lorentzian with a high resonant extinction value when we increase the magnetic field strength and lower the depth of the dipole trap.  
%DIF < We identify operating conditions favorable for quantum information applications
%DIF < in that efficient light-atom coupling and a long coherence time between ground
%DIF < state hyperfine levels concur. 
We present an experimental configuration well-suited for quantum information applications \DIFdelbegin \DIFdel{in that it }\DIFdelend \DIFaddbegin \DIFadd{that }\DIFaddend enables not only efficient light-atom coupling\DIFaddbegin \DIFadd{, }\DIFaddend but also a long coherence time between ground state hyperfine levels.
\end{abstract}

% insert suggested PACS numbers in braces on next line
\DIFdelbegin %DIFDELCMD < \pacs{
%DIFDELCMD < 32.90.+a,        % Other topics in atomic properties and interactions of atoms;
%DIFDELCMD < 37.10.Gh % Atom traps and guides
%DIFDELCMD < %37.10.Vz,	%Mechanical effects of light on atoms, molecules, and ions
%DIFDELCMD < %42.50.Ct      % Quantum description of interaction of light and matter;
%DIFDELCMD <  }
%DIFDELCMD < %%%
\DIFdelend \DIFaddbegin \pacs{
32.90.+a,        % Other topics in atomic properties and interactions of atoms;
37.10.Gh % Atom traps and guides
 }
\DIFaddend 



\maketitle
\section{Introduction}
\DIFdelbegin \DIFdel{Individually controlled }\DIFdelend \DIFaddbegin \DIFadd{Optically trapped, individually addressable }\DIFaddend neutral atoms have been established as a viable platform for advanced applications in quantum information science~\cite{Bernien2017,Lienhard2018}.  
In this approach, a qubit is typically realized by two ground state hyperfine levels of the atom.  
Several strategies have been developed to connect multiple atomic qubits. 
For example, nearby atoms can interact by optical coupling to \DIFdelbegin \DIFdel{highly-excited }\DIFdelend \DIFaddbegin \DIFadd{highly excited }\DIFaddend Rydberg states~\cite{Wilk2010}.
Alternatively, atoms separated by large distances can be connected through an
optical link and the exchange of single photons~\cite{Ritter2012}.
\DIFdelbegin \DIFdel{In both cases, efficient and well-controlled coupling of optical fields to the atoms is essential for using neutral atoms for quantum information applications. 
Depending on }\DIFdelend \DIFaddbegin 

%DIF > \textbf{ 
\DIFadd{While deep optical tweezers are a versatile and convenient tool to prepare~\mbox{%DIFAUXCMD
\cite{Schlosser2001}}%DIFAUXCMD
, move~\mbox{%DIFAUXCMD
\cite{Endres2016}}%DIFAUXCMD
, and hold~\mbox{%DIFAUXCMD
\cite{Nogrette2014} }%DIFAUXCMD
individual atoms, the trapping field can also have undesired consequences like shortened qubit coherence times. 
Thus the sensitivity of }\DIFaddend the \DIFdelbegin \DIFdel{experimental configuration, however, the conditions necessary
for efficient optical couplingcan compromise other qubit properties such as the coherence time. 
}\DIFdelend \DIFaddbegin \DIFadd{qubit coherence to light shifts has been extensively investigated for various experimental configurations~\mbox{%DIFAUXCMD
\cite{Kuhr2003, Kuhr2005, Kim2013,Hutzler2017,Chalopin2018}}%DIFAUXCMD
, including very tightly focused optical tweezers~\mbox{%DIFAUXCMD
\cite{Kaufman2012, Thompson2013}}%DIFAUXCMD
. 
Little explored are light shift induced effects on optical transitions beyond
line shifts~\mbox{%DIFAUXCMD
\cite{Shih2013}}%DIFAUXCMD
; an example is the optically induced breakdown of
the atomic hyperfine coupling~\mbox{%DIFAUXCMD
\cite{Neuzner2015}}%DIFAUXCMD
.
%DIF > }
%DIF > While free atoms exhibit excellent optical and hyperfine coherence properties, optically trapped atoms 
%DIF > In both cases, efficient and well-controlled coupling of optical fields to the atoms is essential for using neutral atoms for quantum information applications.    
%DIF > Depending on the experimental configuration, however, the conditions necessary
%DIF > for efficient optical coupling can compromise other qubit properties such as the coherence time. 
}\DIFaddend 

%DIF < suggest to switch 'good' light-atom coupling to 'efficient' as in abstract (ys)
%DIF < also thinking of alternative to 'good' ground state coherence?
%DIF <  done - ms
%DIF < One approach to achieve efficient light-atom coupling is trapping individual atoms in optical tweezers and placing them at the focal spot of a high numerical aperture lens. 
\DIFdelbegin \DIFdel{We investigate this trade-off for an approach where efficient }\DIFdelend %DIF > \textbf{
\DIFaddbegin \DIFadd{In this article we investigate both the optical and qubit coherence properties of a $^{87}$Rb atom in a deep optical trap. 
We are particularly interested in realizing a ground state qubit with qubit-selective, closed optical transitions that are efficiently coupled to a propagating mode. 
Such a system can potentially be used to sequentially produce large numbers of entangled photons~\mbox{%DIFAUXCMD
\cite{Lindner2009,Schwartz2016, Lee2018}}%DIFAUXCMD
. 
Efficient }\DIFaddend light-atom coupling is achieved by trapping individual atoms in
optical tweezers and placing them at the \DIFdelbegin \DIFdel{focal spot }\DIFdelend \DIFaddbegin \DIFadd{focus }\DIFaddend of a high numerical aperture
lens \DIFdelbegin \DIFdel{. 
}\DIFdelend \DIFaddbegin \DIFadd{for a probe mode.
%DIF >  }
}\DIFaddend In previous work, we used such an arrangement to realize strong extinction of
a coherent beam by a single atom~\cite{Tey2008,Chin2017a}\DIFdelbegin \DIFdel{and }\DIFdelend \DIFaddbegin \DIFadd{, and to }\DIFaddend resolve scattering dynamics for various temporal profiles of the incident light~\cite{Aljunid2013,Leong2016,Steiner2017}. 
\DIFdelbegin \DIFdel{In this work}\DIFdelend \DIFaddbegin \DIFadd{Here}\DIFaddend , we show that conditions for efficient light-atom interaction, i.e.\DIFaddbegin \DIFadd{, }\DIFaddend strong extinction, are compatible with a long qubit coherence time.
%DIF <  of a ground state hyperfine qubit. 
In contrast to \DIFdelbegin \DIFdel{the experiments
in
\mbox{%DIFAUXCMD
\cite{Tey2008,Chin2017a,Aljunid2013,Leong2016,Steiner2017}}%DIFAUXCMD
, here }\DIFdelend \DIFaddbegin \DIFadd{our earlier experiments
\mbox{%DIFAUXCMD
\cite{Tey2008,Chin2017a,Aljunid2013,Leong2016,Steiner2017}}%DIFAUXCMD
, }\DIFaddend we use a
linearly polarized dipole trap, which strongly reduces atomic \DIFdelbegin \DIFdel{motion induced qubit dephasing}\DIFdelend \DIFaddbegin \DIFadd{motion-induced qubit dephasing, }\DIFaddend but affects the light-atom coupling through a tensor light shift. 
We perform transmission spectroscopy to investigate the impact of the tensor light shift on the optical coupling in detail.
\begin{figure} %[ht]
\centering
\DIFdelbeginFL %DIFDELCMD < \includegraphics[width=\columnwidth]{figure/geometry2.pdf}
%DIFDELCMD <   %%%
\DIFdelendFL %DIF >   \includegraphics[width=\columnwidth]{figure/geometry2.pdf}
  \DIFaddbeginFL \includegraphics[width=\columnwidth]{figure1.pdf}
  \DIFaddendFL \caption{\label{fig:geometry} (a) Simplified optical setup. A single atom is held by a linearly polarized FORT and is probed with a circularly polarized beam. 
  (b) Energy level scheme. The two 5\lev{S}{1}{2} Zeeman levels $\ket{F=2,m=-2}\equiv\ket{\uparrow}$ and $\ket{F=1,m=-1}\equiv\ket{\downarrow}$ are used as qubit states.  
  The $\ket{\uparrow}$ state is coupled via a closed optical transition to the 5\lev{P}{3}{2} $\ket{F'=3,m=-3}\equiv\ket{\textrm{aux}}$ state. 
}
\end{figure}

\section{Zeeman and light shift Hamiltonian}
%DIF < inconsistent use of zeeman and light shift(s) with abstract, shift or shifts? 
We consider an optically trapped $^{87}$Rb atom in a magnetic field applied along the quantization axis [z-axis, see Fig.~\ref{fig:geometry}(a)]. 
The magnetic field lifts the degeneracy of the Zeeman levels with the
corresponding Hamiltonian~\DIFdelbegin \DIFdel{$\hat{H}_\textrm{B,F}$ }\DIFdelend \DIFaddbegin \DIFadd{$\hat{H}_{B,F}$ }\DIFaddend for the hyperfine manifold~$F$,
\begin{equation}\label{eq:zeeman}
\hat{H}\DIFdelbegin \DIFdel{_\textrm{B,F} }\DIFdelend \DIFaddbegin \DIFadd{_{B,F} }\DIFaddend =  \hbar \omega_{L} \hat{F}_{z} \,,
\end{equation}
where $ \omega_{L}$ \DIFdelbegin \DIFdel{and $\hat{F}_{z}$ are }\DIFdelend \DIFaddbegin \DIFadd{is }\DIFaddend the Larmor frequency\DIFdelbegin \DIFdel{and }\DIFdelend \DIFaddbegin \DIFadd{, and $\hat{F}_{z}$ }\DIFaddend the z-component of the total angular momentum operator~$\hat{\textbf{F}}$ of the respective hyperfine level.
We use the two 5\lev{S}{1}{2} Zeeman levels\DIFaddbegin \DIFadd{,  }\DIFaddend $\ket{F=2,m=-2}\equiv\ket{\uparrow}$ and $\ket{F=1,m=-1}\equiv\ket{\downarrow}$\DIFaddbegin \DIFadd{,  }\DIFaddend as qubit states~\DIFdelbegin \DIFdel{(}\DIFdelend \DIFaddbegin [\DIFaddend Fig.~\ref{fig:geometry}\DIFdelbegin \DIFdel{b)}\DIFdelend \DIFaddbegin \DIFadd{(b)}]\DIFaddend . 
The choice of these states over the commonly used clock states $\ket{F=2,m=0}$ and $\ket{F=1,m=0}$ is motivated by the possibility to couple $\ket{\uparrow}$ via a closed optical transition to the 5\lev{P}{3}{2} $\ket{F'=3,m=-3}\equiv\ket{\textrm{aux}}$ state. 

The energy levels are further shifted by the light shift induced by the trapping field. 
%DIF < , which is serveral times
For each hyperfine manifold, the \DIFdelbegin \DIFdel{light-shift Hamiltonian~$\hat{H}_\textrm{ls,F}$ }\DIFdelend \DIFaddbegin \DIFadd{light shift Hamiltonian~$\hat{H}_{{\rm{ls}},F}$ }\DIFaddend can be decomposed into a scalar, a vector, and a tensor term,
\begin{equation}\label{eq:Hls}
 \hat{H}\DIFdelbegin \DIFdel{_\textrm{ls,F}}\DIFdelend \DIFaddbegin \DIFadd{_{{\rm{ls}},F}}\DIFaddend = U_0 \left( c_{s} + c_{v} \left(\epsilon^* \times \epsilon\right) \hat{\textbf{F}} + c_{t} \left|\epsilon \cdot \hat{\textbf{F}}\right|^2  \right)\,,
\end{equation}
where $U_0$ is the trap depth, $\epsilon$ is the polarization vector of the trapping field, and $c_{s}$, $c_{v}$, and $c_{t}$ are the coefficients of the scalar, vector, and tensor light shifts~\cite{Deutsch2010}. 

The qubit coherence is greatly affected when the trapping field causes a frequency shift~$\delta$ of the $\ket{\uparrow}$ to $\ket{\downarrow}$  transition. 
Then the qubit frequency changes as the atom oscillates in the trap, which
leads to dephasing on a timescale of \DIFdelbegin \DIFdel{$T^*_2\approx \frac{U_0}{\delta \pi k_B T_\textrm{atom}}$ ($T_\textrm{atom}$ }\DIFdelend \DIFaddbegin \DIFadd{$T^*_2\approx U_0/({\delta \pi k_B
  T_{\rm{atom}}})$, where $T_{\rm{atom}}$ }\DIFaddend is the temperature of the atom and $k_B$ \DIFdelbegin \DIFdel{is }\DIFdelend the Boltzmann constant\DIFdelbegin \DIFdel{)}\DIFdelend ~\cite{Romalis1999}.
In a far off-resonant dipole trap~(FORT), the contribution of the scalar and the tensor term to the shift~$\delta$ is negligible~($c_{s}$ is the same for the two ground state hyperfine manifolds  5\lev{S}{1}{2}  $F=1,2$ and $c_{t}\approx  0$).  
The vector light shift, however, leads to rapid dephasing.
For example, a 1\,mK-deep, circularly polarized trap at 851\,nm shifts the qubit frequency by $\delta=2.6$\,MHz;
\DIFdelbegin \DIFdel{thus }\DIFdelend for a typical atom temperature of $50\,\mu$K, the \DIFaddbegin \DIFadd{corresponding }\DIFaddend dephasing time $T^*_2\approx2\,\mu$s is prohibitively short for quantum information purposes. %we know that we cooled to 50muK in circular trap, but is 50muK 'typical', maybe we can cite the pgc paper here for the temperature? (ys)
Therefore, we use a FORT linearly polarized along the x-axis, for which the vector shift vanishes~$\left(\epsilon^* \times \epsilon=0\right)$. 
In this configuration the light-shift Hamiltonian for the excited state hyperfine manifold 5\lev{P}{3}{2}~$F'=3$ reads
\begin{equation}\label{eq:Hex}
 \hat{H}\DIFdelbegin \DIFdel{_\textrm{ls,F'=3}}\DIFdelend \DIFaddbegin \DIFadd{_{{\rm{ls}},F'=3}}\DIFaddend = U_0 \left( c_{s}  + c_{t} F^2_x  \right)\,,
\end{equation}
with $c_{s}= 0.7417$ and $c_{t}= -0.0716$ for a FORT operating at 851\,nm.
The nonlinear term proportional to $F^2_x$ leads to energy eigenstates that are
superpositions of either even or odd $m_z$ states. 
Consequently, the absorption spectrum and, in particular, the optical coupling
between~$\ket{\uparrow}$ and $\ket{\textrm{aux}}$ depend strongly on the relative strength of Zeeman and light shift. 

\section{Transmission experiment}
To determine the impact of the light shift on the optical coupling, we perform transmission spectroscopy on a single $^{87}$Rb atom in a tightly focused red-detuned FORT~\cite{Schlosser2001}. 
The atom is held between two high numerical aperture lenses (NA=0.75, focal length~$f$=5.95\,mm) with a 2.24\,mK-deep FORT operating at a wavelength 851\,nm~\cite{Chin2017,Chin2017a}. 
%DIF < The dipole trap is formed by 851\,nm light that is tightly focused by a high numerical aperture lens (NA=0.75, focal length~$f$=5.95\,mm, see Fig.~\ref{fig:setup})~\cite{Chin2017,Chin2017a,Chin2017b}. 
\DIFaddbegin \DIFadd{The trapping beam is linearly polarized (with a polarization
 extinction ratio $\approx$34\,dB~\mbox{%DIFAUXCMD
\cite{Chin2017b}}%DIFAUXCMD
) and focused to a waist~$w_0\approx 1.4\,\mu$m.
}\DIFaddend Part of the atomic fluorescence is collected by the same lenses and coupled to single mode fibers connected to avalanche photodetectors, $D_1$ and $D_2$~(Fig.~\ref{fig:setup}). 
\begin{figure} %[ht]
\centering
\DIFdelbeginFL %DIFDELCMD < \includegraphics[width=\columnwidth]{figure/setup.pdf}
%DIFDELCMD <   %%%
\DIFdelendFL %DIF >   \includegraphics[width=\columnwidth]{figure/setup.pdf}
  \DIFaddbeginFL \includegraphics[width=\columnwidth]{figure2.pdf}
  \DIFaddendFL \caption{\label{fig:setup}  Optical setup. 
  %DIF <  for trapping single atoms. 
  D$_{1(2)}$:~single mode fiber connected to avalanche photodetector, (P)BS: (polarizing) beam splitter, $\lambda/4$:~quarter-wave plate, IF: interference filter. 
}
\end{figure}

After loading an atom into the FORT, we cool the atom to 16.4(6)\,$\mu$K by 10\,ms of polarization gradient cooling~\cite{Chin2017b}. 
Then, a bias magnetic field is applied along the quantization axis (z-axis), and the atom is optically pumped into $\ket{\uparrow}$. 
We probe the light-atom interaction with a circularly polarized ($\sigma^-$) beam, driving the transition $\ket{\uparrow}$ to $\ket{\textrm{aux}}$ near 780\,nm. 
The Rabi frequency of the driving field $\Omega = 0.052(3)\Gamma$ is set far below saturation~($\Gamma=2\pi\times 6.07\,$MHz is the spontaneous decay rate). 
During the 1\,ms-long probe pulse, we accumulate the number of detected photons~\DIFdelbegin \DIFdel{$n_\textrm{p}$ }\DIFdelend \DIFaddbegin \DIFadd{$n_{\rm{p}}$ }\DIFaddend at the detector~$D_2$.  
We then obtain the transmission \DIFdelbegin \DIFdel{$T=n_\textrm{p}/n_0$ by comparing $n_\textrm{p}$ }\DIFdelend \DIFaddbegin \DIFadd{$T=n_{\rm{p}}/n_0$ by comparing $n_{\rm{p}}$ }\DIFaddend to the number\DIFaddbegin \DIFadd{~$n_0$ }\DIFaddend of detected photons in a reference measurement \DIFdelbegin \DIFdel{~$n_0$ }\DIFdelend during which the atom is in a state off-resonant with the probe field (F=1). 
\begin{figure} %[ht]
\centering
\DIFdelbeginFL %DIFDELCMD < \includegraphics[width=\columnwidth]{figure/tx_matrix.eps}
%DIFDELCMD <   %%%
\DIFdelendFL %DIF >   \includegraphics[width=\columnwidth]{figure/tx_matrix.eps}
  \DIFaddbeginFL \includegraphics[width=\columnwidth]{figure3.pdf}
  \DIFaddendFL \caption{\label{fig:tx_matrix} Transmission spectra of a single atom in a deep (a,b) and a shallow (c,d) trap. 
  For weak magnetic fields, the light shift of the FORT leads to strong state mixing in the excited state.
  In the shallow trap and with a strong magnetic field applied, the probe field
  couples efficiently to the transition between $\ket{\uparrow}$ and
  $\ket{\textrm{aux}}$ and a high extinction ($\approx23\%$) is observed. }
\end{figure}

Figure~\ref{fig:tx_matrix}(a) shows the observed transmission spectrum as we vary the frequency of the probe field and the amplitude of the bias magnetic field. 
We observe a peak extinction of $\epsilon= 1-T=8.2(3)\%$ for the largest magnetic field applied (\DIFdelbegin \DIFdel{$144\,\mu$T}\DIFdelend \DIFaddbegin \DIFadd{$1.44\,$mT}\DIFaddend ). 
As the magnetic field strength is reduced, the spectrum shows a lower peak extinction and multiple, broad resonances. 
This is in stark contrast to the strong extinction~($\approx22\%$) we observed in our previous experiments with the same optical setup \DIFdelbegin \DIFdel{but }\DIFdelend with a circularly polarized FORT~\cite{Chin2017a}. 
We then repeat the experiment, but this time, after polarization gradient cooling, we lower the trap depth to 0.88\,mK. 
This increases the observed extinction significantly [Fig.\DIFaddbegin \DIFadd{~}\DIFaddend \ref{fig:tx_matrix}(c)]. 
For our largest magnetic field the transmission spectrum consists of a single Lorentzian line with high peak extinction~$\epsilon=23.3(3)\%$.  

To better understand the effect of the tensor light shift on the transmission spectrum, we numerically calculate the dynamics for the 12-level system containing the $F=2$ and $F'=3$ manifolds. 
Aside from the Zeeman and light shifts [Eq.~(\ref{eq:zeeman}-\ref{eq:Hex})], we include a term in the Hamiltonian that
describes the interaction with the $\sigma^-$-polarized probe light detuned
from the natural $F=2$ to $F'=3$ transition frequency $\omega_{0}$ by $\Delta
= \omega_{p}-\omega_{0}$,
\begin{equation}\hat{H}\DIFdelbegin \DIFdel{_\textrm{int}}\DIFdelend \DIFaddbegin \DIFadd{_{\rm{int}}}\DIFaddend =-\frac{\hbar}{2}   \Omega \hat{A}_- + \text{h.c.}\,,
\end{equation}
where $\hat{A}_-$  is the atomic lowering operator. For the total Hamiltonian
\begin{eqnarray}\label{eq:totalhamiltonian}
\hat{H}&=&  \hat{H}_{0} + \hat{H}\DIFdelbegin \DIFdel{_{\textrm{B},F=2} }\DIFdelend \DIFaddbegin \DIFadd{_{B,F=2} }\DIFaddend +
\hat{H}\DIFdelbegin \DIFdel{_{\textrm{B},F'=3}}\DIFdelend \DIFaddbegin \DIFadd{_{B,F'=3}}\DIFaddend \nonumber\\
&&{} + \hat{H}\DIFdelbegin \DIFdel{_{\textrm{LS},F=2} }\DIFdelend \DIFaddbegin \DIFadd{_{{\rm{ls}},F=2} }\DIFaddend + \hat{H}\DIFdelbegin \DIFdel{_{\textrm{LS},F'=3} }\DIFdelend \DIFaddbegin \DIFadd{_{{\rm{ls}},F'=3} }\DIFaddend + \hat{H}\DIFdelbegin \DIFdel{_{\textrm{int}}}\DIFdelend \DIFaddbegin \DIFadd{_{\rm{int}}}\DIFaddend \,,
\end{eqnarray}
with $\hat{H}_{0}= -\Delta \mathbb{1}_{F=2}$,  where $\mathbb{1}_{F=2}$ is the
unity operator acting on the $F=2$ manifold, we numerically solve the
corresponding master equation
\begin{equation}\dot{\rho}=-\frac{i}{\hbar}[\rho,\hat{H}] + \mathcal{L}[\rho]\,,
\end{equation}
with a Lindblad superoperator $\mathcal{L}[\rho]$ to account for
spontaneous emission.

We initialize in $\ket{\uparrow}$ and apply the probe field for a time~$\tau=1\,\textrm{ms}\gg1/\Omega \gg1/\Gamma$. 
Comparing the number of scattered photons during the probe phase,
\begin{equation}
n_p(\Delta)=\int^{\tau}_0 \textrm{Tr}\left(\rho(t) P_{F'=3} \right) \Gamma dt\,,
\end{equation}
with the value expected for a resonantly driven two-level system,
\begin{equation}
n_{2l}=\frac{\Omega^2}{\Gamma^2+2\Omega^2} \Gamma \tau \approx100\,,
\end{equation}
we obtain an
expected reduction $\eta=n_p(\Delta)/n_{2l}$ of the absorption. Here, $P_{F'=3}$ is the projector \DIFdelbegin \DIFdel{on }\DIFdelend \DIFaddbegin \DIFadd{onto }\DIFaddend the $F'=3$ manifold. 
The estimated transmission as a function of probe detuning is then
\begin{equation}T(\Delta) = 1- \epsilon_0 \eta(\Delta)\,,
\end{equation}
where $\epsilon_0$ is the resonant, two-level extinction value which depends on the spatial mode of the probe fields. 
%DIF < We repeat the simulation for various probe detunings~$\Delta$ to get the expected absorption spectrum. 


\begin{figure} %[ht]
\centering
\DIFdelbeginFL %DIFDELCMD < \includegraphics[width=\columnwidth]{figure/tx_over_trapdepth.eps}
%DIFDELCMD <   %%%
\DIFdelendFL %DIF >   \includegraphics[width=\columnwidth]{figure/tx_over_trapdepth.eps}
  \DIFaddbeginFL \includegraphics[width=\columnwidth]{figure4.pdf}
  \DIFaddendFL \caption{\label{fig:tx_over_trapdepth} Resonant extinction for various trap depths. Red circles: measured extinction, red line: full 12-level numerical simulation,  black dashed line: first order approximation \DIFdelbeginFL \DIFdelFL{$\epsilon_0 \eta_\textrm{dme}^{(1)}$}\DIFdelendFL \DIFaddbeginFL \DIFaddFL{$\epsilon_0 \eta_{\textrm{dme}}^{(1)}$}\DIFaddendFL ~[Eq.~(\ref{eq:eta})], blue line: numerically calculated~\DIFdelbeginFL \DIFdelFL{$\epsilon_0\eta_\textrm{dme}$}\DIFdelendFL \DIFaddbeginFL \DIFaddFL{$\epsilon_0\eta_{\textrm{dme}}$}\DIFaddendFL . }
\end{figure}
We find an excellent agreement between the observed spectrum $T(\Delta)$ and the model for $\epsilon_0=24.7\%$, a value consistent with our previous experiment~\cite{Chin2017a}~[Fig.~\ref{fig:tx_matrix}(b,d)]. 
%DIF < how do we arrive at eps0=24.7%? it is adjusted to the first point? or a full fit to the simulation? - ys
%DIF < I think to the largest extinction value
We further test our model by comparing the resonant extinction at various trap depths\DIFaddbegin \DIFadd{, }\DIFaddend but a fixed magnetic field strength of \DIFdelbegin \DIFdel{$144\,\mu$T}\DIFdelend \DIFaddbegin \DIFadd{1.44\,mT}\DIFaddend ~(Fig.~\ref{fig:tx_over_trapdepth}). 
Again\DIFaddbegin \DIFadd{, }\DIFaddend the model matches the experimental data well. 
%DIF < To further understand the scattering process, we estimate the reduction of
%DIF < the optical coupling of $\ket{\uparrow}$ to $\ket{\textrm{aux}}$ from the imperfect overlap $\eta_\textrm{overlap}=|\braket{\psi| \textrm{aux}}|^2$ between  $\ket{\textrm{aux}}$ and the new eigenstate $\ket{\psi}$ that corresponds to $\ket{\textrm{aux}}$ in the limit of vanishing light shift. 
To further understand the scattering process, 
we consider the relevant dipole matrix element: 
%DIF < we compare our observed extinction values to the values expected from the relevant dipole matrix element: 
\DIFdelbegin \DIFdel{For }\DIFdelend \DIFaddbegin \DIFadd{for a }\DIFaddend vanishing light shift\DIFdelbegin \DIFdel{both, }\DIFdelend \DIFaddbegin \DIFadd{, both }\DIFaddend $\ket{\uparrow}$ and $\ket{\textrm{aux}}$ \DIFdelbegin \DIFdel{, }\DIFdelend are energy eigenstates. 
The $\sigma^-$-polarized probe beam couples these two states with a dipole matrix element \DIFdelbegin \DIFdel{$|\braket{\uparrow|\epsilon_{\sigma^-}\textbf{d}| \textrm{aux}}|^2$, where the $\epsilon_{\sigma^-}$ }\DIFdelend \DIFaddbegin \DIFadd{$|\braket{\uparrow|{\bm{\epsilon}}_{\sigma^-}\hat{\textbf{d}}| \textrm{aux}}|^2$, where ${\bm{\epsilon}}_{\sigma^-}$ }\DIFaddend is the polarization vector of the probe beam, \DIFdelbegin \DIFdel{$\textbf{d}$ }\DIFdelend \DIFaddbegin \DIFadd{$\hat{\textbf{d}}$ }\DIFaddend is the electric dipole operator.  
With increasing transverse tensor shift~[Eq.~(\ref{eq:Hex})], the angular
momentum eigenstate $\ket{\textrm{aux}}$ is no longer an energy eigenstate \DIFdelbegin \DIFdel{--
}\DIFdelend \DIFaddbegin \DIFadd{---
}\DIFaddend the corresponding eigenstate~$\ket{\widetilde{\textrm{aux}}}$ of
Eq.~(\ref{eq:totalhamiltonian}) gets admixtures from other $m_z$-states of the same level. 
Thus, the optical coupling strength is reduced by the relative reduction of the dipole matrix element
\begin{equation}
\eta\DIFdelbegin \DIFdel{_\textrm{dme}}\DIFdelend \DIFaddbegin \DIFadd{_{\rm{dme}}}\DIFaddend =\DIFdelbegin \DIFdel{\frac{ |\braket{\uparrow|\epsilon_{\sigma^-}\textbf{d}| \widetilde{\textrm{aux}}}|^2}{|\braket{\uparrow|\epsilon_{\sigma^-}\textbf{d}| \textrm{aux}}|^2}}\DIFdelend \DIFaddbegin \DIFadd{\frac{ |\braket{\uparrow|{\bm{\epsilon}}_{\sigma^-}\hat{\textbf{d}}| \widetilde{\textrm{aux}}}|^2}{|\braket{\uparrow|{\bm{\epsilon}}_{\sigma^-}\hat{\textbf{d}}| \textrm{aux}}|^2}}\DIFaddend .
\end{equation}
%DIF < and  $\ket{\widetilde{\textrm{aux}}}$ is the eigenstate that corresponds to $\ket{\textrm{aux}}$ in the limit of vanishing light shift. 
%DIF < We normalize the matrix element so that $|\braket{\uparrow|\epsilon_{\sigma^-}\textbf{d}| \textrm{aux}}|^2=1$.
For strong magnetic fields, $\hbar \omega_L \gg U_0 c_{t}$, the reduction of the dipole matrix element
%DIF <  $\eta_\textrm{overlap}$
in first order approximation is given by 
\begin{equation}\label{eq:eta}
 \eta\DIFdelbegin \DIFdel{_\textrm{dme}}\DIFdelend \DIFaddbegin \DIFadd{_{\rm{dme}}}\DIFaddend ^{(1)} = 1 -\frac{15}{8} \left(\frac{U_0 c_{t}}{\hbar\omega_L}\right)^2\,.
\end{equation}
%DIF < Figure \ref{fig:tx_over_trapdepth} shows $\epsilon_0\eta_\textrm{overlap}^{(1)}$ as well as the numerically calculated $\epsilon_0\eta_\textrm{overlap}$. 
However, neither \DIFdelbegin \DIFdel{$\epsilon_0 \eta_\textrm{dme}^{(1)}$ }\DIFdelend \DIFaddbegin \DIFadd{$\epsilon_0 \eta_{\rm{dme}}^{(1)}$ }\DIFaddend nor the numerically calculated~\DIFdelbegin \DIFdel{$\epsilon_0\eta_\textrm{dme}$ }\DIFdelend \DIFaddbegin \DIFadd{$\epsilon_0\eta_{\rm{dme}}$ }\DIFaddend reproduces our measured values well~(Fig.~\ref{fig:tx_over_trapdepth}). 
The reason is that the observed spectrum is strongly affected by multiple scattering events. 
When the energy eigenstates are superpositions of $m_z$ states, there is a probability that a scattering event brings the atom out of the $\left\{\ket{\uparrow}, \ket{\textrm{aux}}\right\}$ subspace. 
After such an optical depolarization event, the resonance frequency is shifted, and thus the optical coupling is strongly reduced.
The full numerical simulation takes these spin flips into account, resulting in a good match with the experimental data. 

From the comparison between experiment and theory we learn that (I) it is indeed the tensor light shift that causes the complexity of the transmission spectrum, 
(II) the spin dynamics induced by multiple scattering events are important for the spectral shape of the spectrum, 
and (III) the optical coupling between $\ket{\uparrow}$ and $\ket{\textrm{aux}}$ is close to an ideal two-level system in a shallow trap with a strong magnetic field applied. 

\section{Ground State Qubit}
We characterize the ground state qubit in terms of state readout fidelity and
coherence time to show that efficient optical coupling and a long coherence
time of the qubit can be simultaneously achieved. 
For the following experiments, we choose a trap depth $U_0=k_B\times 0.88\,$mK
and a magnetic field strength of \DIFdelbegin \DIFdel{$144\,\mu$T}\DIFdelend \DIFaddbegin \DIFadd{1.44\,mT}\DIFaddend , in which the highest optical coupling is observed. 
\DIFdelbegin \DIFdel{The state readout fidelity is determined by preparing the atom in a particular state, and then illuminating the atom }\DIFdelend \DIFaddbegin 

\DIFadd{Our state readout method is based on state-selective fluorescence detection~\mbox{%DIFAUXCMD
\cite{Fuhrmanek2011,Gibbons2011}}%DIFAUXCMD
:  
we illuminate the atom }\DIFaddend for $600\,\mu$s with light \DIFaddbegin \DIFadd{that is }\DIFaddend resonant with the 5\lev{S}{1}{2}  $F=2$ to 5\lev{P}{3}{2}  $F'=3$ transition \DIFdelbegin \DIFdel{~\mbox{%DIFAUXCMD
\cite{Fuhrmanek2011,Gibbons2011}}%DIFAUXCMD
. 
The }\DIFdelend \DIFaddbegin \DIFadd{and infer the qubit state from the }\DIFaddend number of photons \DIFdelbegin \DIFdel{detected at }\DIFdelend \DIFaddbegin \DIFadd{registered at photodetectors }\DIFaddend $D_1$ and $D_2$\DIFdelbegin \DIFdel{allows us to infer the qubit state}\DIFdelend \DIFaddbegin \DIFadd{.
The circularly polarized state detection beam propagates perpendicular to the quantization axis and is back reflected with opposite circular polarization~(Fig.~\ref{fig:setup}).
}

\DIFadd{The readout fidelity is determined by preparing the atom in a particular state, 
and then performing the state readout}\DIFaddend .
When the atom is initially in $\ket{\uparrow}$, we detect a mean number of $n_\uparrow = 9.85(8)$~photons. 
For an atom in $\ket{\downarrow}$, the atom ideally scatters almost no photons
because of the large hyperfine splitting of $6.8$\,GHz\DIFdelbegin \DIFdel{, but }\DIFdelend \DIFaddbegin \DIFadd{; }\DIFaddend we occasionally 
register one or two detection events~(mean number of $n_\downarrow = 0.17(1)$~photons).
For this measurement, we indiscriminately prepare the atom in the
5\lev{S}{1}{2} $F=1$  as the same dark state behavior is expected for all three Zeeman levels. 
\begin{figure} %[ht]
\centering
\DIFdelbeginFL %DIFDELCMD < \includegraphics[width=\columnwidth]{figure/state_multi.eps}
%DIFDELCMD <   %%%
\DIFdelendFL %DIF >   \includegraphics[width=\columnwidth]{figure/state_multi.eps}
  \DIFaddbeginFL \includegraphics[width=\columnwidth]{figure5.pdf}
  \DIFaddendFL \caption{\label{fig:statereadout_histo}
  (a)~Histogram of photon detection probability for atoms prepared in $F=1$
  (blue) and $F=2$ (red), respectively. 
  (b)~Rabi oscillation between $\ket{\uparrow}$ and $\ket{\downarrow}$. 
  (c)~Ramsey (red) and spin-echo (blue) when the atom is
  initially prepared in $\ket{\uparrow}$~(solid symbols) or
  $\ket{\downarrow}$~(empty symbols). We fit a decaying exponential to the
  Ramsey signal and a decaying Gaussian to the spin-echo signal to extract the
  $1/e$ time constants, $T^*_2= 38(3)\mu$s and $T_2=446(14)\,\mu$s.  }
\end{figure}

Figure~\ref{fig:statereadout_histo}(a) shows the histogram of $n_\uparrow$ and $n_\downarrow$ after 3000 repetitions of the experiment\DIFdelbegin \DIFdel{. From the histogram }\DIFdelend \DIFaddbegin \DIFadd{, from which }\DIFaddend we determine a threshold value~\DIFdelbegin \DIFdel{$n_\textrm{th}=2$ }\DIFdelend \DIFaddbegin \DIFadd{$n_{\textrm{th}}=2$ }\DIFaddend that optimizes the discrimination between the two distributions. The probabilities for erroneous state assignment are $\chi_\uparrow =1.5\%$ and $\chi_\downarrow =1.4\%$ for an atom prepared in $\ket{\uparrow}$ and $\ket{\downarrow}$, respectively. 
Thus, we achieve a state readout fidelity $F= 1 - (\chi_\uparrow + \chi_\downarrow)/2=98.6(2)\%$, similar to previously reported values~\cite{Fuhrmanek2011,Gibbons2011,Kwon2017,Martinez-Dorantes2017}. 
%DIF < maybe drop 'good' here, we have been using it a lot

To characterize the qubit coherence properties, we apply a microwave field resonant with the $\ket{\uparrow}$ to $\ket{\downarrow}$ transition to drive Rabi oscillations\DIFaddbegin \DIFadd{, }\DIFaddend and perform Ramsey and spin-echo sequences~\cite{Kuhr2003,Andersen2003,Andersen2004,Kuhr2005,Rosenfeld2011}.  
We observe a Rabi frequency of $\Omega_{\textrm{mw}}=2\pi\times 39.6(5)\,$kHz
with a visibility of $0.89(1)$ and little damping in the first $60\,\mu$s [Fig.~\ref{fig:statereadout_histo}(b)]. 
%DIF < We suspect that the imperfect visibility of the oscillation $0.232$ is due to poor state preparation in $\ket{\uparrow}$. 
The dephasing time is determined from a Ramsey experiment, where we apply two resonant microwave pulses for $t_{\pi/2}=\pi/(2 \Omega)$ separated by a free evolution time $\tau$~[Fig.~\ref{fig:statereadout_histo}(c)]. 
We fit an exponential decay to the Ramsey contrast \DIFdelbegin \DIFdel{, and }\DIFdelend \DIFaddbegin \DIFadd{from which we }\DIFaddend extract the dephasing time~$T^*_2= 38(3)\mu$s. 
Subsequently, applying a spin-echo, i.e.\DIFaddbegin \DIFadd{, }\DIFaddend inserting an extra microwave pulse for
$t_{\pi}=\pi / \Omega$ halfway in the free evolution period~$\tau$, the inhomogeneous dephasing is reversed\DIFaddbegin \DIFadd{, }\DIFaddend and we observe a much slower decay of the contrast. 
Fitting to a \DIFaddbegin \DIFadd{heuristically chosen }\DIFaddend decaying Gaussian, we obtain the coherence time~$T_2=446(14)\,\mu$s defined as the $1/e$ decay time of the spin-echo visibility. 
Despite not using the clock states, we achieve a high ratio of state manipulation speed and preserved coherence, $T_2/t_{\pi}\approx 35$. 
%DIF < Although we do not use the clock states, a high ratio of state manipulation speed and preserved coherence is achieved, $T_2/t_{\pi}\approx 35$. 
The Rabi frequency can be further increased by using fast optical Raman transitions~\cite{Yavuz2006,Jones2007}.
Figure~\ref{fig:statereadout_histo}(c) also displays the results of
Ramsey and spin-echo experiments where we initially prepare the atom in state
$\ket{\downarrow}$. 
As expected, the observed values mirror the experiments starting from~$\ket{\uparrow}$.

\section{Conclusion}
We have shown that a combination of a shallow optical trap and strong magnetic fields sufficiently mitigates the effects of the tensor light shift on the optical coupling. 
Under these conditions, we demonstrated high qubit coherence and readout fidelity.
The capability of coupling qubit states selectively to well-defined optical channels enables new ways of building up hybrid light-atom quantum states.
%DIF < This combination of good qubit and good optical coupling enables new ways of building up hybrid light-atom quantum states.  
In particular, we expect that several protocols that were originally developed
for solid state quantum systems -- where qubit state-selective, closed optical transitions are common -- can be realized with a neutral atom in a
dipole trap. This includes the generation of time-bin atom-photon entanglement~\cite{Lee2018} and the sequential generation of entangled photons~\cite{Lindner2009,Schwartz2016}.
%DIF <  and enable ... seems like a weird construction (ys)


\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.
%DIF <  (partly under grant No. ?).
M.\,Steiner acknowledges support by the Lee Kuan Yew Postdoctoral Fellowship.
\end{acknowledgments}

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