Present address:] ICFO-Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain.

# A quantum-network register assembled with optical tweezers in an optical
cavity

Lukas Hartung1<EMAIL_ADDRESS>Matthias Seubert1 Stephan Welte1,2
Emanuele Distante1 [  Gerhard Rempe1 1Max-Planck-Institut für Quantenoptik,
Hans-Kopfermann-Straße 1, 85748 Garching, Germany  2Institute for Quantum
Electronics, ETH Zürich, Otto-Stern-Weg 1, 8093 Zürich, Switzerland

###### Abstract

Quantum computation and quantum communication are expected to provide users
with capabilities inaccessible by classical physics. However, scalability to
larger systems with many qubits is challenging. One solution is to develop a
quantum network consisting of small-scale quantum registers containing
computation qubits that are reversibly interfaced to communication qubits.
Here we report on a register that uses both optical tweezers and optical
lattices to deterministically assemble a two-dimensional array of atoms in an
optical cavity. Harnessing a single-atom addressing beam, we stimulate the
emission of a photon from each atom and demonstrate multiplexed atom-photon
entanglement with a generation-to-detection efficiency approaching 90%.
Combined with cavity-mediated quantum logic, our approach provides a possible
route to distributed quantum information processing.

Quantum networks [1, 2] offer potential for numerous applications, including
secure communication [3], distributed quantum computing [4, 5, 6], precision
sensing [7], and clock synchronization [8]. The envisioned architecture
comprises network nodes with stationary qubits for processing quantum
information, connected via optical fibers for the exchange of photonic qubits.
However, optical losses and unavoidable errors introduce substantial
challenges in practical implementations. The solution is using a scalable
multi-qubit register per node: this allows overcoming losses with multiplexed
communication protocols [9, 10] including repeaters [9, 11, 12, 13], as well
as increasing fidelities through entanglement distillation [14] and quantum
error correction [15].

The goal then is to create a register of individually controllable qubits and
couple each of them to a photonic channel for network connectivity [16]. We
address this twofold challenge by combining cavity quantum electrodynamics
(QED) [17] with the atom-based technologies of optical lattices and optical
tweezers [18]. Specifically, the cavity serves as an atom-photon quantum
interface, the lattice provides minimally perturbing atomic localization, and
the tweezers are used to assemble on demand any desired atomic configuration
by means of tightly focused laser beams that can be maneuvered at will [19,
20, 21]. While large arrays of tweezers can realize registers with hundreds of
qubits [22, 23], no experiment has been performed in the desired network
setting, arguably because the cavity structure restricts the optical access
required for the tweezers. In fact, despite long-standing efforts, only a few
examples of cavity-coupled atoms trapped in tweezers have been reported [24,
25, 26]. All of them lack individual control over the atomic qubit, a
necessary requirement for applications in a quantum network [27].

With an array of optical tweezers and the intra-cavity optical lattice, we
achieve individual control over a set of atomic qubits in a network setting.
We create one- and two-dimensional registers of up to $n{=}6$ atoms and
address each atom separately to generate atom-photon entanglement (APE). We
observe that the APE fidelity remains constant for up to six qubits in the
register, indicating that scalability to larger system sizes is possible. By
implementing a multiplexing scheme, we generate APE with a probability
exceeding $\sim$ $\qty{97}{}$ per attempt, an important step towards the
deterministic distribution of entanglement across networks. All this is
achieved by exploiting the advantages of a Fabry-Pérot cavity that allows for
optical access in a plane perpendicular to its symmetry axis.

Experimental setup. Figure 1A schematically shows our setup to prepare and
individually control two-dimensional arrays of in our case [87]Rb atoms at the
center of the cavity. Laser beams propagating along the $z$-axis are focused
at the cavity plane to $\qty{1.4}{\micro}$ waist radius by an objective
(NA=0.4), and their positions are precisely controlled using two (only one
shown) two-dimensional acousto-optic deflectors (2D-AODs). This enables
selective addressing of individual atoms across a broad wavelength range from
$\qtyrange{770}{810}{\nano}$, covering the $D_{1}$ and $D_{2}$ line of [87]Rb.
Atom arrays are prepared using $\qty{797}{\nano}$ light and driving one pair
of AODs with multiple radio-frequency (RF) tones [21]. This creates a static
two-dimensional array of tweezers which are stochastically loaded with cold
atoms from a magneto-optical trap. Atoms are cooled using pairs of counter-
propagating laser beams aligned at 45 degrees in the $x,z$ plane (not shown)
[28]. By detecting the scattered cooling light using an electron-multiplying
charge-coupled device (EMCCD) camera, we image the initial atoms positions.
Based on the images captured, we use the second pair of AODs to create
additional tweezers that move and arrange the atoms into ordered arrays within
the static tweezers. We observe an increase of about four orders of magnitude
in the success rate of preparing arrays of six atoms compared to stochastic
loading (see supplement [29]), thus laying the foundation for loading many
more atoms than just one or two as in previous experiments.

Figure 1: Setup and experimental sequence. (A) Tweezers and vSTIRAP lasers are
steered via 2D-AODs onto individual atoms at the cavity center to generate and
control two-dimensional atomic arrays. Atoms are imaged via an electron-
multiplying charge-coupled device (EMCCD) camera, and feedback based on the
camera pictures is applied to the 2D-AODs to prepare ordered atomic arrays.
After the preparation, the atoms are transferred to a 2D lattice formed by a
1064 standing wave trap and a 770 intra-cavity standing wave trap. Global
Raman lasers drive transitions between the ground states. (B) Photons emitted
into the cavity mode predominantly exit via the outcoupling mirror and pass
through an optical fiber and an electro-optical modulator (EOM), allowing to
switch the detection basis of the photons. The photons are eventually detected
with efficient superconducting nanowire detectors. (C) Protocol for APE
generation. After optical pumping to $\ket{F{=}2,m_{F}{=}0}$, we apply
sequentially $N$ local vSTIRAP pulses for photon generation (signal photons,
red wiggly arrows). Global $\pi/2$ and $\pi$ pulses set the detection basis
for the atom readout. Eventually, $N$ vSTIRAP pulses generate $N$ readout
photons. The measured time trace in blue shows the signal and readout photons.

Once the preparation is completed, the atoms are transferred from the tweezers
to a two-dimensional optical lattice consisting of a red-detuned standing wave
trap at 1064 orthogonal to the cavity axis ($x$-axis in Fig. 1A) and an intra-
cavity blue-detuned standing wave trap at 770 ($y$-axis in Fig. 1A). During
this process, the atom position may change by up to 328. This is negligible
with regard to the typical atom separation of a few given by the distance
between the tweezers. Compared to trapping atoms in tweezers, the lattice
ensures stronger atom confinement and overcomes limitations in array size
imposed by finite laser power. Crucially, the lattice avoids the emergence of
near-focus extraordinary polarization components, as observed in tightly
focused optical tweezers [30, 31]. These components create large virtual
magnetic fields, reducing the coherence of magnetically sensitive atomic
qubits and posing challenges for entangling atoms and photons in a cavity. The
770 light is also used to stabilize the length of the cavity, which is a high-
finesse Fabry-Pérot resonator with unbalanced mirror transmissivities
$T{=}\qty{4}{ppm}$ and $T{=}\qty{92}{ppm}$ (finesse
$\mathcal{F}=\qty{61(2)e3}{}$). This ensures that light within the cavity
exits predominantly through the higher-transmissivity outcoupling mirror.
Photons exiting from the cavity pass through an optical fiber and are detected
with a polarization-resolving detection setup (Fig. 1B) consisting of an
electro-optic modulator (EOM), a set of waveplates (not shown), a polarizing
beamsplitter (PBS), and two fiber-coupled superconducting nanowire single-
photon detectors (SNSPDs). State initialization and coherent Raman transitions
between the ground states are implemented with large laser beams aligned
orthogonal to the cavity that illuminate the atoms simultaneously with
homogeneous intensity. Throughout the entire experimental sequence, we apply a
bias magnetic field aligned with the cavity axis, inducing a Zeeman splitting
with a Larmor frequency of $\omega_{L}=2\pi\times\qty{100}{\kilo}$.

Figure 2: Images of the atom arrays for up to $\mathbf{n=6}$ atoms with
efficiency and fidelity for each atom. The atoms are arranged along the
$x$-axis, perpendicular to the cavity mode. The atoms are spaced
$\qty{5.5}{\micro}$ apart. From time to time, the atoms jump along the
resonator axis within the blue standing wave trap ($y$-axis). By constantly
evaluating the images and recalculating the necessary RF frequencies, we can
follow the movement of the atoms with our addressing system. The APE fidelity
$\mathcal{F}$ for each individual atom is independent of the number and the
position of the atoms in the cavity. However, the efficiency of the photon
emission process decreases for atoms displaced from the center of the cavity
as the coupling $g$ decreases.

Quantum protocol. Once the atomic array of $n$ atoms has been successfully
prepared, we optically pump all atoms into the state $\ket{F{=}2,m_{F}{=}0}$
(Fig. 1C). We then illuminate the atoms one after the other via the addressing
system with a $\qty{780}{\nano}$ $\pi$-polarized light, red-detuned relative
to the $D_{2}$ line. The resonator frequency is tuned such that, together with
the $\pi$-light, the cavity stimulates off-resonant Raman scattering of a pump
photon into the cavity mode, a process known as vSTIRAP [32]. This transfers
the respective atom into one of the two possible states
$\ket{\uparrow}=\ket{F{=}1,m_{F}{=}+1}$ or
$\ket{\downarrow}=\ket{F{=}1,m_{F}{=}-1}$, depending on the polarization of
the photon, referred to as the signal photon. The signal photon and the atom
are then in the maximally entangled Bell state

$\ket{\Psi^{+}}=\frac{1}{\sqrt{2}}\left(\ket{\uparrow,R}+\ket{\downarrow,L}\right),$
(1)

with $\ket{R}$/$\ket{L}$ denoting a right/left circularly polarized photon. By
addressing the $n$ atoms sequentially, we ideally generate $n$ atom-photon
entangled pairs in $n$ predefined temporal modes, as shown in the measured
time trace of Fig. 1C.

We determine the entanglement fidelity
$\mathcal{F}=\bra{\Psi^{+}}\rho\ket{\Psi^{+}}$ of the generated state with
density matrix $\rho$ by measuring the correlations between the signal photon
polarization and the internal atomic state in three different bases. The
signal photons are measured using the polarization resolving setup described
above (see Fig. 1B). To readout the atomic state, a single-qubit rotation
using a two-photon Raman transition between $\ket{\uparrow}$ and
$\ket{\downarrow}$ (green pulses in Fig. 1C) is applied to set the measurement
basis (see supplement [29]). We then transfer $\ket{\uparrow}$ to
$\ket{F{=}2,m_{F}{=}2}$ and $\ket{\downarrow}$ to $\ket{F{=}2,m_{F}{=}-2}$
with two additional Raman pulses (yellow in Fig. 1C). Finally, we apply atom-
selective vSTIRAP pulses to map each atomic state onto a corresponding state
of a readout photon. By measuring the readout photons in the circular basis,
the internal atomic state can be uniquely determined as $\ket{R}$ implies
$\ket{F{=}2,m_{F}{=}-2}$ and $\ket{L}$ implies $\ket{F{=}2,m_{F}{=}2}$. We use
the EOM to rapidly switch the detection basis of the polarization resolving
setup since the signal photons are measured along any desired polarization
basis while the readout photons are observed along the circular basis.

A multi-qubit register. We measure the APE fidelity for an increasing number
of atomic qubits. Fig. 2 illustrates the shapes and sizes of the prepared atom
arrays and indicates the APE fidelity $\mathcal{F}$ and efficiency $\eta$ for
each atom individually. Because of their finite temperature, the atoms do not
always stay perfectly aligned along the cavity axis. Instead, they can hop
between different lattice sites, causing variations in their position.
However, we continuously monitor the position of the atoms by taking
fluorescence pictures and dynamically adjust the radio-frequency signal
supplying the AODs to address each atom precisely.

When only one atom is prepared in the cavity mode, we measure a fidelity of
$\mathcal{F}=\qty{86.6(5)}{}$ and an efficiency of $\eta=\qty{33.2(2)}{}$. The
fidelity is mostly limited by state preparation and measurement errors
(2.7(2)), and the Zeeman splitting of $\ket{\uparrow}$ and $\ket{\downarrow}$
states, which results in a frequency difference of the $\ket{R}$ and $\ket{L}$
components, inducing a phase chirp within the temporal mode (3.8). The latter
can be minimized by reducing the photon temporal window considered (see
supplement [29]). In addition, state preparation and readout (2.7(3)),
decoherence (on average 1.53.5, depending on the number of qubits),
infidelities in the single-qubit rotations (2.5) and polarization settings
($<\qty{1}{}$) reduce the fidelity further. The APE efficiency is given by
$\eta=\xi P$ where $P$ is the intrinsic probability to generate a photon at
the output of the cavity and $\mathcal{\xi}$ is the product of the state
initialization efficiency in $\ket{F{=}2,m_{F}{=}0}$ (80(5)) and the overall
transmission and detection efficiency (64(6)). The intrinsic photon generation
probability depends on the cavity QED parameters as
$P=(\kappa_{out}/\kappa)\times 2C/(2C+1)$, where $C=g^{2}/(2\kappa\gamma)$ is
the cooperativity with $g$ the atom-cavity coupling constant for the relevant
transition, $\kappa$ and $\kappa_{out}$ the total cavity field decay rate and
the field decay rate through the outcoupling mirror, respectively, and
$\gamma$ the atomic polarization decay rate. For our system
$\left(g,\kappa,\kappa_{out},\gamma\right)=2\pi\times\left(\qty{5.0}{},\qty{2.5}{},\qty{2.3}{},\qty{3}{}\right)\qty{}{\mega}$
resulting in $P=70\%$. Using $\xi=\qty{51(6)}{}$, the expected efficiency is
$\eta=\qty{36(4)}{}$, comparable to the measured value.

When larger atom arrays are prepared, we measure the efficiency and the
fidelity of the APE for each atom individually. For each array size, we
collect a minimum of 3000 clicks per atom per detection basis and extract the
average fidelity $\overline{\mathcal{F}}$. We observe that the fidelity does
not depend on the size of the array (Fig. 3A), remaining constant at
$\overline{\mathcal{F}}=\qty{85.5(4)}{}$ for up to $n=6$ atoms. This proves
minimal errors ($<0.7\%$) associated with increasing the size of the qubit
register and indicates that larger registers can be realized without
increasing the error rate. Conversely, the efficiency is not constant over the
entire array of atoms aligned transversely to the cavity axis. It peaks near
the cavity center and diminishes for outer positions, where the cavity field
and, consequently, $C$ are lower due to the finite cavity mode size
($\qty{30}{\micro}$ waist radius). We confirm this by measuring $\eta$ for a
single atom placed at different positions along the $x$-axis (Fig. 3B). Our
data are well represented by a model that includes the finite cavity mode
size. The number of atoms along the $x$-axis is thus limited, with larger
arrays possible for shorter inter-atomic distances $d_{x}$ along the $x$-axis
or by shuffling atoms in and out of the cavity mode [33, 34]. To determine the
optimal $d_{x}$, we measured the average APE fidelity for two atoms within the
cavity as a function of $d_{x}$, ranging from 3.5 to 16.5 (Fig. 3C). While no
cross-talk is observed, with the fidelity remaining constant within a
measurement error of $0.4\%$ (standard deviation), we note that for small
inter-atomic distances, the array preparation is less efficient due to
interference from neighboring tweezers [35]. We choose
$d_{x}=\qty{5.5}{\micro}$ between adjacent atoms as a compromise between
efficient photon emission and atom preparation.

Figure 3: Average Fidelity $\overline{\mathcal{F}}$, cross-talk and position-
dependent efficiency of APE. (A) Average APE fidelity $\overline{\mathcal{F}}$
for $n$ atoms in the cavity. As more atoms are added, the fidelity remains
constant at $\overline{\mathcal{F}}=\qty{85.5(4)}{}$ within the standard
deviation of 0.7. (B) Efficiency $\eta$ of the photon emission process
depending on the position of one atom along the $x$-axis (transversal to the
cavity). As the atoms are positioned farther away from the center of the
cavity, the coupling $g$ to the cavity and hence $\eta$ decreases. The dashed
blue line is a fit of the data for the emission efficiency $\eta=\xi P$ with a
model that includes the finite cavity waist $w_{0}$, the cavity-QED parameters
$g,\kappa,\kappa_{out},\gamma$ and uses only the overall transmission and
detection efficiency $\xi$ as free fit parameter. This yields
$\xi=\qty{48.6(13)}{}$, comparable with the measured value of $\qty{51(6)}{}$.
(C) Average APE fidelity for two atoms separated by $d_{x}$ along the
$x$-axis. The fidelity remains constant at an average value of $\qty{90.0\pm
0.4}{}$ (dashed line) indicating no cross-talk induced errors within the
measurement error of $\qty{0.4}{}$ (standard deviation).

While the size of the arrays along the $x$-axis is limited, it is possible to
extend it in two dimensions (see Fig. 4), positioning atoms along the cavity
$y$-axis where the size is fundamentally constrained only by the cavity length
of in our case $\qty{485}{\micro}$. Our data consistently show that the
fidelity remains constant as atoms are displaced along the cavity axis, albeit
with a slight drop in efficiency. This results from our trapping scheme, which
uses the intra-cavity blue-detuned trap to confine atoms at the nodes of the
$\qty{770}{\nano}$ standing wave. At the cavity center, the node of the
$\qty{770}{\nano}$ and the antinode of the $\qty{780}{\nano}$ coincide,
ensuring maximum atom-cavity coupling and thus $C$. However, the two standing
waves acquire a different phase as we move along the cavity axis.
Consequently, the effective $C$ for atoms trapped along the $y$-axis
oscillates with a $\qty{32}{\micro}$ beating pattern between the two
wavelengths. This effect can be mitigated by selecting a trapping wavelength
closer to $\qty{780}{\nano}$. Although there is space for positioning many
atoms along the cavity axis, currently, our setup is limited to two-row
configurations due to the small (12) beam waist of the 1064 trap which limits
the overall size of the optical lattice. A larger waist of the 1064 trap, or
multiple parallel standing-wave traps could be used to overcome this
limitation.

Figure 4:  Two-dimensional arrangement of atoms. Upper part: Two atoms are
arranged along the direction of the cavity mode, the $y$ direction in Fig. 1A.
Lower part: Four atoms arranged as a $2\times 2$ square along the $x$ and $y$
directions. On the right, the achieved fidelities and efficiencies of the
atom-photon entangled states are presented. Our ability to arrange the atoms
in two dimensions at constant fidelities and at the expected efficiencies
indicates the scalability of the platform to larger system sizes, the maximum
number of atoms being limited only by the mode volume of the cavity.

Multiplexed atom-photon entanglement generation with a quantum network
register. Despite a notable $\qty{33.2(2)}{}$ probability of generating,
propagating and detecting entanglement with a single atom, practical network
applications demand pushing this probability as close to unity as possible.
Since only one entangled state is needed to establish quantum communication
between two nodes, using multiplexing schemes for registers of $n$ emitters
provide a solution [36, 10]. Given $\eta_{i}$ is the probability of detecting
one photon from the $i$-th atom, the probability of detecting at least one
entangled pair out of $n$ atoms is given by
$\eta_{overall}=1-\prod_{i=1}^{n}(1-\eta_{i})$, approaching unity for large
$n$. As each emitter can be addressed at different times, the communicating
parties can distinguish which emitter has successfully generated an entangled
photon based on the detection time and the communication distance.
Furthermore, when using a single emitter per node, the entanglement
distribution rate over a distance $L$ is limited by the classical
communication time $L/c$. In contrast, employing a large register linearly
increases this rate by a factor of $n$. Fig. 5A displays $\eta_{overall}$ and
thus demonstrates the expected increase in efficiency as we add atoms to the
register. We measure a maximum detection efficiency of $\qty{88.6(1)}{}$ for
$n=6$ atoms, and extract the in-fiber photon generation efficiency
$\eta_{fiber}$ by correcting $\eta_{overall}$ for the finite propagation and
detection efficiencies (70(7)). As shown in Fig. 5A, we find
$\eta_{fiber}=\qty{48(5)}{}$ for a single atom, increasing up to 97.4(6) for
six atoms. The increase in efficiency translates directly into an almost
linear increase in the average number of emitted photons $\overline{n}$ per
attempt (Fig. 5B), demonstrating the increase in rate when using a register.
We achieve $\overline{n}=1.88(1)$ for six atoms, more than a five-fold
increase compared to the case of an individual atom.

Figure 5:  Multiplexed efficiency and average photon number for an increasing
number of atoms. Efficiency to generate and fiber-couple ($\eta_{fiber}$), or
to generate, fiber-couple and detect ($\eta_{overall}$), at least one
entangled photon (A) and average photon number $\overline{n}$ (B) as a
function of the number of atoms. The theory curve in part (A) is calculated
using $\eta_{overall}=1-\prod_{i=1}^{n}(1-\eta_{i})$, with $\eta_{i}$ the
probability to generate a photon from the $i$-th atom and the values
$\eta_{i}$ as in Fig. 2. As a guide for the eye, the blue curve in (B)
connects the data points. For six atoms, we measure an overall (in-fiber)
photon detection efficiency of $\eta_{overall}=\qty{88.6(1)}{}$
($\eta_{fiber}=\qty{97.4(6)}{}$) compared to 33.2(2) (48(5)) for a single
atom. We observe a nearly linear increase in $\overline{n}$ up to 1.88(1)
photons per attempt.

Concluding remarks. The atomic array assembling scheme reported here could
boost the duty cycle of the recently demonstrated two-atom quantum-repeater
node which had rather inefficient probabilistic loading [12]. Moreover, our
system could potentially host many more atoms, as the two-dimensional array of
tweezers could be extended to accommodate 40 atoms along the cavity axis,
limited by the field of view of the addressing system (200), and 7 atoms in
the transversal direction, limited by the cavity waist. The latter number can
be further increased by transporting atoms in and out of the cavity [33, 34],
with atoms outside the cavity serving as a quantum reservoir, similar to what
has been proposed for scalable quantum computation with ions [37]. With a
spacing between atoms around or less than $\qty{10}{\micro}$, smaller than a
typical Rydberg blockade radius, intra-cavity quantum-logic operations between
neighboring qubits seem possible [38]. Alternatively, the all-to-all
connectivity of the atomic qubits in the cavity allows for quantum information
processing between large numbers of qubits in a single step [40, 41, 39], in
comparison to all hitherto implementations of quantum circuits where these
gates are typically decomposed into two-qubit gates and single-qubit rotations
[42].

## Acknowledgments

Funding: This work was supported by the Bundesministerium für Bildung und
Forschung (QR.X, Förderkennzeichen 16KISQ019), by the Deutsche
Forschungsgemeinschaft under Germany’s Excellence Strategy (MCQST, Project No.
390814868), and by the European Union’s Horizon Europe research and innovation
program via the project Quantum Internet Alliance (QIA, Grant Agreement No.
101102140). E.D. acknowledges support from the Max Planck-Harvard Research
Center for Quantum Optics postdoctoral scholarship. S.W. acknowledges
financial support from the SNSF Swiss Postdoctoral Fellowship (Project no.
TMPFP2$\\_$210584). Authors contributions: All authors contributed to the
experiment, the analysis of the results, and the writing of the manuscript.
Competing interests: The authors declare that there are no competing financial
interests. Data availability: The data underlying the figures are deposited at
Zenodo [43]. All other data needed to evaluate the conclusions in the paper
are present in the main text or the Supplementary Material.

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## Supplementary material

###  .1 Preparation of tweezer arrays and addressing

At the beginning of the experimental sequence, we trap [87]Rb atoms in a
magneto-optical trap (MOT) outside the resonator. After the MOT phase, we load
the atoms into a red detuned dipole trap ($\lambda=\qty{1064}{\nano}$) to
transport the atoms over a distance of 14 from the position of the MOT to the
center of the cavity. For this purpose, the dipole trap is focused halfway
between the resonator and the position of the MOT, creating an optical
potential in which the atoms can freely oscillate between the MOT region and
the resonator region. After traveling through the potential exactly once, the
atoms reach the center of the resonator. This half-period oscillation takes
70. Meanwhile, before the atoms reach the resonator, we turn on a two-
dimensional array of optical tweezers midway between the two resonator
mirrors. Each individual tweezer has a waist of 1.40(5) and a trap depth of
about 1 which corresponds to a power of 1.52 per individual tweezer, resulting
in an atomic lifetime of about 20 in the tweezers. Since we are only using the
tweezers to prepare the atomic arrays, we use a rather near-resonant
wavelength of 797, as here little power is needed for the desired trap depth.
Once the atoms reach the center of the cavity in the transport beam, they move
across and are stochastically captured in the tweezers array in the presence
of appropriate laser cooling. We use the scattered cooling light to image the
atoms and assess the state of the preparation. Based on the image evaluation,
another set of tweezers, the moving tweezers, controlled by the second AOD,
arranges the atoms into the desired shape of the array. The arrangement of the
atoms takes another 12, as we arrange the atoms sequentially with a moving
speed of 5/ , because using a faster moving speed leads to atom loss. The
arrangement protocol is implemented to minimize the distance and number of
moves, and in principle the arrangement procedure in a cavity environment is
the same as in a free space experiment once the atoms are trapped in the
initial static array. We track the progress of the atom rearrangement and when
it is complete, we load the atoms into two standing wave traps at 1064 and 770
, respectively. Here, the atomic storage time is $\sim\qty{1}{}$. Once this
step is completed, we start the experimental science sequence. The overall
probability of success for this preparation protocol depends on the number of
atoms. It is as high as 90 for two atoms and 20 for an array of six atoms. In
our previous stochastic loading scheme without optical tweezers, we
immediately transferred the atoms to the optical lattice after transporting
them in the transport trap. In this way of loading atoms, the number of atoms
in the resonator (the average number of atoms loaded could only be influenced
by changing the size of the MOT) and the distance between the atoms are
stochastic. Therefore, the probability of loading three atoms at the correct
distance from each other is already less than 0.5 and working with larger
arrays was not possible due to the low probability of success. Extrapolating
from the measured values for the old loading scheme, we estimate that for six
atoms the probability of success is about four orders of magnitude higher when
the atoms are loaded with the tweezers. The individual addressing of the atoms
is controlled by the same AOD as the moving tweezers. We observe atom-hopping
between different lattice sites along the cavity direction at a rate of
$<\qty{1}{}$. We attribute this to heating caused by the fluctuations of the
intra-cavity $\qty{770}{\nano}$ standing-wave intensity, resulting from the
fluctuations of the cavity length. We overcome this by continuously monitoring
the position of the atoms at a rate of $\qty{3}{}$ and adjusting the AODs
frequency accordingly. As the generation-to-detection efficiency is at the
theoretical maximum, this shows that hopping is not a limitation at present.
In the future, we plan to further reduce the hopping rate by ground-state
cooling the atom [1]. We also plan to implement a faster atom-position
detection by increasing our current imaging setup signal-to-noise ratio.

###  .2 Detailed level scheme

A detailed level scheme of the employed [87]Rb atoms is shown in Fig. S1. The
two hyperfine ground states $\ket{F{=}1}$ and $\ket{F{=}2}$ are split by a
microwave transition at 6.8. The cavity is detuned from the
$\ket{F{=}1}\rightarrow\ket{F^{\prime}{=}1,m_{F}{=}-1}$ transition by
$\Delta_{ac}=2\pi{\times}\qty{200}{\mega}$.

Figure S1:  Detailed level scheme of the employed [87]Rb atoms. (A) At the
start of the experiment, the atom is initialized in the state
$\ket{F{=}2,m_{F}{=}0}$. A vSTIRAP pulse (blue arrow) via the addressing
system triggers the release of a single photon (red wiggly arrows) into the
cavity mode. The cavity, shown schematically on the right, is tuned, such that
it has a detuning of $\Delta_{ac}=2\pi\times\qty{200}{\mega}$ from the
$\ket{F{=}1}\rightarrow\ket{F^{\prime}{=}1,m_{F}{=}-1}$ transition. (B) Level
scheme of the hyperfine ground states $\ket{F{=}1}$ and $\ket{F{=}2}$. As we
apply a guiding field during the experimental sequence, we induce a Zeeman
splitting with a Larmor frequency of $\omega_{L}=2\pi\times\qty{100}{}$.

###  .3 Raman Transitions

To drive transitions between the different Zeeman sublevels of the
$\ket{F{=}1}$ and $\ket{F{=}2}$ manifold, we employ a stimulated Raman
transition as illustrated in Fig. S2. The probe and coupling beam are from the
same laser oscillator to ensure maximum phase stability. The frequency of the
beams is set to be on two-photon resonance with the involved states, i.e. to
$\omega_{\ket{\uparrow}\rightarrow\ket{F{=}2,m_{F}{=}2}}$ in Fig. S2, and the
frequency splitting of 6.8 is generated with a combination of an EOM and AOMs.
The wavelength of the Raman laser is $\lambda=\qty{788.24}{\nano}$ to give
minimum differential light shift on the individual atoms. The beams are
impinging from the side along the $x$-axis, see Fig. 1A (main text), on the
cavity with a waist of $w_{0}=\qty{82}{\micro}$ and are polarized with
$\sigma^{+}+\sigma^{-}$ and $\pi$ polarization. This allows to drive
transitions with $\Delta m_{F}{=}\pm 1$. In Fig. S2, we illustrate the
transition from $\ket{\uparrow}$ to $\ket{F{=}2,m_{F}{=}2}$, but since the
probe beam is polarized with $\sigma^{+}+\sigma^{-}$ polarization, we can also
drive the transitions from $\ket{\downarrow}$ to $\ket{F{=}2,m_{F}{=}-2}$ or
from $\ket{\uparrow}$ or $\ket{\downarrow}$ to $\ket{0}$ by adjusting the
frequency of the control beam (and hence the two-photon frequency) as the
individual $m_{F}$-states are separated in frequency, see Fig. S1.

Figure S2: Raman transition from $\ket{\uparrow}$ to $\ket{F{=}2,m_{F}{=}2}$.
The Raman transitions are driven with two Raman beams (yellow arrows)
$\Omega_{c}$ and $\Omega_{p}$ which are $\pi$ and
$\sigma^{+}+\sigma^{-}$-polarized, respectively. These beams allow to bridge
the hyperfine transition of $6.8$GHz. We show the transition from
$\ket{\uparrow}$ to $\ket{F{=}2,m_{F}{=}2}$ with the single-photon detuning
$\Delta$, the two-photon detuning $\delta$ and the two-photon frequency
$\omega_{\ket{\uparrow}\rightarrow\ket{F{=}2,m_{F}{=}2}}$.

###  .4 Experimental protocol

The complete experimental sequence including the timing is shown in Fig. S3.
It starts by preparing the atoms into the $\ket{5S_{1/2},F{=}2,m_{F}{=}0}$
state. For the preparation, we use a combination of $\pi$-polarized light
resonant with the
$\ket{5S_{1/2},F{=}2,m_{F}{=}0}\rightarrow\ket{5P_{1/2},F^{\prime}{=}2,m_{F}{=}0}$
transition and a repumper resonant with the
$\ket{5S_{1/2},F{=}1}\rightarrow\ket{5P_{3/2},F{=}2}$ transition. The pumping
beams impinge on the atoms along the $x$-axis (see Fig. 1A (main text)) and
initialize all atoms simultaneously with an efficiency of 80(5) and a pumping
fidelity of at least $\qty{98}{}$ in $\ket{F{=}2,m_{F}{=}0}$. We estimate the
lower limit of the fidelity by looking at the minimum of the correlations in
the $ZZ$ basis, see section .5. The bases are defined in table S1.

Photonic bases | Atomic bases
---|---
$\begin{array}[]{@{}c@{}c@{}c@{}c}X_{p}=\\{&\ket{H}&=\frac{1}{\sqrt{2}}\left(\ket{R}+\ket{L}\right),\\\ &\ket{V}&=\frac{1}{\sqrt{2}}\left(\ket{R}-\ket{L}\right)\\}\end{array}$ | $\begin{array}[]{@{}c@{}c@{}c@{}c}X_{a}=\\{&\ket{\uparrow_{x}}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}+\ket{\downarrow}\right),\\\ &\ket{\downarrow_{x}}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}-\ket{\downarrow}\right)\\}\end{array}$
$\begin{array}[]{@{}c@{}c@{}c@{}c}Y_{p}=\\{&\ket{A}&=\frac{1}{\sqrt{2}}\left(\ket{R}+i\ket{L}\right),\\\ &\ket{D}&=\frac{1}{\sqrt{2}}\left(\ket{R}-i\ket{L}\right)\\}\end{array}$ | $\begin{array}[]{@{}c@{}c@{}c@{}c}Y_{a}=\\{&\ket{\uparrow_{y}}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}+i\ket{\downarrow}\right),\\\ &\ket{\downarrow_{y}}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}-i\ket{\downarrow}\right)\\}\end{array}$
$Z_{p}=\left\\{\ket{R},\ket{L}\right\\}$ | $Z_{a}=\left\\{\ket{\uparrow},\ket{\downarrow}\right\\}$
Table S1:  Definition of the atomic and photonic bases. The table displays the
definitions we use in equations (6) and (7).

Here and in the following, the terms $XX$, $YY$, and $ZZ$ basis stand for a
combined atom-photon measurement along the basis $X_{a}\otimes X_{p}$,
$Y_{a}\otimes Y_{p}$ and $Z_{a}\otimes Z_{p}$ respectively.

Figure S3:  Complete experimental sequence. We initially generate $N$ signal
photons (red wiggly arrows) with vSTIRAP pulses (blue arrow) by individually
addressing the $N$ intra-cavity atoms. Subsequently, we set the readout basis
of the atomic spin using $\pi/2$ and $\pi$ pulses (green arrows). Both are
addressing all atoms globally. Eventually, we map the information stored in
the atomic spin to readout photons (red wiggly arrows) with two global Raman
$\pi$ pulses (yellow arrows) and an additional set of N individual vSTIRAP
pulses (blue arrows).

Then, $N$ vSTIRAP pulses sequentially illuminate the atoms and ideally, we
generate $N$ entangled atom-photon pairs. The vSTIRAP pulse and the resonance
frequency of the cavity are set to have the same single-photon detuning with
respect to the excited state $\ket{F^{\prime}=1,m_{F}{=}0}$ and the
transitions from $\ket{F{=}2,m_{F}{=}0}\rightarrow\ket{F{=}1,m_{F}{=}1}$ and
$\ket{F{=}2,m_{F}{=}0}\rightarrow\ket{F{=}1,m_{F}{=}-1}$ are on two-photon
resonance (see Fig. S1). After the emission of the photon, the atoms are in
the $\ket{F{=}1}$ ground state, and the photons emitted into the cavity mode
can be scattered by other atoms in the same ground state. This scattering rate
scales as $g^{2}/\Delta_{ac}$ and therefore it is favorable to choose a large
detuning $\Delta_{ac}$. However, choosing too large a detuning, i.e.
$\left|\Delta_{ac}\right|\gg H_{1^{\prime},3^{\prime}}$, with
$H_{1^{\prime},3^{\prime}}$ being the hyperfine splitting between
$\ket{F^{\prime}{=}1}$ and $\ket{F^{\prime}{=}3}$, increases the probability
of exciting to $\ket{F^{\prime}{=}3}$, resulting in a reduction in fidelity
[2]. It is therefore necessary to set the single photon detuning $\Delta_{ac}$
between the cavity and the atom carefully. We choose a detuning of
$\Delta_{ac}=2\pi\times\qty{200}{\mega}$ with respect to the
$\ket{F^{\prime}{=}1,m_{F}{=}-1}$ state. The rest of the sequence depends on
the measurement basis. For the $Z_{a}$ basis state, we perform two
$\pi$-pulses followed by another vSTIRAP pulse. The $\pi$ pulses map the state
$\ket{\uparrow}$ to $\ket{F{=}2,m_{F}{=}2}$ and the state $\ket{\downarrow}$
to $\ket{F{=}2,m_{F}{=}-2}$. The consecutive vSTIRAP pulse generates a readout
photon transferring the atoms back to the states $\ket{\uparrow}$ and
$\ket{\downarrow}$. A polarization measurement of the readout photon along the
$Z_{p}$ basis projects the atoms onto one specific state, explicitly a
$\ket{L}$ detection projects to $\ket{\uparrow}$ while $\ket{R}$ to
$\ket{\downarrow}$. If we measure the atom along $X_{a}$ or $Y_{a}$, we
introduce in the sequence additional Raman pulses to perform a rotation of an
angle $\pi/2$ with a phase $\theta=0$ or $\theta=\pi/2$ in the atomic qubit
space. This maps $\ket{\uparrow_{x}}\textbackslash\ket{\downarrow_{x}}$ and
$\ket{\uparrow_{y}}\textbackslash\ket{\downarrow_{y}}$ onto
$\ket{\uparrow}\textbackslash\ket{\downarrow}$, depending on the chosen phase.
The $\pi/2$ rotation is composed of three Raman pulses:

  1. 1.

A $\pi$ pulse to transfer the state $\ket{\uparrow}=\ket{F{=}1,m_{F}{=}+1}$ to
$\ket{F{=}2,m_{F}{=}0}\equiv\ket{0}$. The qubit is thus stored in the
$\\{\ket{\downarrow},\ket{0}\\}$-manifold.

  2. 2.

A $\pi/2$ pulse between $\ket{\downarrow}$ and $\ket{0}$ with phase $\theta=0$
or $\theta=\pi/2$, depending on the measurement basis $X_{a}$ or $Y_{a}$
respectively, resulting in the map

$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{\downarrow}+\ket{0}\right)$
$\displaystyle\rightarrow\ket{0}$ (1)
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{\downarrow}-\ket{0}\right)$
$\displaystyle\rightarrow\ket{\downarrow}$ (2)
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{\downarrow}+i\ket{0}\right)$
$\displaystyle\rightarrow\ket{0}$ (3)
$\displaystyle\frac{1}{\sqrt{2}}\left(\ket{\downarrow}-i\ket{0}\right)$
$\displaystyle\rightarrow\ket{\downarrow}.$ (4)

where Eqs. (1,2) correspond to $\theta=0$ and Eqs. (3,4) to $\theta=\pi/2$.

  3. 3.

An additional $\pi$ pulse to transfer the population from $\ket{0}$ back to
$\ket{\uparrow}$. Afterwards, the qubit is again encoded in the
$\\{\ket{\downarrow},\ket{\uparrow}\\}$-manifold.

These rotations, unlike the vSTIRAP pulses, are global pulses, i.e., they are
executed on all atoms simultaneously. After the $\pi/2$ rotation, one proceeds
again as in the $Z_{a}$-basis and reads out the atomic qubit by mapping the
atomic state to the polarization with the scheme described above.

To set $\theta$ we note that the Raman transitions
$\ket{\uparrow}\rightarrow\ket{0}$ and $\ket{0}\rightarrow\ket{\downarrow}$,
have two different two-photon resonance frequencies
$\omega_{\ket{\uparrow}\rightarrow\ket{0}}=\Delta+\omega_{L}$, and
$\omega_{\ket{0}\rightarrow\ket{\downarrow}}=\Delta-\omega_{L}$ respectively,
with $\Delta=2\pi\times\qty{6.8}{\giga}$ the splitting between the
$\ket{F{=}1}$ and $\ket{F{=}2}$ hyperfine ground states (see Fig. S1B). This
means that after the transfer of the population from $\ket{\uparrow}$ to
$\ket{0}$ with pulse (1.), the qubit evolves at a frequency of
$\omega_{\ket{\downarrow}\rightarrow\ket{0}}$ while the RF frequency driving
the Raman laser is set to $\omega_{\ket{\uparrow}\rightarrow\ket{0}}$. To
apply the pulse (2.), the RF frequency is switched to
$\omega_{\ket{\downarrow}\rightarrow\ket{0}}$ after a time $T$ which results
in an accumulated phase of $\theta=2\omega_{L}T$. A carefully chosen $T$
allows to set the phase $\theta$ at will. To determine $T$ corresponding to a
measurement along the $X_{a}$ ($Y_{a}$) basis, we measure the signal photon
along $X_{p}$ ($Y_{p}$), scan $T$, and measure the readout photon along
$Z_{p}$. We then extract the probability $p(i|j)$ of finding the signal photon
in the polarization $i=\\{H,V\\}$ ($i=\\{D,A\\}$) conditioned on the readout
photon in $j=\\{R,L\\}$, see Fig. S4. $p(i|j)$ oscillates as a function of
$\theta$, and a measurement along the $X_{a}$ ($Y_{a}$) basis corresponds to a
$T$ where $p(H|L)$ ($p(D|L)$) is maximum. For the polarization analysis of the
signal photon in $XX$ and $YY$, we need to set the basis of the detection
setup to a linear polarization. However, the readout photon is always detected
in the $\\{\ket{R},\ket{L}\\}$ basis. Therefore, the polarization basis of the
detection setup must be changed between the detection of the signal and the
readout photons. This is achieved by using an EOM, which switches the basis
between linear and circular polarization in less than $\qty{1}{\micro}$. To
switch between the $XX$ and the $YY$ basis, a $\lambda/2$ waveplate (not shown
in Fig. 1B) is inserted in front of the EOM to adjust the detection basis
accordingly.

Figure S4:  Phase scan of the $\pi/2$ pulse in the $YY$ basis. To perform the
$\pi/2$ rotation between $\ket{\uparrow}$ and $\ket{\downarrow}$, we first
transfer the population from $\ket{\uparrow}$ to $\ket{0}$ (see S3). As the
two photon detunings of the two Raman transitions
$\omega_{\ket{\uparrow}\rightarrow\ket{0}}$ and
$\omega_{\ket{0}\rightarrow\ket{\downarrow}}$ differ by $2\times\omega_{L}$,
the qubit in the $\left\\{\ket{0},\ket{\downarrow}\right\\}$ manifold evolves
at a frequency of $2\times\omega_{L}$ with respect to the previously set
frequency $\omega_{\ket{\uparrow}\rightarrow\ket{0}}$. Therefore, to set the
phase for the $\pi/2$ pulse properly, we have to scan the time $T$ at which we
switch the two-photon detuning from
$\omega_{\ket{\uparrow}\rightarrow\ket{0}}$ to
$\omega_{\ket{0}\rightarrow\ket{\downarrow}}$. These scans are performed for
both the $XX$ and $YY$ basis and we set the time $T$ to the maximal amplitude
of the oscillatory signal.

###  .5 Fidelity measurement

In this section, we describe how we determine the fidelity of the prepared
atom-photon states with ideal Bell states. Ideally, after the photon emission
we generate the state

$\ket{\Psi^{+}}=\frac{1}{\sqrt{2}}\left(\ket{R,\uparrow}+\ket{L,\downarrow}\right).$
(5)

and similarly in the $XX$ and $YY$ basis:

$\displaystyle\ket{\Psi^{+}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{H,\uparrow_{x}}+\ket{V,\downarrow_{x}}\right)$
(6) $\displaystyle\ket{\Psi^{+}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{D,\uparrow_{y}}-\ket{A,\downarrow_{y}}\right).$
(7)

We thus see that correlations between linear polarization and superposition
states of $\ket{\uparrow}$ and $\ket{\downarrow}$ arise, e.g., between
$\ket{H}$ and $\ket{\uparrow_{x}}$ or $\ket{D}$ and $\ket{\uparrow_{y}}$.
The fidelity is given by
$\mathcal{F}=\mathrm{Tr}\left(\ket{\Psi^{+}}\bra{\Psi^{+}}\rho\right)$, which
in turn can be simplified to $\mathcal{F}=\bra{\Psi^{+}}\rho\ket{\Psi^{+}}$.
When writing the density operator in the basis of the Pauli matrices
$\rho=\sum_{k,j=0}^{3}S_{k,j}\sigma_{k}\otimes\sigma_{j}$, we obtain the
following expression for the fidelity

$\mathcal{F}=\frac{1}{4}\left(1+S_{xx}+S_{yy}-S_{zz}\right).$ (8)

Here, $S_{xx},S_{yy}$ and $S_{zz}$ are the two-qubit Stokes parameters. The
individual two-qubit Stokes parameters are defined as [3]

$\displaystyle S_{xx}$ $\displaystyle=$ $\displaystyle
P_{\uparrow_{x},\uparrow_{x}}-P_{\uparrow_{x},\downarrow_{x}}-P_{\downarrow_{x},\uparrow_{x}}+P_{\downarrow_{x},\downarrow_{x}}$
(9) $\displaystyle S_{yy}$ $\displaystyle=$ $\displaystyle
P_{\uparrow_{y},\uparrow_{y}}-P_{\uparrow_{y},\downarrow_{y}}-P_{\downarrow_{y},\uparrow_{y}}+P_{\downarrow_{y},\downarrow_{y}}$
(10) $\displaystyle S_{zz}$ $\displaystyle=$ $\displaystyle
P_{\uparrow,\uparrow}-P_{\uparrow,\downarrow}-P_{\downarrow,\uparrow}+P_{\downarrow,\downarrow},$
(11)

where $P_{jj}$ is the probability of measuring the state $\ket{j,j}$ and
$P_{j,j}+P_{j,-j}+P_{-j,j}+P_{-j,-j}=1$. Therefore, to determine the fidelity,
we need to measure the correlations between the polarization of the signal
photon and the internal atomic state in the three pairs of bases, as defined
in table S1. From this, the fidelity can be calculated employing equations (8)
to (11).

###  .6 Temporal shape of the photon

The temporal shape of the photon is a source of infidelity in the
superposition bases $XX$ and $YY$. After the detection of the photon, the
atomic state is projected onto a superposition state of $\ket{\uparrow}$ and
$\ket{\downarrow}$. As these states are energetically separated by
$2\hbar\omega_{L}$, the phase of the superposition states evolves in time. For
example, in the $XX$ basis after the detection of the signal photon in
$\ket{H}$, the atom is projected onto $\ket{\uparrow_{x}}$ which evolves in
time as
$\ket{\uparrow_{x}}(t)=\ket{\uparrow}+\exp\left(-2i\omega_{L}t)\ket{\downarrow}\right)$.
Here, $t$ is the time that passed since the detection of the signal photon. As
the temporal shape of the signal photon (see Fig. S5) has a non-vanishing
width, the phase evolution of the atomic qubit after detection of the signal
photon differs from shot to shot. Therefore, to reduce the influence of the
photon shape on the fidelity, we define a time window $\tau$ and only consider
events for which we have detected the photon within the specified time window.
In the presented data we set the time window to $\tau=\qty{1.25}{\micro}$
which reduces the detection efficiency.

Figure S5: Typical temporal shape of a generated photon. The finite temporal
width of the photon reduces the fidelity of the produced entangled atom-photon
states with ideal Bell states. To minimize this effect, we introduce a photon
acceptance window of $\tau=\qty{1.25}{\micro}$. Photons outside of this
interval are not used. This reduces the photon generation efficiency, i.e. in
the illustrated case by almost $\qty{15}{}$.

###  .7 Analysis of experimental imperfections

In this section, we elucidate the various sources of errors in our experiment
that contribute to inaccuracies in the entanglement generation. One major
source of error stems from the temporal mode of the photon, see section .6. We
calculate the influence of the temporal shape on the fidelity for a photon
acceptance window of $\tau=\qty{1.25}{\micro}$ and a Larmor frequency of
$\omega_{L}=2\pi\times\qty{100}{\kilo}$ and obtain a maximum fidelity of
$\mathcal{F}=\qty{96.2}{}$. This contribution could be reduced by either a
smaller photon acceptance window $\tau$ (resulting in a reduction of the
efficiency), a smaller Larmor frequency (resulting in a smaller Rabi-
frequency), or by increasing the power in the vSTIRAP. The latter, however,
leads to atomic heating, reducing the storage time and the increasing
tunneling to different lattice sites, which results in a lower overall
efficiency. Another contribution comes from the atom preparation and state
readout for which we estimate an infidelity of 2.7(2). The coherence time is
1.1 in the $\left\\{\ket{\uparrow},\ket{\downarrow}\right\\}$-manifold. As the
duration of the sequence depends on the number of atoms that we use, the
influence of the coherence time is the largest for the six-atom case, where we
attribute approximately 4.5 (compared to 3 for the two-atom case) of
infidelity to the limited coherence. We attribute the remaining 2.5 of
infidelity to the $\pi/2$ rotation between $\ket{\uparrow}$ and
$\ket{\downarrow}$. We estimate the errors given by the polarization settings
and the polarization of the vSTIRAP pulse to be below 1.

###  .8 Coherence time scaling analysis

The measured coherence time for the qubit states $\ket{\uparrow}$ and
$\ket{\downarrow}$ is $\sim\qty{1.1}{\milli}$. With this coherence time, we
see a decrease in the fidelity of the first (last) addressed atom of 4.5 (1.5)
for six atoms. However, using dynamical decoupling as in [4], the coherence
time can be extended to $\sim\qty{20}{\milli}$, which allows us to increase
the number of atoms in the register without appreciable decoherence. For
example, for 100 atoms, a maximum fidelity of $\sim\qty{96}{}$ for the atom
that was addressed first, could be expected. This assumes a worst-case
scenario of exponential decay with a coherence time of 20 and considers that
15 are needed to generate a photon and adjust the input RF frequency of the
AODs accordingly to address the next atom. By transferring the atomic qubit
into decoherence-protected superpositions, the coherence time can be further
extended, resulting in a coherence time of $\qty{100}{ms}$ [5]. This would
further increase the maximum fidelity to $\sim\qty{99.2}{}$ for the atom that
was addressed first.

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