††thanks: M. A. N., H. K. and W. B. C. contributed equally to this
work.††thanks: M. A. N., H. K. and W. B. C. contributed equally to this
work.††thanks: M. A. N., H. K. and W. B. C. contributed equally to this work.

# Iterative assembly of 171Yb atom arrays in cavity-enhanced optical lattices

M. A. Norcia  H. Kim  W. B. Cairncross  M. Stone  A. Ryou  M. Jaffe  M. O.
Brown  K. Barnes  P. Battaglino  T. C. Bohdanowicz  A. Brown  K. Cassella
C.-A. Chen  R. Coxe  D. Crow  J. Epstein  C. Griger  E. Halperin  F. Hummel
A. M. W. Jones  J. M. Kindem  J. King  K. Kotru  J. Lauigan  M. Li  M. Lu  E.
Megidish  J. Marjanovic  M. McDonald  T. Mittiga  J. A. Muniz  S.
Narayanaswami  C. Nishiguchi  T. Paule  K. A. Pawlak  L. S. Peng  K. L. Pudenz
D. Rodríguez Pérez  A. Smull  D. Stack  M. Urbanek  R. J. M. van de Veerdonk
Z. Vendeiro  L. Wadleigh  T. Wilkason  T.-Y. Wu  X. Xie  E. Zalys-Geller  X.
Zhang  B. J. Bloom
Atom Computing, Inc

###### Abstract

Assembling and maintaining large arrays of individually addressable atoms is a
key requirement for continued scaling of neutral-atom-based quantum computers
and simulators. In this work, we demonstrate a new paradigm for assembly of
atomic arrays, based on a synergistic combination of optical tweezers and
cavity-enhanced optical lattices, and the incremental filling of a target
array from a repetitively filled reservoir. In this protocol, the tweezers
provide microscopic rearrangement of atoms, while the cavity-enhanced lattices
enable the creation of large numbers of deep optical potentials that allow for
rapid low-loss imaging of atoms. We apply this protocol to demonstrate
deterministic filling (99% per-site occupancy) of 1225-site arrays. Because
the reservoir is repeatedly filled with fresh atoms, the array can be
maintained in a filled state indefinitely. We anticipate that this protocol
will be compatible with mid-circuit reloading, which will be a key capability
for running large-scale error-corrected quantum computations whose durations
exceed the lifetime of a single atom in the system.

##  I Introduction

Individually controlled neutral atoms provide a promising platform for quantum
information processing and simulation, and expanding the size and control of
these systems represent key challenges in the ongoing push to access regimes
beyond the capabilities of classical simulation. Tweezer arrays and optical
lattices have emerged as core technologies for optically trapping and
manipulating cold atoms, each with complementary capabilities. Optical
lattices provide a well-defined potential landscape with features on the sub-
wavelength scale, which can enable tight confinement for imaging and a clean
optical potential for simulations involving itinerant atoms [1]. Optical
tweezers provide a means of selectively manipulating individual atoms [2].
Notably, tweezer arrays have gained prominence for deterministic assembly of
atomic arrays [3, 4, 5] with up to hundreds of atoms [6, 7] and inter-atomic
spacing suitable for enabling interactions between atoms through excitation to
Rydberg states [8, 9, 10, 6, 7, 11, 12]. Recently, optical tweezers and
lattices have been combined to create programmable initial conditions in a
Hubbard-regime system [13, 14], and to enable enhanced scaling of atom number
for metrology [15] and quantum simulation[16].

The merits of optical lattices can be further enhanced through the use of
optical buildup cavities. Optical cavities enhance the lattice depth by
enabling the constructive interference of many overlapped reflections of laser
light, in turn enabling the creation of arrays of many deep traps [17, 18].

Figure 1: Conceptual diagram of repeated loading sequence. A “reservoir”
optical tweezer array (smaller left grid) is repeatedly filled with 171Yb
atoms transported from a spatially separated magneto-optical trap (MOT), and
ultimately transferred into a “target” tweezer array (larger right grid) using
a “rearrangement” tweezer. The rearrangement moves required for the transfer
are determined from low-loss images obtained by transferring atoms from the
tweezers into a cavity-enhanced optical lattice and performing site-resolved
fluorescence detection. The cavity-enhanced lattice allows for the scalable
generation of large numbers of deep traps. The call-out shows example images
prior to the 20th cycle of rearrangement, and after the 70th cycle, with the
final reservoir reloading step omitted. This protocol allows us to load over
1200 atoms into 1225 target sites.

In this work, we combine the capabilities of tweezers and cavity-enhanced
optical lattices to demonstrate an iterative approach to creating large arrays
of individually-controlled atoms. Typically, tweezer rearrangement is
performed by stochastically loading up to a single atom into each trap within
an array [19], imaging the atoms to determine trap occupancy, and then
rearranging atoms within the array to create a deterministically occupied sub-
array [4, 5]. Crucially, the number of atoms contained in the final array with
this approach is no greater than the number initially loaded. Further, because
the initial loading is stochastic, the number of sites in the array must
generally be substantially larger than desired final sub-array (though under
certain conditions, near-deterministic loading can be achieved [20, 21], and
several initial arrays can be used to increase the number of available atoms
[22]). Recently, repeated loading of a “buffer” array from an optical dipole
trap “reservoir” has demonstrated that one can decouple the filling of a six-
site target array from a single loading of a cold reservoir [23]. In this
work, we extend this concept to repeated loading of a reservoir array, from
which we create a deterministically filled target array (typically 99%
occupancy) of over 1200 171Yb atoms in 1225 sites. This is made possible by
combining optical tweezer arrays with a cavity-enhanced optical lattice to
provide both microscopic control and the large number of deep traps required
for rapid, high-fidelity, low-loss imaging of large numbers of atoms.

A single cycle of loading proceeds as follows (See figure 1): Atoms are
collected and cooled in a magneto-optical trap 30 cm below a “science region”
where we create a cavity-enhanced optical lattice, as well as tweezer arrays
projected through a high-numerical-aperture imaging. The atoms are then
transported into the science region, where they are loaded into a set of
tweezers that form the reservoir. The reservoir sits adjacent to the target
tweezer array, which on all loading cycles but the first is already partially
occupied with atoms loaded on previous cycles. The atoms from both reservoir
and target tweezer arrays are then transferred into the trapping potential of
the cavity-enhanced optical lattice, where we perform site-resolved
nondestructive fluorescence imaging to determine which sites of each tweezer
array were occupied. We then transfer the atoms back from the cavity lattice
to the tweezers, and use a single separate “rearrangement” tweezer to move
atoms from the reservoir into empty target tweezers. This defines one loading
cycle. Over the course of multiple cycles, the number of atoms in the target
array increases until the target array is filled. At this point, further
operations may be performed on the atoms, and subsequent loading cycles are
applied to maintain a filled array.

Because we reload the reservoir on each loading cycle from a fresh magneto-
optical trap, we can continue loading the target array indefinitely. In the
near-term, this allows for relatively high data-rates for quantum simulation
and computation with large system sizes, and could also be of benefit for
optical clocks for high statistical precision with low dead-time [24].
Ultimately, the ability to reload new atoms while maintaining both the
presence and coherence of existing atoms will be a key capability for
performing error-corrected quantum computations, where execution of an
algorithm may take much longer than the lifetime of any given atom in the
system. When combined with site-selective hiding [25, 26] and mid-circuit
rearrangement techniques already demonstrated in a similar system [26], we
anticipate that this protocol will be compatible with mid-circuit reloading of
atoms into the array. As a means of maintaining a fully filled array, our
approach represents an alternative to the interleaved use of two atomic
species [27, 28]. Unlike the two-species approach, the method presented here
does not require simultaneous replacement of the entire array, and so may be
expected to ease requirements for scaling to large arrays, and would not
require inter-species gates [29] in order to replenish the array during a
quantum computation.

##  II Trapping and imaging atoms in a 3D cavity enhanced lattice

Figure 2: Cavity-enhanced optical lattices for rapid low-loss imaging. (a) The
intersection of two cavity modes provide three-dimensional confinement for
atoms within the field of view of our high-numerical-aperture imaging system.
Confinement in the $x$ and $y$ directions is provided by a self-intersecting
standing-wave cavity, while confinement in the $z$ direction is provided by a
separate self-intersecting running-wave cavity mode. Imaging light (not shown
for clarity) is incident along the mode of the running-wave lattice. (b)
Observed counts during a 7 ms long image of single atoms within the optical
lattice, showing well-resolved peaks for occupied and unoccupied sites. For
these conditions, per-image survival is 99.8(1)%. (c) Lattice homogeneity
across the target array as characterized by light shifts on the 1S0
$m_{f}=1/2$ to 3P1, $F=3/2$ $m_{f}=-1/2$ transition. Atoms within 13.4 $\mu$m
squares are averaged together for statistics and display clarity. (d, e) Light
shift contributions from each cavity averaged over rows (d) and columns (e).
Red and orange markers and lines represent the measured and predicted profiles
of the XY (red) and Z (orange) cavities, with the prediction centered on the
array.

Rapid low-loss imaging of the locations of atoms is essential to our repeated
loading protocol. Because atoms are heated by the photon scattering required
for imaging, deep optical traps and imaging protocols with low equilibrium
temperatures are required to prevent atom loss. Optical tweezers represent a
common method for achieving deep optical traps – the tight foci can provide
high intensities for moderate powers. However, the total power requirement for
a large array of tweezers presents scaling limitations due to limited
available laser power. Cavity-enhanced optical lattices offer an alternative
path to achieving many deep traps. While the laser intensity in a lattice is
spread over a larger cross-sectional area than in optical tweezers both
because it spans the regions between utilized sites, and because the Gaussian
mode profile must be large compared to the size of the array to achieve
uniform trapping, interference of many reflected paths in a cavity-supported
lattice can enable more power-efficient generation of traps.

We employ two intersecting optical cavities to provide tight confinement in
three dimensions, as shown in figure 2a. The first cavity – which we call the
XY cavity – is a retro-reflected four-mirror cavity whose mode intersects
itself at the location of the atoms. The polarization is perpendicular to the
plane of propagation, providing interference between the two crossed portions
of the cavity mode [30]. Compared to using two crossed non-interfering
lattices [31, 18], and assuming equal cavity finesse, this configuration
provides an eight-fold enhancement in trap depth (defined as the potential
barrier between adjacent lattice sites) per unit of total laser power. The XY
cavity provides tight confinement in the directions perpendicular to the high
numerical aperture optical axis used for the tweezers and imaging. The second
cavity, which we call the Z cavity, is a four-mirror running wave cavity whose
mode intersects itself at the location of the atoms [32] at 15 degree angles
from the XY plane. This provides confinement along the tweezer axis. The
polarization of the Z cavity is also oriented perpendicular to its
intersection plane, enabling complete interference of the crossing modes. Each
cavity has two highly reflecting mirrors, and two matched partially
transmitting mirrors. The finesse of the XY and Z cavities are 2900(100) and
3000(100) respectively, and the design mode waists at the location of the
atoms are 268 $\mu$m and 183 $\mu$m respectively. For imaging, we typically
operate with XY (Z) trap frequencies near 160 kHz (50 kHz), corresponding to
330 $\mu$K (260 $\mu$K) deep traps.

We image atoms using the the narrow-linewidth (180 kHz) 1S0 to 3P1 transition,
which simultaneously provides cooling and scattering of photons for detection
[33, 34, 26]. We use a single beam, detuned several hundred kHz from the
$F=3/2$, $m_{f}=3/2$ state, with projection onto both the X and Z directions.
For applications like array assembly where we wish to determine the presence
of an atom but not the state of its nuclear spin (the qubit we use for quantum
information applications), we simultaneously apply optical pumping on the 1S0
$m_{f}=-1/2$ to 3P1 $F=1/2$, $m_{f}=1/2$ transition with light incident along
the magnetic field ($x$ direction). At our benchmark imaging parameters, we
collect 50 photons in 7 ms, and distinguish occupied from unoccupied sites
with a fidelity of approximately 99.95%, as determined from the overlap error
in a double-Gaussian fit to the bimodal distribution of counts. Atom loss
during such an image is $2(1)\times 10^{-3}$. Spin-selective imaging requires
us to omit the optical pumping, and rely on the combination of frequency and
polarization selectivity within the 3P1 manifold to create a large imbalance
of scattering between the two nuclear spin states, as demonstrated in optical
tweezers in refs. [34, 26]. The omission of the optical pumping does not
modify the photon scattering rate appreciably, but does introduce a finite
spin flip probability of of $4(1)\times 10^{-3}$ for our typical imaging
parameters, measured as an additional apparent loss in repeated images of 1S0
$m_{f}=1/2$. These spin flips are likely due to the fact that the lattice
polarization is perpendicular to the magnetic field, and so can induce spin-
changing transitions within the 3P1 manifold [34].

In order to ensure uniform detuning of the imaging transition across the array
and between different motional states, we operate with a trapping wavelength
that has equal polarizability for the ground and excited states of the imaging
transition – a so-called magic wavelength. For the 1S0 $m_{f}=1/2$ to 3P1
$m_{f}=3/2$ transition, we find a magic condition near 783.8 nm.

We characterize the homogeneity of the optical lattice potential by probing
the 1S0 $m_{f}=1/2$ to 3P1 $F=3/2$, $m_{f}=-1/2$ transition, for which the
excited state has approximately 40% higher polarizability than the ground
state. We infer the trap depth from the transition light shifts, measured
using optical pumping from the 1S0 $m_{f}=1/2$ state, followed by spin-
selective imaging (fig. 2c). By varying the power in each cavity, we can
extract their independent contributions, displayed averaged over rows and
columns in figure 2d,e. Over our 115 $\mu$m square array, peak deviations are
within 20% for the Z lattice and 10% for the XY lattice. Because we operate
with a magic wavelength condition for critical operations, these deviations do
not present a major limitation.

##  III Repeated loading to build up a large array

Figure 3: (a) Timing diagram for our continuous loading sequence. See main
text for a detailed description of the procedure. The beam paths referenced in
the call-out are illustrated in figure 4. (b) Number of atoms and fill
fraction as a function of loading iteration number during initial loading
(left panel) and while maintaining a filled array (right panel). Black points
represent atom number inferred from images preceding the rearrangement step.
Red points represent atom number inferred from additional images inserted
after rearrangement, representing atoms available for computation.

In this section, we describe our continuous loading protocol in detail, and
characterize its performance for loading large arrays.

Our loading cycle (illustrated in figure 3a) begins with collecting atoms from
a pre-cooled atomic beam into a magneto-optical trap (MOT) using a “core-
shell” configuration [35]. In this MOT, each beam consists of a ring of 399 nm
light near resonance with the broad linewidth 1S0 to 1P1 transition
surrounding a central region with only 556 nm light near resonance with the
narrow linewidth 1S0 to 3P1 transition. The shell is primarily responsible for
capturing atoms from the incident beam, while the low Doppler temperature of
the narrow transition used in the core enables cooling to several 10 $\mu$K.
Because atoms do not scatter light from the broad transition except during
initial capture, the core-shell design poses less risk of generating scattered
photons that may be absorbed by atoms in the target array – the total
scattering rate on the narrow transition is much lower, and the large magnetic
field present in the science region dramatically suppresses reabsorption of
photons via the narrow-line transition. In our operating condition, the 399 nm
photon scattering rate is about 2 orders of magnitude lower than the broad-
line stage of a sequential MOT. We find that the core-shell MOT configuration
increases the MOT loading rate by roughly a factor of two, compared to a more
standard sequential MOT configuration in a static magnetic field.

From the core-shell MOT, atoms are loaded into a vertically oriented standing-
wave lattice formed by counterpropagating 532 nm laser beams focused at the
location of the atoms. Loading is achieved by overlapping the lattice with the
center of the MOT, followed by reducing the detuning and intensity of the 556
nm light. We then extinguish the MOT light and transport the atoms vertically
by synchronously applying a frequency offset to the upward-going and downward-
going lattice beams, and translating their foci [26]. This brings the atoms 30
cm vertically into the vicinity of the target array, at which point we
transfer the atoms into a reservoir array formed by 483 nm tweezers (another
magic wavelength for the 1S0 to 3P1 transition [26]). To avoid disturbing
atoms already loaded into the target array, this transfer takes place at a
location displaced 170 $\mu$m horizontally from the target array. No
dissipation is required to transfer atoms from the transport lattice to the
tweezers – loading is achieved by simply increasing the intensity of the
tweezers while the lattice is overlapped and then decreasing the intensity of
the lattice. Once loaded, the reservoir array is translated to be directly
adjacent to the target array by changing the angle of a galvo mirror that is
imaged onto the entrance pupil of the objective.

The target array consists of a rectangular array of tweezer spots formed with
light at 459.5960(5) nm wavelength, a magic wavelength for the 1S0 to 3P0
clock transition [26, 36], and a set of tweezer spots formed by light at
423.31 nm, which is near resonance with a transition between 3P1 to a higher-
lying 6S8S 3S1 state. Both wavelengths provide large light-shifts to atoms in
3P1, which prevents unwanted scattering from atoms loaded in these tweezer
arrays, and can be used interchangeably for this work. For this work, we
operate with a 35x35-site target array with 3.3 $\mu$m spacing, comprised of
two separate rectangular arrays each using one of the two previously mentioned
wavelengths. We have also operated with overlapped full-size arrays to achieve
sufficient trap depth, and obtained similar results. However, the overlapped
condition is more sensitive to alignments, and so we use spatially separated
arrays for all results presented here.

While translating the reservoir array adjacent to the target array, both
arrays are illuminated with light resonant with the 1S0 to 3P1 $F=3/2$,
$m_{f}=3/2$ transition. This induces light-assisted collisions between atoms
in the reservoir tweezers, resulting in either zero or one atom in each
reservoir tweezer with approximately equal probability [19]. Importantly, the
tweezer light illuminating the atoms in the target array prevents them from
scattering the light used to induce collisions, which might otherwise cause
loss of existing atoms in the target array. Next, atoms from both tweezer
arrays are transferred into the optical lattice and imaged, as described
above. The imaging leaves the atoms at a temperature of 20 $\mu$K, which is
too warm to efficiently transfer into the relatively shallow tweezers of the
target array. Following each image, we apply Doppler and Raman sideband
cooling to reach an average of $\sim$ 0.1 motional quanta in each direction
(Appendix A). Once cooled, the atoms are transferred back into the tweezer
arrays, and a single optical tweezer derived from the same laser as the
reservoir light is used to transfer atoms from filled sites of the reservoir
array into empty sites of the target array. This completes a single loading
cycle.

In order to shorten the time required to build up an atomic array, and to
limit the losses associated with background gas collisions, we load the MOT at
the same time as performing operations in the science chamber. This is made
possible by the spatial separation between MOT and science regions, and by the
fact that we operate with static magnetic fields: we can simultaneously
operate with the magnetic field gradients required by the MOT and the large
500 Gauss homogeneous bias field that is desirable for imaging and other
operations required in the science regions. The low scattering rate of 399 nm
photons, combined with the large magnetic field and spatial separation between
the MOT and science regions prevents light scattered in the MOT from being
reabsorbed by atoms in the science region, which could cause heating or
decoherence.

Figure 3b shows examples of the atom number increasing as a function of load
cycles, with 105 sites in the reservoir and 1225 sites in the target array. At
first, the atom number increases approximately linearly as roughly 45 atoms
are transferred into the target array per cycle. The loading rate decreases
slightly as the array fills, as some atoms from the reservoir are used to
counteract loss. Once the number of vacancies in the target array drops below
the typical number of atoms in the reservoir, an equilibrium is reached with
the vacancy fraction set by the per cycle probability of atom loss.

In figure 3b, the atom number is inferred from the image taken before
rearrangement, so the measured fill fraction is sensitive to all losses that
occur between such images. This fill fraction is typically 98%, with the
largest single loss contribution coming from vacuum loss during the typical
300 ms time between images due to our approximately 30 s vacuum lifetime. We
can add additional diagnostic images directly after rearrangement to determine
the atom number available for computation, and observe typical filling
fractions of 99%, again dominated by vacuum loss during the 150 ms typically
allocated for rearrangement. Our experimental control allocates a variable
duration for rearrangement that depends on the number and length of moves to
be performed, as well as the time required to compute the moves. The durations
above are typical for the 1225-site arrays studied in this work. Other sources
of loss or potential loss are described in more detail in Appendix B, and
include the handoff between the tweezer arrays and the lattice, and imaging
loss and infidelity.

##  IV Conclusions and outlooks

We have demonstrated an iterative method for assembling large arrays of
individually controlled neutral atoms, suitable for quantum computation,
simulation, and metrology. Through this approach, we decouple the final size
of the array from the number of atoms that can be loaded at once, and the need
to create many tweezer traps with sufficient depth for loading and imaging
atoms. In particular, our reservoir and lattice traps are much deeper than the
target tweezer traps, allowing for the use of large numbers of sites with
moderate laser power. Our current arrays have percent-level defects, which
could be straightforwardly reduced with improved vacuum.

For quantum computing applications, our iterative loading may be combined with
recently demonstrated mid-circuit measurement techniques [25, 26] in order to
achieve continuous mid-circuit refilling of array defects. This capability
will be critical for mitigating the effect of atom loss during execution of
complex error-corrected circuits that may extend beyond the lifetime of an
individual atom in the array.

##  V Appendix A: Raman Sideband Cooling

Figure 4: Raman sideband cooling (RSC). (a) Diagram of relevant directions and
polarizations. RSC beams RSC1 and RSC2 are combined to cool the $z$ direction,
while RSC1 and RSC3 are combined to cool the $x$ and $y$ directions. Optical
pumping (beam OP) is applied along the $x$ direction, which is along the
applied magnetic field. (b) Level diagram for Raman sideband cooling. Raman
transitions detuned by 35 MHz from 3P1 $F=3/2$ $m_{f}=1/2$ are driven from 1S0
$m_{f}=1/2$ to 1S0 $m_{f}=-1/2$, while reducing the number of motional quanta
along the addressed direction(s) from $n$ to $n-1$. Optical pumping through
3P1 $F=1/2$ $m_{f}=1/2$ returns atoms to 1S0 $m_{f}=1/2$ to enable further
cycles. (c) Sideband spectra in the $x$, $y$ plane. The sideband imbalance
indicates an average number of motional quanta along the differential momentum
vector of the RSC1 and RSC3 beams of $\bar{n}_{xy}=0.08(1)$. (d) Sideband
spectra in the $z$ direction. The sideband imbalance indicates an average
number of motional quanta along the $z$ direction of $\bar{n}_{z}=0.12(1)$.
(e) Heating in the lattice, as measured from the sideband imbalance in the
$x$, $y$ plane, with a fit to an exponential growth profile, returning a time-
constant of 190(40) ms.

Transferring atoms from the lattice into shallow tweezer traps with low loss
requires that the atoms be much colder than the depth of the tweezer traps. By
operating with colder atoms, we can use shallower tweezers, and in turn scale
to larger array sizes. To accomplish this, we use Raman sideband cooling (RSC)
between the two ground nuclear spin states [21, 25]. For the motional Raman
transitions, we use two pairs of beams oriented along the $x$ and $y$
directions, and along the $x$ and $z$ directions to provide cooling along all
three directions (fig. 4a). The Raman transitions transfer atoms from 1S0
$m_{f}=1/2$ to 1S0 $m_{f}=-1/2$, state, and are detuned from the motional
carrier transition by the appropriate trap oscillation frequency in order to
reduce the motional state by one quanta (fig. 4b). Optical pumping is provided
by a beam oriented along $x$ and the magnetic field that addresses the 1S0
$m_{f}=-1/2$ to 1P1 $F=1/2$, $m_{f}=1/2$ transition.

We perform 20 iterations of cooling, with each iteration consisting of a $\pi$
pulse on the red motional sideband for each pair, followed by optical pumping.
The pulse durations are 200 $\mu$s for the $xy$ pair and 100 $\mu$s for the
$xz$ pair (tuned to the $z$ direction sidband), and 50 $\mu$s for optical
pumping. The total cooling sequence lasts 8 ms. In typical sequences, we apply
a 2 ms Doppler cooling pulse prior to the RSC sequence using the same beam as
for the imaging pulse, but with lower intensity and farther red-detuning. The
Doppler cooling results in 10 $\mu$K temperature in the cavity as measured by
a release and recapture protocol [37]. This roughly corresponds to 1 motional
quanta occupancy in the cavity. Subsequent RSC reduces the average motional
quanta to in each direction to $\sim$ 0.1, as evident from the sideband
imbalance following cooling (fig. 4c, d).

We measure heating in the $x$, $y$ plane while holding atoms in the lattice by
performing Raman sideband spectroscopy to extract the average motional quantum
number $\bar{n}_{xy}$ as a function of time. We fit this quantity to a
function representing exponential growth, as expected for parametric heating
due to intensity fluctuations of the lattice [38], and extract an exponential
time-constant of 190(40) ms. This is likely due to conversion of laser
frequency noise to amplitude by the cavity resonance, and could be improved by
optimizing the lock of lasers to the cavity.

##  VI Appendix B: Sources of Per-Cycle Loss

Figure 5: Characterization of per-cycle loss mechanisms. (a), (b). Loss per
pair of handoffs between the 459 nm (a) and 423 nm (b) tweezer arrays and
lattice, measured by performing 25 handoffs with cooling in between. The black
line represents the estimated contribution from vacuum loss, with the red band
representing day-to-day drifts in this value. Tweezer depths are normalized to
their default values, which correspond to roughly 50 $\mu$K. (c) Final vacancy
fraction versus rearrangment tweezer power, normalized to the default tweezer
depth of approximately 150 $\mu$K. For low powers, atoms are not efficiently
transferred from the reservoir to target array. For too high of powers, we
observe increased loss of loaded atoms due to the rearrangement tweezer
passing nearby.

Vacuum loss due to the finite duration of loading cycles is the single largest
source of loss in our system, contributing percent-level per-cycle loss.
Because our loading cycle is of variable length, and because the vacuum level
in our system can fluctuate from day to day, it is difficult to estimate its
exact contribution. However, typical values for the loading cycle duration and
vacuum lifetime are 300 ms and 30 s respectively, so 1% represents a typical
value for this loss. In this section, we describe other loss sources that
contribute to a lesser degree, especially if parameters are not carefully
optimized. In general, the loss mechanisms described below both limit the size
of array we can ultimately load (as the reservoir must be large enough to
replace lost atoms), and limit the final fill fraction of the array.

As described in the main text, typical imaging loss is $2(1)\times 10^{-3}$
where roughly 30 % is the vacuum loss and the rest is the Raman scattering out
of 3P1 due to the trap light , and the discrimination infidelity is typically
at or below the $10^{-3}$ level. If an atom within the target array is lost
during the image after scattering enough photons to be identified as present,
the defect will not be filled during the subsequent rearrangement step,
leading to a defect in the array. If a site within the array is incorrectly
identified as empty, an atom will likely be added to that site from the
reservoir, and subsequently undergo lossy collisions with the original
occupant. Empty sites within the target or reservoir arrays that are
mistakenly identified as full will lead to a previous defect not being
repaired. However, because defects are rare, this mechanism is less
problematic. We currently operate with a discrimination threshold that
balances the correct identification of empty and full sites. In a system where
imaging losses and infidelity become dominant loss sources (a system with
better vacuum), it may be advantageous to bias the threshold to minimize the
more problematic forms of imaging error.

Each image requires handing off atoms from the tweezer arrays into the lattice
and back. We find that this process is sensitive to the alignment of the two
arrays on the scale of a single lattice site, to the depth of the tweezer
arrays, and to the temperature of the atoms. Atom temperature and alignment
are described in Appendix A and C, respectively. We isolate the effect of trap
depth on handoff for cold atoms and well-aligned arrays by performing 25
subsequent handoff pairs between tweezers and lattice in figure 5a, b. We
perform our usual Raman sideband cooling (RSC) each time the atoms are in the
lattice between handoffs. For 459 nm tweezer depths below 75% of our typical
operating conditions, we observe significant loss. Above this power level, we
observe a constant loss rate of 0.0006(1), which is consistent with data taken
with the handoff omitted, and with our typical vacuum losses. The 423 nm
tweezers show a similar behavior, though with a slightly higher loss rate of
0.0015(2) at high powers, likely due to worse alignment or optical
aberrations. If the tweezers are aligned to the trough of the XY cavity
intensity, the loss can be as high as 0.02.

In principle, moving new atoms in the target array may cause loss of existing
atoms in the array, as the rearrangement tweezer moves near the occupied
target sites. We assess this possibility in figure 5c by measuring the final
fill fraction versus the depth of the rearrangement tweezer. For a
rearrangement tweezer that is too shallow, we observe a lower loading rate and
so a lower final fill fraction. For too deep a tweezer, we also see a
reduction in the final fill fraction, as the rearrangement leads to loss of
existing atoms. Near our typical operating conditions, we observe a region
with weak dependency of the fill fraction on the tweezer power.

##  VII Appendix C: Alignment of arrays

Figure 6: Alignment of the tweezer arrays to the cavity lattice is performed
by monitoring atomic survival after repeated handoffs, while scanning the
position offset of the tweezer array. For the representative alignment scan of
the 459 nm tweezer array shown here, we perform 25 handoff pairs between the
tweezers and lattice, with no cooling in between. Data is averaged over the
full utilized field of view – the presence of a visible fringe in the average
image indicates that the sites across the array can be aligned simultaneously.

Precise alignment of the different optical potentials is critical for the
performance of our repeated loading protocol. The alignment of the target
tweezer arrays to the optical lattice is particularly sensitive, as
misalignment here can lead to increased handoff loss that can limit both the
largest array that can be loaded, and the final fill fraction of the array.

Alignment of the arrays involves first matching the spacing and tilt of the
lattice – quantities which do not drift appreciably over time – and then at a
higher frequency aligning the $x$ and $y$ offsets. We obtain the correct
spacings and tilts by populating the lattice with atoms transferred from an
expanded but sparse version of the reservoir array. We image the atoms in the
lattice to determine the lattice grid, and then in the reservoir tweezers to
determine required corrections to the tweezer array. We then use a camera
whose imaging system is corrected for chromatic shifts between the different
tweezer wavelengths to register the target array tweezers to the reservoir
tweezers.

We then optimize the translation alignment by performing repeated handoffs
between the tweezers and the lattice and scanning the position offset of the
tweezer arrays. We observe a periodic modulation in the atomic survival (fig.
6), which we fit to determine the optimal alignment. This process is repeated
as necessary to ensure alignment of the arrays.

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