# Narrow-line imaging of single strontium atoms in shallow optical tweezers

Alexander Urech$\href https://orcid.org/0000-0003-1663-4705{}$ Van der Waals-
Zeeman Institute, Institute of Physics, University of Amsterdam, Science Park
904, 1098XH Amsterdam, The Netherlands  QuSoft, Science Park 123, 1098XG
Amsterdam, The Netherlands  Ivo H. A. Knottnerus$\href
https://orcid.org/0000-0002-7987-1268{}$ Van der Waals-Zeeman Institute,
Institute of Physics, University of Amsterdam, Science Park 904, 1098XH
Amsterdam, The Netherlands  QuSoft, Science Park 123, 1098XG Amsterdam, The
Netherlands  Eindhoven University of Technology, P.O. Box 513, 5600MB
Eindhoven, The Netherlands  Robert J. C. Spreeuw$\href
https://orcid.org/0000-0002-2631-5698{}$ Van der Waals-Zeeman Institute,
Institute of Physics, University of Amsterdam, Science Park 904, 1098XH
Amsterdam, The Netherlands  QuSoft, Science Park 123, 1098XG Amsterdam, The
Netherlands  Florian Schreck$\href https://orcid.org/0000-0001-8225-8803{}$
<EMAIL_ADDRESS>Van der Waals-Zeeman Institute,
Institute of Physics, University of Amsterdam, Science Park 904, 1098XH
Amsterdam, The Netherlands  QuSoft, Science Park 123, 1098XG Amsterdam, The
Netherlands

(August 27, 2024)

###### Abstract

Single strontium atoms held in optical tweezers have so far only been imaged
using the broad ${}^{1}S_{0}$ \- ${}^{1}P_{1}\,$transition. For Yb, use of the
narrow (183 kHz-wide) ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition for
simultaneous imaging and cooling has been demonstrated in tweezers with a
magic wavelength for the imaging transition. We demonstrate high-fidelity
imaging of single Sr atoms using its even narrower (7.4 kHz-wide)
${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition. The atoms are trapped in non-
magic-wavelength tweezers. We detect the photons scattered during Sisyphus
cooling, thus keeping the atoms near the motional ground state of the tweezer
throughout imaging. The fidelity of detection is 0.9991(4) with a survival
probability of 0.97(2). An atom in a tweezer can be held under imaging
conditions for 79(3) seconds allowing for hundreds of images to be taken,
limited mainly by background gas collisions. We detect atoms in an arrary of
36 tweezers with 813.4-nm light and trap depths of 135(20) $\mu$K. This trap
depth is three times shallower than typically used for imaging on the broad
${}^{1}S_{0}$ \- ${}^{1}P_{1}\,$transition. Narrow-line imaging opens the
possibility to even further reduce this trap depth, as long as all trap
frequencies are kept larger than the imaging transition linewidth. Imaging
using a narrow-linewidth transition in a non-magic-wavelength tweezer also
allows for selective imaging of a given tweezer. As a demonstration, we
selectively image (hide) a single tweezer from the array. This provides a
useful tool for quantum error correction protocols.

##  I Introduction

Optical tweezers have emerged as a powerful tool for quantum applications.
They enable state of the art quantum simulation and computation [1, 2, 3, 4],
high fidelity and long coherence time qubits [5, 6, 7, 8], quantum metrology
[9, 10, 11], quantum chemistry [12, 13], among numerous other applications.
Optical tweezers with alkaline-earth(-like) atoms, in particular with
strontium and ytterbium, have been recently realized, offering new
possibilities in expanding these applications [14, 15, 16].

In all strontium tweezer experiments demonstrated so far, the fluorescence of
single atoms on the broad (30 MHz) ${}^{1}S_{0}$ \- ${}^{1}P_{1}\,$transition
at 461 nm was recorded, while simultaneously cooling the atoms on the narrow
(7.4 kHz) ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition at 689 nm [14, 15, 17,
10, 6, 5, 11, 4, 9]. This ‘blue imaging’ method allows for high-fidelity
detection of single atoms in tweezers with high survival probability [14, 15,
17]. However, blue imaging requires repumpers to close the 5s4d ${}^{1}D_{2}$
decay channel, which can only be done at tweezer wavelengths where also the
${}^{1}D_{2}$ state is trapped [17, 6, 9, 5]. Furthermore, the slightly higher
scattering rate obtained in the blue imaging process ($\thicksim$ 75 kHz) can
only be used as long as the tweezers are sufficiently deep. Any excess heating
from the imaging process can then be cooled away after the image. As the trap
depth of the tweezer is reduced, the advantage of fast imaging is lost because
the scattering rate must also be reduced to balance heating from imaging and
cooling. Reducing the tweezer trap depth has the advantages of decreased laser
power requirement per tweezer (allowing for more tweezers using a given laser
source) and increasing metastable state lifetimes (reduced off-resonant
scattering of tweezer light).

A simpler method for imaging alkaline-earth(-like) atoms in tweezers is to use
the narrow ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition for both cooling and
imaging. Single atom detection by fluorescence imaging on a (less) narrow
transition has previously been demonstrated in ytterbium for two different
isotopes. In both cases tweezers with a magic-wavelength for the imaging
transition were used [16, 7].

Here we detect single 88Sr atoms using only the ${}^{1}S_{0}$ \-
${}^{3}P_{1}\,$ transition for simultaneous‘red imaging’ and cooling. We use
optical tweezers that are non-magic for the imaging transition, but magic for
the Sr clock transition (${}^{1}S_{0}$ \- ${}^{3}P_{0}\,$). We detect the
photons scattered during an attractive Sisyphus cooling process [17], thus
keeping the atoms near the motional ground state of the tweezer throughout
imaging.

Attractive Sisyphus cooling is possible at tweezer wavelengths where the
excited state experiences a deeper trap depth than the ground state. More
specifically, at our tweezer wavelength of 813.4 nm, the excited state
($\ket{e}\equiv\,^{3}P_{1}\,(|m_{j}|=1)$) confinement is 1.24 times greater
than the ground state ($\ket{g}\equiv\,^{1}S_{0}$), see Fig. 1(a). This
cooling process can be very efficient with a proper choice of parameters,
leading to a large reduction in energy per scattered photon, and a small
number of scattered photons needed to cool the atom [18]. In addition, due to
the anharmonicity of the tweezer potential, atomic motion in all directions is
coupled, allowing for a single radial cooling beam to remove energy from all
directions [18, 17].

With balanced heating and cooling from the imaging process, the trap depth can
be significantly decreased, reducing the power required per tweezer.
Additionally, red imaging can be performed without repumpers, since optical
pumping to metastable states is much reduced. The only remaining pumping is
due to off-resonant scattering of 813-nm tweezer light when the atom is in
$\ket{e}$, which has a low rate that is even further decreased by using
shallow tweezers.

Imaging in shallow tweezers does limit the scattering rate that can be
achieved without unacceptable atom loss, for both red and blue imaging.
Shallow tweezer imaging therefore requires a longer exposure time for high-
fidelity single atom detection compared to imaging in deeper tweezers. We show
for red imaging that the maximum scattering rate ($\Gamma/2\approx\,23$ kHz
for the ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$ transition, where $\Gamma$=
2$\pi\times$7.4 kHz is the transition linewidth) can be closely approached
with proper modulation of the frequency and intensity of the cooling beam,
while maintaining a lower temperature than when using blue imaging. This
allows for a reduction of the trap depth by a factor of $\thicksim 3$, while
only marginally increasing the imaging duration and maintaining a near unity
detection fidelity and survival probability.

We proceed by presenting an overview of the experimental setup and the
procedure for preparing single atoms in Sec. II. In Sec III, we present our
imaging method along with a description of the attractive Sisyphus cooling
process, optimized parameters, detection fidelity, and survival probabilities.
In Section IV we demonstrate the ability to selectively image (dark out) a
specific tweezer from the array and we conclude in Sec. V.

##  II Overview of experimental setup and procedure

Similar to previously demonstrated strontium tweezer experiments, we load the
optical tweezers with a small and random number of atoms from a magneto-
optical trap (MOT) operating on the narrow ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$
transition [14, 15, 17, 10, 6, 5, 11, 4, 9]. Our procedure for creating the
MOT is similar to the one of [19], but uses a reduced number of MOT beams to
make space for a microscope objective and dynamically moves the MOT from a
loading position into the objective focus, see Appendix A.

We create two dimensional arrays of optical tweezers using a phase-only
spatial light modulator (SLM) to imprint a phase onto an 813.4-nm laser beam
creating an array of foci [1, 2]. This array is then imaged onto the narrow
linewidth MOT through an NA=0.5 microscope objective. An additional
dynamically movable tweezer is created using the same microscope objective and
a pair of crossed acousto-optic deflectors (AODs), see Appendix B.

The tweezer trap depth used throughout this paper is 135(20) $\mu$K unless
otherwise specified. For our $1/e^{2}$ tweezer waist of $\thicksim
0.84\,\mu$m, the ground state radial (axial) trap frequencies are $\omega_{\rm
radial}=\,43(3)$ kHz ($\omega_{\rm axial}=\,6.6(5)$ kHz), respectively. This
trap depth is chosen such that the excited state axial trap frequency (7.3(6)
kHz) is comparable to the linewidth of the ${}^{1}S_{0}$ \-
${}^{3}P_{1}\,$transition.

Figure 1: (a) A sketch of the Sisyphus cooling process at the root of our
imaging technique. The tweezer potential is deeper for the excited state
(${}^{3}P_{1}$) than for the ground state (${}^{1}S_{0}$). The atom is
preferentially excited near the bottom of the potential, then rolls up the
steeper excited state potential before decaying. This leads to a reduction in
energy per scattered photon related to the trap depth mismatch [17, 18] (b) A
simplified schematic of the experimental setup. A high numerical aperture
objective (NA = 0.5) creates the optical tweezers and collects the atomic
fluorescence. A single beam (imaging beam) is used for light assisted
collisions, cooling, and imaging in the tweezer array. The polarization of
both this imaging beam ($\textbf{E}_{\text{img}}$) and the SLM tweezer pattern
($\textbf{E}_{\text{tweezer}}$) are shown. The direction of gravity with
respect to the objective is also shown (g). (c) Averaged fluorescence of
strontium atoms in the 6$\times$6 array of tweezers used throughout the
majority of this work. We collect photons scattered from the ${}^{1}S_{0}$ \-
${}^{3}P_{1}\,$$(|m_{j}|=1)$ transition during the cooling process in order to
image the atoms in the array. The image is the average of 100 experimental
realizations using 500 ms of exposure each. (d) Image obtained by one such
experimental run. Approximately half of the tweezer traps are filled on
average.

Once the tweezers have been loaded and the MOT is switched off, a single (non-
retro-reflected) 689-nm beam is used to address the tweezer array during all
further experimental stages that need 689-nm light (light-assisted collisions,
imaging, cooling, spectroscopy), see Fig. 1(b). This beam, here simply called
imaging beam, is linearly polarized perpendicular to the tweezer propagation
axis to maximize the fluorescence into the microscope. Additionally, we tune
the linear polarization of the tweezer light to match the propagation axis of
the imaging beam. This maximises the $\sigma^{\pm}$ component of the imaging
beam because we operate the tweezers at a 0 G magnetic field making the
tweezer polarization the dominant quantization axis.

To prepare tweezers containing either a single or no atom, we use light-
assisted collisions to induce pairwise loss, leaving either zero or one atom
remaining in each tweezer [20]. The imaging light used in this process is
tuned to a frequency between the Stark shifted resonance of the ${}^{1}S_{0}$
\- ${}^{3}P_{1}\,$$(|m_{j}|=1)$ transition and an electronically excited
molecular state that is further red detuned and that asymptotically
corresponds to the ${}^{3}P_{1}$ state [14, 21].

We perform imaging by collecting the scattered photons from the Sisyphus
cooling process as presented in Sec. III. The fluorescence is collected via
the same microscope objective used to generate the tweezers, and then
separated using a long pass dichroic mirror with 750-nm cutoff. The collected
fluorescence light is sent onto an EMCCD camera [22, 23]. The number of
photons in 5$\times$5 pixel regions of interest (ROIs) around each tweezer
center is summed. We collect photons for 100 ms in order to separate the
single atom signal from the background noise of the camera. This procedure
leads to a histogram with two peaks, corresponding to zero and one atom in a
tweezer, as shown in Fig. 4(a). An atom is assumed to be in a tweezer if the
photon number lies above a threshold located between the two peeks, see Sec.
III(C).

In the previously demonstrated blue imaging technique, the metastable
${}^{3}P_{0}$ and ${}^{3}P_{2}$ states must be repumped to the ground state
during imaging because of decay of ${}^{1}P_{1}$ to those metastable states
[17, 9]. During red imaging, these repumpers are only used to compensate
optical pumping into ${}^{3}P_{0,2}$ by the tweezer light that can happen when
the atom is in the ${}^{3}P_{1}\,(|m_{j}|=1)$ state. However, we find that
this is unnecessary for the shallow traps used in this work. Nonetheless, we
have the ability to repump the metastable states via the ${}^{3}S_{1}$ state
using two lasers at 679 nm and 707 nm for the ${}^{3}P_{0}$ and ${}^{3}P_{2}$
states respectively.

For all the results presented in this paper, we begin an experimental run by
preparing single atoms using the above method followed by an initial image to
determine which tweezers are filled. After this,we perform measurements as
required by the experiment under consideration. The average initial image of
100 preparations for a 6$\times$6 array and an image of a single run is shown
in Fig. 1(c) and (d) respectively.

All plots presented are for 100 repetitions of each experiment unless stated
otherwise and the data is the average of all 36 tweezer sites. Taking into
account the typical tweezer loading efficiency of $50\%$, each data point
consists of approximately 1800 realizations. The error bars for the entire
paper show the standard deviation over the array and are dominated by
variation originating from tweezer depth inhomogeneities across the array.

##  III Imaging via Sisyphus cooling

Our imaging and cooling relies on the attractive Sisyphus cooling technique
first proposed in [24, 18] and more recently observed experimentally in
tweezer arrays [17, 9, 5] as well as in a continuous beam decelerator [25]. We
can keep the scattering rate near maximum with near zero trap loss or heating
in tweezers as shallow as 135(20) $\mu$K by intentionally keeping the imaged
atoms slightly hotter than the coldest possible temperature, and by proper
choice of imaging/cooling parameters.

This section is ordered as follows. The required criterion for attractive
Sisyphus cooling will be outlined in Sec. III(A). Sec. III(B) will present our
characterization of the optimal cooling parameters. We will present our
imaging method in Sec. III(C). We will end with the analysis of the detection
fidelity and survival probability of our imaging process in Sec. III(D).

###  III.1 Sisyphus cooling criteria

Attractive Sisyphus cooling relies on a trap depth mismatch between the
excited and ground state potentials as shown in Fig. 1(a). In addition, three
conditions must be fulfilled for the cooling to work. First, the excited state
of the atom must experience stronger confinement than the ground state.
Second, one must have the ability to excite the atom selectively from the
bottom of the potential, and third the excited atom must have sufficient time
to move away from the center of the potential before decaying [18]. The first
condition is fulfilled in our setup by properly choosing the trapping
wavelength, while the second and third conditions can be fulfilled by using
the narrow linewidth ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition in strontium.

The first condition is needed for the atoms to lose kinetic energy by rolling
up the steeper potential of the excited state before decaying. This allows for
a reduction in potential energy on the order of the differential trap depth
per scattering event. The narrow linewidth of the transition allows for the
atoms to be selectively excited from the bottom of the trap if the
differential trap depth is larger than the linewidth. The lifetime of the
narrow transition is long enough to satisfy the third condition if the trap
frequencies are larger than the linewidth. The atom will then more likely
decay near the motional turning point, away from the center of the trap.

###  III.2 Optimal cooling

To investigate the performance of our cooling/imaging technique and find
optimal cooling parameters, we measure the temperature by the release and
recapture method [26]. We switch the tweezers off, wait a time $t_{\rm
release}$ before turning them back on again, and then image the atoms to
determine their survival fraction. Atoms are lost quicker when they are
hotter. The temperature is determined by comparing the survival fraction for
several values of $t_{\rm release}$ with Monte-Carlo atom trajectory
simulations [26, 17].

To characterize cooling performance, we start by preparing a sample and
detecting which tweezer contain an atom (see Sec. II). We then cool for a time
$t_{\rm cool}$ and perform release and recapture. Next, we cool the array
before taking a final image to see which atoms survived and calculate the
survival fraction.

The results of such measurements under optimal cooling conditions for three
$t_{\rm cool}$ are shown as examples in Fig. 2(a). The first measurement is
taken directly after the first image ($t_{\rm cool}$ = 0 ms, red circles). The
second briefly cools the atoms ($t_{\rm cool}$ = 2 ms, green squares). The
third approaches the asymptotically coldest achievable temperature by using a
long cooling time ($t_{\rm cool}$ = 20 ms, blue triangles).

Comparing this data with release and recapture simulations yields a
temperature of approximately 1.8 $\mu$K (dashed line in same figure) for an
optimally cooled atom ($t_{\rm cool}$=20 ms), which is consistent with a
temperature near the radial motional ground state energy of roughly
$T=\frac{\hbar\omega}{2k_{b}}\thicksim$1.1 $\mu$K for our trap depth [26]. The
release and recapture simulations are based on classical trajectories and
could lead to an overestimation in the temperature as the atom approaches the
motional ground state of the trap. Therefore we take this temperature estimate
as an upper bound.

Figure 2: (a) Temperature measurements using the release and recapture method
for three different cooling times, $t_{\rm cool}$, after taking an image. The
red circles show an atom directly after an image ($t_{\rm cool}$ = 0 ms),
green squares show a briefly cooled atom ($t_{\rm cool}$ = 2 ms), and blue
triangles show an optimally cooled atom ($t_{\rm cool}$ = 20 ms). Error bars
show the standard deviation calculated over the 36 atom array. All other
errors fall well inside these error bars. The dashed (dotted) lines show the
results of Monte Carlo simulations for temperatures of 1.8 $\mu$K (3 $\mu$K)
respectively. (b) Recapture fraction of a single atom versus the cooling
frequency for $t_{\rm cool}$ = 20 ms and $t_{\rm release}$ = 60 $\mu$s. As
explained in Sec. III(c), brief cooling phases are interlaced in the imaging
process, and here we vary the cooling frequency during imaging and during the
period $t_{\rm cool}$. The vertical line shown at -663 kHz indicates the
approximate Stark shifted resonance of the cooling transition. (c) Average
motional quanta $\bar{n}$, obtained by numerical simulation, in dependence of
cooling light detuning at optimal intensity. The cooling transition is
indicated as in (b).

To optimize cooling, we vary cooling light frequency or intensity while
keeping $t_{\rm release}$ = 60 $\mu$s and $t_{\rm cool}$ = 20 fixed. Figure
2(b) shows an example of such a measurement for which the detuning is varied.
We find the highest recapture fraction, and therefore optimal cooling, for a
frequency of $-775$ kHz from the free space resonance and an intensity of
$\thicksim\,88\,\,I_{\text{sat}}$ (Rabi frequency $\thicksim\,$2$\pi\times 50$
kHz).

We compare the experimentally determined optimal parameters and the
temperature with results of a numerical simulation of the cooling process. The
simulation is based on solving the steady state of a Lindblad master equation
for a two-level atom in a pair of 1-D quantum harmonic oscillators (QHO), one
for each internal state $\ket{g}$, $\ket{e}$. The ratio of the QHO frequencies
is given by $\omega_{g}/\omega_{e}=\sqrt{\alpha_{g}/\alpha_{e}}=0.899$, with
$\alpha_{g,e}$ the dynamic polarizabilities at the tweezer wavelength.
Choosing a traveling wave for the Sisyphus cooling laser, the transition
dipole moments between vibrational states of different QHO’s are calculated as
$d_{eg}\bra{m}e^{ikx}\ket{n}$, with $\ket{m}$, $\ket{n}$ the vibrational
states for internal states $\ket{g}$ and $\ket{e}$, respectively, and $d_{eg}$
the transition dipole moment of the ${}^{1}S_{0}$ \-
${}^{3}P_{1}\,$transition.

We find the optimal parameters and the minimum temperature to be in good
agreement (10$\%$) with the experimentally found ones. In Fig. 2(c) we show
the number of average motional quanta after cooling in dependence of detuning
at optimum intensity (all parameters of this simulation are given in Appendix
C). At the minimum we obtain $\bar{n}\,\approx\,0.25$, which is in good
agreement with measurements using sideband spectroscopy done by another group
using the same cooling method [6, 10, 3].

###  III.3 Optimizing the imaging parameters

We now discuss the imaging procedure and optimize its parameters. In a first
approach we record the fluorescence of atoms while cooling. We find that the
parameters that are optimal for cooling lead to a low scattering rate. The
rate increases if the imaging beam frequency is chosen such that the atom is
hotter. In the following we determine the imaging frequency and intensity that
lead to highest scattering rate. We then explore a method to increase the
fraction of atoms that survive imaging: interlacing imaging with brief cooling
stages.

The imaging frequency that leads to maximal scattering is found to be near the
Stark shifted resonance (trap bottom). The scattering rate increases with
imaging beam intensity approaching the theoretical maximum of $\thicksim 23$
kHz at our chosen operating intensity $I\thicksim\,350\,I_{\text{sat}}$.
Deviations of $\pm 50\%$ from this value have barely any effect on the
scattering rate where lower intensities than this range cause a detectable
decrease in the scattering rate away from the saturated regime. Higher
intensities cause unnecessary heating of the atom and excess camera background
noise during detection. To clearly distinguish one atom from zero atoms, we
image for 90 ms (see Sec. III(D)).

The scattering rate is maximized for different conditions than the ones
leading to optimum cooling. The optimum cooling frequency is not close to the
Stark shifted resonance, but approximately 2-3 radial motional sidebands to
the red of the shifted resonance. This behavior is consistent with the
fluorescence being suppressed by the Lamb-Dicke effect. The optimum cooling
intensity ($\thicksim\,88\,I_{\text{sat}}$) is much lower than the intensity
used for imaging. Imaging is therefore accompanied by sub-optimal cooling,
leading to a higher equilibrium temperature than optimum cooling, and
potentially to higher atom loss.

We attempt to increase the fraction of atoms that survive imaging by
interlacing imaging with cooling pulses. In order to determine how much
cooling is needed we execute a single imaging pulse with the full duration
needed for reliable single atom detection (90 ms) followed by cooling. We
estimate the temperature change during cooling by measuring the recapture
fraction for a release time of 60 $\mu$s, see Fig. 3(a). Cooling proceeds
quickly for a few milliseconds, then approaches the steady-state for $t_{\rm
cool}\gtrsim\,8$ ms. This indicates that about 10$\%$ of the total imaging
time should be spent on cooling to maintain a low temperature.

Figure 3: (a) Determination of cooling time scale. After heating the atoms for
90 ms with imaging light only (detuning of -675 kHz and
$I\thicksim\,350\,I_{\text{sat}}$), we cool them for the time $t_{\rm cool}$
and then measure the survival fraction after release and recapture. We use a
fixed release time of $t_{\rm release}$ = 60 $\mu$s. (b) Detected atom
fraction versus imaging frequency. The cooling frequency is fixed at -775 kHz
from free space resonance of the ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition.
We use an optimized imaging duty cycle of 12$\%$ (88$\%$) cooling (imaging)
light. The detuning is plotted with respect to the free space resonance of the
${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$transition. The vertical solid line shows the
approximate Stark shifted resonance of the transition. The point at -675 kHz
shows the highest survival probability of 0.97(2).

To keep the temperature low during the imaging process, we interlace the 90 ms
of imaging time with eight cooling pulses of 1.5 ms duration, i.e. we
alternate eight times between 11 ms of imaging and 1.5 ms of cooling, each
time changing frequency and intensity. This is the standard imaging timing
sequence for all images in this work, unless stated otherwise. The duration of
one cooling pulse was chosen to allow significant cooling while not wasting
time at a low scattering rate for marginal additional cooling, see Fig. 3(a).
The cooling pulse time is much longer than the timescales determining a single
Sisyphus cooling cycle (axial and radial trap period, excited state lifetime,
and inverse scattering rate) and allows the atom to scatter $\leq$ 34 photons.

We now reoptimize the imaging frequency to maximize the fraction of detected
atoms using imaging interlaced with cooling. This fraction is measured by
preparing a sample of single atoms using interlaced imaging with optimized
operating parameters (see Sec. II), and then determining how many atoms are
also detected on a second interlaced image in dependence of the imaging
frequency used for that image (see Fig. 3). For simplicity we use our standard
threshold to distinguish zero and one atoms for all frequencies instead of
optimizing it for every frequency. The best performance is reached for a
detuning of -675 kHz from the free space resonance (-12 kHz from the Stark
shifted resonance), with a detected fraction of 0.97(2). We use this detuning
for all images in this work unless stated otherwise. The benefit of imaging
interlaced by cooling is that the detected fraction is $\thicksim$3$\%$ higher
than what we could obtain without interlaced cooling.

The reduction in the detected fraction for higher and lower detuning is due to
different mechanisms. For blue detuning from the optimum value (right hand
side of the plot in Fig. 3(b)) the reduction is dominated by the probability
of an atom to be lost during imaging as evident from an increased variation in
collected photon number. The blue detuned light can heat the atom out of the
tweezer before sufficient photons can be scattered for detection. In the
region red detuned of the optimum value (left side of the plot) the poor
detected fraction is instead dominated by insufficient scattering rate as
evident from a decreased average number of collected photons. The reduction in
scattering rate also leads to the drastically increased error bars at further
red detuned frequencies. At these frequencies atoms do not scatter enough
photons to clearly separate the single atom signal from the camera noise.

We compare the temperature after our imaging process (which ends in a 1.5 ms
cooling stage) to the one obtained after a long cooling time (20 ms additional
cooling) using release and recapture measurements, see Fig. 2(a). Directly
after the imaging process the data is well described by a simulation assuming
3 $\mu$K (red circles and dotted line). This is not much above the temperature
obtained after long cooling of about 1.8 $\mu$K (blue triangles and dashed
line).

###  III.4 Detection fidelity, survival probability and minimum tweezer depth

To distinguish tweezers containing one or zero atoms on fluorescence images we
use a photon count detection threshold. We now illustrate this method and
determine the optimum detection threshold and the detection fidelity. We
measure the fraction of atoms that survive imaging and discuss its dependence
on tweezer trap depth. Finally, we compare the lifetime of atoms under cooling
and imaging conditions.

Figure 4: (a) A histogram showing fluorescence photon counts from ROIs around
single tweezers during imaging. This histogram combines the results of 1000
experimental runs using a 6$\times$6 tweezer array. The clear separation of
the peaks highlights the uniform scattering over the array. The EMCCD counts
per ROI have been converted to the number of incident photons. We use a bin
size of 0.79 photons (150 EMCCD counts). The dashed line indicates the
threshold separating the 0 atom peak (left) from the one atom peak (right) (b)
The survival fraction versus the time spent under optimized imaging cycles
(red circles) and under optimal cooling (blue triangles). The dashed lines
show fits to the data (see text).

Figure 4(a) shows a histogram of the number of collected photons in a tweezer
ROI, where the average offset from background photons and camera noise is
subtracted. Two distinct peaks are visible: one around zero photons,
corresponding to no atom in the tweezer, and another around 50 photons,
corresponding to the fluorescence count of a single atom. As is standard
procedure [14, 15, 5, 27, 9, 17, 28], we postulate that an atom is present if
the photon count is above a detection threshold, marked as dashed vertical
line in the histogram. It may happen that randomly very few photons are
scattered despite an atom being present in the tweezer or vice versa, leading
to a wrong detection result. The detection fidelity is the probability of the
detection to be correct. We determine it by calculating the overlap between a
skewed Gaussian (fit to the zero atom peak) and a Gaussian (fit to the one
atom peak) following the procedure outlined in [28]. Using the detection
threshold as an optimization parameter, we obtain a maximum detection fidelity
of 0.9991(4) for a 135(20) $\mu$K trap depth.

The duration of images can be decreased to 50 ms with only a small loss in
detection fidelity (fidelity reduced to 0.985). The loss is dominated by
misidentifying a filled tweezer as empty at the optimal threshold, and is
limited by background light on the camera and not by camera electronic noise.
In particular the stray light of the repump lasers contributes to
misidentification (the band pass filter in front of the camera insufficiently
filters their light). In fact, we obtain a better detection fidelity without
using the repump lasers, also because optical pumping to metastable states
happens rarely during imaging for our optical tweezer intensities. This is the
reason why we do not use the repump lasers during our imaging process for
shallow traps.

To determine the probability of an atom to survive the imaging process we
record two images in sequence. The probability to detect an atom on the second
image if it was present on the first is 0.97(2). The trap depth can be reduced
to 99(15) $\mu$K without sacrificing detection fidelity. However, once the
tweezer trap depth is decreased below the $\thicksim$ 135 $\mu$K level, the
chance of recovering the atom on a second image starts to decrease. For
example we measure a decrease in survival probability to 0.926(65) for a trap
depth of 99(15) $\mu$K.

Survival probability is reduced for trap depths below $\thicksim$ 135 $\mu$K
because one of the Sisyphus cooling criteria outlined in Sec. III(A) is not
met. For such low trap depths, the excited state $axial$ trap frequency
becomes lower than the natural linewidth of the transition ($\thicksim\,7.4$
kHz). When the axial trap frequency becomes that small the cooling process
does not sufficiently compensate fluorescence recoil heating in the axial
direction. The fact that Sisyphus cooling works for higher trap depths
highlights the ability of the single radial cooling beam to remove energy from
all directions simultaneously. Imaging at even lower trap depths could be
achieved by using closer to spherically symmetric potentials as those in [11]
or in a 3D lattice [4], allowing reliable imaging at even lower trap depths.
Already the achieved trap depth of 135 $\mu$K for reliable imaging is three
times less than obtained with blue imaging [17], making it possible for us to
obtain three times more tweezers for a given tweezer laser source power.

In Figure 4(b) we show the survival probability over time $t_{\rm hold}$ when
continually imaging or when just cooling. For these long measurements we turn
on the repump lasers and close the atomic beam shutter. We fit the data by
$e^{-(t/\tau)^{\alpha}}$, where $\alpha$ and $\tau$ are fit parameters. For
continual imaging cycles the fit provides $\alpha$=1 and a 1/e lifetime of
$\tau$=79(3) seconds, allowing for hundreds of pictures to be taken of a
single atom. This decay of the survival fraction is equivalent to $p_{1}^{N}$,
where $N$ is the number of elapsed images and $p_{1}=0.9986(4)$ [17]. For
continuous cooling the fit provides $\alpha$=0.8 and $\tau$=116(5) seconds.
The deviation from a pure exponential decay might be due to slowly improving
vacuum quality over the course of each measurement, triggered by the atomic
beam shutter closure. When analysing individual tweezers, we find that some
tweezers have the same lifetime under cooling and imaging conditions.

The finite lifetime can have a variety of origins. We verify that the
temperature of the atoms stays constant under both investigated conditions,
excluding a slow process heating the atoms out of the trap. We find that the
lifetime depends on the vacuum quality, as lifetime degrades over months and
increases to the values stated above only after flashing titanium sublimation
pumps. The decrease in lifetime from cooling to imaging conditions for most
tweezer sites indicates that the small trap depth variation between tweezers
of $3\%$ make it impossible to optimize cooling and imaging for all tweezers.

We observe day to day changes of the survival probability originating from
drifts away from ideal conditions, in particular magnetic field drifts.
Magnetic field drifts on the $\thicksim$ 20 mG level affect the single image
survival probability significantly ($\thicksim\,2$% reduction).

##  IV Site selective imaging

Our imaging technique provides an additional advantage. Using an easily
achievable differential Stark shift, one can tune a certain tweezer out of
resonance with the imaging light used for the rest of the array. This allows
for selective imaging of either the remaining tweezers of the array or of the
single shifted tweezer. The ability to selectively readout a single atom from
the array is a necessary step for error correction in many quantum computation
algorithms [29, 30].

To demonstrate the ability to select (or dark out) an atom from the image, we
use a tweezer created by the crossed AODs to create a deeper potential for a
single tweezer site in a 3$\times$3 tweezer array. To characterize site
selective images, we record four consecutive images in one experimental run,
see Fig. 5. The first image (Fig. 5(a)) is taken directly after single atom
preparation, as described in Sec. II. The AODs are then turned on for the
second and third image (Fig. 5(b,c)), in which we image the single shifted
tweezer and the rest of the array respectively. To record atoms in the shifted
tweezer, we increase the imaging detuning to -2 MHz (i.e. 1.325 MHz to the red
of the usual imaging detuning). In the fourth image we turn off the additional
tweezer and again image the entire array (Fig. 5(d)).

We can image the single site such that it is detected in image two with a
survival probability of 0.96(2), and never appears in image three. Moreover,
the entire atom array survives this ’dark out’ measurement with a probability
of 0.95(2).

The lower survival probability, in comparison to the value obtained in Sec.
III, is due mainly to worse balancing of the trap intensities for the nine
trap array used for this measurement. Additionally, the cooling frequency and
intensity in the deeper single tweezer were not fully optimized. We note that
the AC Stark shift chosen here is too small to fully protect quantum
information of the remaining atoms but provides an initial proof of concept.
However a stronger AC Stark shift could make this feasible.

Figure 5: Site selective imaging using an additional tweezer to apply a
differential Stark shift to one tweezer. (a) A first image is recorded to
check loading. (b-c) After turning on the AOD tweezer on top of the center
tweezer, we image first only the center tweezer and then the rest of the array
by using respectively appropriate imaging and cooling detunings. (d) When the
AODs are turned off, all traps are visible in a final image using the normal
imaging detuning. The images show the average fluorescence from 300
experimental realizations.

##  V Conclusion

In conclusion, we have demonstrated the detection of single Sr atoms in
shallow tweezers with high fidelity (0.9991(4)) and survival probability
(0.97(2)). Detection is based on imaging on the red, narrow linewidth
${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$ $(|m_{j}|=1)$ transition. We show that with
proper frequency and intensity modulation a high scattering rate can be
maintained while keeping the temperature of the atoms low.

Our red imaging technique works for a wide range of trap depths, and for
shallow traps, red imaging is advantageous over blue imaging. We need slightly
($\thicksim$2-fold) increased imaging times (100 ms instead of 50 ms) in
comparison to blue imaging on the broad linewidth ${}^{1}S_{0}$ \-
${}^{1}P_{1}\,$transition in deep traps (450 $\mu$K depth) [17]. However, in
shallow traps blue imaging is limited by the cooling rate, leading to
excessive imaging times in comparison to red imaging [4].

In contrast to blue imaging, red imaging avoids optical pumping of ground
state atoms into the metastable states (via ${}^{1}D_{2}$). Imaging in shallow
traps reduces off-resonant scattering of trap light by metastable state atoms
(${}^{3}P_{0,2}$), leading to longer coherence times. Red imaging in shallow
traps combines both advantages and enables high-fidelity shelving into
metastable states for state specific detection or clock readout.

We show that, with a small additional Stark shift, we can isolate a single
tweezer of the array from the imaging process. This allows us to selectively
image (or hide) a single atom of the array. Through application of a bias
field of $\thicksim$ 50 G, this selective imaging technique could be further
extended to state-selective imaging for hyperfine ground states in the
fermionic isotope. This opens the possibility of imaging more than two
hyperfine ground states without disturbing the others. This will be a useful
tool for quantum simulations or qudit style quantum computing [31, 32, 33].

It should be possible to extend red imaging to situations beyond the specific
one examined here. The small potential wells containing the atoms can also be
created by optical lattices, or other tightly confining dipole traps, making
it possible to use the technique in quantum gas microscopes or 3D lattice
clocks. This detection technique should work at nearly all tweezer wavelengths
where one of the $m_{J}$ states of ${}^{3}P_{1}$ is stronger trapped than the
ground state. In particular, using 515-nm tweezers would be an appealing
option. This is because red imaging avoids the leakage channel through
${}^{1}D_{2}$ from which blue imaging suffers [14, 15]. Tweezers at this
wavelength are also likely to trap most Rydberg states [28]. The large
polarizabilities at this wavelength, small diffraction limit of the tweezer
light, and shallow required trap depth of red imaging would allow for the
creation of $\geq\,1000$ strontium atom tweezer arrays with current laser
technology.

## Acknowledgments

We thank the other members of the Strontium Quantum Gasses group at the
University of Amsterdam for useful discussions and debugging tips. We also
thank the Eindhoven quantum computing group led by Servaas Kokkelmans and
Edgar Vredenbregt for stimulating discussions. This project is supported by
the Netherlands Organization for Scientific Research (NWO) under the
Gravitation grant No. 024.003.037, Quantum Software Consortium, and under
grant No. FOM-153. This work is supported by the Dutch Ministry of Economic
Affairs and Climate Policy (EZK), as part of the Quantum Delta NL programme.

## Appendix A: Experimental Sequence

Figure 6: (a) Detailed sketch of crucial elements of the experimental setup.
Thite arrow from the left indicates the beam of slowed atoms from the oven.
The Zeeman slower and repump beams are shown in blue. The horizontal,
overlapping blue and red MOT beams are shown in purple. An additional
vertically upwards propagating red MOT beam (shown in pale red) provides
confinement against gravity. The initial and final red MOT positions are shown
as intense red spots. The red MOT is moved vertically 1 cm into the focus of
the microscope objective. The 679 nm and 707 nm repump lasers co-propagate
with the cooling/imaging beam. (b) Schematic of the tweezer setup. The main
tweezer array is generated using an SLM. An additional tweezer can be created
using a pair of crossed AODs. The SLM and AOD tweezers are combined using a
polarizing beam splitter. Both systems are imaged through an NA= 0.5 objective
onto the atoms. Fluorescence light from the atoms is separated from the
tweezer light using a long pass dichroic mirror. The fluorescence is then sent
onto an EMCCD camera (Andor Ixon 897). (c) Experimental sequence (numbers in
brackets give time spans in ms). The figure uses acronyms for blue MOT (B
MOT), broadband red MOT (BBR MOT), single frequency red MOT (SFR MOT) and
light assisted collisions (LAC).

We utilize a unique technique for loading our narrow linewidth MOT in order to
create optical access for the microscope objective. First, 88Sr atoms from an
$\thicksim$ $500^{\circ}\mathrm{C}$ oven are slowed using a Zeeman slower
operating on the 2$\pi\times 30$ MHz wide ${}^{1}S_{0}$ \-
${}^{1}P_{1}\,$transition at 461 nm. The slowed atoms are then further cooled
and compressed by a four beam ’blue’ MOT, also using the ${}^{1}S_{0}$ \-
${}^{1}P_{1}\,$transition, in a 3D quadrupole field to milli-Kelvin
temperatures. This blue MOT consists of two sets of retro-reflected beams
($1/e^{2}$ waist of $\thicksim$ 12 mm) that are perpendicular to each other
and horizontal. Refraining from implementing the usual third MOT beam pair
allows us to place the microscope objective along the gravity axis without
complications from that beam pair, as shown in Fig. 6(a). This blue MOT is an
incomplete trap as it provides no confinement against gravity. However the
blue MOT is able to quickly cool and confine the atoms in the horizontal
plane, which comprises the only dimension along which atoms entering from the
Zeeman slower are fast. A significant fraction of atoms are then trapped in
the quadrupole magnetic field (52 G/cm gradient in the axial direction, which
is vertical) of the MOT by optical pumping to low-field seeking states of the
${}^{3}P_{2}$ manifold. This optical pumping is naturally happening when atoms
rapidly scatter MOT light and decay from ${}^{1}P_{1}$ through the 5s4d
${}^{1}D_{2}$ state to ${}^{3}P_{2}$.

After quadrupole trap loading, all blue lasers are switched off and the
magnetically trapped atoms are repumped back to the ground state, using a 497
nm laser resonant with the ${}^{3}P_{2}$-${}^{3}D_{2}$ transition.
Simultaneously the quadrupole field gradient is reduced to 0.63 G/cm in the
vertical direction. The atoms are then loaded into a five beam narrow
linewidth ”red” MOT operating on the ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$
transition. Four of the five red MOT beams are overlapped with the blue MOT
beams and the fifth beam ($1/e^{2}$ waist of $\thicksim$6 mm) is propagating
vertically upwards.

There is no need for a downwards propagating beam because the upward radiation
pressure force is limited by the narrow linewidth and the MOT quadrupole field
to a small phase-space region. This force is counter-balanced by gravity. The
atoms settle into a cloud on the lower part of a shell of equal B-field
magnitude below the quadrupole centre. This shell is defined by the detuning
of the MOT beams being equal to the Zeeman shift induced by the B-field. This
trap scheme again does not need a beam going through the microscope objective.

The red MOT beams are initially frequency modulated in order to create a comb
of frequencies from -60 to -3000 kHz detuning with 20 kHz spacing. The
modulation range and intensity of this broadband red MOT are decreased over
181 ms, while a bias field of $\thicksim$0.6 G against gravity is ramped on,
raising the atoms by 1 cm, from the centre of the vacuum chamber to the focal
plane of the microscope objective, by shifting the center of the quadrupole
field. An additional small bias field produced by three orthogonal coil pairs
is ramped while the MOT position is raised and used to finely position the red
MOT onto the tweezer array. The frequency modulation is then switched off and
single frequency red MOT beams, with a detuning of $-100$ kHz and intensity of
8 $I_{\rm sat}$, are used to load the tweezers.

We optimize all parameters of the experiment up to this point on achieving the
desired red MOT atom number in a reliable way and in a short time. We find
that for MOTs (with our selected detuning) of 5$\times 10^{4}$ to 3$\times
10^{6}$ atoms, the entire tweezer array can be loaded with $\geq$ 1 atom per
site, where on the low end we get slightly below unity filling. On the high
end the high density of atoms in the tweezers leads to less than half of the
tweezers being filled with atoms, presumably because of additional non-
pairwise losses during the light assisted collision step. Because of this
robustness to atom number fluctuations, we load the magnetic reservoir for a
variable amount of time at the end of an experimental sequence
($\thicksim\,200$ ms). The exact time is determined by the need to store data
of the last run on the data analysis computer and the need to prepare the next
experimental sequence in the experiment control computer. This procedure
creates a MOT of approximately 5$\times 10^{5}$ atoms at a temperature
$\leq\,1.5\,\mu$K in the focal plane of our objective.

Slightly before the MOT is switched to single frequency operation, the
tweezers are switched on, see also next Appendix. After the red MOT has
reached its final position we wait for 50 ms to load the tweezers. Then we
switch off the MOT lasers and quadrupole field, and ramp the bias magnetic
fields to 0 G at the position of the tweezers.

Single atom detection only worked properly once we spectrally filtered the
689-nm laser light used to induce fluorescence. The source of all 689-nm light
is an external cavity diode laser (ECDL) that is short-term stabilized on a
reference cavity with a linewidth of 35 kHz, which in turn is long-term
stabilized (in length) on a spectroscopy signal. Light from the ECDL is
amplified by injection locked lasers and then used on the experiment.
Initially we used light from the ECDL directly to inject the amplifying
diodes. This ECDL light is spectrally broadened by servo bumps from the
locking electronics and amplified spontaneous emission and we found it
impossible to prepare and detect single atoms. We then used the light that is
transmitted through and therefore filtered by the reference cavity to inject
the amplifying diodes, allowing us to achieve the single atom preparation and
detection results presented here. The red MOT behavior did not noticeably
change when switching from unfiltered to filtered light.

The light used for light assisted collisions, imaging, and cooling is sent
onto the tweezer array via a single beam with a polarization perpendicular to
both the tweezer propagation axis and the tweezer polarization, and a
$1/e^{2}$ waist of $\thicksim$ 1 mm. We find robust single atom preparation of
$\thicksim$ 50% for a detuning of approximately -100 kHz from the Stark
shifted resonance (-750 kHz from free space resonance). We change the
intensity of the beam from low ($I\,\thicksim\,88\,I_{\text{sat}}$ for 10 ms)
to high ($I\,\thicksim\,700\,I_{\text{sat}}$ for 150-200 ms) then back to low
($I\,\thicksim\,88\,I_{\text{sat}}$ for 10 ms) in order to cool the loaded
atoms into the tweezer, induce light assisted collisions [14, 20], and cool
the single atom before taking the first image of an experimental run.

## Appendix B: tweezer creation

The 813.4-nm laser light used for the optical tweezers in our experiment is
generated by an external cavity diode laser, which is amplified to 1.7 W using
a tapered amplifier (TA). The output of the TA is divided into two optical
paths, a main path to create the tweezer array using an SLM (Meadowlark P1920
1920$\times$1152) and a second path for a movable tweezer using AODs (AA opto-
electronic DTSXY-400-800), see Fig. 6(b). The main output path is sent through
a dispersive prism in order to filter out any amplified spontaneous emission
from the TA and is then sent through an acousto-optic modulator for intensity
control before being coupled into a fibre. The second path is sent without
further filtering into an optical fibre.

The optical tweezers are created by imaging an array of beams through a
microscope objective (NA= 0.5, Mitutoyo 378-848-3). An almost arbitrary and
stationary pattern of tweezers is created using the SLM. In order to calculate
the phase pattern of the desired tweezer pattern we use the weighted
Gerchberg-Saxton algorithm [34]. The phase imprinted onto the incident beam by
the SLM is a sum of phases including the tweezer array pattern phase, a lens
phase to Fourier transform the phase to a real image, a grating phase to
separate the zeroth order, and a factory correction phase.

The sum of these phases creates an array of foci $\thicksim$ 180 cm from the
SLM. This array is imaged through the microscope objective (effective focal
length $f=4$ mm) with a field lens of $f=500$ mm taking care that the array of
beams is conjugated onto the aperture of the microscope objective. In this
work all results shown have been performed with the SLM creating a square
6$\times$6 array of tweezers unless otherwise noted.

Additional balancing of the tweezer trap depths can be achieved by finetuning
the SLM pattern. As a first step we spectroscopically measure the depth of
each tweezer by inducing heating on the ${}^{1}S_{0}$ \- ${}^{3}P_{1}\,$
$(m_{j}=0)$ transition, which is weaker trapped than the ground state. The
loss feature is then fit with a Gaussian, and the center frequency is
extracted for each tweezer. The detuning of this frequency from the free space
resonance is proportional to the tweezer intensities. The amplitude of each
tweezer, in the pattern to be calculated, is then weighted based on these
measured center frequencies. The tweezer phase pattern is then recalculated
using these new weights. This procedure allows us to balance the trap depths
across the 6$\times$6 array to a standard deviation of approximately 3% [26,
27, 14]. This procedure was not executed for the 3$\times$3 array used in Sec.
IV.

We finally characterize the depth of our tweezers using spectroscopic method
explained above. The error in our trap depth determination has two sources.
The choice of either the blue edge frequency or center frequency, based on if
a purely thermally broadened line shape or a purely power broadened line shape
is fit respectively, provides 15 $\mu$K of error [14]. An additional 5 $\mu$K
uncertainty comes from the 3 $\%$ standard deviation of the optimized SLM
pattern. For the tweezers used throughout the paper, we estimate a waist of
$\thicksim$ 0.84 $\mu$m and an optical power on the atoms of $\thicksim$ 2.33
mW per tweezer.

An additional tweezer (or tweezers) can be created using a crossed pair of
acousto-optic deflectors (AODs) whose position in the focal plane can be
controlled through radio frequency tones sent to the AODs. Unlike the SLM’s
slow refresh rate, this tweezer can move at speeds sufficient for sorting
atoms into defect free arrays, or for quickly applying an additional tweezer
for isolating one tweezer from the rest of the array (see Sec. IV).

## Appendix C: Sisyphus cooling simulation parameters

For the Sisyphus cooling simulation we consider a trap depth of
$k_{B}\times$135 $\mu$K = $h\times 2.8$ MHz. For our estimated waist of 0.84
$\mu$m, this gives a radial trap frequency of $43$ kHz for the ground state.
We calculate the ${}^{1}S_{0}$ $(\alpha_{g})$ and
${}^{3}P_{1}\,(|m_{j}|=1,\alpha_{e})$ polarizabilities to be 286 a.u. and 355
a.u. respectively. For the results presented in Fig. 2(c), we use a Rabi
frequency of 2$\pi\times 42$ kHz. We include 15 harmonic oscillator levels in
our calculation.

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