# Repetitive readout and real-time control of nuclear spin qubits in 171Yb
atoms

William Huie  Department of Physics, The University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Lintao Li  Department of Physics, The
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Neville
Chen  Department of Physics, The University of Illinois at Urbana-Champaign,
Urbana, IL 61801, USA  Xiye Hu  Department of Physics, The University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA  Zhubing Jia  Department
of Physics, The University of Illinois at Urbana-Champaign, Urbana, IL 61801,
USA  Won Kyu Calvin Sun  Department of Physics, The University of Illinois at
Urbana-Champaign, Urbana, IL 61801, USA  Jacob P. Covey<EMAIL_ADDRESS>Department of Physics, The University of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA

###### Abstract

We demonstrate high fidelity repetitive measurements of nuclear spin qubits in
an array of neutral ytterbium-171 (171Yb) atoms. We show that the qubit state
can be measured with a spin-flip probability of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.004(4)}$
for a single tweezer and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.012(3)}$
averaged over the array. This is accomplished by near-perfect cyclicity of one
of the nuclear spin qubit states with an optically excited state under a
magnetic field of $B=58$ G, resulting in a spin-flip probability of $\approx
10^{-5}$ per photon during fluorescence readout. The performance improves
further as $\sim 1/B^{2}$. The state discrimination fidelity is
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$
with a state-averaged readout survival of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.994(3)}$,
limited by off-resonant scattering to dark states. We combine our measurement
technique with high-fidelity rotations of the nuclear spin qubit via an AC
magnetic field to explore two paradigmatic scenarios, including the non-
commutivity of measurements in orthogonal bases, and the quantum Zeno
mechanism in which measurements “freeze” coherent evolution. Finally, we
employ real-time feedforward to repetitively and deterministically prepare the
qubit in the $+z$ or $-z$ direction after initializing it in an orthogonal
basis and performing a measurement in the $Z$-basis. These capabilities
constitute an important step towards adaptive quantum circuits with atom
arrays.

##  I Introduction

Measurements play a crucial role in quantum information science to determine
the result of the intended operation, correct errors [1, 2], and prepare
useful many-body states [3, 4, 5, 6, 7, 8, 9, 10]. Ideally, the qubits are not
lost as a result of these measurements and remain in the state corresponding
to the measured outcome. These conditions taken together constitute a quantum
nondemolition (QND) measurement, which has been demonstrated on many quantum
hardware platforms [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25]. The ability to perform measurements in isolated atoms or atom-like
systems in a solid state host is often hampered by the complex, multi-level
structure of the atomic system. Such measurements are typically performed via
optical fluorescence readout in which one qubit state is “bright” while the
other is “dark”. In many cases, imperfect cyclicity of the bright state leads
to leakage between the two states and limits the brightness or contrast of the
qubit, which can be addressed by either using single photon detectors [26, 27,
28, 29, 30] or by coupling the qubit to an optical cavity [31, 32, 33, 34, 35,
36]. However, neither approach is readily compatible with scalable parallel
qubit readout.

FIG. 1:  Overview. (a) The experimental system consists of a glass vacuum cell
with one microscope objective on either side. Atoms are illuminated with two
retro-reflected probe beams that each have an angle with respect to the $xy$\-
and $yz$-planes. The tweezer array lies parallel to the $y$-axis. The DC
magnetic field and the tweezer electric field point in the $y$-direction. An
AC magnetic field produced by the pair of coils shown points in the
$z$-direction. Images recorded on the camera can be analyzed in real-time to
enable or disable AC pulses on the coils. (b) The relevant level structure of
${}^{171}\text{Yb}$, showing the cycling transition from the
${}^{1}\text{S}_{0}$ ${|{m_{F}=-1/2}\rangle}\equiv{|{0}\rangle}$ qubit state
making it bright while ${|{m_{F}=+1/2}\rangle}\equiv{|{1}\rangle}$ is dark
during readout. (c) Energies of the four $m_{F}$ Zeeman states in
${}^{3}\text{P}_{1}$ $F=3/2$ relative to free space versus magnetic field,
taken with a $\approx 1\,\text{mK}$ tweezer depth for a single array site. (d)
A histogram of $12\,\text{ms}$ fluorescence readout
($2500\,\text{repetitions}\times 5\,\text{sites}=12500\,\text{shots}$) of the
nuclear spin qubit, where the dark peak is either zero atoms or an atom in
$|1\rangle$ and the bright peak is an atom in $|0\rangle$. The system was
initialized in $|0\rangle$. The discrimination fidelity is
$\mathcal{F}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$.
Inset: Averaged ($2500\,\text{repetitions}$) camera image of fluorescence from
a 5-site array with spacing of $7.8\,\text{$\mu$m}$.

Arrays of neutral atoms in optical tweezers [37] are rapidly emerging as a
leading platform for myriad quantum science applications ranging from quantum
simulation [38, 39], computing [40, 39], and sensing [41, 42] to networking
[43, 44]. Scalable, lossless readout of hyperfine qubits has been performed
with arrays of neutral alkali atoms in optical tweezers [45, 46, 47], and QND
readout of hyperfine [48, 49] and nuclear spin qubits [50, 51, 52] has been
performed with ensembles of atoms in an optical dipole trap. The use of
optical qubits helps obviate cyclicity limitations and thus optical qubits are
readily compatible with scalable single-atom QND measurements [53, 14, 54, 41,
42], but are hampered by requirements on optical phase stability and atomic
temperature. For spin-encoded qubits on the other hand, a scalable approach to
QND readout in arrays of neutral atoms in optical tweezers remains an
outstanding challenge. Namely, $\approx 1000$ photons must be scattered for
high fidelity detection of single atoms in free space with collection
efficiencies typically at the single-percent level, and thus QND readout with
percent-level atom loss and qubit depolarization requires such events to be at
the $\approx 10^{-5}$ level per photon.

Here, we leverage the unique atomic structure of neutral 171Yb atoms to
directly perform high-fidelity QND measurements of a qubit encoded in the
nuclear spin-1/2 degree of freedom in the electronic ground state. By
performing fluorescence detection via the relatively narrow 3P1 optically
excited state in which the $m_{F}=-3/2$ Zeeman sub-level is sufficiently
isolated at a modest magnetic field of $58$ G, the polarization selection rule
for decay to only the $m_{F}=-1/2$ ground state provides a cyclicity of
$\approx 10^{5}$ with respect to the $m_{F}=+1/2$ ground state, corresponding
to an average qubit depolarization probability of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.004(4)}$
for a single tweezer
(${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.012(3)}$
for array-averaged) during $12$-ms measurements with fidelity of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$
and survival probability of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.994(3)}$.
We demonstrate this technique for an array of 171Yb atoms in optical tweezers
of wavelength $\approx 760\,\text{nm}$ – ideal for subsequent manipulation of
the optical “clock” transition [55, 56]. Unlike the seminal demonstrations of
free space non-destructive qubit readout in alkali atom arrays [45, 46, 47],
our tweezers are relatively shallow ($U_{0}/k_{B}\approx 580\,\text{$\mu$K}$)
and remain on the entire time, obviating the need to chop them out of phase
with the probe light. Moreover, the probe beams are randomly polarized and
have projections onto all three dimensions, allowing the atoms to stay cold in
three dimensions under probe illumination, with a temperature of $T\approx 5$
$\mu$K.

We combine high-fidelity projective measurements with qubit rotations to
explore textbook scenarios including observation of the non-commutivity of
measurements in variable bases, and demonstration of the quantum Zeno
mechanism by studying the interplay of measurement and qubit rotation during
repetitive alternation. Finally, we implement real-time adaptive control [57]
to perform a qubit rotation conditioned on the measurement outcome in order to
deterministically prepare a target state after a projective measurement, and
we show the ability to repetitively do so in alternation with a rotation to an
orthogonal basis. If combined with the ability to perform measurements on only
subsets of qubits [36, 57, 58], this work would aid in the realization of
measurement-based quantum computation [59, 60, 61], non-unitary many-body
state preparation protocols [5, 6, 8, 9, 10], quantum error correction [1, 2],
and the study of measurement-induced phase transitions [62, 63, 64, 65, 66].

##  II Overview of the experimental system

We begin with a laser cooled ensemble of 171Yb atoms suspended in the center
of a glass cell held under ultrahigh vacuum [see Fig. 1(a)]. High-resolution
microscope objectives with diffraction-limited $\text{NA}\approx 0.6$ are
placed on either side of the glass cell. A one-dimensional array of five
optical tweezers with wavelength $\lambda_{T}\approx 760\,\text{nm}$ and
$1/e^{2}$ waist radius of $w_{0}\approx 670$ nm is generated with an acousto-
optic deflector (AOD) [67]. The spacing between adjacent tweezers is
$d=7.8\,\text{$\mu$m}$, corresponding to a frequency difference between
adjacent radio frequency tones sent to the AOD of $1.75$ MHz. The inset to
Fig. 1(d) shows $2500$-shot averaged images of the fluorescence from our
$5$-site array. We use a power of $7\,\text{mW}$ per tweezer, which
corresponds to a depth of $U_{0}/h\approx 12\,\text{MHz}$ ($U_{0}/k_{B}\approx
580\,\text{$\mu$K}$) in the ground state. Appendix A provides further details
of our experimental system.

FIG. 2:  Circuit legend. Simplified equivalent circuit elements. The number
0/1 and the open/filled circle distinguishes “qubit readout” from “atom
readout”, respectively.  FIG. 3:  Characterizing depolarization during qubit
readout. (a) Circuit diagram for studying the dependence of depolarization
during qubit readout on magnetic field. A probe block of variable magnetic
field is placed between two measurements performed at $58\,\text{G}$. The
final “atom readout” pulse is used to post-select on events in which the atom
survived the entire sequence. (b) The measured array-average (diamonds) and
single-site best (squares), SPAM-corrected, and the predicted state
populations (line) under real imaging conditions versus magnetic field. The
error bars reflect the standard deviation of a binomial distribution. (c)
Circuit diagram for directly studying the depolarization probability during
qubit readout at $58\,\text{G}$. (d) The array-averaged histograms from each
image for $17500$ shots, with the results shown together on a 2D histogram.
These results show that the D $\rightarrow$ B and B $\rightarrow$ D
conditional depolarization probabilities during a $12\,\text{ms}$ probe pulse
are
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
(${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.011(3)}$)
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
(${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.018(4)}$)
averaged across the array (best site), respectively, before SPAM correction.

The tweezers are continuously on during the MOT phase; atoms remain in the
tweezer traps after the MOT light and magnetic field gradient have been turned
off. Single atoms are obtained in the tweezers by applying a cooling pulse
using the MOT beams under a field of $1.5\,\text{G}$ in the $y$-direction,
which is also the direction of the tweezer’s polarization [see Fig. 1(a)].
This pulse has a total intensity of $I_{\text{cool}}=1.3\,I_{\text{sat}}$,
where $I_{\text{sat}}$ is the saturation intensity of the intercombination
transition, with a detuning of $\delta/2\pi\approx-150$ kHz
($\approx-0.8\,\Gamma$) from the $F=3/2$, $m_{F}=-1/2$ state. This cooling
pulse drives light-assisted collisions that transform the initially Poissonian
atom number distribution into either just 0 or 1 atom remaining [68]. After
the cooling pulse, we obtain a tweezer loading fraction of $p\approx 0.7$ –
which is correlated with the density of the reservoir from which we load,
suggesting that higher loading fractions are possible [69] – and an atomic
temperature of $T\approx 5\,\text{$\mu$K}$ measured via release and recapture
from the tweezers [70]. See Appendices A and B for further details.

Readout is performed using the same transition as that for the MOT
(1S${}_{0}\leftrightarrow$ 3P1, $F=3/2$) [71, 72], where the transition
linewidth of $\Gamma/2\pi=182$ kHz is well suited for our photon scattering
rate of $\Gamma_{\text{scatt}}\approx 97000\,\text{photons/s}$ during
fluorescence detection. We use two counter-propagating beams that each have an
angle of $\approx 15$ degrees with respect to $xy$-plane and an angle of
$\approx\pm 30$ degrees with respect to the $yz$-plane [see Fig. 1(a)]. This
configuration is used to minimize the effect of surface scatter from the
beams, which do not pass through the faces of the cell used by the microscope
objectives. The polarization of the beams is chosen to have large projections
onto both $\pi$ and $\sigma^{\pm}$. We use an electron-multiplying CCD (EMCCD)
to image atomic fluorescence via the microscope objective opposite the one
used to generate the tweezers. Typical readout pulses are $12\,\text{ms}$ long
with probe intensity of $I_{\text{probe}}\approx 1\,I_{\text{sat}}$ (the total
beam intensity is $\approx 3$ times higher but we assume it is equally divided
among the three polarizations) and detuning
$\delta/2\pi\approx-180\,\text{kHz}$ with respect to the target $m_{F}$ state
within the $F=3/2$ manifold. Our imaging system uses a magnification of
$\approx 9$, and our estimated atom-to-camera collection efficiency is
$\approx 0.04$ based on the calculated scattering rate for these probe
conditions and the number of photons collected on the camera.

Table 1:  Summary of discrimination fidelity, atom survival, and
depolarization probabilities. The discrimination fidelity $\mathcal{F}$, atom
survival probabilities $\eta_{\text{surv}}^{\text{B}}$,
$\eta_{\text{surv}}^{\text{D}}$, $\bar{\eta}_{\text{surv}}$, and
depolarization probabilities
$P_{\text{depol}}^{\text{D}\,\rightarrow\,\text{B}}$,
$P_{\text{depol}}^{\text{B}\,\rightarrow\,\text{D}}$, $\bar{P}_{\text{depol}}$
are listed for the $m_{F}=-3/2$ imaging condition both on a single array site
and averaged over a 5-site array. Symbols with superscripts are state-
dependent, with ‘B’ referring to the ${|{0}\rangle}$ bright state and ‘D’
referring to the ${|{1}\rangle}$ dark state, while those with bars are state-
averaged. Numbers are listed with and without SPAM correction.  Array-averaged
---
| $\mathcal{F}$ | $\eta_{\text{surv}}^{\text{B}}$ | $\eta_{\text{surv}}^{\text{D}}$ | $\bar{\eta}_{\text{surv}}$ | $P_{\text{depol}}^{\text{D}\,\rightarrow\,\text{B}}$ | $P_{\text{depol}}^{\text{B}\,\rightarrow\,\text{D}}$ | $\bar{P}_{\text{depol}}$
Uncorrected | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.960(9)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.9960(1)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.978(5)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
Corrected | — | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.99(1)}$ | — | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.994(3)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.010(6)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.013(2)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.012(3)}$
Best site
| $\mathcal{F}$ | $\eta_{\text{surv}}^{\text{B}}$ | $\eta_{\text{surv}}^{\text{D}}$ | $\bar{\eta}_{\text{surv}}$ | $P_{\text{depol}}^{\text{D}\,\rightarrow\,\text{B}}$ | $P_{\text{depol}}^{\text{B}\,\rightarrow\,\text{D}}$ | $\bar{P}_{\text{depol}}$
Uncorrected | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.992(4)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.966(8)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.9963(2)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.981(4)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.011(3)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.018(4)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.014(2)}$
Corrected | — | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.99(1)}$ | — | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(5)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.003(3)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.005(7)}$ | ${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.004(4)}$

##  III Nondestructive qubit readout

A crucial feature of our nondestructive qubit readout technique is the
electric dipole polarization selection rule associated with our choice of
excited state [see Fig. 1(b)]. The $m_{F}$ states within 3P1 are well resolved
even at low magnetic fields due to the relatively narrow linewidth
($\Gamma/2\pi\approx 182$ kHz) and the large g-factor ($\approx 1.4$ MHz/G),
as shown in Fig. 1(c). Based on the zero-field detunings of $|m_{F}|=1/2$ and
$|m_{F}|=3/2$ and the known ground-state light shift, we estimate the
differential polarizabilities $\alpha=(U_{e}-U_{0})/U_{0}$ at this tweezer
wavelength ($\lambda_{T}\approx 760\,\text{nm}$) to be
$\alpha_{|1/2|}\approx-0.030(3)$ and $\alpha_{|3/2|}\approx 0.25(3)$ (see
Appendix F), which are in good agreement with recent observations [69, 72].
Although the $|m_{F}|=1/2$ states are appealing due to the nearly-zero
differential light shift and the assurance that both nuclear spin ground
states will remain bright [72], the positive differential light shift of the
$|m_{F}|=3/2$ state corresponds to the case where the excited state is deeper
trapped than the ground state – a scenario in which attractive Sisyphus
cooling has been observed for strontium (Sr) [54, 73, 74] and predicted for Sr
and Yb [75, 76].

In this work, we focus on the $m_{F}=-3/2$ excited state which, under ideal
conditions, can decay only to the $m_{F}=-1/2$ ground state. This allows us to
perform “qubit readout” since the $|m_{F}=-1/2\rangle\equiv|0\rangle$ state
will remain bright while the $|m_{F}=+1/2\rangle\equiv|1\rangle$ state is dark
[see Fig. 1(b)]. It is also crucial to be able to perform “atom readout” –
which is state-independent – in order to differentiate a perceived outcome of
$|1\rangle$ in a qubit measurement from cases where the atom may have been
lost. We employ two techniques for performing atom readout (see Fig. 2). One
is to use the $m_{F}=-1/2$ excited state which is connected to both ground
states; the other is to re-initialize the qubit in $|0\rangle$ via optical
pumping and then perform qubit readout. Our optical pumping efficiency is
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.98(1)}$
with state preparation and measurement correction (see Appendices A & I),
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.972(9)}$
without. We focus primarily on the latter technique, mostly to avoid the need
to change the probe frequency by many tens of MHz when going between the
$m_{F}=-3/2$ and $m_{F}=-1/2$ excited states. Nevertheless, we find that
imaging with $m_{F}=-3/2$ and $-1/2$ offer similar performance: the collection
efficiency is similar, and the steady-state temperature under probing is
$T\approx 5$ $\mu$K for both. See Appendix B for further details on their
comparison.

We note that both cases are limited by tweezer-induced off-resonant scatter
during probing and cooling from the steady state population in $6s6p$ 3P1 to
the higher $6s7s$ 3S1 state, which can then subsequently decay to the entire
$6s6p$ 3PJ manifold. In principle, 3P2 and 3P0 could be repumped, but we note
that 3P2 – which is the dominant decay path – is unfortunately strongly anti-
trapped in tweezers with wavelength $\lambda_{T}\approx 760\,\text{nm}$ due to
its proximity to the 3P${}_{2}\leftrightarrow$ 3S1 transition at
$770.2\,\text{nm}$. Without any repumping, we observe the lifetime in
${|{0}\rangle}$ under probing to be $\tau=1.24(8)\,\text{s}$ which is
consistent with our model (see Apppendix E). We choose a probe time of
$12\,\text{ms}$ as an optimal compromise that offers a bright/dark
discrimination fidelity of
$\mathcal{F}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$
corresponding to the histogram in Fig. 1(d), and a probe survival of the
$|0\rangle$ state of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.99(1)}$
with state preparation and measurement (SPAM) correction (see Appendix I), in
good agreement with our measured lifetime. Significant gains are possible by
operating in a shallower tweezer and improving the collection and detection
efficiency. We measure the lifetime in $|1\rangle$ under probe conditions to
be $2.99(6)\,\text{s}$, suggesting that the survival of the $|1\rangle$ dark
state is
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.9960(1)}$.
The state-averaged survival during qubit readout is thus taken to be
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.994(3)}$.

To ensure that the $|0\rangle$ qubit state remains bright while the
$|1\rangle$ qubit state remains dark during probing, we require excellent
isolation of the $|0\rangle\leftrightarrow|3/2,-3/2\rangle$ transition. There
are two effects that limit this isolation: Raman transitions via other excited
states, and mixing between the excited states. We analyze these effects in
Appendices D and G, respectively, showing that both are suppressed
quadratically in magnetic field. The Raman transitions have a spontaneous
contribution and a stimulated contribution. The latter is due to the presence
of all polarization components [see Fig. 1(b)] but the effect is suppressed by
the $\approx 45$ kHz nuclear spin splitting at 58 G. Although choosing our
probe polarizations to contain only $\sigma^{\pm}$ components would have
broken the stimulated Raman condition and removed the $m_{F}=-1/2\rightarrow
m_{F}^{\prime}=-1/2$ channel, the results would not have significantly changed
due to the inevitable $m_{F}=+1/2\rightarrow m_{F}^{\prime}=-1/2$ channel [see
Fig. 1(b)], and doing so would have cost us the ability to use “atom readout”
via the $m_{F}=-1/2$ excited state. The mixing between the excited Zeeman
states is zero with a perfectly linearly polarized tweezer whose polarization
is perfectly aligned with the magnetic field. Finite mixing emerges due to
deviations from this perfect case, but they are suppressed as $\sim 1/B^{2}$
as shown in Appendix G.

We study the qubit depolarization of our $-3/2$ imaging protocol for several
magnetic fields. We place a $12\,\text{ms}$ qubit readout block performed at a
variable magnetic field between two qubit readout blocks performed at
$B=58\,\text{G}$ [see Fig. 3(a)]. We place an atom readout block at the end of
the sequence to post-select on events where the atom survives the entire
sequence. As shown in Fig. 3(b), we see good agreement with the expected $\sim
1/B^{2}$ scaling. The case of 58 G is studied in further detail in Fig. 3(c)
and (d). By plotting all camera counts in each image for all $17500$ shots
with respect to the dark/bright (D/B) threshold [see Fig. 3(d)], we can
directly measure the probability of all four events: B $\rightarrow$ B, D
$\rightarrow$ D, B $\rightarrow$ D, and D $\rightarrow$ B. These results –
with further analysis described in Appendix H – indicate array-averaged and
state-averaged conditional depolarization probabilities of
$\bar{P}_{\text{depol}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.012(3)}$
with SPAM correction;
$\bar{P}_{\text{depol}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
without. Our ability to fully control the tweezer polarization across the
array is currently limited by a slight defocus in the tweezer array optics.
This causes the tweezers to not be exactly parallel, which matters because the
tweezer polarization is along the array axis. Therefore, we see substantially
better depolarization values in the center of our array where the polarization
is better aligned to $B_{\text{DC}}$ (see Appendix G): the state-averaged
corrected and raw depolarizations for our best site are
$\bar{P}_{\text{depol}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.004(4)}$
and
$\bar{P}_{\text{depol}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.014(2)}$,
respectively. This issue can be addressed with straightforward adjustments to
our optics or by rotating our array by 90 degrees, which we leave for future
work. We have explicitly measured the array-averaged spin-flip time scale to
be
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.65(4)\,\text{s}}$,
which is in good agreement with both site-by-site and array-averaged spin-flip
probabilities during readout.

The measured results are listed in Table 1, and include both raw and corrected
values (see Appendix H) for array averages and the best site. As noted above,
there is a
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.01(1)}$
(${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.0040(1)}$)
probability of atom loss in $|0\rangle$ ($|1\rangle$) during readout, which
would manifest mostly as an inflated B $\rightarrow$ D probability (a raw
histogram is shown in Appendix H). However, this issue can be addressed via
post-selection as we have done, or in real time via a second measurement after
a $\pi$-pulse on the qubit.

FIG. 4:  Interleaved readout and qubit rotation. (a) Circuit for qubit
rotations between two qubit readout pulses. The atom readout pulse is included
at the end to post-select on events in which the atom survives the entire
sequence. We start with a state aligned along
$|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$. (b) Rabi oscillations versus time,
showing three groups that span the first second. No obvious contrast decay is
observed. $P_{\text{same}}$ refers to cases where the outcome of the two qubit
measurements are the same. This outcome is equally split between $|0\rangle$
and $|1\rangle$ due to the initial state $|+\rangle$. (c) The 2D scatter plots
of photon counts associated with the early-time data points with rotations of
$0$, $\approx\pi/2$, and $\pi$ as shown in (b). These results indicate that
the $\pi$-pulse fidelity is comparable to the probability of not undergoing a
spin flip during readout.

##  IV Interleaved readout and qubit rotation

We now add qubit rotations to demonstrate the utility of high-fidelity
repetitive qubit readout. Rotations are driven by an AC magnetic field
perpendicular to the DC field. At $B_{\text{DC}}=58$ G, the nuclear spin qubit
splitting is $f\approx 43.5\,\text{kHz}$. We apply up to $B_{\text{AC}}=0.29$
G directly to our shim coil pair in the $x$-direction [see Fig. 1(a)], for
which the Rabi frequency is $\Omega_{\text{RF}}/2\pi\approx 100$ Hz (see
Appendix 1 for further details). Similar results were recently obtained with a
designated antenna loop [72]. The data shown below uses a Rabi frequency of
$\Omega_{\text{RF}}/2\pi\approx 28$ Hz to mitigate transient effects
associated with the AC field. We note that stimulated Raman rotations via
optical transitions offer Rabi frequencies on the $\sim$MHz scale [73, 69].

Rabi and Ramsey coherence. We add qubit rotations between two qubit readout
pulses as shown in Fig. 4(a). We start with a state polarized along
$|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ using a $\pi/2$-pulse, such that
the probabilities of measuring $|0\rangle$ and $|1\rangle$ in the first image
are equal. We plot the probability that both image outcomes are the same,
$P_{\text{same}}$, as shown in Fig. 4(b). No obvious contrast decay is
observed in Rabi oscillations extending out to one second. With the pulse
sandwiched between two qubit readout blocks, we show a scatter plot of counts
in both images for all $200$ shots [see Fig. 4(c)]. We show plots for
rotations of $\theta=0$, $\approx\pi/2$, and $\pi$. In the case of the
$\pi$-pulse, we see that nearly all occurrences are B $\rightarrow$ D and D
$\rightarrow$ B, and thereby calculate the $\pi$-pulse fidelity as
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.997(4)}$
with correction and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.984(8)}$
without (see Appendix I).

We also perform a Ramsey sequence to characterize the $T_{2}^{*}$. We use two
resonant $\pi/2$ pulses separated by a dark time $\tau$ and we vary the phase
of the second pulse to obtain a Ramsey fringe. We plot the fringe contrast
versus $\tau$ to extract $T_{2}^{*}$. See Appendix J. The array-averaged
contrast data is well described by a Gaussian envelope with $1/e$ contrast
occurring at $T_{2}^{*}=0.37(1)\,\text{s}$. We observe similar values at 30
and 90 G (within $\approx 25$%), and we believe that we are limited by ambient
magnetic field noise (see Appendix J). We note that $T_{2}^{*}=0.7(3)$ s has
been realized with molecular nuclear spins at 86 G [77] and that $\approx 2$
mG stability has been realized at $\approx 1000$ G [78]. By adding an echo
pulse in the Ramsey sequence, we observe an extended coherence time of
$T_{2}^{\text{echo}}=1.40(5)\,\text{s}$ (see Appendix J). We also observe a
$1/e$ $T_{1}$ time of $\approx 200$ seconds (see Appendix J). This is somewhat
longer than values reported at low field, but is consistent with observed
trends as the field is increased [69] due to the reduced spectral weight of
magnetic field noise near the qubit frequency – a convenient feature of
operating at higher field. We note that our non-destructive readout technique
makes it straightforward to differentiate between atom loss and spin-flip
events when measuring $T_{1}$.

FIG. 5:  Repetitive readout in variable bases. (a) The $\\{+Z,-Z,+X,-X\\}$
measurement sequence, showing large bit string probabilities only for cases
where the outcome is opposite between both the first two and last two
measurements. The correlation matrix shows strong off-diagonal negative
correlations for these pairs. (b) The $\\{+Z,+X,-Z,-X\\}$ measurement
sequence, showing equal bit string probabilities for all outcomes. The
correlation matrix shows no significant off-diagonal elements. The red lines
show the ideal probability distributions, which are either $0$ or
$1/2^{2}=0.25$ in (a) and $1/2^{4}=0.0625$ in (b). The ideal off-diagonal
correlations are either $0$ or $-1$. Data in black and solid bars post-selects
on detecting an atom in the initial and final atom readout; data in gray and
dotted bars post-selects on only the initial atom readout.

Repetitive readout in variable bases. The non-commutivity of measurements in
different bases is a hallmark feature of quantum behavior and underlies the
textbook examples of cascaded Stern-Gerlach devices and optical polarizers. To
further show the unique capabilities of our repetitive qubit readout
technique, we conduct a version of such experiments by recognizing that
readout in any fixed basis can be combined with qubit rotations to perform
readout in any other basis: We rotate the desired axis of the Bloch sphere
into the measurement direction (we define this without loss of generality to
be in the $z$-basis with a bright count mapped to $+Z={|{0}\rangle}$) and then
rotate back (see Fig. 2). We compare two measurement sequences:
$\\{+Z,-Z,+X,-X\\}$ and $\\{+Z,+X,-Z,-X\\}$. Figure 5(a) and (b) show these
two cases, respectively, where the qubit has again been initialized in
$|+\rangle$. We show histograms of the outcome bit strings, where the anti-
correlation between the first two and latter two measurements is clearly
apparent in the first sequence, while the non-commutativity between each
consecutive measurement in the second sequence leads to the absence of
correlations. We also directly quantify these correlations through a
correlation matrix that shows strong, negative off-diagonal elements in the
former case, but only diagonal elements in the latter case as expected. We
show the results with and without post-selection on detecting an atom at the
end of the sequence; atom loss during the sequence biases measurement outcomes
toward bit strings ending with $1$’s.

The quantum Zeno mechanism. We can also study the interplay between qubit
rotations and projective measurements. The quantum Zeno mechanism describes
the scenario where the measurement rate is large compared to the qubit
rotation rate, such that projection back to the initial, un-rotated state
overwhelms the growth of population in others. This behavior has been observed
in myriad experimental systems [79, 80, 81, 82], and we use our ability to
interleave qubit readout and rotation to access this regime in a unique and
discrete manner.

FIG. 6: The quantum Zeno mechanism. (a) The Zeno circuit. After initializing
in $|+\rangle$ and performing a qubit measurement, we repetitively interleave
a rotation $R(\theta)$ and a readout $N-1$ times. We post-select on the final
atom readout. (b) The probability that the measurement has the same outcome as
the initial measurement, $P_{\text{same}}$, versus the measurement index for
many different rotation angles $\theta$. We see monotonic decay of
$P_{\text{same}}$ to 0.5 for $\theta\in[0,\pi/2]$ and damped alternations to
0.5 for $\theta\in[\pi/2,\pi]$. The lines deviate from the ideal case only by
the finite depolarization probability during readout which is fit to the
$\theta=0$ case as $\bar{P}_{\text{depol}}=0.0127(6)$. (c) The probability
that all outcomes are the same, $P_{\text{all-same}}$, versus total rotation
angle $\theta_{\text{tot}}=\theta\times(N-1)$ for several different numbers of
repetitions $N$. The lines are $P_{\text{all-
same}}=(1-\bar{P}_{\text{depol}})^{N}\text{cos}^{2N-2}(\theta_{\text{tot}}/(2N-2))$.

Specifically, after initializing the atom in ${|{+}\rangle}$ and performing a
first qubit readout, we apply alternating qubit rotations $R(\theta)$ and
readout for $N=10$ times for variable rotation angle $\theta$ [see Fig. 6(a)].
We plot the average probability of finding the qubit to be in the same state
as the first readout, $P_{\text{same}}$, versus the image index for the $N$
images. When $\theta=0$, we expect to always obtain $P_{\text{same}}=1$. The
observed slow decay is due to the small but finite depolarization probability
$\bar{P}_{\text{depol}}$ in each image; $P_{\text{same}}(N=10)\approx 0.9$ is
in good agreement with $(1-\bar{P}_{\text{depol}})^{N}$ for
$\bar{P}_{\text{depol}}\approx 0.01$. When $\theta=\pi$, the measurement
outcome should alternate between $|0\rangle$ and $|1\rangle$ with contrast
limited only by $\bar{P}_{\text{depol}}$ and the $\pi$-pulse fidelity,
consistent with our observations. For intermediate angles, the probabilistic
nature of outcomes in each measurement combined with the averaging over all
trajectories leads to a damping of $P_{\text{same}}$ that asymptotically
approaches $P_{\text{same}}=0.5$. Figure 6(b) shows these expected trends,
where values within $\theta\in[0,\pi/2]$ decay monotonically to
$P_{\text{same}}=0.5$ and values within $\theta\in[\pi/2,\pi]$ undergo damped
alternation.

The quantum Zeno mechanism illustrates how projective measurements can
suppress qubit dynamics when the measurement rate exceeds the coherent qubit
rotation rate. Accordingly, for a total rotation angle $\theta_{\text{tot}}$
applied to the qubit, the Zeno mechanism predicts a strong dependence of the
dynamics on the number $N-1$ of projective measurements during the rotation.
Specifically, the rotation angle $\theta_{\text{tot}}/(N-1)$ between each
measurement leaves the qubit in its initial state with probability
$P_{\text{same}}=\text{cos}^{2}(\theta_{\text{tot}}/(2N-2))$. The probability
that the qubit has remained in the state measured in the first readout during
all $N-1$ subsequent measurements is therefore $P_{\text{all-
same}}=P_{\text{same}}^{N-1}=\text{cos}^{2N-2}(\theta_{\text{tot}}/(2N-2))$.
Figure 6(c) shows this trend, limited primarily by the (uncorrected)
depolarization probability $\bar{P}_{\text{depol}}\approx 0.01$ for each
measurement. The data is in good agreement with
$(1-\bar{P}_{\text{depol}})^{N}\times P_{\text{all-same}}$, and captures the
essence of the Zeno mechanism in which $P_{\text{all-same}}\rightarrow 1$ as
$N\rightarrow\infty$ for fixed $\theta_{\text{tot}}$.

FIG. 7:  Repetitive real-time feedforward. (a) and (c) The circuit for
performing repetitive deterministic state preparation for a single qubit.
After initializing in $|+\rangle$ and performing an initial measurement, we
repeat a loop that contains a first pulse ($\pi/2$), a first measurement, a
second pulse ($0$ or $\pi$), and a second measurement. In (a), we choose the
second pulse based on the first measurement outcome in order to yield the
outcome of the second measurement to match the initial measurement outcome.
This requires feeding forward the result obtained from an exclusive OR (XOR)
gate between the classical bits in the initial measurement and the first
measurement in the loop. The result is shown in (b), where the thick solid
line shows the result averaged over $400$ shots. In (c), we choose to
alternate between obtaining the opposite and the same outcome in the second
measurement as that of the initial measurement, which requires alternating
between exclusive not OR (XNOR) and XOR classical logic on odd- and even-
numbered loops. The result is shown in (d), where now we see full-contrast
alternations. The arrows in (b) and (d) illustrate the role of the
feedforward, with gray vertical lines dividing loop iterations. Loops using
XOR logic have light gray backgrounds and loops using XNOR logic have dark
gray backgrounds. (e) and (f) show a single, representative trajectory for
each circuit and the arrows indicate when a $\pi$-pulse is applied.

##  V Real-time feedforward for active qubit reset

We now add real-time control [57] to our toolbox to deterministically prepare
the qubit in either $|0\rangle$ or $|1\rangle$ after measurement, often called
“active qubit reset” [83, 84, 85, 86]. We perform this study with only a
single atom; a possible extension to arrays is discussed in Section VI. We
initialize the qubit in $|+\rangle$ and then perform a measurement that
projects it to $|0\rangle$ or $|1\rangle$. We then perform six loops, where
each is composed of a $\pi/2$ rotation, a measurement, a conditional rotation
($R(0)$ or $R(\pi)$), and a second measurement. The goal of the conditional
rotation is to rotate the qubit to the same or opposite state as measured in
the initial readout. Hence, the rotation is conditional on both the initial
readout and the first readout in the loop. Details on the real-time
implementation of this circuit are described in Appendix K.

To keep the qubit in the same state as the outcome of the initial measurement
[see Fig. 7(a) and (b)], a classical exclusive OR (XOR) gate is used to
perform a $R(\pi)$ rotation if and only if the two classical bits are
different. Otherwise, no rotation is performed: $R(0)$. We observe alternation
between $P_{\text{same}}\approx 1$ and $P_{\text{same}}\approx 0.5$, where
“same” refers to the initial readout. The first jump is identical to the
$R(\pi/2)$ case of the Zeno study above [see Fig. 6(b)]; however, instead of
staying at $P_{\text{same}}=0.5$ on average after each subsequent rotation,
our feedforward technique deterministically puts the qubit back into the
initial state, such that $P_{\text{same}}$ goes back to unity. The jump then
repeats in each loop.

Alternatively, we can switch between obtaining the initial readout outcome
($P_{\text{same}}=1$) and the opposite outcome ($P_{\text{same}}=0$) on each
iteration of the loop. This can be accomplished by using an exclusive NOR
(XNOR) in odd-numbered loops and an XOR in even-numbered loops [see Fig.
7(c)]. In this case, we observe full-amplitude zig-zags of $P_{\text{same}}$
as shown in Fig. 7(d). Individual trajectories are shown in Fig. 7(e) and (f),
indicating when a $\pi$-pulse is required and showing “quantum jump”-like
behavior between sequential measurements.

Finally, since the qubit is deterministically reset, depolarization during
readout does not accrue upon subsequent measurements. Thus, a high contrast is
maintained for an arbitrary number of loops; we choose six loops (13 total
measurements) as a compromise against atom loss. Note that this data is post-
selected on the final atom readout.

##  VI Concluding discussion

In summary, we have leveraged the unique level structure of 171Yb to perform
repetitive qubit readout, with qubits encoded in the nuclear spin-1/2 ground
state. With a bright/dark discrimination fidelity of $\mathcal{F}\gtrsim
0.99$, atom survival of $\approx 0.99$, and spin-flip probability
$\bar{P}_{\text{depol}}$ of $\approx 0.01$ during a $12\,\text{ms}$ probe
pulse, we show that readout can be repeated 10 times while still maintaining
control over the qubit state at the $\approx 0.9$ level. These numbers would
improve as $\sim 1/B^{2}$. By adding high fidelity qubit rotations with an AC
magnetic field, we study quantum circuits that feature both measurements and
rotations. These include measurements in variable bases as a demonstration of
measurement non-commutivity akin to cascaded Stern-Gerlach devices and optical
polarizers, as well as a manifestation of the quantum Zeno mechanism in which
dynamics is frozen by measurement. Additionally, we use real-time feedforward
to perform repetitive active qubit reset to $|0\rangle$ or $|1\rangle$, and we
show that this deterministic operation circumvents the accrual of
depolarization errors during measurement thereby maintaining excellent
contrast after 13 measurements, limited only by atom loss. The atom loss can
be mitigated by using shallower tweezers and/or shorter readout pulses, or by
working at a tweezer wavelength with larger detuning from the
3PJ$\leftrightarrow$ 3S1 transitions for which we identify $\approx 778$ nm as
a good candidate where 3P1 $|m_{F}|=3/2$ is magic with the ground state (see
Appendix F).

Our work demonstrates global qubit rotations and global QND readout; however,
local operations are often required. Local qubit rotations can be accomplished
by using stimulated Raman pulses [73, 69, 87] instead of an AC magnetic field.
Qubit rotations can then be performed at $\sim$MHz rates rather than $\sim
100$ Hz, and single-site control can be realized with adaptive optical
elements [73, 88]. Local measurements, often referred to “mid-circuit
measurements”, are performed on a subset of “ancilla” qubits while the others
– the “data” qubits – remain unaffected. Local mid-circuit measurement without
crosstalk with the data qubits can be performed using two atomic species [89,
14, 90, 20, 23, 57], separate readout zones [91, 92, 93, 94, 36], and via
“shelving” with other atomic states [84, 95, 58]. Our technique offers QND
readout for all three approaches, and is compatible with shelving techniques
via the optical “clock” transition [96, 97, 98, 87]. Indeed, this is our
primary motivation for operating at the clock-magic wavelength. In this
approach, qubits will be encoded in the metastable 3P0 nuclear spin, and
optical pulses with a phase-stabilized laser [99, 100] will transfer the
ancilla qubit(s) to the ground state for measurement. We also note that our
technique is compatible with the use of local light shifts to perform mid-
circuit measurements.

Finally, we note that having a qubit with excellent optical bright/dark
discrimination is a key prerequisite for time bin remote entanglement
generation [101, 102]. In this scheme, the qubit state becomes entangled with
the state of a single photon in the temporal basis (early or late bin), and
photons from the pair of atoms are coincident on a photonic Bell state
analyzer that haralds the generation of a remote Bell pair between the atoms.
Our work demonstrates that 171Yb is an excellent candidate for high-fidelity
atom-photon entanglement. This could either be performed on the
1S${}_{0}\leftrightarrow\leavevmode\nobreak\ ^{3}$P1 transition at 556 nm used
in this work for short distance distributed or modular computing [103, 104,
105], or via the identical configuration on the
${}^{3}\text{P}_{0}\leftrightarrow{}^{3}\text{D}_{1}$ transition at 1389 nm in
the telecommunication wavelength band that is well suited for long distance
networking [106, 107].

## Acknowledgments

We thank Saeed Pegahan, Mingkun Zhao, Ian Vetter, and Brett Merriman for
contributions to the experimental system. We acknowledge Hannes Bernien, Adam
Kaufman, Kaden Hazzard, and Danial Stack for stimulating discussions, and
Bryce Gadway for critical reading of the manuscript. We acknowledge funding
from the NSF QLCI for Hybrid Quantum Architectures and Networks (NSF award
2016136); the NSF PHY Division (NSF award 2112663); the NSF Quantum
Interconnects Challenge for Transformational Advances in Quantum Systems (NSF
award 2137642); the ONR Young Investigator Program (ONR award
N00014-22-1-2311); the AFOSR Young Investigator Program (AFOSR award
FA9550-23-1-0059); and the U.S. Department of Energy, Office of Science,
National Quantum Information Science Research Centers.

##  APPENDIX A Overview of the experimental apparatus

###  1 Chamber, MOT, and imaging

The experimental apparatus is inspired by Ref. [108] and comprises two main
sections, wherein hot ytterbium atom flux obtained from a single AlfaVakuo
dispenser is first cooled in two dimensions via a 2D magneto-optic trap (MOT)
and then transferred $\approx 40\,\text{cm}$ via a nearly resonant beam
through a differential pumping tube to load a full 3D MOT with $\sim 10^{6}$
atoms over $500\,\text{ms}$. Both MOTs and the push beam are tuned to the
${}^{1}\text{S}_{0}\leftrightarrow{}^{1}\text{P}_{1}$ transition
($\lambda=399\,\text{nm}$, $\Gamma=2\pi\times 29\,\text{MHz}$) and are formed
at the centers of glass cells. Once loaded in the 3D MOT, the atoms are
approximately $1\,\text{mK}$.

Next, another 3D MOT tuned to the
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}$
($\lambda=556\,\text{nm}$, $\Gamma=2\pi\times 182\,\text{kHz}$) transition is
turned on while the $399\,\text{nm}$ MOT beams are turned off and the magnetic
field gradient switched from $\approx 60\,\text{G/cm}$ to $\approx
10\,\text{G/cm}$ accordingly. Initially, this transition is power-broadened
significantly by beam intensities set to $\sim 10^{4}\,I_{\text{sat}}$ to
ensure sufficient atom transfer (roughly $50\%$) between the two MOTs. The
atoms are then cooled further to approximately $5\,\text{$\mu$K}$ by ramping
the beam intensity down to $0.6\,I_{\text{sat}}$ and detuning (relative to
free space) from $\approx-20\Gamma$ to $\approx-1.2\Gamma$ over
$30\,\text{ms}$. The MOT field gradient is then increased to $\approx
14\,\text{G/cm}$ over $10\,\text{ms}$ to compress the atoms into a volume
roughly $150\,\text{$\mu$m}$ in diameter. We estimate that the compressed MOT
holds approximately $5\times 10^{5}$ atoms at this stage.

The atoms are then loaded stochastically into optical tweezers at spacing
$7.8\,\text{$\mu$m}$, $1/e^{2}$ waist $670\,\text{nm}$ (radius), depth
$580\,\text{$\mu$K}$, and wavelength $\approx 760\,\text{nm}$. The tweezers
are generated by a single acousto-optic deflector leading into a
$\text{NA}\gtrsim 0.6$ objective (Special Optics). The tweezer light is
sourced from a M Squared SolsTiS titanium-sapphire laser pumped by a M Squared
Equinox. We choose the tweezer wavelength to give magic trapping for the
ground and excited state manifolds of the
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{0}$ optical clock transition
and near-magic trapping for the $|m_{F}|=1/2$ states in the
${}^{3}\text{P}_{1}\,F=3/2$ manifold. For an array of 5 tweezers, we require
roughly $35\,\text{mW}$ of optical power in the plane of the atoms. We
estimate a loading fraction of approximately $0.7$ (see Appendix H). The atoms
are then cooled using the same $556\,\text{nm}$ beams used for the 3D MOT, now
with intensity $1.3\,I_{\text{sat}}$ and frequency red-detuned from the free-
space
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}{|{F=3/2,\,m_{F}=-1/2}\rangle}$
transition by approximately $0.8\,\Gamma$, which causes atoms to escape from
the tweezers in pairs through light-assisted collisions, leaving only 0 or 1
atom in the trap afterward. This process takes $\approx 120\,\text{ms}$,
although we expect this could be improved significantly with further
optimization. We measure the temperature of the atoms in the tweezers using a
release-and-recapture method to be $\approx 5\,\text{$\mu$K}$, which is close
to the Doppler limit for the transition.

The atoms are imaged using two retro-reflected beams tuned to the
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}$ transition (see Appendix
B) with projections onto all three trap axes. The two beams are collimated
with a $1/e^{2}$ radius of approximately $880\,\text{$\mu$m}$ and have a
$\approx 70^{\circ}$ angle between them. We estimate that each imaging beam
has intensity $0.5\,I_{\text{sat}}$ relative to the probe transition (other
polarization components are not counted). The two probe beams have
polarization overlap, and the polarizations of the retro-reflection beams are
not rotated. Therefore, interference fringes are likely to be present; we do
not wash them out with e.g. dithering mirrors. Imaging performed using either
the $m_{F}=-3/2$ or $m_{F}=-1/2$ transition is done so with laser frequency
red-detuned by approximately $1\Gamma$. Atomic fluorescence is collected
through a second objective identical to the one used to generate the tweezers
but placed on the opposite side of the glass cell [see Fig. 1(a)] and focused
onto an electron-multiplying CCD (EMCCD, Andor iXon Ultra 888) with EM gain
set to 200.

###  2 Tweezer array homogenization

Homogenization of the optical tweezer array is key to the maintenance of the
imaging condition used for readout. Although the atoms in the tweezers are
inherently identical, it is critical – particularly given the nonzero
differential polarizabilities identified in Appendix F – that the trapping
potentials be as uniform as possible to prevent undesired light shifts on the
${}^{3}\text{P}_{1}$ imaging states.

To this purpose, we adopt an iterative procedure based on spectroscopy of the
${}^{3}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}\,F=3/2,\,m_{F}=-3/2$
transition. Since this transition is non-magic in the presence of the chosen
trapping wavelength, the measurement of the transition’s resonance frequency
is linearly related to the trap depth. Thus, spectroscopy is repeatedly
performed for each site in the array, and the amplitudes of the five
generating RF tones sent to the acousto-optic deflector (AOD, AA Opto
Electronic DTSX-400-760) from an arbitrary waveform generator (AWG, Spectrum
Instruments M4i6622) are adjusted to bring the transition resonances measured
across different sites to the same center frequency. Then, the total RF power
sent to the AOD is adjusted to bring each site to the desired trap depth. For
an array of five tweezers, this process generally converges to the $\sim
0.1\%$ level in around 10 iterations. Post-imaging atom survival is
homogenized by this procedure to within $3\%$. Uniformity in the shapes and
depths of the tweezers is monitored by a CCD placed after a dichroic mirror
used to separate atomic fluorescence from tweezer light after they have both
passed through the imaging objective shown in Fig. 1(a). We have observed
similar homogeneity with 10 tweezers.

###  3 Brief overview of experiment control

The many individual components of the apparatus are controlled by means of a
combination of National Instruments PCIe-7820 and PCIe-6738, which
respectively expose 128 digital input/output and 32 analog output-only
configurable voltage channels, housed in a single computer. Communication with
these devices is accomplished by means of low-level field-programmable gate
array (FPGA) programming software provided by Entangleware, Inc. Experimental
sequences are ultimately programmed through a high-level, Python-based
interface. The atomic signal returned from the apparatus via collection on the
EMCCD sensor is sent to a separate computer, which handles both real-time
feedforward (see Appendix K) and the AWG used to control the tweezer array.

##  APPENDIX B Comparison of $m_{F}=-3/2$ and $m_{F}=-1/2$ readout

In this section, we compare aspects of the two methods of performing readout
(i.e. using the $m_{F}=-3/2$ and $m_{F}=-1/2$ ${}^{3}\text{P}_{1}$ $F=3/2$
excited state). As stated in the main text, the principal reason to prefer one
to the other is that the $m_{F}=-3/2$ method is state-selective due to dipole
selection rules and, hence, can be used to convert bright/dark classifications
into qubit state measurements.

The exact imaging conditions used in the two cases shown here differ by only
the strength of the magnetic field at which they are performed. We compare the
methods by exposing the atoms to light from the probe beams at total intensity
$\approx 3\,I_{\text{sat}}$, of which we estimate $\approx 1\,I_{\text{sat}}$
effectively drives the desired transition, but $m_{F}=-3/2$ imaging is
performed at $58\,\text{G}$ with detuning $-1\Gamma$ for $12\,\text{ms}$ while
$m_{F}=-1/2$ imaging is performed at $18\,\text{G}$ with detuning $-0.5\Gamma$
for $20\,\text{ms}$. The lower field strength is used to minimize effects from
the Zeeman splitting between the nuclear spin ground states, and the longer
probe time is because the data was taken before final optimization of the
collection efficiency. Fig. 8(a) shows that the mean numbers of photons
collected from filled tweezer sites (see Appendix H) differs only slightly
with comparable discrimination fidelities. Fig. 8(b) gives temperature
estimates, obtained via a standard release-recapture experiment, for both
cases as well, showing $\approx 5\,\text{$\mu$K}$ for $m_{F}=-3/2$ and
$\approx 4\,\text{$\mu$K}$ for $m_{F}=-1/2$. We do not explicitly measure the
axial temperature and we expect that it could be improved with an axial
cooling beam sent through the objectives. We leave this investigation for
future studies.

FIG. 8:  Comparison of the $m_{F}=-3/2$ and $m_{F}=-1/2$ imaging conditions.
(a) Comparison of photon scatter during imaging. The upper histogram shows
that for imaging using the $m_{F}=-3/2$ excited state with mean number of
photons collected $\mu_{1}=37.1(2)$ at $58\,\text{G}$ magnetic field when an
atom is present (see Appendix H) while the lower shows that for imaging using
$m_{F}=-1/2$ with $\mu_{1}=28.1(2)$ at $18\,\text{G}$. The red vertical lines
show the photon collection threshold for bright/dark discrimination in each
case (see Appendix H). (b) Measurement of atom temperature in the tweezer
under the $m_{F}=-3/2$ (dark green circle) and $m_{F}=-1/2$ (light green
diamond) via release-recapture experiment. Tweezer sites are imaged and the
tweezers are diabatically switched off for variable time before being turned
on and imaged again. Recapture probability is calculated as that of the second
image being bright, post-conditioned on the first being bright as well.
Optical pumping is performed before both images in the $m_{F}=-3/2$ case.
Probabilities have been re-scaled so that the mean of the four shortest-time
probabilities is equal to 1. The lines show predicted probabilities obtained
via Monte Carlo simulation for temperatures $4\,\text{$\mu$K}$ (black) up to
$10\,\text{$\mu$K}$ (red), indicating that the temperature after imaging using
$m_{F}=-3/2$ ($m_{F}=-1/2$) is approximately $5\,\text{$\mu$K}$
($4\,\text{$\mu$K}$).

We also note differences in photon collection efficiency between the two
transitions. In the electric dipole approximation, the direction in which a
given photon will be radiated depends on the associated change in angular
momentum $\Delta m_{F}$ undergone by the atom, and is governed by the
following angular probability distributions:

$f_{|\Delta
m_{F}|}(\theta,\varphi)=\begin{cases}\frac{3}{16\pi}\Big{(}1+\cos^{2}\theta\Big{)}&|\Delta
m_{F}|=1\\\ \frac{3}{8\pi}\sin^{2}\theta&|\Delta m_{F}|=0\end{cases}$ (1)

with polar angle $\theta$ and azimuthal angle $\varphi$.

We then model the emission patterns for the two imaging cases detailed here.
For the $m_{F}=-3/2$ case, we expect the pattern to follow purely that of
$f_{1}$ ($|\Delta m_{F}|=1$). For the $m_{F}=-1/2$ case, we expect it to
follow a mix of both $f_{0}$ ($|\Delta m_{F}|=0$) and $f_{1}$ in a $2:1$ ratio
equal to that of the squared Clebsch-Gordan coefficients for the possible
decay paths from the ${}^{3}\text{P}_{1}$ $F=3/2$, $m_{F}=-1/2$ excited state,
assuming an approximately even mixture of polarization modes in the incident
light. Thus the photon emission for both imaging cases is modeled by the
distributions

$g_{m_{F}}(\theta,\varphi)=\begin{cases}f_{1}(\theta,\varphi)&m_{F}=-3/2\\\
\frac{2}{3}f_{0}(\theta,\varphi)+\frac{1}{3}f_{1}(\theta,\varphi)&m_{F}=-1/2\end{cases}.$
(2)

From these, we then compare photon collection efficiencies between the two
cases via Monte-Carlo integration. Our objectives’ collection areas are
modeled as circular, with bounding curves
$\theta_{\pm}(\varphi)=\arccos[\mp(\text{NA}^{2}-\sin^{2}\varphi)^{1/2}]$,
over which $g_{m_{F}}$ is integrated for both $m_{F}=-3/2$ and $m_{F}=-1/2$.
The ratio between the two cases is found as approximately $1.4$, in favor of
$m_{F}=-1/2$, which is in disagreement with our findings shown in Fig 8. We
attribute to a handful of factors. First, we note that the intensity
distribution in the probe beams across the $\pi,\,\sigma^{\pm}$ polarization
modes may not be uniform. The exact polarization of the beams is difficult to
measure given their angles of entry into the cell, and would manifest as
additional weighting factors on the $f_{\Delta m_{F}}$ components of the
overall photon emission distribution $g_{m_{F}}$. Second, interference between
the two probe beams may also play a role, as noted in Appendix A. Finally, the
non-magic trapping of the $m_{F}=-3/2$ excited state implies that the detuning
of the probe beam varies spatially over the trap, which causes broadening in
the photon distribution. A temperature of $T\approx 5\,\text{$\mu$K}$ gives
rise to a frequency spread of $\Delta f\approx 25\,\text{kHz}$ with a
differential polarizability of $0.25(3)$, which is non-negligible compared to
the probe transition linewidth. However, the excess broadening may suggest
that our atoms are somewhat hotter in the axial direction, to which release-
recapture measurements are not sensitive. We note that it is straightforward
to add an axial cooling beam through the objective, and we leave this study
for future work.

##  APPENDIX C Magnetic field system

###  1 AC magnetic field system

To manipulate the nuclear spin states, we can either use the Raman transition
via the electric dipole coupling [73, 69] or directly using the magnetic
dipole coupling between nuclear states [72]. Here we introduce the second
method implemented in our experiment.

For the 171Yb ground state ($6s^{2}\leavevmode\nobreak\
{}^{1}\textrm{S}_{0}$), the nuclear spin $I=1/2$ gives the hyperfine structure
$F=1/2$ with two nuclear spin states
${|{0}\rangle}\equiv{|{m_{F}=-1/2}\rangle}$ and
${|{1}\rangle}\equiv{|{m_{F}=1/2}\rangle}$. We apply a magnetic field along
the $y$-axis [see Fig. 1(a)], leading to Larmor precession in the atomic spin.
The Hamiltonian of the system is given by

$\hat{H_{0}}=g\mu_{B}\hat{F}\cdot B_{0},$ (3)

where $g\mu_{B}/h$ is around $-750\,\text{Hz/G}$ for the splitting between
${|{0}\rangle}$ and ${|{1}\rangle}$. Thus the Hamiltonian can be simplified
as:

$\hat{H_{0}}=\hbar\begin{pmatrix}\omega_{0}&0\\\ 0&0\end{pmatrix},$ (4)

where $\hbar\omega_{0}$ is the energy splitting between ${|{0}\rangle}$ and
${|{1}\rangle}$. Now considering we add an AC magnetic field along the
$z$-axis, with the quantization axis defined by the magnetic field along the
$y$-axis shown in Fig. 1(a), the Hamiltonian is given by:

$\begin{split}\hat{H}&=\hat{H_{0}}+\frac{1}{2}\hbar\Omega\cos(\omega
t)\hat{\sigma}_{x}\\\
&=\hbar\begin{pmatrix}\omega_{0}&\frac{1}{2}\Omega\cos(\omega t)\\\
\frac{1}{2}\Omega\cos(\omega t)&0\end{pmatrix},\end{split}$ (5)

where $\Omega=\frac{1}{2}g\mu_{B}B_{\text{AC}}/\hbar$ is the Rabi frequency
(defined as the frequency of oscillation in state probabilities, rather than
amplitudes), $B_{\text{AC}}$ is the strength of the driving magnetic field and
$\omega$ is the frequency of the driving signal. Since $\omega_{0}\gg\Omega$,
the factor $1/2$ in $\Omega$ comes from the strength of the counter-rotating
term, which does not significantly contribute to the Rabi oscillation and,
hence, has been neglected.

FIG. 9:  AC magnetic field system (a) The schematic of the RF driver used to
drive one pair of the horizontal shim coils. The driver is capable of giving a
maximum output voltage around $300\,\text{V\textsubscript{pp}}$ and the peak
current about $0.2\,\text{A}$ ranges from $30\,\text{kHz}$ to $2\,\text{MHz}$.
(b) The connection of the RF driver to the shim coils. The isolated output of
the RF driver (green box) is connected in series with the shim coil driver
(teal box). A $1\,\text{$\mu$F}$ polypropylene film capacitor is used to
bypass the RF signal through the shim coil driver.

To achieve a Rabi frequency $\Omega/2\pi\approx 110\,\text{Hz}$, we require
$B_{\text{AC}}\approx 0.29\,\text{G}$. The easiest way of realizing such
magnetic field modulation is by using a pair of “shim” coils [see Fig. 1(a)].
In our system this corresponds to a current modulation amplitude
$I_{\text{mod}}\approx 0.15\,\text{A}$. At $58\,\text{G}$, chosen to match the
magnetic field applied during readout, the modulation must be applied at a RF
frequency equal to the Larmor frequency, $f\approx 43.5\,\text{kHz}$.
Considering an inductance $L_{\text{coil}}\approx 1.5\,\text{mH}$ of the shim
coil, the voltage modulation amplitude is calculated as
$V_{\text{mod}}=I_{\text{mod}}\times 2\pi
fL_{\text{coil}}=64\,\text{V}=128\,\text{V\textsubscript{pp}}$.

Since there is no commercial product that can give both a high voltage and a
high current drive at $\approx 45$ kHz, we built our own driver. The schematic
of this RF driver is shown in Fig. 9(a). The input RF signal is isolated and
divided into two parts with opposite polarity by the signal transformer T1.
Two high current and high speed operational amplifiers (Linear Technology,
LT1210) are driven by the two signals and differentially drive the output
transformer T3. T3 is the main transformer recycled from an ATX (Advanced
Technology eXtended) power supply of retired computers. This transformer has a
turn ratio of about $5:1$ which increases the output voltage by a factor of 5.
The output transformer also isolates the RF driving stage from the shim coils,
which introduce minimum interference with the DC magnetic field control
system. Since the maximum output current of LT1210 is $1.1\,\text{A}$, the
maximum current available after the transformer is $0.2\,\text{A}$.

It is also necessary to use these shim coils with constant DC component in
order to cancel background magnetic fields – for which we also use two other
pairs of coils not shown in Fig. 1(a) – requiring the RF driver to be
connected in series with a DC driver used for all three pairs. This is
possible since the secondary winding of the transformer T3 has very small DC
resistance. However, the magnetic core of T3 will be saturated if the DC
current is larger than $0.3\,\text{A}$, which means the RF driver is only
functional when the shim coils have relatively small DC current. This is not a
problem for our experiment since the background magnetic field is small. As
shown in Fig. 9(b), a $1\,\text{$\mu$F}$ polypropylene film capacitor is
connected in parallel with the shim coil driver, which is used to bypass the
RF signal that passes through the shim coil driver. This capacitor also helps
decrease the interference of the shim coil driver from the RF driver.

###  2 DC magnetic field system

To generate the magnetic field for non-destructive imaging at $58\,\text{G}$,
we repurpose the MOT coils by switching the coils’ electrical connection from
anti-Helmholtz to Helmholtz configuration. To achieve this, we use an H-bridge
to control the current flow direction of the upper MOT coil. The H-bridge is
constructed using eight high-current metal-oxide-semiconductor field-effect
transistors (MOSFETs) connected in parallel. The specific model used is
IXFN300N20X3 from IXYS Corporation. The voltage-controlled resistors are used
to protect each individual device from the back-electromotive force (EMF)
generated by the coils.

To isolate the noise and the computer’s control signal ground from the coil’s
ground connection, two isolated gate drivers (Texas Instruments, UCC21320) are
used to control both sides of the half bridges. The drivers also protect each
side of the half bridge from the dead-zone, which can cause a short in the
coil connection.

The magnetic field stabilization is achieved by stabilizing the current
flowing through the coil. To measure the current with high stability, a high-
stability Hall sensor (Danisense, DS300ID) is used, which has a long-term
stability of better than $0.2\,\text{ppm}$ per month. The secondary current
output of the Hall sensor is converted to voltage through a Kelvin connection,
using a high stability ($0.05\,\text{ppm/\textsuperscript{$\circ$}C}$) metal
film resistor (Vishay, Y16065R00000F9W). For the error signal amplifier, a
low-noise operational amplifier (Analog Devices, AD8675) is DC stabilized by a
zero-drift operational amplifier (Linear Technology, LTC2057), and high gain
stability ($0.2\,\text{ppm/\textsuperscript{$\circ$}C}$) is achieved by a
matched resistor network (Linear Technology, LT5400). The reference signal for
the magnetic field servo is provided by a 20-bit high-stability
($0.05\,\text{ppm/\textsuperscript{$\circ$}C}$) digital-to-analog converter
(Analog Devices, AD5791), with the reference voltage provided by a
temperature-controlled buried Zener diode (Linear Technology, LTZ1000). The
servo output is sent to two insulated gate bipolar transistors (IGBTs)
connected in parallel, with each IGBT in series with the coils. The specific
model used is IXGN200N170, manufactured by IXYS Corporation. Detailed
schematics of the servo and voltage reference can be found in Ref. [109].

The magnetic field servo board and voltage reference board are connected
closely inside a metal box that is grounded. A low noise isolated DC-DC power
supply [110, 111] is used to power the servo, voltage reference and the Hall
sensor, each with a separated ground. The magnetic field control is achieved
by the isolated digital channels that directly control the digital to analog
converter using the serial peripheral interface (SPI) protocol. This approach
avoids any possible magnetic field change introduced by the noise of external
signal ground.

##  APPENDIX D Simulating multi-level dynamics

During the qubit readout process, the relevant energy levels are those in the
1S0 and 3P1 $F=3/2$ manifolds. To estimate the readout fidelity and to
optimize experimental settings, we study the dynamics of atom population
transfer between the 1S0 nuclear spin ground states via coherent and
incoherent processes mediated by the 3P1 $F=3/2$ states.

Since the 3P1 state lifetime is relatively short compared with the imaging
time ($\Gamma^{-1}\sim 10^{-6}$ s versus $\sim 10^{-2}$ s), we ignore the
populations in 3P1 states and only consider the populations in the two nuclear
spin qubit levels in the 1S0 manifold, $|m_{F}=-1/2\rangle\equiv|0\rangle$ and
$|m_{F}=1/2\rangle\equiv|1\rangle$. As mentioned in Sec. III, the qubit
readout ($-3/2$ imaging) is achieved by applying a probe beam that is close to
the ${|{0}\rangle}\leftrightarrow{|{{}^{3}\mathrm{P}_{1},m_{F}=-3/2}\rangle}$
transition, red-detuned by $1\,\Gamma$. Due to the random polarization of the
probe beams ($I_{\sigma^{-}}\approx I_{\sigma^{+}}\approx I_{\pi}=I/3$), off-
resonant scattering processes via other 3P1 states can induce qubit
depolarization between ${|{0}\rangle}$ and ${|{1}\rangle}$. Moreover, the
probe beam can coherently drive stimulated Raman transitions between
${|{0}\rangle}$ and ${|{1}\rangle}$ mediated by the
${|{{}^{3}\mathrm{P}_{1},m_{F}=\pm 1/2}\rangle}$ states.

FIG. 10:  Energy levels and probe beam configuration. The energy splitting
between different Zeeman states in 1S0 and 3P1 manifolds are determined by
external magmetic field (Zeeman splitting) and tweezer depth (vector and
tensor light shifts). For magnetic fields exceeding 5 G, the splitting between
3P1 Zeeman states look nearly identical.

Figure 10 shows the energy levels of 1S0 and 3P1 states and the probe beam
with the corresponding branching ratio of each transition. Since the probe
beam is monochromatic, the Hamiltonian of the stimulated Raman transition
driven by the probe beam reads

$\displaystyle\hat{H}=\frac{\hbar\Omega}{2}\hat{\sigma}_{x}+\frac{\hbar\delta_{01}}{2}\hat{\sigma}_{z}.$
(6)

Here $\Omega=\Omega_{0}\Omega_{1}/2\Delta$ is the effective Raman Rabi
frequency, with Rabi frequencies $\Omega_{0}$ and $\Omega_{1}$ of the dipole
transition between a certain 3P1 Zeeman state and the two ground states. Each
of the two possible intermediate states, ${|{{}^{3}\mathrm{P}_{1},m_{F}=\pm
1/2}\rangle}$, contributes to a Raman Rabi frequency of

$\displaystyle\Omega_{\pm
1/2}=\sqrt{\frac{1}{3}\frac{2}{3}}\frac{\Gamma^{2}}{2\Delta_{\pm
1/2}}\frac{I/3}{2I_{\text{sat}}}$ (7)

where $I_{\text{sat}}$ is the saturation intensity for the
${|{0}\rangle}\leftrightarrow{|{{}^{3}P_{1},m_{F}=-3/2}\rangle}$ transition.
The total Raman Rabi frequency is the sum of $\Omega_{-1/2}$ and
$\Omega_{+1/2}$.

When the atoms are excited to the 3P1 states, subsequent scattering leads to
an incoherent re-distribution of the ground state populations, which can be
described by the following collapse operators

$\displaystyle\Gamma_{00}$ $\displaystyle=R_{00}|{0}\rangle\langle{0}|,$
$\displaystyle\Gamma_{01}$ $\displaystyle=R_{01}|{0}\rangle\langle{1}|,$
$\displaystyle\Gamma_{10}$ $\displaystyle=R_{10}|{1}\rangle\langle{0}|,$
$\displaystyle\Gamma_{11}$ $\displaystyle=R_{11}|{1}\rangle\langle{1}|,$ (8)

where $R_{ij}$ is the scattering rate from qubit state ${|{i}\rangle}$ to
${|{j}\rangle}$. Each scattering rate contains contributions from several
paths via different Zeeman levels in the 3P1 manifold that are allowed by the
selection rule:

$\displaystyle R_{00}$ $\displaystyle=R_{0\rightarrow
0}^{-3/2}+R_{0\rightarrow 0}^{-1/2}+R_{0\rightarrow 0}^{+1/2}$ $\displaystyle
R_{01}$ $\displaystyle=R_{0\rightarrow 1}^{-1/2}+R_{0\rightarrow 1}^{+1/2}$
$\displaystyle R_{10}$ $\displaystyle=R_{1\rightarrow
0}^{-1/2}+R_{1\rightarrow 0}^{+1/2}$ $\displaystyle R_{11}$
$\displaystyle=R_{1\rightarrow 1}^{-1/2}+R_{1\rightarrow
1}^{+1/2}+R_{1\rightarrow 1}^{+3/2}.$ (9)

where $R_{i\rightarrow j}^{m}$ is the rate of the atom starting in
${|{i}\rangle}$ being excited to ${|{{}^{3}\mathrm{P}_{1},m_{F}=m}\rangle}$
and decaying to ${|{j}\rangle}$. After taking the Clebsch-Gordan coefficients
into consideration, the scattering rate via every possible channel in Eqs. 9
can be calculated using rate equations. Equations 10 describe the transitions
with the same initial and final state, representing a dephasing process,

$\displaystyle R_{0\rightarrow 0}^{-3/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{I_{\sigma^{-}}/I_{\text{sat}}}{1+4(\Delta_{-3/2}/\Gamma)^{2}+I_{\sigma^{-}}/I_{\text{sat}}}$
$\displaystyle R_{0\rightarrow 0}^{-1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{2}{3}\frac{2/3\cdot
I_{\pi}/I_{\text{sat}}}{1+4(\Delta_{-1/2}/\Gamma)^{2}+2/3\cdot
I_{\pi}/I_{\text{sat}}}$ $\displaystyle R_{0\rightarrow 0}^{+1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{1}{3}\frac{1/3\cdot
I_{\sigma^{+}}/I_{\text{sat}}}{1+4(\Delta_{+1/2}/\Gamma)^{2}+1/3\cdot
I_{\sigma^{+}}/I_{\text{sat}}}$ $\displaystyle R_{1\rightarrow 1}^{-1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{1}{3}\frac{1/3\cdot
I_{\sigma^{-}}/I_{\text{sat}}}{1+4[(\Delta_{-1/2}+\delta_{01})/\Gamma]^{2}+1/3\cdot
I_{\sigma^{-}}/I_{\text{sat}}}$ $\displaystyle R_{1\rightarrow 1}^{+1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{2}{3}\frac{2/3\cdot
I_{\pi}/I_{\text{sat}}}{1+4[(\Delta_{+1/2}+\delta_{01})/\Gamma]^{2}+2/3\cdot
I_{\pi}/I_{\text{sat}}}$ $\displaystyle R_{1\rightarrow 1}^{+3/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{I_{\sigma^{+}}/I_{\text{sat}}}{1+4[(\Delta_{+3/2}+\delta_{01})/\Gamma]^{2}+I_{\sigma^{+}}/I_{\text{sat}}},$
(10)

and Eqs. 11 describe the transitions between different initial and final
states, which represent an amplitude damping process,

$\displaystyle R_{0\rightarrow 1}^{-1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{1}{3}\frac{2/3\cdot
I_{\pi}/I_{\text{sat}}}{1+4(\Delta_{-1/2}/\Gamma)^{2}+2/3\cdot
I_{\pi}/I_{\text{sat}}}$ $\displaystyle R_{0\rightarrow 1}^{+1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{2}{3}\frac{1/3\cdot
I_{\sigma^{+}}/I_{\text{sat}}}{1+4(\Delta_{+1/2}/\Gamma)^{2}+1/3\cdot
I_{\sigma^{+}}/I_{\text{sat}}}$ $\displaystyle R_{1\rightarrow 0}^{-1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{1}{3}\frac{2/3\cdot
I_{\sigma^{-}}/I_{\text{sat}}}{1+4[(\Delta_{-1/2}+\delta_{01})/\Gamma]^{2}+2/3\cdot
I_{\sigma^{-}}/I_{\text{sat}}}$ $\displaystyle R_{1\rightarrow 0}^{+1/2}$
$\displaystyle=\frac{\Gamma}{2}\frac{2}{3}\frac{1/3\cdot
I_{\pi}/I_{\text{sat}}}{1+4[(\Delta_{+1/2}+\delta_{01})/\Gamma]^{2}+1/3\cdot
I_{\pi}/I_{\text{sat}}}.$ (11)

With the above Hamiltonian and collapse operators, we can therefore use master
equations to extract the simulation results in Fig. 3(b) for depolarization
rates under different imaging conditions, including magnetic field, imaging
time, probe beam intensity and detuning.

At high magnetic field, Fig. 3(b) indicates that the contributions from the
coherent and incoherent parts are similar and they both scale as $1/B^{2}$.
For the incoherent population transfer, the $1/B^{2}$ scaling can be explained
by noticing that $R_{01}$ and $R_{10}$ both go as $1/B^{2}$ when $\Delta_{\pm
1/2}\gg\Gamma$ and $I\sim I_{\text{sat}}$. For the coherent part, the presence
of all polarization components in our probe beam can drive stimulated Raman
transitions between the qubit states, ostensibly at a rate of
$\leavevmode\nobreak\ \Omega_{+1/2}+\Omega_{-1/2}\sim 1/B$. However, this rate
is much smaller than the nuclear spin splitting $\delta_{01}$ at modest
magnetic fields, meaning the coherent population oscillation is negligible.

##  APPENDIX E Simulating the off-resonant scattering rate to 3P2 and 3P0

At typical trapping power for the tweezer wavelength around $760\,\text{nm}$,
the direct off-resonant scattering rates from the 1S0 ground state to the 3P2
and 3P0 metastable states are negligible due to their narrow linewidth.
However, there is still a possibility of the atoms being scattered to these
two states through a two-photon process involving the $6s7s$ 3S1 state when we
probe the atoms on the 1S${}_{0}\leftrightarrow^{3}$P1 transition.

To estimate this off-resonant scattering rate, we can simplify the calculation
into two parts. First, we can calculate the probability of atoms being in the
3P1 state during imaging. Second, we can calculate the off-resonant scattering
rate from 3P1 to the 3P2 and 3P0 states.

The probability of an atom occupying 3P1 during the imaging process is given
by the expression

$\begin{split}P_{{}^{3}\textrm{P}_{1}}=\frac{1}{2}\frac{I_{\text{probe}}/I_{\text{sat}}}{1+4(\Delta/\Gamma_{{}^{3}\textrm{P}_{1}})^{2}+I_{\text{probe}}/I_{\text{sat}}}\end{split}$
(12)

where $\Gamma_{{}^{3}\textrm{P}_{1}}$ and $I_{\text{sat}}$ represent the
linewidth and saturation intensity of the probe transition, respectively.
$I_{\text{probe}}$ and $\Delta$ denote the probe laser intensity and detuning,
respectively.

Starting from 3P1, the off-resonant scattering is dominated by the
3P${}_{1}\leftrightarrow$ 3S1 transition. For the case of large detuning and
negligible saturation, the off-resonant scattering rate is written as:

$\begin{split}\Gamma_{sc}=\frac{3\pi
c^{2}\Gamma^{2}_{{}^{3}\textrm{S}_{1}}}{2\hbar\omega_{0}^{3}}\left(\frac{\omega}{\omega_{0}}\right)^{3}\left(\frac{1}{\omega_{0}-\omega}+\frac{1}{\omega_{0}+\omega}\right)^{2}I_{\text{trap}},\end{split}$
(13)

where $\Gamma_{{}^{3}\textrm{S}_{1}}$ and $\omega_{0}$ are the linewidth and
resonance frequency of the transition. $\omega$ is the laser frequency and
$I_{\text{trap}}$ is the intensity of the $760\,\text{nm}$ tweezer. To match
the experiment, we set the tweezer power to be $7\,\text{mW}$ and the waist
($1/e^{2}$ radius) to be $670\,\text{nm}$.

However, the off-resonant scattering rate given above does not take into
account the atom’s initial $m_{F}$ state and the polarization of the tweezer,
which results in a reduced scattering rate due to the reduction of dipole
matrix elements. In our experiment, the tweezer is linearly polarized with the
polarization parallel to the external magnetic field. Since $J=J^{\prime}=1$,
for the transition starting from the ${|{F,m_{F}}\rangle}$ lower state (3P1)
to the ${|{F^{\prime},m_{F^{\prime}}}\rangle}$ upper state (3S1), the
scattering rate is given by the expression

$\displaystyle\begin{aligned}
\Gamma_{F,m_{F}}&=3\Gamma_{sc}\sum_{F^{\prime},m_{F^{\prime}}}\delta_{m_{F},m_{F^{\prime}}}(2F+1)(2F^{\prime}+1)\\\
&\times\left|\begin{pmatrix}F^{\prime}&1&F\\\
m_{F^{\prime}}&-q&-m_{F}\end{pmatrix}\begin{Bmatrix}1&1&1\\\
F^{\prime}&F&I\end{Bmatrix}\right|^{2},\end{aligned}$ (14)

where $F^{\prime}=1/2\textrm{ or }3/2$ for the 3S1 state and
$q=m_{F^{\prime}}-m_{F}=0$ for a $\pi$-polarized tweezer. The symbols
$(\,\cdots)$ and $\\{\,\cdots\\}$ represent Wigner 3-j and 6-j symbols,
respectively. For our $|m_{F}|=1/2$ and $|m_{F}|=3/2$ imaging methods, the
off-resonant scattering rates are given by $\Gamma_{\text{sc}}/6$ and
$\Gamma_{\text{sc}}/2$, respectively.

FIG. 11:  Off-resonant scattering-limited probe lifetime (a) The calculated
lifetime under $|m_{F}|=3/2$ imaging as a function of probe intensity. The
dashed lines assume the atom will be lost if it scatters to any 3PJ state from
${}^{3}\text{S}_{1}$, while the solid lines assume it will survive only if
scattered to 3P1. (b) The lifetime under $|m_{F}|=1/2$ imaging for the same
condition, which gives a three times higher lifetime due to the lower off-
resonant scattering rate. We set the tweezer {wavelength, power, waist
($1/e^{2}$ radius)} to {$760\,\text{nm}$, $7\,\text{mW}$, $670\,\text{nm}$}
for both calculations. A probe detuning of $-1\,\Gamma$ is used to calculate
the 3P1 state population.

After being off-resonantly excited to the 3S1 state, the atom can decay back
to any of the 3PJ states. Since the 3P2 state cannot be trapped by the
$760\,\text{nm}$ tweezer, atoms in this state will leave the trap immediately.
The 3P0 state is dark to the probe transition and is not re-pumped in our
apparatus, so both processes contribute to atom loss during the fluorescence
imaging. By comparing the branching ratios of all three paths [112], we can
calculate the possibility of decaying to the 3P2 and 3P0 to be around $63\%$ –
dominated by the 3P2 component. The solid curves in Fig. 11(a) and (b) show
the calculation of the $|m_{F}|=3/2$ and $|m_{F}|=1/2$ imaging lifetime
limited by the rate at which atoms decay to either the 3P2 or 3P0 states,
assuming the atom survives if it decays to 3P1.

However, we note that, even if the atom decays to 3P1, is not clear whether
the atom survives after being pumped to the 3S1 state for $\approx
100\,\text{ns}$. Although the 3S1 state is trappable with a tweezer wavelength
of $760\,\text{nm}$, the trap depth is around 70 times higher than that of
both 3P1 and the ground state under the same tweezer power, which could
introduce significant atom loss due to the sudden increase in potential
energy. For comparison, we also plot the case that the atom is fully lost
after being pumped to the 3S1 state, as shown by the dashed curves in Fig.
11(a) and (b). For the $|m_{F}|=3/2$ probe, this calculation gives an imaging
lifetime of $\approx 0.8\,\text{s}$ under typical experimental conditions
($7\,\text{mW}$ tweezer power, $1\,I_{\text{sat}}$ effective probe beam
intensity with $-1\,\Gamma$ detuning), which is in rough agreement with
observation. For the $|m_{F}|=1/2$ probe, this calculation gives a lifetime of
$\approx 2.5\,\text{s}$, which is longer than observation; we find similar
probe lifetimes and survivals in both cases. It is not clear what other decay
mechanism is in play. We measure a lifetime of $8.8(3)\,\text{s}$ for atoms in
tweezers that are not illuminated by any light, which is limited by a
combination of background gas collisions due to finite vacuum pressure as well
as atomic heating due to intensity noise on the tweezer.

We also note that if an atom is off-resonantly excited to the 3S1 state but
eventually decays back to the original 3P1 state and survives, the atom can
still be depolarized between the $m_{F}=-1/2$ and $1/2$ ground states. Using a
similar calculation as the one introduced in the Eq. 14, we can determine the
probability of the depolarization process for the $|m_{F}|=3/2$ readout. After
being off-resonantly pumped to the 3P1 state, this probability is calculated
to be $0.082$, considering decay from ${}^{3}\text{S}_{1}$ to
${}^{3}\text{P}_{1}$. This depolarization rate is significantly smaller than
the off-resonant excitation rate, but would ultimately limit the
depolarization during readout in situations where state mixing and multi-level
dynamics can be neglected at a much higher magnetic field.

##  APPENDIX F Polarizability calculations

The potential that an atom experiences in an optical trap is given by the
product of the state-dependent polarizability $\alpha$ and the spatially
varying intensity profile $I(\mathbf{r})$ of the trap

$U_{\mathrm{trap}}(\mathbf{r})=\frac{\alpha(\omega)}{2\epsilon_{0}c}I(\mathbf{r}),$
(15)

where $\epsilon_{0}$ is the vacuum permittivity, and $c$ is the speed of light
in vacuum. The dependence on the frequency of the trap laser and the atomic
state in the polarizability can be understood from a quantum mechanical
treatment of the induced dipole interaction energy, also known as the AC Stark
shift. Following the derivations in Refs. [113, 114], the polarizability
operator for an atomic state ${|{i}\rangle}$ can be written as a Cartesian
tensor of the form

$\hat{\alpha}_{\mu\nu}(\omega)=\sum_{k}\frac{2\omega_{ki}}{\hbar(\omega_{ki}^{2}-\omega^{2})}\hat{d}_{\mu}|{k}\rangle\langle{k}|\hat{d}_{\nu},$
(16)

where the sum is over all states connected to ${|{i}\rangle}$ via a dipole
transition, $d_{\mu}$ is the projection of the dipole operator along the
$\mu$-th component of the incident electric field, and $\omega_{ki}$ is the
energy difference between the state ${|{k}\rangle}$ and ${|{i}\rangle}$. It is
more insightful to decompose the $\alpha_{\mu\nu}$ into spherical components.
The final result is given as

$\displaystyle\hat{\alpha}(\omega)=\alpha_{S}(\omega)-i\alpha_{V}(\omega)\frac{(\hat{\mathbf{u}}^{*}\times\hat{\mathbf{u}})\cdot\hat{\mathbf{F}}}{2F}$
(17)
$\displaystyle+\alpha_{T}(\omega)\frac{3\\{\hat{\mathbf{u}}^{*}\cdot\hat{\mathbf{F}},\hat{\mathbf{u}}\cdot\hat{\mathbf{F}}\\}-2\hat{\mathbf{F}}^{2}}{2F(2F-1)}.$

The object $\\{\cdot,\cdot\\}$ is the anti-commutator of two operators. The
coefficients $\alpha_{S}$, $\alpha_{V}$, and $\alpha_{T}$ are the scalar,
vector, and tensor polarizabilities, respectively, of the atom for a given
hyperfine state characterized by quantum numbers ${|{nJIFm_{F}}\rangle}$. The
expressions for the individual polarizabilities are

$\displaystyle\alpha_{S}$
$\displaystyle=\frac{1}{\sqrt{3(2J+1)}}\alpha^{(0)}_{nJ}$ (18)
$\displaystyle\alpha_{V}$
$\displaystyle=(-1)^{J+I+F}\sqrt{\frac{2F(2F+1)}{F+1}}\begin{Bmatrix}F&1&F\\\
J&I&J\end{Bmatrix}\alpha^{(1)}_{nJ}$ (19) $\displaystyle\alpha_{T}$
$\displaystyle=(-1)^{J+I+F+1}\sqrt{\frac{2F(2F-1)(2F+1)}{3(F+1)(2F+3)}}\begin{Bmatrix}F&2&F\\\
J&I&J\end{Bmatrix}\alpha^{(2)}_{nJ}$ (20)

where the reduced polarizabilities $\alpha^{(K)}_{nJ}$ are given by

$\displaystyle\begin{aligned} \alpha^{(K)}_{nJ}&=(-1)^{K+J+1}\sqrt{2K+1}\\\
&\quad\times\sum_{n^{\prime}J^{\prime}}(-1)^{J^{\prime}}\begin{Bmatrix}1&K&1\\\
J&J^{\prime}&J\end{Bmatrix}\left|{\langle{n^{\prime}J^{\prime}}|}|d|{|{nJ}\rangle}\right|^{2}\\\
&\quad\times\frac{1}{\hbar}\left(\frac{1}{\omega_{n^{\prime}J^{\prime}nJ}-\omega}+\frac{(-1)^{K}}{\omega_{n^{\prime}J^{\prime}nJ}+\omega}\right).\end{aligned}$
(21)

The reduced dipole matrix elements
$\left|{\langle{n^{\prime}J^{\prime}}|}|d|{|{nJ}\rangle}\right|$ can be
calculated from experimentally determined lifetimes of the relevant states via

$\Gamma_{n^{\prime}J^{\prime}nJ}=\frac{\omega_{n^{\prime}J^{\prime}nJ}^{3}}{3\pi\varepsilon_{0}\hbar
c^{3}}\frac{\left|{\langle{J^{\prime}}|}|d|{|{J}\rangle}\right|^{2}}{2J^{\prime}+1}.$
(22)

A branching ratio will be needed if the excited state decays to several states
of lower energy, such as in the case of decay from 3S1 to 3PJ that leads to
atomic loss from the tweezer.

From (17), we see that the tensor light shift vanishes for states with
$F=0,1/2$. Moreover, the vector light shift vanishes when the tweezer is
linearly polarized. We work mainly with a linearly polarized tweezer, hence
the total light shift experienced by an atomic state is

$U_{\mathrm{trap}}=-\frac{I}{2\epsilon_{0}c}\left(\alpha_{S}+\alpha_{T}\frac{3\cos^{2}\theta-1}{2}\frac{3m_{F}^{2}-F(F+1)}{F(2F-1)}\right)$
(23)

where $\theta$ is the angle between the polarization of the tweezer and the
axis of quantization set by an applied magnetic field. In the case of 171Yb,
the 1S0 and 3P0 states have $F=1/2$, hence they only have a scalar
contribution to the polarizability. On the other hand, the 3P1, $F=3/2$ state
will have all three components, in general.

FIG. 12: Polarizabilities of the atomic states of interest. The
polarizabilities of the 1S0 (black); 3P0 (yellow); 3P1, $F=3/2$, $|m_{F}|=1/2$
(light green); and 3P1, $F=3/2$, $|m_{F}|=3/2$ (dark green) are plotted as a
function of the tweezer wavelength. The shaded band represents the uncertainty
in the total polarizability, which arises from uncertainty in the trap waist
and measured light shifts of the excited states. The magic wavelength
($759\,\text{nm}$, red vertical line) for the ground and clock states is also
near-magic for the $|m_{F}|=1/2$ state. We have included correction factors in
the calculations for the 3P1 states to match the experimentally observed
differential polarizabilities.

To calculate the polarizability of a given state, we perform a sum over states
using measured values of the energy levels and lifetimes wherever possible.
For the 3P1 state, we use the reduced dipole matrix elements given in Ref.
[115]. Our calculations yield
$\alpha_{S}(^{1}\text{S}_{0})=\alpha_{S}(^{3}\text{P}_{0})=186$,
$\alpha_{S}(^{3}\text{P}_{1})=233$ au, and $\alpha_{T}(^{3}\text{P}_{1})=87$
au. The value for $\alpha_{S}(^{1}\text{S}_{0})$ at the clock-magic wavelength
of 759 nm is in excellent agreement with literature [116, 117]. This results
in a predicted differential polarizability of $\alpha_{|1/2|}\approx-0.22$ and
$\alpha_{|3/2|}\approx 0.72$, which is significantly larger than the
experimentally measured values of $\alpha_{|1/2|}\approx-0.030(3)$ and
$\alpha_{|3/2|}\approx 0.25(3)$ (see Sec III). As a result, we
phenomenologically correct our calculated values by using Eq. (23) to generate
a set of linear equations

$\displaystyle\hbar\Delta\omega(|m_{F}|=3/2)$
$\displaystyle\quad\quad=-\frac{I}{2\epsilon_{0}c}\left(\alpha_{S}(^{3}\text{P}_{1})+\alpha_{T}(^{3}\text{P}_{1})-\alpha_{S}(^{1}\text{S}_{0})\right)$
(24) $\displaystyle\hbar\Delta\omega(|m_{F}|=1/2)$
$\displaystyle\quad\quad=-\frac{I}{2\epsilon_{0}c}\left(\alpha_{S}(^{3}\text{P}_{1})-\alpha_{T}(^{3}\text{P}_{1})-\alpha_{S}(^{1}\text{S}_{0})\right)$
(25)

which yields $\alpha_{T}(^{3}\text{P}_{1})=26(6)\,\text{a.u.}$ and
$\alpha_{S}(^{3}\text{P}_{1})-\alpha_{S}(^{1}\text{S}_{0})=20(4)\,\text{a.u.}$
Note that we operate at $\approx 760.2$ nm, for which
$\alpha_{S}(^{1}\text{S}_{0})$ differs from the clock-magic wavelength by
$\approx 0.1$%. The uncertainty in the values are mainly derived from the
uncertainty in the measured differential polarizabilities and the uncertainty
in the measured beam waist of the tweezer. We have ascribed a conservative
estimate for the uncertainties of 10%. Since the trap depth depends on the
beam waist as $1/w_{0}^{2}$, the overall uncertainty is mainly dominated by
the uncertainty in the beam waist.

Using the corrected values, we plot the polarizability as a function of the
tweezer wavelength in Fig. 12. It is interesting to note that the
polarizabilities of all 3P1 Zeeman states converge at $\approx
796\,\text{nm}$. Also, we identify $\approx 778\,\text{nm}$ as a good
candidate for implementing our readout technique because the 3P1 $|m_{F}|=3/2$
states are magic with the ground state.

##  APPENDIX G Effect of vector and tensor light shifts

The vector and tensor light shift can cause additional state-mixing in the
3P1, $F=3/2$ manifold, which opens up a depolarization channel when probing
via the $m_{F}=-3/2$ state that causes the atoms to decay into the dark 1S0,
$m_{F}=1/2$ ($|1\rangle$) state. We estimate the level of state-mixing by
numerically diagonalizing the AC Stark Hamiltonian together with the Zeeman
Hamiltonian, and observe the complex amplitudes of the various eigenstates.

We use a different convention for the coordinate system in Fig 1 for the
following calculations. The applied magnetic field, which defines the axis of
quantization, is oriented along the $+z$-axis, and the tweezer propagates
along the $+y$-axis. Thus, the following parametrization for the polarization
of the tweezer is valid:

$\displaystyle\hat{\mathbf{u}}$
$\displaystyle=\left(\cos\gamma\cos\theta-i\sin\gamma\sin\theta\right)\hat{\mathbf{z}}$
(26)
$\displaystyle\quad+\left(\cos\gamma\sin\theta+i\sin\gamma\cos\theta\right)\hat{\mathbf{x}}$

where $0\leq\gamma\leq\pi/2$ represents the degree of ellipticity of the
tweezer, and $\theta$ is the angle between the linear polarization of the
tweezer and the axis of quantization set by the magnetic field. This
parametrization allows us to consider cases where the tweezer is
simultaneously rotated away from the axis of quantization and contains some
degree of ellipticity. However, we will analyze the two cases separately by
setting the counterpart to zero.

FIG. 13: State-mixing of the 3P1, $F=3/2$, $m_{F}=-3/2$ imaging state due to
the vector and tensor light shifts (a) The probabilities in the $m_{F}=-3/2$
(dark green) and $m_{F}=-1/2$ states (light green) due to the vector
($\gamma=1$ degree and $\theta=0$ degrees; solid) and tensor ($\theta=1$
degree and $\gamma=0$ degrees; dashed) light shifts as the magnitude of the
applied magnetic field is varied. The gray vertical lines at $10\,\text{G}$
(dot-dash) and $60\,\text{G}$ (dotted) show field values chosen for the angle
studies shown in (b). (b) We study the dependence on the angles $\theta$ and
$\gamma$, which describes the misalignment of the tweezer from the axis of
quantization and the degree of ellipticity of the tweezer polarization,
respectively. There is a sharp rise at angles close to zero degree due to the
sudden appearance of the off-diagonal matrix elements.

For the 3P1, $F=3/2$, $m_{F}=-3/2$ state, the dominant state that the vector
and tensor light shift mixes with is the neighboring $m_{F}=-1/2$ state. In
Fig. 13(a), we consider the state-mixing as a function of the magnitude of the
applied magnetic field. We consider $\theta$ or $\gamma=1\leavevmode\nobreak\
\mathrm{deg}$ to illustrate the cases where the tweezer is not purely linear,
or when the tweezer is indeed linear but not perfectly aligned onto the axis
of quantization. At low fields, state-mixing is significant as the light
shifts ($\sim 1$ MHz) are comparable to the Zeeman energy ($\sim 1.4\times
m_{F}$ MHz/G). The mixing can be suppressed by increasing the magnetic field.
A similar trend can be seen when we fix the magnitude of the applied magnetic
field and vary the angles. Whenever the angles are non-zero, the state-mixing
is turned on as there is a competition between the axis of quantization
defined by the applied magnetic field and those defined by the tweezer
polarization. Naturally, it follows that as the applied magnetic field
increases in strength, the effect of state-mixing decreases, which is
confirmed by the results in Figure 13(b).The trend phenomenologically matches
a $\sim 1/B^{2}$ scaling.

For a scattering rate of $\approx 97000\,\text{photons/s}$ and a probe time of
$12\,\text{ms}$, we estimate that $\approx 1200$ photons are scattered by the
atoms, which is consistent with our measured collection of $\approx 36$
photons with a measured atom-to-camera collection efficiency of $0.04$. For a
depolarization probability of $\bar{P}_{\text{depol}}\approx 0.01$ during the
readout, we conclude that the depolarization probability per photon scatter is
$\mathcal{P}_{\text{depol}}\lesssim 1\times 10^{-5}$. Up to a Clebsch-Gordan
coefficient, we expect that this depolarization probability per photon is
exactly equal to the contribution of the $m_{F}=-1/2$ state to the mixed
eigenstate. As shown in Fig. 10, the Clebsch-Gordan coefficients to decay from
3P1, $m_{F}=-1/2$ favor 1S0, $m_{F}=-1/2$ over $m_{F}=1/2$ by a factor of 2,
so we can tolerate twice as much population in 3P1, $m_{F}=-1/2$ than if the
branching ratios were even for a given $\bar{P}_{\text{depol}}$. Therefore,
this analysis suggests that we can tolerate a population of $\approx 2.0\times
10^{-5}$ in 3P1, $m_{F}=-1/2$ to obtain $\bar{P}_{\text{depol}}\approx 0.01$
during the readout, which for $B=58$ G corresponds to $\gamma$ or
$\theta\approx 1-2$ degrees. This estimate is consistent with experimental
observations in which the tweezer polarization was deterministically moved on
the Poincaré sphere using a polarimeter, and corroborates our observation of
larger depolarization probabilities for tweezers at the edges of the array.

##  APPENDIX H Characterization of state preparation and measurement
parameters

In this appendix, we describe several experimentally measured parameters
relevant to state preparation and measurement (SPAM) correction, the procedure
of which is described in Appendix I. These are the tweezer loading probability
$p$, the bright- and dark-state readout survival probabilities
$\eta_{\text{surv}}^{\text{B}}$ and $\eta_{\text{surv}}^{\text{D}}$, the qubit
spin-flip probabilities $P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}$ and
$P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}$, and the $\pi$-pulse
probability $\eta_{\pi}$. We additionally define bright and dark
discrimination fidelities $F_{1}$ and $F_{0}$ which are also relevant to SPAM
correction, but themselves left uncorrected (see Appendix I).

###  1 Definition of base discrimination fidelity

Here we characterize the possible measurement-based error channels in our
atom- and state-readout detection schemes. All such errors derive from a
common source, which is the degree to which the data from a single camera
exposure while a tweezer site is illuminated can be correctly classified (or
not) as holding an atom in a fluorescent state. While it is straightforward to
scatter photons from a single atom in a tweezer and count the number detected
by a sensor, this measurement may be confounded by a number of other
processes. For example, it is possible that the atom is not in a state excited
by a particular laser frequency, the scattered photons may not all be
collected by the sensor, the atom may have exited the trap or gone dark during
the exposure, or the atom simply may not have been loaded in the first place.

We first define a base discrimination fidelity $\mathcal{F}$ as the
probability of correctly classifying a single image as containing a
fluorescent atom, based on the number of collected photons $x$. We assume a
simple Gaussian mixture model of two components governed by an overall
distribution of the form

$\Pi(x)=P_{0}f_{0}(x)+P_{1}f_{1}(x)$ (27)

where the mixture weights $P_{N}$ are the probabilities of $N$ such atoms in a
single tweezer, and $f_{N}$ is a Gaussian distribution with mean $\mu_{N}$ and
variance $\sigma_{N}^{2}$. Individual images are classified as “bright” ($B$,
indicating $N=1$ fluorescent atom) if the total number of photons collected is
greater than or equal to some predetermined threshold value $\theta$, and
“dark” ($D$, indicating $N=0$ fluorescent atoms) otherwise. We therefore
define $\mathcal{F}$ more concretely as

$\displaystyle\mathcal{F}$ $\displaystyle=\text{Pr}(D|N=0)\text{Pr}(N=0)$ (28)
$\displaystyle\phantom{=}+\text{Pr}(B|N=1)\text{Pr}(N=1)$
$\displaystyle=P_{0}\int_{-\infty}^{\theta}\,\text{d}{x}\,f_{0}(x)+P_{1}\int_{\theta}^{\infty}\,\text{d}{x}\,f_{1}(x)$

All the parameters of $\Pi$ are readily obtained by fitting to a measured
histogram of photon counts from a series of images, and $\theta$ is chosen to
maximize $\mathcal{F}$ at the number of photons where $P_{0}f_{0}$ and
$P_{1}f_{1}$ intersect. We also define discrimination fidelities based on the
individual components of the mixture for use in SPAM correction,

$\displaystyle F_{0}$
$\displaystyle=\text{Pr}(D|N=0)=\int_{-\infty}^{\theta}\,\text{d}{x}\,f_{0}(x)$
(29) $\displaystyle F_{1}$
$\displaystyle=\text{Pr}(B|N=1)=\int_{\theta}^{\infty}\,\text{d}{x}\,f_{1}(x).$

In a typical experiment where we initialize atoms in tweezers by loading and
optically pumping to the bright state before imaging, we usually measure
$P_{0}\approx 30\%$, and $P_{1}\approx 70\%$, with $\mu_{0}\approx 1.0$,
$\sigma_{0}\approx 2$, $\mu_{1}\approx 37$, $\sigma_{1}\approx 16$ photons.
Note that we calculate photon counts using a linear transformation from raw
photoelectron counts yielded by our EMCCD (see Appendix A). This
transformation is calibrated to give zero photon counts at some non-zero
number of photoelectrons which, in relatively rare cases, can lead to negative
photon counts.

To estimate the uncertainty in $\mathcal{F}$ under both imaging conditions
described in Appendix B (treated independently), we use a bootstrapping
procedure following Ref. [118]. From a data set of $2500\times 5$ realizations
of the atomic fluorescence signal under a particular condition from all sites
of the array (with optical pumping performed beforehand for $m_{F}=-3/2$
imaging), we then generate $200$ bootstrap data sets, each with $500$
realizations, by sampling from the original set with replacement. The above
calculation is then carried out for each bootstrap set, and the uncertainty in
$\mathcal{F}$ is obtained from the standard deviation over the bootstrap sets.
We find
$\mathcal{F}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.993(4)}$
with
$F_{0}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.997(2)}$
and
$F_{1}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.991(5)}$
for the $m_{F}=-3/2$ imaging condition, and $\mathcal{F}=0.995(3)$ with
$F_{0}=0.997(2)$ and $F_{1}=0.995(4)$ for the $m_{F}=-1/2$ condition. We also
approximate the tweezer loading probability
$p={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.67(4)}$
this way as the fraction of bright shots,

$p=\Pr(B).$ (30)

However, because optical pumping is a process separate from loading the
tweezer, we note that $P(B)$ is more accurately $p\times\eta_{\text{OP}}$; we
decouple these parameters using additional multi-readout sequences that do not
perform optical pumping directly after loading as described below.

The imaging method used for all of the experiments discussed here is state-
selective, such that photons are scattered from an atom only if the atom is in
one of our selected qubit states. Therefore, the mixture weights $P_{0}$ and
$P_{1}$ are generally combinations of probabilities in a space of events
described by two binary degrees of freedom: (a) the internal qubit state of
the atom, and (b) whether an atom whose state is in
${}^{1}\text{S}_{0}$–${}^{3}\text{P}_{1}$ manifold is present in the tweezer.
Here, (a) is the only relevant degree of freedom and (b) represents an error
channel to which experiments are coupled via atom loss, either through off-
resonant scatter to a dark state outside the
${}^{1}\text{S}_{0}$–${}^{3}\text{P}_{1}$ manifold or through heating. Due to
this, it is then impossible to determine whether a single given image
classified as dark has an atom in the non-fluorescent qubit state or no atom
at all.

To resolve this, we append a final, state-independent measurement (called
“atom readout” in the main text) at the end of each experimental sequence,
such that it becomes possible to determine which shots of the sequence contain
atoms. Then filtering out all sequence shots with dark final measurements,
$P_{0}$ and $P_{1}$ are directly converted to state probabilities with atom
loss errors entirely decoupled from all measurements, and bright-dark
classifications on the single-image level are mapped to qubit state
measurements, with bright corresponding to ${|{0}\rangle}$ and dark to
${|{1}\rangle}$. For sufficiently high $\bar{\eta}_{\text{surv}}$, qubit
states can be measured many times in a single shot of an experiment using this
method, at the cost of immediately rejecting a portion of data recorded in an
experiment that grows roughly as $1-\eta_{\text{surv}}^{M}$ where $M$ is the
number of measurements performed. Figure 14 shows the results of the first two
readouts of Fig. 3(c), leaving out post-selection on the final readout.

FIG. 14:  Depolarization probability study without post-selection on atom
readout. Photon counts from the first two readouts in the circuit shown in
Fig. 3(c) used to study depolarization probability without post-selection on
the final atom readout. Here, the population in the
$\text{B}\,\rightarrow\,\text{D}$ depolarization channel is artificially
inflated by atom loss between these two readouts, as is the population in the
$\text{D}\,\rightarrow\,\text{D}$ quadrant due to failed loading of the
tweezer.

Having obtained the discrimination fidelities $F_{0}$ and $F_{1}$ above from a
single-measurement sequence assuming a Gaussian mixture model for the
histograms, we now experimentally verify $F_{0}$ and $F_{1}$ using an
independent three-measurement sequence inspired by Ref. [18] which we describe
below. While the state readout fidelity $\mathcal{F}_{\text{read}}$ and state
initialization fidelity $\mathcal{F}_{\text{init}}$ are generally coupled, it
is possible to effectively decouple them by using two consecutive
measurements. Explicitly, given a bimodal measurement histogram belonging to
the bright (${|{0}\rangle}$) and dark (${|{1}\rangle}$ or empty tweezer)
states as shown in Fig. 1(d), setting a stricter state-assignment threshold
$\theta$ for ${|{0}\rangle}$ (${|{1}\rangle}$) effectively allows one to
initialize the state with perceived increasing probability. Then the resulting
histogram of the second measurement, post-selected on successful
initialization via the first measurement, yields the limiting discrimination
fidelities $\mathcal{F}_{\text{read}}^{\text{B}}$ for a tweezer holding an
atom in ${|{0}\rangle}$ and $\mathcal{F}_{\text{read}}^{\text{D}}$ for either
an empty tweezer or one holding an atom in ${|{1}\rangle}$.

In practice, this translates to examining the results of a experiment similar
to that shown in Fig. 3(c), where the final atom readout is moved to before
the two state readouts. This is done to ensure via post-selection that we
consider only cases where an atom is initially present in the tweezer, but
still include effects arising from imperfect atom survival between readouts.
The method used to analyze results from this sequence additionally include
modified post-selection conditions on the first of the state readouts. Holding
the threshold $\theta_{a}$ used for bright/dark classifications in the first
($a$) measurement fixed at the optimal value identified by Eq. 28 (i.e. fixing
the conditions for detecting the presence/absence of atoms), we allow the
thresholds $\theta_{b}$ and $\theta_{c}$ for the second ($b$) and third ($c$)
measurements to vary. Still post-selecting on a bright initial atom readout,
we then examine the results of the third measurement condition on those of the
second as a function of $\theta_{b}$ and $\theta_{c}$ and identify the
probabilities $\Pr(B_{c}|B_{a}\land B_{b})$ and $\Pr(D_{c}|B_{a}\land D_{b})$
with fidelities $\mathcal{F}_{\text{read}}^{\text{B}}$ and
$\mathcal{F}_{\text{read}}^{\text{D}}$, respectively.

A plot of these quantities as well as their average with $\theta_{c}$ fixed to
its optimum value is shown in Fig. 15. As $\theta_{b}$ is increased
(decreased), the perceived probability in the $\theta_{b}\rightarrow\infty$
($\theta_{b}\rightarrow 0$) limit of successfully initializing the state in
${|{0}\rangle}$ (${|{1}\rangle}$) increases, and the second measurement when
post-selected on the success of the first gives the desired state detection
fidelity. We find that $\mathcal{F}_{\text{read}}^{\text{B}}=0.99(1)$ and
$\mathcal{F}_{\text{read}}^{\text{D}}=0.97(2)$, in good agreement with the
discrimination fidelities $F_{0}$ and $F_{1}$ measured in Appendix 1. We note
that this method of estimating discrimination fidelity is limited by spin-flip
events occurring between readouts. The method additionally relies on full
resolution of the limiting behaviors described above, which requires one to
sample events that occur with vanishing probability with significantly larger
data sets. For these reasons, we choose to use the method based on fitting to
Eq. 27 and account for depolarization events in the SPAM models described in
Appendix I.

FIG. 15:  Validation of discrimination fidelity. Using three consecutive
measurement histograms $a,b,c$, with $a$ being an atom readout, post-selecting
data on measurement $a$ being bright, we estimate the readout fidelity by
examining the behavior of the conditional probabilities
$\mathcal{F}_{\text{read}}^{\text{B}}=\Pr(B_{c}|B_{a}\land B_{b})$ and
$\mathcal{F}_{\text{read}}^{1}=\Pr(D_{c}|B_{a}\land D_{b})$ as well as the
average readout fidelity
$\mathcal{F}_{\text{read}}=\mathcal{F}_{\text{read}}^{\text{D}}\Pr(D_{b})+\mathcal{F}_{\text{read}}^{\text{B}}\text{Pr}(B_{b})$
by sweeping the thresholds $\theta_{b}$ and $\theta_{c}$ used in measurements
$b$ and $c$, respectively, while holding that for measurement $a$ fixed at the
vertical red line, set by the optimization of Eq. 28. We show results for
$\theta_{c}$ fixed to its optimum value with the fidelities as a function of
$\theta_{b}$ only. For large $\theta_{b}$, ${|{0}\rangle}$ is prepared by $b$
with a perceived near-unity probability and subsequently measured in $c$ with
fidelity $\mathcal{F}_{\text{read}}^{\text{B}}=0.99(1)$. For small
$\theta_{b}$, ${|{1}\rangle}$ is perceived to be prepared with near-unity
probability instead and measured with fidelity
$\mathcal{F}_{\text{read}}^{\text{D}}=0.97(2)$. We note that
$\mathcal{F}_{\text{read}}^{\text{D}}$ may not have fully saturated to its
maximal value due to finite sampling of events occurring with vanishing
probability.

###  2 Atom survival probability

The state-averaged probability $\bar{\eta}_{\text{surv}}$ that an atom remains
in its tweezer after readout is estimated in two parts. The ${|{0}\rangle}$
bright state survival $\eta_{\text{surv}}^{\text{B}}$ is measured by
performing two atom readouts $a$ and $b$ – of either sort shown in Fig. 2 – in
quick succession, post-selecting on a bright $a$ readout, and taking the
fraction of those with bright $b$ readout as well,

$\eta_{\text{surv}}^{\text{B}}=\Pr(B_{b}|B_{a})=\frac{\Pr(B_{a}\land
B_{b})}{\Pr(B_{a})}.$ (31)

Uncertainty in this measurement is estimated using a similar bootstrapping
procedure to the one for discrimination fidelity described above, wherein
$2500\times 5$ total realizations of this two-readout sequence are again used
to generate $200$ bootstrap data sets, each with $500$ realizations,
$\eta_{\text{surv}}^{\text{B}}$ calculated for each one, and the standard
deviation across the bootstrap sets taken.

We find that, for the $m_{F}=-3/2$ ($m_{F}=-1/2$) imaging condition taken for
a single array site,
$\eta_{\text{surv}}^{\text{B}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.966(8)}$
($\eta_{\text{surv}}^{\text{B}}=0.96(1)$). Note that the probe time for the
$-1/2$ case is $20\,\text{ms}$ (versus $12\,\text{ms}$ for the $-3/2$ case),
which is the primary explanation for the lower survival.

As described in the main text, the survival probability
$\eta_{\text{surv}}^{\text{D}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.9960(1)}$
of the ${|{1}\rangle}$ dark state, which cannot be directly estimated from the
above sequence, is estimated based on the measured lifetime of the
${|{1}\rangle}$ state under probe conditions. We measure this using a more
complex experimental sequence comprising an initial atom readout ($a$) and a
$\pi/2$-pulse, followed by a state readout ($b$), a hold under probe
conditions, and a final atom readout ($c$). We extract the lifetime
$2.99(6)\,\text{s}$ by fitting the probability $P(B_{c}|B_{a}\land D_{b})$ as
a function of hold time to a decaying exponential. We also use this sequence
to verify the measured value of $\eta_{\text{surv}}^{\text{B}}$ via its
lifetime using the same procedure with $P(B_{c}|B_{a}\land B_{b})$. This
yields a bright state lifetime of $1.24(8)\,\text{s}$, which is in good
agreement with $\eta_{\text{surv}}^{\text{B}}$ above.

###  3 Optical pumping efficiency

Optical pumping is performed using a single beam directed onto the atoms along
an axis perpendicular to that of the tweezers in the horizontal plane. The
polarization of the pump beam is set to be linear, with polarization vector
also lying in the horizontal plane, perpendicular to the quantization axis set
by a $50\,\text{G}$ magnetic field and the polarization of the tweezer light.
The frequency of the beam is set to be near-resonant with the
${}^{1}\text{S}_{0}\leftrightarrow{}^{3}\text{P}_{1}$ $F=3/2$, $m_{F}=-1/2$
transition, such that the ${}^{1}\text{S}_{0}$
${|{m_{F}=+1/2}\rangle}\equiv{|{1}\rangle}$ state is pumped to
${}^{1}\text{S}_{0}$ ${|{m_{F}=-1/2}\rangle}\equiv{|{0}\rangle}$. The pump
beam has $1/e^{2}$ radius $\approx 2\,\text{mm}$ and intensity $\approx
1.3\,I_{\text{sat}}$, and is applied for $80\,\text{$\mu$s}$. We measure the
optical pumping efficiency $\eta_{\text{OP}}$ using the same two-readout
sequence as for bright-state atom survival, now post-selecting on the $b$
readout:

$\eta_{\text{OP}}=\Pr(B_{a}|B_{b})=\frac{\Pr(B_{a}\land B_{b})}{\Pr(B_{b})}.$
(32)

From the same bootstrapping procedure, we find
$\eta_{\text{OP}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.972(9)}$
(single-site best;
$\eta_{\text{OP}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.972(9)}$
array-averaged).

###  4 Qubit depolarization probabilities

As stated in the main text and shown in Fig. 3(c), we use two qubit readouts
$a$ and $b$ followed by a single atom readout $c$ to study the qubit
depolarization probability. Post-selecting on a bright $c$ readout to decouple
from atom loss errors, we can directly measure the probabilities
$P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}$ and
$P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}$ that the qubit state flips
from bright (${|{0}\rangle}$) to dark (${|{1}\rangle}$) or vice-versa between
$a$ and $b$:

$\displaystyle P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}$
$\displaystyle=\Pr(B_{b}|D_{a}\land B_{c})=\frac{\Pr(D_{a}\land B_{b}\land
B_{c})}{\Pr(D_{a}\land B_{c})}$ (33) $\displaystyle
P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}$
$\displaystyle=\Pr(D_{b}|B_{a}\land B_{c})=\frac{\Pr(B_{a}\land D_{b}\land
B_{c})}{\Pr(B_{a}\land B_{c})}.$ (34)

We measure
$P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.011(3)}$
and
$P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.018(4)}$
(single site best;
$P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
and
$P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.025(2)}$
array-averaged).

###  5 $\pi$-pulse fidelity

Finally we estimate the unitary $\pi$-pulse fidelity $F_{\pi}$ given by the
overlap of the ideal $\pi$-pulse $U_{\pi,\text{ideal}}\\!=\\!-i\sigma_{x}$ and
the real $\pi$-pulse applied
$R(\Omega_{0},\delta)\\!=\\!\exp(-iL/2(\Omega_{0}\sigma_{x}+\delta\sigma_{z}))$:

$\displaystyle F_{\pi}$
$\displaystyle=\frac{1}{2}\left|\text{tr}\left[U_{\pi,\text{ideal}}^{\dagger}R(\Omega_{0},\delta)\right]\right|$
(35)
$\displaystyle=\left|\frac{\Omega_{0}}{\Omega}\sin\left(\frac{\theta}{2}\right)\right|,$

where $\theta=\Omega L$ is given by the generalized Rabi frequency
$\Omega=\sqrt{\Omega_{0}^{2}+\delta^{2}}$ and pulse length $L$.

Noting that $F_{\pi}$ is given directly by the state transition probability
$\eta_{\pi}=|\langle 0|R(\Omega_{0},\delta)|1\rangle|^{2}=F_{\pi}^{2}$, we use
the three-readout sequence shown in Fig. 16 to first extract $\eta_{\pi}$. As
shown in the model (Fig. 16) we define the state-averaged $\eta_{\pi}$:

$\displaystyle\eta_{\pi}$
$\displaystyle=\frac{1}{2}\Big{[}\Pr(B_{b}|D_{a}\land
B_{c})+\Pr(D_{b}|B_{a}\land B_{c})\Big{]},$ (36)

and similarly define the (state-averaged) $F_{\pi}$:

$\displaystyle F_{\pi}$
$\displaystyle=\frac{1}{2}\Big{[}\sqrt{\Pr(B_{b}|D_{a}\land
B_{c})}+\sqrt{\Pr(D_{b}|B_{a}\land B_{c})}\Big{]}.$ (37)

Thus we measure
$F_{\pi}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.984(8)}$,
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.997(4)}$
with SPAM correction.

##  APPENDIX I State preparation and measurement correction

FIG. 16:  Example probability graph for SPAM correction. The probability graph
for the three-image sequence from Fig. 4(a) is shown, wherein each node
represents a state of the system and each edge gives the probability of
transitioning to another. Every possible combination of readout results is
represented as the complete path from the “Start” node on the left to any of
the terminal nodes with no outgoing edges on the right, and the total
probability for any such path is found as the product of all the edge weights
along it. Here, ${|{\varnothing}\rangle}$ is used as a shorthand for “empty
tweezer or lost atom,” and we assume that the qubit state has vanishing
probability of flipping during readout. Thus, a model for an experimentally
measurable probability is found by summing the probabilities of all paths that
satisfy a corresponding set of data analysis conditions. For example, the
model for the measured $\pi$-pulse probability
${}^{*}\mkern-2.5mu{\eta_{\pi}}$ is found following Eq. 36 as the sum over
probabilities of all paths passing through a “Bright” Readout 0 node and a
“Dark” Readout 1 node (or vice-versa) as well as a “Bright” Readout 2 node,
divided by that for all paths terminating in a “Bright” Readout 2 node. This
model is set equal to the experimentally measured value of
${}^{*}\mkern-2.5mu{\eta_{\pi}}$ and used for the simultaneous correction of
all relevant parameters as described in Appendix I. Generation of models from
other graphs is done similarly, following Eqs. 30–34.

In experimental sequences, we infer the presence or internal state of atoms by
mapping photon counts onto a binary space and thereby extracting desired
quantities like readout survival probability, qubit depolarization
probability, and $\pi$-pulse fidelity. However, imperfections in tweezer
loading probability, discrimination fidelity, optical pumping efficiency, and
indeed qubit depolarization can produce errors in these measurements. State
preparation and measurement (SPAM) correction attempts to isolate quantities
of interest from sources of error based on a constructed model of possible
subprocesses that may occur in each experimental sequence and estimated
probabilities for each of those subprocesses. Generically, we model a quantity
${}^{*}\mkern-2.5mu{X}$ measured directly from experimental data as a function
of the isolated “true” value of the quantity $X$ as well as some number of
other relevant quantities. A set of models and experimentally measured values
can be used to completely constrain all other such models, and the system can
then be inverted to obtain SPAM-corrected values for all quantities
simultaneously.

Directed, weighted, acyclic graphs provide a convenient language to describe
the various states of the experimental system as it progresses through a given
sequence as well as the measurement outcomes it can produce, and allow for
easy generation of SPAM correction models through well-known graph traversal
algorithms. In these graphs, each node represents a state of the system and
each weighted edge gives the probability of transitioning to another. Every
experimental trajectory is represented as the complete path from a given start
node to any terminal node with no outgoing edges, and the total probability
that the trajectory is realized is the product of the edge weights in the
path. An example graph is shown for the sequence we use to measure the
$\pi$-pulse probability $\eta_{\pi}$ in Fig. 16, and in general models are
generated by summing over the probabilities of trajectories that follow Eqs.
30–36.

We use a number of these graphs, one for each experimental sequence described
in Appendix H, to produce models for all relevant quantities – tweezer loading
probability $p$, depolarization probabilities
$P_{\text{depol}}^{\text{D\,$\rightarrow$\,B}}$ and
$P_{\text{depol}}^{\text{B\,$\rightarrow$\,D}}$, survival probabilities
$\eta_{\text{surv}}^{\text{B}}$ and $\eta_{\text{surv}}^{\text{D}}$, optical
pumping efficiency $\eta_{\text{OP}}$, and $\pi$-pulse fidelity – and solve
the experimentally constrained system numerically. Uncertainties are estimated
by setting each measured value to its own uncertainty bounds independently,
re-solving the system under all $2^{n}$ such combinations, and taking the RMS
deviation of this solution set from the nominal corrected values. We note a
corrected fill fraction of
$p={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0.68(4)}$
(array-averaged) with the rest of the SPAM-corrected values are listed in the
main text or in Table 1.

We note that, here, the state discrimination fidelities $\mathcal{F}$,
$F_{0}$, and $F_{1}$ are deliberately left uncorrected in order to simplify
calculations: Strictly, the treatment of the depolarization probabilities here
is as the probability of a bit-flip occurring after a readout is performed,
which neglects the possibility that the flip occurs during the readout itself.
We note in this scenario, however, that because the atom would not scatter
photons for the complete duration of the imaging pulse, the number of photons
scattered during readout would deviate significantly from the means of either
the $N=0$ or $N=1$ components identified in Eq. 27 and be detectable as a
count somewhere in overall count distribution between the two peaks. Since
such an event would directly affect our measured discrimination fidelities,
SPAM correction using uncorrected fidelities therefore constitutes correction
for both discrimination infidelity and any qubit flip errors that occur during
a single readout.

##  APPENDIX J Estimation of $T_{1}$ and $T_{2}^{*}$

Measurement of the array-averaged $T_{1}$ depolarization lifetimes is often
hampered by atom loss. When using e.g. a state-dependent blow-away pulse to
measure qubit state populations [69], atom loss directly confounds the
measurements from which state populations are inferred. Our ability to de-
couple atom loss from state measurement allows us to measure these quantities
in a straightforward manner. In Figs. 17(a) and 17(b), we show the circuits
used to measure $T_{1}$ and $T_{2}^{*}$, respectively. Both circuits are
similar to that shown in Fig. 3(c) used to measure depolarization probability
during readout, with two state readouts followed by an atom readout for post-
selection, except for the inclusion of a variable holding period and $\pi/2$
pulses in the $T_{2}^{*}$ case.

FIG. 17:  Measurement of array-averaged $T_{1}$, $T_{2}^{*}$, and
$T_{2}^{\text{echo}}$.  (a) The circuit used to measure dark-state survival
and the qubit depolarization time, $T_{1}$. This circuit is identical to that
shown in Fig. 3 except for the addition of a holding time $\tau$ between the
first two readouts. The qubit is also initialized in ${|{0}\rangle}$ here in
order to measure both $P_{\text{same}}$ (blue circles) and the atom survival
fraction (orange triangles), which are plotted below as a function of $\tau$.
The atom survival lifetime is measured to be $8.8(3)\,\text{s}$. $T_{1}$ is
estimated to be $230(50)\,\text{s}$, but our data extends only to
$13\,\text{s}$. (b) The circuit used to measure the qubit dephasing time,
$T_{2}^{*}$. Here, the qubit is rotated to the equator for a time $\tau$
before a $\pi/2$ pulse with phase $\varphi$ relative to the first is applied
and the qubit state is measured. The resulting fringes in $P_{\text{same}}$
are plotted as a function of $\varphi$ in the insets, and the contrast of the
fringes is plotted as a function of the hold time $\tau$ in the main plot. The
decay of the contrast follows a Gaussian profile, the $1/e$ time of which we
measure as $T_{2}^{*}=0.37(1)\,\text{s}$. (c) The same circuit except with a
$\pi$ pulse in the middle of the precession time. This spin echo sequence
gives $T_{2}^{\text{echo}}=1.40(5)\,\text{s}$.

In Fig. 17(a), we initialize in the ${|{0}\rangle}$ qubit state via optical
pumping. This effectively turns the first state readout into an atom readout,
which allows us to simultaneously measure atom survival as the number of shots
that are bright in the final atom readout as a fraction of those bright in the
initial readout as well. The atom survival lifetime is obtained by fitting to