High-Fidelity Control, Detection, and Entanglement of Alkaline-Earth Rydberg Atoms
Ivaylo S. Madjarov,
1,
∗
Jacob P. Covey,
1,
∗
Adam L. Shaw,
1
Joonhee Choi,
1
Anant Kale,
1
Alexandre
Cooper,
1,
†
Hannes Pichler,
1
Vladimir Schkolnik,
2
Jason R. Williams,
2
and Manuel Endres
1,
‡
1
Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Trapped neutral atoms have become a prominent platform for quantum science, where entan-
glement fidelity records have been set using highly-excited Rydberg states. However, controlled
two-qubit entanglement generation has so far been limited to alkali species, leaving the exploitation
of more complex electronic structures as an open frontier that could lead to improved fidelities and
fundamentally different applications such as quantum-enhanced optical clocks. Here we demonstrate
a novel approach utilizing the two-valence electron structure of individual alkaline-earth Rydberg
atoms. We find fidelities for Rydberg state detection, single-atom Rabi operations, and two-atom
entanglement surpassing previously published values. Our results pave the way for novel applica-
tions, including programmable quantum metrology and hybrid atom-ion systems, and set the stage
for alkaline-earth based quantum computing architectures.
Recent years have seen remarkable advances in gen-
erating strong, coherent interactions in arrays of neu-
tral atoms through excitation to Rydberg states, char-
acterized by large electronic orbits [1]. This has led
to profound results in quantum science applications,
such as quantum simulation [2–4] and quantum com-
puting [5–10], including a record for two-atom entan-
glement for neutral atoms [8]. Furthermore, up to 20-
qubit entangled states have been generated in Rydberg
arrays [11], competitive with results in trapped ions [12]
and superconducting circuits [13]. Many of these devel-
opments were fueled by novel techniques for generat-
ing reconfigurable atomic arrays [14–16] and mitigation
of noise sources [8, 17]. While previous Rydberg-atom-
array experiments have utilized alkali species, atoms
with a more complex level structure, such as alkaline-
earth atoms (AEAs) [18–24] commonly used in opti-
cal lattice clocks [25], provide new opportunities for
increasing fidelities and accessing fundamentally dif-
ferent applications, including Rydberg-based quantum
metrology [26–28], quantum clock networks [29], and
quantum computing schemes with optical and nuclear
qubits [30–32].
Here we demonstrate such a novel Rydberg array archi-
tecture based on AEAs, where we utilize the two-valence
electron structure for single-photon Rydberg excitation
from a meta-stable clock state as well as auto-ionization
detection of Rydberg atoms [Fig. 1]. We find leading fi-
delities for Rydberg state detection, coherent operations
between ground- and Rydberg-state, and Rydberg-based
two-atom entanglement [Table I]. More generally, our re-
sults constitute the highest reported two-atom entangle-
ment fidelities for neutral atoms as well as a proof-of-
principle for controlled two-atom entanglement between
∗
These authors contributed equally to this work
†
Permanent address: Institute for Quantum Computing, Univer-
sity of Waterloo, 200 University Ave West, Waterloo, Ontario,
Canada
‡
mendres@caltech.edu
(a)
clock
qubit
detection
3
P
3
S
1
1
S
5s
2
5s5p
5s61s
5p61s
e
R
B
(i)
(ii)
(b)
FIG. 1. Schematic. (a) The relevant level structure (left),
and electronic configuration (right) for strontium
−
88. The
Rydberg-ground state qubit is defined by a metastable ‘clock’
state
|
g
〉
and the 5s61s
3
S
1
m
J
= 0 Rydberg state
|
r
〉
(high-
lighted with a purple box), which we detect by driving to an
auto-ionizing 5p61s state
|
r
∗
〉
. The clock state
|
g
〉
is initial-
ized from the absolute ground state
|
a
〉
. (b) We use atom-by-
atom assembly in optical tweezers to prepare an effectively
non-interacting configuration [(i), blue box and data-points
throughout] and a strongly Rydberg-blockaded pair config-
uration [(ii), red box and data-points throughout] [4]. The
blockade radius
R
B
, where two-atom excitation is suppressed,
is indicated by a dashed circle. Throughout, purple and black
circles indicate
|
r
〉
and
|
g
〉
atoms, respectively. Averaged flu-
orescence images of atoms in configurations (i) and (ii) are
shown.
AEAs. We further demonstrate a high-fidelity entangle-
ment operation with optical traps kept on, an important
step for gate-based quantum computing [1]. As detailed
in the outlook section, our results open up a host of new
opportunities for quantum metrology and computing as
well as for optical trapping of ions.
Our experimental system [23, 33, 34] combines various
novel key elements [35]: First, we implement atom-by-
atom assembly in reconfigurable tweezer arrays [14, 15]
for AEAs [Fig. 1(b)]. Second, we sidestep the typical
arXiv:2001.04455v1 [physics.atom-ph] 13 Jan 2020
2
Table I. Uncorrected and SPAM-corrected fidelities for single-
atom and Rydberg-blockaded pulses. The ‘T’ indicates set-
tings where the tweezers are on during Rydberg excitation.
Quantity
Uncorrected SPAM-corrected
Single-atom
π
-pulse
0
.
9951(9)
0
.
9967(9)
Single-atom 2
π
-pulse
0
.
9951(9)
0
.
998(1)
Blockaded
π
-pulse
0
.
992(2)
0
.
996(2)
Blockaded 2
π
-pulse
0
.
992(2)
0
.
999(2)
Blockaded
π
-pulse, T
0
.
992(2)
0
.
996(2)
Blockaded 2
π
-pulse, T
0
.
987(2)
0
.
994(3)
Bell state fidelity
≥
0
.
983(2)
≥
0
.
995(3)
Bell state fidelity, T
≥
0
.
978(2)
≥
0
.
990(3)
protocol for two-photon excitation to S-series Rydberg
states, which requires significantly higher laser power to
suppress intermediate state scattering, by transferring
atoms to the long-lived
3
P
0
clock state
|
g
〉
[25, 33–36].
We treat
|
g
〉
as an effective ground state from which we
apply single-photon excitation to a
3
S
1
Rydberg state
|
r
〉
[26]. Third, instead of relying on loss through tweezer
anti-trapping as in alkali systems, we employ a rapid
auto-ionization scheme for Rydberg state detection. In
contrast to earlier implementations of auto-ionization de-
tection in bulk gases [37, 38], we image remaining neutral
atoms [33] instead of detecting charged particles.
More generally, our findings improve the outlook
for Rydberg-based quantum computing [1], optimiza-
tion [39], and simulation [2–4, 40, 41]. These applica-
tions all rely on high fidelities for preparation, detection,
single-atom operations, and entanglement generation for
which we briefly summarize our results: We obtain a state
preparation fidelity of 0
.
997(1) through a combination
of coherent and incoherent transfer [35]. The new auto-
ionization scheme markedly improves the Rydberg state
detection fidelity to 0
.
9963
−
0
.
9996 [8, 11]. We also push
the limits of single and two-qubit operations in ground-
to Rydberg-state transitions [3, 8, 9, 11]. For example,
we find
π
-pulse fidelities of 0
.
9951(9) without correct-
ing for state preparation and measurement (SPAM) and
0
.
9967(9) if SPAM correction is applied [35]. Finally, us-
ing a conservative lower-bound procedure, we observe a
two-qubit entangled Bell state fidelity of
≥
0
.
983(2) and
≥
0
.
995(3) without and with SPAM correction, respec-
tively. We note that all values are obtained on average
and for parallel operation in arrays of 14 atoms or 10
pairs for the non-interacting or pair-interacting case, re-
spectively.
We begin by analyzing short-time Rabi oscillations be-
tween
|
g
〉
and
|
r
〉
[Fig. 2(a)] and the auto-ionization
detection scheme [Fig. 2(b)] in an essentially non-
interacting atomic configuration [(i) in Fig. 1(b)]. To de-
tect atoms in
|
r
〉
we excite the core valence-electron from
a 5s to a 5p level [35], which then rapidly auto-ionizes the
Rydberg electron [inset of Fig. 2(b)]. The ionized atoms
e
5p61s
5s61s
Time (ns)
Probability (P
1
)
0
20
40
60
80
100
120
0.0
0.2
0.4
0.6
0.8
1.0
(b)
Time (ns)
Probability (P
1
)
0
50
100
150
0.0
0.2
0.4
0.6
0.8
1.0
(a)
FIG. 2. Rabi oscillations and auto-ionization. (a) Array-
averaged probability,
P
1
, of detecting an atom after a resonant
Rydberg pulse and subsequent auto-ionization as a function of
Rydberg pulse time, showing high-contrast Rabi oscillations
with frequency Ω
R
= 2
π
×
6
.
80(2) MHz. The auto-ionization
pulse time is fixed to 5
μ
s. The points are uncorrected data.
(b)
P
1
as a function of auto-ionization pulse time at a fixed
Rydberg pulse time of 70 ns corresponding to a
π
-pulse (fol-
lowed by a second
π
-pulse). Inset: illustration of the auto-
ionization process.
are dark to subsequent detection of atoms in
|
g
〉
[33],
providing the means to distinguish ground and Rydberg
atoms.
We use a
|
g
〉↔|
r
〉
Rabi frequency of Ω
R
≈
2
π
×
6
−
7
MHz throughout, and observe Rabi oscillations with high
contrast at a fixed auto-ionization pulse length [Fig. 2(a),
Table I]. To quantify the auto-ionization detection, we
perform a
π
-pulse on
|
g
〉 ↔ |
r
〉
, then apply an auto-
ionization pulse for a variable duration [Fig. 2(b)], and
then perform a second
π
-pulse on
|
g
〉 ↔ |
r
〉
before mea-
surement. The detected population decreases to zero with
a 1
/e
time of
τ
A
= 35(1) ns [35]. We can compare
τ
A
to
the lifetime of
|
r
〉
, which is estimated to be
τ
|
r
〉
≈
80
μ
s [42], placing an upper bound on the
|
r
〉
-state detec-
tion efficiency of 0
.
9996(1). A lower bound comes from
the measured
π
-pulse fidelity of 0
.
9963(9) corrected for
preparation and ground state detection errors [35]. These
limits can be increased with higher laser power and faster
switching [35].
To probe our longer-time coherence, we drive the Ryd-
berg transition for as long as 7
μ
s [Fig. 3(a)]. The decay
of the contrast on longer timescales is well modeled by
a Gaussian profile of the form
C
(
t
) =
C
0
exp(
−
t
2
/τ
2
C
).
We find that
τ
C
≈
7
μ
s is consistent with our data, and
corresponds to a 1
/e
coherence of
≈
42 cycles. To our
knowledge, this is largest number of coherent ground-to-
Rydberg cycles that has been published to date [8, 10].
Limitations to short- and long-term coherence are dis-
3
(b)
Time (
μs)
Probability (1-P
{01}
)
0
1
2
3
4
5
6
0.0
0.5
1.0
(a)
Probability (P
1
)
0.0
0.5
1.0
FIG. 3. Long-time Rabi oscillations for single and blockaded atoms. (a) Array-averaged Rabi oscillations for the non-interacting
configuration (i), depicted by the inset. We operate with Ω
R
= 2
π
×
6
.
0 MHz. By fitting with a Gaussian profile, we find a 1
/e
coherence of
≈
42 cycles. (b) Same as in (a) but for the blockaded configuration (ii), depicted by the inset. We plot 1
−
P
{
01
}
,
where
P
{
01
}
is the array-averaged symmetrized probability [35] of detecting one atom of an initial pair (and not both). We
observe a blockade-enhanced Rabi frequency of
̃
Ω
R
= 2
π
×
8
.
5 MHz. We find a 1
/e
coherence of
≈
60 cycles. The points are
uncorrected data.
cussed and modeled in detail in Ref. [35]. The main
contributing factors are laser intensity and phase noise
(which both can be improved upon with technical up-
grades, such as cavity filtering of phase noise [8]), and
finite Rydberg state lifetime.
We now turn to the pair-interacting configuration [(ii)
in Fig. 1(b)] to study blockaded Rabi oscillations [1, 8].
For an array spacing of 3
.
6
μ
m, we anticipate an interac-
tion shift of
V
B
≈
2
π
×
130 MHz for the
n
= 61 Rydberg
state in the
3
S
1
series [42]. In this configuration, simulta-
neous Rydberg excitation of closely-spaced neighbors is
strongly suppressed, and an oscillation between
|
g
〉
and
the entangled
W
-Bell-state
|
W
〉
= (
|
gr
〉
+
e
iφ
|
rg
〉
)
/
√
2
is predicted with a Rabi frequency enhanced by a factor
of
√
2 [1], as observed in our data. We show our results
for long-term coherent oscillations in Fig. 3(b) and find a
1
/e
coherence time corresponding to
≈
60 cycles. Results
for short-term oscillations are shown in Fig. 4(a) and the
fidelity values are summarized in Table I.
We provide a lower bound for the Bell state fidelity
based on measured populations at the (blockaded)
π
-
time and a lower bound on the purity
P
of the two-atom
state [35]. The latter is obtained by measuring the atomic
populations at the (blockaded) 2
π
time, under the as-
sumption that the purity does not increase between the
π
and the 2
π
time. For a detailed discussion and analy-
sis of this bound and the validity of the underlying as-
sumptions, see Ref. [35]. With this approach, we find un-
corrected and SPAM-corrected lower bounds on the Bell
state fidelity of 0
.
983(2) and 0
.
995(3), respectively [Ta-
ble I].
We note that all preceding results were obtained with
the tweezers switched
off
during Rydberg excitation. The
potential application of Rydberg gates to large circuit
depth quantum computers motivates the study of block-
ade oscillations with the tweezers
on
. In particular, we
foresee challenges for sequential gate-based platforms if
tweezers must be turned off during each operation. In
general, the prospects for quickly turning off individual
tweezers while not perturbing the other atoms in the ar-
ray are unclear, especially in two dimensions. In systems
implementing gates between the absolute ground and
clock states for example, blinking traps on and off will
eventually lead to heating and loss, ultimately limiting
the number of possible operations. To remedy this prob-
lem, repulsive traps such as interferometrically-generated
bottles [43] or repulsive lattices [9] have been used in lieu
of standard optical tweezer arrays [14, 15].
Despite finding that our Rydberg state is anti-trapped
(with a magnitude roughly equal to that of the ground
state trapping) at our clock-magic wavelength of
λ
T
=
813
.
4 nm [35], we observe high-fidelity entanglement even
when the tweezers remain on during Rydberg interro-
gation. Certain factors make this situation favorable for
alkaline-earth atoms. One is the ability to reach lower
temperatures using narrow-line cooling, which suppresses
thermal dephasing due to trap light shifts. Furthermore,
a lower temperature allows for ramping down of tweezers
to shallower depths before atoms are lost, further allevi-
ating dephasing. Finally, access to higher Rabi frequen-
cies provides faster and less light-shift-sensitive entan-
gling operations.
We study short-time blockaded Rabi oscillations both
with the tweezers switched
off
[Fig. 4(a)] and left
on
[Fig. 4(b)]. We find similar fidelities for the
π
- and 2
π
-
pulses in both cases [Table I]. Further, we estimate a
4
Time (ns)
Probability (1-P
{01}
)
0
50
100
150
0.0
0.2
0.4
0.6
0.8
1.0
Probability (1-P
{01}
)
0.0
0.2
0.4
0.6
0.8
1.0
(a)
(b)
FIG. 4. Short-time Rydberg-blockaded Rabi oscillations with
tweezers
off
and
on
. (a) Short-time Rabi-oscillations for the
blockade configuration (ii) with the traps
off
, depicted by the
inset. (b) Same as in (a) but with tweezers
on
during Rydberg
interrogation with a
|
g
〉
-state depth of U
/h
≈
0
.
94 MHz. The
points are uncorrected data, and the blockade-enhanced Rabi
frequency is
̃
Ω
R
= 2
π
×
9
.
8 MHz.
lower bound for the Bell state fidelity in the tweezer
on
case, and find uncorrected and corrected values of
≥
0
.
978(2) and
≥
0
.
990(3), respectively. We expect fur-
ther improvements in shorter-wavelength tweezers for
which the Rydberg states of AEAs are trapped [44], and
our observations show promise for Rydberg-based quan-
tum computing in a standard tweezer array [14, 15].
Our work bridges the gap between the fields of
Rydberg atom arrays and optical clocks [25], opening
the door to Rydberg-based quantum-enhanced metrol-
ogy [26, 27], including the programmable generation
of spin-squeezed states [28] in recently demonstrated
tweezer clocks [34, 36], and quantum clock networks [29].
Further, the demonstrated entangling operations provide
a mechanism for two-qubit gates in AEA-based quantum
computation and simulation architectures leveraging
optical and nuclear qubits [30–32]. More generally,
the observed entanglement fidelities could enable gate
fidelities for long-lived ground states approaching fault-
tolerant error correction thresholds [45]. In addition, the
high Rydberg- and ground-state detection-fidelities could
play an important role in applications based on sam-
pling from bit-string probability distributions [39, 46].
Finally, by auto-ionizing the Rydberg electron with
high fidelity and noting that we expect the remaining
ion to stay trapped, we have found a potential new
approach to the optical trapping of ions [47, 48] in up to
three dimensional arrays [16, 49]. Such a platform has
been proposed as a route to ion-based quantum com-
puting [50] as well as for hybrid atom-ion systems [51–53].
We acknowledge discussions with Chris Greene and
Harry Levine as well as funding provided by the In-
stitute for Quantum Information and Matter, an NSF
Physics Frontiers Center (NSF Grant PHY-1733907),
the NSF CAREER award (1753386), the AFOSR YIP
(FA9550-19-1-0044), the Sloan Foundation, and Fred
Blum. Research was carried out at the Jet Propulsion
Laboratory and the California Institute of Technology
under a contract with the National Aeronautics and
Space Administration and funded through the Presi-
dents and Directors Research and Development Fund
(PDRDF). JPC acknowledges support from the PMA
Prize postdoctoral fellowship, and JC acknowledges
support from the IQIM postdoctoral fellowship. HP
acknowleges support by the Gordon and Betty Moore
Foundation. AK acknowledges funding from the Larson
SURF fellowship, Caltech Student-Faculty Programs.
Note added.—
Recently, we became aware of work in
ytterbium tweezer arrays demonstrating trapping of
Rydberg states [54].
Supplementary Materials
APPENDIX A: Summary of experiment
We briefly summarize the relevant features of our
88
Sr
experiment [23, 33, 34]. We employ a one-dimensional ar-
ray of 43 tweezers spaced by 3.6
μ
m. Atoms are cooled
close to the tranverse motional ground state using nar-
row line cooling [33, 34, 36], with an average occupation
number of ̄
n
r
≈
0
.
3 (
T
r
≈
2
.
5
μ
K), in tweezers of ground-
state depth U
0
≈
k
B
×
450
μ
K
≈
h
×
9
.
4 MHz with a
radial trapping frequency of
ω
r
≈
2
π
×
78 kHz.
For state preparation [Fig. 1(a)], we drive from the 5s
2
1
S
0
absolute ground state (labeled
|
a
〉
) to the 5s5p
3
P
0
clock state (labeled
|
g
〉
) with a narrow-line laser [34],
reaching Rabi frequencies of Ω
C
≈
2
π
×
3
.
5 kHz in a mag-
netic field of
≈
710 G [55, 56] (otherwise set to
≈
71 G
for the entire experiment). We populate
|
g
〉
with a
π
-
pulse reaching a loss-corrected fidelity of 0
.
986(2), which
we supplement with incoherent pumping (after adiabati-
cally ramping down the tweezer depth to U
F
= U
0
/
10) to
obtain a clock state population without and with loss cor-
rection of 0
.
997(1) and 0
.
998(1), respectively. This value
is similar to, or higher than, the state preparation fideli-
ties achieved with alkali atoms [9–11, 57].
We treat the long-lived state
|
g
〉
as a new ground state,
from which we drive to the 5s61s
3
S
1
,
m
J
= 0 Ryd-
berg state (labeled
|
r
〉
). The
|
g
〉 ↔ |
r
〉
Rydberg transi-
tion occurs at a wavelength of
λ
R
= 316
.
7 nm and we
use a 1
/e
2
beam radius of 18(1)
μ
m. We readily achieve
a
|
g
〉 ↔ |
r
〉
Rabi frequency of Ω
R
≈
2
π
×
6
−
7 MHz,
corresponding to
≈
30 mW, and up to Ω
R
≈
2
π
×
13
MHz with full optimization of the laser system and beam
5
path. To detect atoms in
|
r
〉
we drive the strong tran-
sition to 5p
3
/
2
61s
1
/
2
(
J
= 1
,m
J
=
±
1), labelled
|
r
∗
〉
.
This transition excites the core ion, which then rapidly
auto-ionizes the Rydberg electron. The ionized atoms are
dark to subsequent detection of atoms in
|
g
〉
with the
high-fidelity scheme described in Ref. [33], providing the
means to distinguish ground and Rydberg atoms. We
switch off the ramped-down tweezers during the Ryd-
berg pulse [3, 4], after which we apply an auto-ionization
pulse while rapidly increasing the depth back to U
0
for
subsequent read-out.
The Rydberg and clock laser beams are linearly polar-
ized along the magnetic field axis, and the auto-ionization
beam is linearly polarized perpendicular to the magnetic
field axis. Accordingly, we excite to auto-ionizing states
with
m
J
=
±
1. The tweezers are linearly polarized along
the axis of propagation of the Rydberg, clock, and auto-
ionization beams – perpendicular to the magnetic field
axis.
APPENDIX B: State preparation
The ground state
|
g
〉
of our Rydberg qubit is the 5s5p
3
P
0
metastable clock state of
88
Sr. We populate this state
in two stages: first, most atoms are transferred via a co-
herent
π
-pulse on the clock transition. Thereafter, any
remaining population is transferred via incoherent pump-
ing.
In our regime where the Rabi frequency of the clock
transition (Ω
C
≈
2
π
×
3
.
5 kHz) is significantly smaller
than the trapping frequency (
ω
r
≈
2
π
×
78 kHz), coher-
ent driving is preferable to incoherent pumping because
it preserves the motional state of an atom, i.e., it does
not cause heating. However, atomic temperature, trap
frequency, trap depth, and beam alignment contribute to
the transfer infidelity of coherent driving. Although we
drive the clock transition on the motional carrier in the
sideband resolved regime, thermal dephasing still plays
an important role. Particularly, each motional state has
a distinct Rabi frequency, a thermal ensemble of which
leads to dephasing [34]. This thermal dephasing is less
severe at higher trapping frequencies; however, this can
only be achieved in our system by using deeper traps,
which would also eventually limit transfer fidelity because
of higher rates of Raman scattering out of the clock state.
We therefore perform coherent transfer initially in deeper
traps (
≈
450
μ
K), followed immediately by an adiabatic
rampdown to one-tenth of that depth. Finally, precise
alignment of the clock beam to the tight, transverse axis
of the tweezer is important to ensure that no coupling
exists to axial motion, which has a much lower trap fre-
quency and thus suffers more thermal dephasing than the
transverse direction.
The remaining population is transferred by simultane-
ous, incoherent driving of the 5s
2 1
S
0
↔
5s5p
3
P
1
, 5s5p
3
P
1
↔
5s6s
3
S
1
, and 5s5p
3
P
2
↔
5s6s
3
S
1
transitions
for 1 ms. This pumping scheme has the clock state as a
unique dark state via the decay of 5s6s
3
S
1
to the clock
state and is in general more robust than coherent driv-
ing. However, due to photon recoil, differential trapping,
and an unfavorable branching ratio of 5s6s
3
S
1
to the
clock state (requiring many absorption and emission cy-
cles), this process causes significant heating, making it
unfavorable as compared to coherent driving. Therefore,
we only use this method as a secondary step to transfer
atoms left behind by the coherent drive.
We measure the fidelity of our state transfer by apply-
ing a 750
μ
s pulse of intense light resonant with the
1
S
0
↔
1
P
1
transition immediately after state transfer. The large
recoil force of this pulse rapidly pushes out atoms in
1
S
0
with a fidelity of
>
0
.
9999 while leaving atoms in the
clock state intact. Upon repumping the clock state back
into our imaging cycle and imaging the remaining atoms,
we obtain a measure of the fraction of atoms that were
successfully transferred to the clock state. With coher-
ent driving alone, we measure a state transfer fidelity of
0.986(2), while adding incoherent pumping increases this
value to 0.998(1). Both of these values are corrected for
loss to quantify state transfer in isolation; however, loss
also contributes to infidelity of the overall state
prepara-
tion
. Taking loss into account, our overall state prepara-
tion fidelity with both coherent driving and incoherent
pumping is
F
SP
= 0
.
997(1).
APPENDIX C: Auto-ionization and Rydberg state
detection fidelity
The auto-ionization beam is resonant with the Sr
+
ionic
transition
2
S
1
/
2
↔
2
P
3
/
2
at
λ
A
= 407
.
6 nm. The 1
/e
2
beam waist radius is
w
A
o
= 16(1)
μ
m with power
P
A
=
2
.
8(4) mW, from which we estimate a Rabi frequency of
Ω
A
≈
2
π
×
3 GHz.
To quantify the Rydberg state detection fidelity of our
auto-ionization scheme, we compare the observed auto-
ionization loss 1
/e
timescale of
τ
A
= 35(1) ns to the
expected lifetime of
|
r
〉
, which is
τ
|
r
〉
≈
80
μ
s [42]. That
is, we compute the probability that an atom in the Ryd-
berg state is auto-ionized before it decays away from the
Rydberg state. This estimate places an upper bound on
the detection fidelity of
|
r
〉
to be 0
.
9996(1), where the
uncertainty is dominated by an assumed uncertainty of
±
20
μ
s in
τ
|
r
〉
. Note that when the auto-ionization pulse
is not applied, there is still a residual detection fidelity of
|
r
〉
of 0
.
873(4) due to anti-trapping of
|
r
〉
in the tweezer
(this value is smaller than the previously reported
<
0
.
98
for alkalis in part because the atoms are colder here than
in previous work [8]). A lower bound on our detection
fidelity is given by the measured
π
-pulse fidelity after
correcting for errors in preparation and ground state de-
tection, which gives 0
.
9963(9).
Although the auto-ionization rate of
|
r
∗
〉
is Γ
A
∼
100’s
of GHz [37, 38] and we drive the ion core transition with
a Rabi frequency Ω
A
≈
2
π
×
3 GHz, the
|
r
〉↔|
r
∗
〉
transi-
tion is inhibited by the continuous quantum Zeno mech-
anism [58, 59]. Accordingly, the effective auto-ionization
rate of
|
r
〉
is given by Γ
eff
A
≈
Ω
2
A
/
Γ
A
. This is in quali-
6
tative agreement with the fact that our measured auto-
ionization loss timescale continues to increase with beam
intensity, despite the ionic transition being driven far be-
yond its bare saturation. Furthermore, the finite rise time
of the acousto-optic modulator (AOM) that we use for
switching the auto-ionization beam is a limiting factor in
achieving faster auto-ionization rates. Therefore, detec-
tion fidelity can be increased further with higher beam
intensity as well as faster beam switching.
APPENDIX D: State preparation and measurement
(SPAM) correction
Our detection basis is binary between zero detected
atoms,
0
, and one detected atom,
1
, for each site in the ar-
ray. Two factors affect our ability to correctly determine
the occupation of a tweezer: the false positive avoidance
probability,
F
0
, and the true positive detection proba-
bility,
F
1
. By fitting the bimodal histogram of detected
photons for each tweezer and using a binary detection
threshold, we determine both
F
0
and
F
1
, with their er-
rors given by averaging over the whole array [33]. We
measure an uncorrected imaging survival probability
S
0
(with no clock or Rydberg excitation) by measuring how
many atoms detected in an image are also detected in a
subsequent image. We also measure an uncorrected clock
state transfer fidelity
K
0
by a technique described in Ap-
pendix B.
With
F
0
,
F
1
,
S
0
, and
K
0
, we compute a detection-
corrected value for survival probability with no clock
or Rydberg excitation,
S
, and a detection and survival-
corrected value for clock state transfer fidelity,
K
, via
Table II. Probabilities entering into SPAM correction calcu-
lations.
Probability
Symbol
Value
False positive avoidance
F
0
0
.
99997(5)
Atom detection
F
1
0
.
9988(7)
Bare survival
S
0
0
.
9979(3)
Corrected survival
S
0
.
9991(7)
Bare
|
g
〉
transfer
K
0
0
.
997(1)
Corrected
|
g
〉
transfer
K
0
.
998(1)
Loss during
|
g
〉
transfer
L
0
.
0008(8)
Rydberg state detection
D
0
.
9996(1)
Table III. Populations in the single-atom four-state basis.
Note that the sum of these populations equals unity for any
P
′
r
, where
P
′
r
is the SPAM-corrected
|
r
〉
population.
State
Symbol
Value
Null (atom lost)
P
l
1
−
S
+
SL
1
S
0
(absolute ground state)
P
a
S
(1
−
L
−
K
)
3
P
0
(clock)
P
g
KS
(1
−
P
′
r
)
3
S
1
(Rydberg state)
P
r
KSP
′
r
S
=
S
0
+
F
0
−
1
F
0
+
F
1
−
1
,
(D1)
K
=
K
0
+
F
0
−
1
S
0
+
F
0
−
1
(1
−
C
)
.
(D2)
Here
C
= 0
.
00104(1) is the probability of clock state de-
cay before the Rydberg pulse due to Raman scattering
from trapping light, estimated by a measure of the life-
time in the clock state at a particular tweezer depth [33]
and the time delay between our state preparation and
Rydberg interrogation. The total clock state preparation
fidelity is then given by
F
SP
=
KS
= 0
.
997(1). While a
fraction of the atoms that are unsuccessfully transferred
to the clock state end up in the absolute ground state
|
a
〉
, some atoms are instead lost entirely due to heating
out of the trap during incoherent pumping. We denote
this loss probability by
L
, such that the probability of
ending up in
|
a
〉
is 1
−
(
K
+
L
). Note that we assume all
loss captured by
S
occurs before the start of the Rydberg
pulse, while loss during the read-out image that leads to
detection infidelity is accounted for by
F
1
.
When considering detection of the Rydberg state, a
further detection fidelity,
D
, is introduced which charac-
terizes the fidelity with which an atom that was in the
Rydberg state is successfully transferred to a dark ionic
state, primarily limited by the finite Rydberg state life-
time. For all calculations we use the theoretical upper
bound of
D
, such as to be maximally conservative in our
SPAM-correction.
To account for imperfect rearrangement, we post-select
our data (both uncorrected and SPAM-corrected) to ac-
Table IV. Populations in the two-atom 16-state basis. Note
that the sum of these populations equals unity for any pulse
fidelities
P
′
rg
,
P
′
gr
, and
P
′
rr
, where
P
′
rg
and
P
′
gr
are the SPAM-
corrected populations of states with one Rydberg excitation,
and
P
′
rr
is the similarly corrected population of the
P
rr
state.
For cases where the initial state
|
gg
〉
is not properly pre-
pared, the factor of
P
c
=
P
′
r
cos
2
(
A
/
(2
√
2)) captures the non-
blockaded Rydberg Rabi excitation, where
A
is either
π
or
2
π
. Note that terms expressed in
{
,
}
are assumed to have a
symmetric partner, e.g.
P
al
≡
P
la
.
States
Symbol
Value
(Lost, Lost)
P
ll
(1
−
S
+
SL
)
2
{
Lost,
1
S
0
}
P
la
(1
−
S
+
SL
)
S
(1
−
L
−
K
)
{
Lost,
3
P
0
}
P
lg
(1
−
S
+
SL
)
SK
(1
−
P
c
)
{
Lost,
3
S
1
}
P
lr
(1
−
S
+
SL
)
SKP
c
(
1
S
0
,
1
S
0
)
P
aa
S
2
(1
−
L
−
K
)
2
{
1
S
0
,
3
P
0
}
P
ag
S
(1
−
L
−
K
)
SK
(1
−
P
c
)
{
1
S
0
,
3
S
1
}
P
ar
S
(1
−
L
−
K
)
SKP
c
(
3
P
0
,
3
P
0
)
P
gg
K
2
S
2
(1
−
P
′
rg
−
P
′
gr
−
P
′
rr
)
(
3
P
0
,
3
S
1
)
P
gr
K
2
S
2
P
′
gr
(
3
S
1
,
3
P
0
)
P
rg
K
2
S
2
P
′
rg
(
3
S
1
,
3
S
1
)
P
rr
K
2
S
2
P
′
rr
7
count only for instances where a target number of atoms
(either one atom or two neighboring atoms in the case
of blockaded oscillations) have no further neighboring
atoms detected within a two-tweezer distance in any of
the images following rearrangement. We do not correct
for false negative detections of neighboring atoms that
may increase infidelity due to interactions. False nega-
tives of such neighbors are rare because of an already
small rate of false negative detection, but are even fur-
ther suppressed by rearrangement and the fact that we
use two images to post-select on such events
1. Correcting the single-atom pulse fidelities
Having defined the relevant states, their preparation
probabilities, and detection fidelities, we now turn to con-
verting the detection basis of
0
and
1
into the basis of
|
g
〉
and
|
r
〉
to perform SPAM-correction. The bare, measured
value
P
1
shown in Figs. 2 and 3 of the main text gives the
probability of measuring
1
in the detection basis, which
corresponds to positively detecting the combined popula-
tion in
|
a
〉
and
|
g
〉
plus the probability of a false positive
when no neutral atom is present. This can be quantified
by writing
P
1
in terms of the values in Tables II and III:
P
1
= (
P
a
+
P
g
+
P
r
(1
−
D
))
F
1
+ (
P
l
+
P
r
D
)(1
−
F
0
)
.
(D3)
We put in all the quantities from Table III and solve for
P
′
r
, the desired quantity, to obtain:
P
′
r
=
SF
1
+ (1
−
S
)(1
−
F
0
)
−
LS
(
F
0
+
F
1
−
1)
−
P
1
KSD
(
F
0
+
F
1
−
1)
.
(D4)
For the single-atom short-time Rabi oscillations shown
in Fig. 2 of the main text, we observe the bare values
of
P
1
(
π
) = 0
.
0049(9) and
P
1
(2
π
) = 0
.
9951(9), yielding
pulse fidelities of
F
SPAM
(
π
) =
P
′
r
(
π
) = 0
.
9967(9) and
F
SPAM
(2
π
) = 1
−
P
′
r
(2
π
) = 0
.
998(1), respectively.
2. Correcting the two-atom pulse fidelities
For the atomic configuration (ii), there are 16 possible
states for each atom pair. Similarly to Table III, we can
write populations of each of these states in terms of the
survival and transfer fidelities in Table II, as shown in
Table IV.
We now write the experimentally measured quantities
P
10
,P
00
, and
P
11
in terms of the values in Tables II
and IV. For notational simplicity we define
̄
F
0
≡
(1
−
F
0
),
and similarly for
F
1
and
D
:
P
10
=
P
ll
(
̄
F
0
F
0
)
+
P
la
(
̄
F
0
̄
F
1
)
+
P
al
(
F
1
F
0
)
+
P
lg
(
̄
F
0
̄
F
1
)
+
P
gl
(
F
1
F
0
)
+
P
lr
(
̄
F
0
F
0
D
+
̄
F
0
̄
D
̄
F
1
)
+
P
rl
(
̄
F
0
DF
0
+
F
1
̄
DF
0
)
+
P
aa
(
F
1
̄
F
1
)
+
P
ag
(
F
1
̄
F
1
)
+
P
ga
(
F
1
̄
F
1
)
+
P
ar
(
F
1
DF
0
+
F
1
̄
D
̄
F
1
)
+
P
ra
(
F
1
̄
D
̄
F
1
+
̄
F
0
D
̄
F
1
)
+
P
gg
(
F
1
̄
F
1
)
+
P
gr
(
F
1
DF
0
+
F
1
̄
D
̄
F
1
)
+
P
rg
(
F
1
̄
D
̄
F
1
+
̄
F
0
D
̄
F
1
)
+
P
rr
(
F
1
̄
DF
0
D
+
̄
F
0
D
̄
F
1
̄
D
+
̄
F
0
F
0
D
2
+
F
1
̄
F
1
̄
D
2
)
,
(D5)
P
00
=
P
ll
(
F
2
0
)
+
P
la
(
F
0
̄
F
1
)
+
P
al
(
̄
F
1
F
0
)
+
P
lg
(
F
0
̄
F
1
)
+
P
gl
(
̄
F
1
F
0
)
+
P
lr
(
F
2
0
D
+
F
0
̄
F
1
̄
D
)
+
P
rl
(
F
2
0
D
+
̄
F
1
̄
DF
0
)
+
P
aa
(
̄
F
1
2
)
+
P
ag
(
̄
F
1
2
)
+
P
ga
(
̄
F
1
2
)
+
P
ar
(
̄
F
1
F
0
D
+
̄
F
1
2
̄
D
)
+
P
ra
(
̄
F
1
2
̄
D
+
F
0
D
̄
F
1
)
+
P
gg
(
̄
F
1
2
)
+
P
gr
(
̄
F
1
F
0
D
+
̄
F
1
2
̄
D
)
+
P
rg
(
̄
F
1
2
̄
D
+
F
0
D
̄
F
1
)
+
P
rr
(
̄
F
1
2
̄
D
2
+
F
0
D
̄
F
1
̄
D
+
F
2
0
D
2
+
̄
F
1
̄
DF
0
D
)
,
(D6)
8
P
11
=
P
ll
(
̄
F
0
2
)
+
P
la
(
̄
F
0
F
1
)
+
P
al
(
F
1
̄
F
0
)
+
P
lg
(
̄
F
0
F
1
)
+
P
gl
(
F
1
̄
F
0
)
+
P
lr
(
̄
F
0
2
D
+
̄
F
0
F
1
̄
D
)
+
P
rl
(
̄
F
0
2
D
+
F
1
̄
D
̄
F
0
)
+
P
aa
(
F
2
1
)
+
P
ag
(
F
2
1
)
+
P
ga
(
F
2
1
)
+
P
ar
(
F
1
̄
F
0
D
+
F
2
1
̄
D
)
+
P
ra
(
F
2
1
̄
D
+
̄
F
0
DF
1
)
+
P
gg
(
F
2
1
)
+
P
gr
(
F
1
̄
F
0
D
+
F
2
1
̄
D
)
+
P
rg
(
F
2
1
̄
D
+
̄
F
0
DF
1
)
+
P
rr
(
F
2
1
̄
D
+
̄
F
0
DF
1
̄
D
+
̄
F
0
2
D
2
̄
D
+
F
1
̄
D
̄
F
0
D
)
.
(D7)
Note that
P
01
= 1
−
P
10
−
P
00
−
P
11
. Thus, with the
three above equations, we can solve for
P
′
rg
,
P
′
gr
, and
P
′
rr
in terms of the measured
P
00
,
P
{
10
}
,
P
[10]
and
P
11
,
where
P
{
10
}
=
P
10
+
P
01
and
P
[10]
=
P
10
−
P
01
. We
perform this analysis on the measurements both with the
traps off and with the traps on (denoted with a T, as
in Table I). These experimentally measured values are
shown in Table V, with which we compute the SPAM-
corrected values shown in Table I of the main text.
Table V. Experimentally measured (uncorrected) values used
to calculate
P
′
rg
,
P
′
gr
, and
P
′
rr
at both the
π
- and 2
π
-times.
The ‘T’ superscript indicates the values for which the traps
were on.
Variable
Value
P
{
10
}
(
π
)
0.992(2)
P
[01]
(
π
)
0.01(1)
P
00
(
π
)
0.0032(7)
P
11
(2
π
)
0.992(2)
P
[01]
(2
π
)
0.004(2)
P
00
(2
π
)
0.0036(7)
P
T
{
10
}
(
π
)
0.992(2)
P
T
[01]
(
π
)
0.004(10)
P
T
00
(
π
)
0.0032(7)
P
T
11
(2
π
)
0.987(2)
P
T
[01]
(2
π
)
-0.003(2)
P
T
00
(2
π
)
0.0030(6)
APPENDIX E: Bell state fidelity
1. Bounding the Bell state fidelity
Characterizing the state of a quantum system is of fun-
damental importance in quantum information science.
Canonical tomographic methods addressing this task re-
quire a measurement of a complete basis set of opera-
tors. Such measurements are often expensive or not ac-
cessible. More economic approaches can be employed to
assess the overlap with a given target state. For exam-
ple the overlap of a two-qubit state with a Bell state
is routinely determined by measuring the populations in
the four computational basis states (yielding the diago-
nal elements of the density operator), in addition with a
measurement that probes off-diagonal elements via par-
ity oscillations [8, 60]. To access the latter it is however
necessary to perform individual, local operations on the
qubits. Here, we present a bound on the Bell state fidelity
that can be accessed with only global control and mea-
surements in the computational basis and elaborate on
the underlying assumptions.
Specifically, we are interested in the overlap
F
of the
experimentally created state
ρ
with a Bell state of the
form
|
W
φ
〉
=
1
√
2
(
|
gr
〉
+
e
iφ
|
rg
〉
). This is defined as
F
= max
φ
〈
W
φ
|
ρ
|
W
φ
〉
=
1
2
(
ρ
gr,gr
+
ρ
rg,rg
+ 2
|
ρ
gr,rg
|
)
.
(E1)
Here we denote matrix elements of a density operator
ρ
in the two-atom atomic basis by
ρ
i,j
=
〈
i
|
ρ
|
j
〉
, with
i,j
∈ {
gg,gr,rg,rr
}
. Clearly a measurement of
F
re-
quires access to the populations in the ground and Ry-
dberg states
ρ
i,i
as well as some of the coherences
ρ
i,j
with
i
6
=
j
. While the former are direct observables (in
particular, we identify
ρ
i,i
with our measured values
P
i
),
the latter are not. We can however bound the fidelity
F
from below via a bound on
|
ρ
gr,rg
|
. Namely, it can be
shown via Cauchy’s inequality
|
ρ
a,b
|
2
≤
ρ
a,a
ρ
b,b
and the
normalization of states
∑
i
ρ
i,i
= 1 that
|
ρ
gr,rg
|
2
≥
1
2
(tr
{
ρ
2
}
−
1) +
ρ
gr,gr
ρ
rg,rg
(E2)
where tr
{
ρ
2
}
=
∑
i,j
|
ρ
i,j
|
2
is the purity.
Evaluating the bound given by equation Eq. (E2) re-
quires access to the purity (or a lower bound thereof).
One can bound the purity from below by the populations
in the ground and Rydberg states as
tr
{
ρ
2
}
≥
∑
i
(
ρ
i,i
)
2
.
(E3)
In general Eq. (E3) is a very weak bound. In particular,
it does not distinguish between a pure Bell state
|
ψ
φ
〉
and the incoherent mixture of the two states
|
gr
〉
and
|
rg
〉
. However, if the state
ρ
is close to one of the four