An Atomic Array Optical Clock with Single-Atom Readout
Ivaylo S. Madjarov,
1
Alexandre Cooper,
1
Adam L. Shaw,
1
Jacob P. Covey,
1
Vladimir Schkolnik,
2
Tai Hyun Yoon,
1,
∗
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
Currently, the most accurate and stable clocks use optical interrogation of either a single ion or
an ensemble of neutral atoms confined in an optical lattice. Here, we demonstrate a new optical
clock system based on an array of individually trapped neutral atoms with single-atom readout,
merging many of the benefits of ion and lattice clocks as well as creating a bridge to recently
developed techniques in quantum simulation and computing with neutral atoms. We evaluate single-
site resolved frequency shifts and short-term stability via self-comparison. Atom-by-atom feedback
control enables direct experimental estimation of laser noise contributions. Results agree well with
an
ab initio
Monte Carlo simulation that incorporates finite temperature, projective read-out, laser
noise, and feedback dynamics. Our approach, based on a tweezer array, also suppresses interaction
shifts while retaining a short dead time, all in a comparatively simple experimental setup suited for
transportable operation. These results establish the foundations for a third optical clock platform
and provide a novel starting point for entanglement-enhanced metrology, quantum clock networks,
and applications in quantum computing and communication with individual neutral atoms that
require optical clock state control.
I. INTRODUCTION
Optical clocks — based on interrogation of ultra-
narrow optical transitions in ions or neutral atoms —
have surpassed traditional microwave clocks in both rel-
ative frequency stability and accuracy [1–4]. They en-
able new experiments for geodesy [2, 5], fundamental
physics [6, 7], and quantum many-body physics [8], in
addition to a prospective redefinition of the SI sec-
ond [9]. In parallel, single-atom detection and control
techniques have propelled quantum simulation and com-
puting applications based on trapped atomic arrays; in
particular, ion traps [10], optical lattices [11], and opti-
cal tweezers [12, 13]. Integrating such techniques into an
optical clock would provide atom-by-atom error evalu-
ation, feedback, and thermometry [14]; facilitate quan-
tum metrology applications, such as quantum-enhanced
clocks [15–18] and clock networks [19]; and enable novel
quantum computation, simulation, and communication
architectures that require optical clock state control com-
bined with single atom trapping [20–22].
As for current optical clock platforms, ion clocks
already incorporate single-particle detection and con-
trol [23], but they typically operate with only a single
ion. Research towards multi-ion clocks is ongoing [24].
Conversely, optical lattice clocks (OLCs) [1, 2, 4] interro-
gate thousands of atoms to improve short-term stability,
but single-atom detection and control remains an out-
standing challenge. An ideal clock system, in this context,
would thus merge the benefits of ion and lattice clocks;
namely, a large array of isolated atoms that can be read
out and controlled individually.
∗
Permanent address: Department of Physics, Korea University,
Seoul 02841, Republic of Korea
†
mendres@caltech.edu
Here we present a prototype of a new optical clock plat-
form based on an atomic array which naturally incorpo-
rates single-atom readout of currently
≈
40 individually
trapped neutral atoms. Specifically, we use a magic wave-
length 81-site tweezer array stochastically filled with sin-
gle strontium-88 (
88
Sr) atoms [25]. Employing a repeti-
tive imaging scheme [25], we stabilize a local oscillator to
the optical clock transition [26, 27] with a low dead time
of
≈
100 ms between clock interrogation blocks.
We utilize single-site and single-atom resolution to eval-
uate the in-loop performance of our clock system in terms
of stability, local frequency shifts, selected systematic ef-
fects, and statistical properties. To this end, we define an
error signal for single tweezers which we use to measure
site-resolved frequency shifts at otherwise fixed parame-
ters. We also evaluate statistical properties of the in-loop
error signal, specifically, the dependence of its variance
on atom number and correlations between even and odd
sites.
We further implement a standard interleaved self-
comparison technique [28, 29] to evaluate systematic fre-
quency shifts with changing external parameters – specif-
ically trap depth and wavelength – and find an oper-
ational magic condition [30–32] where the dependence
on trap depth is minimized. We further demonstrate
a proof-of-principle for extending such self-comparison
techniques to evaluate
single-site-resolved
systematic fre-
quency shifts as a function of a changing external param-
eter.
Using self-comparison, we evaluate the fractional short-
term instability of our clock system to be 2
.
5
×
10
−
15
/
√
τ
.
To compare our experimental results with theory predic-
tions, we develop an
ab initio
Monte Carlo (MC) clock
simulation (Appendix A), which directly incorporates
laser noise, projective readout, finite temperature, and
feedback dynamics, resulting in higher predictive power
compared to traditionally used analytical methods [1].
arXiv:1908.05619v3 [physics.atom-ph] 29 Oct 2019
2
Our experimental data agree quantitatively with this
simulation, indicating that noise processes are well cap-
tured and understood at the level of stability we achieve
here. Based on the MC model, we predict a fractional
instability of (1.9–2.2)
×
10
−
15
/
√
τ
for single clock oper-
ation, which would have shorter dead time than that in
self-comparison.
We further demonstrate a direct evaluation of the
1
/
√
N
A
dependence of clock stability with atom num-
ber
N
A
, on top of a laser noise dominated background,
through an atom-by-atom system-size-selection tech-
nique. This measurement and the MC model strongly
indicate that the instability is limited by the frequency
noise of our local oscillator. We note that the measured
instability is comparable to OLCs using similar trans-
portable laser systems [33].
We note the very recent, complementary results of
Ref. [34] that show seconds-long coherence in a tweezer
array filled with
≈
5
88
Sr atoms using an ultra-low noise
laser without feedback operation. In this and our sys-
tem, a recently developed repetitive interrogation pro-
tocol [25], similar to that used in ion clocks, provides
a short dead time of
≈
100 ms between interrogation
blocks, generally suppressing the impact of laser noise
on stability stemming from the Dick effect [35]. Utiliz-
ing seconds-scale interrogation with such low dead times
combined with the feedback operation and realistic up-
grade to the system size demonstrated here promises a
clock stability that could reach that of state-of-the-art
OLCs [2, 4, 36, 37] in the near-term future, as further
discussed in the outlook section.
Concerning systematic effects, the demonstrated
atomic array clock has intrinsically suppressed interac-
tion and hopping shifts: First, single atom trapping in
tweezers provides immunity to on-site collisions present
in one-dimensional OLCs [38]. While three-dimensional
OLCs [36] also suppress on-site collisions, our approach
retains a short dead time as no evaporative cooling is
needed. Further, the adjustable and significantly larger
interatomic spacing strongly reduces dipolar interac-
tions [39] and hopping effects [40]. We experimentally
study effects from tweezer trapping in Sec. IV and de-
velop a corresponding theoretical model in Appendix E,
but leave a full study of other systematics, not specific
to our platform, and a statement of accuracy to future
work. In this context, we note that our tweezer system is
well-suited for future investigations of black-body radia-
tion shifts via the use of local thermometry with Rydberg
states [14].
The results presented here and in Ref. [34] provide the
foundation for establishing a third optical clock plat-
form promising competitive stability, accuracy, and ro-
bustness, while incorporating single-atom detection and
control techniques in a natural fashion. We expect this
to be a crucial development for applications requiring
advanced control and read-out techniques in many-atom
quantum systems, as discussed in more detail in the out-
look section.
II. FUNCTIONAL PRINCIPLE
The basic functional principle is as follows. We gener-
ate a tweezer array with linear polarization and 2
.
5
μ
m
site-to-site spacing in an ultra-high vacuum glass cell
using an acousto-optic deflector (AOD) and a high-
resolution imaging system (Fig. 1a)[25]. The tweezer ar-
ray wavelength is tuned to a magic trapping configu-
ration close to 813
.
4 nm, as described below. We load
the array from a cold atomic cloud and subsequently in-
duce light-assisted collisions to eliminate higher trap oc-
cupancies [25, 41]. As a result,
≈
40 of the tweezers are
stochastically filled with a single atom. We use a recently
demonstrated narrow-line Sisyphus cooling scheme [25]
to cool the atoms to an average transverse motional oc-
cupation number of ̄
n
≈
0
.
66, measured with clock side-
band spectroscopy (Appendix B 7). The atoms are then
interrogated twice on the clock transition, once below (
A
)
and once above (
B
) resonance, to obtain an error signal
quantifying the frequency offset from the resonance cen-
ter (Fig. 1b,c). We use this error signal to feedback to
a frequency shifter in order to stabilize the frequency of
the interrogation laser — acting as a local oscillator —
to the atomic clock transition. Since our imaging scheme
has a survival fraction of
>
0.998 [25], we perform multi-
ple feedback cycles before reloading the array, each com-
posed of a series of cooling, interrogation, and readout
blocks (Fig. 1d).
For state-resolved readout with single-shot, single-atom
resolution, we use a detection scheme composed of two
high-resolution images for each of the
A
and
B
interro-
gation blocks (Fig. 1e) [25]. A first image determines if
a tweezer is occupied, followed by clock interrogation. A
second image, after interrogation, determines if the atom
has remained in the ground state
|
g
〉
. This yields an in-
stance of an error signal for all tweezers that are occupied
at the beginning of both interrogation blocks, while un-
occupied tweezers are discounted. For occupied tweezers,
we record the
|
g
〉
occupation numbers
s
A,j
=
{
0
,
1
}
and
s
B,j
=
{
0
,
1
}
in the images after interrogation with
A
and
B
, respectively, where
j
is the tweezer index. The
difference
e
j
=
s
A,j
−
s
B,j
defines a single-tweezer error
variable taking on three possible values
e
j
=
{−
1
,
0
,
+1
}
indicating interrogation below, on, or above resonance,
respectively. Note that the average of
e
j
over many inter-
rogations,
〈
e
j
〉
, is simply an estimator for the difference
in transition probability between blocks
A
and
B
.
For feedback to the clock laser,
e
j
is averaged over all
occupied sites in a single
AB
interrogation cycle, yielding
an array-averaged error ̄
e
=
1
N
A
∑
j
e
j
, where the sum
runs over all occupied tweezers and
N
A
is the number of
present atoms. We add ̄
e
times a multiplicative factor to
the frequency shifter, with the magnitude of this factor
optimized to minimize in-loop noise.
III. IN-LOOP SPECTROSCOPIC RESULTS
We begin by describing results for in-loop detection se-
quences. Here, feedback is applied to the clock laser (as
3
-20
-10
0
10
20
Frequency (Hz)
0.0
0.5
1.0
Probability
A
B
-0.8
0.0
0.8
Mean error
1
2
3
4
A
B
-1
+1
0
Error (
e
j
)
0
NaN
NaN
Feedback
f
FIG. 1. Atomic array optical clock. (a) We interrogate
≈
40
88
Sr atoms, trapped in an 81-site tweezer array, on the ultra-
narrow clock transition at 698 nm and use high-resolution fluorescence imaging at 461 nm to detect population changes in
the clock states (labeled
|
g
〉
and
|
e
〉
) with single-atom resolution. This information is processed by a central processing unit
(CPU) and a feedback signal is applied to the clock laser frequency using an acousto-optic modulator (AOM). (b) Tweezer-
averaged probability to remain in
|
g
〉
as a function of frequency offset measured with an in-loop probe sequence (circles). Dashed
horizontal lines indicate state-resolved detection fidelities (Appendix B 5). To generate an error signal, we interrogate twice:
once below (
A
) and once above (
B
) resonance. (c) Tweezer-averaged error signal as a function of frequency offset (circles).
The shaded areas in (b) and (c) show results from MC simulations. (d) Simplified experimental sequence, consisting of tweezer
loading and
N
-times-repeated
AB
feedback blocks followed by an optional probe block, with
N
= 10 throughout. (e) To detect
the clock state population in block
A
, we take a first image before interrogation to identify which tweezers are occupied and
a second image after interrogation to detect which atoms remain in
|
g
〉
(images 1 and 2). The same procedure is repeated for
block
B
(images 3 and 4). We show fluorescence images with identified atoms (circles) (Appendix B 4) and examples of single
tweezer error signals
e
j
.
described before) and probe blocks, for which the interro-
gation frequency is varied, are added after each feedback
cycle. Using a single probe block with an interrogation
time of 110 ms (corresponding to a
π
-pulse on resonance)
shows a nearly Fourier-limited line-shape with full-width
at half-maximum of
≈
7 Hz (Fig. 1b). We also use these
parameters for the feedback interrogation blocks, with
the
A
and
B
interrogation frequencies spaced by a to-
tal of 7
.
6 Hz. Using the same in-loop detection sequence,
we can also directly reveal the shape of the error sig-
nal by using two subsequent probe blocks spaced by this
frequency difference and scanning a common frequency
offset (Fig. 1c). The experimental results are in agree-
ment with MC simulations, which have systematic error
denoted as a shaded area throughout, stemming from
uncertainty in the noise properties of the interrogation
laser (Appendix A 3).
Importantly, these data also exist on the level of indi-
vidual tweezers, both in terms of averages and statisti-
cal fluctuations. As a first example, we show a tweezer-
resolved measurement of the repetition-averaged error
signal
〈
e
j
〉
for all 81 traps (Fig. 2a) as a function of fre-
quency offset.
Fitting the zero-crossings of
〈
e
j
〉
enables us to detect
differences in resonance frequency with sub-Hz resolution
(Fig. 2b). The results show a small gradient across the ar-
ray due to the use of an AOD: tweezers are spaced by 500
kHz in optical frequency, resulting in an approximately
linear variation of the clock transition frequency. This ef-
fect could be avoided by using a spatial light modulator
for tweezer array generation [42]. We note that the total
frequency variation is smaller than the width of our in-
terrogation signal. Such “sub-bandwidth” gradients can
still lead to noise through stochastic occupation of sites
with slightly different frequencies; in our case, we pre-
dict an effect at the 10
−
17
level. We propose a method
to eliminate this type of noise in future clock iterations
with a local feedback correction factor in Appendix D 3.
Before moving on, we note that
e
j
is a random variable
with a ternary probability distribution (Fig. 2c) defined
for each tweezer. The results in Fig. 2a are the mean of
this distribution as a function of frequency offset. In ad-
dition to such averages, having a fully site-resolved signal
enables valuable statistical analysis. As an example, we
extract the variance of ̄
e
,
σ
2
̄
e
, for an in-loop probe se-
quence where the probe blocks are centered around res-
onance.
Varying the number of atoms taken into account (via
post-selection) shows a 1
/N
A
scaling with a pre-factor
dominated by quantum projection noise (QPN) [1] on top
of an offset stemming mainly from laser noise (Fig. 2d). A
more detailed analysis reveals that, for our atom number,
the relative noise contribution from QPN to
σ
̄
e
is only
≈
26% (Appendix C). A similar conclusion can be drawn
on a qualitative level by evaluating correlations between
tweezer resolved errors from odd and even sites, which