# Reducing the instability of an optical lattice clock using multiple atomic
ensembles

Xin Zheng1 Jonathan Dolde1 Shimon Kolkowitz1,2<EMAIL_ADDRESS>1Department of Physics, University of Wisconsin-Madison, Madison, WI 53706 USA
2Department of Physics, University of California, Berkeley, CA 94720, USA

###### Abstract

The stability of an optical atomic clock is a critical figure of merit for
almost all clock applications. To this end, much optical atomic clock research
has focused on reducing clock instability by increasing the atom number,
lengthening the coherent interrogation times, and introducing entanglement to
push beyond the standard quantum limit. In this work, we experimentally
demonstrate an alternative approach to reducing clock instability using a
phase estimation approach based on individually controlled atomic ensembles in
a strontium (Sr) optical lattice clock. We first demonstrate joint Ramsey
interrogation of two spatially-resolved atom ensembles that are out of phase
with respect to each other, which we call “quadrature Ramsey spectroscopy,”
resulting in a factor of 1.36(5) reduction in absolute clock instability as
measured with interleaved self-comparisons. We then leverage the rich
hyperfine structure of 87Sr to realize independent coherent control over
multiple ensembles with only global laser addressing. Finally, we utilize this
independent control over 4 atom ensembles to implement a form of phase
estimation, achieving a factor of greater than $3$ enhancement in coherent
interrogation time and a factor of 2.08(6) reduction in instability over an
otherwise identical single ensemble clock with the same local oscillator and
the same number of atoms. We expect that multi-ensemble protocols similar to
those demonstrated here will result in reduction in the instability of any
optical lattice clock with an interrogation time limited by the local
oscillator.

††preprint: APS/123-QED

##  I Introduction

Thanks to their remarkable precision and accuracy, optical atomic clocks are
rapidly advancing the frontiers of timekeeping, quantum science, and
fundamental physics [1]. State-of-the-art optical clocks have now reached the
level of $10^{-18}$ in both stability and accuracy [2, 3, 4, 5, 6], enabling
novel emerging applications such as relativistic geodesy, searches for
ultralight dark matter, and gravitational wave detection [7, 8, 9, 10, 11, 12,
13, 14, 15]. Instability is a critical figure of merit for optical atomic
clocks, determining the sensitivity of clock comparisons to target signals in
fundamental physics applications and the achievable precision in applications
to timekeeping, metrology, and navigation [1, 12]. The fundamental limit to
the instability of an optical clock with unentangled atoms is given by the
quantum projection noise (QPN),

$\sigma_{\rm QPN}(\tau)=\frac{1}{2\pi\nu CT}\sqrt{\frac{T+T_{\rm d}}{N\tau}},$
(1)

where $\nu$ is the clock transition frequency, $C$ is the contrast, $T$ is the
coherent interrogation time, $T_{\rm d}$ is the dead time, $\tau$ is the
averaging time, and $N$ is the number of atoms being interrogated.

Equation 1 implies that the clock instability can be reduced through longer
coherent interrogation times. However, in optical clocks both conventional
Ramsey and Rabi spectroscopy are limited by frequency noise of the local
oscillator (LO), or clock laser, used to interrogate the atoms. In the case of
Ramsey spectroscopy, the interrogation time is limited by the requirement that
the phase accumulated by the atoms must remain within $[-\pi/2,\pi/2]$ in
order to avoid phase slips. In addition, for atomic clock operations with duty
cycles less than 1, noise on the LO during the dead time leads to Dick noise,
an aliased noise at harmonic frequencies of $1/T_{\rm c}$ ($T_{\rm c}=T+T_{d}$
corresponds to the cycle time) that compromises the clock instability [16,
17]. In particular, the Dick-limited clock instability for Ramsey spectroscopy
can be approximated as (see Appendix E for details)

$\sigma_{\rm Dick}(\tau)=\frac{\sigma_{\rm LO}}{\sqrt{2{\rm
ln}2}}\sum_{m=1}^{\infty}\bigg{|}\frac{{\rm sin}(\pi mT/T_{\rm c})}{\pi
mT/T_{\rm c}}\bigg{|}\sqrt{\frac{T_{\rm c}}{m\tau}},$ (2)

where $\sigma_{\rm LO}$ is the flicker frequency noise floor limited
instability of the LO. From Eq. 2, we can see that for an atomic clock limited
by Dick noise, the instability no longer scales with atom number, but can
still be reduced by increasing the interrogation time. These factors have
motivated considerable efforts into the development of optical LOs with longer
coherence times [18, 19, 20, 21], the use of synchronous differential
comparisons to push the interrogation time beyond the coherence time of the LO
[22, 23, 24, 25], the use of multiple seperate optical atomic clocks to
actively perform feedback or feedforward on the LO [26], as well as the use of
entangled atomic states to push beyond the QPN limit for the same LO coherence
time [27, 28], although the latter case will only be useful in scenarios where
the clock is QPN-limited.

Figure 1: Quadrature Ramsey spectroscopy with a multi-ensemble clock. (a) Two
ensembles of strontium atoms separated by 1 cm are prepared using a movable
optical lattice. A magnetic field gradient $\Delta B(z)$ is applied to
introduce a differential phase shift ($\Delta\phi$) of $\pi/2$ between the two
ensembles for a given Ramsey interrogation time $T$. (b) Theory curves showing
the expected Ramsey fringes for the two atom ensembles with a differential
phase of $\pi/2$. The 4 regions (I, II, III, and IV) correspond to quadratures
for decoding the clock laser detuning. (c) Parametric plot of the measured
excitation fractions from the two ensembles ($P_{1},P_{2}$) for a $\pi/2$
phase difference for the full randomly sampled range of detunings
corresponding to $\Delta f_{i}\in[-1/2T,1/2T]$, with the light blue curve a
fit to the expected ellipse. Each labeled quadrature corresponds to the
matching region in Fig. 1(b).

A proposed alternative and complementary path to reducing the instability of
optical atomic clocks is to more efficiently allocate the available atomic
resources. For example, in Ref. [29], Rosenband and Liebrandt argued that if
the atoms in a clock are split up into multiple ensembles and probed with
different interrogation times in parallel, the clock instability can be
reduced exponentially with atom number rather than with $1/\sqrt{N}$ as in Eq.
1 for uniform interrogation. Similarly, in a recent theoretical work [30], Li
et al. argued that through joint interrogation of two atom ensembles that are
90 degrees out of phase, it should be feasible to extend the interpretable
phases accumulated by the atoms from $[-\pi/2,\pi/2]$ to $[-\pi,\pi]$, thereby
doubling the achievable Ramsey interrogation time for a given LO and reducing
the corresponding clock instability. There have been several prior and recent
works on combining multiple separate optical atomic clocks to realize a single
compound clock with reduced instability and longer coherent interrogation
times [23, 26]. For example, a simplified 3 ensemble version of the Rosenband
and Liebrandt scheme was recently demonstrated by Kim et al. [26] using two
independent interleaved Yb optical lattice clocks to realize zero-dead-time
operation and feed-forward on the interrogation of an additional Al+ single
ion clock, achieving lifetime limited interrogation times. However, the use of
multiple independent optical clocks is extremely resource intensive, and any
compound clock would still benefit from reductions in the instability of the
constituent clocks, motivating work into realizing similar gains within
individual optical clocks through the use of multiple atomic ensembles.

In this work, we demonstrate a multi-ensemble strontium optical lattice clock
that takes advantage of up to 4 spatially-resolved atom ensembles within a
single vacuum chamber to increase the coherent interrogation time and reduce
the absolute clock instability. We first experimentally demonstrate joint
interrogation on 2 out-of-phase atom ensembles [30], which we call “quadrature
Ramsey spectroscopy”, which enables the measurement of phase shifts of up to
$[-\pi,\pi]$, and reduces the clock instability by a factor of 1.36(5), as
measured in a self-comparison. We then demonstrate a novel approach which
leverages the rich hyperfine structure of 87Sr ($I=9/2$) to enable independent
coherent control of 4 atom ensembles in order to implement a simple form of
phase estimation algorithm [35, 36]. We prepare 4 atom ensembles in pairs of
two distinct nuclear spin states, ($\ket{{}^{1}S_{0},m_{F}=5/2}$ and
$\ket{{}^{3}P_{0},m_{F}=3/2}$ (where $m_{F}$ is the hyperfine sublevel), and
demonstrate independent coherent control of each ensemble pair on two
different clock transitions through frequency multiplexing. Combining the 4
ensemble scheme and the quadrature Ramsey technique, we further improve the
interrogation time by a factor of roughly 1.7, achieving a factor of 2
reduction in instability over a single ensemble clock with the same atom
number, duty cycle, and LO.

##  II Experimental apparatus and quadrature Ramsey spectroscopy

The experimental apparatus used here is similar to the multiplexed strontium
optical lattice clock presented in prior works [25, 31]. A movable, one-
dimensional optical lattice is loaded with multiple atom ensembles, for
example two atom ensembles at a 1 cm separation along the lattice axis
($\hat{z}$) as shown in Fig. 1(a), followed by nuclear spin-polarization
through optical pumping and in-lattice-cooling. A bias magnetic field applied
along $\hat{x}$ defines the quantization axis, with a typical magnitude of 5.5
G. A tunable magnetic field gradient along $\hat{z}$ ($\Delta B(z)$), ranging
from 0 to 45 mG/cm, is applied to introduce a differential frequency shift,
dominated by the first-order Zeeman shift, and thus a differential phase shift
for a Ramsey sequence of duration $T$ of $\Delta\phi\approx\gamma\Delta
B(z)T$, where $\gamma$ is the differential gyromagnetic ratio for the
addressed clock transition, between the two ensembles. To take full advantage
of the magnetic field gradient, we probe on the magnetically sensitive
$\ket{{}^{3}P_{0},m_{F}=7/2}\rightarrow\ket{{}^{1}S_{0},m_{F}=5/2}$ (noted as
$\ket{e,7/2}\rightarrow\ket{g,5/2}$) transition, which has a magnetic field
sensitivity of roughly 564 Hz/G. Synchronous differential Ramsey spectroscopy
is performed to characterize the differential phase $\Delta\phi$ [25], and the
magnetic field gradient is fine-tuned so that $\Delta\phi=\pi/2$, where the
Ramsey fringes for the two ensembles are 90 degrees out-of-phase as shown in
Fig. 1(b) (see Appendix B for details) in order to realize quadrature Ramsey
spectroscopy. A typical parametric plot taken at 40 ms free evolution time is
shown in Fig. 1(c), where the differential phase has a typical long-term
differential instability below $10^{-19}$ as demonstrated in prior work [25].

Figure 2: Increasing the interrogation time and reducing clock instability
with quadrature Ramsey spectroscopy. (a) Schematic of clock operation using
Ramsey spectroscopy with two atom ensembles. The differential phase,
$\Delta\phi$, is tuned to either $0$ or $\pi/2$ to compare the clock
performance between standard Ramsey spectroscopy and quadrature Ramsey
spectroscopy. The clock laser detuning ($\Delta f$) is decoded using the
excitation fractions from both clocks, and is then fed back to the LO. (b)
Instability as a function of Ramsey free evolution time ($T$) for both
standard Ramsey (SR) with $\Delta\phi=0$ (red) and quadrature Ramsey (QR) with
$\Delta\phi=\pi/2$ (blue). Each point corresponds to the extracted Allan
deviation of the self-comparison at 1 s. The shaded region represents a
conservative estimate of the region where phase slips do not occur for SR with
experiments taken over an hour for at least 2000 runs. Dashed line is a fit to
the data based on Dick noise limited instability after accounting for finite
pulse length, assuming the LO noise is dominated by flicker frequency noise
(see Appendix E). Inset: A representative plot of the measured frequency
difference between the two interleaved servos with an interrogation time of
$T=40$ ms for both SR (red) and QR (blue). A phase slip occurs for SR, leading
to a frequency offset of $1/T$ (dashed line). (c) Lowest measured Allan
deviations for clock self comparisons at SR at $T=20$ ms (red) and QR at
$T=40$ ms (blue). The Allan deviation for SR at $T=40$ ms (black), where a
phase slip occurred as shown in the inset of (b), is shown for reference. The
line corresponds to a fit assuming only white frequency noise ($1/\sqrt{\tau}$
scaling). Due to the finite attack time of the servos we only fit to data
after 40 s. Each measurement averages for approximately 1 hour.

A standard Ramsey spectroscopy sequence begins with a $\pi/2$-pulse that
prepares the atoms in a 50:50 superposition of $\ket{g}$ and $\ket{e}$. The
atoms then evolve freely for time $T$ and accumulate a phase $\theta$,
$\ket{g}+\ket{e}^{i\theta}$, in which $\theta=2\pi T\Delta f$ and $\Delta f$
is the detuning of the clock laser from atomic resonance. A final
$\pi/2$-pulse converts $\theta$ into a population difference and generates the
error signal for clock feedback (see Fig. 2a). However, phases beyond
$[-\pi/2,\pi/2]$ result in population differences that are no longer unique,
producing an erroneous error signal. The interrogation times for conventional
Ramsey spectroscopy are therefore typically constrained to well below the LO
coherence time in order to avoid a failure in the feedback servo due to phase
slips [32], an example of which is shown in the inset to Fig. 2(b).

Clock operation with Ramsey spectroscopy is performed on the
${}^{1}S_{0}\rightarrow{}^{3}P_{0}$ clock transition. Because we lack a second
independent optical lattice clock to perform direct comparisons against, we
instead perform a self-comparison, where two independent atomic servos using
the same experimental sequences are interleaved and compared in order to
determine the clock instability [33]. Ramsey spectroscopy on the
$\ket{e,7/2}\rightarrow\ket{g,5/2}$ transition is performed simultaneously on
two spatially resolved atom ensembles, each containing roughly 4000 atoms. The
interrogation time in our system is primarily limited by frequency noise of
the LO, which limits our measured atom-laser coherence time to roughly 100 ms
[25]. The typical dead time per cycle in our system is about 1.5 s, which is
primarily limited by atom preparation (roughly 1 s), and camera readout
(roughly 250 ms), so there is a total of roughly 3 s of dead time for each
clock in the self-comparison. The process of loading multiple ensembles
contributes only roughly 100 ms of extra dead time (see Appendix C for
details) and is not a limiting factor for the duty cycle in our apparatus.

In order to perform a fair comparison between the quadrature Ramsey scheme and
standard Ramsey spectroscopy using the same total atom number and experimental
sequence, the magnetic field gradient $\Delta B(z)$ is tuned such that the
differential phase between the two ensembles is either $\Delta\phi=0$
(standard Ramsey) or $\Delta\phi=\pi/2$ (quadrature Ramsey) (Fig. 2(a)). In
the case of $\Delta\phi=0$, the measurements from the two ensembles are simply
averaged together and standard clock operation using Ramsey spectroscopy is
performed, with the measured phase given by

$\theta={\rm sin}^{-1}(2(P-P_{0})/C),$ (3)

where $P=(P_{1}+P_{2})/2$ is the average of the excitation fractions from both
clocks, and $P_{0}\approx 0.5$ and $C\approx 0.95$ are the measured half-
maximum and contrast of the Ramsey fringe, respectively. In this case the two
ensembles are equivalent to a single ensemble clock with $2N$ total atoms. In
the case of $\Delta\phi=\pi/2$, quadrature Ramsey spectroscopy is employed,
with the excitation fractions ($P_{1},P_{2}$) used as inputs for the phase
decoder $\mathcal{R}(P_{1},P_{2})$, which determines the quadrature of the
acquired phase according to,

$\begin{split}\theta&=\mathcal{R}(P_{1},P_{2})\\\
&=\begin{cases}(-\pi-\theta_{1}-\theta_{2})/2&\text{if $P_{1}<P_{0}$,
$P_{2}<P_{0}$, (I)}\\\ (\theta_{1}-\theta_{2})/2&\text{if $P_{1}\leq P_{0}$,
$P_{2}\geq P_{0}$, (II)}\\\ (\theta_{1}+\theta_{2})/2&\text{if $P_{1}\geq
P_{0}$, $P_{2}\geq P_{0}$, (III)}\\\ (\pi-\theta_{1}+\theta_{2})/2&\text{if
$P_{1}>P_{0}$, $P_{2}<P_{0}$, (IV)}\end{cases},\end{split}$ (4)

in which $\theta_{1}={\rm sin}^{-1}(2(P_{1}-P_{0})/C)$, $\theta_{2}={\rm
cos}^{-1}(2(P_{2}-P_{0})/C)$. We perform a series of self-comparisons for both
$\Delta\phi=0$ and $\pi$ cases by varying the interrogation time $T$, with
each measurement averaging down for roughly an hour. Each self-comparison
yields Allan deviations which are linearly fit assuming only white frequency
noise (that averages down as $\tau^{-1/2}$), which is on the order of $1\times
10^{-14}/\sqrt{\tau}$ level and is dominated here by the Dick noise from the
LO. The extracted Allan deviations are plotted as a function of $T$ in Fig.
2(b). In the case of $\Delta\phi=0$, we find that phase slips begin to occur
somewhere between the measurements at $T=20$ and $T=30$ ms, indicated by the
cutoff of the shaded region in Fig. 2(b), which we conservatively place near
$T=30$ ms. Phase slips lead to the sudden accumulation of frequency
differences between the two servos at multiples of the Ramsey fringe linewidth
in Fig. 2(b). As a result, outside the shaded region in Fig. 2(b) the self-
comparison no longer averages down and no meaningful Allan deviation can be
defined. In the case of quadrature Ramsey spectroscopy with
$\Delta\phi=\pi/2$, we are able to extend the interrogation time to $T=50$ ms
before we begin to observe phase slips in the self-comparison. The optimal
clock performance for quadrature Ramsey spectroscopy in our apparatus is
achieved at $T=40$ ms with an instability of $9.5(3)\times
10^{-15}/\sqrt{\tau}$, a factor of $1.36(5)$ smaller than the instability
measured at $T=20$ ms with conventional Ramsey interrogation (Fig. 2(c)).

Figure 3: Independent control over multiple atomic ensembles with only global
addressing. (a) Right: The lattice is loaded with 4 evenly-spaced atomic
ensembles with 0.25 cm nearest-neighbor separation, and a linear magnetic
field. Left: Relevant energy levels and the pulse sequence used to prepare the
ensembles in different hyperfine states. All four ensembles are initially
prepared in the stretched hyperfine ground state
$\ket{{}^{1}S_{0},m_{F}=9/2}$, or $\ket{g,9/2}$. A first $\pi$-pulse (i)
transfers the population into $\ket{{}^{3}P_{0},m_{F}=7/2}$ (or
$\ket{e,7/2}$), followed by a $\pi/2$-pulse (ii) on the
$\ket{e,7/2}\rightarrow\ket{g,5/2}$ transition that prepares a 50:50
superposition. A second $\pi$-pulse (iii) prepares a superposition of two
nuclear spins ($\ket{e,3/2}$ and $\ket{e,5/2}$). During the free evolution
time ($T_{p}$), an applied field gradient (about $45$ G/cm) introduces a phase
shift of $0$, $\pi$, $2\pi$, and $3\pi$ for each atom ensemble, respectively.
Final $\pi$-pulse (iv) and $\pi/2$-pulse (v) prepare the the ensembles into
opposite states, with ensembles (1, 3) on $\ket{g,5/2}$ and ensembles (2, 4)
on $\ket{e,3/2}$. (b) Global Rabi spectroscopy of the 4 ensembles after state
preparation at a biased magnetic field of about 5.5 G. (c) Independent
coherent Rabi oscillations for ensembles (1, 3) (orange, red) on the
$\ket{g,5/2}\rightarrow\ket{e,7/2}$ transition and for ensembles (2, 4) (blue,
light blue) on the $\ket{e,3/2}\rightarrow\ket{g,1/2}$ transition,
respectively. Each plot is an average of 4 independent scans. The solid curves
correspond to fits to the expected sinusoidal Rabi oscillations, and are used
to bound the cross-talk.

##  III Hyperfine state initialization and individual coherent control over
ensemble pairs through global addressing

While quadrature Ramsey spectroscopy with two ensembles enables a doubling of
the coherent interrogation time, it cannot be naively extended beyond this by
adding more ensembles because Ramsey fringes are inherently $2\pi$ periodic.
In order to take advantage of additional ensembles and further extend the
coherent interrogation time, different interrogation times are required for
different samples, as proposed in Refs. [29, 34], which constitute a form of a
phase estimation algorithm [35, 36]. One approach would be to address
different spatially resolved ensembles in multiple optical latices with
different focused clock beams, or even to have different ensembles in
different vacuum chambers. However, both approaches are challenging and would
suffer from added differential phase [37, 38] and intensity noise on the clock
beams, as well as additional frequency shifts due to the differing
environments of each ensemble. It is therefore desirable to have all of the
atomic ensembles share the same lattice, vacuum chamber, and clock beam.
Fortunately, the magnetic field gradient that enables quadrature Ramsey
spectroscopy can be combined with the rich hyperfine level structure of 87Sr
($I=9/2$) to achieve independent clock interrogation with multiple ensembles
using only global addressing of all the ensembles with a single clock beam.

The basic principle and relevant pulse sequence is shown in Fig. 3(a). Similar
to the two-ensemble case, we start with a sample of 4 evenly spaced atom
ensembles, each containing roughly 2000 atoms, in the stretched state
$\ket{g,9/2}$. We first transfer the population into $\ket{e,7/2}$ with a
clock $\pi$-pulse (i), then create a 50:50 superposition between $\ket{e,7/2}$
and $\ket{g,5/2}$ using a clock $\pi/2$-pulse(ii). We then transfer half of
the population into $\ket{e,3/2}$ (iii), and let the sample evolve for
$T_{p}$, such that each ensemble accumulates phase shifts of 0, $\pi$, $2\pi$,
and $3\pi$ respectively due to the applied linear magnetic field gradient (at
about 45 mG/cm). We then transfer the populations back to $\ket{g,5/2}$ (iv),
and convert the accumated phases into population in either the $\ket{g,5/2}$
and $\ket{e,3/2}$ states with a final $\pi/2$ clock pulse (v). Ensemble 1 and
3 are left in $\ket{g,5/2}$, while ensemble 2 and 4 are left in $\ket{e,3/2}$
(Fig. 1(a)). This state preparation sequence typically takes about 250 ms
yielding an overall state preparation fidelity of about 90 % limited by the
clock $\pi$-pulse fidelity ($>95$ % per pulse). Cleanup pulses are then
applied to remove unwanted residual populations in each of ensemble pairs.

To demonstrate that the ensembles are now in two independently addressable
pairs, Rabi spectroscopy is performed by scanning the frequency of the global
clock laser applied to the 4 atom ensembles (Fig. 3(b)). The $\pi$-pulse
duration is roughly 3 ms, which power broadens the linewidth to roughly 300
Hz. The transitions remain resolved under a large bias magnetic field of 5.5
G. Relevant transitions for the subsequent clock interrogations are the
$\ket{e,7/2}\rightarrow\ket{g,5/2}$ (noted as $7/2-5/2$) and
$\ket{e,3/2}\rightarrow\ket{g,1/2}$ ($3/2-1/2$) transitions, while both
ensemble pairs share the transition $\ket{g,e/2}\rightarrow\ket{e,3/2}$
($5/2-3/2$). The detuning from the magnetic field insensitive $5/2-3/2$
transition (-22.4 Hz/G) roughly indicates the ratio of the magnetic
sensitivities between the two clock transitions of interest. In particular,
the $7/2-5/2$ transition is roughly a factor of $1.7\times$ more magnetically
sensitive than the $3/2-1/2$ transition. To characterize the cross-talk
between the two clock transitions, independent Rabi oscillations were
performed on the $7/2-5/2$ ($3/2-1/2$) transition for ensemble pairs 1 and 3
(2 and 4), as shown in Fig. 3(c). A fit to the oscillations bounds the cross-
talk to below 3 %.

Figure 4: Phase estimation using 4 atom ensembles with independent control
through frequency multiplexing. (a) Basic principle and pulse sequence for
clock operation using phase estimation with four ensembles. We interrogate on
the $\ket{e,7/2}\rightarrow\ket{g,5/2}$ transition for clock pair (1, 3) for a
free evolution time of $T_{A}$, during which the magnetic field gradient is
chosen such that the phase difference between ensembles 1 and 3 is $\pi/2$.
Asynchronously, we interrogate on the $\ket{e,3/2}\rightarrow\ket{g,1/2}$
transition for $T_{B}=1.7\times T_{A}$, which is chosen such that the phase
difference between ensembles 2 and 4 is also $\pi/2$ due to the difference in
magnetic sensitivities for the two transitions. An estimator extracts the LO
detuning using information from all 4 clocks, which is then used to feed back
on the LO. (b) Comparison of inferred phase shifts from the same set of 1000
experimental runs ($T=85$ ms) for various phase estimators: 1 ensemble (top,
standard Ramsey using only ensemble 2), 2 ensembles (middle, quadrature Ramsey
using ensembles 2 and 4), and 4 ensembles (bottom, quadrature Ramsey with
phase estimation using all 4 ensembles). The dashed lines indicate the
$[-\pi/2,\pi/2]$ and $[-\pi,\pi]$ regions. The numbers above the histograms
represent the summed occurrences within each region. The y-axes are shown in
log scale for clarity. (c) Achievable interrogation time and clock instability
by probing using 2 ensembles with $\Delta\phi=0$ (standard Ramsey, SR), 2
ensembles with $\Delta\phi=\pi/2$ (quadrature Ramsey, QR), and 4 ensembles
with $\Delta\phi=\pi/2$ (quadrature Ramsey with phase estimation as shown in
panel (a), PE & SR). The shaded regions represent conservative bounds on where
phase slips do not occur for each case with experiments taken over an hour for
at least 2000 runs. The dashed line is a fit to the data based on Dick noise
limited instability after accounting for finite pulse length, assuming the LO
noise is dominated by flicker frequency noise (see Appendix E).

##  IV Phase estimation with 4 atomic ensembles

Having achieved independent control with global addressing through a
combination of spatial and frequency multiplexing, we proceed to demonstrate a
combination of quadrature Ramsey spectroscopy with a simple phase estimation
algorithm using 4 atomic ensembles. The basic principle is shown in Fig. 4(a),
in which we first apply a global $\pi/2$-pulse on the $3/2-1/2$ transition for
ensemble pair (2, 4), and subsequently apply another global $\pi/2$-pulse on
the $7/2-5/2$ transition for (1, 3). As the two transitions are spectrally
resolved, each pulse only impacts the targeted pair, aside from a negligibly
small probe shift (see Appendix G.2). We let ensembles (1, 3) freely evolve
for time $T_{A}$, where the magnetic field gradient $\Delta B(z)$ is fine-
tuned such that the differential phase shift between ensembles (1, 3) is
$\pi/2$, and a second global $\pi/2$-pulse is applied to the $7/2-5/2$
transition and the phase accumulated by the ensembles (1, 3) is stored in the
populations of the clock states and is shelved there temporarily. After a
total free precession time of $T_{B}$ we apply the second $\pi/2$-pulse on the
$3/2-1/2$ transition for ensembles (2, 4). The free precession time $T_{B}$ is
chosen such that the differential phase of ensembles (2, 4) is again $\pi/2$,
with $T_{B}=1.7\times T_{A}$ due to the difference in magnetic moment for the
two clock transitions.

The populations of all 4 ensembles are then simultaneously read out using
global imaging pulses, where the excitation fractions are used as input to the
phase estimator. In particular, we obtain two phases from the two quadrature
Ramsey measurements: $\theta_{A}=\mathcal{R}(P_{1},P_{3})$ and
$\theta_{B}=\mathcal{R}(P_{2},P_{4})$. We constrain $\theta_{A}$ within
$[-\pi,\pi]$, and we have $\theta_{A}=2\pi\Delta fT_{A}$, in which $\Delta f$
is the LO detuning. In the limit that the LO noise is dominated by low
frequency noise and for a low duty cycle, we can assume that the frequency of
the LO did not significantly change during the second half of the
interrogation period $T_{B}$, and we approximate the estimated phase from
ensembles (2, 4) as $\theta_{\rm est,B}=1.7\times\theta_{A}$. In our
estimator, we compare $\theta_{\rm est,B}$ with
$\theta_{B}=\mathcal{R}(P_{2},P_{4})$, $\mathcal{R}(P_{2},P_{4})-2\pi$, and
$\mathcal{R}(P_{2},P_{4})+2\pi$, and select the $\theta_{B}$ that yields the
smallest difference, which we then use to feedback on the LO according to
$\Delta f=\theta_{B}/(2\pi T_{B})$.

The use of two quadrature Ramsey sequences with different interrogation times
allows us to perform phase estimation and discriminate accumulated phases for
ensembles (2,4) beyond $\pm\pi$. A representative plot from 1,000 experimental
clock sequences of the inferred phase shifts from the longest interrogation
time ($T=85$ ms) without phase slips for the 4 ensemble approach is shown in
Fig. 4(b). The distributions of inferred phases acquired by ensemble 2 using
information from only a single ensemble with standard Ramsey (top, red), 2
ensembles using quadrature Ramsey (middle, yellow), and 4 ensembles using
phase estimation (bottom, blue) are compared. Instances of inferred phases
beyond $\pm\pi/2$ and $\pm\pi$ in the case of 2 and 4 ensembles respectively
can be observed, which would result in corresponding phase slips for the
single ensemble or 2 ensemble decoders for this interrogation time.

As shown in Fig. 4(c), the clock self-comparison Allan deviation at 1 s from
all 3 approaches (standard Ramsey, quadrature Ramsey with 2 ensembles, and
quadrature Ramsey combined with phase estimation with 4 ensembles) are
compared using identical experimental sequences and atom numbers, with the
only differences being the applied field gradients and how the accumulated
phases are extracted by the estimator. We are able to extend the coherent
interrogation time $T_{B}$ out to 85 ms (with $T_{A}=50$ ms) without observing
phase slips, and achieve a factor of 1.2(1) reduction in instability when
comparing the 4-ensemble phase estimation scheme to the 2-ensemble quadrature
Ramsey scheme, and a factor of 2.08(6) reduction with respect to standard
Ramsey. We note that the instabilities are slightly degraded over those shown
in Fig. 2(b) for the same interrogation times, which we attribute to the added
time required for the hyperfine state preparation (250 ms), which contributes
a total additional dead time of 500 ms in the self-comparisons.

##  V Discussion and outlook

While the multiple ensemble schemes employed in this work improve the Ramsey
interrogation time and reduce the clock instability, they will also contribute
additional systematic shifts that could impact the achievable clock accuracy.
Examples of potential effects include Zeeman shifts arising from the magnetic
field gradient applied to generate the $\pi/2$ phase shift for quadrature
Ramsey spectroscopy, probe shifts from the off-resonant clock pulses in the 4
ensemble phase estimation approach, and the servo error associated with the
quadrature Ramsey decoder. Although a full systematic evaluation is beyond the
scope of this work, we expect that none of these effects would significantly
degrade achievable clock accuracy. Estimates of the magnitude of the
associated systematics and potential mitigation strategies are discussed in
Appendix G.

We note that all the instabilities and Allan deviations quoted in this work
are the instabilities extracted from the self-comparisons, and that the actual
single clock instabilities are expected to be at least a factor of $2$ lower
due to the product of the factor of $\sqrt{2}$ from comparing two clocks and
the factor of $\sqrt{2}$ from the doubling in cycle time for each clock due to
the interleaved nature of the self-comparison. While the instabilities
demonstrated here are not competitive with the best optical lattice clocks
demonstrated to date [6], this is primarily due to the relatively short
coherence time of our LO ($100$ ms), which is compounded by the need for a
self-comparison as we do not have another independent clock to compare with,
resulting in very low duty-cycles ($<2$ %). These factors are not fundamental,
and we anticipate that our multiple ensemble techniques can be extended to
other existing Sr optical lattice clocks in a straight forward fashion and
that similar reductions in clock instability would be expected.

For example, with a state-of-the-art atom-laser coherence time on the order of
600 ms [6], a factor of 2 improvement in the coherence time using quadrature
Ramsey spectroscopy with 2 ensembles should be fairly straightforward to
implement, offering roughly a factor of $\sqrt{2}$ reduction in instability.
However, given the resulting higher duty-cycle (50 %), more of the frequency
drift of the local oscillator will take place during the Ramsey interrogation
time rather than during the dead time, so the 4 ensemble phase estimation
scheme employed here would likely be insufficient. We anticipate that 6
ensembles split into three quadrature Ramsey pairs, with one pair interrogated
for $2T$ in parallel to two pairs each interrogated back-to-back for $T$,
would be sufficient to achieve a comparable enhancement of roughly a factor of
4 in coherent interrogation time and a factor of 2 reduction in instability.
However, this would require the use of an additional hyperfine clock
transition or an alternative mechanism for achieving independent control over
three ensemble pairs.

It is also natural to consider scaling up to many more atomic ensembles, which
would require an alternative approach to achieving independent control that
does not require a unique clock transition for each pair of ensembles. This
could be achieved in optical lattice clocks through the introduction of larger
magnetic field gradients to spectroscopically resolve each ensemble on the
same clock transition [39], targeted light shifting beams to bring individual
ensembles in and out of resonance with the global clock beam [40, 41], or the
introduction of an additional clock transition, such as the
${}^{1}S_{0}\rightarrow{}^{3}P_{2}$ transition in 87Sr [42]. The resulting
scaling of clock instability with the number of ensembles $M$ will depend on
the noise spectrum of the LO, the number of atoms available, and the clock
duty cycle, but both the coherent interrogation time and clock instability can
be expected to continue to improve with the addition of more ensembles as long
as the coherent interrogation time is limited by the LO. While we leave the
determination of the optimal algorithm and number of ensembles for a given
optical atomic clock for future work, a naive extension of the 6 ensemble
algorithm discussed in the previous paragraph can be expected to provide a
scaling of the instability with $1/\sqrt{M}$ for large $M$, so long as the
clock remains Dick noise limited. If combined with non-destructive mid-circuit
measurement [43, 44], the use of multiple independently controlled atom
ensembles could ultimately offer exponential reductions in clock instability
with atom and ensemble number using the scheme put forward by Rosenband and
Leibrandt [29]. Finally, we note that similar multi-ensemble phase estimation
algorithms using maximally entangled atomic Greenberger-Horne-Zeilinger (GHZ)
states have also been proposed as a way to achieve Heisenberg-limited clock
performance [35].

##  VI Conclusion

In conclusion, in this work we experimentally demonstrated an optical lattice
clock with enhanced Ramsey interrogation times and reduced absolute clock
instability by harnessing multiple atomic ensembles in a single clock. We
first demonstrated quadrature Ramsey spectroscopy, a simple protocol for
jointly interrogating two atom ensembles that are 90 degrees out of phase in
order to extend the phase interrogation window from $[-\pi/2,\pi/2]$ to
$[-\pi,\pi]$, extending the coherent interrogation time without phase slips by
roughly a factor of $2\times$ and measuring a corresponding reduction in clock
instability by a factor of 1.36(5). We then leveraged the hyperfine structure
of 87Sr and a linear magnetic field gradient to achieve independent state
preparation of multiple ensembles with only global addressing. We finally
demonstrated a combination of quadrature Ramsey spectroscopy and enhanced
phase estimation with 4 ensembles to further extend the coherent interrogation
time by a factor of 1.7 compared to the 2-ensemble scheme, resulting in a
factor of 2.08(6) reduction in clock instability compared to a standard single
ensemble clock with the same atom number, LO, and duty-cycle. Our multi-
ensemble approach could be extended to other existing optical lattice clocks
with state of the art LOs [18, 19, 20, 21], and similar reductions in
instability would be anticipated.

###### Acknowledgements.

We thank Jeff Thompson, Adam Kaufman, and Andrew Jayich for fruitful
discussions and insightful comments on the manuscript. We acknowledge
technical contributions from H.-M. Lim and N. Ranabhat. This work was
supported by the NIST Precision Measurement Grants program, the Northwestern
University Center for Fundamental Physics and the John Templeton Foundation
through a Fundamental Physics grant, the Wisconsin Alumni Research Foundation,
a Packard Fellowship for Science and Engineering, the Army Research Office
through agreement number W911NF-21-1-0012, the Sloan Foundation, and the
National Science Foundation under Grant No. 2143870.

Authors’ note: While completing this work we became aware of a recent related
and complementary work in Ref. [47].

##  Appendix A Atomic sample preparation

The experiment beings with a standard two-stage magneto-optical trapping
(MOT), trapping and cooling 87Sr atoms to a temperature of $1~{}\mu$K. We then
load multiple ensembles into a movable one-dimensional (1D) optical lattice as
described and demonstrated in prior work [25], followed by spin-polarization
and in-lattice-cooling both axially (sideband cooling) and radially (Doppler
cooling) on the narrow-linewidth ${}^{1}S_{0}\rightarrow{}^{3}P_{1}$
transition at 689 nm. The typical lattice trap depth for loading and cooling
is at $70~{}E_{\rm rec}$, where $E_{\rm rec}/h\approx 3.5$ kHz is the recoil
energy of a lattice photon at the magic wavelength around 813.4 nm. The
lattice is then adiabatically ramped down to the operational trap depth at
$15~{}E_{\rm rec}$ for state preparation and clock interrogation, after which
it is ramped back up to $70~{}E_{\rm rec}$ for read-out.

##  Appendix B Calibration of differential phase shift for quadrature Ramsey
spectroscopy

A magnetic field of roughly 5.5 G is applied along $\hat{x}$ to define the
quantization axis using 3 pairs of orthogonal Helmholtz bias coils,
($B_{x},B_{y},B_{z}$). The coil set-points are optimized to zero out $B_{y}$
and $B_{z}$ at the waist of the lattice, $z=0$. To introduce the $\pi/2$ phase
shift between a pair of ensembles at $z=\pm 0.5$ cm, we employ a magnetic
field gradient which results in a differential Zeeman shift between the two
ensembles. The magnetic field gradient can be tuned by scanning the offset
voltage applied to the $B_{z}$ bias coils, as shown in Fig. 5. To take full
advantage of the magnetic field gradient, we interrogate on the
$\ket{{}^{3}P_{0},m_{F}=7/2}\rightarrow\ket{{}^{1}S_{0},m_{F}=5/2}$ clock
transition, which has a magnetic field sensitivity of about $\gamma=564$ Hz/G.
The differential phase shift is dominated by the differential first-order
Zeeman shift as,

$\Delta f_{\rm ZS,1^{st}}=\gamma\Delta B,$ (5)

where $\Delta B$ is the magnetic field gradient. This corresponds to a typical
differential frequency shift of 25 Hz for a magnetic field gradient of $\Delta
B=45$ mG/cm.

Figure 5:  Magnetic field gradient ($\Delta B$) adjustment by tuning the
offset voltage of the $B_{z}$ bias coils. The B field gradient is measured via
spectroscopy on the ${}^{1}S_{0}\rightarrow{}^{3}P_{1}$ transition. Each data
point has a $1\sigma$ standard deviation of roughly 0.7 mG/cm. The solid line
is a linear fit to the data. The fit residuals are shown in the lower panel.

The magnetic field gradient is then fine tuned such that the differential
phase $\Delta\phi=\pi/2$ for each interrogation time $T$. In particular, we
scan the $B_{z}$ offset set-points and map out $\Delta\phi$ using synchronous
Ramsey interrogation [25], in which the two ensembles are prepared in a 50:50
superposition of ${}^{1}S_{0}$ and ${}^{3}P_{0}$, freely evolve for $T$ and
project using a final global $\pi/2$-pulse with its phase randomly drawn from
$[0,2\pi]$. An ellipse fitting is then fit to the data using the least-squares
method to extract $\Delta\phi$, which has a typical uncertainty of 0.02 rad,
limited by quantum projection noise. The resulting $\Delta\phi$ as a function
of $B_{z}$ offset voltage is fit using a quadratic function, and the intercept
of the fit with $\pi/2$ is used as the $B_{z}$ offset set-point for a specific
Ramsey evolution time $T$, as shown in Fig. 6.

Figure 6:  Differential phase shift adjustment by fine tuning the $B_{z}$
coils offset voltage. The differential phase shift $\Delta\phi$ is measured
via synchronous Ramsey spectroscopy with free evolution time of $T=30$ ms on
the magnetically sensitive
$\ket{{}^{1}S_{0},m_{F}=5/2}\rightarrow\ket{{}^{3}P_{1},m_{F}=7/2}$
transition. The solid line is a quadratic fit to the data. The residuals are
shown in the lower panel.

##  Appendix C Quadrature Ramsey clock interrogation

Before clock interrogation, the atomic population is transferred from
$\ket{{}^{1}S_{0},m_{F}=9/2}\rightarrow\ket{{}^{3}P_{0},m_{F}=7/2}$ (note as
$\ket{g,9/2}\rightarrow\ket{e,7/2}$) via a $\pi$-pulse, followed by a clean-up
pulse on resonant with the 461 nm ${}^{1}S_{0}\rightarrow{}^{1}P_{1}$
transition to blow out residual populations in the ground state. Quadrature
Ramsey spectroscopy is then performed by interrogating the
$\ket{e,7/2}\rightarrow\ket{g,5/2}$ clock transition, which has a first-order
Zeeman shift sensitivity of 564 Hz/G. To generate an error signal for
frequency feedback, the phase of the final $\pi/2$-pulse is detuned by $\pi/2$
with respect to the phase of the first $\pi/2$-pulse. After spectroscopy, the
lattice is adiabatically ramped back up to full depth for readout. The
populations in the ground and excited clock states of both ensembles are read-
out in parallel with imaging pulses along the lattice axis, with scattered
photons collected on an EMCCD camera (Andor, iXon-888). The excitation
fraction is extracted through $P=(N_{e}-N_{bg})/(N_{g}+N_{e}-2N_{bg})$, where
$N_{g},N_{e}$ and $N_{bg}$ are the ground state population, excited clock
state population, and background counts without atoms, respectively. The
excitation fractions from both ensembles are used as inputs to the phase
estimator to extract the clock laser detuning, which is used to feed back to
the acousto-optical modulator on the clock laser path that addresses the
atomic resonance.

Due to the lack of an additional independent optical lattice clock to compare
against, in order characterize the clock instability we perform a self-
comparison. Two aysnchronous independent servo, sharing the same LO are
interleaved and compared to extract the Allan deviation. A typical
experimental timing sequence for the self-comparison is shown in Fig. 7. The
typical dead time is about 1.5 s within each servo, and thus 3 s per clock
when accounting for the other servo. The primary contribution comes from
sample loading, which takes about 1 s to cool, trap, and optically pump the
atoms, with an additional 100 ms required to prepare multiple ensembles rather
than a single one. The camera imaging and data processing takes about 250 ms,
while an additional 250 ms is spent on the nuclear spin state preparation when
performing phase esitimation with 4 atom ensembles, which is not shown in Fig.
7

Figure 7:  Experimental sequence for the self-clock comparisons. Two
independent atomic servos (I) and (II) are interleaved and compared to
determine the clock comparison instability. The typical exerimental cycle is
shown in the bottom (not to scale), which takes about 1.5 s, and thus leads to
roughly 3 s dead times in each servo. An additional 250 ms is spent on nuclear
spin state initialization when probing 4 atom ensembles (not shown in the
figure).

##  Appendix D Hyperfine state initialization of 4 atom ensembles

The experimental sequence used to initialize 4 atom ensembles is shown in Fig.
3(a) in the main text. We elaborate on some of the experimental details here.
First, we start with a sample of spin-polarized atoms in the $\ket{g,9/2}$
state with 4 ensembles spaced evenly over a 1 cm extent. A first $\pi$-pulse
on the $9/2-7/2$ transition transfers the population into $\ket{e,7/2}$,
followed by a clean-up pulse on the 461 nm ${}^{1}S_{0}\rightarrow{}^{1}P_{1}$
transition. We then prepare the atoms in a 50:50 superposition of the
$\ket{e,7/2}$ and $\ket{g,5/2}$ state with a $\pi/2$ clock pulse, and
coherently transfer the population in $\ket{g,5/2}$ into $\ket{e,3/2}$ via a
$\pi$ pulse on resonance with the $5/2-3/2$ transition. This creates a
superposition of two nuclear spin states in the excited clock state manifold,
$\ket{e,7/2}$ and $\ket{e,3/2}$, which freely evolve for roughly $T_{p}=40$ ms
under a linear magnetic field gradient tuned to about 45 mG/cm to maximize the
differential phase shifts between the adjacent atom ensembles. We then
coherently transfer the populations and close the interferometer via a
$\pi$-pulse on the $7/2-5/2$ transition and a final $\pi/2$-pulse on $5/2-3/2$
transition, respectively. Pulses on resonance with either the
${}^{1}S_{0}\rightarrow{}^{1}P_{1}$ transition or the
${}^{3}P_{0,2}\rightarrow{}^{3}S_{1}$ transitions are combined with clock
$\pi$ pulses to clean up residual populations leftover in unwanted states in
each pair of ensembles. $T_{p}$ and the laser phase of the last $\pi/2$-pulse
are fine tuned such that the 4 atom ensembles accumulate $0$, $\pi$, $2\pi$,
and $3\pi$ phase shifts respectively, enabling preparation of two ensemble
pairs in distinct nuclear spin states, pair (1, 3) in $\ket{g,5/2}$ and pair
(2, 4) in $\ket{e,3/2}$, using only global pulses without the need for
individual ensemble addressing.

Relevant transitions, their differential magnetic moments, and the measured
Rabi frequencies used for state initialization and clock interrogation are
summarized in Table 1.

Table 1:  Relevant transitions used in state initialization of 4 atom ensembles. Transition | B field sensitivity | Rabi frequency
---|---|---
$\ket{g,9/2}\rightarrow\ket{e,7/2}$ | 194 Hz/G | 110 Hz
$\ket{e,7/2}\rightarrow\ket{g,5/2}$ | 564 Hz/G | 125 Hz
$\ket{g,5/2}\rightarrow\ket{e,3/2}$ | -23 Hz/G | 142 Hz
$\ket{e,3/2}\rightarrow\ket{g,1/2}$ | 340 Hz/G | 166 Hz

##  Appendix E Scaling of the Dick-noise-limited clock instability

The instability of a clock limited by Dick noise is given by [16]

$\sigma^{2}_{\rm
Dick}(\tau)=\frac{1}{\tau}\sum_{m=1}^{\infty}\bigg{(}\frac{g_{m}^{c2}}{g_{0}^{2}}+\frac{g_{m}^{s2}}{g_{0}^{2}}\bigg{)}S_{\rm
LO}^{f}(m/T_{c}),$ (6)

where $T_{c}$ is the cycle time, $m$ is the $m$-th harmonic, $S_{\rm
LO}^{f}(m/T_{c})$ is the one-sided power spectral density of the relative
frequency fluctuations of the LO at Fourier frequencies $m/T_{c}$, and the
parameters $g_{0}$, $g_{m}^{s}$, and $g_{m}^{c}$ are defined as:

$\begin{split}g_{m}^{s}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi){\rm sin}(2\pi
m\xi/T_{c})d\xi,\\\ g_{m}^{c}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi){\rm
cos}(2\pi m\xi/T_{c})d\xi,\\\
g_{0}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi)d\xi,\\\ \end{split}$ (7)

with $g(\xi)$ being the sensitivity function of Ramsey spectroscopy sequence.

The LO noise is modeled as

$S_{\rm LO}^{f}(f)=\sum_{k=-2}^{0}b_{k}f^{k},$ (8)

in which $b_{0}$ corresponds to white frequency noise, $b_{-1}$ corresponds to
flicker frequency noise, and $b_{-2}$ corresponds to random walk frequency
noise.

Here we derive the scaling of Dick-limited clock instability as function of
interrogation time $T$ for an arbitrary duty-cycle. As pointed out in the
references [16, 17], for an LO dominated by its flicker frequency noise floor,
$S^{f}_{\rm LO}(f)$ = $\sigma_{\rm LO}^{2}/(2{\rm ln}2)f^{-1}$, where
$\sigma_{\rm LO}$ is the LO flicker frequency instability, the Dick noise
limited clock instability can be approximated as:

$\sigma_{\rm Dick}(\tau)\approx\frac{\sigma_{\rm LO}}{\sqrt{2{\rm
ln}2}}\sum_{m=1}^{\infty}\sqrt{\frac{g^{s2}_{m}+g_{m}^{c2}}{g_{0}^{2}}}\sqrt{\frac{T_{c}}{m\tau}},$
(9)

where $T_{c}$ is the cycle time.

In the limit that $T_{\pi/2}\ll T$ (where $T_{\pi/2}$ is the duration of
$\pi/2$-pulse), we can approximate the sensitivity function of a Ramsey
spectroscopy with free evolution time $T$ as

$g(\xi)=\begin{cases}1&\quad 0<\xi<T,\\\ 0&\quad T<\xi<T_{c}.\\\ \end{cases}$

We then have the parameters $g_{0,1}$ as

$\begin{split}g_{0}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi)d\xi\\\
&=\frac{1}{T_{c}}\int^{T}_{0}d\xi\\\ &=T/T_{c},\end{split}$ (10)

$\begin{split}g^{s}_{m}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi){\rm sin}(2\pi
m\xi/T_{c})d\xi\\\ &=\frac{1}{T_{c}}\int^{T}_{0}{\rm sin}(2\pi
m\xi/T_{c})d\xi\\\ &=-\frac{1}{2\pi m}\bigg{(}{\rm cos}(2\pi
mT/T_{c})-1\bigg{)},\end{split}$ (11)

$\begin{split}g^{c}_{m}&=\frac{1}{T_{c}}\int^{T_{c}}_{0}g(\xi){\rm cos}(2\pi
m\xi/T_{c})d\xi\\\ &=\frac{1}{T_{c}}\int^{T}_{0}{\rm cos}(2\pi
m\xi/T_{c})d\xi\\\ &=\frac{1}{2\pi m}{\rm sin}(2\pi mT/T_{c}),\end{split}$
(12)

and thus after some algebraic manipulation we have:

$\sqrt{\frac{g_{m}^{s2}+g_{m}^{c2}}{g_{0}^{2}}}=\bigg{|}\frac{{\rm sin}(\pi
mT/T_{c})}{\pi mT/T_{c}}\bigg{|}.$ (13)

Plugging this back into Eq. 9, we arrive at

$\sigma_{\rm Dick}(\tau)=\frac{\sigma_{\rm LO}}{\sqrt{2{\rm
ln}2}}\sum_{m=1}^{\infty}\bigg{|}\frac{{\rm sin}(\pi mT/T_{c})}{\pi
mT/T_{c}}\bigg{|}\sqrt{\frac{T_{c}}{m\tau}}.$ (14)

It’s worth noting that taking only the lowest order term $m=1$, we have Eq. 14
as

$\sigma_{{\rm Dick},m=1}(\tau)=\frac{\sigma_{\rm LO}}{\sqrt{2{\rm
ln}2}}\bigg{|}\frac{{\rm sin}(\pi T/T_{c})}{\pi
T/T_{c}}\bigg{|}\sqrt{\frac{T_{c}}{\tau}},$ (15)

which matches the approximation given in Eq. 12 of Ref. [17].

We can thus fit our data (as taken in Fig. 2(b) and Fig. 4(c) in the main
text) to the scaling in Eq. 14. However, we need to bound the maximum $m_{\rm
max}$ in the actual numerical fitting given limited computational resources,
and due to the finite bandwidth of the Ramsey sensitivity function. The cutoff
$m_{\rm max}$ can be roughly bounded by the multiples of sine waves within the
minimum Ramsey free evolution time $T_{\rm min}$ as

$m_{\rm max}={\rm mod}(nT_{c}/T_{\rm min}),$ (16)

in which we conservatively choose $n=2$. In our actual data taking, we have a
typical $\pi/2$-pulse duration of $T_{\pi/2}=1.5$ ms, and a minimum $T_{\rm
min}=2$ ms. To account for the finite pulse duration, we can take the average
of sensitivity function $g^{\prime}(\xi)$ over the entire duration of
$(0,T+2T_{\pi/2})$, where we have

$\begin{split}&g^{\prime}(\xi)=\\\ &\begin{cases}{\rm sin}(\Omega\xi)&\quad
0<\xi<T_{\pi/2},\\\ 1&\quad T_{\pi/2}<\xi<T+T_{\pi/2},\\\ {\rm
sin}(\Omega(T+2T_{\pi/2}-\xi))&\quad T+T_{\pi/2}<\xi<T+2T_{\pi/2},\\\
\end{cases}\end{split}$

where $\Omega T_{\pi/2}=\pi/2$, with $\Omega$ being the Rabi frequency. For
$T_{\pi/2}=1.5$ ms and $T=2$ ms, this yields an effective duration of
$T^{\prime}_{\rm min}=3.9$ ms, giving $m_{\rm max}=1700$ in our numerical
fitting.

In order to fit to the instability data shown in both Fig. 2(b) and Fig. 4(c),
we use a simplified expression for the Dick noise that assumes only flicker
frequency noise, and treat the instability of the LO as a free parameter. The
extracted LO flicker frequency instability of
$\sigma_{\textrm{FF}}=3.2(2)\times 10^{-15}$ is roughly a factor of
$2-3\times$ larger than the value provided by the manufacturer of the LO
(Menlo Systems). This discrepancy likely arises in part from not including the
contributions of random walk and white frequency noise in the fit. In
addition, the LO is now in a relatively noisy vibration and thermal
environment, and we believe its noise spectrum is now degraded (see next
section for details).

##  Appendix F Numerical calculations of single clock instability

The expected clock instability is given by the quadrature sum of QPN and Dick
noise,

$\sigma^{2}=\sigma_{\rm QPN}^{2}+\sigma_{\rm Dick}^{2},$ (17)

where $\sigma_{\rm Dick}$ is given in Eq. 6. The LO frequency noise spectrum
is modeled with Eq. 8, and the coefficients $b_{m}$ ($m=0,-1,-2$) are
extracted from the measured cavity instabilities provided by the manufacturer
by beating the LO against another independent cavity-stabilized laser, and are
given as

$\begin{split}\sigma_{\rm WH}(\tau)&=5.3^{+2.2}_{-2.0}\times
10^{-16}/\sqrt{\tau},\\\ \sigma_{\rm FF}(\tau)&=1.3^{+0.2}_{-0.1}\times
10^{-15},\\\ \sigma_{\rm RW}(\tau)&=1.0^{+0.1}_{-0.1}\times
10^{-15}\sqrt{\tau},\\\ \end{split}$ (18)

where $\sigma_{\rm WH}$, $\sigma_{\rm FF}$, and $\sigma_{\rm RW}$ correspond
to the estimated fractional white, flicker, and random walk frequency noise
instabilities, respectively. These instabilities (at $\tau=1$ s) are then
converted into $b_{m}$ as

$\begin{split}b_{0}=2\sigma_{\rm WH}^{2}&=5.7^{+3.0}_{-2.5}\times
10^{-31}~{}{\rm Hz}^{-1},\\\ b_{-1}=\frac{\sigma^{2}_{\rm FF}}{2{\rm
ln}2}&=1.3^{+0.3}_{-0.3}\times 10^{-30},\\\ b_{-2}=\frac{6\sigma^{2}_{\rm
RW}}{(2\pi)^{2}}&=1.6^{+0.4}_{-0.3}\times 10^{-31}~{}{\rm Hz}.\\\ \end{split}$
(19)

Figure 8:  Numerical calculations of the expected clock instability using the
manufacturer’s measurements of the LO instability as inputs into Eq. 19 . The
shaded region represents upper and lower bound limits from the uncertainty in
the LO noise estimates. The experimental data are taken from Fig. 4(c) in the
main text, divided by $\sqrt{2}$ to infer the single clock instability. The
discrepancy between experimental data and numerical calculations is likely
because the noise spectrum of our LO is degraded in a relatively noisy
vibration and thermal environment.

The QPN limited instability ($\sigma_{\rm QPN}$) for Ramsey spectroscopy is
given in Eq. 1 in the main text. In our case, we typically have 4000 atoms per
ensemble, interrogation times of less than 100 ms, and a dead time of
$1.75\times 2$ s, and are primarily limited by Dick noise due to our low duty-
cycle ($<2\%$). The calculated single clock instability is shown in Fig. 8,
where the shaded regions represent uncertainties in the calculation, primarily
arising from the estimated frequency noise spectrum of the LO. The scatter
points represent single clock instability corresponding to the data in Fig.
4(c) in the main text, divided by $\sqrt{2}$ assuming equal contribution from
either atomic servo. The measured instabilities are slightly greater than the
numerical calculations using LO frequency noises provided by the manufacturer,
which likely because our LO is now in a relatively noisy vibration and thermal
environment, and we believe its noise spectrum is now degraded.

##  Appendix G Potential systematics associated with multiple ensemble schemes

In this section, we consider potential sources of additional systematic shifts
arising from the quadrature Ramsey spectroscopy and phase estimation
approaches using multiple ensembles, and strategies to mitigate such effects.

###  G.1 Magnetic field gradient

In this work we employ a magnetic field gradient to introduce the $\pi/2$
phase shift between the ensemble pairs, which is on the order of 20 mG/cm
depending on the interrogation time $T$ and magnetic moment of clock
transition. The first-order Zeeman shift is typically cancelled out by
averaging between opposite spin states. While the magnetic field gradient
itself contributes to the second-order Zeeman shift as well as potential
tensor Stark shifts due to spatially varying B-field vectors, across a single
atom ensemble over a finite extent of about hundreds of microns, it can, in
principle, be averaged and cancelled out by performing spatially-resolved
read-out with the use of high-resolution imaging, as was demonstrated in Refs.
[15, 45, 46]. Besides, with the atom-laser coherence times approaching second-
scale afforded by the state-of-the-art LO, a smaller magnetic field gradient
and a magnetically less sensitivity clock transition can be used to generate
the $\pi/2$ phase shift for quadrature Ramsey spectroscopy. For example, with
the $5/2-3/2$ transition, which has a magnetic field sensitivity of $-22.4$
Hz/G, only a B field gradient of 3 mG/cm is needed at $T=3$ s interrogation
time for a pair of ensembles separated by 1 mm [15].

###  G.2 Systematic shifts from off-resonant clock pulses

The additional $\pi/2$ clock pulses employed in the 4 ensemble phase
estimation scheme will lead to additional light shifts on the off-resonant
ensmebles. Given the measured sensitivity of -13(2) Hz (W cm-2)-1 from prior
work [48], we estimate a probe light shift of $6\times 10^{-16}$ caused by the
on-resonant clock pulse with a Rabi frequency of 140 Hz. This sets an upper
bound limit below $1\times 10^{-15}$ after accounting for the detuning and
Clebsh-Gordan coefficient for the off-resonant clock pulses. While this is a
sizable systematic shift for the state-of-the-art clocks, given the large Rabi
frequency employed in this work, we note this effect can be reduced by either
lengthening the clock pulses, as the clock laser intensity scales with
$1/T^{2}$, or through the adoption of more sophisticated composite pulse
sequences such as Hyper-Ramsey spectroscopy [49].

###  G.3 Servo errors from phase decoder

The phase decoder we employ here for quadrature Ramsey spectrscopy assumes a
differential phase shift of $\Delta\phi=\pi/2$. Inaccuracy in $\Delta\phi$
could introduce a frequency offset in the phase decoder (Eq. 4 in the main
text). In the limit of $\delta\phi\ll\Delta\phi$, the phase shift error
$\delta\phi$ will be converted into a decoder error of $\delta\phi/2$, and
thus result in a frequency offset of $\delta\phi/2\pi T$. For a typical phase
error of 10 mrad, this would lead to an error on the order of $1\times
10^{-17}$ at an interrogation time of $T=100$ ms, which could be further
suppressed well below the $10^{-18}$ level using longer interrogation times
with a better LO. In addition, averaging interleaved measurements on opposite
hyperfine transitions, as is typically employed to cancel the first order
Zeeman shift, will to lowest order also cancel this error. Finally,
measurements of $\delta\phi$ can be performed with higher precision if
necessary by taking advantage of the ability to probe well beyond the LO
coherence time using synchronous differential comparisons [25, 15, 31].

## References

  * [1] A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, P. O. Schmidt, Optical Atomic Clocks, Rev. Mod. Phys. 87, 637–701 (2015)
  * [2] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm and E. Peik, Single-Ion Atomic Clock with $3\times{10}^{-18}$ Systematic Uncertainty, Phys. Rev. Lett. 116, 063001 (2016)
  * [3] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Schäffer, K. Beloy, D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, T. H. Yoon and A. D. Ludlow, Atomic Clock Performance Enabling Geodesy Below the Centimetre Level, Nature 564, 87–90 (2018)
  * [4] T. Bothwell, D. Kedar, E. Oelker, J. M. Robinson, S. L. Bromley, W. L. Tew, J. Ye and C. J. Kennedy, JILA SrI Optical Lattice Clock with Uncertainty of $2.0\times 10^{-18}$, Metrologia 56, 065004 (2019)
  * [5] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume and D. R. Leibrandt, ${}^{27}{\rm Al}^{+}$ Quantum-Logic Clock with a Systematic Uncertainty Below ${10}^{-18}$, Phys. Rev. Lett. 123, 033021 (2019)
  * [6] E. Oelker, R. B. Hutson, C. J. Kennedy, L. Sonderhouse, T. Bothwell, A. Goban, D. Kedar, C. Sanner, J. M. Robinson, G. E. Marti, D. G. Matei, T. Legero, M. Giunta, R. Holzwarth, F. Riehle, U. Sterr and J. Ye, Demonstration of $4.8\times 10^{-17}$ Stability at 1 s for Two Independent Optical Clocks Nat. Photon. 13, 714–719 (2019)
  * [7] C. W. Chou, D. B. Hume, T. Rosenband and D. J. Wineland, Optical Clocks and Relativity, Science 329, 1630–1633 (2010)
  * [8] A. Derevianko and M. Pospelov, Hunting for Topological Dark Matter With Atomic Clocks, Nat. Phys. 10, 933–936 (2014)
  * [9] T. Takano, M. Takamoto, I. Ushijima, N. Ohmae, T. Akatsuka, A. Yamaguchi, Y. Kuroishi, H. Munekane, B. Miyahara and H. Katori, Geopotential Measurements with Synchronously Linked Optical Lattice Clocks, Nat. Photon. 10, 662-666 (2016)
  * [10] S. Kolkowitz, I. Pikovski, N. Langellier, M. D. Lukin, R. L. Walsworth and J. Ye, Gravitational Wave Detection with Optical Lattice Atomic Clocks, Phys. Rev. D 94, 124043 (2016)
  * [11] J. Grotti, S. Koller, S. Vogt, S. Häfner, U. Sterr, C. Lisdat, H. Denker, C. Voigt, L. Timmen, A. Rolland, F. N. Baynes, H. S. Margolis, M. Zampaolo, P. Thoumany, M. Pizzocaro, B. Rauf, F. Bregolin, A. Tampellini, P. Barbieri, M. Zucco, G. A. Costanzo, C. Clivati, F. Levi and D. Calonico, Geodesy and Metrology with a Transportable Optical Clock, Nat. Phys. 14, 437–441 (2018)
  * [12] M. S. Safronova, D. Budker, D. DeMille, D. F. Jackson Kimball, A. Derevianko and C. W. Clark, Search for New Physics with Atoms and Molecules, Rev. Mod. Phys. 90, 025008 (2018)
  * [13] M. Takamoto, I. Ushijima, N. Ohmae, T. Yahagi, K. Kokado, H. Shinkai and H. Katori, Test of General Relativity by a Pair of Transportable Optical Lattice Clocks, Nat. Photon. 14, 411–415 (2020)
  * [14] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell, D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko and J. Ye, Precision Metrology Meets Cosmology: Improved Constraints on Ultralight Dark Matter from Atom-Cavity Frequency Comparisons, Phys. Rev. Lett. 125, 201302 (2020)
  * [15] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robinson, E. Oelker, A. Staron and J. Ye, Resolving the Gravitational Redshift Across a Millimetre-Scale Atomic Sample, Nature 602, 420–424 (2022)
  * [16] G. J. Dick, LO Induced Instabilities in Trapped Ion Frequency Standards, Proc. Precise Time and Time Interval Meeting (ed. Sydnor, R. L.) 133–147 (1987)
  * [17] G. Santarelli, C. Audoin, A. Makdissi, P. Laurent, G. J. Dick and A. Clairon, Frequency Stability Degradation of an Oscillator Slaved to a Periodically Interrogated Atomic Resonator, IEEE Trans. Ultra. Ferro. Freq. Cont. 45, 887–894 (1998)
  * [18] T. Kessler, C. Hagemann, C. Grebing, T. Legero, U. Sterr, F. Riehle, M. J. Martin, L. Chen and J. Ye, A Sub-40-mHz-Linewidth Laser Based on a Silicon Single-Crystal Optical Cavity, Nat. Photon. 6, 687–692 (2012)
  * [19] W. Zhang, J. M. Robinson, L. Sonderhouse, E. Oelker, C. Benko, J. L. Hall, T. Legero, D. G. Matei, F. Riehle, U. Sterr and J. Ye, Ultrastable Silicon Cavity in a Continuously Operating Closed-Cycle Cryostat at 4 K, Phys. Rev. Lett. 119, 243601 (2017)
  * [20] J. M. Robinson, E. Oelker, W. R. Milner, W. Zhang, T. Legero, D. G. Matei, F. Riehle, U. Sterr and J. Ye, Crystalline Optical Cavity at 4 K with Thermal-Noise-Limited Instability and Ultralow Drift, Optica 6, 240–243 (2019)
  * [21] D. Kedar, J. Yu, E. Oelker, A. Staron, W. Milner, J.M. Robinson, T. Legero, F. Riehle, U. Sterr, and J. Ye, Frequency stability of cryogenic silicon cavities with semiconductor crystalline coatings, Optica 10, 464 (2023)
  * [22] M. Takamoto, T. Takano, and H. Katori, Frequency Comparison of Optical Lattice Clocks Beyond the Dick Limit, Nat. Photon. 5, 288–292 (2011)
  * [23] M. Schioppo, R. C. Brown, W. F. McGrew, N. Hinkley, R. J. Fasano, K. Beloy, T. H. Yoon, G. Milani, D. Nicolodi, J. A. Sherman, N. B. Phillips, C. W. Oates and A. D. Ludlow, Ultrastable Optical Clock with Two Cold-Atom Ensembles, Nat. Photon. 11, 48–52 (2017)
  * [24] E. R. Clements, M. E. Kim, K. Cui, A. M. Hankin, S. M. Brewer, J. Valencia, J.-S. Chen, C.-W. Chou, D. R. Leibrandt and D. B. Hume, Lifetime-Limited Interrogation of Two Independent ${}^{27}{\mathrm{Al}}^{+}$ Clocks Using Correlation Spectroscopy, Phys. Rev. Lett. 125, 243602 (2020)
  * [25] X. Zheng, J. Dolde, V. Lochab, B. N. Merriman, H. Li and S. Kolkowitz, Differential Clock Comparisons with a Multiplexed Optical Lattice Clock, Nature 602, 425–430 (2022)
  * [26] M. E. Kim, W. F. McGrew, N. V. Nardelli, E. R. Clements, Y. S. Hassan, X. Zhang, J. L. Valencia, H. Leopardi, D. B. Hume, T. M. Fortier, A. D. Ludlow and D. R. Leibrandt, Improved Interspecies Optical Clock Comparisons through Differential Spectroscopy, Nat. Phys. 19, 25–29 (2022)
  * [27] E. Pedrozo-Peñafiel, S. Colombo, C. Shu, A. F. Adiyatullin, Z. Li, E. Mendez, B. Braverman, A. Kawasaki, D. Akamatsu, Y. Xiao and V. Vuletić, Entanglement on an Optical Atomic-Clock Transition, Nature 588, 414–418 (2020)
  * [28] J. M. Robinson, M. Miklos, Y. M. Tso, C. J. Kennedy, T. Bothwell, D. Kedar, J. K. Thompson and J. Ye, Direct Comparison of Two Spin Squeezed Optical Clocks Below the Quantum Projection Noise Limit, arXiv:2211.08621 (2022)
  * [29] T. Rosenband and D. R. Leibrandt, Exponential Scaling of Clock Stability with Atom Number, arXiv:1303.6357 (2013)
  * [30] W. Li, S. Wu, A. Smerzi and L. Pezze, Improved Absolute Clock Stability by the Joint Interrogation of Two Atomic Ensembles, Phys. Rev. A 105, 053116 (2022)
  * [31] X. Zheng, J. Dolde, H. M. Lim and S. Kolkowitz, A Lab-Based Test of the Gravitational Redshift with a Miniatured Clock Network, arXiv:2207.07145 (2022)
  * [32] D. B. Hume and D. R. Leibrandt, Probing Beyond the Laser Coherence Time in Optical Clock Comparisons, Phys. Rev. A 93, 032138 (2016)
  * [33] T. L. Nicholson, M. J. Martin, J. R. Williams, B. J. Bloom, M. Bishof, M. D. Swallows, S. L. Campbell and J. Ye, Comparison of Two Independent Sr Optical Clocks with $1\times 10^{-17}$ Stability at $10^{3}$ s, Phys. Rev. Lett. 109, 230801 (2012)
  * [34] J. Borregaard and A. S. Sorensen, Efficient Atomic Clocks Operated with Several Atomic Ensembles, Phys. Rev. Lett. 111, 090802 (2013)
  * [35] E. M. Kessler, P. Kómár, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye and M. D. Lukin, Heisenberg-Limited Atom Clocks Based on Entangled Qubits, Phys. Rev. Lett. 112, 190403 (2014)
  * [36] R. Kaubruegger, D. V. Vasilyev, M. Schulte, K. Hammerer and P. Zoller, Quantum Variational Optimization of Ramsey Interferometry and Atomic Clocks, Phys. Rev. X 11, 041045 (2021)
  * [37] L.-S. Ma, P. Jungner, J. Ye and J. L. Hall, Delivering the Same Optical Frequency at Two Places: Accurate Cancellation of Phase Noise Introduced by an Optical Fiber or Other Time-Varying Path, Optica 19, 1777–1779 (1994)
  * [38] S. Falke, M. Misera, U. Sterr and C. Lisdat, Delivering Pulsed and Phase Stable Light to Atoms of An Optical Clock, Appl. Phys. B. 107, 301-–311 (2012)
  * [39] M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. Blatt, T. Zanon-Willette, S. M. Foreman and J. Ye, Nuclear Spin Effects in Optical Lattice Clocks, Phys. Rev. A 76, 022510 (2007)
  * [40] T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower and M. Saffman, Randomized Benchmarking of Single-Qubit Gates in a 2D Array of Neutral-Atom Qubits, Phys. Rev. Lett. 114, 100503 (2015)
  * [41] T. M. Graham, Y. Song, J. Scott, C. Poole, L. Phuttitarn, K. Jooya, P. Eichler, X. Jiang, A. Marra, B. Grinkemeyer, M. Kwon, M. Ebert, J. Cherek, M. T. Lichtman, M. Gillette, J. Gilbert, D. Bowman, T. Ballance, C. Campbell, E. D. Dahl, O. Crawford, N. S. Blunt, B. Rogers, T. Noel and M. Saffman, Multi-Qubit Entanglement and Algorithms on a Neutral-Atom Quantum Computer, Nature 604, 457–-462 2022)
  * [42] J. Trautman, D. Yankelev, V. Klüsener, A. J. Park, I. Bloch and S. Blatt, ${}^{1}S_{0}-{}^{3}P_{2}$ Magnetic Quadrupole Transition in Neutral Strontium, Phys. Rev. Research 5, 013219 (2023)
  * [43] K. Singh, C. E. Bradley, S. Anand, V. Ramesh, R. White and H. Bernien, Mid-Circuit Correction of Correlated Phase Errors Using an Array of Spectator Qubits, arXiv:2208.11716 (2022)
  * [44] T. M. Graham, L. Phuttitarn, R. Chinnarasu, Y. Song, C. Poole, K. Jooya, J. Scott, A. Scott, P. Eichler and M. Saffman, Mid-circuit Measurements on a Neutral Atom Quantum Processor, arXiv:2303.10051 (2023)
  * [45] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom, J. Ye, A Fermi-Degenerate Three-Dimensional Optical Lattice Clock, Science 358, 90–94 (2017)
  * [46] G. E. Marti, R. B. Hutson, A. Goban, S. L. Campbell, N. Poli and J. Ye, Imaging Optical Frequencies with 100 $\mu$Hz Precision and 1.1 $\mu$m Resolution, Phys. Rev. Lett. 120, 103201 (2018)
  * [47] A. L. Shaw, R. Finkelstein, R. B.-S. Tsai, P. Scholl, T. H. Yoon, J. Choi and M. Endres, Multi-ensemble metrology by programming local rotations with atom movements, arXiv:2303.16885 (2023)
  * [48] X. Baillard, M. Fouché, R. Le Targat, P. G. Westergaard, A. Lecallier, Y. Le Coq, G. D. Rovera, S. Bize and P. Lemonde, Accuracy Evaluation of an Optical Lattice Clock with Bosonic Atoms, Opt. Lett. 32, 1812–1814 (2007)
  * [49] V. I. Yudin, A. V. Taichenachev, C. W. Oates, Z. W. Barber, N. D. Lemke, A. D. Ludlow, U. Sterr, Ch. Lisdat and F. Riehle, Hyper-Ramsey Spectroscopy of Optical Clock Transitions, Phys. Rev. A 82, 011804(R) (2010)