coherence times in atom-light interactions.
Figure 3 shows a progression of Ramsey fringes
with free evolution times from 100 ms to 6 s,
beyond what has been demonstrated in 1D
lattice clocks ( 10, 15, 40). The ◯ lattice beam
was operated at a depth (>80Erec) sufficient to
prevent atoms from tunneling along the clock
laser axis during the 6-s free evolution period.
Spectroscopy was performed on a spin-polarized
sample, prepared by first exciting jg;; 9=2i→
je; ; 9=2i, then removing all ground state atoms
by means of resonant 1S0↔1P1 light. Our longest
observed coherence time approached the limit
of our clock laser based on its noise model ( 41)
and the 12.7-s dead time between measurements.
For the demonstration of Ramsey fringes at longer free evolution times, maintaining atom-light
phase coherence will require a substantial reduction in fundamental thermal noise from the
optical local oscillator. Additionally, the observation of narrower lines will require magnetic
field control below the 100-mG level. The contrast of the observed Ramsey fringes was likely
limited by lattice light causing both dephasing
over the atomic sample and excited-state population decay. The external harmonic confinement of the lattice beams limited the size of the
Mott insulating region in the center of the trap
to ~2 × 104 atoms. We maximized the number
of singly occupied sites in the center of the trap
by operating with 1 × 104 atoms at a temperature of 15 nK.
The combination of large numbers of atoms
and a long atom-light coherence time enabled
by this system opens the possibility of improving
the quantum projection noise (QPN) limit on
clock stability by more than an order of mag-
nitude over the current state of the art ( 10, 40).
The QPN limit for Ramsey spectroscopy is
sQPNðtÞ¼ 1 2pnTR
TR þ Td
where n is the clock frequency, TR is the free
evolution time, Td is the dead time, and t is the
total averaging time. Typically, optical clocks
operate at a stability above this limit owing to
the Dick effect ( 42). However, operation at or near
the QPN limit has recently been demonstrated in
a number of different systems; common-mode
rejection of clock laser noise has been achieved
through synchronous interrogation of two independent atomic ensembles ( 6, 43) and two separated ions in a single ion trap ( 44), whereas
interleaved interrogation of two clocks with zero
dead time has enabled continuous monitoring of
the clock laser phase ( 40).
Here we demonstrate synchronous clock comparison between two spatially separated atomic
ensembles in the 3D lattice. The regions, represented by the blue and orange spheres in Fig.
1A, are rectangular cross sections separated
from one another by > 6 mm, each containing
3000 atoms. After a TR = 4 s Ramsey sequence,
we performed absorption imaging on the 1S0 ↔
1P1 transition with a 1-mm imaging resolution to
independently measure the excitation fractions
P1 and P2 in the two respective regions. Although
most clock shifts are common-mode–suppressed
between the two regions, the slightly elliptical
polarization of the z lattice beam produces a
differential vector Stark shift of a few millihertz
between the two regions. We measured this ef-
fect by probing the jg; 1=2i → je; 1=2i clock transi-
tion. The shift manifests itself as a fixed phase
between the signals P1 and P2. Plotting P1 versus
P2 (Fig. 4A, inset) allows us to extract this dif-
ferential shift in a manner independent of the
clock laser noise through either ellipse fitting
( 45) or Bayesian estimation ( 46). The QPN is un-
correlated between the two independently de-
tected regions and thus limits the precision of the
extracted phase shift. In 2.2 hours of averaging,
we measured a shift of 5. 56( 15) mHz, correspond-
ing to an instability of 3:1 ; 10; 17= ffiffi t p (Fig. 4A)
and a measurement precision of 3. 5 × 10–19 (Fig.
4B). To observe the scaling of measurement pre-
cision with the number of atoms, we reduced the
number of atoms by a factor of 3 and observed
an increase in QPN noise by
line in Fig. 4A).
By using larger lattice beam waists, our design
can accommodate even greater numbers of atoms,
which, combined with reduced preparation time
of degenerate gases and increased atomic coherence in the lattice, should enable the operation
of synchronous comparisons at better than the
10; 18= ffiffi t p level, leading to a new generation of
precision measurement tools, including space-based gravitational wave detectors ( 47). Reaching such performance is extremely challenging
for 1D optical lattice clocks, because collisional
effects force a compromise between interrogation time and the number of atoms that can be
simultaneously interrogated ( 10, 15).
With quantum-degenerate atoms frozen into
a 3D lattice, we can further advance the state
of the art in coherent atom-light interactions
with the next generation of ultrastable optical
reference cavities based on crystalline materials
( 48–50). Quantum-degenerate clocks also provide
a promising platform for studying many-body
physics. Future studies of dipolar interactions
will not only be necessary for clock accuracy, but
will also provide insight into long-range quantum spin systems in a regime distinct from those
explored by polar molecules ( 51, 52), Rydberg
gases ( 53, 54), and highly magnetic atoms ( 55–58).
When clocks ultimately confront the natural linewidth of the atomic frequency reference, degenerate Fermi gases may be useful for engineering
longer coherence times through Pauli blocking
of spontaneous emission ( 59) or collective radiative effects ( 39, 60).
REFERENCES AND NOTES
1. A. Derevianko, M. Pospelov, Nat. Phys. 10, 933–936 (2014).
2. K. Van Tilburg, N. Leefer, L. Bougas, D. Budker, Phys. Rev. Lett.
115, 011802 (2015).
3. A. Arvanitaki, J. Huang, K. Van Tilburg, Phys. Rev. D 91, 015015
4. Y. V. Stadnik, V. V. Flambaum, Phys. Rev. A 93, 063630 (2016).
5. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, P. Schmidt,
Rev. Mod. Phys. 87, 637–701 (2015).
6. T. L. Nicholson et al., Phys. Rev. Lett. 109, 230801 (2012).
7. N. Hinkley et al., Science 341, 1215–1218 (2013).
8. B. J. Bloom et al., Nature 506, 71–75 (2014).
9. I. Ushijima, M. Takamoto, M. Das, T. Ohkubo, H. Katori,
Nat. Photonics 9, 185–189 (2015).
10. T. L. Nicholson et al., Nat. Commun. 6, 6896 (2015).
11. C. Grebing et al., Optica 3, 563 (2016).
12. J. Lodewyck et al., Metrologia 53, 1123–1130 (2016).
13. G. K. Campbell et al., Science 324, 360–363 (2009).
14. N. D. Lemke et al., Phys. Rev. Lett. 107, 103902 (2011).
15. M. J. Martin et al., Science 341, 632–636 (2013).
Fig. 4. Synchronous clock comparison. (A) The Allan deviation of the differential frequency
shift between two independent regions of the 3D lattice, each having 3000 atoms (blue
filled circles). The blue solid line shows an instability of 3.1 × 10; 17= ffiffi t p . The frequency difference
is determined through a Bayesian estimation algorithm that determines the eccentricity of
the ellipse of the parametric plot of P1 versus P2 (inset, 420 runs). When the number of atoms
is reduced to 1000 in each region, the instability increases by
(red filled circles and
red dashed line). The error bars represent 95% confidence intervals, assuming white frequency
noise. (B) A histogram of the measured frequency differences for an averaging time of
2.2 hours and 3000 atoms. The fitted Gaussian gives a fractional frequency difference of
129. 6( 3. 5) × 10–19.