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Bothwell, T., Kennedy, C. J., Aeppli, A., Kedar, D., Robinson, J. M., Oelker, E., Staron, A. and Ye, J. (2022) Resolving the gravitational redshift across a millimetre-scale atomic sample. Nature, 620, pp. 420-424. (doi: 10.1038/s41586-021-04349-7) This is the Author Accepted Manuscript. There may be differences between this version and the published version. You are advised to consult the publisher’s version if you wish to cite from it.
https://eprints.gla.ac.uk/264685/
Deposited on: 4 February 2022
Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk

1 Resolving the gravitational redshift within a millimeter atomic

2

sample

3 Tobias Bothwell1,*, Colin J. Kennedy1,2, Alexander Aeppli1, Dhruv Kedar1, John M. Robinson1,

4

Eric Oelker1,3, Alexander Staron1, and Jun Ye1,*

5

1JILA, National Institute of Standards and Technology and University of Colorado

6

Department of Physics, University of Colorado

7

Boulder, Colorado 80309-0440, USA

8

2Present Address: Honeywell Quantum Solutions, Broomfield, Colorado 80021

9

3Present Address: Physics Department, University of Glasgow

10
11 Einstein’s theory of general relativity states that clocks at different gravitational

12 potentials tick at different rates – an effect known as the gravitational redshift1. As

13 fundamental probes of space and time, atomic clocks have long served to test this 14 prediction at distance scales from 30 centimeters to thousands of kilometers2–4.

15 Ultimately, clocks will study the union of general relativity and quantum mechanics

16 once they become sensitive to the finite wavefunction of quantum objects oscillating in

17 curved spacetime. Towards this regime, we measure a linear frequency gradient

18 consistent with the gravitational redshift within a single millimeter scale sample of

19 ultracold strontium. Our result is enabled by improving the fractional frequency 20 measurement uncertainty by more than a factor of 10, now reaching 7.6×10-21. This

21 heralds a new regime of clock operation necessitating intra-sample corrections for

22 gravitational perturbations.

23

Modern atomic clocks embody Arthur Schawlow's motto to "never measure anything but

24 frequency." This deceptively simple principle, fueled by the innovative development of laser

25 science and quantum technologies based on ultracold matter, has led to dramatic progress in
26 clock performance. Recently, clock measurement precision reached the mid-19th digit in one
1

27 hour5,6, and three atomic species achieved systematic uncertainties corresponding to an error 28 equivalent to less than 1 s over the lifetime of the Universe7–10. Central to this success in 29 neutral atom clocks is the ability to maintain extended quantum coherence times while using 30 large ensembles of atoms5,6,11. The pace of progress has yet to slow. Continued improvement 31 in measurement precision and accuracy arising from the confluence of metrology and quantum 32 information science12–15 promises discoveries in fundamental physics16–20.

33

Clocks fundamentally connect space and time, providing exquisite tests of the theory of

34 general relativity. Hafele and Keating took cesium-beam atomic clocks aboard commercial

35 airliners in 1971, observing differences between flight-based and ground-based clocks 36 consistent with special and general relativity21. More recently, RIKEN researchers compared

37 two strontium optical lattice clocks (OLCs) separated by 450 m in the Tokyo Skytree, resulting 38 in the most precise terrestrial redshift measurement to date22. Proposed satellite-based 39 measurements23,24 will provide orders of magnitude improvement to current bounds on 40 gravitational redshifts3,4. Concurrently, clocks are anticipated to begin playing important roles 41 for relativistic geodesy25. In 2010 Chou et al.2 demonstrated the precision of their Al+ clocks by

42 measuring the gravitational redshift resulting from lifting one clock vertically by 30 cm in 40

43 hours. With a decade of advancements, today’s leading clocks are poised to enable local

44 geodetic surveys of elevation at the sub-centimeter level on Earth, complimenting spatial 45 averaging techniques26.

46

Atomic clocks strive to simultaneously optimize measurement precision and systematic

47 uncertainty. For traditional OLCs operated with one-dimensional (1D) optical lattices, achieving

48 low instability has involved the use of high atom numbers at trap depths sufficiently large to

49 suppress tunnelling between neighboring lattice sites. While impressive performance has been

2

50 achieved, effects arising from atomic interactions and AC Stark shifts associated with the 51 trapping light challenge advancements in OLCs. Here we report a new operational regime for 52 1D OLCs, both resolving the gravitational redshift across our atomic sample and 53 synchronously measuring a fractional frequency uncertainty of 7.6×10-21 between two 54 uncorrelated regions. Our system employs ~100,000 87Sr atoms at ~100 nK loaded into a 55 shallow, large waist optical lattice, reducing both AC Stark and density shifts. Motivated by our 56 earlier work on spin-orbit coupled lattice clocks27,28, we engineer atomic interactions by 57 operating at a ‘magic’ trap depth, effectively removing collisional frequency shifts. These 58 advances enable record optical atomic coherence (37 s) and expected single clock stability 59 (3.1×10-18 at 1 s) using macroscopic samples, paving the way toward lifetime limited OLC 60 operation.

61

Central to our experiment is an in-vacuum optical cavity (Fig. 1a and Methods) for

62 power enhancement of the optical lattice. The cavity (finesse 1300) ensures wavefront

63 homogeneity of our 1D lattice while the large beam waist (260 μm) reduces the atomic density 64 by an order of magnitude compared to our previous system10. We begin each experiment by 65 trapping fermionic 87Sr atoms into the 1D lattice at a trap depth of 300 lattice photon recoil

66 energies (Erec), loading a millimeter scale atomic sample (Fig. 1a). Atoms are simultaneously

67 cooled and polarized into a single nuclear spin before the lattice is adiabatically ramped to an 68 operational depth of 12 Erec. Clock interrogation proceeds by probing the ultranarrow 1S0 ( )→ 69 3P0 ( ) transition with the resulting excitation fraction measured by fluorescence spectroscopy.

70 Scattered photons are collected on a camera, enabling in-situ measurement with 6 μm

71 resolution, corresponding to ~15 lattice sites.

3

72

Quantum state control has been vital to recent advances in atom-atom and atom-light

73 coherence times in 3D OLCs and tweezer clocks5,11,15. Improved quantum state control is

74 demonstrated through precision spectroscopy of the Wannier-Stark states of the OLC29,30. The

75 1D lattice oriented along gravity has the degeneracy of neighboring lattice sites lifted by the

76 gravitational potential energy. In the limit of shallow lattice depths, this creates a set of

77 delocalized states. By ramping the lattice depth to 6 Erec, much lower than in traditional 1D 78 lattice operations7,8,10, clock spectroscopy probes this delocalization (Fig. 1d). The ability to

79 engineer the extent of atomic wavefunctions through the adjustment of trap depth creates an

80 opportunity to control the balance of on-site p-wave versus neighboring-site s-wave atomic 81 interactions. We utilize this tunability by operating at a ‘magic’ trap depth31, where the

82 frequency shifts arising from on-site and off-site atomic interactions cancel, enabling a

83 reduction of the collisional frequency shifts by more than three orders of magnitude compared 84 our previous work10.

85

Extended atomic coherence times are critical for both accuracy and precision. An

86 aspirational milestone for clock measurement precision is the ability to coherently interrogate

87 atomic samples up to the excited state’s natural lifetime. To evaluate the limits of our clock’s

88 atomic coherence, we perform Ramsey spectroscopy to measure the decay of fringe contrast

89 as a function of the free-evolution time. By comparing two uncorrelated regions within our

90 atomic sample, we determine the contrast and relative phase difference between the two sub-

91 ensembles (Fig. 2). The contrast decays exponentially with a time constant of 37 s (quality 92 factor of 3.6×1016), corresponding to an additional decoherence time of 53 s relative to the 3P0 93 natural lifetime (118 s)32. This represents the longest optical atomic coherence time measured

94 in any spectroscopy system to date.

4

95

We utilize Rabi spectroscopy in conjunction with in-situ imaging to microscopically

96 probe clock transition frequencies along the entire vertically oriented atomic ensemble. With a

97 standard interleaved probing sequence using the

to

transitions for

98 minimal magnetic sensitivity, we reject the first order Zeeman shifts and vector AC Stark shifts. 99 The in-situ imaging of atoms in the lattice allows measurement of unprocessed frequencies 100 across the entire atomic sample (Fig. 1a and Methods). The dominant differential perturbations 101 arise from atom-atom interactions (residual density shift contributions after we operate at the 102 ‘magic’ trap depth) and magnetic field gradients giving rise to pixel-specific 2nd order Zeeman 103 shifts. Using the total camera counts and mF-dependent frequency splitting, we correct the 104 density and 2nd order Zeeman shift at each pixel. These corrections result in the processed 105 frequencies per pixel shown in Fig. 3a, with error bars representing the quadrature sum of 106 statistical uncertainties from the center frequency, the density shift correction, and the 2nd order 107 Zeeman shift correction. Additional systematics are described in the Methods. This approach 108 demonstrates an efficient method for rapid and accurate evaluation of various systematic 109 effects throughout a single atomic ensemble. Unlike traditional 1D OLCs where systematic 110 uncertainties are quoted as global parameters, we now microscopically characterize these 111 effects.

112

This new microscopic in-situ imaging allows determination of the gravitational redshift

113 within a single atomic sample, probing an uncharacterized fundamental clock systematic. Two

114 identical clocks on the surface of a planet separated by a vertical distance will differ in

115 frequency ( as given by

116

(Eq. 1),

5

117 with the clock frequency, the speed of light, and the gravitational acceleration. The 118 gravitational redshift at Earth’s surface corresponds to a fractional frequency gradient of -1.09× 119 10-19/mm in the coordinate system of Fig. 1a. Measurement of a vertical gradient across the 120 atomic sample consistent with the gravitational redshift provides an exquisite verification of an 121 individual atomic clock’s frequency control.

122

Our intra-cloud frequency map (Fig. 3a) allows us to evaluate gradients across the

123 atomic sample. Over 10 days we performed 14 measurements (ranging in duration from 1-17

124 hours) to search for the gravitational redshift across our sample. For each we fit a linear slope 125 and offset after taking into account density shift and 2nd order Zeeman corrections, reporting

126 the slope in Fig. 3b. From this measurement campaign we find the weighted mean (standard 127 error of the weighted mean) of the frequency gradient in our system to be -1.00(12)×10-19/mm.

128 We evaluate additional differential systematics (see Methods) and find a final frequency 129 gradient of -9.8(2.3)×10-20/mm, consistent with the predicted redshift.

130

The ability to resolve the gravitational redshift within our system suggests a level of

131 frequency control beyond previous clock demonstrations, vital for the continued advancement 132 of clock accuracy and precision. Previous fractional frequency comparisons15 have reached 133 uncertainties as low as 4.2×10-19. Similarly, we perform a synchronous comparison between

134 two uncorrelated regions of our atomic cloud (Fig. 4a). By binning ~100 pixels per region, we 135 substantially reduce instability caused by quantum projection noise33. Analyzing the frequency

136 difference between regions from 92 hours of data, we find a fractional frequency instability of

137 4.4×10-18/ ( is the averaging time in seconds), resulting in a fractional frequency uncertainty 138 of 7.6×10-21 for full measurement time, nearly two orders of magnitude lower than the previous

139 record. From this measurement we infer a single region instability of 3.1×10-18/ . Dividing the
6

140 fractional frequency difference by the spatial separation between each region’s center of mass 141 gives a frequency gradient of -1.30(18)×10-19/mm. Correcting for additional systematics as 142 before results in a gradient of -1.28(27)×10-19/mm, again fully consistent with the predicted 143 redshift.

144

In conclusion, we have established a new paradigm for atomic clocks. The vastly

145 improved atomic coherence and frequency homogeneity throughout our sample allow us to

146 resolve the gravitational redshift at the submillimeter scale, observing for the first time the

147 frequency gradient from gravity within a single sample. We demonstrate a synchronous clock

148 comparison between two uncorrelated regions with a fractional frequency uncertainty of 149 7.6×10-21, advancing precision by nearly two orders of magnitude. These results suggest that

150 there are no fundamental limitations to inter-clock comparisons reaching frequency 151 uncertainties at the 10-21 level, offering new opportunities for tests of fundamental physics.

7

152
153 Fig. 1: Experimental system and quantum state control. a, A millimeter length sample of 154 ~100,000 87Sr atoms are trapped in a 1D optical lattice formed within an in-vacuum cavity. The 155 longitudinal axis of the cavity, z, is oriented along gravity. We probe atoms along the 1S0 → 3P0 156 transition using a clock laser locked to an ultrastable crystalline silicon cavity6,34. b, Rabi 157 spectroscopy with a 3.1 s pulse time. Open purple circles indicate data with a corresponding 158 Rabi fit in green. c, Neighboring lattice sites are detuned by gravity, creating a Wannier-Stark 159 ladder. Clock spectroscopy probes the overlap of Wannier-Stark states between lattice sites 160 that are m sites away with Rabi frequency Ωm. d, Rabi spectroscopy probes Wannier-Stark 161 state transitions, revealing wavefunction delocalization of up to 5 lattice sites. The number of 162 lattice sites is indicated above each transition, with blue(red) denoting Wannier-Stark 163 transitions to higher(lower) lattice sites.
8

164 165
166 Fig. 2: Atomic coherence. We use Ramsey spectroscopy with a randomly sampled phase for 167 the second pulse to determine the coherence time of our system11. a, We measure the 168 excitation fraction across the cloud, shown in purple for a single measurement, and calculate 169 the average excitation fractions in regions p1 and p2, separated by 2 pixels. b, Parametric plots 170 of the excitation fraction of p1 versus p2 in purple for 6 s, 30 s and 50 s dark time demonstrate 171 a phase shift between the two regions and contrast decay. Using a maximum likelihood 172 estimator, we extract the phase and contrast for each dark time with the fit, shown in green. c, 173 Contrast decay as a function of time in green is fit with an exponential decay in gold, giving an 174 atomic coherence decay time of 36.5(0.7) s and a corresponding quality factor of 3.6×1016. 175 After accounting for the finite radiative decay contribution, we infer an additional decoherence 176 time constant of 52.8(1.5) s.
177 178
9

179
180 Figure 3: Evaluating frequency gradients. a, For each measurement we construct a 181 microscopic frequency map across the sample, with raw frequencies shown in green. The 2nd 182 order Zeeman correction is shown as a dashed gold line. Processed frequencies shown in 183 purple include both density shift corrections and 2nd order Zeeman corrections, with 184 uncertainties arising from the quadrature sum of statistical, density shift correction, and 2nd 185 order Zeeman correction uncertainties. To this we fit a linear function, shown in black. b, Over 186 the course of 10 days, we completed 14 measurements. For each measurement, we create a 187 corrected frequency map and fit a linear slope as in a. This slope is plotted for each 188 measurement, as well as a weighted mean (black) with associated statistical uncertainty 189 (dashed black) and total uncertainty as reported in Table 1 (dotted black). The expected 190 gravitational gradient is shown in red. All data is taken with Rabi spectroscopy using a 3.1 s π191 pulse time except for 08/13 which used a 3.0 s pulse time. The reduced chi-square statistic is 192 3.0, indicating a small underestimation of error variances entirely consistent with the additional 193 systematic uncertainties in Table 1.
194 195 196
10

197
198 Figure 4: In-situ synchronous clock comparison. a, The cloud is separated as in Fig. 2a. 199 The gravitational redshift leads to the higher clock(blue) ticking faster than the lower one(red). 200 The length scale is in millimeters. b, Allan deviation of the frequency difference between the 201 two regions in a over 92 hours. Purple points show fractional frequency instability fit by the 202 solid green line, with the quantum projection noise limit indicated by the dashed black line. We 203 attribute the excess instability of the measurement relative to QPN to detection noise. The 204 expected single atomic region instability is shown in gold.
205 206 207 208 209
11

210

Methods

211 In-Vacuum Cavity

212

Central to our system is an in-vacuum lattice buildup cavity oriented along gravity (Fig.

213 1a). Two mirrors with radius of curvature of 1 m are separated by ~15 cm, achieving a mode

214 waist of 260 μm. Our over-coupled cavity has a finesse at the lattice wavelength (813 nm) of

215 ~1300 and a power buildup factor of ~700 (ratio of circulating to input intensity). This enables

216 lattice depths in excess of 500 Erec (lattice photon recoil energy) using a diode-based laser

217 system. The dimensional stability of the cavity combined with the simplified diode laser system 218 enables robust operation compared with our previous Ti:Sapphire retro-reflected design10. The

219 cavity mirrors are anti-reflection coated at the clock wavelength of 698 nm.

220

One cavity mirror is mounted to a piezo for length stabilization while the other mirror is

221 rigidly mounted for phase reference for the clock laser. Grounded copper shields between

222 atoms and mirrors prevent DC Stark induced shifts due to charge buildup on the mirrors and 223 piezo35,36. Each shield (5 mm thick) has a centered hole of 6 mm diameter to accommodate

224 the optical lattice beam, with shielding performance verified through evaluation of the DC Stark

225 shift systematic.

226 Atomic sample preparation

227

87Sr atoms are cooled and loaded into a 300 Erec optical lattice using standard two stage

228 magneto-optical trapping techniques10. Once trapped, atoms are simultaneously nuclear spin

229 polarized, axially sideband cooled, and radially doppler cooled into a single nuclear spin state

230 at temperatures of 800 nK. The lattice is then adiabatically ramped to the operational trap

231 depth of 12 Erec, where a series of pulses addressing the clock transition prepares atoms into
12

232

. Clock spectroscopy is performed by interrogating the

to

233

transition, the most magnetically insensitive 87Sr clock transition6.

234 Imaging

235

The clock excitation fraction is read out using standard fluorescence spectroscopy

236 techniques6,10,37. Photons are collected on both a photo-multiplier tube for global readout and

237 electron multiplying charge coupled device camera for an in-situ readout of clock frequency.

238 Camera readout is performed in full vertical binning mode, averaging the radial dimension of

239 the lattice. This provides 1D in-situ imaging for all synchronous evaluations.

240

We use a 25 μs fluorescence probe with an intensity of I/Isat ~ 20 (Isat being the

241 saturation intensity), ensuring uniform scattering across the atomic sample. Before imaging,

242 the optical lattice is ramped back to 300 Erec to decrease imaging aberration resulting from the

243 extended radial dimension at 12 Erec.

244 Analysis

245

Standard clock lock techniques and analysis are used6,11,33, with differences in

246 excitation fraction converted to frequency differences using Rabi lineshapes. Each dataset is

247 composed of a series of clock locks, tracking the center of mass frequency of the atomic

248 sample. A clock lock is four measurements probing alternating sides of the Rabi lineshape for

249 opposite nuclear spin transitions. Frequency corrections based on excitation fraction become

250 ambiguous when the excitation fraction measured is consistent with the Rabi lineshape at

251 multiple detunings. To avoid erroneous frequency corrections, we remove clock locks with

252 excitation fractions above (below) .903×C (.116×C), where C is the Rabi contrast. From each

13

253 clock lock, a pixel specific center frequency fi and frequency splitting between opposite mF 254 states Δi are calculated, creating an in-situ frequency map of the 1D atomic sample. This 255 allows rejection of vector shifts on a pixel-by-pixel basis and probes the magnetic field induced 256 splitting of mF transitions. The atom weighted mean frequency is subtracted from every lock 257 cycle to reject common mode laser noise.

258

For each dataset we approximate the atomic profile with a Gaussian fit, identifying a

259 center pixel and associated Gaussian width (σ). All analysis is performed within the central

260 region of ±1.5σ which demonstrates the lowest frequency instability. Identifying a center pixel

261 for data processing ensures rejection of any day-to-day drift in the position of the cloud due to

262 varying magnetic fields modifying MOT operation on the narrow line transition. The density

263 shift coefficient (see Density Shift section) is derived from the average center frequency per 264 pixel. Using this coefficient, we correct fi and Δi for the density shift. 2nd order Zeeman

265 corrections using these updated frequencies are then applied.

266

Gradient analysis is based on the processed center frequencies per pixel. A linear fit to

267 the frequencies as a function of pixel is performed using least squares, with uncertainty per 268 pixel arising from the quadrature sum of statistical frequency uncertainty, statistical 2nd order

269 Zeeman uncertainty, and density shift correction uncertainty.

270

For the two-clock comparison (Fig. 4b), all data from 8/14-8/22 was taken with the same

271 duty cycle and π-pulse time (3.1 s). Data was processed relative to a fit center pixel as

272 discussed and concatenated. Two equal regions extend from the center of the sample to a

273 width of ±1.5σ, with two empty pixels between regions to ensure uncorrelated samples. Each

274 region is processed for the atom weighted mean frequency, enabling a synchronous frequency

275 comparison between two independent clocks.
14

276 Atomic Coherence

277

We use a Ramsey sequence to measure the atomic coherence. We prepare a sample

278 in the

state and apply a π/2 pulse along the

to

279 transition. After waiting for a variable dark time, we apply a second π/2 pulse with a random 280 phase relative to the first. We then measure the excitation fraction.

281

Two regions, p1 and p2, are identified using the same technique as in the synchronous

282 instability measurement. For each experimental sequence, we find the average excitation

283 fraction in p1 and p2. A mean frequency shift across the sample primarily due to a magnetic

284 field gradient creates a differential phase as a function of time between p1 and p2. We create a

285 parametric plot of the average excitation in p1 and p2 for each dark time and use a maximum 286 likelihood estimator to fit an ellipse to each dataset, calculating phase and contrast15,38. To 287 estimate uncertainty in the contrast for each dark time a bootstrapping technique is used11.

288 Fitting the contrast as a function of dark time with a single exponential returns an effective

289 atomic coherence time.

290 Systematics

291 Imaging

292

We calibrate our pixel size using standard time of flight methods: we observe an atomic

293 sample in freefall for varying times to determine an effective pixel size along the direction of

294 gravity. Immediately after our 10-day data campaign we measured our effective pixel size to be

295 6.04 μm. Due to thermal drift of our system, the pixel size can vary by up to 0.5 μm /pixel over

296 months which we take as the calibration uncertainty.

15

297

Spatial correlations may limit imaging resolution. We measure these correlations by

298 placing atoms into a superposition of clock states. Any measured spatial correlation is due to 299 the imaging procedure. In our system we find no correlations between neighboring pixels11.

300 The optical resolution of our imaging lens is specified at 2 μm.

301

Lattice tilt from gravity will modify the measured gradient. We find the lattice tilt in the

302 imaging plane to be 0.11(0.06) degrees, providing an uncertainty orders of magnitude smaller

303 than the pixel size uncertainty. We are insensitive to lattice tilt out of the imaging plane.

304 Zeeman Shifts

305

First order Zeeman shifts are rejected by probing opposite nuclear spin states6. The 2nd

306 order Zeeman shift is given by

( )

307 where

is the splitting between opposite spin states and the corresponding 2nd order

308 Zeeman shift coefficient. For stretched spin state operation (

), =-2.456(3)×10-7 Hz-1.

309 Using known atomic coefficients39 we find the 2nd order Zeeman coefficient for the

310

to

transition to be =-1.23(8)×10-4 Hz, with the uncertainty arising from

311 limited knowledge of atomic coefficients.

312

The 2nd Order Zeeman corrections are made for every clock lock (analogous to the in-

313 situ density shift corrections). For a typical day (8/13) the average 2nd order Zeeman gradient

314 is -7.0×10-20/mm, corresponding to a splitting between opposite nuclear spin states of 12.7

315 mHz/mm (0.291 mG/mm). We include an error of 4×10-21/mm in Table 1 to account for the

316 atomic uncertainty in the shift coefficient.

16

317 DC Stark

318

Electric fields perturb the clock frequency via the DC Stark effect. We evaluate

319 gradients arising from this shift by using in-vacuum quadrant electrodes to apply bias electric 320 fields in all three dimensions. We find a DC Stark gradient of 3(2)×10-21/mm.

321 Black Body Radiation Shift

322

The dominant frequency perturbation to room temperature neutral atom clocks is black

323 body radiation (BBR). Similar to our previous work10, we homogenize this shift by carefully

324 controlling the thermal surroundings of our vacuum chamber. Attached to the vacuum chamber

325 are additional temperature control loops, with each vacuum viewport having a dedicated

326 temperature control system. This ensures our dominant BBR contribution – high emissivity

327 glass viewports – are all the same temperature to within 100 mK.

328

To bound possible BBR gradients, we introduce a 1 K gradient between the top and

329 bottom of the chamber along the cavity axis by raising either the top or bottom viewports by 1

330 K. We compare these two cases and find no statistically significant changes in the frequency

331 gradient across the entire sample. Accounting for uncertainty in linear frequency fits for each 332 case, we estimate an uncertainty of 3×10-21/mm. This finding is supported with a basic thermal

333 model of the vacuum chamber.

334 Density Shift

335

Atomic interactions during Rabi spectroscopy lead to clock frequency shifts as a

336 function of atomic density40. For each gradient measurement, we evaluate the density shift

337 coefficient

by fitting the average frequency per pixel versus average camera counts per

338 pixel to an equation of the form
17

339

(

.

340 Here B is an arbitrary offset. Once

is known, we remove the density shift at each pixel.

341

Residual density shift corrections may lead to error in our linear gradient. To bound this

342 effect, we compare the density shift coefficient and gradient from our data run with a separate 343 dataset at 8 Erec. With the trap depth at 8 Erec we found a linear gradient of s=-1.08×10-18/mm 344 and a density shift coefficient of =-1.39×10-6 Hz/count. During our data run we had an 345 average density shift coefficient of =-2.43×10-8 Hz/count. We bound the uncertainty in our

346 gradient from density shift as

|=1.7×10-20/mm.

347 Lattice Light Shifts

348

Lattice light shifts arise from differential AC Stark shifts between the ground and excited

349 clock states. An approximate microscopic model of the lattice light shift ( ) in our system is 350 given by41

351

(

(

) *

+ ,

352 where is the trap depth in units of Erec ,

the differential electric dipole polarizability,

353

the differential multi-polarizability, and

(

the detuning between lattice

354 frequency and effective magic frequency . Our model has no dependence on the

355 longitudinal vibrational quanta since we are in the ground vibrational band. We neglect higher

356 order corrections from hyperpolarizability due to our operation at depths <60 Erec. At our 357 temperatures thermal averaging of the trap depth is a higher order correction (<5%) that is also

358 neglected.

18

359

We model the linear differential lattice light shift across the atomic cloud as

360

(

(
[

)
] ,

361 where z is the coordinate corresponding to the axis of the cavity along gravity. To evaluate our 362 differential lattice light shift at our operational depth we need and . We modulate our 363 lattice between two trap depths (u1=14 Erec, u2=56 Erec) and find our detuning from scalar 364 magic frequency to be =7.4(0.6) MHz. To evaluate at our operational depth ( we 365 measure the linear gradient across the atomic cloud at + 250 MHz and - 250 MHz, the 366 difference given by

367

(

)

(

) [

]

,

368 where = 500 MHz. We find

0.0383/mm, which when combined with =7.4 MHz,

369 gives us a fractional frequency gradient of -5×10-21/mm. Accounting for error in our lattice 370 detuning and linear gradient gives us an uncertainty of 1×10-21/mm.

371 Other Systematics

372 For a 3.1 s π-pulse the probe AC Stark shift7 is -3(2)×10-21. A frequency scan of the

373

to

transition limits the variation of excitation fraction across the atomic

374 sample to 1% or below, bounding any possible probe AC Stark gradient across the sample to 375 <1×10-22.

376 Known Redshift

19

377

The gravitational acceleration (rounding to 4 digits) within our lab was evaluated by a

378 USGS survey42 to be a=-9.796 m/s2.

379 Systematic Budget
Systematic Gradient (Fig. 3) BBR Density Lattice light shift DC Stark Pixel Calibration 2nd Order Zeeman Other Corrected Gradient Known Redshift
380

Slope (10-20/mm) -10.0 0 -0.5 0.3 0 0 -9.8 -10.9

Uncertainty (10-20/mm) 1.2 0.3 1.7 0.1 0.2 .8 .4 <.1 2.3 <.1

381 Table 1: Gradient Systematic Budget. Fractional frequency gradients and corresponding 382 uncertainties. Fractional frequencies denoted with ‘-‘ are corrected on a pixel-by-pixel basis 383 during initial data processing (Fig. 3a). The corrected gradient has known systematics 384 removed with uncertainty given by the quadrature sum of all correction uncertainties.

385

386 Acknowledgements

20

387

We acknowledge funding support from Defense Advanced Research Projects Agency,

388 National Science Foundation QLCI OMA-2016244, DOE Quantum System Accelerator, NIST,

389 NSF Phys-1734006, and Air Force Office for Scientific Research. We are grateful for theory

390 insight from Anjun Chu, Peiru He, and Ana María Rey. We acknowledge stimulating discussion

391 and technical contributions from John Zaris, James Uhrich, Josephine Meyer, Ross Hutson,

392 Christian Sanner, William Milner, Lindsay Sonderhouse, Lingfeng Yan, Maya Miklos, Yee Ming

393 Tso, and Shimon Kolkowitz. We thank James Thompson, Cindy Regal, John Hall, and Serge

394 Haroche for careful reading of the manuscript.

395 Authors' note: While performing the work described here, we became aware of 396 complementary work where high measurement precision was achieved for simultaneous 397 differential clock comparisons between multiple atomic ensembles in vertical 1D 398 lattices separated by centimeter scale distances using a hertz-linewidth clock laser43.

399 Author Contributions

400 All authors contributed to carrying out the experiments, interpreting the results, and writing the 401 manuscript.

402 *tobias.bothwell@colorado.edu, ye@jila.colorado.edu

403 Competing interests

404 The authors declare no competing interests. 405 Data and Code Availability 406 The experimental data and code analysis are available from the corresponding authors upon 407 reasonable request.
408 21

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498

499

24

a Counts

E lattice

Si cavity

b 1.0
0.8

Excitation Fraction

0.6

B

g

0.4

0.2

z

E clock
c

… |e,0
|e,-m

… |e,+m

Ω Ω-m
…

Ω+m
…

|g,0 |g,-m

|g,+m

0.0 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0
Frequency (Hz)
d 1.0

0.8

-1

+1

-2

+2

0.6

Excitation Fraction

0.4

0.2

-3

+3

-5 -4
0.0 −4

+4 +5 −2 Frequen0cyL(okrHemz) i2psum 4

Excitation Fraction Contrast
p2

a 1.0

b 1 6s

30 s

p1

p2

0.8

0.6

c 00 1.0

p1

1

0.8

0.4

0.6

50 s

0.4 0.2

0.2

τ=36.5(0.7) s

0.0

−0.5 0.0

0.5

z (mm)

0.0 0

20

40

60

Dark Time (s)

a 1.00

0.75

0.50 1
0.25
0 0.00

−0.25

−0.50

−0.75

−1.00 −0.4 −0.2 0.0 0.2 0.4

z (mm)

b3

Fractional Frequency (10-18)
Fractional Frequency Gradient (10-19/mm)
000000088888880000000///////88888882122212///////152125112111117034698CBBBAAA
Gravitational Acceleration (m/s2)

2

−1

−2

−3

20 10 0 −10 −20 −30
Measurement Date

z (mm) Fractional Frequency Uncertainty

a
-0.5 0.0 0.5

b
10-18
10-19

4.4×10−18/√ τ 3.1×10−18/√ τ 2.6×10−18/√ τ

10-20

102

103

104

105

Time (s)