An atomic array optical clock with single-atom readout
Ivaylo S. Madjarov, Alexandre Cooper, Adam L. Shaw, Jacob P. Covey, Vladimir Schkolnik, Tai Hyun Yoon, Jason R. Williams, Manuel Endres
AAn Atomic Array Optical Clock with Single-Atom Readout
Ivaylo S. Madjarov, Alexandre Cooper, Adam L. Shaw, Jacob P. Covey, Vladimir Schkolnik, Tai Hyun Yoon, ∗ Jason R. Williams, and Manuel Endres † Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Currently, the most accurate and stable clocks use optical interrogation of either a single ion oran ensemble of neutral atoms confined in an optical lattice. Here, we demonstrate a new opticalclock system based on an array of individually trapped neutral atoms with single-atom readout,merging many of the benefits of ion and lattice clocks as well as creating a bridge to recentlydeveloped techniques in quantum simulation and computing with neutral atoms. We evaluate single-site resolved frequency shifts and short-term stability via self-comparison. Atom-by-atom feedbackcontrol enables direct experimental estimation of laser noise contributions. Results agree well withan ab initio
Monte Carlo simulation that incorporates finite temperature, projective read-out, lasernoise, and feedback dynamics. Our approach, based on a tweezer array, also suppresses interactionshifts while retaining a short dead time, all in a comparatively simple experimental setup suited fortransportable operation. These results establish the foundations for a third optical clock platformand provide a novel starting point for entanglement-enhanced metrology, quantum clock networks,and applications in quantum computing and communication with individual neutral atoms thatrequire optical clock state control.
I. INTRODUCTION
Optical clocks — based on interrogation of ultra-narrow optical transitions in ions or neutral atoms —have surpassed traditional microwave clocks in both rel-ative frequency stability and accuracy [1–4]. They en-able new experiments for geodesy [2, 5], fundamentalphysics [6, 7], and quantum many-body physics [8], inaddition to a prospective redefinition of the SI sec-ond [9]. In parallel, single-atom detection and controltechniques have propelled quantum simulation and com-puting applications based on trapped atomic arrays; inparticular, ion traps [10], optical lattices [11], and opti-cal tweezers [12, 13]. Integrating such techniques into anoptical clock would provide atom-by-atom error evalu-ation, feedback, and thermometry [14]; facilitate quan-tum metrology applications, such as quantum-enhancedclocks [15–18] and clock networks [19]; and enable novelquantum computation, simulation, and communicationarchitectures that require optical clock state control com-bined with single atom trapping [20–22].As for current optical clock platforms, ion clocksalready incorporate single-particle detection and con-trol [23], but they typically operate with only a singleion. Research towards multi-ion clocks is ongoing [24].Conversely, optical lattice clocks (OLCs) [1, 2, 4] interro-gate thousands of atoms to improve short-term stability,but single-atom detection and control remains an out-standing challenge. An ideal clock system, in this context,would thus merge the benefits of ion and lattice clocks;namely, a large array of isolated atoms that can be readout and controlled individually. ∗ Permanent address: Department of Physics, Korea University,Seoul 02841, Republic of Korea † [email protected] Here we present a prototype of a new optical clock plat-form based on an atomic array which naturally incorpo-rates single-atom readout of currently ≈
40 individuallytrapped neutral atoms. Specifically, we use a magic wave-length 81-site tweezer array stochastically filled with sin-gle strontium-88 ( Sr) atoms [25]. Employing a repeti-tive imaging scheme [25], we stabilize a local oscillator tothe optical clock transition [26, 27] with a low dead timeof ≈
100 ms between clock interrogation blocks.We utilize single-site and single-atom resolution to eval-uate the in-loop performance of our clock system in termsof stability, local frequency shifts, selected systematic ef-fects, and statistical properties. To this end, we define anerror signal for single tweezers which we use to measuresite-resolved frequency shifts at otherwise fixed parame-ters. We also evaluate statistical properties of the in-looperror signal, specifically, the dependence of its varianceon atom number and correlations between even and oddsites.We further implement a standard interleaved self-comparison technique [28, 29] to evaluate systematic fre-quency shifts with changing external parameters – specif-ically trap depth and wavelength – and find an oper-ational magic condition [30–32] where the dependenceon trap depth is minimized. We further demonstratea proof-of-principle for extending such self-comparisontechniques to evaluate single-site-resolved systematic fre-quency shifts as a function of a changing external param-eter.Using self-comparison, we evaluate the fractional short-term instability of our clock system to be 2 . × − / √ τ .To compare our experimental results with theory predic-tions, we develop an ab initio Monte Carlo (MC) clocksimulation (Appendix A), which directly incorporateslaser noise, projective readout, finite temperature, andfeedback dynamics, resulting in higher predictive powercompared to traditionally used analytical methods [1]. a r X i v : . [ phy s i c s . a t o m - ph ] O c t Our experimental data agree quantitatively with thissimulation, indicating that noise processes are well cap-tured and understood at the level of stability we achievehere. Based on the MC model, we predict a fractionalinstability of (1.9–2.2) × − / √ τ for single clock oper-ation, which would have shorter dead time than that inself-comparison.We further demonstrate a direct evaluation of the1 / √ N A dependence of clock stability with atom num-ber N A , on top of a laser noise dominated background,through an atom-by-atom system-size-selection tech-nique. This measurement and the MC model stronglyindicate that the instability is limited by the frequencynoise of our local oscillator. We note that the measuredinstability is comparable to OLCs using similar trans-portable laser systems [33].We note the very recent, complementary results ofRef. [34] that show seconds-long coherence in a tweezerarray filled with ≈ Sr atoms using an ultra-low noiselaser without feedback operation. In this and our sys-tem, a recently developed repetitive interrogation pro-tocol [25], similar to that used in ion clocks, providesa short dead time of ≈
100 ms between interrogationblocks, generally suppressing the impact of laser noiseon stability stemming from the Dick effect [35]. Utiliz-ing seconds-scale interrogation with such low dead timescombined with the feedback operation and realistic up-grade to the system size demonstrated here promises aclock stability that could reach that of state-of-the-artOLCs [2, 4, 36, 37] in the near-term future, as furtherdiscussed in the outlook section.Concerning systematic effects, the demonstratedatomic array clock has intrinsically suppressed interac-tion and hopping shifts: First, single atom trapping intweezers provides immunity to on-site collisions presentin one-dimensional OLCs [38]. While three-dimensionalOLCs [36] also suppress on-site collisions, our approachretains a short dead time as no evaporative cooling isneeded. Further, the adjustable and significantly largerinteratomic spacing strongly reduces dipolar interac-tions [39] and hopping effects [40]. We experimentallystudy effects from tweezer trapping in Sec. IV and de-velop a corresponding theoretical model in Appendix E,but leave a full study of other systematics, not specificto our platform, and a statement of accuracy to futurework. In this context, we note that our tweezer system iswell-suited for future investigations of black-body radia-tion shifts via the use of local thermometry with Rydbergstates [14].The results presented here and in Ref. [34] provide thefoundation for establishing a third optical clock plat-form promising competitive stability, accuracy, and ro-bustness, while incorporating single-atom detection andcontrol techniques in a natural fashion. We expect thisto be a crucial development for applications requiringadvanced control and read-out techniques in many-atomquantum systems, as discussed in more detail in the out-look section.
II. FUNCTIONAL PRINCIPLE
The basic functional principle is as follows. We gener-ate a tweezer array with linear polarization and 2 . µ msite-to-site spacing in an ultra-high vacuum glass cellusing an acousto-optic deflector (AOD) and a high-resolution imaging system (Fig. 1a)[25]. The tweezer ar-ray wavelength is tuned to a magic trapping configu-ration close to 813 . ≈
40 of the tweezers arestochastically filled with a single atom. We use a recentlydemonstrated narrow-line Sisyphus cooling scheme [25]to cool the atoms to an average transverse motional oc-cupation number of ¯ n ≈ .
66, measured with clock side-band spectroscopy (Appendix B 7). The atoms are theninterrogated twice on the clock transition, once below ( A )and once above ( B ) resonance, to obtain an error signalquantifying the frequency offset from the resonance cen-ter (Fig. 1b,c). We use this error signal to feedback toa frequency shifter in order to stabilize the frequency ofthe interrogation laser — acting as a local oscillator —to the atomic clock transition. Since our imaging schemehas a survival fraction of > A and B interro-gation blocks (Fig. 1e) [25]. A first image determines ifa tweezer is occupied, followed by clock interrogation. Asecond image, after interrogation, determines if the atomhas remained in the ground state | g (cid:105) . This yields an in-stance of an error signal for all tweezers that are occupiedat the beginning of both interrogation blocks, while un-occupied tweezers are discounted. For occupied tweezers,we record the | g (cid:105) occupation numbers s A,j = { , } and s B,j = { , } in the images after interrogation with A and B , respectively, where j is the tweezer index. Thedifference e j = s A,j − s B,j defines a single-tweezer errorvariable taking on three possible values e j = {− , , +1 } indicating interrogation below, on, or above resonance,respectively. Note that the average of e j over many inter-rogations, (cid:104) e j (cid:105) , is simply an estimator for the differencein transition probability between blocks A and B .For feedback to the clock laser, e j is averaged over alloccupied sites in a single AB interrogation cycle, yieldingan array-averaged error ¯ e = N A (cid:80) j e j , where the sumruns over all occupied tweezers and N A is the number ofpresent atoms. We add ¯ e times a multiplicative factor tothe frequency shifter, with the magnitude of this factoroptimized to minimize in-loop noise. III. IN-LOOP SPECTROSCOPIC RESULTS
We begin by describing results for in-loop detection se-quences. Here, feedback is applied to the clock laser (as -20 -10 0 10 20Frequency (Hz)0.00.51.0 P r o b a b ili t y A B-0.80.00.8 M ea n e rr o r e j ) 0 NaN NaN Feedback f FIG. 1. Atomic array optical clock. (a) We interrogate ≈ Sr atoms, trapped in an 81-site tweezer array, on the ultra-narrow clock transition at 698 nm and use high-resolution fluorescence imaging at 461 nm to detect population changes inthe clock states (labeled | g (cid:105) and | e (cid:105) ) with single-atom resolution. This information is processed by a central processing unit(CPU) and a feedback signal is applied to the clock laser frequency using an acousto-optic modulator (AOM). (b) Tweezer-averaged probability to remain in | g (cid:105) as a function of frequency offset measured with an in-loop probe sequence (circles). Dashedhorizontal lines indicate state-resolved detection fidelities (Appendix B 5). To generate an error signal, we interrogate twice:once below ( A ) and once above ( B ) resonance. (c) Tweezer-averaged error signal as a function of frequency offset (circles).The shaded areas in (b) and (c) show results from MC simulations. (d) Simplified experimental sequence, consisting of tweezerloading and N -times-repeated AB feedback blocks followed by an optional probe block, with N = 10 throughout. (e) To detectthe clock state population in block A , we take a first image before interrogation to identify which tweezers are occupied anda second image after interrogation to detect which atoms remain in | g (cid:105) (images 1 and 2). The same procedure is repeated forblock B (images 3 and 4). We show fluorescence images with identified atoms (circles) (Appendix B 4) and examples of singletweezer error signals e j . described before) and probe blocks, for which the interro-gation frequency is varied, are added after each feedbackcycle. Using a single probe block with an interrogationtime of 110 ms (corresponding to a π -pulse on resonance)shows a nearly Fourier-limited line-shape with full-widthat half-maximum of ≈ A and B interrogation frequencies spaced by a to-tal of 7 . (cid:104) e j (cid:105) for all 81 traps (Fig. 2a) as a function of fre-quency offset.Fitting the zero-crossings of (cid:104) e j (cid:105) enables us to detectdifferences in resonance frequency with sub-Hz resolution(Fig. 2b). The results show a small gradient across the ar-ray due to the use of an AOD: tweezers are spaced by 500kHz in optical frequency, resulting in an approximately linear variation of the clock transition frequency. This ef-fect could be avoided by using a spatial light modulatorfor tweezer array generation [42]. We note that the totalfrequency variation is smaller than the width of our in-terrogation signal. Such “sub-bandwidth” gradients canstill lead to noise through stochastic occupation of siteswith slightly different frequencies; in our case, we pre-dict an effect at the 10 − level. We propose a methodto eliminate this type of noise in future clock iterationswith a local feedback correction factor in Appendix D 3.Before moving on, we note that e j is a random variablewith a ternary probability distribution (Fig. 2c) definedfor each tweezer. The results in Fig. 2a are the mean ofthis distribution as a function of frequency offset. In ad-dition to such averages, having a fully site-resolved signalenables valuable statistical analysis. As an example, weextract the variance of ¯ e , σ e , for an in-loop probe se-quence where the probe blocks are centered around res-onance.Varying the number of atoms taken into account (viapost-selection) shows a 1 /N A scaling with a pre-factordominated by quantum projection noise (QPN) [1] on topof an offset stemming mainly from laser noise (Fig. 2d). Amore detailed analysis reveals that, for our atom number,the relative noise contribution from QPN to σ ¯ e is only ≈
26% (Appendix C). A similar conclusion can be drawnon a qualitative level by evaluating correlations betweentweezer resolved errors from odd and even sites, which F r e q u e n c y ( H z ) -1 0 150 e j P ( e j ) ( % )
01 10-1 σ ( - ) N A -0.610.0 < e j > F r e q u e n c y ( H z ) T w ee z e r i n d e x Odd-sites error E v e n - s i t e s e rr o r -1 FIG. 2. Site-resolved error signal. (a) Repetition-averagedsingle-tweezer error signal (cid:104) e j (cid:105) as a function of frequency offsetmeasured with an in-loop sequence. (b) Fitted zero-crossingsas a function of tweezer index for our usual interrogation trapdepth of U = 245(31) E r where E r = h × .
43 kHz (circles).Solid lines correspond to theory predictions, with the shadedarea resulting from systematic uncertainty in trap depth (Ap-pendix E). (c) Ternary probability distribution for e j for a se-lected tweezer. The vertical dashed line shows the mean. (d)Variance of the error signal as a function of atom number, cal-culated through post-selection. Solid line is a fit with a 1 /N A function plus an offset. Purple region is a MC simulation. (e)Plot of correlations between the error signals of even and oddsites. show a strong common mode contribution indicative ofsizable laser noise (Fig. 2e). IV. SELF-COMPARISON FOR EVALUATIONOF SYSTEMATIC SHIFTS FROM TWEEZERTRAPPING
We now turn to an interleaved self-comparison [28, 29],which we use for stability evaluation and systematic stud-ies. The self-comparison consists of having two feedbackloops running in parallel, where feedback is given in analternating fashion to update two independent AOM fre-quencies f and f (Fig. 3a). This is used for a lock-intype evaluation of clock frequency changes with varyingparameters. As a specific example, we operate the clockwith our usual interrogation trap depth U during blocksfor feedback to f and with a different trap depth U during blocks for feedback to f . The average frequencydifference f − f now reveals a shift of the clock op-eration frequency dependent on U (Fig. 3b). For opti-mal clock operation, we find an “operationally magic” Feedback f Feedback f -3036 -71 MHz-26 MHz-7 MHz20 MHz65 MHz110 MHz Relative trap depth ( U / U ) FIG. 3. Systematic evaluation of clock shifts with tweezerdepth and wavelength. (a) Illustration of interleaved self-comparison, where two independent AOM frequencies ( f and f ) are updated in an alternating fashion. Respective interro-gation blocks are set to two independent tweezer depths U and U . (b) Average frequency difference f − f as functionof U /U , with U fixed to our usual interrogation depth, formultiple frequency offsets of the trapping laser (see legend forcolor coding). We fit the data with a model for light shifts inoptical tweezers (colored lines) with only a single free param-eter (for all data simultaneously), accounting for an unknownfrequency offset (Appendix E). Operational magic intensitiesare found at the minima of these curves (gray squares andconnecting line), which minimize sensitivity to trap depthfluctuations. The trap laser frequency is tuned such that theminimum coincides with our nominal depth. (c) Combiningthis technique with the single-tweezer resolved error (cid:104) e j (cid:105) , wecan extract a frequency dependence with trap depth for eachtweezer (colored squares). Solid lines show the expected de-pendence for the outermost and central tweezers. The datacorresponds to the − U /U = 10. The color coding of the inset definesthe color coding of its containing sub-figure. condition that minimizes sensitivity to trap depth fluc-tuations [30–32] by performing two-lock comparisons fordifferent wavelengths (Fig. 3b) (Appendix E). We notethat this type of standard self-comparison can only re-veal array-averaged shifts.In this context, an important question is how such lock-in techniques can be extended to reveal site-resolved sys-tematic errors as a function of a changing external pa-rameter. To this end we combine the tweezer resolved er-ror signal (cid:104) e j (cid:105) with interleaved self-comparison (Fig. 3c).Converting (cid:104) e j (cid:105) to frequencies (using measured errorfunctions, such as in Fig. 2a) yields frequency estimators δf ,j and δf ,j for each tweezer during f and f feedbackblocks, respectively. These estimators correspond to therelative resonance frequency of each tweezer with respect Averaging time τ (s) -15 -16 -17 Number of atoms N A N A FIG. 4. Stability results. (a) Fractional Allan deviation σ y obtained via self-comparison as a function of integration time τ (circles). Fitting a 1 / √ τ behavior past an initial lock onsettime (red solid line), we find 2 . × − / √ τ . The shaded areadenotes MC results. The purple solid line shows the quantumprojection noise limit obtained from MC by switching off allother noise sources. (b) Based on atom-by-atom feedbackcontrol, we perform a series of self-comparisons with fixedatom number N A . Shown is the Allan variance σ y at one sec-ond (from a 1 / √ τ fit) as a function of N A . Inset: Allan vari-ance σ y as a function of 1 /N A . Solid lines show a fit with afunctional form σ y = σ ∞ + σ N A , where σ N A scales as 1 / √ N A . to the center frequency of the individual locks. Plottingthe quantity δf ,j − δf ,j + f − f then shows the ab-solute frequency change of each tweezer as a function oftrap depth (Fig. 3c). V. SELF-COMPARISON FOR STABILITYEVALUATION
We use the same self-comparison sequence to evaluatethe fractional clock instability by operating both lockswith identical conditions (Fig. 4a). This approach fol-lows previous clock studies, where true comparison toa second, fully independent clock system was not avail-able [28, 29]. We plot the Allan deviation σ y [43] of y = ( f − f ) / ( ν √
2) in Fig. 4a, where ν is the clocktransition frequency and the √ / √ τ behavior after alock onset time, where τ is the averaging time in sec-onds. Fitting this behavior yields σ y = 2 . × − / √ τ ,in excellent agreement with MC simulations (Fig. 4a).Self-comparison evaluates how fast averaging can beperformed for systematic studies — such as the oneshown in Fig. 3 — and reveals the impact of various noisesources on short-term stability; however, by design, thistechnique suppresses slow drifts that are common to the f and f interrogation blocks. We performed a separatestability analysis by locking f to the left half of the ar-ray and f to the right half of the array [36], a methodwhich is sensitive to slow drifts of gradients, and found nolong-term drift of gradients to within our sensitivity (Ap-pendix D 2).Having shown good agreement between our data andMC simulations, we are able to further use the simulationto predict properties of our clock that are not directlyexperimentally accessible. One of these properties is thetrue stability of the local oscillator frequency, computeddirectly by taking the Allan deviation of the simulatedlaser frequency time traces under feedback. This allowsto simulate the stability of single clock operation, whichhas shorter dead time than the double clock scheme thatwe use to evaluate stability in experiment. Following thisprotocol, our simulations predict (1.9–2.2) × − / √ τ forthe local oscillator stability during single clock opera-tion (Appendix A). In this context, we note the resultsof Ref. [34], where stability is evaluated by convertinga spectroscopic signal into a frequency record (withouta closed feedback loop). Based on interrogation with anultra-low noise laser system, they achieve a short-termstability of 4.7 × − / √ τ with ≈ / √ N A through a reduction in readout-noise as long as atoms are uncorrelated. However, in thepresence of laser noise — which is common mode to allatoms — a limit to stability exists even for an infinitenumber of atoms [1]. Intriguingly, we can directly ex-tract such contributions by performing a series of self-comparisons where we adjust the atom number one-by-one (Fig. 4b). To this end, we restrict the feedback op-eration to a subset of atoms in the center of the arraywith desired size, ignoring the remainder. We are ableto achieve stable locking conditions for N A ≥ N A and fit the re-sults with a function σ y = σ ∞ + σ N A , where σ N A scalesas 1 / √ N A . We find σ N A = 6 . × − / √ N A · τ and σ ∞ = 2 . × − / √ τ , the latter being an estimator forthe limit of our clock set by laser noise, in agreement withMC simulation. VI. OUTLOOK
Our results merge single-particle readout and controltechniques for neutral atom arrays with optical clocksbased on ultra-narrow spectroscopy. Such atomic arrayoptical clocks (AOCs) could approach the sub-10 − / √ τ level of stability achieved with OLCs [2, 4, 36, 37] by in-creasing interrogation time and atom number. Reachingseveral hundreds of atoms is realistic with an upgrade totwo-dimensional arrays, while Ref. [34] already demon-strated seconds-long interrogation. A further increase inatom number is possible by using a secondary array forreadout, created with a non-magic wavelength for whichhigher power lasers exist [41, 44]. We also envision a sys-tem where tweezers are used to “implant” atoms, in astructured fashion, into an optical lattice for interroga-tion and are subsequently used to provide confinementfor single-atom readout. Further, the lower dead time ofAOCs should help to reduce laser noise contributions toclock stability compared to 3d OLCs [36], and even zerodead time operation [36, 37] in a single machine is con-ceivable by adding local interrogation. Local interroga-tion could be achieved through addressing with the mainobjective or an orthogonal high-resolution path by us-ing spatial-light modulators or acoustic-optic devices. Forthe case of addressing through the main objective, atomswould likely need to be trapped in an additional latticeto increase longitudinal trapping frequencies.Concerning systematics, AOCs provide fully site-resolved evaluation combined with an essential mitiga-tion of interaction shifts, while being ready-made for im-plementing local thermometry using Rydberg states [14]in order to more precisely determine black-body inducedshifts [1]. In addition, AOCs offer an advanced toolsetfor generation and detection of entanglement to reach be-yond standard quantum limit operation — either throughcavities [16, 45] or Rydberg excitation [15] — and forimplementing quantum clock networks [19]. Further, thedemonstrated techniques provide a pathway for quantumcomputing and communication with neutral alkaline-earth-like atoms [8, 20, 22]. Finally, features of atomicarray clocks, such as experimental simplicity, short deadtime, and three-dimensional confinement, make these sys-tems attractive candidates for robust portable clock sys-tems and space-based missions [31]. ACKNOWLEDGMENTS
We acknowledge funding provided by the Institute forQuantum Information and Matter, an NSF Physics Fron-tiers Center (NSF Grant PHY-1733907). This work wassupported by the NSF CAREER award (1753386), by theAFOSR YIP (FA9550-19-1-0044), and by the Sloan Foun-dation. This research was carried out at the Jet Propul-sion Laboratory and the California Institute of Tech-nology under a contract with the National Aeronauticsand Space Administration and funded through the Pres-idents and Directors Research and Development Fund(PDRDF). JPC acknowledges support from the PMAPrize postdoctoral fellowship, AC acknowledges supportfrom the IQIM Postdoctoral Scholar fellowship, and THYacknowledges support from the IQIM Visiting Scholarfellowship and from NRF-2019009974. We acknowledgegenerous support from Fred Blum.
APPENDIX A: Monte Carlo simulation1. Operation
We compare the performance of our clock to MonteCarlo (MC) simulations. The simulations include the ef-fects of laser frequency noise, dead time during loading and between interrogations, quantum projection noise,finite temperature, stochastic filling of tweezers, andexperimental imperfections such as state-detection infi-delity and atom loss. The effects of Raman scatteringfrom the trap and of differential trapping due to hyper-polarizability or trap wavelength shifts from the AOD arenot included as they are not expected to be significant atour level of stability.Rabi interrogation is simulated by time evolving aninitial state | g (cid:105) with the time-dependent Hamiltonianˆ H ( t ) = (cid:126) (Ω σ x + ( δ ( t ) ± δ o ) σ z ), where Ω is the Rabifrequency, δ o is an interrogation offset, and δ ( t ) is theinstantaneous frequency noise defined such that δ ( t ) = dφ ( t ) dt , where φ ( t ) is the optical phase in the rotatingframe. The frequency noise δ ( t ) for each Rabi interro-gation is sampled from a pre-generated noise trace (Ap-pendix A 2, A 3) with a discrete timestep of 10 ms. Deadtime between interrogations and between array refillingis simulated by sampling from time-separated intervals ofthis noise trace. Stochastic filling is implemented by sam-pling the number of atoms N A from a binomial distribu-tion on each filling cycle, and atom loss is implementedby probabilistically reducing N A between interrogations.To simulate finite temperature, a motional quantumnumber n is assigned to each of the N A atoms be-fore each interrogation, where n is sampled from a 1dthermal distribution using our experimentally measured¯ n ≈ .
66 (Appendix B 7). Here, n represents the mo-tional quantum number along the axis of the interro-gating clock beam. For each of the unique values of n that were sampled, a separate Hamiltonian evolution iscarried out with a modified Rabi frequency given byΩ n = Ω e − η L n ( η ) [46], where η = πλ clock (cid:113) (cid:126) mω is theLamb-Dicke parameter, L n is the n -th order Laguerrepolynomial, and Ω is the bare Rabi frequency valid inthe limit of infinitely tight confinement.At the end of each interrogation, excitation probabili-ties p e ( n ) = |(cid:104) e | ψ n (cid:105)| are computed from the final statesfor each n . State-detection infidelity is simulated by defin-ing adjusted excitation probabilities ˜ p e ( n ) ≡ f e p e ( n ) +(1 − f g )(1 − p e ( n )), where f g and f e are the ground andexcited state detection fidelities (Appendix B 5), respec-tively. To simulate readout of the the j -th atom on the i -th interrogation, a Bernoulli trial with probability ˜ p e ( n j )is performed, producing a binary readout value s j,i . Anerror signal ¯ e = N A (cid:80) j ( s j,i − − s j,i ) is produced ev-ery two interrogation cycles by alternating the sign of δ o on alternating interrogation cycles. This error signalproduces a control signal (using the same gain factor asused in experiment) which is summed with the gener-ated noise trace for the next interrogation cycle, closingthe feedback loop.
2. Generating frequency noise traces
Using a model of the power spectral density of our clocklaser’s frequency noise (Sec A 3), we generate random fre- S ν ( f ) ( H z / H z ) Frequency f (Hz)10 -1 -2 -3 -2 -1 FIG. 5. Frequency noise spectrum of the clock laser. Powerspectral density of the frequency noise of our clock laser mea-sured from a beat signal with a reference laser over a 42-hourperiod (red trace). Our theoretical estimate of the thermalnoise contribution is plotted in yellow. Plotted also are ourbest- (purple) and worst- (blue) case models for total fre-quency noise, as used in Monte Carlo simulations. quency noise traces in the time domain [47] for use in theMonte Carlo simulation. Given the power spectral den-sity of frequency noise S ν ( f ), we generate a complex one-sided amplitude spectrum A ν ( f ) = e iφ ( f ) (cid:112) S ν ( f )∆ f ,where φ ( f ) is sampled from a uniform distribution in[0 , π ) for each f and ∆ f is the frequency discretiza-tion. This is converted to a two-sided amplitude spec-trum by defining A ν ( − f ) = A ∗ ν ( f ). Finally, a time trace ν ( t ) = F{ A ( f ) } ( t ) + ν l ( t ) is produced by taking a fastFourier transform (FFT) of A ( f ) and adding an experi-mentally calibrated linear drift term ν l ( t ).
3. Frequency noise model
The power spectral density of the frequency noise ofour clock laser is modeled by the sum of contributionsfrom random walk frequency modulation (RWFM) noise( f − ), flicker frequency modulation (FFM) noise ( f − ),and white frequency modulation (WFM) noise ( f ), suchthat S ν ( f ) = αf − + βf − + γf . We obtain these pa-rameters through an estimation of the thermal noise ofour reference cavity and a fit of a partially specified fre-quency noise power spectral density obtained via beatingour laser with a reference laser (Fig. 5). Due to remaininglarge uncertainty in the white noise floor of our laser, wedefine a worst- and best-case noise model. The range be-tween these models is the dominant source of uncertaintyin our Monte Carlo simulations.FFM noise results from thermal mechanical fluctua-tions of the reference cavity [48, 49]. By estimating thenoise contribution from the ultra-low expansion spacer,fused silica mirrors, and their reflective coating, we esti-mate a fractional frequency instability of σ y = 1 . × − at 1 s, which corresponds to a frequency noise power spec-tral density of βf − = 0 .
34 Hz / Hz at f = 1 Hz.As a worst case noise model, we assume a cross-overfrequency from FFM to WFM noise at 1 Hz (Fig. 5),such that γ = βf − = 0 .
34 Hz / Hz, and we estimatea frequency noise power spectral density of αf − =0 .
05 Hz / Hz at 1 Hz for RWFM noise. As a best case noise model, assuming no cross-over from FFM to WFMnoise (such that γ = 0 .
00 Hz / Hz) we estimate a fre-quency noise power spectral density for RWFM noise of αf − = 0 .
08 Hz / Hz at f = 1 Hz. We note that the dif-ference in predicted clock stability between the best andworst case model is relatively minor. This indicates thatdominant contributions to clock instability stem fromfrequencies where we have experimental frequency noisedata and where both models exhibit similar frequencynoise. This is confirmed by an analytical Dick noise anal-ysis [35] (not shown). APPENDIX B: Experimental details1. Experimental system
Our strontium apparatus is described in detail inRefs. [25, 41]. Strontium-88 atoms from an atomic beamoven are slowed and cooled to a few microkelvin temper-ature by a 3d magneto-optical trap operating first on thebroad dipole-allowed S ↔ P transition at 461 nm andthen on the narrow spin-forbidden S ↔ P transitionat 689 nm. Strontium atoms are filled into a 1d array of81 optical tweezers at λ T = 813 . S ↔ P opticalclock transition. The tweezers have Gaussian waist radiiof 800(50) nm and an array spacing of 2 . µ m. Duringfilling, cooling, and imaging (state detection), the trapdepth is 2447(306) E r . Here E r is the tweezer photonrecoil energy, given by E r = h / (2 mλ T ), where h isPlanck’s constant and m is the mass of Sr. The tweezerdepth is determined from the measured waist and theradial trapping frequency found from sideband measure-ments on the clock transition (discussed in more detail inAppendix B 7). After parity projection, each tweezer hasa 0.5 probability of containing a single atom, or otherwisebeing empty. Thus, the total number of atoms N A aftereach filling cycle of the experiment follows a binomialdistribution with mean number of atoms ¯ N A = 40 .
2. Clock laser system
Our clock laser is based on a modified portable clocklaser system (Stable Laser Systems) composed of an ex-ternal cavity diode laser (Moglabs) stabilized to an iso-lated, high-finesse optical cavity using the Pound-Drever-Hall scheme and electronic feedback to the laser diodecurrent and piezoelectric transducer. The optical cavityis a 50 mm cubic cavity [50] made of ultra-low expan-sion glass maintained at the zero-crossing temperature of40 . ◦ C with mirror substrates made of fused silica witha finesse of
F > ,
000 at 698 nm. The clock laser lightpasses through a first AOM in double-pass configura-tion, injects an anti-reflection coated laser diode (SacherLasertechnik GmbH, SAL-0705-020), passes through asecond AOM, and goes through a 10 m long fiber to themain experiment with a maximum output optical power P r o b a b ili t y FIG. 6. Rabi oscillations on the clock transition with π -pulselength of 110 ms. Each point is probed directly after stabiliz-ing the clock laser with a feedback sequence as described inthe main text. The shaded area denotes Monte Carlo results. of 20 mW. The first AOM is used for shifting and stabi-lizing the frequency of the clock laser, whereas the secondAOM is used for intensity-noise and fiber-noise cancella-tion. The clock laser light has a Gaussian waist radius of600 µ m along the tweezer array. This large width is cho-sen to minimize gradients in clock intensity across thearray arising from slight beam angle misalignments.
3. Bosonic clock transition
Optical excitation of the S ↔ P clock transmis-sion in a bosonic alkaline-earth-like atom is facilitatedby applying a bias magnetic field B [26]. This field cre-ates a small admixture of P into P , and results ina Rabi frequency of Ω R / π = α √ I | B | , where I is theintensity of the clock probe beam and α is the couplingconstant. For Sr, α = 198 Hz/T(mW/cm ) / [26]. Theprobe beam induces an AC Stark shift ∆ ν P = kI , where k = −
18 mHz/(mW/cm ) for Sr [26]. The magneticfield gives rise to a quadratic Zeeman shift ∆ ν B = βB ,where β = − . for Sr [26].We choose B ≈ µ T, for which ∆ ν B ≈ − . I ≈ , for which ∆ ν P ≈− . B and I are experimen-tally calibrated by measuring ∆ ν B and ∆ ν P via two-clock self-comparison (Sec. IV) where the value of thesystematic parameter in the second rail is varied. We fitthe measured frequency shift to a quadratic model forthe magnetic shift and to a linear model for the probeshift (not shown) and extrapolate both fits to the knownzero values of the systematic parameters, thus extracting∆ ν B and ∆ ν P .We note that our measured π -time of 110 ms (Fig. 6) islonger than what would be expected from the calibratedbeam intensity. This is likely explained by spectral im-purity of the interrogating light, which has servo-inducedsidebands at ≈
600 kHz. These sidebands are spectrallyresolved enough so as to not affect clock interrogation,but still contribute to the probe light shift of the transi-tion frequency.
4. Interrogation sequence
We confirm the presence of atoms in each tweezer us-ing fluorescence imaging for 30 ms on the 461 nm tran-sition while cooling on the 689 nm transition and re-pumping atoms out of the metastable P , states. Thisimaging procedure initializes the atoms in the S elec-tronic ground state | g (cid:105) . We then further cool the atomsfor 10 ms using attractive Sisyphus cooling [25] on the689 nm transition and adiabatically ramp down to a trapdepth of 245(31) E r for 4 ms. We apply a weak biasmagnetic field of B ≈ µ T along the transverse di-rection of the tweezer array to enable direct optical ex-citation of the doubly-forbidden clock transition at 698nm [26, 51]. After interrogating the clock transition for110 ms (Fig. 6), we adiabatically ramp the trap depthback up to 2447(306) E r to detect the population ofatoms in | g (cid:105) using fluorescence imaging for 30 ms withoutrepumping on the P ↔ S transition. This interroga-tion sequence is repeated a number of times before thearray is refilled with atoms.
5. Clock state detection fidelity
Based on the approach demonstrated in Ref. [25], weanalyze the fidelity of detecting atoms in the S ( | g (cid:105) )and P ( | e (cid:105) ) states under these imaging conditions. Wediagnose our state-detection fidelity with two consecu-tive images. In the first image, we detect atoms in | g (cid:105) byturning off the P ↔ S repump laser such that atomsin | e (cid:105) in principle remain in | e (cid:105) and do not scatter pho-tons [25]. Hence, if we find a signal in the first image, weidentify the state as | g (cid:105) . In the second image, we turn the P ↔ S repump laser back on to detect atoms in both | g (cid:105) and | e (cid:105) . Thus, if an atom is not detected in the firstimage but appears in the second image we can identify itas | e (cid:105) . If neither of the images shows a signal we identifythe state as “no-atom”.The inaccuracy of this scheme is dominated by off-resonant scattering of the tweezer light when atoms areshelved in | e (cid:105) during the first image. By pumping atomsinto | e (cid:105) before imaging, we observe that they decay backto | g (cid:105) with a time constant of τ p = 370(4) ms at our imag-ing trap depth of 2447(306) E r . This leads to events in thefirst image where | e (cid:105) atoms are misidentified as | g (cid:105) atoms.Additionally, atoms in | g (cid:105) can be misidentified as | e (cid:105) ifthey are pumped to | e (cid:105) in the first image. We measurethis misidentification probability by initializing atoms in | g (cid:105) and counting how often we identify them as | e (cid:105) . Usingthis method, we place a lower bound for the probabilityof correctly identifying | e (cid:105) as f e ≡ e − t/τ p = 0 . | g (cid:105) as f g = 0 . -30 -20 -10 0 10 20 300.00.51.0 Frequency (kHz) P r obab ili t y FIG. 7. Clock sideband thermometry. Array-averaged ra-dial sideband spectrum of the optical clock transition, takenwith a carrier Rabi frequency of ≈
360 Hz. A narrow carrierstands in between two broader sidebands on the red and bluedetuned sides. Sideband broadening is due mainly to smallinhomogeneities in the array. A suppressed red sideband indi-cates significant motional ground state population. The solidline is a simultaneous fit to two skewed Gaussians. From theratio of the area under the red sideband to that under theblue sideband, we obtain ¯ n ≈ .
66. The carrier is probed foran interrogation time of 1.4 ms while the sidebands are probedfor 3.3 ms.
6. Stabilization to the atomic signal
The clock laser is actively stabilized to the atomic sig-nal using a digital control system. The frequency devia-tion of the clock laser from the atomic transition is esti-mated from a two-point measurement of the Rabi spec-troscopy signal at δ o / π = ± . e is converted into a frequency correction by multi-plying it by a factor of κ = 3 Hz. We choose κ to bethe largest value possible before the variance of the errorsignal in an in-loop probe sequence begins to grow. Feed-back is performed by adding the frequency correction tothe frequency of the RF synthesizer (Moglabs ARF421)driving the first AOM along the clock beam path.
7. Sideband thermometry on the clock transition
We perform sideband thermometry on the clock transi-tion (Fig. 7) using the same beam used to interrogate theatoms for clock operation. Using a standard technique oftaking the ratio of the integrated area under the first redand blue sidebands [52], we obtain ¯ n ≈ .
66 along the di-rection of the interrogation beam, oriented along one ofthe tight radial axes of our tweezers. From the sidebandseparation, we measure a trap frequency of ω ≈ π × . S ↔ P transition for 10 ms [25] in a trap ofdepth 2447(306) E r and adiabatically ramping down toour clock interrogation depth of 245(31) E r .We note that the clock transition is sufficiently narrowto observe sub-kHz inhomogeneities of trap frequenciesbetween tweezers. This precision afforded by the clocktransition allows for detailed knowledge about inhomo- geneities in the array, and we envision using it for finecorrections and uniformization of an array in the future.However, for the purpose of thermometry, we broadenthe clock line to a degree that these inhomogeneities areunresolved on an array-averaged level so we may obtaina spectrum that can be easily fit and integrated. Specif-ically, we use a much higher magnetic field of ≈
75 mTto obtain a carrier Rabi frequency ≈
360 Hz at the sameoptical intensity.
8. Evaluating Allan deviations
Repeated interrogation introduces a bimodal distribu-tion in the time between feedback events due to the pe-riodic refilling of the array. To account for this variation,we approximate that all feedbacks are equally spaced intime with ∆ t ≈
835 ms. This introduces a slight error∆ τ ≈
100 ms for all τ , though this error is inconsequen-tial for fitting the long time Allan deviation behavior. Wefit all Allan deviations from τ = 10 s to τ = 100 s, using σ y = A/ √ τ , with free parameter A = σ y ( τ = 1 s). APPENDIX C: Statistical properties of the errorsignal1. Probability distribution function
In the absence of additional noise and given N A atoms,the probability of finding N g atoms in the ground stateafter a single clock interrogation block is given by the bi-nomial distribution P B ( N g ; N A , p ), where p is the prob-ability of detecting an atom in its ground state fol-lowing clock interrogation. The probability of measur-ing a given error signal ¯ e = ∆ N g /N A is thus given bythe probability of measuring the difference atom num-ber ∆ N g = N Ag − N Bg , where N Ag ( N Bg ) is the numberof atoms detected in the ground state after the A ( B )interrogation blocks. It can be shown that the proba-bility distribution for ∆ N g is given by the convolutionof two binomial distributions, P ∗ (∆ N g ; N A , p A , p B ) = (cid:80) N P B ( N ; N A , p B ) P B ( N − ∆ N g ; N A , p A ). This discretedistribution has support on {− N A , − N A + 1 , · · · , N A } with 2 N A + 1 non-zero values. Thus, the probabil-ity distribution for ¯ e is given by P (¯ e ; N A , p A , p B ) = P ∗ (¯ eN A ; N A , p A , p B ). In the absence of statistical corre-lation between the A and B interrogation blocks, thisdistribution has a mean µ ¯ e = ( p B − p A ) and a variance σ e = ( p A (1 − p A ) + p B (1 − p B )) /N A .
2. Additional noise
In the presence of noise, such as laser noise or finitetemperature, the excitation probability p A and p B fluc-tuates from repetition to repetition. These fluctuationscan be accounted for by introducing a joint probability0density function π ( p A , p B ), so that P (¯ e ; N A ) = (cid:90) dp A dp B (cid:18) π ( p A , p B ) × P (¯ e ; N A , p A , p B ) (cid:19) = (cid:104) P (¯ e ; N A , p A , p B ) (cid:105) , (C1)where (cid:104)·(cid:105) denotes statistical averaging over π ( p A , p B ). As-suming the mean of P (¯ e, N A ) to be zero, which is equiv-alent to (cid:104) p A (cid:105) = (cid:104) p B (cid:105) ≡ (cid:104) p (cid:105) , and the variance of p A and p B to be equal, σ p A = σ p B ≡ σ p , it can be shown thatthe variance of P (¯ e, N A ) is given by σ e = 2( (cid:104) p (cid:105) (1 − (cid:104) p (cid:105) ) − σ p ) /N A + 2( σ p − C ) , (C2)where C is a correlation function between p A and p B defined as C = (cid:104) p A p B (cid:105) − (cid:104) p A (cid:105)(cid:104) p B (cid:105) .
3. Experimental data
We can directly extract the correlation function C through the results of images (2) and (4) for valid tweez-ers (Fig. 1e). We explicitly confirm that C is indepen-dent of the number of atoms used per AB interrogationcycle and extract C = − . C , σ p is not directly experimentally accessible as it ismasked by QPN. The fit to the variance of the errorsignal (Fig. 2d) yields σ e = 0 . /N A + 0 . .
169 combined with theknowledge of C to extract σ p = 0 . /N A term of 0 . (cid:104) p (cid:105) = 0 .
41 to extract σ p = 0 . σ ¯ e,QP N = (cid:112) (cid:104) p (cid:105) (1 − (cid:104) p (cid:105) ) /N A /σ ¯ e , which for N A = 40 . σ ¯ e,QP N = 0 . , as quoted in the maintext. APPENDIX D: Exploiting single-site resolvedsignals1. Atom number dependent stability
To study the performance of our clock as a functionof atom number, we can choose to use only part of ourfull array for clock operation (Fig. 4b). We preferentiallychoose atoms near the center of the array to minimizeerrors due to gradients in the array e.g. from the AOD.Due to the stochastic nature of array filling, we generallyuse different tweezers during each filling cycle such as toalways compute a signal from a fixed number of atoms.When we target a large number of atoms, some repeti-tions have an atom number lower than the target due tothe stochastic nature of array filling, resulting in a meanatom number slightly smaller than the target as well as
Averaging time τ (s) -15 -16 -17 FIG. 8. Spatially-resolved clock comparison. The fractionalAllan deviation from an asynchronous clock comparison be-tween the left and right half of our array. Fitting a 1 / √ τ be-havior past an initial lock onset time, we find 3 . × − / √ τ ,slightly higher than the result measured for a self-comparisonof the full array (Fig. 4). Importantly, we see no upturn fortimes approaching 10 s and below the 10 − level, indicatingthat slowly-varying drifts of gradients across the array do notcontribute to instability up to our sensitivity. a small fluctuation in atom number. The data points inFig. 4b show the mean atom numbers used for clock oper-ation, with error bars around these means (denoting thestandard deviation of atom number) being smaller thanthe marker size.
2. Clock comparison between two halves of thearray
We use the ability to lock to a subset of occupied trapsto perform stability analysis that is sensitive to slow driftsof gradients across the array (such as from external fieldsor spatial variations in trap homogeneity). In this case,we lock f to traps 1-40 and lock f to traps 42-81, suchthat noise sources which vary across the array will show adivergence in the Allan deviation at long enough times.As shown in Fig. 8, we perform this analysis for timesapproaching τ = 10 s and down to the σ y = 1 × − level, and observe no violation of the σ y ∝ / √ τ be-havior. Thus, we conclude that such temporal variationsin gradients are not a resolvable systematic for our cur-rent experiment. However, this analysis will prove usefulwhen using an upgraded system for which stability at the σ y = 10 − level or lower becomes problematic. In prin-ciple, the lock could be done on a single trap position ata time, which would allow trap-by-trap systematics to beanalyzed. In situ error correction
Single-site resolution offers the opportunity both to an-alyze single-atom signals, as discussed in the main text,and to modify such signals before using them for feed-back. As an example, the AOD introduces a spatial gra-dient in trap frequencies across the array, leading to aspatial variation in zero-crossings of the error signal (asshown in Fig. 2b) and subsequently leading to an increasein the Allan deviation at the σ y ≈ − level due to1stochastic trap loading. While this effect is not currentlysignificant in our experiment, it and other array inho-mogeneities may be visible to future experiments withincreased stability.Therefore, we propose that this problem can be cor-rected (for inhomogeneities within the probe bandwidth)by adjusting the error signal e j of each tweezer by acorrection factor before calculating the array-averaged ¯ e that will produce feedback for the local oscillator. For in-stance, consider the modification ¯ e f = N A (cid:80) j ζ j e j − f ,j ,where ¯ e f is the tweezer-averaged error in Hz, ζ j is atweezer-resolved conversion factor such as could be ob-tained from Fig. 2a, and f ,j is the tweezer-resolvedzero-crossing of the error signal. This new formulationmitigates inhomogeneity without any physical change tothe array. While physically enforcing array uniformity isideal, this is a tool that can simplify the complexity ofcorrecting experimental systematics. APPENDIX E: Tweezer-induced light shifts
Several previous studies have analyzed the polar-izability and hyperpolarizability of alkaline-earth-likeatoms, including Sr, in magic wavelength optical lat-tices [30–32, 53]. In their analyses, these studies includethe effect of finite atom temperature by Taylor expand-ing the lattice potential in powers of √ I ( I is the latticeintensity) in the vicinity of the magic wavelength [53].We repeat this derivation for an optical tweezer insteadof an optical lattice.The Gaussian tweezer intensity (assumed tohave azimuthal symmetry) is given by I ( ρ, z ) = I ( w /w ( z )) e − ρ /w ( z ) , where w is the beam waist, I = 2 P /πw is the maximum intensity, P is the beampower, w ( z ) = w (cid:112) z/z R ) , and z R = πw /λ T isthe Rayleigh range. The trapping potential is deter-mined from this intensity I ( ρ, z ) by the electric dipolepolarizability α E , the electric quadrupole and magneticdipole polarizabilities α qm = α E + α M , and thehyperpolarizability effect βI .By considering a harmonic approximation in the x - and y -directions as well as harmonic and anharmonic termsin the z -direction, we arrive at the following expressionfor the differential light shift of the clock transition in anoptical tweezer, where ρ = (cid:112) x + y and n ρ (= n x + n y )and n z are vibrational quantum number along the radial and axial directions, respectively: hν LS = − (cid:20)(cid:18) ∂∂ν ˜ α E (cid:19) δ L + (cid:18) w z R (cid:19) (cid:18) n ρ + 12 (cid:19) ˜ β + √ (cid:18) w z R (cid:19) (cid:18) n z + 12 (cid:19)(cid:18) n ρ + 12 (cid:19) ˜ β + 38 (cid:18) w z R (cid:19) (cid:18) n z + n z + 12 (cid:19) ˜ β (cid:21) u + (cid:20) √ (cid:18) w z R (cid:19)(cid:18) n ρ + 12 (cid:19) + (cid:18) w z R (cid:19) (cid:18) n z + 12 (cid:19)(cid:21) × (cid:20)(cid:18) ∂∂ν ˜ α E (cid:19) δ L + ˜ α qm (cid:21) u / + (cid:20) √ (cid:18) w z R (cid:19)(cid:18) n ρ + 12 (cid:19) + (cid:18) w z R (cid:19) (cid:18) n z + 12 (cid:19)(cid:21) ˜ βu / − ˜ βu , (E1)where ˜ α E = ∆ α E ( E R /α E ), ∆ α E = α E e − α E g is thedifferential E α qm = ∆ α qm ( E R /α E ),where ∆ α qm is the differential E M β = ∆ β ( E R /α E ) , where ∆ β is the differential hyper-polarizability; u = I/ ( E R /α E ) is the tweezer depth.We use this formula to predict the light shifts stud-ied in the main text (Fig. 3). 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