Frequency measurements of superradiance from the strontium clock transition

Matthew A. Norcia, Julia R. K. Cline, Juan A. Muniz, John M. Robinson, Ross

B. Hutson, Akihisa Goban, G. Edward Marti, Jun Ye, and James K. ThompsonJILA, NIST, and University of Colorado, 440 UCB, Boulder, CO 80309, USA

(Dated: November 29, 2017)

We present the first characterization of the spectral properties of superradiant light emitted fromthe ultra-narrow, 1 mHz linewidth optical clock transition in an ensemble of cold 87Sr atoms. Sucha light source has been proposed as a next-generation active atomic frequency reference, with thepotential to enable high-precision optical frequency references to be used outside laboratory environ-ments. By comparing the frequency of our superradiant source to that of a state-of-the-art cavity-stabilized laser and optical lattice clock, we observe a fractional Allan deviation of 6.7(1) × 10−16

at 1 second of averaging, establish absolute accuracy at the 2 Hz (4 × 10−15 fractional frequency)level, and demonstrate insensitivity to key environmental perturbations.

I. INTRODUCTION

The development of atomic clocks has lead to a wealthof applications from technology to fundamental physics.Atomic clocks are currently the most precise and accu-rate absolute frequency references [1–6] and are used todefine the second [7–9]. Precision timekeeping enablesthe synchronization needed for communication and navi-gation systems [10, 11], even through direct optical linksacross countries [12], and for very-long-baseline radiotelescopes [13]. Precision frequency metrology has beenproposed as a means of studying quantum many-bodyphysics [14, 15], searching for exotic physics [16–18] andexploring fundamental quantum limits imposed by grav-ity [19–21].

Atomic clocks typically function by combining a sta-ble oscillator (e.g. a crystal oscillator or laser) and anatomic frequency reference, which is an atom or collec-tion of atoms with a well-suited transition between inter-nal states (the “clock transition”) to which the oscillatoris stabilized. The oscillator provides short-term stability,while the atomic frequency reference provides long-termstability and absolute accuracy.

The atomic frequency reference can either be activeor passive [22]. In a passive reference, as sketched inFig. 1(a), radiation from the oscillator is applied to theatoms in a spectroscopic sequence that maps the oscilla-tor’s frequency to internal state populations in the atoms.The populations are then measured to infer and stabilizethe oscillator’s frequency [6, 23, 24]. Examples includecaesium and rubidium clocks in the microwave domain[25, 26], and strontium, ytterbium, and aluminum ionclocks in the optical domain [6].

In an active reference, radiation is collected directlyfrom the clock transition, as depicted in Fig. 1(b). Whilepassive frequency references have so far demonstratedsuperior long-term stability and absolute accuracy com-pared to their active counterparts, active references canhave substantial advantages in terms of detection band-width and dynamic range, which can be especially im-portant in situations where the system performance islimited by large excursions of the oscillator frequency

(discussed further in Section II). Hydrogen masers arecurrently used as active frequency references in the mi-crowave domain to provide complimentary short-termstability to microwave atomic clocks [22, 27]. Until now,there have been no high-precision active atomic frequencyreferences in the optical domain.

Operating a frequency reference in the optical domaininstead of the microwave domain provides a huge fun-damental advantage that has allowed passive optical fre-quency references to overtake their microwave counter-parts in both precision and accuracy [6, 23, 28, 29]. Here,we demonstrate the first active optical frequency refer-ence to realize this advantage, achieving a measured frac-tional frequency stability at short times surpassing thatof hydrogen masers by roughly two orders of magnitude.

II. EXPERIMENTAL SYSTEM

Our system, also described in [30], consists of an en-semble of up to several 105 87Sr atoms confined withina high finesse optical cavity by a near-magic wavelength[31] optical lattice that is supported by the cavity. Thislattice confines the atoms tightly along the cavity axis (inthe so-called Lamb-Dicke regime) so as to eliminate first-order Doppler shifts, while imparting near equal shiftsto the ground and excited states of the lasing transition.The cavity is locked at frequency fc, close to the fre-quency fa of the dipole-forbidden optical clock transitionbetween the ground state 1S0 and the excited state 3P0,as shown in Fig. 1(b). The wavelength of the clock tran-sition is near 698 nm, and it has a radiative linewidthof 1.1(3) mHz [32]. The cavity detuning from the clocktransition is δc = fc− fa. At 698 nm, the cavity’s finesseis 2.5 × 104, and its linewidth is κ = 2π × 145 kHz. Intypical operation, δc is nominally zero (δc � κ), but canbe varied for characterization purposes.

We prepare a superradiant state by initializing theatoms in one or more ground states, and then applying acoherent drive to the clock transition using a laser thatis coupled through the optical cavity (we will call thisthe state preparation pulse). Both the state preparationpulse and the optical cavity are nominally on resonance

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1040

7v1

[ph

ysic

s.at

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8 N

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017

2

Reference laser

Atomic medium

Reference laser

a)

b)

Detection laser

FIG. 1. Passive and active optical frequency references. (a)In a passive optical frequency reference, the “clock transition”in the atomic medium is driven by radiation from a laser atfrequency f`. The laser frequency is then determined relativeto the clock transition frequency through a population mea-surement of the two clock states using a fluorescence signal (I)obtained from a separate atomic transition. The fluorescencesignal provides information about the atomic response at fre-quency f`, which can be used to stabilize the frequency ofthe laser. (b) In an active optical frequency reference, light iscollected directly from the atomic clock transition. This lightcan be compared to the reference laser by forming a hetero-dyne beat-note. The power spectral density of the resultingsignal (I) provides information about the atomic system atall offset frequencies from f` within a large detection band-width set by the detection electronics. In our system, weuse 87Sr atoms emitting into a high-finesse optical cavity on aroughly 1 mHz linewidth clock transition as the active atomicmedium. The clock transition frequency, fa, is nearly equalto a cavity resonance frequency, fc, up to a tunable offsetδc = fa − fc. The cavity linewidth κ = 2π × 145 kHz greatlyexceeds the atomic linewidth, placing the system in a highlydamped, “superradiant” regime.

with the clock transition. This prepares each atom ina superposition of the ground and excited states withthe correct relative phase between atoms to radiate col-lectively into the cavity mode, and a population inver-sion set by the pulse duration. A pulse that prepares amaximally inverted state lasts about 100 µs. As soon asthe state preparation pulse is turned off, the atoms emitcollectively into the cavity. We refer to this process assuperradiant emission, which describes the collective ra-diation of an ensemble of emitters into a highly dampedoptical mode. After the atoms emit a superradiant pulse,a new ensemble of atoms is laser cooled and trapped andthe sequence is repeated with a typical total cycle timeTc = 1.1 s.

In order to study the spectral properties of the superra-diant light, we form a heterodyne beat-note between thelight emitted from the strontium atoms and light froma state-of-the-art stable laser system that we refer to asthe reference laser, as shown in Fig. 1(b). The referencelaser system is described in [33, 34] and is stabilized toan optical reference cavity with a thermal noise floor of

1 × 10−16 from 1 to 1000 seconds, and a linewidth of26 mHz. Further details on the experimental setup anddata processing can be found in Appendix I.

a)

b)

c)

FIG. 2. Superradiant pulses in the time and frequency do-mains with different initial nuclear spin state populations.Atomic state population is initialized (a) with all atoms in asingle stretched state, (b) with atoms in both stretched states,and (c) with all ten spin states populated. Columns representthe output power measured in the time domain (left), the ini-tial spin state population (inset), and power spectrum of out-put power measured relative to stable reference laser up to anarbitrary frequency offset (right), for single trials of the ex-periment. In the frequency domain, we observe a single peakin the power spectrum when we populate only one stretchedstate, two peaks when we populate both stretched states, andmany peaks when we populate all ten Zeeman sublevels.

Fig. 2 shows characteristic superradiant pulses in thetime and frequency domain, for different initial nuclearspin state populations. The ground and excited statesof the lasing transition in 87Sr each have ten Zeemansublevels (F = 9/2), which are non-degenerate in thepresence of an applied magnetic field. Because the statepreparation pulse is polarized along the magnetic field,it maintains the initial spin projection mf of the groundstate. Further, because the state preparation pulse estab-lishes coherence between the ground and excited states,the radiated light is also polarized along the magneticfield and we do not observe lasing between different val-ues of mf .

In row (a) of Fig. 2, we optically pump the atoms intoone stretched state (|mf | = 9/2) prior to the state prepa-ration pulse. If we measure the output power versus time,we observe the immediate onset of superradiant emissionthat lasts a few 10’s of ms. The power spectral density(PSD) of the beat-note between the superradiant lightand the reference laser shows a single peak with width oforder 10 Hz, corresponding to the finite duration of thepulse.

Instead of populating a single spin state, we may pop-

3

ulate both stretched states simultaneously and observelasing on both transitions at the same time, as shown inFig. 2(b). In this configuration, we are preparing a spinmixture, in contrast to a coherent superposition betweenthe two stretched states — half of the atoms are in onestretched state, and the other half are in the other. Be-cause the Rabi frequency of the state preparation pulseexceeds the Zeeman splitting between the two transitions,both experience a similar initial rotation. In the time do-main, we see a beating at the frequency of the Zeemansplitting between the two transitions. In the beat-note’sPSD, we see two distinct peaks at frequencies that we la-bel f9/2 and f−9/2, corresponding to light emitted fromthe two stretched transitions. Because these two peaksshift in equal and opposite directions in response to mag-netic fields, this configuration can be used to derive amagnetic-field insensitive signal, as described in Fig. 3(b)and the subsequent discussion.

Finally, we can populate all ten nuclear spin states, andobserve interference between the fields radiated by atomspopulating the ten nuclear spin states as in Fig. 2(c).This manifests as a series of short bursts of power as theten transitions come into phase with one-another, sepa-rated by dark periods when their radiated fields cancel.In the beat-note’s PSD, we see peaks associated withdifferent transitions, with heights that reflect the rela-tive strengths of the Clebsh-Gordan coefficients of thetransitions.

The data shown in each panel of Fig. 2 represent asingle run of the experiment. In contrast to a passiveclock, where one would have to run the experiment atleast once for every point on the frequency axis in orderto gain information about the atomic response at thatfrequency, we get all of this information from a singlepulse. For example, resolving the ten nuclear spin tran-sitions of Fig. 2(c) in a passive clock would require a scanof the clock laser frequency, as in [35].

Furthermore, in a standard passive clock, dynamicrange and precision are fundamentally linked – one onlygains useful information if the frequency of the local oscil-lator is within the bandwidth of the measurement (1/T ,where T is the interrogation time) from the atomic tran-sition. T can be reduced in order to increase the band-width, but this also increases the width of the spectro-scopic feature, degrading precision. Large frequency ex-cursions are thus difficult to measure with high precisionin a passive clock. With an active source however, aslong as the cavity that surrounds the atoms is withina few cavity linewidths of the atomic transition (severalhundred kHz in our case), the output of the active clockprovides a frequency reference with no reduction in pre-cision.

The large dynamic range of the active reference mayprove valuable for the operation of high precision opticalfrequency references outside of laboratory environments,and in mobile applications. In these contexts, vibrationsand temperature fluctuations could cause large and rapidperturbations to the frequency of the local oscillator laser

which would be very difficult to suppress using a passiveatomic frequency reference. The large bandwidth anddynamic range of the active reference could provide sub-stantial advantages in these conditions.

III. STABILITY AND ACCURACY

To quantify the stability of our superradiant source,we rely on the commonly-used metric of the frac-tional Allan deviation [36], which is defined as σ(τ) =√

12f2

a

⟨(fn+1 − fn

)2⟩, where each fn is the measured fre-

quency averaged over the n-th time interval of durationτ , and fa is the atomic transition frequency. In each runof the experiment, the frequency is determined from thecenter of a Lorentzian fitted to one or more peaks in thepower spectrum of the beat-note. For τ > Tc, each fnis obtained by averaging the fitted centers of the Fourierpeaks obtained in several subsequent trials.

The process of extracting the center frequency from afit suppresses the effects of frequency variations that oc-cur on timescales short compared to the duration of thesuperradiant pulse. As such, our reported Allan devia-tions should not be extrapolated to measurement dura-tions shorter than the cycle time Tc. For τ ≥ Tc how-ever, where we are interested in average frequencies overlong periods of time, this filtering is unimportant andour reported Allan deviations can be compared directlyto those calculated using other measurement protocols.The impact of the pulse duration being shorter than thecycle time is discussed in Appendix III.

The Allan deviation of the frequencies of our super-radiant pulses, σ(τ), is shown in Fig. 3(a), after sub-tracting a linear drift associated with the reference laser.In this measurement, we populate both stretched states,as in Fig. 2(b) and compute the Allan deviation fortwo quantities: the average frequency of the two lasingtransitions fm = (f9/2 + f−9/2)/2 (black, blue points),which provides an indictation of the stability of our sys-tem if used as an optical frequency reference, and halfthe difference of the two lasing transition frequenciesfd = (f9/2 − f−9/2)/2 (red points), which provides ameans of assessing the fundamental limits of our system.For the black points, σ(τ) = 1.04(4) × 10−15/

√τ/s for

small τ . Our most favorable measurement of this value(blue points) was σ(τ) = 6.7(1)× 10−16/

√τ/s, obtained

for a different, shorter set of data. This corresponds to astandard deviation of 300 mHz.

For typical hydrogen masers, the short term stability isless than 10−12 at 1 second of averaging, while cryogenichydrogen masers can achieve stabilities lower than 10−13

at 1 second [22, 37]. The recently-obtained record forshort-term stability in an atomic frequency reference is6 × 10−17/

√τ/s, achieved by interleaved interrogation

of two ensembles of Ytterbium at NIST [2]. Our valueis encouragingly close, given the maturity and carefulengineering of current optical lattice clocks.

4

c)

b)a)

FIG. 3. Stability and accuracy of the superradiant laser. (a) Short-term stability of superradiant light source expressed asfractional Allan deviation σ(τ). We populate both stretched states and compute the Allan deviation for two quantities: theaverage frequency of the two lasing transitions fm (black, blue points) and half the difference of the two lasing transitionfrequencies fd (red points). During a longer dataset, we measure a short-term fractional instability of σ(τ) = 1.04(4) ×10−15/

√τ/s for the average frequency (black points), and a value of σ(τ) = 6.7(1) × 10−16/

√τ/s for a shorter data set

taken under particularly quiet conditions (blue points). (b) Real-time insensitivity to changing magnetic fields is obtained bysimultaneously fitting the frequencies of two radiating transition. Perturbations to the two individual transitions ∆f±9/2 (blackpoints) are suppressed to below the half-percent level in the sum frequency ∆fm (red points). (c) Lattice-induced frequencyshifts. Frequency difference between the superradiant laser and an optical lattice clock for different lattice laser detunings andpower levels. Different colored points correspond to different lattice depths: U = 1025, 760, 570 Erec for green, red and bluepoints respectively, where Erec is the lattice recoil energy. The horizontal axis shows ∆flat = flat − fwm

magic, where fwmmagic is

the expected value for the magic wavelength as measured by our wavemeter. A simultaneous fit (see text) to all data pointsreturns a nominal frequency offset consistent with zero, represented by the horizontal dashed line, with the uncertainty ofthe fit represented by the red band. The vertical dashed line represents the fit magic wavelength, relative to fwm

magic, with fituncertainty given by the green band.

The stability of fm is fundamentally bounded by pho-ton shot noise. In order to determine how close weare to this limit, we consider the short-term stabilityof the difference frequency fd (red points of Fig. 3(a)).This constitutes a synchronous comparison of the atomspopulating each of the two stretched states, similar tosynchronous comparisons used to extract the quantum-limited stability of passive optical lattice clocks in thepresence of local oscillator noise [2, 3, 38].

The frequency difference fd is insensitive to perturba-tions that affect the measurement of both lasing tran-sitions in the same manner – for example noise on thereference laser, cavity pulling effects, and shifts associ-ated with atomic density. For the data set correspondingto the black points of Fig. 3(a), fd exhibits a short-term

stability of σ(τ) = 4.5(2) × 10−16/√τ/s. This result

can be compared to a recent differential measurement be-tween two regions of a strontium Fermi-degenerate three-dimensional optical lattice clock that obtained a frac-tional Allan deviation of 3.1× 10−17/

√τ/s [3].

In these passive clocks, atomic quantum projectionnoise dominates over shot noise associated with the pho-tons used to measure atomic state populations, as manyphotons can be scattered and collected from each atom inthe fluorescence detection process. In contrast, in our ac-

tive system each atom currently emits only a single pho-ton per measurement. Because our detection efficiency(η = 0.06, see Appendix I and III) is significantly lowerthan unity, photon shot noise is a larger source of noisethan atomic projection noise. By deliberately attenuat-ing the output of our active reference, we have verifiedthat the short term-instability in fd is dominated by pho-ton shot noise.

Much of the difference between the measured Allandeviations for fm and fd can be attributed to noise onthe reference laser, which is exacerbated by the fact thatthe duration of our pulses is shorter than the cycle time ofthe experiment [39] (see Appendix III for details). Otherlikely contributions include fluctuations in atom numberand cavity frequency.

For longer averaging windows (large τ), the measuredAllan deviations flatten out. The reference laser has anoise floor due to thermal perturbations of the length ofthe reference cavity at around σ(τ) = 1 × 10−16 from 1to 1000 seconds [33, 34], which contributes to the floorof the average frequencies fm (black points). The Allandeviation for the difference frequencies, fd, (red points) isinsensitive to noise on the reference cavity. These pointsalso hit a floor at σ(τ) ' 2×10−16, perhaps due to slowlyvarying magnetic fields in the lab. To our knowledge, the

5

fact that the asymptotic values of these two quantities isso similar is a coincidence.

In Fig 3(b), we demonstrate the insensitivity of fmto magnetic fields. The clock transition frequenciesf±9/2 have a low but finite magnetic field sensitivity of108.4×mf (Hz/G) [40], which is opposite in sign for thetwo transitions. For demonstration, we deliberately adda randomized magnetic field between different runs of theexperiment. While this leads to large shifts of the indi-vidual peaks, ∆f9/2 and ∆f−9/2, the shift to fm, called∆fm, is suppressed by at least two orders of magnitudeover our range of applied magnetic fields. In contrast topassive clocks, where the spin polarization is alternatedbetween experimental trials to achieve a magnetic fieldinsensitive signal, our method provides real-time rejec-tion of magnetic field noise, making it effective at elimi-nating errors from time-varying magnetic fields.

In order to assess the absolute accuracy of the super-radiant light source, we perform a comparison with theoptical lattice clock described in [3]. For this compar-ison, we account for known shifts to the frequencies ofboth the lattice clock and superradiant system at the Hzlevel, as detailed in Appendix II.

The largest such shift is due to the differential ACStark shift from the optical lattice. Because the cavitymust be on resonance with the clock transition at 698 nmand the lattice frequency must be on resonance with an-other mode of the cavity near 813 nm, we do not havecomplete control of the lattice frequency. This preventsus from operating at exactly the magic wavelength, atwhich the shift to the clock transition vanishes. The clos-est longitudinal TEM00 mode to the magic wavelengthmay be up to half a free spectral range (1.85 GHz) away.

To account for this, we take data with the lattice atfour frequencies near the magic wavelength and at threelattice depths at each of these frequencies (Fig. 3(c)). Wethen perform a fit of the form: ∆f = a(flat − fmagic) ×U + foffset to all of the data points. Here, flat and U arethe measured lattice frequencies and trap depths, anda, fmagic, and foffset are fit parameters that representthe sensitivity of the clock transition to lattice detun-ing, the magic wavelength frequency, and the systematicfrequency offset between the superradiant laser and theoptical lattice clock, which is our primary quantity ofinterest.

After accounting for other known shifts, dominated byatomic collisions (see Appendix II), the measured fre-quency difference between our superradiant light and thevalue measured by the optical lattice clock is 1 ± 2 Hz(fractionally 2(4)× 10−15).

IV. CAVITY PULLING

Next, we characterize the sensitivity of the frequency ofthe superradiant light to changes in the cavity resonancefrequency. Long-range exchange interactions mediatedby the optical cavity give rise to a frequency shift that

a)

b)

FIG. 4. (a) Sensitivity of superradiant light frequency, fm,to detuning of the cavity from resonance, δc, for differentinitial population inversions, as sketched on the side insets.An offset frequency f0 set by the detuning of the refer-ence laser from the strontium transition is subtracted foreach data set. When tuned to minimize sensitivity to cav-ity detuning (red line, points) we observe a pulling coefficientP = ∆fm/δc = 2(3) × 10−6. Changing the duration of thestate preparation pulse leads to an increased absolute value ofP (black, blue points). (b) Atomic inversion changes through-out the superradiant pulse, leading to a chirp in the frequencyof the emitted light for finite cavity detuning. The emittedpower is shown in the upper panel for the first 45 ms of thepulse. The pulling coefficient at different time intervals iscomputed from frequency measurements made on subsets ofthe pulse (as indicated by colored vertical bands) for the reddata set shown in (a). The pulling coefficient of the entirepulse can be made insensitive to cavity detuning by choosingan initial inversion for which the time-averaged (weighted byinstantaneous power) frequency shift is zero.

depends dispersively on cavity detuning, and linearly onatomic population inversion [41, 42]. These interactionslead to so-called one-axis-twisting dynamics, and a many-body energy gap. The key effect of these dynamics for ourpurposes here is that small perturbations of the cavity

6

detuning from atomic resonance, δc, cause the frequencyof the laser to be “pulled” by an amount Pδc, relative toits value when the cavity is on resonance. We refer toP as the pulling coefficient. While the cavity detuningis normally stabilized near zero |δc| < 5 kHz � κ usingpiezoelectric actuators (PZTs) on the cavity mirrors, wenext use the PZTs to introduce a finite and controllabledetuning.

Because the pulling coefficient is proportional to thepopulation inversion, it changes signs as the inversionchanges from positive to negative during a superradiantpulse. When we fit the center frequency of a Fourierpeak, we are sensitive to a value of P that is averagedover different values of inversion. By preparing statesfor which the average inversion during the superradiantpulse is near zero, we can make the average P very small.

In Fig 4(a), we vary the duration of the state prepa-ration pulse in order to prepare states with differing ini-tial inversion, and measure the frequency shift that re-sults from a shift in cavity frequency. We can tune theduration of the state preparation pulse to give a lowpulling coefficient of P = 2(3)× 10−6 (red points). Pulsedurations 13% longer and 7% shorter yield pulling co-efficients roughly an order of magnitude higher (P =−2.5(5)× 10−5 and P = 3.0(3)× 10−5 for the black andblue points, respectively). For the data used to measurestability and accuracy, we tuned the initial inversion nearthe point of minimum sensitivity to cavity frequency de-viations. Further, we nominally tuned the cavity on res-onance with the atomic transition (δc = 0) for each pointin Fig. 3c. As such, we do not expect cavity pulling tohave contributed a systematic shift to our accuracy as-sessment.

We directly observe that as the inversion changes dur-ing a superradiant pulse, the frequency of the output lightindeed chirps in time. A typical single-shot measurementof optical power versus time is shown in the upper panelof Fig. 4(b), with both stretched states populated as inFig. 2(b). In the lower panel of Fig. 4(b), we calculatethe pulling coefficient from 5 ms long subsets of the datacorresponding to the red points of Fig. 4(a). The pullingcoefficient switches sign during the pulse as inversion goesfrom positive to negative. The average of the pulling co-efficient measurements (weighted by the output power) isconsistent with the pulling coefficient for the entire pulse.

For any of these configurations, the output frequencyof the laser is highly insensitive to the cavity frequency.This is a key promise of superradiant lasers – because theoutput frequency is determined by the atomic transitionfrequency rather than the cavity resonance frequency, asuperradiant laser is largely insensitive to technical andfundamental thermal fluctuations in cavity length thatlimit today’s most stable lasers [43–47].

It is important to note that minimizing P by carefullysetting the initial inversion is only possible in pulsed op-eration. In steady-state, the existence of an equilibriumcondition requires positive population inversion, and thusleads to a finite pulling coefficient [48–50]. For our sys-

tem under realistic steady-state operating conditions, wewould expect a pulling coefficient of order 10−5, which isstill a very promising number. In order to limit frequencyshifts associated with cavity detuning to the 1 mHz level,we would thus require a cavity that is stable at the 100 Hzlevel. While our current cavity exhibits short-term fluc-tuations in its resonance frequency of order 5 kHz, andslow drifts of several 10s of kHz over a typical data takingsession of a few hours, relatively straightforward techni-cal improvements should enable stabilization of the cav-ity frequency to the 100 Hz level.

V. CONCLUSION

We have demonstrated the first optical high-precisionactive atomic frequency reference to date. We havedemonstrated a short-term stability that already greatlysurpasses that of existing active atomic frequency refer-ences (masers), which operate at microwave frequencies.Our system is highly insensitive to drifts in the opticalcavity frequency and to fluctuating magnetic fields. Ma-jor systematic frequency shifts have been characterizedto the Hz level.

In the near future, it will be advantageous to ex-tend the pulsed mode of operation demonstrated hereto steady-state operation. Doing so could dramaticallyimprove the stability of the frequency reference, particu-larly at short timescales where stability is limited by thefinite duration of the pulses and by photon shot noise.For example, if our system were operated in steady-statewith a similar output power to what we demonstrate inpulsed mode, and continued to be dominated by pho-ton shot noise out to an averaging time of one second,we would expect to see an improvement in the τ = 1 sAllan variance by a factor of roughly 103 compared topulsed operation with a single 100 ms long pulse each sec-ond. The rapid improvement arises because the Fourier-limited linewidth would be reduced by 10, and simultane-ously 10 times more photons would be collected, allowingthe narrower line to be split more precisely.

ACKNOWLEDGMENTS

We acknowledge useful input from N. Poli, A. D. Lud-low, and S. L. Campbell. This work was supportedby NSF PFC grant number PHY 1734006, DARPAQuASAR and Extreme Sensing, and NIST. J.R.K.C. ac-knowledges financial support from NSF GRFP.

APPENDIX I: EXPERIMENTAL SETUP ANDDATA PROCESSING

Our experiment starts by loading about 4 × 105 87Sratoms into a near-magic-wavelength one-dimensional op-tical lattice created by the intracavity fields of the cav-

7

ity. The trap depth varies, but nominally we operatewith a trap depth of U0 = 180 µK ≈ 1000Erec. Cool-ing into the lattice is performed by the narrow linewidth7.5 kHz transition at 689 nm, following a similar proce-dure as detailed in [51]. Typically, the measured tem-perature of the atoms in the lattice is around 12 µK≈ 65Erec. To prepare the atoms in a near-equal mixtureof the two stretched ground states, we optically pumpthe atoms using a π-polarized 689 nm laser tuned to the1S0 F = 9/2 → 3P1 F

′ = 7/2 transition.

698nm Reference

laser

AOM

AOM

Reference Path

IQdemod

FFT

Fiber NoiseCancellation

Ultra-stableReference cavity

Sr atoms

FIG. 5. Light from a state-of-the-art stabilized cavity lasersystem is transported by phase stabilized a 75 m optical fiberand is combined with the superradiant pulse light to originatean optical beat-note detected at the photodiode. An IQ de-modulation step is performed before data acquisition (FFT).The reference path for the phase stabilization is referenced tothe back cavity mirror.

The superradiant emission exits the cavity equallythrough each cavity mirror and it is collected in an op-tical fiber on one side of the cavity. The total detectionefficiency is η = 0.06, which accounts for real and effec-tive losses, including the fact that half of the photons exitthrough the undetected end of the cavity. Light from thereference laser is also coupled to the optical fiber to forman optical beat-note on a fast photodetector, as shownin Fig. 5. The two fields differ in frequency by approxi-mately 60 MHz, which is within the detection bandwidth

of the photodetector. The observed photo-electron shotnoise is approximately 8 dB above the electronic noisefloor of the detector. We further mix the signal downusing an IQ demodulator to translate the signal close to40 kHz. The I and Q outputs are directly sampled intoour data acquisition system.

Acoustic noise in the 75 m long optical fiber that de-livers the reference laser light leads to fluctuations of thereference laser frequency of order 10 kHz, which precludethe observation of a beat-note. We retro-reflect a smallfraction of the reference light to actively cancel this noiseusing the technique demonstrated in ref. [52], which re-duces contribution of this noise to below at least the1 Hz level between measurements. Because the cavitycan move with respect to the optical table upon whichthe superradiant pulse is overlapped with the referencelaser, it is important to reflect the path-length stabiliza-tion light from the cavity mirror opposite to the end ofthe cavity from which superradiant light is detected, asshown in Fig 5.

APPENDIX II: SHIFTS ON THE CLOCKTRANSITION.

Table I summarizes the corrections and uncertain-ties associated with the measurement of the absolute fre-quency. After accounting for these shifts, the correctedfrequency difference between the superradiant light andthe strontium transition frequency measured by the op-tical lattice clock is 1± 2 Hz.

Corrections and Uncertainties

Effect Value orShift (Hz)

Uncertainty(Hz)

Measured Offset (foffset) 0.7 0.9

Cavity Drift, Magic Wave-length

0 0.7

Collisions 1 1

Hyperpolarizability 0.3 0.3

Optical lattice clock 0 0.3

2nd order Zeeman -0.04 0.01

DC Stark Shift 0 +0, -0.1

Differential Blackbody Ra-diation

0 0.06

TABLE I. Summary of shifts and uncertainties in the fre-quency comparison measurements with the optical latticeclock.

Lattice configuration

The intracavity lattice is driven by an ECDL laser,whose power is increased by a tapered amplifier. The lat-

8

tice light is linearly polarized along the direction of theapplied bias magnetic field. This configuration greatlysuppresses the vector shift to the clock transition due tothe lattice. As we use both stretched states, mf = ±9/2of the clock states (F = 9/2), the lattice stark shift, inthis case equal for both mf states, for a lattice depth U is∆fAC = −[κs +2κtF (2F −1)]U , where κs and κt are thedifferential scalar and tensor polarizabilities respectively[4, 53]. The magic wavelength is then defined as the onethat cancels the scalar and tensor shifts for this config-uration. From the fitting presented in the main text,we found the magic wavelength to be 813.4275(2) nm,limited by the resolution of the wavemeter (calibratedusing our narrow-line cooling laser at 689 nm), in goodagreement with known results [3, 4, 54, 55]. The sensi-tivity to trap depth and detuning from the magic wave-length is described by the fitting parameter a men-tioned in the main text. Our measured value of a is−14.8(3) (µHz/Erec)/MHz, comparable to the previouslymeasured value for the sensitivity of the clock transitionto the magic wavelength [3, 54]

Collisional frequency shifts

Collisions between atoms in the lattice can lead to fre-quency shifts of the emitted light that scale with atomicdensity.

In order to determine the density-dependent frequencyshift, we toggle between measuring the frequency ofthe superradiant light at an atomic density of ρ0 ≈2×1012 atoms/cm3, and a density of approximately ρ0/4.In order to modulate the density at fixed atom number,we spread out the atoms by adding frequency sidebandson the trapping light at ±1 free spectral range of thecavity, as demonstrated in [56]. As adjacent longitudinalmodes of the cavity have opposite symmetry around thecenter of the cavity (the nodes of one mode correspondsto the antinodes of the other mode), adding the frequencysidebands with the appropriate power ratio results in ashallower axial modulation of the trapping potential atthe center of the cavity, while preserving the radial con-finement (i.e. roughly converting the standing wave lat-tice into a smooth optical dipole trap.). After loading theatoms in the lattice, the sidebands are applied to allowthe atoms to spread axially across the optical potential.After 30 ms, the sidebands are turned off, restoring thelattice potential and pinning the atoms in place. Coolingis performed for 20 ms to return the atoms to their orig-inal temperature. Only 10% of the atoms are lost duringthis process, and the final temperature of the atomic sam-ple is unchanged as determined from time of flight mea-surements. This allows us to investigate the effects ofdensity with minimal change to the atom number, whichwould lead to additional cavity-mediated pulling effects.

We attribute frequency shifts that occur when theatomic density is toggled at fixed atom number toatomic collisions. For spin-polarized fermionic samples,

these shifts can strongly depend on population inversion[14, 57]. In our case, the atomic sample is prepared in twospin projection states, such that even for the fermionicspecies the s-wave collisions are allowed, and we expectthat they are the dominant source of frequency shifts.This conclusion is supported by measuring the collisionsshifts in a fully spin polarized sample and observing asignificantly reduced density-dependent frequency shift.

The observed collisional shift between ρ0 and ρ0/4 of1.7(4) Hz, independent of population inversion. For thefrequency comparison and stability measurements pre-sented in Fig. 3(a)-(c), the density is reduced to roughlyρ0/5 by using a lower atom number, without spreadingout the atoms. In Table I we present a conservativeatomic collision shift of 1 ± 1 Hz. The uncertainty onthis number reflects both uncertainty in collisional shiftsat a density of ρ0 and the uncertainty in the relative totalatom number between when the density shifts were mea-sured and when the absolute frequency measurementswere performed.

Other shifts

Other known sources of frequency shifts that can af-fect the comparison are estimated to be negligible at thelevel of our frequency comparison measurement. Theseinclude hyperpolarizability, 2nd order Zeeman shifts, DCStark shifts and differential blackbody radiation shiftsbetween the clock and superradiant source. These aresummarized below and shown in Table I.

As our lattice can be relatively deep compared to usualoptical lattice clock experiments, the hyperpolarizabilityeffect that scales quadratically with the lattice intensitycould be important. Using the measured coefficient inRef. [4], we expect a shift of 0.3(3) Hz for the deepestlattice used in this experiment.

The second-order Zeeman shift changes the transitionfrequency of both stretched states by the same amount(-0.233 Hz/G2)B2 [40] and does not cancel in the averagefrequency fm. The half frequency difference fd betweenthe peaks generated by the mf = ±9/2 states providesan excellent estimate of the magnetic field on each trial.We estimate a shift of -0.04(1) Hz.

The uncertainty in the optical lattice clock’s lattice-induced shift was 0.3 Hz at the time of the frequencycomparison. This was due to the 100 MHz uncertaintyof the wavemeter used to measure the frequency of thelattice laser.

Blackbody radiation shifts the transition frequency inboth the superradiant light source and in the passive op-tical clock [4, 58–61]. We measured the temperature ofboth vacuum chambers to be well within 1 K of eachother. However, it was challenging to measure the tem-perature of the clock chamber very close to the atoms,so we allow for a temperature difference of up to 2 K.Accounting for the variation of the blackbody radiationshift with respect to temperature [4] of 0.03 Hz/K, the

9

uncertainty on the differential shift due to blackbody ra-diation is 0.06 Hz. We assume that similar characteri-zation of the environment could in-principle be done toestablish the absolute magnitude of the shift as was donein [4, 59] or by placing the atoms in a low temperatureenvironment as was done in [58].

Static electric fields can shift the atomic transition dueto differential DC polarizability of the excited and groundstates. It has been reported that dielectric surfaces undervacuum can accumulate surprising amounts of charge,and lead to electric fields that can cause significant per-turbation to the frequency of atomic clocks [62]. Thiseffect could be relevant in our experiment, due to theceramic cavity spacer and mirrors that are in close prox-imity to the atoms. Furthermore, the piezo tubes thatcontrol the position of the cavity mirrors are driven withup to 120 V, although they are arranged in a configura-tion were the electric field generated by the piezos at thecavity center should be approximately canceled. For thedata presented in Fig. 3(c), the value of the piezo voltagefor each lattice configuration can be taken into accountfor the fitting, as an additional quadratic term on thevoltage, without significant changes on the fit quality.

The magnitude of the DC stark shift scales with thesquare of the electric field at the location of the atoms.The magnitude of the shift due to stray fields can be de-termined by applying an additional field. As the strengthof the applied field is scanned, one observes a parabolicfrequency shift of the clock transition with a minimumwhen the applied field just cancels the component of thestray field along the direction of the applied field. Thiscan be done along three different directions to determinethe frequency shift due to the total stray field [4, 62].

To generate an applied field, we used a simple elec-trode placed on the top vacuum chamber viewport, abovethe horizontally-oriented cavity. We applied a voltage tothis electrode relative to the vacuum chamber (which isgrounded), and varied this voltage over roughly ±500 V.We repeated this measurement with different electrodepositions on the top viewport in order to vary the direc-tion of the applied field, and observed frequency offsets ofthe parabolic variation with applied voltage always wellbelow 0.1 Hz, as determined by comparing the frequencymeasured at the minimum of the parabola to its value atzero applied voltage. We thus assign a correction and un-certainty associated with DC Stark shifts of -0.05(10) Hz

APPENDIX III: PHOTON SHOT NOISE ANDREFERENCE LASER FREQUENCY NOISE

ALIASING IN THE SUPERRADIANT PULSEDLASER

The instability in the difference frequency fd (redpoints in Fig. 3(a)) is dominated by photon shot noise.For a fixed duration of pulse, the photon shot noise lim-ited uncertainty in our frequency measurements shouldscale as δf ∝ 1/

√M , where M is the number of detected

photons. We verify this scaling by attenuating the su-perradiant emission while holding reference power fixed,and measuring the increase in the short-time Allan devi-ation of fd. We find that for a factor of 4 attenuation,the short-time Allan deviation increases by a factor ofof 1.9(2), consistent with photon shot noise. Under thesame conditions, the short-time Allan deviation of theaverage frequencies fm increased by a factor of 1.4(2),confirming that this quantity was not purely limited byphoton shot noise. As the total detection efficiency isonly η = 0.06, see Appendix I, the photon shot noiserepresents a more stringent limit than atom shot noise inour setup.

Frequency noise on the reference laser contributes toour measured Allan deviation for the average frequencyfm in a manner that depends both on the noise spec-trum of the laser, and on the protocol used to compareits frequency to that of the superradiant source. For aperiodic measurement protocol, as is used here and intypical passive atomic clocks, the contribution to the Al-lan variance at a time τ = nTc, where Tc is the cycle timeof the experiment, can be quantified as [22, 63]

σ2(nTc) =

∫ ∞0

df2 sin4(πfnTc)

n2 sin2(πfTc)S1s

∆f (f)|R(f)|2, (1)

Here, S1s∆f (f) is the single-sided laser frequency noise

power spectral density and R(f) = F [r(t)](f) quantifiesthe sensitivity of a single measurement to laser noise at agiven frequency, as computed from the Fourier transformof the sensitivity function r(t) described below. R(f)depends on the manner in which the laser frequency ismeasured, for example on whether it is measured usinga Ramsey sequence, a Rabi sequence, or a beat-note asperformed here.

For long averaging times τ , corresponding to n � 1,the reference laser’s contribution to the Allan variancetakes the form

σ2(τ) = limn→∞

σ2(nTc) =1

τ

∞∑m=0

S1s∆f (m/Tc)|R(m/Tc)|2.

(2)which samples discrete frequency components of the ref-erence laser’s noise spectrum. This effect is often referredto as Dick noise aliasing [39].R(f) can be calculated by taking the Fourier transform

of a quantity r(t), the time-domain sensitivity functionwhich represents the sensitivity of the frequency mea-surement procedure to a discrete phase jump at timet [22, 63]. To estimate r(t) for our measurement se-quence, we numerically calculate the sensitivity of ourdata-analysis procedure by applying phase-jumps to asimulated signal that matches the typical profile and du-ration of our superradiant pulses.

The result of this calculation is shown in Fig. 6 (a)(blue trace). Intuitively, this indicates that our measure-ment protocol is most sensitive to phase jitter that occursin the middle of the superradiant pulse.

10

For comparison, the sensitivity function for a Ramseysequence of duration 40 ms that begins at t =10 ms isshown as the red trace in Fig. 6 (a) [64]. Unlike our het-erodyne frequency measurement, the Ramsey sequencehas equal sensitivity to a phase jump any time during itsduration.

The frequency-domain sensitivity function |R(f)|2 forboth the heterodyne detection and Ramsey sequence areshown in Fig. 6 (b) (blue and red traces, respectively).The key result here is that R(f) falls off at a frequencygiven roughly by the inverse of the duration of a sin-gle measurement sequence. Shorter superradiant pulsesor Ramsey sequences lead to appreciable sensitivity athigher frequencies, and an increase in the area under thefrequency-domain sensitivity curve.

Fig. 6 (c) illustrates the effect of this increased sensi-tivity due to short measurement sequences. For a contin-uous measurement, the Allan deviation of the referencelaser is known, and is represented by the green line. Us-ing the sensitivity functions for our superradiant pulsesand for 40 ms long Ramsey sequences, we compute theblue and red lines, respectively.

From the above, we estimate that at 1 s, the referencelaser’s frequency noise contributes σ(1 s) = 4.4 × 10−16

to the total Allan deviation. For the data with the bestobserved short term Allan deviations (blue points in Fig.3(a) in the main text), the difference between the Allandeviation for the frequency average, fm, and difference,fd, can be explained by the reference laser’s frequencynoise.

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