High-performance coherent population trapping clock with polarization modulation
Peter Yun, François Tricot, Claudio Eligio Calosso, Salvatore Micalizio, Bruno François, Rodolphe Boudot, Stéphane Guérandel, Emeric de Clercq
aa r X i v : . [ phy s i c s . a t o m - ph ] O c t High-performance coherent population trapping clock with polarization modulation
Peter Yun , ∗ Fran¸cois Tricot , Claudio Eligio Calosso , Salvatore Micalizio ,Bruno Fran¸cois , Rodolphe Boudot , St´ephane Gu´erandel , and Emeric de Clercq LNE-SYRTE, Observatoire de Paris, PSL Research University,CNRS, Sorbonne Universit´es, UPMC Univ. Paris 06,61 avenue de l’Observatoire, 75014 Paris, France Istituto Nazionale di Ricerca Metrologica, INRIM, Strada delle Cacce 91, 10135 Torino, Italy and FEMTO-ST, CNRS, UFC, 26 chemin de l’ ´Epitaphe 25030 Besan¸con Cedex, France (Dated: October 4, 2016)We demonstrate a vapor cell atomic clock prototype based on continuous-wave (CW) interro-gation and double-modulation coherent population trapping (DM-CPT) technique. The DM-CPTtechnique uses a synchronous modulation of polarization and relative phase of a bi-chromatic laserbeam in order to increase the number of atoms trapped in a dark state, i.e. a non-absorbing state.The narrow resonance, observed in transmission of a Cs vapor cell, is used as a narrow frequencydiscriminator in an atomic clock. A detailed characterization of the CPT resonance versus numerousparameters is reported. A short-term frequency stability of 3 . × − τ − / up to 100 s averagingtime is measured. These performances are more than one order of magnitude better than industrialRb clocks and comparable to those of best laboratory-prototype vapor cell clocks. The noise bud-get analysis shows that the short and mid-term frequency stability is mainly limited by the powerfluctuations of the microwave used to generate the bi-chromatic laser. These preliminary resultsdemonstrate that the DM-CPT technique is well-suited for the development of a high-performanceatomic clock, with potential compact and robust setup due to its linear architecture. This clockcould find future applications in industry, telecommunications, instrumentation or global navigationsatellite systems. I. INTRODUCTION
Microwave Rb vapor-cell atomic clocks [1], based onoptical-microwave double resonance, are today ubiqui-tous timing devices used in numerous fields of industry in-cluding instrumentation, telecommunications or satellite-based navigation systems. Their success is explained bytheir ability to demonstrate excellent short-term frac-tional frequency stability at the level of 10 − τ − / , com-bined with a small size, weight, power consumption and arelatively modest cost. Over the last decade, the demon-stration of advanced atom interrogation techniques (in-cluding for instance pulsed-optical-pumping (POP)) us-ing narrow-linewidth semiconductor lasers has conductedto the development in laboratory of new-generation vaporcell clocks [2–5]. These clocks have succeeded to achievea 100 times improvement in frequency stability comparedto existing commercial vapor cell clocks.In this domain, clocks based on a different phe-nomenon, named coherent population trapping (CPT),have proven to be promising alternative candidates.Since its discovery in 1976 [6], coherent population trap-ping physics [7–10] has motivated stimulating studies invarious fields covering fundamental and applied physicssuch as slow-light experiments [11], high-resolution laserspectroscopy, magnetometers [12, 13], laser cooling [14]or atomic frequency standards. Basically, CPT occurs byconnecting two long-lived ground state hyperfine levels of ∗ E-mail: [email protected], permanent e-mail:[email protected] an atomic specie to a common excited state by simulta-neous action of two resonant optical fields. At null Ra-man detuning, i.e. when the frequency difference betweenboth optical fields matches perfectly the atomic ground-state hyperfine frequency, atoms are trapped through adestructive quantum interference process into a nonin-teracting coherent superposition of both ground states,so-called dark state, resulting in a clear decrease of thelight absorption or equivalently in a net increase of thetransmitted light. The output resonance signal, whoseline-width is ultimately limited by the CPT coherencelifetime, can then be used as a narrow frequency dis-criminator towards the development of an atomic fre-quency standard. In a CPT-based clock, unlike the tra-ditional double-resonance Rb clock [15], the microwavesignal used to probe the hyperfine frequency is directlyoptically carried allowing to remove the microwave cavityand potentially to shrink significantly the clock dimen-sions.The application of CPT to atomic clocks was firstlydemonstrated in a sodium atomic beam [16, 17]. In1993, N. Cyr et al proposed a simple method to pro-duce a microwave clock transition in a vapor cell withpurely optical means by using a modulated diode laser[18], demonstrating its high-potential for compactness.In 2001, a first remarkably compact atomic clock pro-totype was demonstrated in NIST [19, 20]. Further in-tegration was achieved later thanks to the proposal [21]and development of micro-fabricated alkali vapor cells[22], leading to the demonstration of the first chip-scaleatomic clock prototype (CSAC) [23] and later to the firstcommercially-available CSAC [24]. Nevertheless, this ex-treme miniaturization effort induces a typical fractionalfrequency stability limited at the level of 10 − τ − / ,not compliant with dedicated domains requiring betterstability performances. In that sense, in the frame ofthe European collaborative MClocks project [25], signif-icant efforts have been pursued to demonstrate compacthigh-performance CPT-based atomic clocks and to helpto push this technology to industry.In standard CPT clocks, a major limitation to reachbetter frequency stability performances is the low con-trast ( C , the amplitude-to-background ratio) of the de-tected CPT resonance. This low contrast is explainedby the fact that atoms interact with a circularly po-larized bichromatic laser beam, leading most of theatomic population into extreme Zeeman sub-levels of theground state, so called ”end-states”. Several optimizedCPT pumping schemes, aiming to maximize the num-ber of atoms participating to the clock transition, havebeen proposed in the literature to circumvent this issue([26, 27], and references therein), but at the expense ofincreased complexity.In that sense, a novel constructive polarization mod-ulation CPT [28] pumping technique, named double-modulation (DM) scheme, was recently proposed. Itconsists to apply a phase modulation between both op-tical components of the bichromatic laser synchronouslywith a polarization modulation. The phase modulationis needed to ensure a common dark state to both po-larizations, allowing to pump a maximum number ofatoms into the desired magnetic-field insensitive clockstate. This elegant solution presents the main advan-tage compared to the push-pull optical pumping [29–31] or the lin ⊥ lin technique [32, 33] to avoid any opti-cal beam separation or superposition and is consequentlywell-adapted to provide a compact and robust linear ar-chitecture setup.In this article, we demonstrate a high-performanceCW-regime CPT clock based on the DM technique. Op-timization of the short-term frequency stability is per-formed by careful characterization of the CPT resonanceversus relevant experimental parameters. A short-termfrequency stability at the level of 3 . × − τ − / up to100 s, comparable to best vapor cell frequency standards,is reported. A detailed noise budget is given, highlightinga dominant contribution of the microwave power fluctu-ations. Section II describes the experimental setup. Sec-tion III reports the detailed CPT resonance spectroscopyversus experimental parameters. Section IV reports bestshort-term frequency stability results. Noise sources lim-iting the stability are carefully analysed. In section V,we study the clock frequency shift versus each param-eter and estimate the limitation of the clock mid-termfrequency stability. II. EXPERIMENTAL SET-UPA. Optical set-up
Our setup is depicted in Fig. 1. A DFB laser diodeemits a monochromatic laser beam around 895 nm, thewavelength of the Cs D line. With the help of a fiberelectro-optic phase modulator (EOPM), modulated at4 . . ≈
160 MHz) in the CPT clock cell. Adouble-modulated laser beam is obtained by combiningthe phase modulation with a synchronized polarizationmodulation performed thanks to a liquid crystal polar-ization rotator (LCPR). The laser beam is expanded to9 mm ×
16 mm before the vapor cell. The cylindrical Csvapor cell, 25 mm diameter and 50 mm long, is filled with15 Torr of mixed buffer gases (argon and nitrogen). Un-less otherwise specified, the cell temperature is stabilizedto about 35 ◦ C. A uniform magnetic field of 3 . µ T isapplied along the direction of the cell axis by means ofa solenoid. The ensemble is surrounded by two magneticshields in order to remove the Zeeman degeneracy.
LCPREOPM Magnetic shieldPD2PD3Ref cellDFB /2 PD1/2 /4 BeamExpander
Cs+BG B AOM1/2P1/4 /2/2 AOM2
FIG. 1. (Color online) Optical setup for DM CPT, wherethe abbreviations stand for: DFB distributed feedback diodelaser, EOPM electro-optic phase modulator, LCPR liquidcrystal polarization rotator, AOM acousto-optic modulator,P polarizer, λ/ B. Fiber EOPM sidebands generation
We first utilized a Fabry-Perot cavity to investigatethe EOPM sidebands power ratio versus the coupling4 .
596 GHz microwave power ( P µw ), see Fig. 2. We choose P µw around 26 dBm to maximize the power transfer effi-ciency into the first-order sidebands. The sidebands spec-trum is depicted in the inset of Fig. 2.
15 20 25 30050100 P o w e r r a t i o ( % ) P w (dBm)
Carrier -1st 1st -2nd 2nd -3rd 3rd -4th 4th
F-P cavity length (A.U.) P o w e r r a t i o ( % ) -4th-3th-2nd-1st Carrier 1st 2nd3th4th FIG. 2. (Color online) Fractional power of laser sidebandsat the EOPM output as a function of 4 .
596 GHz microwavepower. Inset, the laser sidebands spectrum with P µw =26 .
12 dBm obtained by scanning the FP cavity length, noticethe log scale of the y-axis.
C. Laser power locking -160-150-140-130-120-110-100-90 Unlocked Locked R I N ( d B / H z ) f (Hz) FIG. 3. (Color online)Power spectral density of the laser RINwith and w/o power locking.
Since the laser intensity noise is known as being oneof the main noise sources which limit the performancesof a CPT clock [10, 31], laser power needs to be care-fully stabilized. For this purpose, a polarization beamsplitter (PBS) reflects towards a photo-detector a partof the laser beam, the first-order diffracted by AOM1following the EOPM. The output voltage signal is com-pared to an ultra-stable voltage reference (LT1021). Thecorrection signal is applied on a voltage variable powerattenuator set on the feeding RF power line of the AOM1with a servo bandwidth of about 70 kHz. The out-looplaser intensity noise (RIN) is measured just after the firstPBS with a photo-detector (PD), which is not shownin Fig. 1. The spectrum of the resulting RIN with and w/o locking is shown in Fig. 3. A 20 dB improvement at F M = 125 Hz (LO modulation frequency for clock oper-ation) is obtained in the stabilized regime.It is worth to note that the DFB laser diode we used,with a linewidth of about 2 MHz, is sensitive even to thelowest levels of back-reflections [34], e.g., the coated col-limated lens may introduce some intensity and frequencynoise at the regime of 0 . D. Laser frequency stabilization -3000 -2000 -1000 0 1000 20001.21.51.8 -3000 -2000 -1000 0 1000 20000.300.450.60 P S R e f c e ll s i gna l ( V ) L (MHz)LF’=4 3F=4 3 C l o ck c e ll s i gna l ( V ) FIG. 4. (Color online) Spectrum of the Cs D line in thevacuum reference cell and in the clock cell recorded withthe bichromatic laser. The two absorptions from left toright correspond to the excited level | P / , F ′ = 3 i and | P / , F ′ = 4 i , respectively. For the reference cell signal:laser power 0 .
74 mW, beam diameter 2 mm, cell temperature22 ◦ C, AOM frequency 160 MHz. Inset, the atomic levels in-volved in D line of Cesium. Our laser frequency stabilization setup, similar to[31, 35], is depicted in Fig. 1. We observe in a vacuumcesium cell the two-color Doppler-free spectrum depictedin Fig. 4. The bi-chromatic beam, linearly polarized, isretro-reflected after crossing the cell with the orthogonalpolarization. Only atoms of null axial velocity are reso-nant with both beams. Consequently, CPT states builtby a beam are destroyed by the reversed beam, leadingto a Doppler-free enhancement of the absorption [35].The laser frequency detuning ∆ L = 0 in Fig. 4 corre-sponds to the laser carrier frequency tuned to the cen-ter of both transitions | S / , F = 3 i → | P / , F ′ = 4 i and | S / , F = 4 i → | P / , F ′ = 4 i in D line of Ce-sium, where F is the hyperfine quantum number. For La s e r f r equen cy no i s e ( H z / H z ) f (Hz) Unlocked Locked FIG. 5. (Color online) The laser frequency noise measured onthe error signal of the Doppler-free spectrum with and w/ofrequency locking. this record, the microwave frequency is 4 .
596 GHz, halfthe Cs ground state splitting, and the DFB laser fre-quency is scanned. The frequency noise with and w/olocking are presented in Fig. 5. The servo bandwidthis about 3 kHz and the noise is found to be reduced byabout 25 dB at f = 125 Hz (local oscillator modulationfrequency in clock operation). E. Polarization modulation T r an s m i ss i on li gh t ( V ) D r i v e v o l t age ( V ) Time (ms)
FIG. 6. (Color online) LCPR response.
We studied the response time of the LCPR (FPR-100-895, Meadowlark Optics). As illustrated in Fig. 6, themeasured rise (fall) time is about 100 µ s and the polar-ization extinction ratio is about 50. In comparison, theelectro-optic amplitude modulator (EOAM) used as po-larization modulator in our previous investigations [28]showed a response time of 2 . µ s (limited by our highvoltage amplifier) and a polarization extinction ratio of63. Here, we replace it by a liquid crystal device because its low voltage and small size would be an ideal choice fora compact CPT clock, and we will show in the followingthat the longer switching time does not limit the contrastof the CPT signal. F. Microwave source and clock servo-loop
LCPR EOPMDFB PD1
Cs+BG B ×3÷9 ×16DDS ÷8 LO FIG. 7. (Color online) Electronic architecture of the DM CPTclock: DDS direct digital synthesizer, DAC digital-to-analogconverter, ADC analog-to-digital converter, FPGA field pro-grammable gate array, LO local oscillator.
The electronic system (local oscillator and digital elec-tronics for clock operation) used in our experiment isdepicted in Fig. 7. The 4 .
596 GHz microwave sourceis based on the design described in [36]. The localoscillator (LO) is a module (XM16 Pascall) integrat-ing an ultra-low phase noise 100 MHz quartz oscilla-tor frequency-multiplied without excess noise to 1 . .
596 GHz signal is synthesized by a few frequencymultiplication, division and mixing stages. The fre-quency modulation and tuning is yielded by a directdigital synthesizer (DDS) referenced to the LO. Theclock operation [2, 37] is performed by a single field pro-grammable gate array (FPGA) which coordinates the op-eration of the DDS, analog-to-digital converters (ADC)and digital-to-analog converter (DACs):(1) the DDS generates a signal with phase modulation(modulation rate f m , depth π/
2) and frequency modula-tion ( F M , depth ∆ F M ).(2) the DAC generates a square-wave signal to drivethe LCPR with the same rate f m , synchronous to thephase modulation.(3) the ADC is the front-end of the lock-in amplifier.Another DAC, used to provide the feedback to the localoscillator frequency, is also implemented in the FPGA.The clock frequency is measured by comparing the LOsignal with a 100 MHz signal delivered by a H maser ofthe laboratory in a Symmetricom 5125A Allan deviationtest set. The frequency stability of the maser is 1 × − at 1 s integration time. III. CLOCK SIGNAL OPTIMIZATIONA. Time sequence and figure of merit
Frequencytrigger F M Polarization H Phase L switchDetection onoff D F M -D F M Σ + H Π (cid:144) L Σ - H L onoff t F M = (cid:144) F M t m = (cid:144) f m t d t w Time ™ FIG. 8. Time sequence. F M modulation frequency of the4 .
596 GHz signal, f m polarization and phase modulation fre-quency, t d pumping time, t w detection window. As illustrated in Fig. 8, the polarization and phasemodulation share the same modulation function. Aftera pumping time t d to prepare the atoms into the CPTstate, we detect the CPT signal with a window of length t w . In order to get an error signal to close the clockfrequency loop, the microwave frequency is square-wavemodulated with a frequency F M , and a depth ∆ F M . Inour case, we choose F M = 125 Hz, as a trade-off betweena low frequency to have time to accumulate the atomicpopulation into the clock states by the DM scheme anda high operating frequency to avoid low frequency noisein the lock-in amplification process and diminish the in-termodulation effects. -100 -50 0 50 100440450 S i gna l ( m V ) (kHz) FIG. 9. Zeeman spectrum. The center peak is the 0 − t d = 2 ms, t w = 2 ms, f m =500 Hz, P L = 163 µ W, P µw = 26 .
12 dBm, T cell = 35 . ◦ C. A typical experimental CPT signal, recorded with thistime sequence, showing all the CPT transitions allowed between Zeeman sub-levels of the Cs ground state is re-ported in Fig. 9. The Raman detuning ∆ is the differ-ence between the two first sideband spacing and the Csclock resonance. The spectrum shows that the clock lev-els (0 − m = 0) sub-levels. The distortion of neighbouringlines is explained by magnetic field inhomogeneities.It can be shown that the clock short-term frequencystability limited by an amplitude noise scales as W h /C [10], with W h the full width at half maximum (FWHM) ofthe clock resonance, and C the contrast of the resonance.Usually, the ratio of contrast C to W h is adopted as afigure of merit, i.e. F C = C/W h . The best stabilityshould be obtained by maximizing F C .The stability of the clock is measured by the Allanstandard deviation σ y ( τ ), with τ the averaging time.When the signal noise is white, with standard deviation σ N , and for a square-wave frequency modulation, the sta-bility limited by the signal-to-noise ratio is equal to [38] σ y ( τ ) = 1 f c σ N S ℓ r τ , (1)with f c the clock frequency, and S ℓ the slope of the fre-quency discriminator. In CPT clocks, one of the mainsources of noise is the laser intensity noise, which leadsto a signal noise proportional to the signal. Thereforeit is more convenient to characterize the quality of thesignal of a CPT atomic clock by a new figure of merit, F S = S ℓ /V wp , where S ℓ is the slope of the error sig-nal (in V/Hz) at Raman resonance (∆ = 0), and V wp is the detected signal value (in V) at the interrogatingfrequency (the clock resonance frequency plus the modu-lation depth ∆ F M ), see Fig. 10. Note that an estimationof the discriminator slope is also included in F C , sincethe contrast is the signal amplitude A divided by thebackground B . F C then equals ( A/W h ) /B , ( A/W h ) is arough approximation of the slope S ℓ , and B an approx-imation of the working signal V wp . In our experimentalconditions A/W h ∼ S ℓ / F M to maximize F S . Here for simplicity, wefirst recorded the CPT signal, then we can numericallycompute optimized values of ∆ F M and F S .In the following, we investigate the dependence of F C and F S on several parameters including the cell temper-ature ( T cell ), the laser power ( P L ), the microwave power( P µw ), the detection window duration ( t w ), the detectionstart time ( t d ), and the polarization and phase modula-tion frequency ( f m ). -4 -2 0 2 4470480490500 -15015 (kHz) V wp at = F M C P T s i gna l ( m V ) E rr o r s i gna l ( m V ) C=5.6%FWHM=385HzF S =S /V wp =4.5*10 -4 /Hz S ddError
FIG. 10. (Color online) Signal of the clock transition and errorsignal. Working parameters: t d = 3 ms, t w = 1 ms, f m =250 Hz, P L = 163 µ W, P µw = 26 .
12 dBm, T cell = 35 . ◦ C. B. Cell temperature and laser power
30 32 34 36 38 40 423504004503456100200300400
Tcell ( o C) F S F C C ( % ) F W H M ( H z ) F S & F C ( - / H z ) FIG. 11. (Color online) Figures of merit F S , F C , contrast C and width of the clock transition as function of cell tem-perature T cell . All other working parameters is the same asFig. 10. From the figures of merit shown in Fig. 11, the op-timized cell temperature is around T cell = 35 . ◦ C for P L = 163 µ W. The narrower linewidth observed athigher T cell , already observed by Godone et al. [39],can be explained by the propagation effect: the higherthe cell temperature, the stronger the light absorptionby more atoms, and less light intensity is seen by theatoms at the end side of the vapour cell. This leads toa reduction of the power broadening and a narrower sig-nal, measured by the transmitted light amplitude. Theoptimum temperature depends on the laser power as de-picted in Fig. 12. Nevertheless, the overall maximum of F S is reached with P L = 163 µ W at T cell = 35 . ◦ C.The figures of merit F S , F C , the contrast C and thewidth are plotted as a function of the laser power inFig. 13 for T cell = 35 . ◦ C. The laser powers maximizing F S and F C are P L = 163 µ W and P L = 227 µ W, re-
30 32 34 36 38 40250300350400450 F S ( - / H z ) Tcell ( o C)P L ( W): 109 163 227 298 FIG. 12. (Color online) F S as a function of cell temperature T cell for various laser powers. All other working parametersis the same as Fig. 10. spectively. The Allan deviation reaches a better value at P L = 163 µ W, which justifies our choice of F S as figureof merit. And in the following parameters investigation,we only show the F S as figure of merit for clarity. C ( % ) F W H M ( H z ) F C F S & F C ( - / H z ) PL ( W) F S FIG. 13. (Color online) F S , F C , C and width of the clocktransition as a function of laser power P L with T cell = 35 . ◦ C.All other working parameters is the same as Fig. 10.
C. Microwave power F S , C and width versus the microwave power areshown in Fig. 14. The behaviour of F S is basically inagreement with the fractional power of first ( ±
1) side-bands of Fig. 2. The optimized microwave power isaround 26 .
12 dBm. C ( % ) F W H M ( H z ) P w (dBm) F S ( - / H z ) FIG. 14. (Color online) F S , C and width of the clock transi-tion as function of microwave power P µw with T cell = 35 . ◦ C.All other working parameters is the same as Fig. 10.
D. Detection window t w and pumping time t d As we can see on Fig. 15, a short detection window t w would generate a higher contrast signal and higherfigures of merit. However, we found that a longer time,e.g., t w = 1 ms, results in a better Allan deviation at one-second averaging time. It is due to the conflict betweenthe higher signal slope ( ∝ F S ) and the increased numberof detected samples which help to reduce the noise, seeEq.(5). C ( % ) F W H M ( H z ) t w (ms) F S ( - / H z ) FIG. 15. (Color online) F S , C and width of the clock transi-tion as a function of t w . With t d = 4 ms − t w and all otherworking parameters is the same as Fig. 10. The same parameters are plotted versus the pumpingtime t d in Fig. 16. The figures of merit and C firstly in-crease and then decrease at t d = 1 ms, because when thedetection window t w ≥ t d = 2 ms, F S in-creases again and reach a maximum. Thus we can saythat, in a certain range, a longer t d will lead to a greateratomic population pumped into the clock states as de- picted in Fig. 16, yielding higher figures of merit. Thebehaviour of the linewidth versus the pumping time isnot explained to date. Nevertheless, note that here thesteady-state is not reached and the width behaviour re-sults certainly from a transient effect. F S ( - / H z ) C ( % ) F W H M ( H z ) td (ms) FIG. 16. (Color online) F S , C and width of the clock transi-tion versus t d . All other working parameters is the same asFig. 10. E. Polarization-and-phase modulation frequency f m Figure 17 shows F S , C , and width versus the polariza-tion (and phase) modulation frequency f m The maximaof F S is reached at low frequency f m . In one hand, thisis an encouraging result to demonstrate the suitabilityof the LCPR polarization modulator in this experiment.In an other hand, the higher F M rate would be betterfor a clock operation with lock-in method to modulateand demodulate the error signal, to avoid the low fre-quency noises such as 1 /f noise. Therefore, we chose F M = 125 Hz and f m = 250 Hz. We have noticed thatthe behavior of C is not exactly the same than the oneobserved in our previous work [40, 41] with a fast EOAM,where the signal amplitude was maximized at higher fre-quencies. This can be explained by the slower responsetime of the polarization modulator and the lower laser in-tensity used. The linewidth reaches a minimum around1 . IV. FREQUENCY STABILITYA. Measured stability
The high contrast and narrow line-width CPT sig-nal obtained with the optimized values of the parame-ters is presented in Fig. 10, with the related error sig-nal. The Allan standard deviation of the free-runningLO and of the clock frequency, measured against the Hmaser, are shown in Fig. 18. The former is in correct
246 3403603804000 1 2 3 4 50100200300400500 C ( % ) F W H M ( H z ) F S ( - / H z ) f m (kHz) FIG. 17. (Color online) The F S , C and width of the clocktransition as function of f m . All other working parameters isthe same as Fig. 10. agreement with its measured phase noise. In the 1 Hzto 100 Hz offset frequency region, the phase noise spec-trum of the free-running 4 .
596 GHz LO signal is given indBrad /Hz by S ϕ ( f ) = b − f − with b − = −
47, signa-ture of a flicker frequency noise [36]. This phase noiseyields an expected Allan deviation given by σ y (1 s) ≈ q × b − f c ≈ × − [42], close to the measuredvalue of 2 × − at 1 s. The measured stability of theCPT clock is 3 . × − τ − / up to 100 s averaging timefor our best record. This value is close to the best CPTclocks [31, 33], demonstrating that a high-performanceCPT clock can be built with the DM-CPT scheme. Atypical record for longer averaging times is also shown inFig. 18. For averaging times τ longer than 20 s, the Allandeviation increases like √ τ , signature of a random walkfrequency noise. -2 -1 -14 -13 -12 -11 -10 -14 1/2 A ll an de v i a t i on (s)(a) LO unlocked(b) LO locked (Quiet environment )(c) LO locked 3.2 10 -13 -1/2 FIG. 18. (Color online) The LO frequency stability (a) freerunning, (b) and (c) locked on the atomic resonance. (b)best record in quiet environment, (c) typical record. Theslope of the red (blue) dashed fitted line is 3 . × − τ − / (1 . × − τ / ), respectively. B. Short-term stability limitations
We have investigated the main noise sources that limitthe short-term stability. For a first estimation, we con-sider only white noise sources, and for the sake of sim-plicity we assume that the different contributions can in-dependently add, so that the total Allan variance can becomputed as σ y ( τ ) = Σ i σ y,p i ( τ ) + σ y,LO ( τ ) , (2)with σ y,LO ( τ ) the contribution due to the phase noise ofthe local oscillator, and σ y,p i ( τ ) the Allan variance of theclock frequency induced by the fluctuations of the param-eter p i . When p i modifies the clock frequency during thewhole interrogation cycle, σ y,p i ( τ ) can be written σ y,p i ( τ ) = 1 f c (cid:16) σ p i (cid:17) Hz (cid:0) δf c /δp i (cid:1) τ , (3)with (cid:16) σ p i (cid:17) Hz the variance of p i measured in 1 Hz band-width at the modulation frequency F M , (cid:0) δf c /δp i (cid:1) is theclock frequency sensitivity to a fluctuation of p i . Here,the detection signal is sampled during a time window t w with a sampling rate 2 F M = 1 /T c , where T c is a cycletime. In this case Eq.(3) becomes σ y,p i ( τ ) = 1 f c (cid:16) σ p i (cid:17) tw (cid:0) δf c /δp i (cid:1) T c τ , (4)with (cid:16) σ p i (cid:17) tw the variance of p i sampled during t w ; (cid:16) σ p i (cid:17) tw ≈ S p i ( F M ) / (2 t w ) with S p i ( F M ) the value of thepower spectral density (PSD) of p i at the Fourier fre-quency F M ( assuming a white frequency noise around F M ). When p i induces an amplitude fluctuation with asensitivity (cid:0) δV wp /δp i (cid:1) , Eq.(4) becomes σ y,p i ( τ ) = 1 f c S p i ( F M ) (cid:0) δV wp /δp i (cid:1) S ℓ T c t w τ , (5)with S ℓ the slope of the frequency discriminator in V/Hz.We review below the contributions of the different sourcesof noise.Detector noise: the square root of the power spectraldensity (PSD) S d of the signal fluctuations measured inthe dark is shown in Fig. 19. It is N detector = 64 . / √ Hz in 1 Hz bandwidth at the Fourier frequency125 Hz. According to Eq.(5) the contribution of thedetector noise to the Allan deviation at one second is0 . × − .Shot-noise: with the transimpedance gain G R =1 . × V A − and the detector current I = V wp /G R =32 . µ A, Eq.(5) becomes σ y,sh ( τ ) = 1 f c (2 eIG R ) S ℓ T c t w τ , (6) -8 -7 -6 -5 -4 -3 -2 N o i s e ( V / H z ) f (Hz) =100Hz Nolight Analyzer noise floor FIG. 19. (Color online) The RIN after the cell, detector noiseand analyser noise floor. Working parameters is the same asFig. 10. with e the electron charge. The contribution to the Allandeviation at one second is 0 . × − .Laser FM-AM noise: it is the amplitude noise in-duced by the laser carrier frequency noise. The slopeof the signal V wp with respect to the laser frequency f L is S F M − AM = 0 .
16 mV MHz − at optical resonance.According to Eq.(5) with data of laser-frequency-noisePSD of Fig. 5 at 125 Hz, we get a Allan deviation of0 . × − at one second.Laser AM-AM noise: it is the amplitude noise inducedby the laser intensity noise. The measured signal sen-sitivity to the laser power is S AM = 3 .
31 mV µ W − at f L = 163 µ W, combined with the laser intensity PSDof Fig. 3 it leads to the amplitude noise S AM × P L × RIN (125 Hz) = 30 . / √ Hz, and an Allan deviation of0 . × − at one second.LO phase noise: the phase noise of the local oscilla-tor degrades the short-term frequency stability via theintermodulation effect [43]. It can be estimated by: σ y LO (1 s ) ∼ F M f c q S ϕ (2 F M ) . (7)Our 4 .
596 GHz microwave source is based on [36]which shows an ultra-low phase noise S ϕ (2 F M ) = −
116 dB rad Hz − at 2 F M = 250 Hz Fourier frequency.This yields a contribution to the Allan deviation of0 . × − at one second.Microwave power noise: fluctuations of microwavepower lead to a laser intensity noise, which is alreadytaken into account in the RIN measurement. We showin the next section that they also lead to a frequencyshift (see Fig. 22). The Allan deviation of the microwavepower at 1 s is 2 . × − dBm, see inset of Fig. 22. Witha measured slope of 7 . . × − , which is thelargest contribution to the stability at 1 s. Note that inour set-up the microwave power is not stabilized. The other noise sources considered have much lowercontributions, they are the laser frequency-shift effect, i.e. AM-FM and FM-FM contributions, the cell tem-perature and the magnetic field. Table I resumes theshort-term stability noise budget.
TABLE I. Noise contributions to the stability at 1 s.Noise source noise level σ y (1 s ) × Detector noise 64 . / √ Hz 0 . . / √ Hz 0 . . / √ Hz 0 . . / √ Hz 0 . −
116 dBrad /Hz 0 . P µw . × − dBm@1 s 2 . . . × − Laser FM-FM ∼
100 Hz@1 s 2 . × − T cell . × − K@1 s 3 . × − B . . × − Total 2 . The laser intensity noise after interacting with theatomic vapor is depicted in Fig. 19. It encloses thedifferent contributions to the amplitude noise, i.e. de-tector noise, shot-noise, FM-AM and AM-AM noises.The noise spectral density is 100 nV / √ Hz at the Fourierfrequency 125 Hz, which leads to an Allan deviation of0 . × − at 1 s. This value is equal to the quadraticsum of the individual contributions. The quadratic sumof all noise contribution leads to an Allan deviation atone second of 2 . × − , while the measured stabilityis 3 . × − τ − / (Fig. 18). The discrepancy could beexplained by correlations between different noises, whichare not all independent. The dominant contributionis the clock frequency shift induced by the microwavepower fluctuations. This term could be reduced by mi-crowave power stabilization or a well chosen laser power(see Fig. 14), but to the detriment of the signal ampli-tude. V. FREQUENCY SHIFTS AND MID-TERMSTABILITY
We have investigated the clock-frequency shift with re-spect to the variation of various parameters. For eachfrequency measurement, the LO frequency is locked onthe CPT resonance. With a 100 s averaging time, themean frequency is measured with typical error bar lessthan 10 − , i.e. , 0.01 Hz relative to the Cs frequency f Cs = 9 .
192 GHz.0 A. f vs T cell The resonance frequency of the microwave resonanceis shifted by collisions between Cs atoms and buffer-gasatoms [38]. This collisional shift is temperature depen-dent but can be reduced by using a well-chosen mixtureof gas. Here we use a N -Ar mixture with 37% of Ar,optimized for cancelling the temperature coefficient at31 . ◦ C and which should allow to reduce the sensitiv-ity of the cock frequency at the level of − .
16 Hz K − at35 ◦ C [44]. The record of the frequency shift versus T cell is presented in Fig. 20. The temperature sensitivity ismeasured to be 0 .
47 Hz K − at T cell = 35 ◦ C. This valueis in disagreement with the expected value. Nevertheless,it is important to note that the latter is valid at null laserpower and that the laser power shift is also temperature-dependent [45]. In this experimental test, we measurethe result of the collisional shift and of the laser powershift together. The Allan deviation of T cell is shown inthe inset of Fig. 20. With a typical temperature fluctu-ation of 5 . × − K at 1000 s, the contribution of celltemperature variations to the clock fractional frequencystability is about 2 . × − at 1000 s.
30 32 34 36 38 40 4290989100910291049106 (s)0.47 Hz/ o C@35 o C C l o ck s h i ft ( H z ) Tcell ( o C) -1 -5 -4 -3 T c e ll () ( o C ) FIG. 20. (Color online) Clock frequency as a function of thecell temperature T cell . The inset shows the Allan deviationof T cell . All other working parameters is the same as Fig. 10. B. f vs P L The clock frequency shift versus P L is presented inFig. 21. The coefficient of the light power shift is14 . − at P L = 163 µ W and P µw = 26 .
12 dBm.This shift is difficult to foresee theoretically because it re-sults of the combination of light shifts (AC Stark shift) in-duced by all sidebands of the optical spectrum, but also ofoverlapping and broadening of neighboring lines. The in-set of Fig. 21 shows the typical fractional fluctuations thelaser power versus the integration time. They are mea-sured to be 1 × − at 1000 s, impacting on the clock fractional frequency stability at the level of 2 . × − at 1000 s. Since the power distribution in the sidebandsvary with the microwave power, the laser power shift isalso sensitive to the microwave power feeding the EOPM.This is clearly shown in Fig. 21. As previously observedin CPT-based clocks [46, 47] and double-resonance Rbclocks [48], it is important to note that the light-powershift coefficient can be decreased and even cancelled atspecific values of P µw . Consequently, it should be possi-ble to improve the long-term frequency stability by tun-ing finely the microwave power value [49, 50], at the ex-pense of a slight degradation of the short-term frequencystability. L =163 W 1.4Hz/mW@P L =163 W C l o ck s h i ft ( H z ) PL ( W) P w : 26.12dBm27.02dBm -1 -6 -5 -4 -3 P L () / P L (s) FIG. 21. (Color online) Clock frequency as a function of laserpower P L for different values of P µw . Inset: fractional Allandeviation of the laser power. All other working parameters isthe same as Fig. 10. C. f vs P µW The frequency shift versus the microwave power atfixed laser power is shown on Fig. 22. At constant op-tical power, only the power distribution among the dif-ferent sidebands changes. This is a different laser powereffect. The shift scales as the microwave power in theinvestigated range, with a sensitivity of 7 . − at P L = 163 µ W. In this range, the power ratio of bothfirst ( ±
1) sidebands changes by about 10%. The insetof Fig. 22 shows the Allan deviation of the microwavepower in dBm. The typical microwave power standarddeviation of 5 × − dBm at 1000 s yields a fractionalfrequency stability of about 4 . × − at 1000 s. D. f vs ∆ L The frequency of the laser beam incident on the clockcell can be tuned by setting the driving frequency ofAOM2 in the laser frequency stabilization setup. For1 -7.7Hz/dBm @PL=163 W C l o ck s h i ft ( H z ) P w (dBm) P L ( W): 34 66 109 163 227 298 -4 -3 P w () ( d B m ) (s) FIG. 22. (Color online) Clock frequency as a function ofthe microwave power P µw . Inset: Allan deviation of the mi-crowave power in dBm, log scale. All other working parame-ters is the same as Fig. 10. each AOM driving frequency, the laser carrier frequencyis stabilized onto the reference cell, the clock frequencyis locked onto the CPT resonance and measured againstthe hydrogen maser. The observed shift results from acombination of AC Stark shift, CPT resonance distortionand effect of neighboring lines. The recorded frequencyshift is shown on Fig. 23 versus the laser detuning. Theshift is well-fitted by a linear function with a slope of − . − . Having no second similar DFB diodelaser set-up, we did not measure the laser frequency sta-bility. Taking into account that we have the same diodelaser and laser frequency stabilization setup than the onereported in [35], we expect similar performances, i.e. , astandard deviation of ∼ . × − at 1000 s. E. f vs f m For the sake of completeness, we have measured theclock shift versus polarization (and phase) modulationfrequency f m . Results are reported in Fig. 24. Theother parameters are fixed. The shift coefficient is3 .
17 mHz Hz − at f m = 250 Hz. As f m is synchronizedto the LO, which exhibits in the worst case (unlocked)a frequency stability at the level of 7 × − at 1000 s(see Fig. 18), the effect of the polarization and phasemodulation frequency on the clock shift is negligible, i.e. . × − at 1000 s second. F. f vs B The clock frequency shift versus the magnetic fieldstrength is shown on Fig. 25. The experimental shift -30 -20 -10 0 10 20 30910191029103
Experimental data Linear fitting-26.6mHz/MHz C l o ck s h i ft ( H z ) L (MHz)
FIG. 23. (Color online) Clock frequency as a function of laserfrequency detuning ∆ L . Experimental data are fitted by alinear function. All other working parameters is the same asFig. 10. C l o ck s h i ft ( H z ) f m (kHz)3.17mHz/Hz FIG. 24. (Color online) Clock frequency as function of polar-ization(phase) modulation frequency f m . All other workingparameters is the same as Fig. 10. is in good agreement with the theoretical prediction ofthe quadratic Zeeman shift [38], f = 4 . × − B ,with B in µ T. In order to measure the time evolu-tion of the magnetic field, we locked the LO frequencyto the magnetic-field sensitive Zeeman CPT transition | F = 3 , m F = − i → | F = 4 , m F = − i . The latter ex-hibits a sensitivity 7 .
01 kHz µ T − . The Allan standarddeviation of the Zeeman frequency ( f Z ) is shown in theinset of Fig. 25. The measured frequency deviation isabout 0 . τ = 1000 s. For a mean magneticfield of 3 . µ T, this yields a fractional frequency stabilityof 0 . × − at 1000 s.2 Experimental Theoretical C l o ck s h i ft ( H z ) B0 ( T) -2 -1 -2 -1 f Z () ( H z ) (s) FIG. 25. (Color online) Clock frequency as function of mag-netic field B . Inset: Allan deviation of the Zeeman frequency.All other working parameters is the same as Fig. 10. G. Mid-term stability
With the shift coefficients and the Allan standard devi-ation of the involved parameters, we can estimate the var-ious contributions to the mid-term clock frequency sta-bility. They are listed in Table II for a 1000 s averagingtime. Their quadratic sum leads to a frequency stabil-ity of 4 . × − at τ = 1000 s, in very good agreementwith the measured stability 4 . × − (see Fig. 18).Again, the main contribution to the instability comesfrom the microwave power fluctuations, before the laserpower and frequency fluctuations. Thus in the future, itis necessary to stabilize the microwave power to improveboth the short-and-mid-term frequency stability. TABLE II. Noise contributions to the stability at 1000 s.Parameter coefficient σ y (1000 s ) × T cell .
47 Hz K − . × − P L . − . P µw − . − . L − . − . B .
29 Hz µ T − . × − Total 4 . VI. CONCLUSIONS
We have implemented a compact vapor cell atomicclock based on the DM CPT technique. A detailed char-acterization of the CPT resonance versus several exper-imental parameters was performed. A clock frequencystability of 3 . × − τ − / up to 100 s averaging timewas demonstrated. For longer averaging times, the Allandeviation scales as √ τ , signature of a random walk fre-quency noise. It has been highlighted that the main lim-itation to the clock short and mid-term frequency stabil-ity is the fluctuations of the microwave power feeding theEOPM. Improvements could be achieved by implement-ing a microwave power stabilization. Another or comple-mentary solution could be to choose a finely tuned laserpower value minimizing the microwave power sensitivity.This adjustment could be at the expense of the signalreduction and a trade-off has to be found. Nevertheless,the recorded short-term stability is already at the levelof best CPT clocks [31, 33] and close to state-of-the artRb vapor cell frequency standards. These preliminaryresults show the possibility to a high-performance andcompact CPT clock based on the DM-CPT technique. ACKNOWLEDGEMENTS
We thank Moustafa Abdel Hafiz (FEMTO-ST), DavidHolleville and Luca Lorini (LNE-SYRTE) for helpful dis-cussions. We are also pleased to acknowledge CharlesPhilippe and Ouali Acef for supplying the thermal insu-lation material, Michel Abgrall for instrument Symmet-ricom 5125A lending, David Horville for laboratory ar-rangement, Jos´e Pinto Fernandes, Michel Lours for elec-tronic assistance, Pierre Bonnay and Annie G´erard formanufacturing Cs cells.P. Y. is supported by the Facilities for Innovation, Re-search, Services, Training in Time & Frequency (LabeXFIRST-TF). This work is supported in part by ANRand DGA(ISIMAC project ANR-11-ASTR-0004). Thiswork has been funded by the EMRP program (IND55Mclocks). The EMRP is jointly funded by the EMRPparticipating countries within EURAMET and the Eu-ropean Union. [1] J. Camparo, The rubidium atomic clock and basic re-search, Physics Today, pp 33–39 (November 2007).[2] S. Micalizio, C. E. Calosso, A. Godone and F. Levi,Metrological characterization of the pulsed Rb clock withoptical detection, Metrologia , 425-436 (2012).[3] T. Bandi, C. Affolderbach, C. Stefanucci, F. Merli, A.K. Skrivervik and G. Mileti, coninuous-wave double-resonance rubidium standard with 1 . × − τ − / sta- bility, IEEE Ultrason. Ferroelec. Freq. Contr. , 11,1769–1778 (2014).[4] S. Kang, M. Gharavipour, C. Affolderbach, F. Gruet,and G. Mileti, Demonstration of a high-performancepulsed optically pumped Rb clock based on a compactmagnetron-type microwave cavity, J. Appl. Phys. ,104510 (2015).[5] A. Godone, F. Levi, C. E. Calosso and S. Micalizio, High-performing vapor cell frequency standards, Rivistadi Nuovo Cimento , 133-171 (2015).[6] G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, An exper-imental method for the observation of R. F. transitionsand laser beat resonances in oriented Na vapour, Il NuovoCimento
36 B , 5 (1976).[7] E. Arimondo, Coherent population trapping in laserspectroscopy, Progress in Optics , 257-354 (1996).[8] K. Bergmann, H. Theuer, and B. W. Shore, Coherentpopulation transfer among quantum states of atoms andmolecules. Rev. Mod. Phys., , 1003-1025 (1998).[9] R. Wynands and A. Nagel, Precision spectroscopy withcoherent dark states, Appl. Phys. B 68 , 1 (1999).[10] J. Vanier, Atomic clocks based on coherent populationtrapping: A review, Appl. Phys. B , 421-442 (2005).[11] M. Bajcsy, A. S. Zibrov and M. D. Lukin, Stationarypulses of light in an atomic medium, Nature
26, 6409-6411 (2004).[13] E. Breschi, Z. D. Gruji, P. Knowles and A. Weis, A high-sensitivity push-pull magnetometer, Appl. Phys. Lett. , 2112-2124 (1989).[15] J. Vanier and C. Mandache, The passive opticallypumped Rb frequency standard: The laser approach,Appl. Phys. B Lasers Opt. , 565-593 (2007).[16] J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard,and C. R. Willis, Ultrahigh-resolution spectroscopy andfrequency standards in the microwave and far-infraredregions using optical lasers, Opt. Lett. , 298-300 (1981).[17] J. E. Thomas, P. R.Hemmer, S. Ezekiel, C. C. Leiby,,R. H. Picard, and C. R. Willis, Observation of Ramseyfringes using a stimulated, resonance Raman transitionin a sodium atomic beam, Phys. Rev. Lett. , 867-870(1982).[18] N. Cyr, M. Tetu, and M. Breton, All-optical microwavefrequency standard - a proposal, IEEE Trans. Instrum.Measur. , 640-649 (1993).[19] J. Kitching, N. Vukicevic, L. Hollberg, S. Knappe,R.Wynands, and W. Weidemann, A microwave frequencyreference based on VCSEL-driven dark line resonances inCs vapor, IEEE Trans. Instrum. Measur. , 1313-1317(2000).[20] J. Kitching, L. Hollberg, S. Knappe, and R. Wynands,Compact atomic clock based on coherent populationtrapping, Electron. Lett. , 1449 (2001).[21] J. Kitching, S. Knappe, and L. Hollberg, Miniaturevapor-cell atomic-frequency references, Appl. Phys. Lett. , 553 (2002).[22] L. Liew, S. Knappe, J. Moreland, H. Robinson, L. Holl-berg, and J. Kitching, Microfabricated alkali atom vaporcells, Appl. Phys. Lett. , 2694 (2004).[23] S. Knappe, V. Shah, P. D. D. Schwindt, L. Hollberg, J.Kitching, L.-A.Liew, and J. Moreland, A microfabricatedatomic clock, Appl. Phys. Lett. ,231106 (2014).[29] Y.-Y. Jau, E. Miron, A. B. Post, N. N. Kuzma, and W.Happer, Push-Pull Optical Pumping of Pure Superposi-tion States, Phys. Rev. Lett. , 160802-1-4 (2004).[30] X. Liu, J.-M. M´erolla, S. Gu´erandel, C. Gorecki, E. deClercq, and R. Boudot, Coherent population trappingresonances in buffer-gas-filled Cs-vapor cells with push-pull optical pumping, Phys. Rev. A , 013416 (2013).[31] M. Abdel Hafiz and R. Boudot, A coherent populationtrapping Cs vapor cell atomic clock based on push-pulloptical pumping, J. Appl. Phys. , 124903 (2015).[32] T. Zanon, S. Gu´erandel, E. de Clercq, D. Holleville, N.Dimarcq and A. Clairon, High Contrast Ramsey Fringeswith Coherent-Population-Trapping Pulses in a DoubleLambda Atomic System, Phys. Rev. Lett. , 2982-2985 (2016).[36] B. Fran¸cois, C. E. Calosso, M. Abdel Hafiz, S. Mical-izio, and R. Boudot, Simple-design ultra-low phase noisemicrowave frequency synthesizers for high-performing Csand Rb vapor-cell atomic clocks, Rev. Sci. Instrum. ,094707 (2015).[37] C. E. Calosso, S. Micalizio, A. Godone, E. K. Bertacco, F.Levi. Electronics for the pulsed rubidium clock: Designand characterization. IEEE Trans. Ultrason. Ferroelectr.Freq. Control , 1731-1740 (2007).[38] J. Vanier and C. Audoin, The quantum physics of atomicfrequency standards (Adam Hilger, Bristol, 1989).[39] A. Godone, F. Levi, S. Micalizio, and J. Vanier, Dark-line in opticall-thic vapors: inversion phenomena and linewidth narrowing, Eur. Phys. J. D , 5-13 (2002).[40] P. Yun, S. Gu´erandel, and E. de Clercq , Coherent pop-ulation trapping with polarization modulation, J. Appl.Phys. 119, 244502 (2016).[41] P. Yun, S. Mejri, F. Tricot, M. Abdel Hafiz, R. Boudot,E. de Clercq, S. Gu´erandel, Double-modulation CPT ce-sium compact clock, 8th Symposium on Frequency Stan-dards and Metrology 2015, J. of Physics: Conference Se-ries , 012012 (2016). [42] Enrico Rubiola, Phase noise and frequency stability inoscillators, Cambridge University Press, 2009.[43] C. Audoin, V. Candelier, and N. Dimarcq, A limit to thefrequency stability of passive frequency standards due toan intermodulation effect, IEEE Trans. Instrum. Meas. , 121 (1991).[44] O. Kozlova, S. Gu´erandel, and E. de Clercq, Tempera-ture and pressure shift of the Cs clock transition in thepresence of buffer gases: Ne, N , Ar, Phys. Rev. A ,062714 (2011).[45] O. Kozlova, J-M. Danet, S. Gu´erandel, and E. de Clercq,Limitations of long-term stability in a coherent popula-tion trapping Cs Clock, IEEE Trans. Instrum. Meas. ,1863 (2014).[46] F. Levi, A. Godone, J. Vanier, The light-shift effect in thecoherent population trapping cesium maser, IEEE Trans.Ultrason. Ferroelectr. Freq. Control , 466 (2000). [47] M. Zhu and L.S. Cutler, Theoretical and experimen-tal study of light shift in a CPT-based Rb vapor cellfrequency standard, in Proceedings of the 32nd PreciseTime and Time Interval Systems and Applications Meet-ing, p. 311, ed. by L.A. Breakiron (US Naval Observatory,Washington, DC, 2000).[48] C. Affolderbach, C. Andreeva, S. Cartaleva, T. Ka-raulanov, G. Mileti, and D. Slavov, Light-shift suppres-sion in laser optically pumped vapour-cell atomic fre-quency standards, Appl. Phys. B , 841-8 (2005).[49] V. Shah, V. Gerginov, P. D. D. Schwindt, S. Knappe, L.Hollberg and J. Kitching, Continuous light- shift correc-tion in modulated coherent population trapping clocks,Appl. Phys. Lett.94