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Quantum-Limited Spectroscopy G.-W. Truong,1,2 J. D. Anstie,2,1 E. F. May,3 T. M. Stace,4, and A. N. Luiten2,1, ∗ † 1School of Physics, The University of Western Australia, Perth, Western Australia 6009, Australia 2Institute for Photonics and Advanced Sensing (IPAS) and School of Chemistry and Physics, The University of Adelaide, Adelaide, SA 5005, Australia 3Centre for Energy, School of Mechanical and Chemical Engineering, The University of Western Australia, Crawley, Western Australia 6009, Australia 4ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane 4072, Australia (Dated: January 5, 2015) Spectroscopy has anillustrious history delivering serendipitous discoveries andproviding a strin- 5 gent testbed for new physical predictions, including applications from trace materials detection, to 1 understanding the atmospheres of stars and planets, and even constraining cosmological models. 0 Reaching fundamental-noise limits permits optimal extraction of spectroscopic information from 2 an absorption measurement. Here we demonstrate a quantum-limited spectrometer that delivers high-precision measurements of the absorption lineshape. These measurements yield a ten-fold n improvement in the accuracy of the excited-state (6P ) hyperfine splitting in Cs, and reveals a a 1/2 breakdown in the well-known Voigt spectral profile. We develop a theoretical model that accounts J for this breakdown, explaining the observations to within the shot-noise limit. Our model enables 2 ustoinferthethermalvelocity-dispersionoftheCsvapourwithanuncertaintyof35ppmwithinan hour. This allows us to determine a value for Boltzmann’s constant with a precision of 6ppm, and ] h an uncertainty of 71ppm. p - m Spectroscopy is a vital tool for both fundamental and 3.7ppm after 200 scans. These values are 7 times bet- ∼ o applied studies. It is critical to drive improvements in ter than the previous best results [10–12], and also yield t both precision and accuracy in order to gain maximum a 10-fold improvement in the uncertainty of the excited a . physicalinformationaboutthequantumabsorbers–even state hyperfine splitting in Cs [13, 14]. Our measure- s c as they are perturbed by the probe. The immediate moment of the homogenous broadening component of the i tivationforthisworkhasarisenoutofacalltothescien- line shape has a precision within a factor of two of the s y tificcommunitytodevelopnewtechniquestore-measure best ever measurement of natural linewidth. Modest im- h Boltzmann’s constant, k , in preparation for a redefini- provementsintheprobelaserperformancewoulddeliver B p tion of the kelvin [1]; however, advances in absorption a new result for the excited state lifetime of Cs in a sys- [ lineshape measurement and theory will find applications tem that is experimentally and theoretically much sim- 1 in accurate gas detection and monitoring [2], studies of pler than that typically used for lifetime studies [15, 16]. v planetary atmospheres [3], thermometry in tokamaks [4] At thermal equilibrium, the velocity distribution of 6 and understanding distant astrophysical processes [5, 6]. atomicabsorbersinavapourcellisrelatedtothetemper- 4 3 The accurate measurement of the natural linewidth and ature through the Boltzmann distribution. This simple 0 transition frequencies, which can be directly related to and fundamental relationship forms an excellent founda- 0 the atomic lifetime, level structure and transition proba- tionforatypeofprimarythermometryknownasDoppler 1. bilities, is important for testing atomic physics [7–9]. broadening thermometry (DBT) [17, 18]. DBT differs 0 Here we present measurements of a transmission line- fromthecurrentleadingtechniqueforprimarythermom- 5 shape of cesium (Cs) vapor with a quantum-limited etry, which measures the temperature-dependent speed 1 transmission uncertainty of 2ppm in a 1 second mea- of sound in a noble gas contained in a well-characterized : v surement,whichistoourknowledge,afactorof16times acoustic resonator. Until recently, the best determina- Xi superior to anything previously demonstrated [10]. This tion of kB was made in 1988 with a total uncertainty extremeprecisionallowsustodirectlydetectsubtleline- of 1.7 parts-per-million (ppm) [19]. Refinements to this r a shape perturbations that have not been previously ob- techniqueoverthelasttwentyyearshavereduceditsun- served. This observation prompted the development of certaintytothe1ppmlevel[20,21],improvingtheuncer- a theoretical model, which now allows us to discriminate tainty in the CODATA-recommended value for kB from betweentheinternalatomicstatedynamicsandtheirex- 1.8ppm (CODATA-02) to 0.91ppm (CODATA-10). De- ternal motional degrees. Using the model we are able to spitethesesuperbmeasurements,itisimportantthatdif- estimatethevelocitydispersionoftheatomswithapreci- ferent techniques are employed to measure kB to reduce sion(standarderror) of53ppmduringasingle linescan, the possibility of underestimating systematic uncertain- taking 30seconds. Thisisconsistentwiththethesam- ties that may be inherent to any single technique [22]. ∼ plestandarddeviation(also53ppm)overmultiplescans, The DBT approach is one of the most theoretically demonstrating the excellent reproducibility of the spec- transparent of the new approaches to primary thermom- trometer. The measurement precision averages down to etry. The spectrum of a gas sample is measured with 2 PDB P W GT C SPRT 1 OC Cs Cell SPRT 2 PDA Blocker AOM M Shield FIG. 1. Schematic of a high-accuracy linear absorption spectrometer based on a probe laser (P) that is tightly locked at a tunablefrequencydifferenceawayfromamasterlaser(M),whichisitselfstabilisedtoanatomictransitioninCs. Thefrequency stability of the master laser is compared to a reference frequency provided by an optical frequency comb (C). The probe is spectrally filtered with an optical cavity (OC), and actively power-stabilised with an acousto-optic modulator (AOM). The probe is then divided with high-precision into two beams using a Glan-Taylor (GT) polarizing prism followed by a Wollaston beamsplitter (W). One beam passes through a Cs cell embedded inside a thermal and magnetic shield and is measured by photodiode A (PD A); the other beam is measured directly by photodiode B (PD B). Temperature and thermal gradients are measured independently with calibrated standard platinum resistance thermometers (SPRTs). high precision, and a theoretical model is fitted to the controlled while the frequency of the incident light is set measured data. If the model includes all of the relevant to an absolute accuracy of 2kHz. The temperature of physics then one can accurately extract the contribu- the vapour cell is monitored directly using a calibrated tion to spectral broadening arising solely from the ther- capsule-type standard platinum resistance thermometer mal distribution of atomic/molecular speeds. Ammonia (CSPRT),allowingustoquantifytheperformanceofour probed by a frequency-stabilized CO laser at 1.34µm Doppler thermometer. Further technical details about 2 wasthefirstthermometricsubstanceemployedinaDBT the apparatus are given in the supplementary informa- experiment [23]. Subsequently, an extended-cavity diode tion (SI). laser at 2µm was used to probe a ro-vibrational tran- We measure the transmission through the vapour cell sition of CO2 [24]. In these first experiments, the line- astheprobelaserfrequencyscansacrosstheCsD1tran- shapewasassumedtobeaGaussianoraVoigtprofile(a sition (6S – 6P ), shown in Fig. 2(a). The relative 1/2 1/2 GaussianconvolvedwithaLorentzian). Since2007,with noise in the measurement of the atomic transmission is the ambitious goal of approaching 1 ppm accuracy, sub- just 2 ppm in a 1s measurement (at the highest opti- stantialexperimentalandtheoreticalimprovementshave cal powers used). The two dips seen on Fig. 2(a) come been made to DBT using ammonia [25, 26], oxygen [27], from the hyperfine splitting in the excited state and our ethyne [28], and water [29]. However, one key challenge excellent signal-to-noise ratio enables us to determine peculiar to molecular absorbers is the need to account this splitting to be f =1167.716(3) MHz consistent HFS for complex collisional effects on the line shape [26]. We with previous measurements (f =1167.680(30) MHz HFS avoid this by using a dilute atomic vapour [30, 31] with [13, 14]), but 10 times more accurate. a strong dipole transition for which a tractable, micro- The absorption line-shape, (f), of an atomic tran- at scopic theory has been developed [32, 33]. T sition with a rest-frame transition frequency f , is 0 ThespectrometerispicturedinFig.1. Theprobelaser given by a convolution of the natural (half) linewidth is spectrally and spatially filtered using a combination ΓL = 1/(4πτ) due to the finite lifetime τ of the tran- of an optical cavity of moderate finesse ( 305) and sition, with a Gaussian distribution having a e−1-width single-modefibre. ThisreducesthespontanFeo≈usemission ΓD =f0 2kBT/(mCsc2) due to Doppler broadening of content of the probe beam from 1.6% to below 0.01%. It atoms of mass m at temperature T. By fitting a line- p is then delivered into a vacuum chamber in which the shape to the measured transmission data, with Γfit as D vapourcellismountedinathermalandmagneticshield. a fitting parameter, we extract the Doppler component, The temperature of the shield can be controlled to a few from which we infer the temperature. Systematic er- millikelvinandgradientsaresuppressedtothesamelevel. rors in Γfit are quantified by converting the indepen- D The light is split into two output signals using a combi- dent PRT temperature measurements into an expected nation of a Glan-Taylor polarising prism and Wollaston Doppler width, ΓPDRT, using the CODATA values for kB beam-splitter. The ratio of powers in the output beams and mCs, which we compare with ΓfiDt. is stable to better than 10 6. One output directly illu- If the natural lineshape of the transition is a − minates a photodiode to give the incident power, whilst Lorentzian, then the resulting convolution is the well- the other passes through the vapour cell and is then de- known Voigt profile, which is commonly used to model tected. The ratio of these photodiode signals gives us dipole resonances. However, our extremely low-noise thetransmissionratio, . Theincidentpowerisactively transmission measurements reveal deviations from the T 3 1.0 200 (a) (cid:84)0.8 sion, 0.6 (cid:76)m 0 smis 0.4 (cid:72)pp (cid:45)200 n a 1 Tr 0.2 (cid:45) T 0.0 PRD(cid:45)400 6000 7000 8000 9000 10000 11000 (cid:71) (b) (cid:144) (cid:76)m 200 fit(cid:71)D(cid:45)600 100 p p (cid:72)al 0 (cid:45)800 u 0 0.001 0.002 0.003 resid (cid:45)(cid:45)210000 NormalisedIntensity(cid:72)I(cid:144)Isat(cid:76) FIG.3. FractionaldeviationbetweenthefittedDopplerwidth (c) 6 7 8 9 10 11 (cid:84)et 1. and that inferred from PRT temperature measurements at 296 K. Green points are fits using the power-independent ctor, 0.9999 Voigt-only profile. Blue points are fits using the first-order efa intensity-dependent correction. Red points are fits using the pr second-orderintensity-dependentcorrection. Solidcurvesare on 0.9998 fits using our central estimate of Γ¯L = 2.327 MHz; dashed etal curvesareforΓL =2.320MHz(upper)andΓL =2.334MHz (lower) representing Γ¯ σ : the shaded regions indicates (d) 10 6000. 7000. 8000. 9000. 10000. 11000. thesensitivityofΓfiDttoLΓ±LinΓLthisrange. Errorbarsrepresent (cid:76)m 5 standarderrorsoffitstotheaverageof200scansforfixedΓL. p p (cid:72)al 0 u d esi (cid:45)5 this Voigt-only fit have characteristic M-shaped features r nearresonance,withamplitude 200ppmatthehighest (cid:45)10 ∼ 6 7 8 9 10 11 probe powers, as shown in the green trace in Fig. 2(b). Frequency(cid:72)GHz(cid:76) These features demonstrate the breakdown of the Voigt profile,indicatingadditional,unaccountedphysics,which causes a spurious linear dependence of Γfit on probe in- FIG. 2. (a) Measured transmission at 296 K for the highest D tensity, as shown in the green curve on Fig. 3. In what intensity, I/I = 0.0028, averaged over 200 scans [points] sat and theory [solid]. (b) Residuals including Voigt only [dark- follows, we describe how we sequentially include addi- green] and first-order correction [light-blue]. (c) Etalon pre- tionalphysicsinourtransmissionmodeltoremovethese, factor, Eq. 1, with 6 etalons [solid] and data divided by and other, systematic effects, leaving only the shot-noise second-order power-corrected Voigt profiles [points], assum- limit of our detection apparatus. ingΓ =2.327 MHz. (d)Residualsafterincludingfirst-order L In earlier work we showed that the M-shaped features correctionwith0etalons[light-blue]and6etalons[dark-blue], in Fig. 2(b) arise from atomic population dynamics in- and second-order with 6 etalons [dotted-red]. duced by the probe laser, which are significant even at exceedingly low intensities [32]. We subsequently calcu- lated corrections to the Voigt profile up to linear-order Voigt profile. Some of these deviations are technical in in the probe intensity [33], to account for optical pump- origin e.g. instrumental broadening due to the lineshape ing. Including these first-order intensity-dependent cor- of the probing laser, residual spontaneous emission from rectioninourmodelsuppressesthefeaturessubstantially, the probe laser, unwanted optical etalons, and photode- as shown in the blue trace in Fig. 2(b). tector linearity; however, there is an important funda- Far from the resonances, oscillations are evident in mental effect resulting from frequency-dependent optical Fig. 2(b). These arise from low-finesse etalons in the op- pumping, which perturbs the atomic natural line-shape ticsusedtodeliverthelighttotheatoms. Thedominant away from a Lorentzian. All of these effects, whether etalonhasanamplitudearound40ppm,correspondingto technicalorfundamental,causesystematicperturbations interference between surfaces with power reflectivities of to the line shape, and must be taken into account to . 10−9. Although obviously technical in nature, reduc- model the lineshape accurately. ing the size of etalons beyond this already fantastically To demonstrate these deviations, we fit a model con- low level is an experimental challenge. Instead, we in- sisting of two Voigt profiles separated by f , to the cludeetalonsinourtransmissionmodel,sothatthetotal HFS raw transmission data shown Fig. 2(a). The residuals of transmission is given by = , where includes at et et T T T T 4 Source u (k ) (ppm) 1000 r B Statistical (from Fig. 4) 5.8 Lorentz Width (Γfit)a 65 L (cid:76)m 500 Lorentz Width (Γind)b 190 p L p (cid:72) Laser Gaussian noise 16 1 Optical pumping 15 (cid:45) 0 T Etalons (misidentification) 15 PRD (cid:71) Etalons (unresolved) 3 (cid:144) Spontaneous Emission 3.6 fit(cid:71)D (cid:45)500 Temperature 1.9 Temp. Gradient 1.2 (cid:45)1000 0 200 400 600 800 1000 PD Linearity 1 ScanNumber Zeeman Splitting <1 Total (fit ΓL) 71 FIG. 4. Deviation between Γfit (using second-order inten- D Total (independent ΓL) 191 sity correction) and ΓPDRT, for each scan at 296 K, with Γ =2.327 MHz. Colours and vertical lines demark differ- L ent incident intensities (lowest at left, corresponding to nor- TABLE I. Error budget for the determination of the Boltz- malised intensities in Fig. 1). Dashed lines are the mean of mannconstantat296K.Opticalpumpingshiftsarereported the corresponding set of scans, error bars are the estimated atI/Isat =3 10−3 assumingsecond-ordercorrectionstothe parameter error ( 1σ) for each selected scan, and are con- Voigtprofile.×aEstimatedfromfitstoscandatawithfreeLW. sistentwithsampl±estandarddeviation. Etalonsareincluded bEstimatedfrompublisheduncertaintyinCslifetime,plusun- in each fit, with parameters fixed by fits to the average of all certainty in the independent estimate of the laser linewidth. scanswithinapowerlevel,i.e.etalonparametersarenotfree ur denotes fractional uncertainty. to vary within a particular power level. n etalons and a slowly-varying quadratic background: where ν = Γ /Γ , ∆ is the detuning from the res- D L α+βf +γf2 onance, and the generalised Voigt profile is given by (f)= . (1) Tet nj=1 1+ajsin(2πf/fj +φj) Vν(n)(∆)=π−3/2 ∞ dxe−(∆−x)2/ν2(1+x2)−n,andthe Etalons are of pQarticu(cid:0)lar concern for DBT si(cid:1)nce they conventional VoigRt−∞profile is Vν(∆) = Vν(1)(∆). For a cell of length L with linear absorption coefficient α, the introduce systematic errors in Γfit. We note that it is D optical depth on resonance is p = αL, q I is the lin- onlybecauseofourextremetransmissionsensitivitythat 1 ∝ earintensity-dependentcoefficient[32],andq I2 are thesefeaturesarerevealed,andthusgivetheopportunity 2,3 ∝ quadratic intensity-dependent coefficients. to suppress this systematic error. The red dotted trace in Fig. 2(d) shows the residuals We add etalons to the model until the residuals far after including second-order intensity corrections. The from resonance are consistent with a white-noise back- residuals are suppressed to 2 ppm, consistent with the ground. Inpractice,itisnecessarytoincludeupton=6 ∼ detection noise floor, giving confidence that the trans- etalons for high power scans, with the smallest resolved mission model accounts for systematic effects. etalons having amplitude 7 ppm. The solid curve in ∼ Figure 3 compares the fractional deviation between Fig.2(c)shows forthehighestpowerdata;thepoints Tet Γfit and ΓPRT as a function of incident intensity for the showtherawtransmissiondatadividedbythefitted , D D Tat three possible models for the atomic line shape: Voigt- demonstrating that is uncorrelated with the atomic Tet only (green), first-order (blue) and second-order (red) transmission profile shown in Fig. 2(a). intensity-dependent corrections. Naturally, the true Γ The dark blue trace in Fig. 2(d) includes first-order D shouldbeintensityindependent. ThesimplestVoigt-only power-dependent corrections, and 6 etalons. Away from theory clearly exhibits a linear intensity-dependence, resonance, the residuals are then consistent with the leading to 750 ppm discrepancy even at intensities noise-floor of our apparatus. However, around the deep- ∼ as low as I/I =2.8 10 3 where I =2.5mW/cm2 est resonance we observe 10 ppm features. To elimi- sat × − sat ∼ is the saturation intensity for the transition [13]. From nate these we include second-order intensity-dependent [33], we expect the first-order correction to have a resid- corrections to the Voigt profile (see SI): ual quadratic dependence on intensity, which is consis- Ta(ti) =e−pVν(∆) 1+q1(e−pVν(∆)−1)Vν(2)(∆)/Vν(∆) toerndterwcitohrrtehcteiobnlu(erecdurcvuerovne)Fiisg.se3e.nFtionaslulyp,ptrheessseaclolnind-- +q(cid:16)2 (e−pVν(∆)−1)Vν(2)(∆)/Vν(∆) 2 tTehnesistiymduelptaennedoeunsceretmoobvealloowfathllesmysetaemsuarteimcefenattpurreecsifsrioonm. +q3((cid:0)e−2pV(∆,ν)−1)Vν(3)(∆)/Vν(∆)(cid:1) ,(2) the residuals together with the elimination of intensity- (cid:17) 5 dependence in the recovered Γfit gives a high degree of the line-shape parameters of a gas at thermal equilib- D confidence that all relevant physics is properly included rium. Our reproducibility is exactly consistent with the in the theoretical model. quantum-limits of measurement [34] which gives us con- Thermometry: In this section we demonstrate the fidence that we have captured all of the relevant physics. powerofourspectroscopictechniquebyapplyingittothe Withourunprecententedsensitivity,wehavemeasured problem of primary thermometry, and determine a value the transmission spectrum of Cs at 895nm at the shot- for Boltzmann’s constant using atomic spectroscopy. noise limit. From these measurements, we derived a The atomic vapour is brought to thermal equilibrium value for Boltzmann’s constant with an uncertainty of at a chosen temperature by suspending it within a ther- 71 ppm, which is consistent with the recommended CO- mal isolator with an independent means for temperature DATAvalue,andthehyperfinefrequencysplittingofthe measurement accurate to 0.55mK (1.9ppm). After sys- 62P level with an uncertainty ten times better than 1/2 tematically removing the effects of optical pumping and ever previously reported: etalons, the largest source of uncertainty in ΓfiDt comes kB =1.380545(98) 10−23 J/K from uncertainty in Γ . We obtain a value for Γ di- × L L rectlybyfittingittothedata,whichgivesΓfit =2.327(7) fHFS =1167.716(3) MHz. L MHz. Thisisconsistentwith,butalsomoreprecisethan, Our current performance permits a measurement of the an independent estimate, Γind =2.331(19) MHz, given Boltzmann constant with a precision of 6ppm after only L by the sum of the natural linewidth of the Cs 6P1/2 a few hours of data acquisition. Our total uncertainty is level, Γat =2.287(6) MHz [13], and the laser linewidth, dominated by the imprecision in the literature value of L Γlas = 0.044(18) MHz. From Fig. 3 we estimate that a the excited state lifetime of the Cs D1 transition (known L 1kHz change in ΓL leads to ∼ 5 ppm change in ΓfiDt, so to 0.26% [13]). A future experiment designed to fur- that the 6.5 kHz uncertainty in ΓL contributes 32.5 ppm ther reduce the amplitude of etalons, and using a more uncertainty in ΓfiDt [65 ppm in ur(kB)]. We note that if broadly tuneable ( 30GHz) and higher power probe ∼ the probe laser linewidth were decreased to the kilohertz laserwithsub-kHzlinewidthwouldeliminateprobelaser level, then our measurement would yield a state-of-the- effects, allow better estimation of etalon contamination, art value for the excited state lifetime of Cs. and lower the contribution of optical pumping. This We now briefly describe additional sources of uncer- opens the door to the best measurement of the Cs D1 tainty, which are quantified in Table I (see SI for a com- transition lifetime, and a ppm-level measurement of k B prehensivedescription). Statisticalerrorarisingfromthe thus making an important contribution to efforts to re- scatter in Fig.4 contributes 2.9 ppm to ur(ΓfiDt) [5.8ppm define the kelvin. to ur(kB)]. This is also consistent with the estimated The authors wish to thank the NIST Precision Mea- standard error for each point in Fig.4, shown as er- surement Grants Program, the Australian Research ror bars. The probe laser has gaussian noise of width Council through the CE110001013, FT0991631 and 0.88(39)MHz, contributing 8 ppm to ur(ΓfiDt) [16ppm to DP1094500 grants for funding this work. The authors ur(kB)]. Residualeffectsofopticalpumpingaftersecond- also wish to acknowledge the South Australian Gov- order power corrections contribute 7.5 ppm to ur(ΓfiDt), ernment who have provided generous financial support [15 ppm to ur(kB)]. Misidentification of etalon parame- through the Premier’s Science and Research Fund. We ters contributes 7.5 ppm to ur(ΓfiDt), [15 ppm to ur(kB)]. thank Michael Moldover and Greg Strouse of NIST, and Possible unresolved etalons contribute 3 ppm to ur(kB). MarkBallicofromNMIAustralia,fortheircontributions Residualspontaneousemissionfromthelaser,afterfilter- to the SPRT thermometry. ing by the optical cavity, contributes 3.6ppm to u (k ). r B To determine a value for k , we take a weighted B mean of Γfit/ΓPRT extracted from fits using all the D D second-ordercorrecteddatashownonFig.3(redpoints). We find k =1.380545(98) 10 23J/K, where the 71 ∗ theoretical questions: [email protected] B − ppm uncertainty is calcula×ted in Table I. This is † experimental questions: [email protected] [1] B. Fellmuth, C. Gaiser, and J. Fischer, Measurement consistent with the recommended CODATA value of Science and Technology 17, R145 (2006). 1.3806488(13) 10−23 J/K. [2] G. Stewart, W. Johnstone, J. R. P. Bain, K. Ruxton, × Inconclusion,wehavedevelopedanatomicabsorption and K. Duffin, J. Lightwave Technol. 29, 811 (2011). spectrometerthatoperateswithatransmissionmeasure- [3] P. Drossart, Comptes Rendus Physique 6, 817 (2005). ment precision of just 2ppm. This has revealed several [4] J.A.Koch,R.E.Stewart,P.Beiersdorfer,R.Shepherd, M.B.Schneider,A.R.Miles,H.A.Scott,V.A.Smalyuk, new phenomena. In combination with a theory that cor- and W. W. Hsing, Review of Scientific Instruments 83, rectlytreatstheinteractionbetweenlightandaneffusive 10E127 (2012). vapour we are able to explain all of the observed effects [5] B. Grefenstette, F. Harrison, S. Boggs, S. Reynolds, to a level of precision never before demonstrated. The C. Fryer, K. Madsen, D. Wik, A. Zoglauer, C. Ellinger, power of our technique is demonstrated by measuring D. Alexander, et al., Nature 506, 339 (2014). 6 [6] A. Hayato, H. Yamaguchi, T. Tamagawa, S. Katsuda, [20] L.Pitre,F.Sparasci,D.Truong,A.Guillou,L.Risegari, U.Hwang,J.P.Hughes,M.Ozawa,A.Bamba,K.Kinu- andM.Himbert,InternationalJournalofThermophysics gasa,Y.Terada,A.Furuzawa,H.Kunieda, andK.Mak- 32, 1825 (2011). ishima, The Astrophysical Journal 725, 894 (2010). [21] M. de Podesta, R. Underwood, G. Sutton, P. Morantz, [7] C. S. Wood, S. C. Bennett, D. Cho, B. P. Masterson, P. Harris, D. F. Mark, F. M. Stuart, G. Vargha, and J.L.Roberts,C.E.Tanner, andC.E.Wieman,Science G. Machin, Metrologia 50, 354 (2013). 275, 1759 (1997). [22] E. R. Cohen and J. W. M. DuMond, Rev. Mod. Phys. [8] P.Jo¨nsson,A.Ynnerman,C.FroeseFischer,M.R.Gode- 37, 537 (1965). froid, and J. Olsen, Phys. Rev. A 53, 4021 (1996). [23] C. Daussy et al, Phys. Rev. Lett. 98, 250801 (2007). [9] T. Brage, C. F. Fischer, and P. Jo¨nsson, Phys. Rev. A [24] G. Casa et al, Phys. Rev. Lett. 100, 200801 (2008). 49, 2181 (1994). [25] M. Triki, C. Lemarchand, B. Darquié, P. L. T. Sow, [10] L. Moretti, A. Castrillo, E. Fasci, M. D. De Vizia, V. Roncin, C. Chardonnet, and C. Daussy, Phys. Rev. G. Casa, G. Galzerano, A. Merlone, P. Laporta, and A 85, 062510 (2012). L. Gianfrani, Phys. Rev. Lett. 111, 060803 (2013). [26] C. Lemarchand, M. Triki, B. Darquié, P. L. T. Sow, [11] C. Lemarchand, K. Djerroud, B. Darquié, O. Lopez, S. Mejri, C. Chardonnet, C. J. Bordé, and C. Daussy, A. Amy-Klein, C. Chardonnet, C. Bordé, S. Briaudeau, Journal of Physics: Conference Series 397, 012028 and C. Daussy, International Journal of Thermophysics (2012). 31, 1347 (2010). [27] A. Cygan, D. Lisak, R. S. Trawin´ski, and R. Ciurylo, [12] K. Djerroud et al, Comptes Rendus Physique 10, 883 Phys. Rev. A 82, 032515 (2010). (2009). [28] J.Hald,L.Nielsen,J.C.Petersen,P.Varming, andJ.E. [13] D. Steck, “Cesium D line data (unpublished) Pedersen, Opt. Express 19, 2052 (2011). http://steck.us/alkalidata,” (2009). [29] L.Gianfrani,JournalofPhysics: ConferenceSeries397, [14] T. Udem, J. Reichert, R. Holzwarth, and T. Ha¨nsch, 012029 (2012). Physical Review Letters 82, 3568 (1999). [30] G.-W.Truong,J.D.Anstie,E.F.May,T.M.Stace, and [15] R. J. Rafac, C. E. Tanner, A. E. Livingston, and H. G. A. N. Luiten, Phys. Rev. A 86, 030501 (2012). Berry, Phys. Rev. A 60, 3648 (1999). [31] G.-W.Truong,E.F.May,T.M.Stace, andA.N.Luiten, [16] J. M. Amini and H. Gould, Phys. Rev. Lett. 91, 153001 Phys. Rev. A 83, 033805 (2011). (2003). [32] T. M. Stace, G.-W. Truong, J. Anstie, E. F. May, and [17] C. Bordé, Phil. Trans. of the Royal Society A 363, 2177 A. N. Luiten, Phys. Rev. A 86, 012506 (2012). (2005). [33] T. M. Stace and A. N. Luiten, Phys. Rev. A 81, 033848 [18] C. Bordé, Comptes Rendus Physique 10, 866 (2009). (2010). [19] M. Moldover, J. Trusler, T. Edwards, J. Mehl, and [34] T. M. Stace, Phys. Rev. A 82, 011611 (2010). R. Davis, Physical Review Letters 60, 249 (1988). Supplemental Information: Quantum-Limited Spectroscopy (Dated: January 5, 2015) APPARATUS 105 Master laser Slave laser Light Source: A dual-beam linear absorption spec- Heterodyne beat trometer operating at 895nm was built using a pair z) 104 H of extended cavity diode lasers (ECDLs). The first of V ( tthoes6eP, theFma=ste3r ltarsaenrs,itiisonlocokfedthteoDth1el6inSe1/2inFCs=a4t SRA 103 1/2 5 894.6054nm. This provided an optical reference fre- 1 quency with a square-root-Allan-variance (SRAV) of 0 2kHz for timescales between 1s and 30s, as shown by 102 2 ∼ 0.01 0.1 1 10 100 1000 the grey circles on Fig.1. These are the relevant time- n Integration time (seconds) scales for the spectroscopy, reflecting the time to take a a J single scan. The other laser is frequency locked to the FIG. 1. Square-root-Allan-variance (SRAV) of the absolute 2 masteratauser-selectablefrequencyoffsetbystabilising frequency stabilities of the master laser (grey circles), slave the heterodyne beat frequency against a tunable radio laser(olivetriangles)andtheheterodynebeat(redtriangles). ] frequency (RF) reference oscillator. This frequency lock h contributed no additional frequency instability to this p - slave laser (red triangles). The resulting slave laser fre- m quencystabilityisshownbytheolivetraceonFig.1. The from higher-order spatial modes of the optical cavity ( ∼ o RF reference oscillator was step tuned under computer 20% of the main mode) was heavily attenuated by cou- t control to ensure that the transmission data was syn- pling the beam into a single-mode optical fibre prior to a . chronous with the selected frequency. The wavelength of introduction into the evacuated chamber. s c theslavelaserwascenteredonthetwinabsorptionpeaks The combined cavity and single-mode filtering pro- si at 894.5793nm, corresponding to the 6S1/2 F = 3 to videsareductioninthespontaneousemissionbyafactor y 6P1/2 F =3andF =4transitions. Theoffsetfrequency 2 /π 195. We numerically modeled the effect of the h wastunedinincrementsof50MHzovera6500MHzspan reFsidua≈lspontaneousemission(0.008%)onthemeasured p that captured the atomic absorption features. Cs spectrum and found that the perturbation to the fit- [ The slave laser output was coupled into a Fabry-Perot ted atomic Doppler width νD was only 1.8ppm (equiv- 1 resonator whose length was actively controlled to keep it alent to 3.6ppm in u (k )). This systematic error can v r B resonantwiththeslavelaserlight. Theoutputofthecav- easily be reduced to below the 1ppm level by a cavity 6 4 ity was coupled into an acousto-optic modulator (AOM) finesse of over 1000. 3 which was used to actively stabilise the power in the de- Instrumental linewidths: The probe laser spectral 0 flectedbeam. TheoutputoftheAOMwascoupledintoa lineshape can be expressed to a high degree of accuracy 0 single-modefibreforspatialfilteringandfordeliveryinto as a Voigt function [1] with a Lorentzian half-width-at- . 1 the vacuum chamber. This fibre contained an auxiliary halfmaximumofΓlasandGaussiane 1-halfwidthofΓlas. 0 detector on a fibre coupler which provided the signal to L − D When this instrumental function is convolved with the 5 stabilise the input power into the chamber. The rest of atomic absorption profile (which, to first order, is also 1 thelight inthefibrewas senttoa Cssamplecell embed- : a Voigt function with parameters ΓL and ΓD), the ap- v ded in a thermal and magnetic shield. parently observed atomic lineshape is also a Voigt func- Xi Spontaneous Emission: In a separate experiment tion with Lorentzian and Gaussian components given by theslavelaserlightwastunedtothecentreoftheatomic r Γ =Γat+ΓlasandΓ = (Γat)2+(Γlas)2,respectively. a resonance and passed through a heated Cs cell for which L L L D D D the optical depth was 30 i.e. e 30 10 12. A Thepowerspectraldenqsity(PSD),S (f),oftheprobe Tat ≈ − ≈ − ν measurement of the residual transmission demonstrated laser frequency noise was directly measured by using the thatthelaseroutputcontained1.6%broadbandemission frequency-dependentabsorptiononthesideofanatomic at its operating point. transition. TworegionsofthePSDcanthenbeidentified The laser light was filtered using the fundamental [1];athigherFourierfrequenciesweseeadominantwhite mode of a scannable Fabry-Perot cavity ( =305 and frequency process, while at lower frequencies we observe F free-spectral range 17.6GHz). This was necessary be- steeper noise which in our case has a dominantly flicker cause broadband emissions contaminate the measure- (1/f) character. White noise components (with ampli- ment by creating a transmission offset associated with tude S ) produce a Lorentzian lineshape with HWHM 0 thenon-resonantphotons. Lowlevelfeedthroughoflight givenbyπS /2. Theintegratedprobelasernoiseatlower 0 2 frequencies give rise to a Gaussian lineshape which has the DMM range). It is clear that the transmission mea- an estimated e-fold width of Γlas = 0.88(39)MHz when surement are shot-noise limited in several bands of pow- D integrated over the 40ms observation time for each fre- ersthatliejustnearthemaximumofeachrange. Atthe quency point in the scan. highest powers used in the experiments ( 0.6µW), shot ∼ Optical power control: Thespectrallypurifiedlight noise contributed more than 50% of the total measure- is passed through an acousto-optic modulator (AOM) ment noise. and the deflected beam is coupled into a non-polarising The linearity of the photodetection chain was mea- fibersplitter,whichsplitsthepowerintheratio9:1. The sured in two ways. In the first approach, the ratio of lower power beam is measured on a photodiode and a the transmitted and incident detector signals were mea- feedbackloopadjuststheAOMinputtomaintainacon- sured over the full range of common-mode input power stant optical power in the fibre. The higher power out- variation. This provided a reliable calibration curve for put is out-coupled and used for spectroscopy. The po- which the deviations from linearity at high powers could larisation of this beam is fixed by passing it through a then be corrected. For low powers, this method did not Glan-Taylorprism, andthenaWollastonbeamsplitteris provide sufficient signal-to-noise for calibration. Instead, used to divide it into a reference and sample beam. The we monitored the ratio of absorption coefficients for the samplebeampassesthroughtheCsvapourandcommon- two optical transitions. The second approach relies on mode amplitude variations in this signal are removed by the fact that the Clebsch-Gordon coefficients for these dividing it by the reference beam signal. transitions require that the linear (i.e. low-power) ratio Conventional thermometry: The Brewster-angled of absorption coefficients should be 1:3 [3]. Forming this Cs sample cell is embedded in a cylinder of copper in- ratio, andenforcingthattheextrapolationtozeropower side a multilayered thermostat, which is itself inside an agrees with the 1:3 condition, we have a guaranteed, in evacuated chamber. The temperature of the thermo- situ and independent way to determine any nonlinear- stat can tuned using a thermo-electric cooler attached ity in the detector response at low powers. In this way, to the outer-most layer of shielding. The temperature we were able to resolve and remove the nonlinearities in at the cell was monitored using a capsule-type standard the photodetection scheme. The technique allowed us to platinumresistancethermometer(CSPRT)thatwascal- measure offsets corresponding to <8pW allowing us to ibrated to the ITS-90 temperature scale with a temper- ensure linearity over a dynamic range of 105 :1. A more ature precision of better than 0.5ppm in the range be- complete description of this method is found in Ref.[4]. tween 273K and 300K. However, our resistance meter (HP Model 3458A) limited our temperature measurement uncertainty to 1.2ppm. Measurements usinga sec- ERROR BUDGET ond calibrated CSPRT located in the copper cylinder at a distance 50mm away from the primary CSPRT re- Here we give a detailed description and derivation of ∼ vealednotemperaturegradientsabovethe1.2ppmlevel. the uncertainties reported in Table I of the main text. The ITS-90 temperature scale, T , itself is known to u (X) denotes the fractional uncertainty in X. 90 r deviatefromthetruethermodynamictemperature,T by Statistical: For each power level, the statistical con- T T = 3.2 0.4mK at T = 296K[2]. When the tribution is computed by fixing the etalon parameters 90 90 − ± uncertainty in this correction is included, the total un- (tovaluesobtainedfromfitstocoaveragedscans, seebe- certainty due to conventional thermometry is 1.9ppm. low), and extracting Γfit from each scan independently. D MagneticShielding: Themagneticenvironmentwas These are shown in Fig.4, and we find a statistical error nulled around the Cs cell using a dual-layer shield made of 2.9ppm in ΓfiDt, [which represents 5.8 ppm in ur(kB)]. from high permeability metal. Numerical modeling and Lorentz Width: Our probe laser has a Lorentzian experimentalcharacterisationofthisshielddemonstrated linewidth, which adds to the atomic linewidth, ΓL = that the apparent broadening of the atomic absorption ΓaLt+ΓlLas. To determine ΓL, there are two alternatives: line due to Zeeman splitting of the otherwise degenerate either to treat ΓL as a free fitting parameter (together levels contributed less than 1ppm uncertainty in kB. with ΓD), or to fix ΓaLt to the best reported literature Optical detection: Reverse-biased silicon photodi- value and independently estimate Γlas. L odes were used to measure the optical power in each of In the former route we fit to the average of 200 in- the reference and sample beams. The photosignals were dependent scans for each probe power level we retrieve measured by digital multimeters (DMMs) recording the a value of Γfit = 2.327(7)MHz. In the latter route, L voltagegeneratedbythephotocurrentpassingthrougha we take Γat = 2.287(6) MHz for the natural linewidth L load resistor. The noise contributions to each measure- of the Cs 6P level [3], and an independent estimate 1/2 mentoftheatomictransmissionfromshotnoise,Johnson of the laser noise Γlas = 0.044(18) MHz (see above), L noise and the DMMs are shown in Fig2 as a function whichtotalsΓind =2.331(19)MHz. Thesetwoestimates L of input signal level to each photodiode (the stair-case for Γ are consistent within their measurement uncer- L structure revealed in the figure arises from a change in tainty, andsowetakethevaluewithsmalleruncertainty 3 I/I 200 10-4 10-3 sat 10-2 10-1 m 0 Optical Power ( W) (cid:76)m 0.01 0.1 1 10 p p 10 (cid:72) (cid:45)200 1 4 (cid:45) moltage (V) 142 T D fitPRT(cid:71)(cid:144)(cid:71)DD(cid:45)(cid:45)640000 Noise v 0.12 J (cid:45)800 4 0 0.001 0.002 0.003 S 2 NormalisedIntensity(cid:72)I(cid:144)Isat(cid:76) 0.01 FIG.3. FractionaldeviationbetweenthefittedDopplerwidth 1000 and that inferred from PRT temperature measurements at m) 296 K. Green points are fits using the power-independent p Voigt-only profile. Blue points are fits using the first-order p 100 e ( intensity-dependent correction. Red points are fits using the s oi second-orderintensity-dependentcorrection. Solidcurvesare on N 10 S T fits using our central estimate of Γ¯L = 2.327 MHz; dashed nsmissi 1 (tclhuoerwvseeersn)asrirteeipvfrioteryseΓonLftiΓn=figt2Γt.¯o3L2Γ0±MσinΓHLtzh:(istuhpreapnesgrh)ea.daEneddrrrΓoerLgbi=oanr2ss.r3ien3pd4riecMsaetHnezst a D L Tr 12.5 Hz noise bandwidth J D standarderrorsoffitstotheaverageof200scansforfixedΓ . L 0.1 0.001 0.01 0.1 1 10 Photovoltage (V) component sums in quadrature with Γ , and so con- D FIG.2. Top: Calculatedshotnoise(S)andJohnsonnoise(J) tributes 8 ppm to u (Γfit) [16ppm to u (k )]. r D r B voltagesasafunctionofthephotosignal. TheDMM(D)input Optical Pumping: The second-order power- noisewasempiricallydeterminedandfoundtodependonthe dependentcorrectionssuppressthefitresidualsin(shown measurementrangesetting(indicatedbytheshadedregions). in Fig. 2(d) of the main text) to the measurement noise Bottom: Measured noise in the atomic transmission channel floor, so there is no statistically significant signature of (circles)comparedtothecalculated(withnofreeparameters) contributions from the DMM, Johnson noise, shot noise and opticalpumping. Nevertheless,anyresidualintensityde- total (T) are also shown for comparison. pendence in the second-order intensity-corrected Γfit is D expected to scale as (I/I )3. We therefore estimate the sat error arising from uncorrected third-order optical pumping by fitting a cubic through the highest intensity red ΓL =2.327(7)MHz. In particular, the solid curves in datapointsinFig.3. Thisgivesamaximumdeviationof Fig.3assumethecentralvalueΓL =2.327MHz,andthe 15 ppm in ur(ΓfiDt) at the maximum probe intensity, and dashedcurvescorrespondtotheupperandlowerbounds smaller contributions at lower intensities. Forming the oftheunceraintyrange,ΓL =2.320MHzand2.334MHz. weightedaverageoverallintensitiesinFig.3contributes Wenotethattherelativeprecisioninourmeasurement 7.5 ppm to u (Γfit), [15 ppm in u (k )]. r D r B ofΓL iswithinafactoroftwooftheprecisionofthebest Etalons: Etalonsenterintotheuncertaintybudgetin measurementsoftheCsD1transitionlifetime,τCs [5,6], two ways. Firstly, etalon parameters may be misiden- and matches the precision of the best synthesised value tified during fitting, and so contaminate Γfit. Secondly D of τCs [3] formed by combining these measurements. unresolved etalons masked by measurement noise may We also see from Fig. 3 that a 1kHz variation in introduce residual systematic shifts to Γ . D ΓL leads to ∼ 5 ppm change in ΓfiDt, which is consis- In Fig.3 we fit ΓfiDt with free etalon parameters. The tent with a 1:1 tradeoff between increases in ΓL and de- weighted average of the standard errors is 8ppm in creases in ΓfiDt, when ΓD ≈ 215 MHz. This implies that ur(ΓfiDt), which is a convolution of the statistical noise our 6.5 kHzuncertaintyinΓfit contributes32.5ppmto discussed above and uncertainties arising from possible ± L ur(ΓfiDt) (equivalent to 65 ppm to ur(kB)). The ±19kHz misidentification of the etalon parameters during fits. uncertainty in ΓiLnd represents 190 ppm in ur(kB). When the statistical errors in ΓfiDt are deconvolved from Laser Gaussian Noise: The probe-laser spectrum thestandarderrorsinFig.3,wefindthesystematicerror has a Gaussian contribution [1]. This was measured and due to free etalon parameters in the fitting routine to be found to have a width of 0.88(39)MHz. The Gaussian (82 2.92)1/2 =7.5 ppm in u (Γfit), [15 ppm in u (k )]. − r D r B 4 1000 0.1 V(cid:72)1(cid:76) (cid:76)m 500 10(cid:45)4 Ν p p (cid:72) 10(cid:45)7 (cid:45)1 0 (cid:72)(cid:76)iVΝ VΝ(cid:72)2(cid:76) PRTD 10(cid:45)10 (cid:71) (cid:144) fit(cid:71)D (cid:45)500 10(cid:45)13 V(cid:72)3(cid:76) Ν 10(cid:45)16 (cid:45)1000 0 200 400 600 800 1000 (cid:45)1000 (cid:45)500 0 500 1000 ScanNumber (cid:68) FIG. 4. Deviation between Γfit (using second-order Voigt FIG. 5. Generalised Voigt profiles versus detuning, ν =100. D correction) and ΓPRT, for each scan at 296 K, with D Γ =2.327 MHz. Colours and vertical lines demark differ- L ent incident intensities (lowest at left, corresponding to nor- malised intensities in Fig. 1). Dashed lines are the mean of the corresponding set of scans, error bars are the estimated the Rabi frequency experienced by the atom as it moves parametererror( 1σ)foreachselectedscan,andareconsis- through regions of different intensity. We write the Rabi ± tent with sample standard deviation. frequency in this form since in what follows we will ex- pand the atomic response in powers of Ω2/(1+∆2). For simplicity of presentation, we will assume that the equilibrium atomic level populations areP = 1/2,1/2,0 , We bound the error arising from unresolved etalons in eq { } i.e. we ignore degeneracies in the ground state manifold. twoways: bysynthesisingdatawithfalseetalons,andby Accounting for degeneracies is a simple modification to computing the fractional shift in Γ after the addition D of the jth etalon in the fit function. Both approaches the relative fractions for P1 and P2 at equilibrium. give worst-case shifts in Γfit that are comparable to the The light is absorbed as it propagates through the D amplitude of the etalon. The largest unresolved etalons atomic medium. While Eq. 1 is defined with respect to are smaller than the measurement noise floor, so these the rest-frame of a particular atom, the axial evolution cannot contribute more than 3 ppm to ur(kB). of the field in the lab-frame is governed by [7] LINESHAPE CORRECTIONS AT SECOND ∂ lnΩ2 =2π3/2K ∞ d∆ e−(∆−∆vz)2/ν2P(∆ ), ORDER IN INTENSITY ∂z vzπ3/2(1+∆2 ) vz Z−∞ vz (2) Here we derive Eq. 2 of the main text, which gives the where P =P3 P1, K =ρ0µ2/(√πε0~v0), µ is the tran- − second-order intensity-dependent correction to the Voigt sition dipole moment and v = 2k T/m is the mean 0 B profile. We follow closely the derivation in [7]. thermal atomic velocity for an atom of mass m. The p Webeginbycomputingthespectraldependenceonthe notation ∆ indicates that a given atom is subject to vz atomic populations for a three level atom, consisting of a detuning which depends on its axial velocity via the twogroundstates: oneopticallyactive, 1 ,andtheother Dopplershift. Ifthefieldintensityisweak,sothatonres- | i optically inactive, 2 , and an excited state 3 . Transi- onance populations are negligibly perturbed from ther- | i | i tions between states 1 and 3 are optically driven, and mal equilibrium, then P = 1/2, and the integral yields | i | i − sptoaptuela|3tiiocnarnatreeleaqxuatotioenitshaerreo[f7]the ground states. The Ωre2c(ozv)er= Ω20=e−eπ−−π3/−23K/2zKVzνV(∆ν()∆.)S,iwncheicTh(izs)t=heΩc2o(nzv)e/nΩt20io,nwael T Ω2f(t)/2 0 2β+ Ω2f(t)/2 Beer’s law result for a Voigt atomic lineshape. P˙ =MP, M = 1+0∆2 0 2(1 1+β∆)2 , (1) For larger Ω2, perturbations to P become significant,  Ω2f(t)/2 0 2 Ω−2f(t)/2  and we need to compute corrections to P arising from − 1+∆2 − − 1+∆2  thedynamicsdescribedbyEq.1. Sinceweareconcerned   withsmallbutsignificantpumpinducedperturbationsto where P = P ,P ,P , ∆ is the detuning between 1 2 3 { } the thermal population, we do a perturbative analysis of the laser frequency and the atomic transition, β is the Eq. 1 around the limiting case where Ω=0. branchingratiofromstate 3 to 1 ,Ωisthepeakatomic | i | i Rabi frequency (proportional to the electric field am- ByinspectionofEq.1weseethatΩ2onlyeverappears plitude) and f(t) represents the temporal profile of aspartoftheratioΩ2/(1+∆2),soitmustappearinthe p