Accurate Determination of an alkali-inert gas diffusion coefficient using coherent transient emission from a density grating
A. Pouliot, G. Carlse, T. Vacheresse, H. C. Beica, U. Shim, S. B. Cahn, A. Turlapov, T. Sleator, A. Kumarakrishnan
aa r X i v : . [ phy s i c s . a t o m - ph ] F e b Accurate determination of an alkali–inert gas diffusion coefficient using coherenttransient emission from a density grating
A. Pouliot, ∗ G. Carlse, H. C. Beica, T. Vacheresse, and A. Kumarakrishnan † Department of Physics and Astronomy, York University,4700 Keele St. Toronto ON Canada, M3J 1P3.
U. Shim, ‡ S. B. Cahn, § A. Turlapov, ¶ and T. Sleator Department of Physics, New York University (Dated: February 17, 2021)We demonstrate a new technique for the accurate measurement of diffusion coefficients for alkalivapor in an inert buffer gas. The measurement was performed by establishing a spatially periodicdensity grating in isotopically pure Rb vapor and observing the decaying coherent emission fromthe grating due to the diffusive motion of the vapor through N buffer gas. We obtain a diffusioncoefficient of 0 . ± .
002 cm / s at 50 ◦ C and 564 Torr. Scaling to atmospheric pressure, we obtain D = 0 . ± . / s. To the best of our knowledge, this represents the most accuratedetermination of the Rb–N diffusion coefficient to date. Our measurements can be extended to dif-ferent buffer gases and alkali vapors used for magnetometry and can be used to constrain theoreticaldiffusion models for these systems. PACS Numbers:I. INTRODUCTION
During the last forty years, there have been signifi-cant improvements in the sensitivity of vapor cell mag-netometers used for the detection of small magnetic fieldsand magnetic anomalies. The development of spin–exchange relaxation—free (SERF) atomic magnetome-ters [1] has allowed these sensors to reach sensitivitiesbelow 1 fT / Hz / , competing with, and often surpass-ing, superconducting (SQUID) magnetometers [2] to bethe most precise magnetometers in the world [3]. Atomicmagnetometers operate by optically pumping alkali va-por into a specific internal state, thereby aligning theindividual magnetic dipole moments of atoms. This netmagnetic moment will oscillate at the Larmor frequency,which is uniquely determined by the external magneticfield. In a conventional, time–domain magnetometer, theLarmor frequency is measured by observing the absorp-tion of a weak probe laser [4]. SERF magnetometersachieve high precision by preserving the alignment overextended time scales. This is achieved by using high al-kali densities and specific concentrations of buffer, andquenching gases. Under these conditions the optically–pumped alkali vapor slowly diffuses with minimal deco-herence due to radiation trapping and spin–disruptingcollisions.To optimize these devices, it is necessary to develop de-tailed models of optical pumping of the D1 and D2 lines ∗ [email protected] † [email protected] ‡ Current address: ONTO Innovation Inc. 59-2 Sukwoo–dongHwasung City, Gyeonggi–Do, South Korea 445-170 § Current address: Physics Department, Yale University ¶ Current address: Institute of Applied Physics, Russian Academyof Sciences, Nizhniy Novgorod, 603000, Russia in alkali atoms [5–7], and make precise measurementsof collisional cross sections [8–13] and diffusion coeffi-cients [14, 15] for relevant alkali and inert gas mixtures.Other motivations for precise diffusion measurements in-clude spin–polarized, high–resolution imaging using no-ble gases [16], and mesospheric magnetometry involvingsodium vapor for the monitoring of the Earth’s oceancurrents and interior dynamics [17].Previous measurements of diffusion coefficients haveinvolved analyzing transient signals associated with theoptical pumping of alkali vapors [12, 14, 18–21], measur-ing the amplitude decay of spin echoes in a magnetic gra-dient [15, 22], and analyzing the spectrum of transmittedprobe light well below the shot–noise limit to directly ob-serve atomic motion [23]. The diffusion coefficient mayalso be determined indirectly via measurements of col-lision cross–sections [24]. Diffusion coefficients can beinferred from these cross–sections, however such an ap-proach would rely on the accuracy of the intermolecu-lar potentials used by the Chapman–Enskog formalism[25, 26].Table I summarizes representative values of the Rb–N diffusion coefficient. The smallest uncertainty prior tothis work was achieved by reference [15] (2.5%). The dis-crepancy between measurements utilising different tech-niques emphasizes the necessity of a variety of methods,subject to different systematic effects, in arriving at amore reliable measurement. Additionally, accurate mea-surements of the diffusion coefficient constrain theoreticalmodels of many particle systems.In this paper, we present a contrasting technique thatdirectly measures diffusion from the decay time of a long–lived, coherent transient signal with a simple functionalform. The signal arises from a density grating which is in-sensitive to magnetic fields and magnetic field gradients.The timescale of the decay also shows a characteristic de-pendence on the grating spacing, which can be varied and TABLE I. Representative measurements of the Rb − N diffusion coefficient at atmospheric pressure D . Uncertainties areprovided where available.Reference Technique D (cm / s) D rescaled to 50 ◦ C a (cm / s)Wagshul et al. [12] Optical pumping relaxation 0.28 at 150 ◦ C b ◦ C . ± .
004 at 60 ◦ C 0 . ± . ◦ C 0.32Franz et al. [19] Optical pumping relaxation 0.16 at 32 ◦ C 0.18Erickson [20] Optical pumping relaxation 0.30 at 180 ◦ C 0.18This work Dephasing of density grating 0 . ± . ◦ C 0 . ± . a Rescaled using D ∝ T / [25, 26] b Average of values taken at various pressures and rescaled to 1 atm measured precisely. The characteristic dependence alsoprovides a good systematic check of the accuracy. As aresult, this technique appears to be suitable for accurateand precise measurements of diffusion coefficients.The rest of the paper is organized as follows: In Sec. IIwe contrast a traditional, time–domain magnetometerwith one based on spatially–periodic atomic coherences.We demonstrate that for specific excitation polarizationsand energy level schemes, it is also possible to realizedensity gratings with the same periodicity that are in-sensitive to magnetic fields. We explain how the charac-teristic decay times of these gratings can be exploited formeasurements of diffusion coefficients. In Sec. III we de-scribe the experimental details. The diffusion coefficientmeasurement is presented in Sec. IV. We conclude with adiscussion of the impact of this work on the developmentof magnetometers.
II. POPULATION AND COHERENCEMAGNETOMETRY
Fig. 1 shows a schematic of a well–understood, time–domain,“population” magnetometer [4]. Here, a contin-uous wave (CW) diode laser is amplified by a taperedwaveguide amplifier (TA) and used to generate a strongpump and a weak probe that are aligned at a small anglethrough a vapor cell containing an alkali sample suchas rubidium. These beams are amplitude–modulatedby acousto–optic modulators (AOMs). In this example,the circularly–polarized pump laser is tuned to the Rb FIG. 1. (Color online) Schematic of time–domain populationmagnetometer. F = 3 → F ′ = 4 transition, and is used to opticallypump atoms into the F = 3, m F = 3 ground state mag-netic sublevel, resulting in spin–polarization. If a mag-netic field is applied perpendicular to the quantizationaxis defined by the pump laser, the transfer of populationacross the ground state manifold and back is modulatedat the Larmor frequency ω L = ( ~µ F · ~B ) / ~ . This popu-lation evolution is detected as a periodic variance in thedifferential transmission of the two orthogonal circularcomponents of the probe beam.The decay time of the signal—which is modulated atthe Larmor frequency—is limited first by the transit timeof the atoms through the pumping volume. If the pump-ing volume is extended to encompass the entire vapor cell,then the measurement time will be limited by the effectof wall collisions that decohere the Larmor oscillations.Although the measurement time can be extended usingwall coatings, commonly–available coatings degrade atthe high temperatures required for SERF magnetometry[27]. A simpler way to extend the measurement timeis to add a high concentration of a buffer gas—such asN or a noble gas—whose principle requirement is a lowspin–destruction cross–section. Collisions between thealkali atoms and the buffer gas will result in diffusivemotion and effectively increase the transit time. Underthese conditions the measurement time is limited by radi-ation trapping which scrambles the atomic polarization.The addition of a small concentration of quenching gas—such as N —with a broad range of resonant energies canensure that collisional de–excitations dominate sponta-neous emission while preserving the spin–polarization.In this regime, spin–exchange collisions between rubid-ium atoms, which result in a transfer to atomic statesthat precess with the opposite phase, limit the time scale.Even so, this effect can be avoided by increasing the alkalidensity until the collisional frequency is large enough tore–initialize the phase of the Larmor precession resultingin the so–called SERF regime.Other transit time limited experiments involving theconfiguration in Fig. 1 have been utilized for precise mea-surements of atomic g –factor ratios [28, 29]. However,this type of magnetometer is not ideally suited for diffu-sion measurements since the signal decay must be mod-elled by a complex function which is sensitive to variousmechanisms of spin–depolarization in addition to diffu-sion. Further, the magnetic field response, which is alsosensitive to various systematic effects, cannot be decou-pled from the signal decay. In general, the transit time ofatoms— τ transit = f R /D , where R is the beam radius, D is the diffusion coefficient, and f is a form factor—issensitive to the volume of the pump–probe overlap re-gion. In devices of this type, the geometry of the overlap-ping region and its corresponding form factor contributessubstantial errors into the diffusion measurement. TheRb–N diffusion coefficient has been inferred by experi-ments such as this by illuminating an entire vapor cell, ofsimple geometry, with a circularly–polarized lamp sourceand monitoring the transmission [14]. These measure-ments rely on knowledge of the form factor of the cellgeometry and still must deconvolve magnetic field effectsand all sources of spin de–polarization.Fig. 2 shows a schematic of a “coherence” magne-tometer. Here, a spatially–modulated coherence grat-ing is created between adjacent magnetic sublevels of the F = 3 ground state in Rb by an excitation pulse thatconsists of two perpendicular linear–polarized travelingwaves, with wave vectors ~k and ~k , aligned at a smallangle θ (a few mrad). The grating is formed along thedirection ~ ∆ k = ~k − ~k as shown in Fig. 3 and has a spa-tial periodicity of ∼ λ/θ , where λ = 2 π/k and k is themagnitude of the wavevector k = | ~k | = | ~k | . The grat-ing can be detected by applying a read–out pulse alongthe direction ~k , and observing the coherent emissionscattered along the phase–matched direction ~k . Thissignal, called the magnetic grating free induction decay(MGFID) [30], exhibits a Gaussian decay with a timeconstant τ = 2 /kuθ , where u is the most probable speedassociated with the Maxwell–Boltzmann velocity distri-bution. This decay corresponds to the thermal motionof atoms causing the grating to dephase. The scatteredelectric field from the grating is then given by E ( t ) = E e − ( kuθ ) t . (1)If the excitation pulses have opposite circular polariza-tions, they will excite coherences between magnetic sub-levels separated by ∆ m = 2. The signal scattered from FIG. 2. (Color online) Schematic of the coherence magne-tometer. this coherence grating will have the same time constantas for the case of perpendicular linear polarizations. Thedephasing time of the grating has been used to measurethe velocity distributions of warm vapor [31, 32], coldatomic gases [32, 33], and atomic beams [34].In the presence of a magnetic field, the functional formof the coherence can have a complicated dependence, pa-rameterized by the Larmor frequency. This behavior hasbeen described in references [32, 33] based on the formal-ism presented in reference [35]. While Eq. 1 assumes thethermal trajectory of the atoms is uninterrupted over thelength scale of the grating, in the presence of a high con-centration of buffer gas the mean–free–path of Rb atomsis reduced by collisions and may become much less thanthe grating spacing. In this limit, the motion of Rb atomsbecomes a random walk that can be modelled by the dif-fusion equation [36–38]. This condition is represented by δu Γ Col ≪ kθ . (2)Here, δu is the average velocity change per collisionand Γ Col is the effective collisional rate. When Eq. 2 issatisfied, the evolution of the ground state density matrix ρ can be described by the diffusion equation, ∂ρ ( x, t ) ∂t = − D ∇ ρ ( x, t ) . (3)Here, D is the diffusion coefficient, which is inverselyproportional to the perturber pressure. D can be accu-rately converted to its value at atmospheric pressure D ,using the relationship D P = DP . Here, P is atmo-spheric pressure and P is the buffer gas pressure in the FIG. 3. (Color online) Upper figure shows directions of theexcitation pulses, read–out pulse, and signal. The lower figureshows the relative timing of the pulse envelopes along ~k and ~k and the signal envelopes recorded on the detector. experiment [25, 26]. If the x –axis is along ~ ∆ k , the spatialdependence of the coherence ρ may be written as e ikθx .This results in ∂ρ ( x, t ) ∂t = − ( θk ) Dρ ( x, t ) . (4)The solution to Eq. 4 is a decaying exponential with atime constant ( kθ ) D . The MGFID is therefore given by E ( t ) = E e − ( θk ) Dt . (5)Under these conditions, the coherent scattering fromthe grating is preserved but the signal exhibits an ex-ponential decay with a characteristic time constant τ =1 / ( D ( kθ ) ). Since ( kθ ) − represents the characteristiclength scale in this problem, namely the grating spacing,the scaling law for τ is representative of a random walk.Therefore, the coherence magnetometer offers a directapproach for measuring diffusion rates [37]. However,this method is prone to inaccuracies since the scatteredsignal has a small amplitude and is sensitive to magneticfield gradients.As a result, we have exploited an interesting aspectof the lin–perp–lin excitation, namely that it simultane-ously produces a density grating with the same periodas the coherence grating. Accordingly, we are able torecord decays with much improved signal–to–noise ratiosand with greater accuracy due to the insensitivity of thedensity grating to magnetic fields and field gradients. Itshould be noted that the density gratings used in thiswork can be modelled without atomic recoil or matter–wave interference effects [39–43]. By recording the decaytime as a function of angle, we rely on Eq. 5 to measurethe diffusion coefficient with a statistical uncertainty of1%. The novelty of the technique and the high precisionare the central results of this paper.The density grating forms as the result of a spatially–periodic light intensity modulation in the combined(standing–wave) excitation field [44]. This standing–wave potential channels atoms into the nodes of the op-tical potential since our excitation pulses, which are res-onant with the unperturbed rubidium resonance, are ef-fectively blue–detuned with respect to the center of thecollisionally–broadened line. This is because collisionsred–shift the center of the atomic resonance by ∼ III. EXPERIMENTAL DETAILS
The experiment relies on a home–built external–cavitydiode laser (ECDL) [48] that seeds a TA with ∼
15 mWof light to realize an output of 2 W [49]. The ECDL is fre-quency stabilized with respect to the F = 2 → F ′ = 2 , Rb using saturated absorption in a 5cm–long vapor cell. The output of the TA is split intotwo beams, each amplitude–modulated by an 80 MHzAOM as shown in Fig. 4. The AOMs are driven by anRF network consisting of an RF generator, RF ampli-fiers, TTL switches and pulse generators. By adjustingthe power and timing of the RF pulses, the power andpulse sequence of the diffracted AOM output may be var-ied. The upshifted beams from these AOMs, which areat a frequency of 53 MHz below the F = 2 → F ′ = 3resonance, are aligned along directions ~k and ~k (seeFig. 3) through a 10 cm, quartz, vapor cell containingisotopically pure Rb and 564 ± . The pres-sure in the sealed cell was spectroscopically determinedas reported in Sec. IV A. The isotopic purity of the cellsimplifies its magnetic response but is not required forthe eventual diffusion measurement which is insensitiveto magnetic fields. The cell is insulated and maintainedat a temperature of 50 ± ◦ C using a resistive heater.As shown in Fig. 4, the cell is placed in a constant mag-netic field, transverse to the direction of laser propaga-tion. The B–field is produced by a pair of “racetrack”coils with an elliptical cross–section. The undiffractedbeam from the k AOM, at a frequency 80 MHz below thefrequency of the diffracted beam, bypasses the cell andis combined with the beam along ~k on a beam splitterdownstream from the cell. The outputs of the beam-splitter, that contain a heterodyne signal with a beatfrequency of 80 MHz, are incident on a balanced detec-tor. This detector consists of two Si:PIN photodiodeswith 1 ns risetimes that are biased to produce signalswith opposite polarity. The combined, 80 MHz signalfrom the photodiodes is amplified and mixed down toDC to generate the in–phase and out–of–phase compo-nents. These signal envelopes are further amplified andrecorded on a 12–bit analog to digital converter (ADC)with a bandwidth of 125 MHz corresponding to a two–channel acquisition rate of 250 MS / s. The total ampli-tude is obtained by adding the two signal componentsin quadrature. The experiment is operated at approxi-mately 1 kHz repetition rates using digital delay gener-ators. The time base of these generators is slaved to a10 MHz rubidium atomic clock [50] with an Allan devi-ation floor value of 3 × − at one hour. The pulsesfrom the delay generators are coupled to the AOMs us-ing TTL switches. The RF generator that produces the80 MHz AOM drive frequency is also phase locked to thesame 10 MHz output of the rubidium clock. This prac-tice ensures phase noise makes a negligible contributionto the error of this measurement.The same setup was used to generate signals from thecoherence grating (MGFID) and the density grating (as FIG. 4. (Color online) Schematic of the experimental set–up. The laser (ECDL) seeds the TA, the output of which is split byhalf wave plate ( λ/ B ≈ ◦ phase shift provided by thephase–shifter (PS). The in–phase and out–of–phase components of the signal are obtained in this manner. These mixed–downsignals are recorded and analyzed on the PC oscilloscope. shown in Fig. 2. The quartz vapor cell was replaced witha pyrex cell containing a natural abundance of Rb and Rb isotopes and no buffer gas to record reference MG-FID signals. The important difference between this celland the previously mentioned cell is the absence of abuffer gas. For studies of the population magnetome-ter, which were carried out in the quartz cell containingisotopically pure Rb vapor, the k beam was circularly–polarized and served as the pump, while the k beam waslinearly polarized and attenuated to serve as the probe.The signal was recorded by measuring the differential ab-sorption of the oppositely polarized circular componentsof the probe beam that are split by the λ/ ~k and ~k was measured using ascanning knife edge profiler with a rotation frequency of ∼
10 Hz. The separation between the beams was mea-sured at two locations separated by ∼ µ m, more than two orders ofmagnitude greater than the mean–free–path for 564 Torrof N ( ∼
280 nm).
IV. RESULTS AND DISCUSSION
Fig. 5 shows representative signals of the coherenttransients described in this paper. Fig. 5(a) shows thepopulation magnetometer signal recorded in the pyrexvapor cell without a buffer gas. The duration of the pumppulse was 300 ns and the duration of the weak probe (a) (b)(c) (d)
FIG. 5. (Color online) Representative responses of the coherence and population magnetometers. Field strengths are varied toclearly demonstrate the magnetic field response over the varied timescales. (a) Decay of the spin–polarization of a populationmagnetometer in the pyrex vapour cell containing natural abundance of Rb and no buffer gas. With a magnetic field of 0 . .
433 MHz) is consistent with this field. (b) Gaussian decay of the total signal of the coherencemagnetometer in the same cell. The magnetic field is zeroed to minimize Larmor oscillations in the decay and infer the mostprobable speed. The excitation pulse width is 70 ns and θ = 2 mrad. The fit gives u = 246 m / s, which corresponds to atemperature of 30 ◦ C, agreeing with the cell temperature. Inset shows the MGFID signal in a magnetic field of 13 . . .
25 G with564 Torr of N buffer gas. The excitation pulse width is 1 µ s, which ensures that the signal from the population grating issmall. The Larmor oscillation of 1 . µ s), θ = 5 .
23 mrad, and magnetic field of 0 .
17 G. The magnetic field responseis visible at early times but the coherent scattering from the density grating dominates at later times. pulse was 100 µ s. Here, the signal represents the inten-sity of the differential absorption of the two polarizationcomponents of the probe pulse. The decay time is prin-cipally limited by the estimated transit time across the3 mm × µ s). Thefrequency of the Larmor oscillations is consistent withthe applied magnetic field. The data illustrates the dif-ficulty of using this signal for measurements of diffusion.Firstly, the magnetic field response must be deconvolved.Secondly, the decay time is sensitive to the probe volume,which must be quantified. These requirements add sig-nificant uncertainty to any diffusion measurement. Fig.5(b) shows the MGFID of the coherence magnetometerrecorded in the same cell as in 5(a), with k and k exci-tation pulse widths of 70 ns. Each point was recorded by varying the delay of an intense, 70 ns read–out pulse, thenintegrating and adding the two components in quadra-ture to obtain the total intensity. These measurementswere carried out by annulling the ambient magnetic fieldto avoid Larmor oscillations from the coherence grating.A Gaussian fit to the intensity gives a temperature of30 ◦ C which is consistent with the cell temperature. The1 /θ dependence of the decay time has been verified in ref-erence [37]. The inset shows the in–phase component ofthe MGFID signal in a magnetic field (13 . . .
25 G, from isotopically pure Rbvapor in 564 Torr of N gas. Here, the k and k excita-tion pulse widths were 1 µ s. The delay time of a 100 ns,intense, read–out pulse, was varied to record the signaldecay. As in Fig. 5(b), the total intensity of the scatteredsignal is displayed. The frequency of the Larmor oscilla-tions (1 . µ sand the magnetic field is reduced to 0 .
17 G. The coher-ent scattering is recorded in the same manner as in Figs.5(b) and 5(c) by varying the delay time of an intense100 ns read–out pulse. The coherent scattering from themagnetic field–dependent coherence grating is visible atearly time delays while the scattering from the densitygrating dominates at later times. It is evident that thesignal from the density grating can be observed on muchlonger timescales since it has a greater amplitude thanthe signal from the coherence grating for suitably longexcitation pulses.Fig. 6(a) shows the signal from a long–lived densitygrating recorded by varying the delay time of an intense100 ns read–out pulse. This data was recorded over atimescale of ∼ θ isless than 0 .
001 mrad by sampling the beam profile on aprofile analyzer. The signal exhibits the expected expo-nential decay curve represented by the fit line. Here, theangle θ between ~k and ~k was adjusted to be 1 . ± . τ is363 µ s with a fit error of ± µ s. The data also showsthat the timescale of the decay is comparable to thoseobtained in all previous diffusion measurements. Thevalue of D (at 564 Torr) extracted from the fit on thebasis of Eq. 5 is 0 . ± .
008 cm / s. Fig. 6(b) showsthe normalized fit residuals which demonstrate that themodel based on an exponential fit agrees with the data.Fig. 6(c) shows the decay time constant measuredas a function of angle θ and plotted as a function of1 /θ . This trend demonstrates one of the key advan-tages of this technique, namely the ability to change thelength scale on which diffusion occurs. We note that thislength scale (the grating spacing) is significantly smallerthan the beam diameter of 3 mm. Over the range ofangles this ratio of the beam size to grating spacingvaries from 27 to 360, suggesting that the transit time (a)(b)(c) [ μs ] S i gn a l [ a r b . un it s ] - - Delay time [ μs ] N o r m a li ze d r e s i du a l (cid:2) [ mrad - ] T i m ec on s t a n t [ μ s ] FIG. 6. (Color online) (a) Exponential signal decay of a den-sity grating at 564 Torr N . The excitation pulse width is80 µ s and the delay of a 100 ns read–out pulse is varied.Here, each excitation pulse has a single–photon, average Rabifrequency Ω ≈ θ = 1 . ± .
01 mrad and the decay time τ = 363 ± µ s,which together give D = 0 .
246 cm / s. (b) Normalized fitresiduals of the exponential decay. (c) Fit to the decay timeof the density grating as a function of 1 /θ showing a lineardependence. The slope gives D = 0 . ± .
002 cm / s , whichrepresents a statistical error of 1%. Scaling to atmosphericpressure gives D = 0 . ± . / s. correction is negligible. The linear dependence confirmsthe characteristic scaling law expected for a diffusion–dominated system (Eq. 5). The slope of this line gives D = 0 . ± .
002 cm / s, which represents a statisticaluncertainty of 1%. We scale this to atmospheric pressureand obtain D = 0 . / s. The combined error in k , θ and τ , computed in quadrature, is 0.9%, which con-sistent with the observed statistical error. Additionally,the variation in cell temperature is 0.5%. These factorscontribute to an overall uncertainty of 1.3% (assuming a T / scaling law for D [25, 26]), giving an absolute errorof ± . / s. We now address the most challeng-ing source of systematic uncertainty, associated with thepressure in the cell. A. Pressure Measurement
The cell pressure provided by the manufacturer wasmeasured with a capacitance manometer with a preci-sion of ±
5% but the uncertainty in the pressure at thetime of heating and separation from the gas manifold isestimated to be around ± Rb cell. The cell was maintained at atemperature of 47 ◦ C as measured by an array of thermo-couples. These parameters are similar to the conditionsat which the diffusion measurement was recorded (50 ◦ C).At this temperature, the increase in rubidium densitycompared to room temperature, as measured by the areaunder the absorption spectra, was consistent with esti-mates from rubidium vapor pressure curves [47, 52].The spectra from the isotopically purified Rb cellwere recorded on a photmultiplier tube (PMT) for arange of logarithmically stepped laser powers from 1 − µ W at which there are no detectable optical pump-ing effects. The laser diode was scanned at rates of21 −
106 Hz with 10 , − ,
000 samples per sweep. Thespectrum was obtained by averaging 560-1660 sweeps anda representative sample is shown in Fig. 8. To providea frequency calibration, a reference scan was simulta-neously recorded in a vapour cell containing a naturalisotopic abundance of Rb at room temperature. A pho-todiode was used to record the laser intensity variationduring the scan. This information was used to model fre-quency dependant intensity variations of the laser diode,such as those caused by etalon effects and the diode gaincurve.To infer the N gas concentration from the measuredspectrum, we perform a fit to the collisionally broadenedand shifted profile of Rb. The profile is modeled as the
RbPBS PMTPDND1 87RbN2+ PDNPBSND2
Reference SpectrumBackgroundBroadenedSpectrum
FIG. 7. (Color online) Schematic of experimental set up tomeasure the pressure in the isotopically purified Rb cell. Adiode laser is scanned over ∼
60 GHz by scanning the cur-rent. The power of the laser is attenuated using two 13 dBND filters to avoid spectral distortions due to optical pump-ing. The frequency scan is calibrated using the spectrum ofa low–pressure reference cell. Power fluctuations in the laseroutput are calibrated by recording the laser intensity on aphotodiode. The pressure–broadened spectrum of the Rbcell is recorded using a PMT. sum of two Voigt profiles [47] separated by the knownhyperfine splitting of the Rb ground states (6.823468GHz) [53]. Both Voigt profiles share the same Lorentzianwidth and shift parameters, which are defined by a singlepressure parameter that uses the relationships measuredin reference [46]. The ratio of peak heights of the twoVoigt profiles are assumed to be fixed. We also use ascalable background term based on the measured laserintensity. To ensure that the fit parameters are stronglyconstrained, datasets obtained by increasing and decreas-ing the laser frequency during the scan are fit simulta-neously. The pressure is extracted from fits such as theone shown in Fig. 8. This fit is superimposed on an illus-tration showing the Doppler-broadened Rb spectrumfrom a cell without buffer gas.Fig. 9 shows the inferred pressure obtained with laserpowers ranging from 1 − µ W and an effective temper-ature of 47 ◦ C. We find that the fit errors typically rangefrom 0.07% to 0.2%. The scatter in the data can be at-tributed to temperature fluctuations of ± . ◦ C over thetime in which the data was acquired. The average valuewas determined to be 559 ± ◦ C. Using theideal gas law, this value can be scaled to the cell temper-ature during the diffusion measurement (50 ◦ C) giving apressure of 564 ± ◦ C (650 ±
65 Torr). It is significant that thestatistical uncertainty in the pressure measurement hasreduced the dominant systematic uncertainty in the mea-surement of the diffusion coefficient from 10% to 0.4%,resulting in a negligible contribution to the overall error.The final value, D = 0 . ± . /s , reported - -
10 0 10 200.00.20.40.60.81.0
Frequency ( - ) [ GHz ] A b s o r b ti on c o e ff . [ a r b . ] FIG. 8. (Color online) Absorption spectrum of the Rb cellnear the D2 transition, ν denotes the vacuum resonance fromthe F = 2 ground state. Background subtracted data is shownin light blue with the fit shown in red, noise in data has beenemphasized to distinguish the data from the fit. The pressureinferred from this fit is 561 . ± .
43 Torr, corresponding to aLorentzian width of 10 . ± .
008 GHz, and a shift of − . ± .
003 GHz. A representation of the low–pressure, Doppler–broadened spectrum associated with the D2 transitions fromthe F = 1 and F = 2 ground states in Rb (solid black) isalso shown to demonstrate the broadening and shift causedby collisions (the amplitude change is arbitrary)
P(cid:0)(cid:1)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7) [ T o rr ] FIG. 9. (Color online) Distribution of pressure measurementstaken at varying input laser powers (between 1-2 µW ) andlaser scan rates of 106 Hz or 21 Hz, at an effective temperatureof 47 ◦ C. The fit errors in the pressure parameter of the Voigtprofiles are shown as error bars. The additional spread isattributed to the ± .
5% temperature fluctuations in the cell.The mean value of 559.0 Torr is indicated by a pink line. Thestandard deviation of ± . in Table I has been rescaled using D = D ′ P ′ P , where D ′ is the diffusion coefficient measured at 50 ◦ C and P ′ isthe pressure inside the cell at this temperature, inferredfrom the pressure measurement at 47 ◦ C. Here, D and P represent the values at atmospheric pressure. Conclusions
We have demonstrated a distinctive and accurate mea-surement of the diffusion coefficient of Rb in N relevant to magnetometry. Ideally, the systematic effect due tothe N concentration should be measured at buffer gasconcentrations of several atmospheres so that spectra canbe fit to a smooth lineshape as in reference [45, 51]. Sincethe buffer gas pressure in our isotopically purified ru-bidium cell could not be changed, we fit the pressure–broadened spectrum to a function appropriate for ourpressure range where the hyperfine splitting is signifi-cant. From Table I it is clear that our measurement dis-agrees with the previous most precise measurement ob-tained using spin echoes [15]. Although the diffusion co-efficients measured by modelling optical pumping curves[12, 14, 19, 20] are in good agreement with our results,we note that these measurements do not report errorbounds.The disagreement between our measurement and refer-ence [15] could point to unaccounted systematic effects ineither technique. We also note that our inferred diffusioncoefficients depend on the low–pressure measurements ofthe broadening and shift parameters in reference [46],which are in disagreement with the same parameters de-termined at much higher pressure [45, 51]. However, wenote that using the values in references [45, 51] wouldalso result in a discrepancy with respect to the diffusioncoefficient measurement in reference [15].Systematic errors in this work that have not been ex-plicitly accounted for include the residual effects of wallcollisions at the cell windows and the temperature scalinglaw used to rescale D to compare with other works inTable I, which depends on the nature of the intermolecu-lar potential [25, 54]. This measurement can be improvedfurther by reducing the uncertainty in the angle measure-ment using a more stable spatial profiler, and by increas-ing the number of measurements to improve statistics.It should also be possible to carry out this experimentin a gas manifold in which the pressure of the buffer gascan be varied and measured precisely using both spec-troscopy and a capacitance manometer. This techniquecan also be extended to other buffer gases and alkali va-pors used in magnetometry for further comparisons withtheoretical models. V. ACKNOWLEDGEMENTS
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