Table Of ContentQuantitative Atomic Spectroscopy for Primary Thermometry
Gar-Wing Truong,1,∗ Eric F. May,2 Thomas M. Stace,3 and Andr´e N. Luiten1
1Frequency Standards and Metrology Research Group, School of Physics,
The University of Western Australia, Perth, WA 6009, Australia
2Centre for Energy, School of Mechanical and Chemical Engineering,
The University of Western Australia, Perth, WA 6009, Australia
3School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
(Dated: January 14, 2011)
QuantitativespectroscopyhasbeenusedtomeasureaccuratelytheDoppler-broadeningofatomic
transitionsin85Rbvapor. ByusingaconventionalplatinumresistancethermometerandtheDoppler
1
1 thermometrytechnique,wewereabletodeterminekB witharelativeuncertaintyof4.1×10−4,and
with a deviation of 2.7×10−4 from the expected value. Ourexperiment, using an effusive vapour,
0
2 departs significantly from other Doppler-broadened thermometry (DBT) techniques, which rely on
weakly absorbing molecules in a diffusive regime. In these circumstances, very different systematic
n
effectssuch as magnetic sensitivity and optical pumpingare dominant. Using themodel developed
a
recentlybyStaceandLuiten,weestimatetheperturbationduetoopticalpumpingofthemeasured
J kB value was less than 4×10−6. The effects of optical pumping on atomic and molecular DBT
3 experimentsis mapped overa wide range of beam size and saturation intensity,indicating possible
1
avenuesfor improvement. We also compare theline-broadening mechanisms, windows of operation
and detection limits of some recent DBT experiments.
]
s
c
i I. INTRODUCTION acoustic or dielectric constant gas thermometry, it suf-
t
p fersfromverydifferenttypesofsystematicerrors[10]. In
o An ability to measure absolute thermodynamic tem- contrast to those techniques which measure the macro-
.
s perature is currently confined to an exclusive list of na- scopicquantityRT,theDBT methodcandirectly probe
ic tional standards laboratories. Such measurements re- the microscopic thermal energy kBT by measuring the
s quire primary thermometry expertise and highly spe- characteristicspectralwidth of molecular/atomictransi-
y
cialisedequipment, making it impracticalfor wider-scale tions,therebyavoidingtheuncertaintiesintheAvogadro
h
p use. Instead, for practical purposes, temperature scales constant NA.
[ such as ITS-90 [1], are used as an approximation to the ThefirstDBT measurementsperformedbyDaussyet.
true thermodynamic temperature, even though they are al. and Casa et. al. have used molecular gases at 1-130
3 knowntohaverelativeuncertaintiesatthe10−4 level[2]. Pa and determined k with a relative uncertainty [11–
v B
9 Accordingly, there is now a global push to develop more 13] of order 10−4. More recent work has improved the
2 convenient methods of primary thermometry that could statistical uncertainty [14] to 4 10−5. In this paper,
×
2 make thermodynamic temperature measurements more we present results from an atomic Rubidium (Rb) DBT
5 broadlyavailable[3]. Asecondmotivatingfactordriving system at 3 10−5 Pa. The use of an alkali metal atom
8. recent renewed interest in primary thermometry is the absorberpre×sentssome distinct spectroscopic differences
0 call by the Bureau des Internationale de Poids et Mea- thatallowustoexplorenewexperimentalregimeswithin
0 sures’to remeasurethe Boltzmannconstant, k , using a the DBT method. In doing so, we have identified some
B
1 wide range of techniques in preparation for the redefini- advantages in using an atomic system and encountered
:
v tion of the kelvin in 2011 [4, 5]. new challenges whose resolutions demand a deep under-
Xi At present, the recommended value for kB is derived standing anddetailedunification oflight-matterinterac-
primarily from one acoustic measurement performed by tions and gas dynamical theories [15]. Whilst this work
r Moldover et. al. [6]. It would be preferable if k could has been motivated by primary thermometry,we believe
a B
be remeasured using different methods with compara- thatithas widerimplications inallfields requiringhigh-
ble uncertainties so that any systematic errors can be resolution or precision, quantitative spectroscopy.
identified. Recently, Bord´e suggested that the known[7]
The primary difference between a low-pressureatomic
temperature dependence of the spectral linewidth of ab-
vaporsystem,likeRb,andmolecularexperimentsisthat
sorption features, previously exploited for measurments
atomicmotioniseffusive,socollisionsareextremelyrare.
of stellar atmospheric temperatures[8], can be used as
An advantage of moving to this effusive regime is the
a new technique for accurate primary thermometry[9].
avoidance of pressure-induced systematic changes, such
This Doppler-broadening thermometry (DBT) method
ascollisionalline-shapeperturbations[16]. Thisremoves
has since been experimentally realized and while cur-
the need to extrapolate results to an equivalent zero-
rently less precise than other primary methods such as
pressurevalue. Further,byusingasealedvapor-pressure
referencegascell,weavoidanyunintendedpressuredrifts
associated with a more sophisticated variable-pressure
∗ [email protected] gas chamber. In this study, we chose Rb which has a
2
high optical cross section, ensuring that the absorption
!"#$ %#4
signal-to-noise is satisfactory in spite of the very low va-
/$
por density. This choice also gives the ability to operate 3<=>8?=@ )3
in a range where high quality lasers are readily available
$"# /7
and where silicon detectors, with their extremely good #& 0,,6789: %/)
linearity [17] and quantum efficiency, can be used. At %
."
room temperature, it was possible to use a smaller cell %#’ ;!
%()
than those in the molecular experiments for the same 12#0
*+,,-
absorptiondepth,whichwasconvenientforthermalcon-
trol.
%#5
Onthe other hand, thereare alsodisadvantagesto us- %# 0%#
ing atomic vapors. Care is required to ensure low levels
of residual magnetic fields which can lead to the broad-
ening of spectral lines by lifting the degeneracy of Zee- FIG.1. (ColorOnline)Aschematicdiagramoftheopticalcir-
cuit and electronic control systems. ECDL: extended cavity
man levels. In the transition used in this experiment
(D2 line of 85Rb), optical pumping between unresolved diode laser; LCD: variable optical depth liquid crystal dis-
play; OC: optical cavity; PZT: annular piezo-electric stack;
hyperfine transitions contributed the greatest potential
PDH: Pound-Drever-Hall top-of-resonance feedback control
for systematic error by perturbing the lineshape. In ad-
loop; RL:reference laser; VNDF:variable neutral density fil-
dition, the comparability of the natural lifetime, beam
ter;BE: beam expanderlenses; PRT:ITS-90calibrated plat-
transit time and the Rabi period in our situation leads inum resistance thermometer; TI: thermal isolator; (F)PD:
to optical pumping effects that requires either an accu- (fast) photodetector.
rate model of the gas kinetics [15] or the use of a low
power probe beam to avoidlineshape perturbations. We
followedthesecondroutehereandshowherethatthere-
tween measurements of the temperature, frequency and
sulting perturbation is at the 4 parts-per-million (ppm)
absorbedpower. AnLCDvariableopticalattenuatorwas
level. Wenote,however,thatitshouldbepossibletoop-
used to keep the intensity transmitted through the cav-
erate at higher power levels and use the model of Stace
ity, which was monitored by the photodetector PD1, at
and Luiten [15] to correct the results.
a constant. This suppressed variations in the amount of
The remainder of this paper is organised as follows.
opticalpumpingassociatedwithpowerchangesoccurring
In Sec. II, we describe the experimental setup and data
throughout the scan.
acquisition for a Rb DBT experiment. The theoretical
Theopticalfrequencydifferencebetweentheprobeand
absorption coefficient lineshape for unperturbed atoms
a reference laser (RL) was measured using a fast pho-
and the derivation of a suitable lineshape profile for ex-
todetector (FPD) and a high bandwidth counter. The
tracting the quantity k T from the acquired spectra are
B reference beam was derived from a tunable Ti:Sapphire
then described in Sec. III. Our experimental results are
laser locked to a temperature-controlled, ultra-low ex-
presentedin Sec. IV, followedby a discussionaboutsys-
pansionopticalcavity. Thestabilityofthereferencelaser
tematic uncertainties in Sec. V. In Sec. VI, we discuss
( 1 kHz) contributed negligible uncertainty to the opti-
therelativemeritsandchallengesoftheatomicapproach ∼
cal frequency measurement. A variable neutral density
inmoredetail. Wewillalsodiscusshowtheuseofatomic
filter (VNDF) allowedmeasurementsto be made atinci-
vapors mightbe extended to a wider temperature range.
dent probe powers below that set by the LCD intensity
control system, without affecting intensity or frequency
stabilities. A photodetector (PD2) located just before
II. METHOD AND EXPERIMENT
the absorption cell was used to correct for any residual
probe power changes. A beam expander (BE) was used
A schematic of the experimental setup used in this to give the largest beam possible through the cell (2 cm
work is shown in Fig. 1. The kBT product was de- in diameter)to maximise the input power for a givenin-
terminedfromspectroscopicmeasurementsofunresolved tensitytoachieveahighsignal-to-noiseratio. Thepower
hyperfine transitions (from the 52S1/2, F =2 52P3/2, transmittedthrougha10cmlongRbabsorptioncellwas
→
F = 1,2,3 states) in the D2 line of 85Rb at 780 nm. measured on a third photodetector (PD3). The Rb cell
A custom-built extended cavity diode laser (ECDL) was wasencasedinapassivethermalisolator(TI)whosetem-
locked to a tunable optical Fabry-P´erot cavity (OC) us- perature was monitored using a using a platinum resis-
ing the Pound-Drever-Hall (PDH) technique [18], which tance thermometer (PRT) calibrated to ITS-90 with 30
suppressed acoustic and other fluctuations of the laser mK uncertainty. The maximum temperature difference
frequency. We tuned the laser’s wavelength by chang- across the isolator was measured to be less than 7 mK
ing the length of the cavity. A sweep generator (Sweep) using auxiliary thermometers (not shown). The thermal
was used to displace one of the OC cavity mirrors by relaxation time constant of the isolator ( 2 hours) was
∼
driving a piezo-electric stack actuator (PZT). A step- muchlargerthanthetimerequiredforasinglesweep(10
wise sweep was used to reduce the asynchronicity be- mins).
3
time. Various phenomena including collisional, transit
)T(f) fit 01..50 tLiomreenatnzdiapn.owIterwbarsoiamdpenoirntganctaninptehritsurebxptehreimweidntthtoofkteheips
T(f)- 0.0 these perturbations at negligible levels.
3( -0.5 The Gaussian component has the form of G(ν) =
0 -1.0
1 exp( [(ν ν )/∆ν ]2), where the 1/e-half-width of the
0 D
− −
Gaussian component is related to the thermal energy
1.0 k T as:
B
mc2 ∆ν 2
D
0.9 kBT = (1)
T(f) 2 (cid:18) ν0 (cid:19)
n, and arises from the thermal motion of the atoms. Here
o 0.8
si c is the speed of light, m is the mass of the absorb-
mis fit ing atom and ν0 is the absolute transition frequency.
ns 0.7 data For Rb, the latter two parameters are known with rel-
Tra ative uncertainties of 5 × 10−8 and 5 × 10−11, respec-
tively [19, 20], which are negligible for this experiment.
0.6
Whilst it was possible to compute the absorption coeffi-
cientV(ν,∆ν )=L(ν) G(ν)byperformingaconvolu-
D
⊗
tionintegral,thiswasacomputationallyinefficientproce-
7.5 8.0 8.5 9.0 9.5 dure. Instead,anumericalroutinedescribedbyHumlicek
Beat Freq. (GHz) [21] implemented in the scientific data analysis program
Igor Pro[22] was used to generate the Voigt function.
FIG. 2. (Color online) The lower panel shows a typicalspec- The relative accuracy of this algorithm was better than
trum(bluecircles) obtained from Rbvapourat295 K (∼550 3 10−5 and no consideration of these numerical errors
×
points) and the fit (red) using Eqn. (2). The Lorentzians was required in the following analysis.
(green)indicateboththelocationoftheunderlyinghyperfine The observed transmission spectrum was determined
transitions and their relative strengths (in arbitrary units). by Beer’s Law [7], T(ν) = exp( V(ν,∆ν )L), where L
D
The upperpanel displays theresiduals to thefit. is the optical path length of th−e probe inside the ab-
sorbing gas. Since this experiment concerns frequency
differences, it was convenient to shift the frequency ori-
The outputs of PD2 and PD3 that monitored the in-
gin by subtracting away the absolute optical frequency
cident probe and the transmitted powers, respectively, of the 87Rb F = 2 1 hyperfine transition. We denote
were recorded using digital multimeters with 5-digit res- →
thesefrequencydifferenceswithf toavoidconfusionwith
olution. A third multimeter was used to make four-wire
the absolute frequencies ν. We performed least-squares
resistance measurements of the PRT. A computer-based
regressionof the transmission spectra to Eqn. (2) to de-
data acquisition system centrally controlled and logged
termine ∆ν .
D
the three digital multimeters and the high bandwidth
counter outputs. Before each set of measurements at a 3
fixedprobepowerlevel,theDCvoltagesonPD2andPD3 T(f)=Af+Bexp C SiVi(f fi fcav,∆νD) (2)
− − − !
caused by stray lights, were recorded in the absence of i=1
X
the probe beam. These backgrounds were removed from
Here,T(f)isthesumofthree,unresolvedhyperfinetran-
the detector readings before reconstructing the Doppler-
sitions, indicated schematically in Fig. 2; the V (f
i
broadened absorption spectrum by dividing the output −
f f ) are the Voigt functions corresponding to each
i cav
of PD3 by that of PD2. −
hyperfine transition, each centered on frequency f and
i
with relative strength S ; and A, B and C are three of
i
the five parameters adjusted in the regression. The S
i
III. EXTRACTING kBT FROM SPECTRA were fixed at values derived from the Clebsch-Gordan
coefficients while the f were fixed at values determined
i
In a low density gas cell, the absorption coefficient of by Arimondo et. al. [20] Of the adjustable parameters,
an isolated spectral line is the convolution between a AandB accountfor,respectively,aresidualbackground
Lorentzian and a Gaussian function, known as a Voigt andimperfectbackgroundnormalisation(toavalueof1)
profile [7]. The Lorentzian component can be written as of the spectra. C is the on-resonance absorption depth
L(ν)= (1+[(ν ν )/Γ]2)−1, where ν is the optical fre- determined by the number density and the cell length.
0
−
quency and ν is the atomic transition frequency. This Thetwootheradjustableparameters,f and∆ν were
0 cav D
component arises from the finite lifetime of the upper contained within, and were common to the three Voigt
state and has a full-width-at-half-maximum (FWHM) functions,withf allowingforthearbitrary(butfixed)
cav
of Γ = 1/(2πτ), where τ is the total upper state life- frequency offset of the reference laser from the lowest
4
TABLEI.ValuesofthefixedconstantsusedinEqn. 2forthe
0.4 20 40 80 relative transition strengths, Si, and their frequency offsets
[20], fi. The relative transition strengths are exact numbers,
whilst there is a relative uncertainty of order 1×10−5 asso-
10 ciated with the differences between the hyperfine transition
0.2
frequencies.
L
z
H
M Fupper =i Si fi (MHz)
H 0.0 1 1/3 4372.399(85)
opt 2 35/81 4401.771(37)
G 1
- 3 28/81 4465.172(53)
G
-0.2
2 2
Probe Intensity (W/m ) 0.01
2 3 4 5 6 7 8 9 2 3 4
-0.4 80 40 20 4 10
-23
1.384x10
-0.4 -0.2 0.0 0.2 0.4 K)
J/
DΝD-DΝD,opt HMHzL ant ( 1.382
st
n
o
FIG.3. (ColorOnline)The(χ2−χ2 )-surfaceasafunctionof C
changein Lorentzian and Gaussianopwt idths,denoted Γ−Γopt ann 1.380
and ∆νD −∆νD,opt respectively, show that there is a high m
z Data
degree of correlation between these fit parameters. The sub- olt CODATA value
scripts opt denote the optimum quantities determined from B
Weighted Mean
theleast-squaresfit,whichwere2.75MHzand308.0MHz,for 1.378 One Std. Dev. band
the Lorentz and Gaussian widths, respectively. The thicker,
red contour is generated from the covariance matrix and is
2 3 4 5 6 7 8 9 2 3
intended toguide the eye. 0.1 Probe Beam Power (µW) 1
frequency hyperfine line. The Boltzmann constant was FIG. 4. (Color Online) The value of kB (dots) extracted
obtained from the Gaussian width ∆νD using Eqn. (1). from fitting spectra measured at several probe powers. The
Inprinciple, itis possible to performthe regressional- weighted mean (blue dashed) and weighted one-standard-
lowingeveryparameterinEqn. 2tobeadjustable. How- deviationband(bluesolid)areshown. Thegreen,dot-dashed
ever,inthiscase,thereturnedparametersfromtheleast- line is theCODATA [6] value.
squaresfitshowstrongcorrelationsbetweenanumberof
thefreeparameters. Thisleadsalargeruncertaintyrange
mann constant, based on a weighted mean of the mea-
in those parameters. In particular, we find strong corre-
surements at the various probe powers, is
lations in the S , and also in Γ and ∆ν which have a
i D
correlationcoefficient of -0.96. Operating in an intensity k =1.38104(59) 10−23J/K.
B
range where it is possible to fix the Lorentz width and ×
the strengths Si to known values, we avoid an artificial The uncertainty in this measurement was limited pri-
inflation of the uncertainties associated with the fitted marily by residual amplitude noise at frequencies higher
Doppler width due to these correlations. Ourtheoretical than the bandwidth of the LCD feedback control loop.
model (see Section V) guides us on the upper intensity. This white noise was exacerbated by a lower than ex-
pected common-mode rejection of amplitude noise re-
sulting from some asynchronicity in the data acquisition
IV. RESULTS scheme.
The broadening of the Lorentzian linewidth by mag-
The results from several k determinations are shown netic fields contributed a smaller systematic uncertainty
B
in Fig. 4 as a function of incident probe power. These of 9.8 10−5, which was summed in quadrature give a
data were produced from 24 spectra taken at intensi- total r×elative uncertainty of 4.1 10−4. All other sys-
×
ties at least 500-fold below the saturation intensity [23] tematic uncertainties were kept at negligible levels and
(I 20 W/m2). The total uncertainties at each power will be discussed in the following section. However, as
sat
∼
levelareindicatedbytheerrorbarsandweredetermined the random uncertainties and measurement timing are
fromthestandarddeviationofthedatasetstakenatthe addressed in future work, these will become increasingly
same power level. Our final determination of the Boltz- important. Our value of k has a relative deviation of
B
5
2.7 10−4 from the current CODATA value. To our
× TABLE II. Different experimental regimes accessed by vari-
knowledge, this is the most accurate demonstration of
ous choices of absorbers. The relative noise reported in the
primary thermometry using spectroscopy of atomic ab-
bottom two rows are given for a 1000 s bandwidth.
sorbers.
Rb NH3 [11] CO2 [12]
Doppler Width (MHz) 308 50 275
V. ESTIMATION OF SYSTEMATIC Pressure (Pa) 10−5 8 130
UNCERTAINTIES Pressure Broadening (MHz) 0.02 1 4
Optical Pumping, F 0.021 8×10−8 3×10−7
The greatest source of systematic uncertainty in this Probe Power (µW) 0.1 0.1 50
experiment was caused by the Earth’s magnetic field. 1000 sShot Noise Floor (ppm) 0.7 0.8 1.2
This externalfieldlifted the frequency degeneracyofthe
1000 sAchieved Reproducibility (ppm) 4000 1000 200
Zeeman sub-levels in the Rb hyperfine transitions. The
fieldalongthecellwasmeasuredtobe0.21(1)Gandnu-
merical simulations shows that this causes an apparent
cause of this early onset of optical saturation behaviour
broadening of the Lorentzian linewidth by 0.5%, which
which allows us to confirm that a negligible amount of
in turn, contributes a relative uncertainty of 9.8 10−5
× optical pumping occurred in the experiments reported
in the measured value of k . Whilst magnetic shields
B here. In contrast to previous models [17, 25, 26] that
can be used to reduce the field strength by a factor of
approximate the atom as a two-level system under uni-
40 000 times, this was not attempted because the effect
form illumination, we developed a theory for multi-level
was smaller than the observed random uncertainties of
atoms probed by a beam with finite spatial extent [15].
the experiment.
Thisbettercapturesthephysicalpropertiesandsituation
TheRbspectrawererecorded(non-contiguously)over
of this Rb experiment. We find that the saturation-like
a one month period. Whilst no drift would be expected
behaviour comes from an optical pumping process that
from the use of a permanently sealed Rb vapor-pressure
evacuated the population in the laser-coupled ground
referencecellwithnobuffergas,checksforlong-termsta-
state,andnotfromaRabi-floppingmechanismresponsi-
bility were performed by periodically re-acquiring spec-
ble forsaturationinatwo-levelsystem[7,27].We, there-
tra at previously investigated probe power levels. We
fore, quantify the perturbation away from thermal equi-
detected no long term drift larger than the short-term librium by introducing a figure-of-merit,denoted F, de-
reproducibility of the experiment. A long-term drift of
finedastherelativedepletionofthelaser-coupledground
the reference laser’s wavelength only changes the fitted
state, i.e.
offsetfrequencyf butdoesnotchangetheRbspectrum
0
and is therefore inconsequential. F 1 ρ1 (3)
We also considered the systematic uncertainty caused ≡ − ρ
1,th
by holding Γ and S constant. Fig 3 demonstrates
i
that the minimum of the χ2 surface lies along (Γ where ρ1,th and ρ1 are the thermal and pumped frac-
− tional occupation of the ground state. In the absence of
Γ )/(∆ν ∆ν ) 1, so the propagated relative
opt D − D,opt ≈ optical pumping (ρ = ρ ), F = 0; whilst conversely,
uncertainty in ∆ν due to the uncertainty of Γ is re- 1 1,th
D
when the ground state is fully depleted, (ρ = 0) and
duced by a factor ∆ν /Γ 100. Since the unper- 1
∼ D ≈ F = 1. Using the full model of Stace and Luiten [15],
turbed Lorentzian width is known [24] with a relative
uncertainty of 3 10−4, this contributes a negligible un- wecomputedthe figure-of-meritto be F 0.021forthe
≈
× largest powers in this experiment. In Appendix A, we
certainty in ∆ν . Similarly, there is no uncertainty as-
D show how F may be calculated approximately using a
sociated with the transition strengths because they are
simple method based on conservation of energy consid-
exact numbers. However, these assumptions are valid
erations. At F = 0.021, the perturbation in k due to
only in the limit of zero probe beam power because the B
optical pumping is 4 ppm (see Appendix B), which is
measurement process itself can perturb the actual val-
well below the current statistical uncertainty. We also
ues of Γ and S away from their theoretical values. For
i
showed experimentally that we were sufficiently close to
example, excess probe beam power will lead to power-
thezero-probe-powerlimitbydemonstratingtheabsence
broadening of the underlying Lorentzian linewidth, in-
of any statistically significantslope in the deduced value
troducing an intensity dependence in the fitted Doppler
of k as a function of power on Fig. 4.
width and, thus, in k . Similarly, any optical pumping B
B
between the hyperfine states that produces alignment or
polarizationofthesamplewillperturbtheeffectiveratios
S /S . We did indeed observe such effects at intensities VI. COMPARISON TO MOLECULAR DBT
i j
greaterthan 1 W/m2 (at least30 times largerthan in-
∼
tensitiesreportedhere),whichisstillmorethananorder In this section, we will highlight the differences be-
of magnitude below the saturation intensity [23]. tween a low-pressureatomic approachand high-pressure
We have developed a new theory [15] to explain the molecular DBT experiments by examining reported re-
6
sults,simulatedspectraandthe resultspresentedinSec- 104
tion IV. Table II shows a comparison of some recent Rb 0.98
DBT experiments using molecules and atoms. 103 10-2
A. Shot-noise limits 102 10-4
10
Using the probe powers shown in Table II, row 5,
wesimulatedshot-noiselimitedDoppler-broadenedspec- Gt Gt=2 10-1
tra consisting of 1000absorbancesamples spanning f 1 10-6
0
3∆νD. An approximate estimate of the shot-noise lim±- 10-3
itedrelativeuncertaintyofkB wasfoundby fitting these 10-1 10-8
spectratoourmodel. Theresultsaresurprisinglysimilar
10-5
for all experiments (row 6). The results depend mostly
on the total scan time and not details such as the fre- 10-2 10-10
qpuroebnecypsopwaecrinrgespobrettewdeeinnRpoefin.ts[1.2D],etshpeiteshtohte-nhoiigsehlCimOi2t 10-3 10-11 10-9 10-7
is similar to the other experiments because of the low 10-6 10-5 10-4 10-3 10-2 10-1
absorption depth in that experiment. The last row of
HW(cid:144)GL2
TableIIshowsthattheapproximateactualrepeatability
calculated from reported results, when scaled to 1000 s
FIG. 5. (Color Online) A contour plot of F computed from
timescale,issignificantlyworsethantheshot-noiselimit
the model of Stace and Luiten[15], as a function of normal-
in all cases. Djerroud et. al. [14] have reported recent
ized probe intensity (Ω/Γ)2 and normalised coherent evolu-
improvementstotheworkofDaussyet. al. byachieving
tion time (Γt). The red and blue dots indicate the condi-
a reproducibility of 100 ppm at 1000 s. In our case, we
tions of this and the molecular DBT experiments, respec-
found that backgroundlight levels have a significant im-
tively,demonstratingtheexploration ofvastlydifferentspec-
pactonexperimentalrepeatability,soweintendinfuture troscopic regimes. The dot-dashed lines are contours of con-
experiments to employ synchronous detection of a mod- stant input power. The solid red line indicates the experi-
ulated carrier. Preliminary results suggest this improves mental conditions that would lead to a 1 ppm perturbation
the repeatability to 150 ppm. in kB.
5 shows a contour plot of F as a function of the non-
B. Optimization of Probe Beam Power and
Geometry dimensional quantities Γt and (Ω/Γ)2. The vertical axis
correspondstotheratioofthecoherentinteractiontime,
t, to its lifetime (1/Γ): for an effusive vapor like Rb, t is
Current approaches in DBT experiments range in
pressures[12, 14] from 10−5 to 100 Pa. In the Rb exper- the average beam transit time while for a diffusive gas,
like those used by the molecular experiments, t is the
iment described here, the mean free path is 120 m and
meantimebetweencollisions. Thehorizontalaxisispro-
thegasdynamicsisgovernedbyaneffusiveflowofatoms,
portionaltothebeamintensity,andisexpressedinterms
i.e. the system is essentially collisionless, leading to neg-
ofthe Rabifrequency,Ω,andnaturaldecayrate,Γ. The
ligible pressure-induced self-broadening [7]. In contrast,
conventional intensity can be related to these parame-
the molecular DBT experiments are dominated by colli-
ters by the expression I =2(Ω/Γ)2I , where I is the
sionsandisinthediffusiveregime,causing 0.01∆ν of sat sat
∼ D conventional2-levelsaturationintensity[23]. TheF con-
pressure-relatedbroadening[11,28]. Moreover,thephys-
tours,whichwerecalculatedwiththemodelofStaceand
icaldetailsofthemolecule-moleculecollisionsarecritical
Luiten[15], can be closely approximated by the simpler
indetermining the observedlineshape profile. Forexam-
expressions:
ple, the degree of elasticity of molecular collisions per-
turbs the lineshape away from a Voigt profile [16]. This
requires higher-order forms such as the Galatry profile F (Ωt/2)2 if Γt≫2 . (4)
and a greater number of adjustable parameters required ≈( Ω2t/(2Γ) if Γt 2
≪
for the regressionof spectra.
In contrast to the collisional effects of the molecu- Thus, contours of constant F have a slope of 1
−
lar DBT systems the critical linewidth perturbation for decade/decade and 1/2 decade/decade when Γt 2
− ≫
vapours with strong light interaction is associated with and Γt 2, respectively. Eqn. 4 was derived using a
≪
optical pumping. This is clearly demonstrated in row 4 model of a three level atom with two ground states and
of Table II. To illustrate the very different operational anexcited state. The scales ofFig. 5 obscurethe details
spectroscopic regimes interrogated in this work and by ofthemorecomplexcaseΓt 2,whichareunimportant
≈
Daussy et. al. [11, 14] and Casa et. al. [12, 13], Fig. to this wide range overview.
7
In Fig. 5, the solid red line indicates the parameters
wheretherewouldbea1ppmperturbationtothedeter-
mination of k (derived from the analysis in Appendix
B
B). For collisionless gases, t is proportional to the beam
radius r and (Ω/Γ)2 r−2. Therefore, changing the
∝
probebeamdiameterforafixedinputpowerleadtolines
with a slope of 1/2decade/decade. Two such contours
−
(dot-dashed) are drawn through the location of the Rb
experimentreportedhereaswellasforthemolecularex-
periments. Contours of constant absorbed power follow
linesofconstantF sincetheabsorptioncoefficientisap-
proximatelyproportionaltothe populationperturbation
in the ground state, i.e.
V(f ,∆ν ) ρ =ρ (1 F). (5)
i D 1 1,th
∝ − FIG.6. (Color online) Approximatetemperaturewindows of
operationforvariousvapor-pressurecells,oflengthrestricted
It is clear from Fig. 5 that there are two regimes
to 10 cm. The species in each cell is shown on the vertical
of coherent interaction time, probe intensity and opti-
axis. The temperatures of selected ITS-90 fixed points are
cal pumping which are delineated by the line Γt = 2.
indicated by the solid vertical red lines. The subscripts TP,
For Γt 2, the contours of F and those for fixed in- MP and FP denote Triple Point, Melting Point and Freezing
≪
putpowerareparallel,demonstratingthatthemeasured Point, respectively.
results will be independent of the probe beam geometry.
However, this does not hold when Γt 2, indicating
≫
thatthe resultwilldependonprobebeamshape. Inthis of fundamental parameters,the optical depth can be ex-
situation, Fig. 5 guides the experimenter into an opti- pressed as[27]
misation of probe beam radius and intensity for a given
perturbation to kB. By increasing the beam radius, it αL=σ0ρ0L∝Γλ2PL, (6)
is possible to increase the total input power whilst re-
whereσ ,ρ ,P andLaretheopticalcross-section,num-
maining on the red line. This is advantageous as it will 0 0
ber density, pressure of the absorber and optical path
leadtoanimprovedSNRofthe measurementforagiven
length, respectively. We can more clearly see the trade-
linewidth perturbation. The limit to this procedure is
offsbetweentheparametersthataffectthe opticaldepth
set by the size of the gas cell. For molecular experi-
bywritingan“equationofstate”fortheopticaldepthus-
ments where the coherent interaction time is limited by
ing the conditions of the Rb experiment as the reference
themeantimebetweencollisions(andnotthebeamtran-
values,
sit time), an increase in probe power will require a cor-
responding increase in pressure to preserve the amount αL
of optical pumping. Of course, this compensation may
0.7 ∼
be forgone given the much smaller magnitude of optical (cid:18) (cid:19)
(λ/nm)2 Γ/s−1 P/Pa L/m
pumpinginthisregime. However,thesmallopticalcross . (7)
sectionsofthemoleculartransitionswhichreducetheim- 7802 4 107 3.5 10−5 10−1
(cid:18) (cid:19)(cid:18) × (cid:19)(cid:18) × (cid:19)(cid:18) (cid:19)
pact of optical pumping effects give rise to other issues
ToachievecomparableSNRtoatomicabsorbersinwhich
associated with SNR and pressure-broadening.
Γ is large,molecular-basedapproachesmust compensate
byincreasingtheproductofP andLbecauseofthemuch
longer lifetimes. Increasing either of these parameters
C. Absorption Depth and SNR canleadtothepotentialforunwantedinaccuraciesinthe
experiment relating to pressure broadening or thermal
Whilst Fig. 5 demonstrates the impact of experimen- gradients in the apparatus.
tally controllable variables such as the coherent interac-
tion time (which can be adjusted by changing the beam
radius in an effusive gas system, and by the pressure in D. Windows of Operation
a diffusive gas) and the probe beam intensity, there are
other considerations pertaining to the atomic/molecular In this experiment, we used a commercially available
propertiesthatindependently affectSNR.Onesuchcon- Rb vapor cell of a standard length (10 cm), which gave
sideration is the optical depth. Numerical simulations an absorption depth of 50% at room temperature.
∼
showthatinthe presenceofwhite amplitude noise(such However, operating along the metal’s sublimation curve
as detector noise) there is an optimum optical depth, means that the equilibrium vapor density increases ex-
αL 3 at which the variance in the Doppler width ob- ponentially with temperature, and thus the absorption
≈
tained from repeated simulations is minimised. In terms depth is much more sensitive to temperature for atomic
8
systemsthaninthemolecularexperiments. Therangeof rium ground-state population was perturbed by 0.2%
temperaturesoverwhichwecanuseRbasathermomet- which shifted the measured k value by 4 ppm. The
B
ricsubstanceiscomparativelysmallerthanformolecules. small cross-sections of the transitions used in molecular
The lower temperature bound is provided by the dimin- DBTexperimentsmeansthattheopticalpumpingeffects
ishing absorption depth resulting from the exponential arerelativelysmall(perturbationtoequilibriumground-
decrease in vapor pressure. Conversely, the upper tem- state population of order 10−7). However, these small
peratureboundissetwhenthevapordensitybecomesso cross-sections mean that an equivalent signal-to-noise in
large that the probe beam is completely absorbed over molecularexperimentsrequireslongpathlengthsand/or
a small range of frequency detuning. Fig. 6 shows ap- higher pressures, both of which have associated system-
proximate temperature windows over which a selection atic effects on measurements of k . The method devel-
B
of metal gases at their vapor pressure might be able to oped here for quantifying optical pumping effects can be
operate[23,30–36]. Attheupperandlowertemperature used to search for candidate absorbers optimally suited
boundsofeachwindow,theirreproducibilityofthefitted to DBT experiments in either the diffusive or effusive
Doppler width in repeated simulations is double that for regimes.
the optimal optical depth (i.e. αL 3). This condition
≈
restrictsthe opticaldepth to arange0.5<αL<90. We
have restricted attention to 10 cm long cells for conve- Appendix A: Approximate method for calculating F
nient thermal control. Some fixed points of the ITS-90
[10] scale are also shown, demonstrating that metallic Whilst the approach of Stace and Luiten [15] cap-
vapor-pressureDBTcanbe usedtoverifythethermody- turesthedetailsofthegas-dynamicalinteractionbetween
namic temperature at those fixed points. atoms and the beam volume, which can be used to esti-
Foraparticularchoiceofabsorber,itisalsopossibleto mate the population perturbation using no phenomeno-
fillthereferencecellusinganexternalatomicreservoirat logicalparameters, it relies on a knowledge of the dipole
a lower temperature, resulting in a lower vapor number matrix element in order to calculate the Rabi frequency.
density. Oncethevaporcellissealed,nofurtherincrease Itisusefultocomparethisagainstasimpler,independent
in number density is possible, thereby restricting optical analyticcalculation. We providesuchacalculationhere,
depth to an optimum level when at high temperatures. based on energy conservation and using an a posteriori
This would permit the exploration of temperatures and knowledge of the optical depth.
pressures away from the sublimation phase boundary. Assuming that the input intensity is sufficiently weak
so that an atom never absorbs more than one photon
during a single transit of the beam, the rate of atoms
VII. CONCLUSIONS scatteredoutofthelaser-coupledgroundstateisapprox-
imately the same as the rate of scattered photons. This
is given by
Doppler broadening thermometry provides a signifi-
cantly different approach to the determination of the I
0
Boltzmann constant in preparation for the re-definition R=a (1 I/I0), (A1)
E −
γ
of the kelvin. Whilst previous DBT experiments have
studied molecularabsorbersatup to 10Pain a diffusive where a=πr2 is the cross-sectionalarea of a beam with
regime,wehaveusedRbat3 10−5Pawherethegasdy- radius r, E is the photon energy and I and I are the
× γ 0
namics are effusive. Our approach avoids problems such on-resonance output and input intensities.
as pressure broadening and poor signal-to-noise. Fur- Using kinetic theory [37], the flux of atoms crossing
thermore, the Rb cell is compact in size, enabling more the beam surface (of area A = 2πrl, where l is the cell
convenient temperature control. We used this system to length) is
determine k with a relative uncertainty of 4 10−4.
B
The present experiment was limited by amplitud×e noise 1 Γ
ΦA= v¯ρ A, (A2)
0
intheprobebeam,whichinfutureworkwillbeovercome 6 ∆ν
D
by using a control loop with larger bandwidth. We esti-
where v¯= 2kT/m is the mean atomic speed and ρ is
mate that all current DBT experiments are far from the 0
the total number density. A factor of Γ/∆ν is used to
shotnoiselimitandthattheuseofsynchronousdetection D
p
count only those atoms which are on-resonance.
methods could address this situation.
The value of F (Eqn. 3 from Section V) is then the
We also compared sources of systematic uncertainty
ratio of the rate of scattering to the rate of new atoms
for DBT experiments in the diffusive regime of molec-
impinging on the beam, i.e.
ular DBT and the effusive regime of atomic DBT. For
atomic vapors such as Rb, the shift in the determined F =R/(ΦA)
value of k due to the Earth’s magnetic field is approxi-
B
3r ∆ν I I
mately100ppm. Opticalpumpingeffectsaredetermined = D 0 1 (A3)
by the combination of beam transit time, upper-state v¯ρ0l Γ (cid:18)Eγ(cid:19)(cid:18) − I0(cid:19)
lifetime and probe intensity. In this work, the equilib- 0.022 for this experiment.
≈
9
0.006 the perturbation term r(∆). We performed an expan-
Ν»100 ′′ ′′
sion in the small parameter ǫ (P P )/P , where
0.004 Α0HDL Ptui′n′i≡ng∂2∆P.i/T∂∆he2mtheeandienrgivsatoifvethi≡es wreimt3hari−nesinpg1ecvtatro1iatbhleesdien-
DL this section are as defined in Ref. [15]. Written in this
DLH,r 0.002 form,itisclearthattheabsorptioncoefficientinthepres-
H enceofsomeopticalpumpingisaVoigtfunction,denoted
0
Α
0.000 V(∆,ν) with an additional perturbation term.
Consider the expansion of a characteristic width for
-0.002 rHDL´105 the perturbed absorption coefficient,
-400 -200 0 200 400 d∆
1/e,pert
∆ (ǫ) ∆ (0)+ δǫ,(B2)
D 1/e,pert ≈ 1/e,pert dǫ
(cid:12)ǫ=0
such that α(∆1/e,pert)/α(0) 1/e. Differen(cid:12)tiating this
≡ (cid:12)
FIG. 7. (Color Online) An example of the unperturbed ab- definition and using ∆1/e,pert(0) = ∆1/e, we(cid:12) obtain the
sorption coefficient (α0(∆)) and the additive perturbation perturbation to the linewidth
term (r(∆)), plotted for ν ≈30MHz/3MHz=100. The per-
turbationtermisverticallyexpandedby105 forclarity. Note d∆1/e,pert α0(0)[r(0)/e r(∆1/e)]
= −
that ν in thisAppendixhasa differentmeaning to that used dǫ [α (0)+ǫr(0)][α′(∆ )+ǫr′(∆ )]
in thepreceding sections of this paper. (cid:12)ǫ=0 0 0 1/e 1/e
(cid:12)
(cid:12) a0(0)[r(0)/e r(∆1/e)]
(cid:12) ≈ a a′(∆− )
0 0 1/e
This approximation agrees with the more detailed anal-
1/e 1
ysis, differing only by 10−3. ≈−16√πν3α′(∆ )
0 1/e
1
. (B3)
Appendix B: Linewidth Perturbation ≈−32√πν2
To evaluate δǫ, we have used the low probe power ap-
Using the approach and notation of Stace and Luiten
proximation of Stace and Luiten (P 0 << P ) and
[15], the generalexpressionfor the absorptioncoefficient 3 1
≈
∆ ν ∆ν /Γ 100 >> 1. We also used the
(Eqn. (22)ofRef. [15])canbeexpanded(Appendix A1) 1/e D
≈ ≡ ≈
approximate form for the steady-state population den-
in the case of small optical pumping. The expression for
sity P that ignores atomic coherence. This quantity is
a perturbed absorption coefficient lineshape is 1
maximal on resonance (i.e. when ∆=0):
κcP z P′′ P′′ ν2+1/4
α 1 3 − 1 π V(∆,ν) d2 1 Ω∗2t∗
≈ 2ω ( P1 p 8ν − ) Max[P1′′]=P1′′(0)= d∆2 2(1−exp(−1+4∆∗2))
+P3′′−P1′′4(∆/ν)2−3exp ∆2 = 4t∗Ω(cid:18)∗2exp( t∗Ω∗2) (cid:19)(cid:12)(cid:12)(cid:12)∆=0
P1 64√πν3 (cid:18)−ν2+1/4(cid:19)(cid:19) <−4t∗Ω∗2 − (cid:12)
α (∆)+ǫr(∆), (B1) −
0
≈ 0.4 for this experiment.
≈−
where α (∆) is the unperturbed lineshape, r(∆) is de-
0
fined as the exponential term in the parentheses and ∆ Substitution of the relevant quantities into Eqn. (B2)
is the detuning in units of the Gaussian width. Fig. (7) gives2ppmperturbationtothelinewidthand,therefore,
shows a plot of the unperturbed lineshape along with 4 ppm in the determination of k for this experiment.
B
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