Table Of ContentFemtosecond frequency comb measurement of absolute frequencies and hyperfine
coupling constants in cesium vapor
Jason E. Stalnaker∗,1,2 Vela Mbele†,2,3 Vladislav Gerginov‡,2 Tara M.
Fortier,2 Scott A. Diddams,2 Leo Hollberg §,2 and Carol E. Tanner4
1Department of Physics and Astronomy, Oberlin College, 110 N. Professor St., Oberlin, OH 44074
2Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway MS 847, Boulder CO 80305
3School of Physics, University of Witwatersrand, Private Bag 3, WITS, 2050, RSA
4Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556-5670
0 (Dated: January 18, 2010)
1
We report measurements of absolute transition frequencies and hyperfine coupling constants for
20 the8S1/2,9S1/2,7D3/2,and7D5/2statesin133Csvapor. Thestepwiseexcitationthrougheitherthe
6P1/2or6P3/2intermediate stateisperformed directly with broadbandlaser light from astabilized
n femtosecond laseroptical-frequencycomb. Thelaserbeam issplit, counter-propagatedandfocused
a into a room-temperature Cs vapor cell. The repetition rate of the frequency comb is scanned and
J
wedetectthefluorescenceonthe7P1/2,3/2 →6S1/2 branchesofthedecayoftheexcitedstates. The
8 excitationstothedifferentstatesareisolated bytheintroductionofnarrow-bandwidthinterference
1 filters in the laser beam paths. Using a nonlinear least-squares method we find measurements
of transition frequencies and hyperfine coupling constants that are in agreement with other recent
] measurementsforthe8Sstateandprovideimprovementbytwoordersofmagnitudeoverpreviously
h
published results for the 9S and 7D states.
p
-
m PACSnumbers: 42.62.Fi,42.62.Eh,32.30.-r,32.10.Fn
o
t
a I. INTRODUCTION butionofnuclearchargeandmagnetization[17],andnu-
. merous applications in laser cooling, Bose-Einstein con-
s
c densation [18], atomic clocks [19], and magnetometers
i The basic ideas of using repetitive pulses and mode- [20].
s
y locked lasers for high-resolution precision spectroscopy The present work focuses on measuring multiple tran-
h were originally developed in the 1970’s, with particular sition frequencies and the hyperfine structure of select
p focus on Doppler-free two-photon transitions [1–6]. The
low-lying states in cesium via two-photon excitation.
[ adventoffullystabilizedopticalfrequencycombs[7,8]in
The relatively simple experimental setup uses a room-
1 2000renewedinterestinmanyspectroscopicapplications temperature Cs vapor cell and a self-referenced optical
v ofmode-lockedlasers(forareviewseeRef.[9]). Inmany frequency comb to excite Doppler-free two photon tran-
1 instances the use of direct frequency-comb spectroscopy
sitions from the Cs ground state to the 8S , 9S ,
7 (DFCS) offers many advantages over spectroscopy done 1/2 1/2
7D , and 7D states. Our results are consistent with
0 withcwlasers. Inparticular,thespectralversatilitypro- 3/2 5/2
previous measurements for the well-studied 8S state
3 1/2
vided by modern frequency combs allows the simultane-
. [21, 22] and provide significantly reduced uncertainties
1 ous study of multiple atomic transitions with a single
for the 9S , 7D , and 7D states [23].
0 experimentalsetup. Additionally,DFCS allowsfor arel- 1/2 3/2 5/2
Thispaperisstructuredasfollows: SectionIIcontains
0
atively simple determination of the absolute frequency
1 a description of our experimental system and measure-
byprovidingadirectlinktotheS.I.second. Inthiswork
: ments. In Section III we examine the features of the flu-
v weexploitthisversatilitytostudymultipletransitionsin
orescencespectra anddiscuss the theoreticalmodel used
i cesium (Cs).
X to describe the results. Section IV describes the analy-
r Precision spectroscopy of Cs plays an important role sis of the two-photon transitions and the recipe used in
a in atomic and fundamental physics, including measure- extracting the transition energies. This section also in-
ments ofatomicparityviolation[10–12],searchesforthe cludesourresultsandthedetailsofthesystematiceffects.
permanentelectricdipolemomentofanelectron[13,14], We conclude our work in Section V.
photonrecoilmeasurementsthat helpdetermine the fine
structure constant [15, 16], measurements of the distri-
II. EXPERIMENTAL SETUP
The relevant Cs energy levels are shown in Fig. 1.
∗Correspondingauthor: [email protected] The excitationpathwaysto the four states8S , 9S ,
1/2 1/2
†Present address: P. O. Box 755, Durban, KWAZULU-NATAL,
7D , or 7D proceed through a stepwise excitation
4000,RSA. 3/2 5/2
through either the 6P or 6P states. We detect the
‡Present address: Physikalisch-Technische Bundesanstalt, Bunde- 1/2 3/2
sallee100,38116,Germany two-photon excitation as blue fluorescence emission as
§Presentaddress: P.O.Box60157, Sunnyvale,CA94088 the atoms decay from the 7P1/2,3/2 states to the ground
2
9 2S 6 BS 1 BS 2
1/2 4 7 2D5/2 5 Self-Referenced
4
3 3 Ti:S Frequency Comb
2 f 1 GHz
1 rep(cid:1)
6
58.8 7 2D3/2 45 H Maser PD F1 F2
8 2S1/2 nm 4 3
77694.6 nm37 2P1/2,3/2 697.5 nm698.65 72n.5m nm 2 Synthesizer Mixer L VC F3L
1. PMT
1
n
m 5
6 2P
3/2 4 Preamp
Computer
3 Filter
2
m
852.3 n 6 2P1/2 34 FpoIGne.n2t:sBarloe:ckBdSi1a,gBraSm2onfonthpeoleaxrpizeirnigmbeneatamlsseptliuttpe.rsT;hFe1c,oFm2-,
894.6 n m Fm3ulitniptelirefrerteunbcee;fiVltCe:rsC;sLvlaepnosr; cPeDll.photodiode: PMT photo-
6 2S 4
1/2
seconds.
3
The laser beam was split by use of a 50-50 nonpolar-
FIG. 1: (Color online) Energy levels of 133Cs(I = 7/2) that izing beam-splitting cube (BS 2). The two output laser
beams were linearly polarized along the same axis. The
arerelevanttothiswork. Theexcitationpathwaysareshown
insolidredalongwiththetransitionwavelengths. Thedecay beams were filtered by use of either narrow-band inter-
channelsusedinthedetectionareshowninbluedashedlines. ference filters or a combination of long-pass and short-
Thetotalangularmomentum ofthehyperfinecomponentsis pass filters to select specific transition pathways. The
listed next to each state. counter-propagating beams were then focused by lenses
of 25 cm focal length to a spot size of ∼ 100 µm inside
thevaporcell. Focusingthebeamstoasmallerwaistsize
state.
increases the two-photon excitation rate, but also leads
The experimental set-up is shown in Fig. 2. We ex-
to broadeningof the resonancesdue to the finite transit-
cite the cascaded two-photon transitions using a mode-
time of the atoms through the beams. The transit-time
locked Ti:Sapphire laser that generates a broad optical
broadening for a waist of this size is about 2 MHz for
frequency comb. The portion of the output spectrum
room-temperature Cs atoms. The average optical inten-
that was used extended from 650 nm to 1000 nm, cov- sities in the vapor cell were tens of W/cm2 spread over
ering all of the excitation transitions of interest in Fig.
≈ 10 nm bandwidth, corresponding to ≈ 4000 optical
1. The laser is described in detail in Ref. [24] and here
frequencies.
we only briefly mention the information relevant for the
An evacuated Pyrex Cs vapor cell was used in the ex-
experiment. TheKerr-lensmode-lockedTi:Sapphirering
periment. The cell has a diameter of 2.5 cm and is a
laser produces an output spectrum consisting of discrete
wedged shape with an optical path length ranging from
equally spaced coherent optical modes. The frequency
0.5cm-1cm. Windowswerefuseddirectlytothecellat
of each optical mode, ν , can be related to two mi-
n an angle. The cell was carefully cleaned and filled on an
crowave frequencies, the laser repetition rate, f , and
rep oil-freevacuumsystemandwasevacuatedtoabasepres-
the carrier-envelopeoffset frequency, f , by
0 sureof∼10−5 Pa. AsmallamountofCswasintroduced
intothecellpriortoremovingitfromthevacuumsystem.
ν =nf +f , (1)
n rep 0
Thecelltemperaturewasvariedbyresistiveheaterwires
where n is an integer mode number ∼ 4 × 105. The wound around the cell. We note that we observed shifts
carrier-envelope-offsetfrequencyisstabilizedbyuseofan in the transition frequencies as large as 0.5 MHz in a Cs
f-to-2f interferometer, thus “self-referencing” all modes vaporcellthatwasnotcleanedandvacuumprocessedas
of the comb [7]. For the measurements described here carefully as the cell used in these measurements.
f was stabilized to 45 MHz. The repetition rate of the Most of the measurements in this experiment were
0
laser is ≈ 1 GHz. The repetition rate was phase-locked takenwiththeCscelltemperatureat297K,withfluctu-
to a signal generator that was referenced to a hydrogen ations less than 1 K. Data were taken at higher temper-
maser. With the comb stabilized in this way, the frac- aturestoevaluatesystematicshifts arisingfrompressure
tional instability of every optical frequency of the comb shifts as described in Section IVB.
is ≈ 2×10−13τ−12, where τ is the averaging period in Asetofsixcoilswasusedtozeroouttheearth’smag-
3
netic field in the region of the vapor cell. The field was u.]
reduced to a level of < 1µT over the interaction region a.
inside the vapor cell. e [
c
n
Two-photonresonancesignalsweremonitoredbymea- e
c
suring the 7P → 6S (J = 1/2, 3/2) cascade fluo- s
J 1/2 e
rescence at 459 nm and 455 nm. The fluorescence was or
u
collectedatarightangleto the twocounter-propagating Fl
beams and was detected by a photomultiplier tube. The 1/2
S
detectorwasshieldedfrombackgroundlightandthescat-
2
tered light from the excitation laser beams by a 457.9 6
nm filter with a 10 nm pass band (F3). The detector
P(cid:0)
output was amplified and filtered (low-pass at 1 kHz, 6 2
7
dB/octave) before being sampled and recorded with a
0.772 0.773 0.774
computer-controlled data acquisition board (DAQ).
The DAQ used in these measurements is similar to f -1000 MHz [MHz]
rep
that used in Refs. [16, 17, 25]. The computer controlled
the scan of the repetition rate by stepping the frequency FIG. 3: Fluorescence from the 7PJ →6S1/2 decay as a func-
of the signal generator to which the repetition rate was tionoflaserrepetitionratewhennofilters(F1andF2inFig.
phase locked. Data scans were collectedby changing the 2) are used to restrict the laser’s spectral bandwidth.
frequencyofthesignalgeneratorfrom1000.771725MHz
to 1000.774725 MHz in 1 Hz steps. This 3 kHz range 4- 4 3 - 3 4- 4
in the repetition rate results in a 2-3 GHz variation in m on n 4- 4 3 - 3
tAmmhtueealetntiawpcalohen-dp1shatomhHtezopnulesntsfcereeparqntutdaheinenatcvyflye,urianodgreteephsdeecnefmondrcienea1gnwsowa.nserTtdehhieegevittasirlztauaenadtstisietwtdiiocitanahtl. scence froSExcitati1/2 Calculatio 4 - 33 - 4 4 - 33 - 4
eachrepetitionratefrequency. Spectrawerecollectedfor ore28 n
bothincreasinganddecreasingfrequencyscans,resulting Flu a f
in 6002 data points. 26 S1/226 P3/2 (cid:2)eriment b e l
TIHIIE.OROEBTSIECRAVLECDOSNPSEICDTERRAATAINODNS 27 P(cid:2)26 S1/2(cid:2)Exp c d ghij k m
Figure 3 shows the spectrum collected as the repeti- 0.772 0.773 0.774
tion rate of the frequency comb was scanned without frep - 1000 MHz [MHz]
using the filters marked as F1 and F2 in Fig. 2. The
spectrum consists of a multitude of narrow Doppler-free FIG. 4: The calculated spectrum (upper trace) and exper-
imental signal (lower trace) of the 7P → 6S fluorescence
peaksandaDoppler-broadenedbackground. Thenarrow
peaks are due to excitation to the 8S, 9S and 7D states whentheCsatomsareexcitedtothe8S1/2statethroughthe
by two counter-propagatingphotons, while the Doppler- 6P3/2state. The groups of peaks in the calculated spectrum
arelabeledbyF-F′′todenotethehyperfinegroundstateand
broadened background is due to excitation from two co-
hyperfinefinalstateofthetransition. Thepeaksineachgroup
propagating photons. The observed spectrum is quite
correspondtothedifferenthyperfinecomponentsoftheinter-
complicatedbecauseofthenumerousdiscretefrequencies mediate state. The peaks in the experimental spectrum that
present in the comb combined with the multiple atomic arelabeledwithletterswereusedinextractingthetransition
energy levels and transition pathways accessible. frequencies, as described in thetext.
The spectrum is greatly simplified and the Doppler
background is eliminated by introducing filters F1 and
F2, as shown in Fig. 2. The presence of the filters re- citationstothe9S ,7D ,and7D states,aswellas
1/2 3/2 5/2
stricts the transitions and enables the selection of indi- the excitation to the 8S state through the 6P state
1/2 1/2
vidual excitation pathways. Figure 4 shows the spec- andremovestheDoppler-broadenedbackground. There-
trumobservedwhenoneofthecounter-propagatinglaser sultingspectrumconsistsofgroupsoftwoorthreepeaks.
beams is restrictedto be between845nm to 855nm and The peaks labeled d, e, and f form one such spectral
the other between 780nm to 800 nm. With this combi- groupandareshowninmoredetailinFig.5. Eachspec-
nation, the first beam can excite the atoms on only the tralgroupistheresultofexcitationfromagivenground-
6S → 6P transition at 852 nm (the D transition) state hyperfine level to a given excited-state hyperfine
1/2 3/2 2
and the second beam can only excite the atoms on the level, and the individual peaks in the group correspond
6P →8S transition at 795 nm. This eliminates ex- to specific transition pathways through every allowable
3/2 1/2
4
hyperfine level of the intermediate state. For example, peaks arising from the same ground, excited, and inter-
the d, e, and f peaks in Fig. 5 all correspond to ex- mediate states are not the same in the different spectral
citation from the the 6S (F = 4) ground state to the groups.
1/2
8S (F′′ =4)excitedstate. Thethreepeakscorrespond
1/2
to excitation through the 6P (F′ =3, 4, 5) intermedi- We can understand and calculate the spectra using
3/2
ate states. We see the groups recur as a different pair of the standard two-photon transition probabilities deriv-
optical modes is resonant with a given transition. This ablefromsecond-ordertime-dependentperturbationthe-
repeating pattern occurs at intervals of δf ≈1.2 kHz, ory. We start by considering a transition of an atom
rep
corresponding to a change in the two-photonoptical fre- in the ground state with total angular momentum F,
quency of ≈ 1 GHz (i.e., the laser mode spacing). For a 6S1/2F ,toanexcitedstatewithtotalangularmomen-
givengroupofthreepeakstherearethreedifferentveloc- (cid:12)tumF′′,(cid:11)|n′′L′J′′′ F′′iviaexcitationbytwocwlaserswith
ity classes that satisfy the resonance condition for each (cid:12)angularfrequencies ω1 and ω2, intensities I1 and I2, and
of the corresponding hyperfine components of the inter- wave vectors ~k and ~k . Here n′′ refers to the principal
1 2
mediate level. For anothergroupofthree peaks,the res- quantum number of the excited state. We further as-
onant condition for the components of the intermediate sume that ω is close to the resonant frequency for the
1
level is satisfied by a different pair of comb modes, and 6S1/2 → 6PJ′ transition, with J′ = 1/2 or 3/2. The
by different velocityclasses. Thus, the amplitudes ofthe probability of the atom to make a transition is [26]
P(cid:0)6S1/2F, n′′L′J′′′F′′(cid:1)∝2F1+1hω6S1/2F:n′′L′J′′′F′′ −(cid:16)ω1+k~1·I~v1(cid:17)I−2 (cid:16)ω2+k~2·~v(cid:17)i2+(cid:18)γn′′2L′J′′′(cid:19)2
2
× (cid:12) hn′′L′J′′′F′′MF′′|eˆ2·d~|6PJ′F′MF′ih6PJ′F′MF′|eˆ1·d~6S1/2FMF (cid:12) , (2)
MFX,MF′′(cid:12)(cid:12)(cid:12)FX′,MF′ ω6S1/2F:6PJ′F′−(cid:16)ω1+~k1·~v(cid:17)−iγ6P2J(cid:12)(cid:12)′ (cid:11)(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where~visthevelocityoftheatom,M ,M′ ,andM′′ are ization, specified by the vector component in the spheri-
F F F
the projections of the total angularmomenta F, F′, and calbasisq,thematrixelementsforthedifferentmagnetic
F′′ alongthe axisofquantization,d~is the electricdipole sublevelscanbe relatedto reducedmatrix elements that
operator, eˆ is the unit vector along the direction of do not depend on the magnetic sublevels or the total
1(2)
the polarization for the first(second) laser beam, γ angularmomentum(F,F′,orF′′)byuseoftheWigner-
nLJ
is the homogeneous line width of the state |nL i, and Eckart theorem and standard Clebsch-Gordan relations
J
ωnLJF:n′L′J′F′ is the resonant angular frequency of the as [27]
transition|nL Fi→|n′L′ F′i. Foragivenlightpolar-
J J′
hn′L′J′F′MF′ |dq|nLJFMFi=(−1)F′−MF′ −FM′′ 1q MF (n′L′J′F′||d||nLJF)
(cid:18) F F (cid:19)
=(−1)F′−MF′+I+1+J+F (2F +1)(2F′+1) −FM′′ 1q MF
(cid:18) F F (cid:19)
p J′ F′ I
× F J 1 (n′L′J′||d||nLJ). (3)
(cid:26) (cid:27)
Here I = 7 is the nuclear spin and the and laserbeamsarecounter-propagatingthenexaminationof
2
terms are the usual 3-J and 6-J symbols. Thus, Eq.2 revealsthat for a giventransitionfrom the ground
(cid:0) (cid:1)
the transitionprobabilities for different ground-statean- state with angular momentum F, through an interme-
g(cid:8)ular(cid:9)momentaF andexcited-stateangularmomentaF′′ diate state with angular momentum F′, to a final state
can be related to a common factor by use of the appro- with F′′ the transition probability will be a maximum
priateClebsch-Gordancoefficients. Ifweassumethetwo
5
when IV. ANALYSIS AND RESULTS
v v
ω1 1+ c +ω2 1− c =ω6S1/2F:n′′L′J′′′F′′ (4) A. Analysis
(cid:16) (cid:17) (cid:16) (cid:17)and
v We calculated the excitation probabilities by numer-
ω1 1+ c =ω6S1/2F:6PJ′F′. (5) ically integrating Eq. 2 over a Gaussian velocity distri-
(cid:16) (cid:17) bution for optical modes near the resonant condition. A
If we now consider the effect of having a number of dis-
total of ≈ 15 optical modes were used (10 for the first
cretefrequencies,comingfromthedifferentmodesofthe
stage excitation and 5 for the second stage excitation).
comb, we see that, as a result of the detuning from the
The calculated spectra agree quite well with the experi-
intermediate state, only two near-resonant comb modes
mental spectra (Fig. 4). Slight variations in the relative
will contribute substantially to the transition probabil-
amplitudesofthepeaksareattributedtoslightdeviation
ity. Because the spacing of the frequencies for the comb,
of the polarization of the excitation light from linear.
f ≈ 1 GHz, is much larger than the homogeneous
rep
linewidths of the states, γ ≈ 5 MHz, we are justified Theresonantfrequenciesforatransitionfromaground
2π state with a specific F to an excited state with F′′ are
in considering only the two modes closest to the reso-
s
nance conditions of Eqs. (4) and (5). Since these two al 0.04
frequencies are coming from the comb, they are related sidu -00..0004
to the offset frequency and repetition rate of the comb Re
1.0
by two integers as nits] e f
ω1 =2π(f0+n1frep) (6) y U 0.8
ar
ω2 =2π(f0+n2frep). (7) bitr 0.6
Ar
Sseuebstthiatuttifnogr athgeisveenreplaatiironosf imntoodeEs,qsn.1(4a)ndannd2,(a5n)dwae ence [ 0.4
fixed value of the offset frequency there exists a unique c
repetition-rate frequency and velocity that will simul- ores 0.2 d
u
taneously satisfy both resonance conditions. Therefore, Fl 0.0
givenanatomicsamplewithabroadenoughvelocitydis-
0.77318 0.77320 0.77322 0.77324 0.77326
tribution,therewillexistarepetition-ratefrequencythat
will result in a stepwise resonant excitation of the two- f - 1000 MHz [MHz]
rep
photon transition for a specific velocity class. This leads
tothe narrowDoppler-freepeaksseeninFig.4. Because FIG. 5: The 7P → 6S fluorescence when the Cs atoms are
thevelocityclassexcitedandtheresonantrepetition-rate excitedtothe8S1/2statethroughthe6P3/2state. Thepeaks
frequencybothdependontheenergyoftheintermediate correspond to the d, e, and f peaks in Fig. 4. This data set
state,thetransitionresonanceforeachdifferentinterme- was taken at a temperature of 319 K and was used in evalu-
diatestateoccursatdifferentrepetitionrates. Becausea ating systematic effects related to temperaturedependence.
differentvelocityclass contributesto the differentpeaks,
the amplitudes of the peaks vary greatly, depending on
the number of atoms that are present in the resonant related to the center-of-gravity frequencies and the hy-
velocity class. perfine coupling constants through (see e.g. [28])
1 1
ν6S1/2F:n′′L′J′′′F′′ =ν6S1/2:n′′L′J′′′ − 2A6S1/2κ(F,J)+ 2An′′L′J′′′ κ(F′′,J′′)
3κ(F′′,J′′)2+ 3κ(F′′,J′′)−I(I+1)J′′(J′′+1)
+Bn′′L′J′′′ 4 2I(42I−1)J′′(2J′′−1) , (8)
where κ(F,J)=F(F +1)−I(I+1)−J(J +1). Spectra were calculated with different values of the two-
Therepetitionratesatwhichthecalculatedresonances photon center-of-gravity frequency and hyperfine cou-
occurdepend onthe center-of-gravityfrequencies forthe pling constants of the final excited state. The interme-
two stages of the excitation and the hyperfine A and B diate state energy and coupling constants used in the
constants of the ground, intermediate, and final states. calculated spectra were fixed to the values determined
6
from Gerginov et al. [16, 25]. The two-photon tran- where CG and CL are the Gaussian and Lorentzian
i i
sheixyttiporeanrcfitcneedentcferorou-mopfl-igntrhgaevcieotxynpsfterareniqmtuseennAtcanie′l′sLd′Jν′a′′6tSaa1/nb2d:yn′B′lLen′Ja′′′s′′Lt-′Ja′s′nq′duwaterhreees apamneadpkl,Litouardneednst,zCiraBenspiwsecidatitvhcesol,yn,sftiγaGnistatbnhadeselγ-inlLieneacreeonffttseheret.oGfaWtuhesesiaiattnh-
comparison of the experimental resonant repetition-rate tributethedeviationoftheexperimentallineshapefrom
frequencies with the resonant repetition-rate frequencies the expected Lorentzian line shape to residual misalign-
determined from the calculated spectra. ment of the two beams, which we discuss further be-
The resonant repetition rate for each of the peaks in low. Wedesignatetheresonantrepetition-ratefrequency
boththeexperimentalandcalculatedspectraweredeter- for the ith peak determined by fitting the calculated
mined from a least-squares fit. Fitting the spectra was spectra for a given ν6S1/2:n′′L′J′′′, An′′L′J′′′, and Bn′′L′J′′′
nfeerceenstsapreyaikns.orTdehretcoaalccucloautnedt fsopretchtreaowveerrleapasosfutmheedditfo- by ficalc ν6S1/2:n′′L′J′′′,An′′L′J′′′,Bn′′L′J′′′ . The resonant
repetitio(cid:16)n-rate frequencies determined(cid:17)from the experi-
haveaLorentzianlineshape. However,theexperimental
mental spectra we designate as fexpt. The best fit val-
spectra were found to be well described by a line shape i
uesfortwo-photonfrequencyandthehyperfineconstants
of a Gaussian coupled to a Lorentzian,
weredeterminedbyminimizingtheweightedχ2 function
Xi CiGe−(f2−γfG2i)2 +CiL(f −f(cid:0)i)γ22L+(cid:1)2 γ2L 2!+CB, (9)
(cid:0) (cid:1)
2
χ2 ν6S1/2:n′′L′J′′′,An′′L′J′′′,Bn′′L′J′′′ = fiexpt−ficalc(cid:16)ν6S1/2:σne′′xLp′Jt′′′,An′′L′J′′′,Bn′′L′J′′′(cid:17) (10)
(cid:16) (cid:17) Xi i
ottwσhhnieirexteehpefitwrtieopssutaptolrhdeaetcmhtesexettptaoeenexrνcdsp6taSer(1trνd/hi2m6e:uSne1n′m/′ncL2te:i′Jnan′r′′l′ti′,mLadiA′Ju′na′′nttm,ya′′.LAivn′J′anF′′l′fo,′uLiereax′J′npN′o′dt,fdBpatehenntaee′d′krLmχs′JB′′2i′ann.n′te′dHoLd′Je′bab′r′lyee)l scence from SExcitation1/2 Calculation 4-44-33-43-3 4-4 3-4 3-3 4-4
N − 3. We consistently observed larger values of the ore28 a 4-3
χ2. We attribute this uncertainty to residual misalign- Flu l
ment, which will be discussed in more detail below. In S1/2P1/2 (cid:3)ent b h
pχore2dcfetuerndtcovtaiaoclnuceos.uonWtthefoatrthtiehtnsismdlaeitrnegirmemruχimn2evctaohlruereeu,snpwcoeenrndtoasrimntotaytlihizneettehhxee- 22P6 (cid:3)2S6 1/2(cid:3)Experim c defg i j k m
two-photonopticalfrequency andthe hyperfine coupling 7 26
constants by finding the values of ν6S1/2:n′′L′J′′′, An′′L′J′′′, 0.772 0.773 0.774
and Bn′′L′J′′′ where the normalized χ2 function increases frep - 1000 MHz [MHz]
by one from the point at which it is a minimum. This
amounts to increasing the uncertainty to make the fre- FIG. 6: The calculated spectrum (upper trace) and exper-
imental signal (lower trace) of the 7P → 6S fluorescence
quencies extracted from the different peaks self consis-
tent. whentheCsatomsareexcitedtothe8S1/2statethroughthe
6P1/2state. The peaks in the experimental spectrum that
arelabeledwithletterswereusedinextractingthetransition
frequencies as described in thetext.
B. Results for the 8S1/2state
Thespectralrangeofthecomballowedthe 8S state performedonlyforthetransitionsthatoccurredthrough
1/2
to be studied by use of transitions through either the the 6P3/2states.
6P (895 nm and 761 nm) or 6P (852 nm and 795 Figure 6 shows the spectrum for the
1/2 3/2
nm) states. This redundancy provided a check for con- 6S →6P →8S transition along with the cal-
1/2 1/2 1/2
sistency of the measurement and the parameter extrac- culated spectrum. The spectrum was collected by use
tion method. A complete study of the systematics was of a 890 nm filter with a 10 nm passband for F1 and
7
a combination of a 755 nm filter with a 40 nm pass gated by varying the optical powers. Neutral density
band with a long-pass filter at 715 nm for F2. The filters were used to reduce the power in both beams
13 peaks, labeled a through m in Fig. 6, were used by approximately the same amount. Data were col-
to extract the transition frequencies. The peaks were lected at three optical powers ranging from the maxi-
acquired with both increasing and decreasing scans of mum power, which was used for collection of the full
f , resulting in 26 individual peaks. By use of the pro- spectrumshowninFig.4,to60%ofthemaximumpower.
rep
cedure described above, the extracted center-of-gravity The three peaks used in this analysis correspond to
two-photon frequency and the hyperfine A constant for the6S (F =4)→6P (F′ =3,4,5)→8S (F′′ =4)
1/2 3/2 1/2
the 8S state were found to be 729009798.844(38) transitions. Because the final state is the same for all
1/2
MHz and 219.137(19) MHz. The χ2 was 31 for 26 three peaks,it is not possible to extractboththe center-
peaks. Here the uncertainties are determined from the of-gravity frequency and the hyperfine A constant from
χ2 function, as described in Section IVA, and do not these data. In order to place a limit on possible effects
include any systematic uncertainties. due to the ac Stark shifts on the center-of-gravity fre-
The6S →6P →8S transitionisshowninFig.4. quency,wefixedthehyperfineAconstanttothevaluede-
1/2 3/2 1/2
For this spectrum, one of the counter-propagating laser terminedfromtheanalysisofthe fullspectrumandthen
beams was filtered with an interference filter centered at extracteda center-of-gravityfrequency at eachpowerby
850 nm with a 10 nm pass band. This resulted in an findingtheminimumintheχ2 functionasdefinedbyEq.
average intensity of ≈ 60 W/cm2 in the interaction re- (10). The center-of-gravity frequencies as a function of
gion,spreadover≈4000opticalmodes. Thislightserved power were fitted to a straight line. We used the slope
to excite the first step of the two-photon transition at of the line to look for any possible dependence of the
852 nm. Based on the transmission profile of the filter center-of-gravity frequency on the power. We find the
we estimate that we have ≈ 10 mW/cm2 of light in the slope to be −43(123)kHz/P, where P is the normalized
optical mode resonant with the 6S →6P transition. operating power. Combining the value of the slope with
1/2 3/2
The second stage of the transition was selected by use its uncertainty we arrive at a maximum possible shift of
of a combination of a 800 nm short-pass filter and a 166kHz atthe the nominaloperating power,P =1. We
780 nm long-pass filter and provided ≈ 70 W/cm2 over take this value as an estimate of the systematic uncer-
≈ 8000 modes. Using the procedure described above we tainty from the ac Stark effect. Because we fix the hy-
fitthe 14peakslabeledinthefigureandfindacenter-of- perfineAconstantthisapproachprovidesanupperlimit
gravity frequency of 729009798.863(29)MHz and a hy- on the center-of-gravityfrequency. Therefore, we do not
perfine coupling constan 219.133(17)MHz, where again, attempt to apply a correctionto the data. To determine
theuncertaintiesaredeterminedpurelyfromtheχ2func- the possible effect on the hyperfine splitting we fix the
tion. The value of the χ2 was 64 for 28 peaks (14 peaks center-of-gravityfrequencytothe value determinedfrom
fromboththeincreasinganddecreasingfrequencyscans) the analysis of the full spectrum and extract the hyper-
prior to the normalization. As described in Section IVA fineAconstantfromthedata. FittingthedatafortheA
the uncertainties are increased to make the χ2 consis- constant we found a slope of −22(72) kHz/P, resulting
tent with the expected value. The resulting frequencies in an uncertainty of 94 kHz.
arein goodagreementwiththe value extractedfromthe
For two-photon transitions between S states with
transition through the 6P state.
1/2 ∆M = 0 there is no linear Zeeman shift. However, the
F
Weanalyzedindetailthesystematiceffectsthatcould laser polarization is not perfectly linear and these states
lead to a shift of the measured frequencies from the arestep-wiseresonantthroughtheintermediatePstates,
true transition frequencies using the transitions to the which are magnetically sensitive. In addition, the power
8S1/2state through the 6P3/2state. Systematic shifts on the first stage of the transition is near saturation,
might be more significant for excitation through the sothere maybe someopticalpumping that couldcouple
6P3/2state, because of the smaller hyperfine structure withanexternalmagneticfieldtoleadtoasymmetricline
shifts of the 6P3/2state, relative to the 6P1/2state. In shapes. To investigate the effect of Zeeman-shifts due to
addition, the signal-to-noise ratio of the spectrum ex- imperfect cancelationof stray magnetic fields, data were
cited through the 6P3/2state was higher than that ex- collected without the compensation coils and also with
cited throughthe 6P1/2state, as an artifact of the filters thecurrentthroughthecompensationcoilsreversed,thus
used in the measurements. doublingtheresidualmagneticfield. Again,thepeaksla-
We considered four possible sources of systematic ef- beled d, e, and f in Fig. 4 were used. The value of the
fects: ac Stark shifts, Zeeman shifts, pressure shifts, hyperfine A constant was fixed and the center-of-gravity
and errors arising from misalignment of the counter- frequency was extracted as described above. We see no
propagating laser beams. The associated uncertainties evidence for a systematic shift as the magnetic field is
are summarized in Tab. I and described below. increased(Fig. 7). The standarddeviationof the data is
To evaluate the systematics we focused on the peaks 70kHz,whichwetakeastheuncertaintyassociatedwith
labeled d, e, and f in Fig. 4. A typical data set and themagneticfield. AsimilaranalysisforthehyperfineA
fit to the peaks is shown in Fig. 5. The effect of ac constant(withthecenter-of-gravityfrequencyfixed)also
Stark shifts on the extracted frequencies was investi- shows no evidence of any shift with the magnetic field.
8
Using the standarddeviationofthe points shownin Fig. z] z]
7unacseratnainestytimtoatteheofAthceonusntcaenrttadiunetytowepoassssiibglne ama4g0nkeHtizc Gravityhift [MH00..01 Gravityhift [MH 00..01
fielFdosr.ananalysisofthe shifts due to collisions,the pres- er-of-ncy S-0.1 er-of-ncy S -0.1
sure of the Cs vapor was changed by heating the vapor Centeque Centeque
cell as described above. Data were taken at tempera- Fr 0.0 0.5 1.0 Fr 0 1 2
Normalized Power Normalized Magnetic Field
tures of 297, 318.6, and 345.5 K, corresponding to Cs
vapor pressures of 0.15, 1.2, and 11 mPa. As described z]
aabfioxveed,whyepdeertfienrmeAinecotnhsetacennttfeorr-oefa-cghrapvrietyssfurreeq.uWenecyseweintho Gravityhift [MH 00..01
ipnrdeiscsautrieon(Foifg.a7n)y.vTahrieatsitoanndoafrtdhededvaitaatiwonithoftthheeCfrsevqaupeonr- er-of-ncy S -0.1
cwiehsichexwtreactatekde aasttthheeudnicffeerrteaninttyteimnptehreatcuenretseri-sof3-g1rakvHitzy, CentFreque 0 5 10
frequency due to temperature or pressure effects. For Pressure [mPa]
the hyperfine A constant, the scatter in the data taken
at different temperatures gives a standard deviation of FIG. 7: The dependence of the center-of-gravity frequencies
18 kHz, which we adopt as the uncertainty. for the6S1/2→6P3/2→8S1/2stateon thepower (upperleft),
the magnetic field (upper right), and the Cs vapor pressure
The final systematicconsideredwaspossible errordue
(bottom). The straight line for the power dependence is a
to imperfect alignment of the counter-propagating laser
weighted linear fit to the data. The vertical axis has been
beams. The effect of misalignment of the two beams
offset and centered so that the nominal operating point is at
is to broaden and shift the peaks through a first-order
zero for the power-dependence plot. The magnetic field and
Doppler shift. As can be seen from Eq. (2), the position
pressure data are shown scattered about the weighted mean
of the peaks depends on the difference in the magnitude of thedata. The horizontal axis for the power dependenceis
and direction of the two vectors ~k and ~k as well as plotted as a function of the fraction of the maximum power.
1 2
the velocity class excited. Additional first-orderDoppler Thehorizontalaxisforthemagneticfielddependenceisplot-
shifts come into play if the wavefronts of ~k and ~k are ted as a function of theresidual magnetic field, ≈50µT.
1 2
notpreciselycounter-propagating,whichcanresultfrom
divergence of the counter-propagating beams and mis-
matchofthespatialmodes. Foraspectrumthatcontains
The study of this transition confirmed that while the in-
many peaks coming from many different velocity classes
dividual peaks do shift, there was no shift in the mean
the shift due to the misalignment largely averages out.
value of the transition frequency, within the uncertainty
This effect was investigated theoretically by modifying
of our measurement.
the program to include the effect of misalignment of the
twobeams. This wasdoneby changingthe~k2·~v termin The non-zero second-order Doppler shift (v/c)2ν0 is
Eq. (2) to include an additional perpendicular velocity about 300 Hz for this transition and is negligible com-
component. For the 8S state three spectra were calcu- pared to the other uncertainties.
lated with different angles of misalignment. Analysis of Table I summarizes the correctionsandassociatedun-
thecalculatedspectrumandanestimateofthemaximum certainties on the extracted center-of-gravity and hyper-
possible misalignment of the beams gives a conservative fine A constants. We see no evidence for any shift that
limit on possible effects from angular misalignment of is larger than the uncertainty associated with evaluat-
22 kHz for the center-of-gravity frequency and 20 kHz ing the effect. As our final value we take the mean
for the hyperfine A constant. If we attribute the scatter of the 8S frequencies determined from the 6P and
1/2 1/2
in the experimental results to angular misalignment, we 6P intermediate states, weighted by their respec-
3/2
concludethatforallthespectra,theeffectiveangularde- tive statistical uncertainties, and apply the corrections
viation due to either misalignment or to divergence and andsystematicuncertainties(propagatedinquadrature)
mode mismatch of the beams is less than 0.04 radians. listed in Table I. We find final values for the center-of-
We note that for the calculated data the misalignment gravityfrequencyandhyperfineAconstantof8S state
1/2
of the counter-propagating laser beams leads to no sig- of
nificant shift in the extracted frequency, it does lead to
a larger spreadof the data and an increasedvalue of χ2,
as observed. This additional spread of the data beyond ν =729009798.86(19)MHz
6S1/2:8S1/2
what is expected from the statistical noise is taken into
A =219.14(11)MHz.
accountby conservativelyincreasingthe uncertainties to 8S1/2
achieve the expected χ2, as described in Section IVA.
Inaddition,theeffectofmisalignmentwasinvestigated These results are in good agreement with previous mea-
experimentally on the 6S →6P →7D transition. surements of the 8S state, summarized in Table II.
1/2 1/2 3/2 1/2
9
C. Results for the 9S1/2state 3 - 3 4 - 4
m on n
tcanwum8lFoa5-ifit0pgeluhtdneromerstpo8wfinesciltthttherrorauwanmwss1.iittt0hhiToenhnasmips61eS0spcp1atn/ers2mcus→tbmrpau6naomPsdbs3twb/fao2aairnn→sedFcd9o2fSolf.lo1re/rTc2Ftthaeh1edneadmbcnaytdeshauceasasuced6ra5eoeld8--f uorescence fro29 SExcitati1/2 Calculatio 3 - 3 4 - 4 3 - 4 4 - 3 l 3 - 4 4 - 3 3 - 3
ment of the 9S state through the 6P intermediate Fl
statewasnotpo1s/s2iblebecausethecomple1/m2entaryradia- S1/2P3/2 (cid:4)ent
gtiroaTnvhiatetyp6f3era5eqk.5usenanm-cpywwaaesnrdoenuthtsheeedheytdopgeedrefiotfneetrhmAeinlacesoetnrhsebtaacnnetdnwtaesird-dtohef--. 22P6 (cid:4)26 S1/2(cid:4)Experim abc de fg hi j k mn o p
scribedabove. Theextractedcenter-of-gravityfrequency 7 26
0.772 0.773 0.774
is806761363.380(58)MHzandthehyperfineAconstant frep - 1000 MHz [MHz]
is 109.932(30)MHz. The χ2 was 83 for 32 peaks.
The systematic effects due to ac-Stark shifts, Zee- FIG. 8: The calculated spectrum (upper trace) and exper-
man shifts, pressure shifts, and misalignment effects imental signal (lower trace) of the 7P → 6S fluorescence
were investigated in the same way as they were for the whentheCsatomsareexcitedtothe9S1/2statethroughthe
8S1/2state. The systematic effects were studied by fo- 6P3/2state. The peaksin theexperimental spectrum labeled
cusing on the three peaks labeled h, i, and j in Fig. 8, withletterswereusedinextractingthetransitionfrequencies,
as described in the text.
correspondingtotransitionsfromtheF =3groundstate
hyperfineleveltotheF′′ =3excitedstatehyperfinelevel
through the F′ =2, 3, and 4 intermediate hyperfine lev-
z]0.1 z] 0.1
etfrhloseT.m6hSe≈1l/a2s4→e0r6WinPt/3ec/n2mssi2ttyatgofeoro2fb0toWhteh/ttcrrmaann2ssiiattinioodnnstwhwaesavsasvprairaretiaeioddn.owFvoaersr er-of-Gravityncy Shift [MH0.0 er-of-Gravityncy Shift [MH 0.0
≈4000opticalmodes. Alinearfitofthecenter-of-gravity Centeque-0.1 Centeque -0.1
frequency to the power gave a dependence of −42(127) Fr 0.0 0.5 1.0 Fr 0 1 2
Normalized Power Normalized Magnetic Field
kHz/P where P is the normalized operating power. We
therefore assign a 169 kHz uncertainty to the center-of- z] 0.1
gSrimaviiltayrlyf,refqouretnhceyhydpueerfitonepAoscsoibnlsetalingthwteshfoifutnsd(Failgi.ne9a)r. Gravityhift [MH 0.0
WAdeeApcoetsnnawdskteeiatn7nhc9tetdkhcuHeoezn8tsSuoisntpceeonsrstttsaaitwbienli,ettyhnaoctzo-edSrbitsoaecrestkrlhonsepahubeif,nltesc2.ed1re(tp5a8einn)tdykeHnincze/tPhoef. Center-of-Frequency S -0.1 0 5 10
1/2 Pressure [mPa]
thefrequencyontheexternalmagneticfieldwasobserved
(Fig. 9). Based on the scatter (standard deviation) of
FIG. 9: The dependence of the center-of-gravity frequencies
the data we arrive at an uncertainty of 18 kHz for the
for the6S1/2→6P3/2→9S1/2stateon thepower (upperleft),
center-of-gravity frequency and 7 kHz for the hyperfine the magnetic field (upper right), and the Cs vapor pressure
A constant. (bottom). The straight line for the power dependence is a
Spectra were taken at four different temperatures of weighted linear fit to the data. The vertical axis has been
the vapor cell corresponding to Cs vapor pressures of offset and centered so that the nominal operating point is at
0.15, 1.06, 3.9, and 7.6 mPa. The dependence of the zero for the power-dependence plot. The magnetic field and
center-of-gravity frequency is shown in Fig. 9. There is pressure data are shown scattered about the weighted mean
ofthedata. Thepowerisplottedasafunctionofthefraction
evidenceofapossibleshiftinthefrequencywithpressure.
of the maximum power. The magnetic field is plotted as a
However,even atpressuresthat are50 times largerthan
function of theresidual magnetic field, ≈50µT.
the nominal operating pressure, we see only a 67 kHz
shift. Weagainusethestandarddeviationofthedatato
place a conservative limit on possible uncertainties due
to pressure shifts and find a 34 kHz uncertainty in the a 58 kHz uncertainty for the center-of-gravity frequency
optical frequency due to temperature dependent effects. and 20 kHz for the hyperfine A constant.
A similar analysis for the hyperfine A constant gives an
The uncertainties to the center-of-gravity frequency
uncertainty due to temperature dependent effects of 15
andthe hyperfineAconstantaresummarizedinTableI.
kHz.
An analysis of the frequency dependence on misalign- We arrive at a final value for the center-of-gravityfre-
ment done in the same way as described above yielded quencyforthe6S →9S transitionandthehyperfine
1/2 1/2
10
z]0.1 z] 0.1
uorescence from 27 DExcitation3/2 Calculation 4 –F''3 –F''l 4 –F'' 3 –F'' Center-of-GravityFrequency Shift [MH-00..100.0 0.5 1.0 Center-of-GravityFrequency Shift [MH -00..10 0 1 2
Fl m n o Normalized Power Normalized Magnetic Field
227 P6 S1/2(cid:5)226 S6 P1/21/2 (cid:5)(cid:5)Experiment 0.772 abcdef ghfrijekp 0- .1707030 MHz [MHz] 0.774 p q r Center-of-GravityFrequency Shift [MHz] -000...101 0 1 2 3
Pressure [mPa]
FIG. 10: The calculated spectrum (upper trace) and exper-
imental signal (lower trace) of the 7P → 6S fluorescence FIG.11: Thedependenceofthecenter-of-gravity frequencies
whentheCsatomsareexcitedtothe7D3/2statethroughthe for the 6S1/2→6P1/2→7D3/2transition on the power (upper
6P1/2state. The peaks in theexperimental spectrum labeled left),themagneticfield(upperright),andtheCsvaporpres-
sure(bottom). Thestraight linefor thepowerdependenceis
withletterswereusedinextractingthetransitionfrequencies
a weighted linear fit to the data. The vertical axis has been
as described in thetext.
offset and centered so that the nominal operating point is at
zero for the power-dependence plot. The magnetic field and
A constant of the 9S state of pressure data are shown scattered about the weighted mean
1/2
ofthedata. Thepowerisplottedasafunctionofthefraction
ν =806761363.38(19)MHz of the maximum power. The magnetic field is plotted as a
6S1/2:9S1/2 function of theresidual magnetic field, ≈50µT.
A =109.93(9)MHz.
9S1/2
This is in good agreement with the previous measure-
ments of this transition, summarized with other data in Thesystematic effectswerestudiedbyfocusing onthe
TableII,andimprovesonthecenter-of-gravityfrequency six peaks labeled g, h, i, j, k, and l in Fig. 10, corre-
by two orders of magnitude. sponding to transitions from the F = 3 ground state.
The laser intensity on both stages of the transition was
varied. For the 6S →6P stage, the change in inten-
1/2 1/2
D. Results for the 7D3/2state sityrangedfrom≈19W/cm2 to≈3W/cm2,spreadover
≈ 4000 optical modes within the 10 nm (3.8 THz) filter
The center-of-gravity frequency, magnetic dipole, and bandwidth. The slope of the center-of-gravity frequency
electric quadrupole hyperfine coupling constants for the asafunctionoflightpoweryieldsadependenceof22(27)
7D state were determined from the spectrum in Fig. kHz(Fig.11),correspondingtoanuncertaintyof49kHz.
3/2
10. The filters used in collecting this spectrum were a Wefinduncertaintiesduetoac-Starkshiftsof14kHzfor
890 nm filter with a 10 nm passband for F1 and a 670 the A constant and 17 kHz for the B constant.
nm filter with a 30 nm passband for F2. This spectrum No effect of the external magnetic field was observed
wastakenby exciting Cs throughthe 6P intermediate (Fig. 11), within the measurement uncertainties. Based
1/2
state. Use of the 6P state eliminates possible transi- on the scatter of the data, we assign uncertainties of 17
1/2
tions to the 7D state due to angular momentum se- kHz, 3 kHz, and 17 kHz for the center-of-gravity fre-
5/2
lection rules. The groups consist of six peaks, corre- quency, the hyperfine A constant and the hyperfine B
sponding to the F → F′ = 3 → F′′ = 2,3,4 and constant, respectively.
F → F′ = 4 → F′′ = 3,4,5 transitions. The 18 peaks Spectra were taken at four different temperatures of
labeled a through r in Fig. 10 were used to extract the the vapor cell, corresponding to a Cs vapor pressure of
frequency and hyperfine coupling constants. 0.15,0.2,0.3,and2.0mPa. Aswiththe9S state,there
1/2
The 7D state has J > 1/2 and, consequently, an elec- appearstobeasmalldependenceofthecenter-of-gravity
tricquadrupolecouplingdenotedbythehyperfineB con- frequency as a function of the pressure in the cell, lead-
stant. Minimizing the χ2(ν, A, B) function defined by ing to a 28 kHz shift when the pressure is increased by
Eq. (10) with respect to the center-of-gravity frequency a factor of ≈ 10. However, since we fix the A and B
and the hyperfine A and B constants, we find a value constants,thisservesasamaximalshiftinthe center-of-
for the center of gravity of 780894762.250(28) MHz, a gravity frequency and we do not attempt to correct for
hyperfine A constant value of 7.386(6) MHz, and a hy- this shift. We take the standard deviation as a conser-
perfine B constant value of −0.182(50). The χ2 was 78 vative estimate of the error and assign an uncertainty of
for 36 peaks. 12 kHz to the center-of-gravity frequency due to possi-
