Two-step pulse observation to improve resonance contrast for coherent population trapping atomic clock

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by  Yuichiro Yano

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Two-step pulse observation to improve resonance contrast for coherent
population trapping atomic clock
YuichiroYano · ShigeyoshiGoka · MasatoshiKajita
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Received:date/Accepted:date

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Abstract We studyresonancecontrastbyatwo-steppulse
]
h observationmethodtoenhancethefrequencystabilityofco-
p herent population trapping (CPT) atomic clocks. The pro-
-
posed method is a two-step Raman–Ramsey scheme with
m
low intensity duringresonanceobservationand highinten-
o
sityaftertheobservation.Thismethodreducesthefrequency
t
a
variationin the lightintensity and maintainsa high signal-
.
s to-noiseratio.Theresonancecharacteristicswerecalculated
c
i by density matrix analysis of a Λ-type three-level system
s
that was modeled on the 133Cs D1 line, and the character-
y
h istics were also measured using a vertical-cavity surface-
p emittinglaserandaCsvaporcell.
[ Fig.1 ProposedRamseypulsesequence.

PACS PACScode1·PACScode2·more
1
v
clocksisoftenlimitedbyacombinationoflightsourceAM
6
noise and FM-AM conversion noise in the atomic absorp-
9 1 Introduction
6 tion. Therefore, the contrast, which is given by the reso-
5 nance signal amplitude divided by the background signal
Coherent population trapping (CPT) atomic clocks are in
0
level,isusedastheSNRoftheCPTresonance.TheQvalue
great demand for many applications, such as telecommu-
.
1 is given by clock transition frequencydivided by the reso-
nications, navigation systems, and the synchronization of
0
nance linewidth. The Q value is degeneratedby the power
networks[1],andsuchclocksarerequiredfortheirhighfre-
7
broadeningeffect.Therefore,itisnecessarytoimprovethe
1 quencystability.
: The frequencystability in atomic clocks can be classi- resonancecontrastandreducepowerbroadeningtoenhance
v
theshort-termfrequencystability.Thelong-termfrequency
i fiedintotwotypesbasedontheaveragingtimeoftheAllan
X stabilityismainlylimitedbythelightshift(theacStarkshift
deviation,whicharetheshort-termandlong-termfrequency
r inducedbythelaserlight)becauseitchangeswiththelight
stability. The short-term stability is determined as the in-
a
powerfluctuationortheagingofopticalelements.
verse of the productof the signal-to-noiseratio (SNR) and
theQvalue[2].TheSNRoflaser-pumpedvaporcellatomic TheRaman–Ramsey(RR)schemeforenhancingthefre-
quencystabilityofCPTatomicclockshasbeeninvestigated
YuichiroYano( )·ShigeyoshiGoka byanumberofresearchers[3,4,5,6,7].Thisschemesimul-
(cid:0)
DepartmentofElectrical&Engineering,TokyoMetropolitanUniver-
taneouslyreduceslinebroadeningandthelightshift[6,7,8].
sity
Recently,a short-termstability of 3.2×10−13τ−1/2 has been
1-1Minami-Osawa,Hachioji-shi,Tokyo,Japan192-0397
PresentE-mailaddress:[email protected] obtained,andstabilityaslowas3×10−14hasbeenachieved
at an averaging time of 200 s using a Cs vapor cell with
MasatoshiKajita
National Institute of Information and Communications Technology theRRscheme[9].Also,whencoldRbatomswereused,a
(NICT) short-termfrequencystabilityof4×10−11τ−1/2wasobtained,
2 YuichiroYanoetal.
andalong-termfrequencystabilityof3×10−13wasachieved
foranaveragingtimeof5h[10].
In ourpreviouspaper,we investigatedthe light shiftin
the RR scheme both theoretically and experimentally with
the aim of enhancing the long-term frequency stability of
CPTatomicclocks[7].Theresultsshowedthatthelightshift
intheRRschemeislowerthanthatundercwillumination.
We also found that the fluctuation of the clock frequency
duetothelightintensityfluctuationisreducedbyreducing Fig.2 Ramseypulsesequence:τistheexcitationtime,T isthefree
evolutiontime,andτ istheresonancesignalobservationtime.
the observation time, which is defined as the time interval m
betweentheriseofthepulsedlaserandtheobservation.
Themainreasonforthisisthattheatomsevolvetowards
asteadystateduringobservation.Inaddition,wefoundthat
the light shift in the RR scheme is a nonlinear function of
thelightintensity,andreducedclockfrequencyvariationis
obtainedunderahighlightintensitybeyondasetthreshold
value. To reduce the light shift, it is preferable to perform
measurementswiththeobservationtimeasshortaspossible
andthelightintensityslightlyhigherthanthethreshold.
Ahighersignalcanalsobeobtainedwhensettingbotha
shortdetectiontime andhighlightintensity[11].Certainly,
thesignalincreaseswithincreasinglightintensity;however
thenoisealsoincreasesowingtotheshortdetectiontimeand
the AM noise in the highintensity.Thus,the contrastdoes Fig.3 (a)Excitationschemewiththeσ+lightfieldontheD lineof
1
notincreasewithincreasinglightintensity.Whilealowlight Cs. (b) Closed Λ-type three-level model used tocalculate CPT phe-
nomenon:δ andδ arethedetuningsoftheprobelaserandcoupling
intensityleadsto a weaksignal,a detectedsignalwithlow p c
laser,respectively.Ω andΩ areRabifrequencies.Γ andΓ arethe
p c 31 32
laser intensity decreasesmore slower than one with a high
relaxationtermsbetweenanexcitedstateandthetwogroundstates.γ
f
intensity.Consequently,the signalcan bestrongerata low isthedecoherenceratebetweentheexcitedstateandthegroundstates,
intensityforlargedetectiontimes. andγsisthedecoherenceratebetweenthetwogroundstates.
Weproposedatwo-steppulseobservationmethodtore-
solvethisissue(Fig.1).ThismethodisanRRschemewith 2 Theory
low lightintensity duringobservation[12]. We investigated
the lightshiftbytwo-step pulseobservation.Asa result, it 2.1 Two-steppulseobservation
was shown that the clock frequency variation in the light
intensityusingthismethodislowerthanthatoftheconven- Theobservationscheme forthetwo-step pulseobservation
tionalmethodowingto thereducedrepumpingratetoward methodis shown in Fig. 2. Ip is the total light intensity ir-
thesteadystateduringtheobservationtime. radiatingtothegascell.TheRamsey–CPTresonanceisob-
servedbymeasuringthetransmittedlightintensityatobser-
In this work, we discuss the improvementof the reso- vationtimeτ afterthepulserise.Thetwo-steppulseobser-
m
nance contrast when using the two-step pulse observation vationmaintainsalowlightintensityofrI untiltheobser-
p
method via numericalcalculation and experiment. A com- vationpoint,whererisdefinedastheobservationintensity
parisonwiththeRRschemeisperformedbycalculationus- ratio,andtakesavaluesfrom0to1.Whenr = 1isset,the
ingthedensitymatrixanalysis.Thecharacteristicswerealso two-steppulseobservationmethodistreatedwiththesame
measuredusingavertical-cavitysurface-emittinglaser(VC- scheme as that used for the conventional RR scheme. Af-
SEL)andaCsvaporcell.WeinvestigatedthecontrastandQ terthemeasurementistaken,thelightintensityisconverted
valueasfunctionsofthefreeevolutiontimeandobservation intoI andtheatomsarepreparedforthenextmeasurement.
p
time because they are importantparametersthat determine Alaserpulsewithadurationof(τ−τ )irradiatestheatoms
m
the contrastand Q value. Fromthe results, it is shownthat forpumpingofthesteadydarkstate.Afterthefreeevolution
this method reduces the repumping to the steady state and timeT,alaserpulseirradiatestheatomsagainwithlightin-
providesalargercontrastthantheconventionalmethod. tensityrI .ThelightintensityI(t)forthisschemeiswritten
p
Two-steppulseobservationtoimproveresonancecontrastforcoherentpopulationtrappingatomicclock 3
as usedtoobtaintheabsorptionindex,andtheresonancecon-
trastcanbederivedas
rI (0<t≤τ )
 p m
I(t)=Ip (τm <t≤τ) . (1) Contrast(%)=k~ωnatomL∆ΩIm(ρ)·100, (5)
0 (τ<t≤τ+T) 3rIp
where k is a proportional constant that is determined by
2.2 Theoreticalmodel the sum of the laser intensities not interacting with alkali
atoms such as higher-order sidebands, ~ is Planck’s con-
Figure3(a)showstheexcitationschemeusingtheσ+ light stant,ωistheangularfrequencyofthelaser,cisthespeed
fieldforthe133Cs-D line.IntheCPTphenomenon,thetwo of light, n is the number of atoms per unit volume, L
1 atom
groundstatesof62S arecoupledsimultaneouslytoacom- is the optical length of the gas cell, and ∆ΩIm(ρ) is the
1/2
mon excited state of 62P . In this system, the dynamical difference between the maximum and minimum values of
1/2
behaviorofthedensitymatrixρisgovernedbythequantum Ω Im(ρ )+Ω Im(ρ ).
13 13 23 23
Liouvilleequation,
∂ i
ρ(t)= [ρ,H]+Rρ, (2) 2.3 CalculatedRamsey–CPTresonancespectrumwith
∂t ~
two-steppulseobservation
where H is theHamiltonianmatrixforthisthree-levelsys-
temandRrepresentstherelaxationterms.Usingtherotating- Thecalculatedresonancespectrafordifferentfreeevolution
wave approximation with the simplified Λ-type model de- timesT areshowninFig.4.Theexcitationtimeτandobser-
pictedinFig.3(b),Eq.(2)canberewrittenas vationtimeT were1.0msand10µs,respectively.Theden-
sitymatrixwascalculatedusingEq.(10)intheappendixA.
Ω TheRamsey–CPTamplitudedecreasedwithincreasingfree
p
ρ˙11 =i (−ρ13+ρ31)+Γ31ρ33+γs(ρ22−ρ11), evolutiontimebecauseofthecoherencerelaxationbetween
2
Ω ground states. The resonance linewidth decreased with in-
ρ˙22 =i c(−ρ23+ρ32)+Γ32ρ33−γs(ρ22−ρ11), creasingT sincetheRamseyschemewasused[14].
2
Ω Ω Figure5showsthecalculatedresonancespectrafordif-
ρ˙33 =i 2p(ρ13−ρ31)+i 2c(ρ23−ρ32)−Γ3ρ33, ferentobservationtimesτm andobservationintensityratios
(3)
Ω Ω r. T and τ are 0.5 and 1.0 ms, respectively.The total light
ρ˙12 =iρ12(δp−δc)−i 2cρ13+i 2pρ32−γsρ12, intensityIp is2.4mW/cm2.TheRamseyfringewasclearly
ρ˙13 =iρ13δp−iΩcρ12+iΩp(ρ33−ρ11)−γfρ13, voabnseisrhveeddffoorraalsamrgaelloobbsseerrvvaatitoionntitmimee.T(τhme∼res0o)n.aHnocweeavmepr,lii-t
2 2
ρ˙23 =iρ23δc−iΩ2pρ21+iΩ2c(ρ33−ρ22)−γfρ23, stuudbestoanfttihaellyRdamecsreeyasferdinwgeithstrionncrgelaysidnegpeτnmd.eTdhoisniτsmb,eacnadusiet
atomsevolvedtowardasteadystateduringthetimefromthe
whereΩ andΩ areRabifrequenciesandδ andδ arefre-
p c p c pulserisetotheobservationpoint.Theresonanceamplitude
quencydetunings,Γ andΓ arerelaxationtermsbetween
31 32 ofthe Ramseyfringein thetwo-steppulseobservationcan
anexcitedstateandthetwogroundstates,withthetraceof
be retained for a longer observation time. Because the re-
thedensitymatrixsatisfyingtheclosed-systemcondition
pumpingrateisproportionaltothelightintensity,asmaller
Tr(ρ)=ρ +ρ +ρ =1. (4) repumpingrateisobtainedwhensettingasmallerr.There-
11 22 33
fore,evenifalongobservationtimeisset,alargeresonance
InFig.3(b),|1iand|2icorrespondtothetwogroundstates
amplitudecanbeobtained.Sincethelinewidthofthecenter
|F = 3,m = 0i and |F = 4,m = 0i in the 62S state,
F F 1/2 fringe is independentof the observationtime, the two-step
respectively,and |3i correspondsto the state in 62P . We
1/2 pulse observation enhances the SNR without degenerating
assume Γ = Γ (= γ ). The ground-state relaxation rate
31 32 f theQvalue.
γ isaminusculequantity,γ ≪ γ .Inthiscalculation,the
s s f
totalemissionratewassetat370MHz,whichwasobtained
experimentallyfromtheabsorptionlinesofaCscellwithNe 3 Experimentalsetup
buffer gas at a pressure of 4.0 kPa. The obtained emission
rate of 370 MHz is consistent with the value estimated in Figure 6 shows the experimental setup. The measurement
Ref. [13]. For bichromaticlight, we havethe relationδ = systemisbasedonapreviousRamsey–CPTobservationsys-
p
−δ =∆ /2. tem.
c 0
Thenumericalcalculationmethodforthedensitymatrix A single-mode VCSEL (Ricoh Company Ltd., Japan)
ρ is described in the appendix A. The density matrix ρ is wasused to simultaneouslyexcite the two groundstates to
4 YuichiroYanoetal.
1.0
u.) 0.01 µs
) a. 100 µs
a.u. T (ms) nd ( 200 µs
sity ( 010...602 kgrou0.5
en 1.4 ac
Int al / B
n
g
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Si
0.0
Frequency detuning (kHz)
-30 -20 -10 0 10 20 30
Fig.4 (Coloronline)Calculatedresonancespectrumfordifferentfree s Frequency detuning (kHz)
evolutiontimesT:theexcitationtimeτis1.0ms,theobservationtime (a) r=1.00
τ is10µs,theobservationintensityratioris1.0. 1.0
m
u.) 0.01 µs
a. 100 µs
the common excited state. The wavelength of the VCSEL d ( 200 µs
n
usedtoexcite133CsattheD linewas895nm.TheVCSEL u
1 o
r
wasdrivenbyadcinjectioncurrentusingabiasTandwas g0.5
k
c
modulated at 4.6 GHz using an analog signal generator to a
B
generatefirst-ordersidebandsaroundthelasercarrier. al /
Forpulseexcitation,anacousto-opticalmodulator(AOM) n
g
wasusedtomodulatethelightintensity.Afirst-orderdiffrac- Si
0.0
tion beam was irradiated to the gas cell. The AOM had a -30 -20 -10 0 10 20 30
nominal rise and fall time of 65 ns. The total light inten- Frequency detuning (kHz) 0
sity incidenton the gascell was adjusted using the control (b) r=0.25
voltageoftheAOMandwascalibratedbyanopticalpower 1.0
meter.APyrexgascellcontainingamixtureof133Csatoms u.) 0.01 µs
a. 100 µs
and Ne buffer gas at a pressure of 4.0 kPa was used. This d ( 200 µs
cellwascylindricalwithadiameterof20mmandanoptical n
u
o
lengthof 22.5mm.Its temperaturewas maintainedat 42.0 r
g0.5
◦C. The gas cell and Helmholtz coil were covered with a ck
a
magneticshieldtopreventexternalmagneticfieldsfromaf- B
fectingthemagneticfieldinsidethecell.TheHelmholtzcoil al /
n
producedaninternalmagneticfieldinthegascell.Theaxis g
Si
ofthe10µTmagneticfieldwassetparalleltothedirection 0.0
-30 -20 -10 0 10 20 30
ofthelaserlight(C-axisdirection).
Frequency detuning (kHz)
(c) r∼0.00
Fig.5 (Coloronline)Calculatedresonancespectrumfordifferentob-
4 Results
servationtimeτ andr:thefreeevolutiontimeTis500µs,theexcita-
m
tiontimeτis1.0ms,theobservationintensityratioris1.0.
4.1 ContrastandQvalueasfunctionsoffreeevolutiontime
T
atoms,thebuffergas,andthecellwalls.Thecontrastexpo-
Thecontrastasafunctionofthefreeevolutiontimeisshown nentiallydecreasedwithincreasingfreeevolutiontimefrom
inFig.7.Theexcitationtimeandobservationtimewereset 0.2 ms to 1.1 ms. The variation was 0.79% per 1 ms. The
to 1.0 ms and 5.0 µs in both the experiment and the cal- calculatedresonancecontrasthadthesametendencyasthe
culation, respectively. The total light intensity I was 2.4 measurementvalues.
p
mW/cm2 in both the experiment and the calculation. The The Q value as a function of the free evolution time is
coefficient kn was set to 8.55 × 1016 m−3. γ /2π was showninFig.8.TheQvaluelinearlyincreasedwithincreas-
atom s
127Hz. Themeasuredcontrastundercontinuousillumina- ing free evolution time because the linewidth is inversely
tionwas2.4%andthelinewidthwas2.4kHz.Thecontrast proportionaltothefreeevolutiontimeT.Allthemeasured
decreasedwith increasing free evolutiontime owing to the Qvalueswerelargerthanthecalculatedvalues.Oneofthe
coherencerelaxationinducedbythecollisionwithotherCs reasonsforthisdifferenceisthattheexperimentalfreeevo-
Two-steppulseobservationtoimproveresonancecontrastforcoherentpopulationtrappingatomicclock 5
Oscillo- with increasing free evolution time, and the Q value pro-
scope portionally increased with increasing free evolution time.
RF AOM Pulse Sample Therefore,thereisanoptimalfreeevolutiontimeT formax-
Generator Driver Generator /Hold imizingtheproductoftheSNRandtheQvalue.Theoptimal
Magnetic shield freeevolutiontimewas1.0msinthisexperiment.Themax-
RF frequency B imumvaluewasaboutfivetimeslargerthanthatundercon-
tinuous illumination, and it is expected that the short-term
Cs + Ne
Bias T VCSEL AOM PD stabilitywasimprovedbyafactoroffive.
Cell
Linearpolarizer Helmholtz coil
4.2 Contrastasafunctionofobservationtime
Current 60 kHz Function Lock-in
Driver Generator Amp.
Figure 9 shows the contrast as a function of the observa-
Fig.6 Schematicofexperimentalsetup.PD:photodiode. tiontimefordifferentr.Thecontrasthadamaximumvalue
at the pulse rise (τ ∼ 1/Γ) and exponentially decreased
m
with increasing observation time. At a very short observa-
2.0
tiontime(τ < 1/Γ),thecontrastwassmallbecauseofthe
m
limitation of the optical transition response. When the ob-
)1.5
% servation time was larger than 1/Γ, the contrast decreased
(
st Calc. withincreasingobservationtimebecauseoftherepumping
ra1.0 Expt. tothesteadydarkstate[11].Ontheotherhand,therewaslit-
nt
o tlechangeintheQvaluewiththeobservationtime.There-
C0.5
fore,itisnecessarytoreducetheobservationtimetoobtain
goodshort-termfrequencystability.
0
0 0.4 0.8 1.2 1.6 In the conventional observation method (r = 1.0), the
Free evolution time T (ms)
contrast substantially decreased after a pulse rise. A larger
Fig.7 ContrastasafunctionofthefreeevolutiontimeT. decrease in contrast was observedwhen a shorter observa-
tiontimewasset.Thedecreaseincontrastwas0.078%/µsat
τ =5.0µsinthisexperiment.Thiswastwoordersofmag-
3.0 m
nitude larger than that with increasing free evolution time.
) The contrast under the RR scheme was dominated by the
7
102.0 observationtime.

·( Calc. Inthetwo-steppulseobservation(0.0<r<1.0),higher
ue Expt. contrast was obtained. Because there was little repumping
al1.0 underasmallr,asmallerdecreaseincontrastwasobserved
v
Q forasmallr.Whenr wasnegligiblesmallandapproached
zero,thecontrastincreasedandgraduallyapproachedacon-
0
0 0.4 0.8 1.2 1.6 trast curve with r ∼ 0. Therefore, the contrast curve with
Free evolution time T (ms) r ∼ 0isanasymptoticsolutionandistheupperlimitofthe
Fig.8 QvalueasafunctionofthefreeevolutiontimeT. contrast improvementin two-step pulse observation. How-
ever,thecontrastalsotendedtodecreasewhensettingr∼0.
In such a case, the decrease in contrast is strongly depen-
lution time is long because of the relationship betweenthe denton the relaxation between the groundstates γ , and it
s
atomicmotionandthe interactionregion.Forexample,the increasedwithincreasingγ .Therefore,thedecreaseincon-
s
atomsfallinginthedarkstatemovedoutsidetheinteraction trastunderr∼0wasinducedbytherelaxationoftheatomic
regionduringthefirstpulse,andtheyreturnedtotheinterac- stateduringtheobservationtime.Thedecreasewasreduced
tionregionasaresultofthecollisionswithbuffergasatoms toabouthalfofthatintheconventionalmethod.Therefore,
afterthefreeevolutiontime,wheretheywereirradiatedby thecontrastincreasedwithdecreasingr,andthecontrastin
thelightagain.Becausetheeffectivefreeevolutiontimewas thetwo-steppulseobservation(r=0.3)wastwicethatinthe
long in the experiment,the measuredQ valueswere larger conventionalmethodatτ =25µs.Thedecreaseincontrast
m
thanthecalculatedvalues. waslowerforalargerobservationtime,andthecontrastin
The short-term frequencystability is determined as the thetwo-steppulseobservationwas4.5timeslargerthanthat
productoftheSNRandtheQvalue.Thecontrastdecreased in the conventionalmethod at τ = 115 µs. These results
m
6 YuichiroYanoetal.
effectivemethodforrealisingCPTatomicclockswithgood
2.0 r fi 0
short-termandlong-termfrequencystability.
)1.5 ropt
%
st ( 0.30 AppendixA:Numericalcalculation
a1.0
r
nt Here,thebehaviorofthedynamicdensitymatrixisderived
o 0.50
C0.5 from the solution of Eq. (3). The easiest way to solve the
time-dependentbehaviorofthedensitymatrixistoperform
1.00
0.0 numericalintegrationbythefinite-differencemethod.How-
0 20 40 60 80 100 120
Observation time tm (m s) ever, finite-difference methods require a long computation
time because the time step must be smaller than the relax-
Fig.9 (Coloronline)Contrastasafunctionoftheobservationtimefor
ationtime Γ−1 to satisfythe stabilitycondition.Inthiscal-
differentvaluesoftheobservationintensityratior:thedashedlineisa
contrastwithr thatsuppresesthevariationofthelightshiftwiththe culation,we showhowto calculate the time dependentbe-
opt
observationtime[12]. haviorofthedensitymatrixusingitseigenvector.Sincethe
numericalsolutiondoesnotrequirenumericalintegration,it
canreduceboththecomputationtimeandtheerror.
showthatalargecontrastcanbeobtainedbytwo-steppulse
The total number of calculation elements is 9 because
observationwithoutreducingtheobservationtime.
thedensitymatrixisa3×3Hermitianmatrix.Forthesakeof
Thelightshiftisdependentontheobservationtimeτ .
m simplicity of the differential equation, a vector comprising
Our previous paper showed that the variation of light shift
thenineelementsρisdefinedasfollows:
withobservationtimeissuppressedbyadjustingtheobser-
vationintensityratior[12]. Inthiscalculation,thevalueof
rthatsuppressesthelightshiftvariationwas0.04.Thecon-  ρ11 
1eal5tirvd2gaCej0hsurtot,sµfnsttsoehhc.rdeilfTutttmhohsviieasoaarxnrnveiifaamaoltpuiruopeemnr,oowcdpfahirrnffieaienstsreiemtlhvonueawclleouetarbeins,stehebnoraoovuntahsmttlhiygooaorbnteoefditonohcrtbaeortnnnas=it1nirt0ae0y%sd.t0r.aa0atn;tidohτolmrowwi=s- ρ:=IIIRRRmmmeeeρρ((((((23ρρρρρρ23211211332332)))))), (6)
Usingρ,Eq.(3)isrewrittenas†
We theoreticallyandexperimentallydiscussedcontrastim-
∂ρ
provementby two-step pulse observation.From the results =H˜ρ. (7)
∂t
forthe contrastandQ valueasfunctionsofthe freeevolu-
tion time, the contrast decreased with increasing free evo- IntheRaman–Ramseyscheme,H˜ duringtheexcitation
lutiontime,whereasQ valueproportionallyincreasedwith timeisdifferentfromthatduringthefreeevolutiontime.The
the free evolution time. The results show that there is an matrixH˜ isdenotedasH˜onandH˜offwhenthepulseisonand
optimalvalueatwhichtheshort-termfrequencystabilityis off,respectively.Also,thevectorattheboundary(vectorat
maximized. From the results for the contrast as a function pulseriseρonandfallρoff)isdefinedasfollows:
oftheobservationtime,thecontrastsignificantlydecreased
ρ =ρ(n(τ+T)),
on
withincreasingobservationtime.Also,therateofdecrease
ρ =ρ(n(τ+T)+τ ), (8)
ofthecontrastwithincreasingobservationtimewastwoor- m m
dersofmagnitudelargerthanthatwithincreasingfreeevo- ρoff =ρ(n(τ+T)+τ),
lutiontime,meaningthatthecontrastunderpulseexcitation
wherenisaninteger.Becauseρ isthevectorT safterthat
on
was strongly dependenton the observation time. Two-step
atthepulsefallρ ,andρ isthevectorτsafterthatatthe
off off
pulseobservationsubstantiallyreducedtherateofdecrease
pulserise ρ , the relationshipbetweenρ andρ canbe
on off on
ofthecontrastwithincreasingobservationtime.Asaresult,
expressedasfollows:
goodcontrastrelativetothatintheconventionalmethodwas
obtained. These results show that two-step pulse observa-
ρ =exp(H˜ T)ρ ,
tionleadsto improvedshort-termstability. Inaddition,our on off off
previouspapershowedthattwo-steppulseobservationalso ρm =exp(H˜mτm)ρon, (9)
reducesthelightshift[12].Two-steppulseobservationisan ρ =exp(H˜ (τ−τ ))ρ ,
off on m m
Two-steppulseobservationtoimproveresonancecontrastforcoherentpopulationtrappingatomicclock 7
†
−γ γ Γ 0 0 0 0 Ω 0
 s s 31 p 
H˜ =−Ωγ000000ps/2−−Ω000000γc/s2−(ΓΩΩ3Γ1pc00003+//222Γ32) δ−−p−ΩΩ−0000γcp/s/δ22c −−Ωδ00000γcp/f2Ω−δp00000γc/f2−(−δΩ−Ωpc0000γp−//s22δc)Ω−−Ωc0000γδ/ppf2Ω−−ΩΩp000γδ/cccf2
whereexpistheexponentialfunctionofthematrix. Because αL is small, taking into account the fact that the
By rearranging Eq. (9), the following equation can be conventional resonance contrast is no more than 10%, we
obtained: canassumethat
(cid:16)E−exp(H˜mτm)exp(H˜offT)exp(H˜on(τ−τm))(cid:17)ρm =0, (10) Itr ≈ I(1−αL). (16)
whereEistheidentitymatrix.FromEq.(10),becauseρ is FromEqs.(14)and(16),theresonanceamplitudeI is:
m pp
notanullvector,ρ istheeigenvectoroftheeigenvalue0of
m ~ωn L
(cid:16)E−exp(H˜mτm)exp(H˜offT)exp(H˜on(τ−τm))(cid:17).Sinceallthe Ipp = atom ∆ΩIm(ρ). (17)
3
matrix elements are known, the vector at the pulse rise ρ
m
canbederivedbycalculatingtheeigenvectorofthematrix. The contrast is defined as the signal amplitude divided by
Notethatthevectorisnormalizedtosatisfythenormaliza- thebackgroundsignallevel.Thebackgroundlevelispropor-
tionconditionofEq.(4). tionaltorandkisaproportionalconstantthatisdetermined
bythesumofthelaserintensitiesnotinteractingwithalkali
atomssuchashigher-ordersidebands.
AppendixB:Contrastderivationfromdensitymatrix
I ~ωn L
Contast(%)= pp ·100=k atom ∆ΩIm(ρ)·100 (18)
Here, the relationshipbetweenthe contrastand the density rIp 3rIp
matrixofEq.(5)isderived.
Thus,thecontrastcanbederivedusingthedensitymatrix.
Thephotonnumbersdensityn is:
p
I
np = ~ωc, (11) Acknowledgements
andthenumberdensityofabsorbedphotonsperunittimenv This work was supported by a Grant-in-Aid for JSPS Fel-
is: lows (No. JP26·6442). The authors are grateful to Ricoh
1 CompanyLtd.forprovidinguswiththeCsD1VCSEL.The
n = n ΩIm(ρ). (12)
v atom researchofM.K.wassupportedbyaGrant-in-AidforSci-
3
entific Research (B) (Grant No. JP25287100), a Grant-in-
FromEqs.(11)and(12),thetimerateofabsorbedphotons
Aid for Scientific Research (C) (Grant No. JP16K05500),
is
andaGrant-in-AidforExploratoryResearch(GrantNo.JP
n ~ωcn ΩIm(ρ) 15K13545)fromtheJapanSocietyforthePromotionofSci-
v = atom . (13)
n 3I ence(JSPS).
p
The absorption index α, which is the number rate of ab-
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