Table Of ContentObservationofspontaneous quantum fluctuations inphoton absorptionby atoms
Takahisa Mitsui, Kenichiro Aoki
ResearchandEducationCenterforNaturalSciencesandHiyoshiDept. ofPhysics,KeioUniversity,Yokohama223–8521,Japan
Fluctuationsinlightabsorptionbyatomsareobservedbyapplyinglaserlightonrubidiumatomsandmea-
suringthetransmittedlightintensityfluctuations. Thesefluctuationsarespontaneousnoise,whicharegeneric
tophotonatominteractions. Bymakinguseofthesub-shotnoiserandomsignaldetectiontechnology,wehave
measuredthespectraatsub-shotnoiselevelstorevealtheirrichnature,whichhadpreviouslybeenunobserved.
Theeffectsofatomstransitingthelaserbeam,RabifloppingintheopticaltransitionsandLarmorprecessionof
3 themagneticmomentareobservedinthespectra.Thepropertiesofthefluctuationsreflectnotonlythequantum
1 behaviorofatoms,butalsothatoflight.
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2
Quantumaspectsofphotons,atomsandtheirinteractionshavebroughtaboutnewexcitingphenomena,suchasBoseEinstein
n
condensationingases[1], quantumteleportation[2],quantumkeydistribution[3]andrecentdevelopmentsinatomicclocks[4].
a
J Mostinvestigationsintothepropertiesofphotonatominteractionsmakeuseofcoherenceofalargenumberofatomstoobtain
3 theiraverageproperties.Whilethisapproachgivesriseto relativelylargesignalswhicharemeasurable,italsoaveragesoutthe
2 quantumfluctuationsintrinsic to the system. Spontaneousnoise in the light transmitted throughatomic vaporshould contain
alltheinformationabouttheinteractionsbetweenasingleatomandphotons,yetthenatureofitsspectrumisunknown. Here,
]
h wereportonthemeasurementsofthisspectrum,whichcontainfluctuationsthatarisegenericallyinphotonatominteractions.
p Themeasuredspectrarevealtheeffectsofatomstransitingthelaserbeam,RabifloppingandLarmorprecessionofthemagnetic
- moment. The main reason these spectra had not previously been seen is that the size of the fluctuations in the transmitted
m
light is small compared to the background(ratio ∼ 10−7 in our experiments) and the fluctuations are buried underneath the
o
photonshotnoise. Ourmeasurementsweremadepossiblebystatisticallyreducingtheshotnoisebyfourordersofmagnitude
t
a through methods we developed. Additionally, the effects of the laser noise are reduced through stabilization and differential
s. detection. Thepropertiesofthefluctuationsreflectnotonlythe quantumbehaviorofatoms, butalso thatoflight. Measuring
c these fluctuations opens a way to direct measurements of quantum properties of photon atom interactions. Similarly to the
i
s thermalandshotnoisethatplayimportantrolesincurrentscience,weexpectthespontaneousnoisearisingfromphotonatom
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interactionstoalsoplaysuchacrucialroleinthenearfuture.
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Ourexperimentalprincipleis simple; we applythe minimalamountof externalperturbationto measurespontaneousnoise
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[ caused by photonatom interactions, just by shining light on atomic vapor. We furtherapply a static magnetic field, but only
when studying its effects. In our experiment, a laser beam (power P, beam waist w), which has the resonant frequency for
1
atomic transition levels is shone on sample cells containing rubidium (Rb) vapor. A large fraction of the beam traverses the
v
cellandwemeasuretheintensityofthetransmittedbeamusingphotodiodes,tofinditsfluctuations. Weobtainthefluctuation
5
0 spectrumafternoisereduction,asdescribedbelow.Inthiswork,forclarityandsimplicity,werestrictourattentiontocircularly
4 polarizedlightinresonancewiththe85RbD2 transitionfromthehyperfinelevel52S1/2(F = 3)to53P3/2[5]. Wehavefurther
5 investigatedfluctuationsindifferentatomicleveltransitionsandtheeffectsofusinglightwithlinearpolarization. Thephysics
.
1 ofthespectrainthosecasescanalsobeunderstoodinafashionsimilartowhatispresentedhere.
0 InFig.1,atypicalmeasuredfluctuationspectruminthetransmittedlightintensityisshown. Throughsystematicanalysesof
3
thesetypesoffluctuationspectra,wehaveidentifiedthreekindsofphysicsthatunderliethisspectrum. Arelativelybroadpeak
1
belowfewhundredkHziscausedbytheintensitychangesduetoatomstransitingthelaserbeam. Weshallcallthisthetransit
:
v noise. The distinctfeaturethatexistsfromaround1MHz to fewtimes 10MHz, whichwe callRabinoise, is dueto the Rabi
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X flopping[6–8]. Whentheatomsareinthegroundstate,theyabsorbphotons,reducingthetransmittedlightintensityand when
theyareintheexcitedstate,theyincreasethetransmissionthroughstimulatedemission. Theatomsalsodecayspontaneously,
r
a givingrisetofurtherrandomfluctuationsintheintensity. Thenoisepeakat1.7MHzoccurswhenamagneticfieldisappliedto
theatoms,inadditiontothecircularlypolarizedlaserbeam.TheatomsinthegroundstateperformLarmorprecessionunderthis
magneticfield,causingtheabsorptioncoefficienttovary.Quantummechanically,thisnoisearisesduetothetransitionsbetween
theZeemansublevelsandhenceisherecalledZeemannoise. Thebehaviorofthesethreetypesofspontaneousnoiseandtheir
underlyingphysicsisexplainedinmoredetailbelow. Theshotnoiselevelforthisspectrum,2eI,isindicatedinFig.1,where
e is the electronchargeand I the photocurrent. It can be seen fromthis that the analysisof these fluctuationsis notpossible
withoutthe eliminationof the shotnoise. Also forcomparison,the spectrum forRb vaporbufferedwith nitrogenatpressure
2.7×104Pa,whichhasthesameshotnoiselevel,isshowninFig.1. Inthiscase,thecollisiontimescaleismuchshorterthan
thetransitandRabifloppingtimescales,sothattheircorrespondingspontaneousnoisearedrasticallyreduced.
Webrieflyexplaintheexperimentalschemeandthenoisereductionweemploy,whichisnecessaryforobtainingthespectra
in thiswork. The basic underlyingscheme tomeasurethe intensityfluctuationsofthe transmittedlightisshownin Fig. 2(a).
In this scheme, the intensity of the light transmitted through the cell is measured using a photodetector (PD). The resulting
2
] 10-22 transit noise Shot noise level
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Frequency [Hz]
FIG. 1: (Color online) A typical measured spectrum of intensity fluctuations. The beam was transmitted through Rb gas, measured as
photocurrent fluctuations(P = 615µW,w = 0.96mm,red). Transitnoise, ZeemannoiseandRabinoiseareseen atprogressivelyhigher
frequencies, respectively. Aspontaneous noisespectrumforthebufferedRbgaswhosefluctuationsaremuchsmallerisalsoshown(P =
650µW,w = 0.53mm,blue). Inthiscase,Zeemannoisefor87Rbatomsisalsovisiblesincetheirtransitionsalsooccur, duetocollisional
broadening. Theshotnoiselevel(black),commontobothspectra,isalsoindicated.
(a)
cell
laser PD ADC FFT
(b)
computer
laser PD1 ADC1 FFT
multiplier
cell BS
averager
PD2 ADC2 FFT
(c)
cellA
laser PD
ToADC1
PD
cell B
PD
ToADC2
PD
FIG.2:Theschematicsofourmeasurementsystem:(a)Astraightforwardschemeformeasuringintensityfluctuationsoflightpassingthrough
agascellthatformsthebasisofoursetup. (b)Ameasurement systemwithshotnoisereduction. (c)Themeasurementsystemusedinthis
work,whichincorporatesdifferentialdetectioninadditiontotheshotnoisereduction.BS:Beamsplitter.FFT:FastFouriertransform.
photocurrent is digitized using an Analog-to-digital converter (ADC) and then Fourier transformed to obtain its fluctuation
spectrum. This setup, however, is insufficient for obtaining spectra such as the one shown in Fig. 1, due to the unavoidable
existenceofextraneousnoise. Anymeasurementincludesvariouskindsofunwantednoise,ofwhichshotnoiseandthesignal
inducedbythelasernoiseareimportantinthecurrentsetup.
Shot noise inevitably occurs in any photoconversion and is often regarded as a limiting factor in the precision, “standard
quantumlimit”,forthesetypesofmeasurements[8].Indeed,previousmeasurementsofspontaneousnoisemadeuseofthesmall
amountofZeemannoisethatcouldbeseenabovetheshotnoiselevelinFig.1[12–15]. Inthiswork,forthefirsttime,wehave
beenabletoobservewhatliesunderneaththisshotnoiselevel,touncoverthetransitandRabinoiseinadditiontotheZeeman
noise,andclarifytheirunderlyingphysics. Toaccomplishthis,weusedmultiplePD’s(PD1,2)tomakemultiplesimultaneous
measurementsD =S+N (j =1,2)ofthespontaneousnoiseS (Fig.2(b)). N isanyuncorrelatednoisethatarisesineach
j j j
3
PD,includingtheshotnoise.WecomputetheaveragedcorrelationoftheFouriertransformsofthemeasurementsD˜ (j =1,2)
j
toobtainthespontaneousnoisespectrum,
hD˜1D˜2i−→h|S˜|2i (N −→∞) . (1)
Here, N is the number of averagings. hS˜N˜ji and hN˜1N˜2i average to zero, statistically, in the limit of infinite number of
averagings. It is crucial here that the shot noise arising in differentPD’s are uncorrelateddue to its quantum nature. In this
work, we haveachieveda reductionby fourordersof magnitude. Thismethodforachievingsub-shotnoisein measurements
hasprovensuccessfulinobtainingsurfacethermalfluctuationspectra[9–11].
Another unwanted effect is the noise that occurs generically in a laser source, even with stabilization, especially at higher
powers. This noise, throughphotonatom interactions, give rise to extraneouscontributionsto the spectra[13]. In the current
experiment,thelasernoiseinducedsignalinthefluctuationspectrahavebeenreducedtosuchaleveltobenegligibleforsmaller
light powers even with the setups of Fig. 2(a),(b), yet substantial signal remains for largerpowers in this case. To eliminate
this signal, whose origin is laser noise, we employ two independentgas cells. The spontaneous noise in these two cells are
independent,eventhoughtheyaresetupidentically. Ontheotherhand,thelasernoiseinducedsignalsarecausedbythesame
lightsourceandareidentical,sothattheycanberemovedthroughdifferentialdetection.Incorporatingthisdifferentialdetection
inadditiontotheshotnoisereductionsystemdescribedabove,wearriveatthemeasurementsystemusedinthiswork,shownin
Fig.2(c).
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ct
pe 10-1
s
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e
w
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10-2
104 105 106
Frequency [Hz]
FIG.3:(Coloronline)Transitnoisespectra:Beamsizesarew=2.1(red),0.64(blue),0.34(magenta),0.17(cyan)mmandthecorresponding
fallofffrequency increasesinthisorder. Resultsfromthe theoreticalcalculationforw = 0.64mm(blackdashed) agreesreasonablywell
withtheexperimentalresults,butfallsoffataslightlysmallerfrequency. Here,thespectrawerenormalizedtomakethefallofffrequency
differenceclearer(a.u.:arbitraryunits).
Semiconductorlaser, DFB-0780-080, Sacher lasertechnik (wavelength 780nm) was used as the light source. The spectral
width of this source is about 3MHz, the same order as the spectral width of Rb-D2 transitions, which is 6MHz. We fur-
ther narrowed the spectral width to less than 1kHz with an optical feedback system using a confocal cavity (SA–300, Tech-
nical Optics)[16] and additional electrical feedback. Analog to digital conversion was performed by PicoScope5203 (ADC,
8bit,500Ms/s,PicoTechnology). Fouriertransformsandaveragingswerecomputedonapersonalcomputer. Thedataacquisi-
tiontimeisaround10seconds,butthetotalmeasurementtimeisaboutfiveminutes,duetolimitationsinthedataanalysisspeed.
RbatomssealedinvacuumPyrexglasscellswereusedassamplesandwereheatedasnecessary. Lightsourcewascircularly
polarizedusinga1/4wavelengthplateandastaticmagneticfieldgeneratedbyaHelmholtzcoilwasapplied,whenrequired.
ThetransitnoisespectraisshownforbeamswithvariousbeamsizesinFig.3. Sincethetransittimescalevariesas1/w,the
falloffinthespectrumoccursathigherfrequenciesforsmallerw. Aroughestimateforthisfallofffrequencycanbeobtained
as 2w/v2D, where v2D is the average thermal velocity in two dimensions. For w = 1mm at 320K, 2w/v2D ∼ 8×10−6s,
consistentwith the resultsin Fig.3. Thetheoreticallycomputedspectrumis seen to agreewellwith theexperimentalresults.
Thereisasmalldeviationintheobservedspectrafromthetheoreticalspectra,shiftingthetransitnoisespectratowardshigher
frequency.Thiscanbeunderstoodastheresultsoftwocauses:Anatominthelaserbeamabsorbsandemitsphotons,muchlike
atwolevelsystem[8]. However,theatomcanalsospontaneouslydecaytoastatewhichisnotinresonancewiththeincoming
light,depopulatingthistwolevelsystem. Thiseffectivelyshortensthetransittime,leadingtoahigherfrequencyforfalloffin
the transitnoise spectrum. This suggeststhatthe deviationshouldbe largerforleveltransitionsthatare easierto depopulate,
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10-22
]
z
H
2/
A 10-23
[
m
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ct
e
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s
r
e
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P
10-25
104 105 106 107 108
Frequency [Hz]
FIG. 4: (Coloronline) Dependence of the spectrum, in particular the Rabi noise, on the light power. P = 57(red), 96(green), 195(blue),
429(magenta),870(cyan)µW,allforw = 0.19mm. ForlargerP,thespectrumhasalargermagnitudeandthepeakintheRabinoiseisata
higherfrequencyandsharper. TheoreticalcomputationsfortheRabinoiseareshown(blackdashed)forP = 57, 195, 870µW,whichare
consistentwiththemeasurements.
which we have further experimentally confirmed. Another reason is that the absorption by atoms reduce the light intensity,
modifyingthebeamprofileandeffectivelyshorteningthetransittime.Apartfromtheoverallmagnitude,theshapeofthetransit
noisespectrumisexperimentallyseentodependonlyweaklyonP,asisalsoevidentinFig.4.
ThespectrafordifferentlaserpowersareshowninFig.4,forthesamebeamsize. Zeemannoiseisnotpresentsincenostatic
magneticfieldhasbeenapplied.Rabinoisedevelopsapronouncedpeakforhigherlaserpowersandthepeakfrequencyincreases
withthelaserpower.ThepeakfrequencyisaroundtheRabifrequency,µE/h,thoughthefullbehaviorismorecomplex.Here,
µ is the dipolemomentofthe atom, E is the maximumelectricfield strengthand h is thePlanck’sconstant. Thepeakshave
widthsmainlyforthreereasons:First,theelectricfieldstrengthwithinthelaserbeamdependsonthelocation.Second,theatoms
havethermallydistributedvelocities,detuningthelightfrequenciesthroughDopplershifts. Third,apartfromRabiflopping,the
atomscan spontaneouslydecay, contributingto a finite width. The theoreticalpredictionincludingall these aspects hasbeen
computed and shown in Fig. 4 and is seen to agree well with the measured spectrum. The spectrum manifests the quantum
propertiesoftheunderlyingquantumfieldcorrelations,whichcanbeobtainedusingthequantumregressiontheorem[7,8,17].
Semi-classicalapproximation,suchasapplyingtheBlochequations,doesnotsufficehere,sinceifwetreatphotonsclassically,
thequantumfluctuationsintrinsictothefieldcorrelations areaveragedout. Therefore,notonlythequantumpropertiesofthe
atom,butalsothequantumpropertiesofphotonsarereflectedinthespectra.
In Fig. 5, static uniform magnetic fields with different strength are applied to the atoms, with other conditions fixed. As
the magnetic fields become stronger, we see that the Rabi noise is suppressed and the peaks in Zeeman noise become more
pronounced. Thepeakfrequenciesarewelldescribedby µB, whereµisthemagneticmomentofthe85RbatomandB isthe
magnetic field. Peaks at frequencies 2µB are also clearly visible. It is importantto note that in these measurements, only a
staticmagneticfieldandasinglelaserbeamisapplied,withnoadditionalexternalperturbationsdrivingthesystemawayfrom
equilibrium. The observationof Zeemaneffectin spontaneousnoise haspreviouslybeenattempted[12, 13] andseen[14, 15].
However,theshotnoisereductionwasnotappliedtheresothatthetheZeemannoisesignalswerenotmuchabovetheshotnoise
level.Inourexperiment,weobtainedZeemannoisewithhighcontrast,byaveragingouttheshotnoise. Thespectraobtainedin
thisworkfurtherrevealedthetransitandRabinoisewithin,whosemechanismcouldbeunderstood.Whileoutsidethescopeof
thiswork,theclearsignalsduetoZeemannoisehaveallowed ustomeasurepowerbroadeningandlightshiftsintheZeeman
frequency[18,19].
In thiswork, measurementsof spontaneousnoise spectrarequiredno externalperturbationexceptforthose essential to the
noise itself. We havebeen able to revealthe structureof the spectrum below the shotnoise level, which had previouslybeen
unseen.Thebasicaspectsofquantumphysicsunderlyingthisstructurewerethenelucidated.Weexpectthesespectratoprovide
anideallaboratoryforstudyingvariousquantumpropertiesofphotonatominteractions,withprecisionandwithoutambiguity.
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a.
[
m
u
r
t
c
e
p
s
er 100
w
o
P
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10
-4
10
4 5 6 7
10 10 10 10
Frequency [Hz]
FIG. 5: Spectra with Zeeman noise under static magnetic fields. Magnetic field strengths are B = 0(orange), 2B0(blue), 3B0(green),
5B0(red)withB0 = 3.4Gfrombottomtotop. SpectraforB = 2B0,3B0,5B0 wererescaledbyfactorsof 102,104,106 respectively, to
clearlyseparatetheplots. P =137µW,w=0.96mm.
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