Phase shift spectra of a fiber-microsphere system at the single photon level

0.36 MB·English

by  Akira Tanaka

Phase shift spectra of a
fiber–microsphere system at the single
photon level
AkiraTanaka1,2,TakeshiAsai1,2,KiyotaToubaru1,2,Hideaki
Takashima1,2,MasazumiFujiwara1,2,RyoOkamoto1,2,
1 andShigekiTakeuchi1,2,
∗
1
0 1ResearchInstituteforElectronicScience,HokkaidoUniversity,Sapporo001–0020,Japan
2 2TheInstituteofScientificandIndustrialResearch,OsakaUniversity,Mihogaoka8-1,
Ibaraki,Osaka567-0047,Japan
n
a *[email protected]
J

7
2 Abstract: We succeeded in measuring phase shift spectra of a micro-
sphere cavity coupled with a tapered fiber using a weak coherent probe
]
h light at the single photon level. We utilized a tapered fiber with almost
p no depolarization and constructed a very stable phase shift measurement
- schemebasedonpolarizationanalysisusingphotoncounting.Usingavery
t
n weak probe light (n¯ =0.41), we succeeded in observing the transition in
a
the phase shift spectrum between undercouplingand overcoupling(at gap
u
distances of 500 and 100 nm, respectively). We also used quantum state
q
[ tomographytoobtaina’purityspectrum’.Evenintheovercouplingregime,

the average purity was 0.982 0.024(minimum purity: 0.892), suggesting
1 ±
that the coherence of the fiber–microsphere system was well preserved.
v
8 Based on these results, we believe this system is applicable to quantum
9 phase gates using single light emitters such as diamond nitrogen vacancy
1 centers.
5
. © 2011 OpticalSocietyofAmerica
1
0 OCIS codes: (060.5565) Quantum communications; (140.3948) Microcavity devices;
1 (270.5565)Quantumcommunications.
1
:
v
i Referencesandlinks
X
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1. Introduction
Microsphereresonatorscoupledwithtaperedopticalfibers[1,2]havebeenattractinginter-
estbecauseoftheirultrahighqualityfactors[3],smallmodevolumes[4],polarizationselective
coupling[5],andhighlyefficientsingle-spatial-modeinput-output[6].Followingthepioneer-
ingdemonstrationsofcouplingbetweena microcavityanda taperedfiber[1, 2],applications
tolasers[7,8,9],biosensors[10],andnonlinearoptics[11]havebeenreported.Whenasingle
lightemitterisdepositedonthecavity,theinteractionbetweenthelightemitterandphotonsis
enhancedduetoconfinementinthesmallmodevolume.Thus,thissystemisanidealtestbed
forcavityquantumelectrodynamics(QED)experiments[12].Examplesofapplicationsinclude
a nonlinearsign shift gate [13, 14, 15] for photonic quantumcomputation[16] and quantum
memory [17] for long-distancequantum communication.We are currently interested in real-
izing such devices using a fiber–microspheresystem with a coupled single light emitter, e.g.
nitrogen-vacancy(NV)centersindiamond[18].
To characterize such photonic quantum devices, the probe input light has to be very weak
(i.e.,singlephotonlevel).Itisalsoessentialtoanalyzethephasechangeoftheprobelightto
evaluate the coherencepropertiesof quantum devices. The first step toward performingsuch
evaluationsisobservingthephaseshiftofanemptymicrospherecavitycoupledwithatapered
fiberusingaprobelightatthesinglephotonlevel[19].
A phase shift spectrum was recently obtained by interfering the bright coherent signal
outputfromafiber–microspheresystemwithareferencelight[20].However,itistechnically
verydifficulttostabilizetheopticalphasebetweensignalandreferencelightsthathavedifferent
opticalpaths,especiallywhenusingaveryfaintprobelight.
Inthisletter,wereportthemeasurementofphaseshiftspectraofafiber–microspheresys-
tematthesinglephotonlevel.Torealizestablephaseshiftmeasurementsofafiber–microsphere
system, we utilized a taperedfiber thathas almost no depolarization[5, 21]. We assume that
therearetwoorthogonalpolarizationmodes,XandY,inthetaperedfiber.Inthiscase,wecan
detectthephaseshiftduetotheresonanceofthemicrospherecavityasachangeinthepolar-
izationusingaprobelightthatisasuperpositionofXandY;forexample,polarizationmodeX
servesasareferencewhenonlypolarizationmodeYcouplestothecavitymode.Sinceneither
polarizationsexhibitanysignificantdepolarizationinthetaperedfiber[5],itisnotnecessaryto
performanystabilization,unlikein thepreviousexperiment[20].We succeededinobserving
a suddentransitioninthephaseshiftspectrumbetweenundercouplingandovercoupling[19]
usingaveryweakprobelightwithn¯=0.41,wheren¯istheaveragenumberofphotonsper10
ns(i.e.,typicaldecaytimeofadiamondNVcenter[18]).
Furthermore,weneedtocharacterizethedepolarizationofanoutputphotonfromafiber–
microsphere system. For this purpose, we performedfrequency-dependentquantum state to-
mography [22] of an output photon and derived the purity spectrum, which is important for
characterizingthedepolarizationofthesystem.Thismethodinvolvesreconstructinganoutput
stateasadensitymatrixrˆ(w )fromtheoutputspectraforthreedifferentpolarizationbasesfor
apolarizationinputconsistingofacombinationoftwoorthogonalpolarizationmodes,Xand
Y.Quantumstatetomographyisabletoaccuratelyestimatethecoherenceevenwhenthereare
statisticalfluctuations,whichissignificantwhenaveryweakprobelightisused.Fromrˆ(w ),
we obtainedpurityspectrathatindicatethedecoherence(i.e. thedepolarization)in the fiber–
microspheresystem.Theaveragepuritywas0.982 0.024(minimumpurity:0.892),indicating
±
that our fiber–microsphere system can maintain coherence and is thus suitable for quantum
coherentdevices.Suchanobservationofthepurityspectrumwillbeessentialinfuturecavity
QEDexperiments.
2. ExperimentalSetup
Ataperedfiberwasfabricatedfromacommercialsingle-modefiber(Thorlab,780HP).The
fiber was heated using a ceramic heater [23] and stretched at about 1330 C. To generate a
◦
large evanescent field strength, a tapered fiber was fabricated with a diameter of 410 nm in
the tapered region, as measured by scanning electron microscopy (Keyence, VE9800). The
total transmittance of the tapered fiber used in the experiment was 30% (the values of the
transmittancegivenbelowwerecompensatedbythisvalue).Amicrospherecavitywithastem
[24] was fabricated by melting the tip of a tapered fiber (fused silica) by focusing a carbon
dioxidelaserbeam(outputpower:4–8W;wavelength:1.55m m)onthetip.Opticalmicroscopy
observationsrevealedthatthemicrospherecavityhasadiameterof43.3m m.
Fig.1.showsthemeasurementsetup.Theprobelightisgeneratedbyatunablelaserdiode
(NewFocus,Velocity6312),whosefrequencywassweptbyafunctiongeneratorabouttheres-
onancefrequencyofthemicrospherecavity.Wesetthesweepingrangeto150and300MHz;
these frequencies correspond to the two coupling conditions. To calibrate the frequency, we
measuredtheabsorptionspectrumof a rubidiumgascell.The laseroutputwasattenuatedby
threeneutraldensityfilterstoaweakwithn¯ 1.Afterapolarizingfilter,ahalf-waveplatewas
≤
used to prepare a combinationof X and Y modesat the couplingregion in the tapered fiber.
Fig.1.Experimentalsetup.HWP:halfwaveplate;QWP:quarterwaveplate;PBS:polariz-
ingbeamsplitter;SMF:singlemodefiber;PD:photodiode;PZT:piezoelectrictransducer;
SPCM:singlephotoncountingmodule.
Twoquarter-waveplatesandahalf-waveplatewereusedforprecompensationofthebirefrin-
genceinasingle-modefiber.Theprobewasthencoupledtothesingle-modefiber,whichwas
connectedtothetaperedfiber.Theintensityoftheprobelightwasmeasuredattheconnector.
After thetaperedfiber,we used anadditionalpolarizationcontrollerfor postcompensationof
thebirefringence.Themicrospherewassetclosetothefiberinthetaperedregion.Tocontrol
thecouplingwiththecavity,thestemofthemicrospherewasfixedontoametaljigofathree-
axispiezoelectrictransducer(PI,NanoCube)toenablethedistancebetweenthecavityandthe
taperedfibertobevariedin20nmsteps.Theoutputwassenttothepolarizationmeasurement
region,whichconsistsofahalf-waveplate,aquarter-waveplate,apolarizingbeamsplitter,two
fiber-coupledsingle-photoncountingmodules(ParkinElmer,SPCM-AQR-14)andtwophoton
counters(StanfordResearchSystems,SR400).Theanalogoutputsofthephotoncounters(per
1 ms) were sent to an oscilloscope (Tektronix, DPO 4104). Photon counting spectra, which
indicatethecouplingofindividualpolarizationswith themicrospherecavity mode,weredis-
playedontheoscilloscope.Thetypicalphotoncountwas800countsper1msandthedarkcount
ofSPCMswerebelow0.3countsper1ms.Theoriginofthesignallevelontheoscilloscopewas
setwhentheprobelaserwasoff.
3. Gapdistancedependenceoftransmittancespectrum
Fig.2(a)showsatransmittancespectrumofthetaperedfiberobtainedwhenthemicrosphere
cavity was set close to the fiber. The input light power to the fiber–microsphere system was
10.5 pW, which corresponds to n¯ =0.41. The horizontal axis represents the detuning D f of
the probe light from the offset frequency f and the vertical axis indicates the transmittance
0
of the input probe light for X polarization. The offset frequency f was set to the frequency
0
atwhich theminimumwas obtained;the minimumtransmittanceisapproximately40%after
compensation. A transmittance of unity was defined as the transmittance obtained when the
microsphere was set very far from the tapered fiber. The dip in the spectrum is due to the
coupling[19] betweenthe guidedwave modeof the tapered fiber andthe whisperinggallery
modeofthemicrospherecavity[25].ThesolidcurveinFig.2(a)showsafitofthetransmittance
spectrumtothetransmittancederivedfromthecoupled-modetheorypresentedin[19],which
isrepresentedbyT(w )inthefollowingequation:
AAX((ww )) = 1−g (cid:20)1y−xxyee−ifif (cid:21)= T(w )e−iq (w ), (1)
0X p − − p
whereA (w )andA (w )correspondrespectivelytothecomplexamplitudeoftheprobelight
0X X
Fig.2.(a)Minimumtransmittance,(c)phaseshift,and(e)purityspectraobtainedatagap
distance of500 nm; (b),(d),and (f)spectraobtainedat agapdistance of100 nm. Solid
curvesin(a)to(d)aretheoreticalfitsbasedoncoupled-modetheory.
atanangularfrequencyw =2p (f +D f)inthetaperedfiberinX-polarizationbeforeandafter
0
fiber–microspherecoupling,g isthetotalphotonlossrateduetocoupling,f istheround-trip
phase in the cavity, and q (w ) is the phase shift of the probe light for X-polarization at w .
Wedefinedx=√1 g e r L andy=cosk ,wherer istheabsorptioncoefficientofthecavity
−
medium, L is the ca−vity length, and k is the coupling efficiency from the fiber to the cavity,
whichisdefinedastheoverlapintegralbetweenthetwoguidedwavemodes.
Fig. 3showstheminimumtransmittanceT oftheprobelightasafunctionofthedis-
min
tance d between the tapered fiber and the microsphere cavity. In this experiment, we used a
1 m Winputprobelightanda calibratedphotodiode,whichhasplacedafterthe taperedfiber.
The horizontalaxisrepresentsthe distanced betweenthetaperedfiberandthe surfaceofthe
microspherecavity. Theleft verticalaxisis the minimumtransmittanceT at the resonance
min
frequency(indicatedbytheredtriangles).Sincethespectrumdidnotchangewhenthemicro-
spherewasmovedtowardthetaperedfiber,webelievethatthemicrospherewasincontactwith
thetaperedfiberatd=0.Asthemicrospherewasmovedtowardthetaperedfiber(fromapprox-
imately800nm),T decreasesford>300nm.T hasaminimumneard=d 300nm.As
min min c
≃
d isfurtherreduced,T tendstoincrease,aspredictedbytheory[19].Theregionwhered is
min
larger(smaller)thand iscalledtheunder(over)couplingregime[19,26].Fig.2(a)wasobtained
c
atd=500nm,wherethereisundercoupling.Tostudytheeffectofthecouplingcondition,an-
othertransmittancespectrumwasobtainedatd=100nm(overcoupling)(seeFig.2(b)).Note
thatthescan rangeofFig. 2(b)(200MHz)ismuchlargerlargerthanFig.2(a)(60MHz).The
fullwidthathalfmaximum(FWHM)ofthedipinFig.2(b)isthreetimesgreaterthanthatin
Fig.2(a),duetothelargedampingofthecavitymodeinthetaperedfiberintheovercoupling
regime.
Forreference,wecalculatedthequalityfactorQ=2p f /d f fromtheFWHMd f ofthe
0
resonancedip.Qindicatesthedegreeofconfinementoftheprobelightinthecavity.Thecal-
culated Q for differentd is indicated by the green trianglesin Fig. 3. As a result, the quality
factor(rightverticalaxis)decreasescontinuouslyfrom3.0 107atd=800nmto1.1 106at
× ×
d=0nm,aspredictedbytheory.Theeffectofthesmallparasiticdipat 50MHzinFig.2(b)
−
isdiscussedbelow.
Fig. 3. Dependences of (red) minimum transmittance and (green) quality factor on gap
distance.BlackarrowsindicategapdistancesusedtomeasurethespectrainFig.2.
4. Phaseshiftspectrumatthesinglephotonlevel
We then measured phase shift spectra for both coupling conditions. As stated earlier, the
phase shiftis measuredusing referencepolarizationmodeY, whichdoesnotcouplewith the
cavity mode; in contrast, X mode couples with the cavity mode, giving a phase shift q (w )
X
describedbyeq.(1).WerepresenttheX(Y)-polarizedcomplexamplitudeoftheinputelectric
fieldbyA .Here,weconsiderthecasewhenonlyA coupleswiththecavityresonance
0X(0Y) 0X
modewithatransmittanceT T (w )andaphaseshiftq (w ).T isthetotaltransmittanceof
all X X all
bothpolarizationmodes.Theoutputelectricfieldofthefiber–microspheresystemthenobeys
thefollowingtransformation:
A (w ) A (w )= T T (w )eiqX(w )A (w ),
0X X all X 0X
→
A (w ) A (w )p= TpA (w ) (2)
0Y Y all 0Y
→
p
Thephaseshiftq (w )isobtainedasfollows:
X
S (w )
q (w )=Tan 1 3 ArgA +ArgA (3)
X − (cid:18)S (w )(cid:19)− 0X 0Y
2
where S (w )=I (w ) I (w ) and S (w )=I (w ) I (w ) are Stokes parametersat the fre-
2 P M 3 R L
quencyw .I (w )andI−(w )arerespectivelytheinte−nsitiesofdiagonal/antidiagonalpolariza-
P M
tionmodes,andI (w )andI (w )arerespectivelytheintensitiesofright/leftcircularpolariza-
R L
tion modes, where X and Y are taken to be horizontaland vertical polarization modes. Note
thatI (w )isgivenby[27]
P
A (w )2
IP(w )= | P2h |
1 A (w )+A (w ) 2
X Y
= (4)
2h (cid:12) √2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where h = m /e is vacuumimpedance.(cid:12)Similarly,I (w )(cid:12), I (w ), and I (w ) are foundus-
0 0 M R L
ingA (w )=p(A (w ) A (w ))/√2,A (w )=(A (w ) iA (w )/√2,andA (w )=(A (w )+
M X Y R X Y L X
iA (w )/√2. Figs.2−(c)and(d)arephaseshiftspectra−obtainedforundercoupling(Fig.2(a))
Y
andovercoupling(Fig.2(b)),respectively.Theverticalaxisindicatesthephaseshiftq (w ).We
X
subtractedaconstantphaseshiftduetothesmallbirefringenceofthefiber(0.9radinFigs.2(c)
and(d)).Thephase shiftasymptoticallyapproaches0rad in thehighlydetunedregionin the
undercouplingcondition,whereasitapproaches p intheovercouplingcondition( 2.9radat
± ±
100MHz).Thus, the transition of the phase shift spectrum from undercouplingto overcou-
±
pling[20,21]isclearlyobservedusingaprobelightwithn¯=0.41.Notethattheparasiticdip
at 50MHzinFig.2(b)causedasmallstep-likechangeat-50MHzinFig.2(d).Thechange
−
is small(0.3rad)maybebecauseitisstill in theundercouplingconditionwith smallcoupling
efficiencyk .
5. Purityspectrumofthefiber–microspheresystem
We next investigated the depolarization, or dephasing, in this system using quantum state
tomography [22]. This method has the advantage that it estimates the most probable state,
whichisphysicallymeaningfulevenwhentherearestatisticalfluctuationsduetofinitephoton
counting,whichiscriticalwhenn¯ 1.Thisenablesthe’purity’tobeestimated,whichcanbe
≤
usedtoevaluatethedepolarization.Thedensitymatrixofasinglepolarizationqubitisobtained
bymeasuringStokesparametersS ,S ,S ,andS oftheoutputprobelight.Thedensitymatrix
0 1 2 3
rˆ forthesingle-photonoutputstateisgivenby
1 S S S
rˆ = Iˆ+ 1Zˆ+ 2Xˆ 3Yˆ . (5)
2(cid:18) S S −S (cid:19)
0 0 0
Here,IˆistheidentitymatrixandXˆ,Yˆ,andZˆ arethePaulispinmatricesinthebasisofX-and
Y-polarizationmodesofasinglephoton.Inquantumstatetomography,thedensitymatrixrˆ(w )
is estimated using the maximum-likelihoodestimation methodwith the differentialevolution
algorithminMathematica7.0(scalingfactor=1.5)undertheconstraints(a)Trrˆ(w )=1and
(b)rˆ(w ) 0;theformerconstraintstatesthatthetotalprobabilityisunityandthelattercon-
≥
straintstatesthattheprobabilitiesarenotnegative.Weperformedthistomographyfordifferent
frequenciesw and obtained the frequency-dependentr (ˆw ). From r (ˆw ), the purity spectrum
p(w )isobtainedasfollows.
p(w )=Tr[rˆ(w )2]. (6)
Thispurityisunitywhenacompletelypolarizedphotonexperiencesnodepolarization,whereas
itis0.5whenthephotoniscompletelydepolarized.Whentheabsolutedetuningfromthereso-
nancefrequencyisrelativelylarge(D f >20MHzinFig.2(e)),thepuritywas0.999 0.004
| | ±
(basedonanaverageof200points)forundercouplingand0.998 0.004forovercoupling.This
±
highpurityismainlyduetotheoptimizationprocedureandtheintrinsicnondepolarizationof
thetaperedfiber.Inundercoupling,thepurityis0.992 0.016withinthebandwidthaboutthe
resonance(D f <15MHz)(theminimumis0.913).In±overcoupling,thepurityis0.982 0.024
withintheb|and|widthabouttheresonance(D f <50MHz)(theminimumis0.892.N±otethat
| |
thepurityishighforalldetunings.Thepossiblereasonsforthesmalldegradationinthepurity
spectra aboutthe resonanceis consideredto be the intrinsicdepolarizationofthe system and
spectraljitter.Thus,usingquantumstatetomography[22],wehavedemonstratedthatphotons
thatpassthroughthefiber–microspheresystemexperienceatinydepolarization.Basedonthis
experimentalresult,whichindicateshighpurityforbothcouplingconditions,weconsiderthat
thefiber–microspheresystemissuitableforapplicationsinvolvingcoherentquantumdevices.
Our recent experiments have demonstrated that it is possible to evaluate photonic quantum
devicesusing,forexample,diamondNVcenterswithasimilarsetup[12]
6. Conclusion
Wesucceededinmeasuringphaseshiftspectraofamicrospherecavitycoupledwithatapered
fiber usinga weakcoherentprobelightatthe single photonlevel.We utilizeda taperedfiber
with almostnodepolarizationand constructeda verystable phase shiftmeasurementscheme
basedonpolarizationanalysisusingphotoncounting.Usingaveryweakprobelight(n¯=0.41),
we succeeded in observing the transition in the phase shift spectrum between undercoupling
and overcoupling(at gap distances of 500 and 100 nm, respectively).We also used quantum
state tomography to obtain a ’purity spectrum’. Even in the overcoupling regime, the aver-
age purity was 0.982 0.024 (minimum purity: 0.892), suggesting that the coherence of the
±
fiber–microspheresystemwaswellpreserved.Basedontheseresults,webelievethissystemis
applicabletoquantumphasegatesusingsinglelightemitterssuchasdiamondnitrogenvacancy
centers.
Acknowledgements
Thecurrentworkwassupportedinpartbytheprogram“R&DSupportSchemeforFundingSe-
lectedITProposals”oftheMinistryofPublicManagement,HomeAffairs,PostsandTelecom-
munications, a Grant-in-Aid from the Japan Society for the Promotion of Science, the 21st
CenturyCOEProgram,CRESTProject,JapanScienceandTechnologyAgency,JSPS-Grantin
AidforScientificResearchonInnovativeareas‘QuantumCyberneticsf,FundingProgramfor
World-LeadingInnovativeR&DonScienceandTechnology,andSpecialCoordinationFunds
forPromotingScienceandTechnology.