On the Biphoton Wavelength
P. H. SoutoRibeiro
InstitutodeFsia,
UniversidadeFederaldoRiode Janeiro,
CaixaPostal 68528,RiodeJaneiro,RJ,21945-970,Brazil
Reeivedon30January,2001
Wereportonanexperimentshowingthatthewavelengthofabiphotonislearlydependentonthe
measurementshemeandonthewayitisdened. Itisshownthatitantakeanyvalue,depending
ontheontrolof theinterferometerphasedierenes. It ispossibleto identifythe interfereneof
thesingleandtwo-photonwavepaketsaspartiularasesofthemostgeneralinterfereneproess.
Thevariablewavelengthhasnoimpliationontheenergyoftheindividualphotonsneitheronthe
total energyofthebiphoton.
I Introdution
Theuseofinterferometryformeasuringthewavelength
of a radiation eld is probably oneof its older
appli-ations. Nowadays the onept of interferometry has
beenextended. It has beome possibleto observe
ex-perimentallytheinterfereneforonepartileandeven
formultipartilewaveelds. Thetwo-photoneld
pro-dued in the parametri down-onversion, have been
extensivelyutilized in many ofthe so-alled
multipar-tileinterferometryexperiments[1,2℄.
Inthisnewtypeofinterferene,itisnotpossibleto
ignorethequantumaspetsoftheeletromagnetield.
In quantum interferometry, it is also possibleto
asso-iateinterferenepatternstothewavelengthofaeld.
For single photon elds, lassial and quantum
inter-pretations of the interferene experiments lead to the
same wavelength. For multi-photon or multi-partile
elds however, the wavelength an be dependent on
thewaythemeasurementisperformed,andalassial
interpretation is no longer possible. Thinking of
two-photon wavepakets, for example, if we anmakethe
twophotonstraveltogetherthroughaninterferometer
astheywereontainedinonepaket,weanmeasurea
wavelengthorrespondingtoanentitywiththeenergy
twotimeslargerthanthesinglephotonone[3,4℄. This
oneptisquitegeneralinquantumphysisanditan
beextendedto anypartile oreld and theDeBroglie
wavelengthwillbeassoiatedtothetotalenergyofthe
system.
Inthis paperwestudythetwophotoninterferene
fromthepointofviewofthemeasurementofthe
wave-length. We present an experiment whose
ongura-tionis apable to produequantum interferene
with-out the use of material interferometers, in the sense
It onsists of a transverse version of interferometers
of the type of Mandel's[5℄ and Zeilinger's[6℄. It is
alsosimilartotheinterferometerpresentedbyKlyshko
et al. in Ref.[7℄, but without the double-slits and
with the possibility of deteting signal and idler
pho-tons in ompletely independene, asit will be shown.
This is the main dierene from previous transverse
interferometers[8, 9, 10, 11℄. Another experiment
re-entlyperformedbyFonseaetal.[12℄,utilizesthesame
priniple for measuring a non-loal wavelength for a
two-photon wavepaket. The onguration presented
here is similar to that presented by White et al.[13℄
for produing polarization entangled states with high
intensities, however in our ase the polarization state
is not entangled. Twin photons from the parametri
down-onversionproessareused. Thesephotonshave
beenalledbiphotonsasareferenetotheirstrong
or-relationat thequantum level. Theinterferenefringes
are obtainedby measuringoinideneountsand the
frequeny of theosillationof the patterns are
assoi-atedtowavelengthsforthebiphotons. Itisshownthat
this frequenyanbearbitrarily varied,depending on
the way the measurements are performed. It is also
shown,that themeasuredwavelengthsanbeassigned
tosingleandtotwo-photonwavepakets,fortwokinds
of measurement. Attempts to onneting the
osilla-tionfrequenyandthebiphotonwavelengthsforother
kindsofmeasurementsindiate thatspeialaremust
betakenindeningandmeasuringwavelengthsin
mul-tipartileinterferometry. Ontheotherhand,this
inter-ferometeranbeusedintheprodutionand
manipula-tion ofposition entangledstatesand has potentialfor
appliationsin measurementsofdirationindexes. A
single mode quantum theory is enough to explain the
behaviorof the frequeny of the patterns and it is in
II The two-photon interferene
Let us analyze the situation skethed in Fig. 1. The
pump laser passes throughtwo nonlinear rystals,
la-beled rystal 1 and rystal 2. Twin photons an be
produedin eah oneofthe rystals. Signal and idler
photonsproduedinrystal1arediretedtodetetors
A andBrespetively, sothat oinidenebetween
sig-nal andidlerhannelsanbemeasured. Supposethat
degeneratephotonpairsproduedinrystal2,analso
be direted to the same detetors. This ondition is
simplyfullledbytiltingrystal2relativelytothe
ver-tial axis.
Figure1. Outlineoftheexperiment.
Thesituation desribedaboveissuitablefor
quan-tuminterferene.Notethatthetimeofemissionof
pho-tonpairsannotbespeied,sineitisaspontaneous
emissionandtheoherenelengthofthepumplaseris
largerthanthedistanebetweenrystals. Inthisase,
oinideneountsproduedbyphotonpairsoriginated
in rystal 1are indistinguishablefrom thoseof rystal
2. Interferenefringesintheoinideneountingrate
anbeobserved,aslongasthephasedierenebetween
thesetwoprobabilitiesisvaried. Aseahrystalworks
likeanextendedsoure,nointerfereneisobservedfor
theindividualintensities.
Thisinterfereneproessanbedesribedina
sim-pliedform withtheuseof amonomodequantum
ap-proah. It isenoughto explainthemain propertiesof
the oinidenepatterns, inludingthe eetive
wave-length. However,fortakingintoaountforthedegree
of oherene and its onsequenes in the visibility of
the fringes, a multi-mode theory would be neessary.
In this work, wewill restrit ourselves to the simpler
ase.
The quantum state of the eld produed by both
rystalsisgivenby:
j i= 1 p 2 (j1i i1 j1i s1 j0i i2 j0i s2 +j0i i1 j0i s1 j1i i2 j1i s2 ): (1)
The eletri eld operators for signal and idler
E (+) A =a 1s e i( 1s +kr 1s ) +a 2s e i( 2s +kr 2s ) ; (2) E (+) B =a 1i e i( 1i +kr 1i ) +a 2i e i( 2i +kr 2i ) ; where jx
isthephaseoftheeldattheemissionpoint,
withj=1,2 andx =s,iandkisthewavenumberfor
bothsignalandidlermodes. r
jx
isthedistanebetween
rystalj andthedetetoratthex side.
The oinideneountingrate anbe easily
alu-lated:
C = jE (+) A E (+) B j ij 2 (3)
= 2[1+os(
1i + 1s +kr 1i +kr 1s 2i 2s kr 2i kr 2s )℄:
From the phase mathing onditions we havethat
1i + 1s = 1p and 2i + 2s = 2p , where 1p and 2p
are the pump laser phases at rystals 1and 2
re-spetively. Wesee thatoneonditionforobserving
in-terfereneisthat
1p
2p
=onst. Thatistosaythe
oherenelengthofthepumplasermustbelargerthan
thedistane betweenrystals.
Eq. 3anbeputin theform:
C=2f1+os[k(Æ
i +Æ
s
)+℄g; (4)
where
=
1p
2p +k(r
1i +r
1s r
2i r
2s ); (5)
r
1i = r
1i +Æ 1i ; r 1s = r
1s +Æ 1s ; r 2i = r
2i +Æ 2i ; r 2s = r
2s +Æ 2s ; Æ i = Æ 1i Æ 2i ; Æ s = Æ 1s Æ 2s :
With theaidofEqs. 3and4itislearlyseenthat
theinterferenefringes aresensible to phasesthat
de-pend on thepaths from rystals1and 2 todetetors.
ItisworthnotingthatphaseÆ
i
anbevaried
indepen-dently fromÆ
s
. Displaing signaloridler detetorone
anvary eah one of these phases. Considerthe ase
whereÆ
i =Æ
s
. Inthis ase,thevariablephasein Eq.
4anbewrittenas:
C=2f1+os[k(1+)Æ
s
+℄g: (6)
Theparameterintheaboveequationanassume
any value and we demonstrate experimentally in this
III The Experimental Set-up
The experimental set-up shown in Fig. 1 was
imple-mentedwith a.w. He-Cd laseroperatingat 442nm.
Theoutput powerwasabout 200mW.It was used to
pump two1mlongLiIO
3
rystals. Thedistane
be-tween rystalswas around 2 m and the distane
be-tweenrystal1andbothdetetorsAandBwasnearly
1.5m. Crystalswereutforollineardegenerate
down-onversion,thatistosaytheoptialaxisat37.3degrees
relativelytothe input/outputfaes. Inorderto make
degeneratebeamsemergefromtherystalatangles
dif-ferentfromzero,itwasneessarytotiltitslightly
rela-tivelytothevertialdiretion.Inourase,thediretion
ofpropagationofthetwinbeamswerenearly7degrees
with the pump beam diretion. The output angle of
the beams produed in rystal 2 were slightly bigger,
in ordertoahievesuperposition atthedetetorswith
thebeamsoriginatedatrystal1.
The detetors were avalanhe photodiodes inside
photon ounting modules (SPCM-AQ/ EG&G). The
inoming light passes through a small vertial slit
(about0.5mm) andanAR oatedlens with25.4mm
foal length,before reahing theativedetetion area
of about 0.2 mm. The modules are mounted on
X-Y translation stages, sothat the transverse detetion
planeanbesannedwithupto5m resolution. The
outputpulsesweresenttoounters(SR-400andSR-620
/SRS),wheresinglerateswereountedandthe
oini-denelogiwasimplemented. Counterswereontrolled
byamiroomputerwhihwasusedtosavedata.
IV Experimental Results
We havearried outoinidene interferene patterns
for several values of the parameter in Eq. 6. For
doingso,wehavesimplyhangedtherelative
displae-mentbetweendetetorsA andB.
For=0, detetorBwaskept xed,while
dete-torAwassannedtransversallyinthehorizontalplane.
The single photonand the oinidene ountingrates
were thenregistered. The resultsare shownin Fig. 2.
Whilethesinglesshowanearlygaussianprole,the
o-inidenesshowinterferenefringes. Thevisibilityand
thewavevetoroftheoinidenefringeswereobtained
byanonlinearurvettingwiththeusualfuntionfor
thedouble-slit interferene. Thewavevetorfor=0
will be alled k
0
. It isassoiatedto thesingle photon
interferene. Notethattheabsolutevalueofk
0
depends
onthegeometry. Intheanalogywith adouble-slit
ex-periment,eahrystalorrespondtooneslit. However,
thelightisnotemittedinthediretionthatwould
or-respond toaentral(zeroorder)maximum. It is
emit-tedinadiretionorrespondingtoahigherorder. This
meansthattheassoiatedwavelengthismultipliedbya
largeinteger,whihis notimportantinthis work. For
traryunits. Theproedureisrepeatedkeepingdetetor
A xed and sanning detetor B. The result shows a
prolesimilartothatofFig. 2,showingthesymmetry
between sans with oneof the detetors xed. These
patternsanbeinterpretedassinglephotonwavepaket
interfereneones. Thisisaonsequeneofthefatthat
onlysignaloridlerpathsarehanged,whenonlysignal
oridlerdetetorsaremovedindividually.
Figure2.Singlesandoinideneprolefor=0.Detetor
AissannedanddetetorBisxed.
For = + 1, detetors A and B were
simultane-ously displaedwith thesame veloity. This is
equiv-alent to sayingthat the detetorswere displaedwith
equal steps. The resultis shown in Fig. 3. The
on-vention usedestablishes that thepositive impliesin
additive phase shifts in signal and idler sides. In the
experiment, this ondition was ahieved bydisplaing
both detetors towards the pump beam. The
visibil-ity and the wavenumber, that gives thefrequeny of
the osillations,wereset as free parametersin the
t-tings. As a result, we observethat the wave number
for the urve in Fig. 3 is two times the one in Fig.
2. k
+1
(signal side)=k
+1
(idler side)=2k
0
. This was
predited by Eq. 6 for =+ 1and itworks alsofor
= -3. The pattern of Fig. 3 anbe interpreted as
atwo-photonwavepaketinterferene. Wewilladdress
thispointagainin nextsetion.
Figure3. Coinidene prolefor =+1. DetetorAand
Baresannedsimultaneouslywiththesameveloity.
For = + 1
2
so that detetors get together to the position
orre-spondingtotheoinidenepeakdetetioninprevious
measurements. With this proedure it is possible to
take are for the symmetry of the interferene urve.
Theresultsareshownin Fig. 4. InFig. 4athe
oini-dene ountsare plotted as afuntion of thedetetor
ApositionandinFig. 4b,thesameoinideneounts
are plotted as a funtion of the detetor B position.
This is neessary now, beause the displaements are
dierent, we do not have a ommon oordinate
any-more, as in previous ases. The tting of the urves
leadto k
+ 1
2
(signalside)= 3
2 k
0
, whih orrespondsto
=+ 1
2 andk
+ 1
2
(idlerside)=3k
0
whihorresponds
to
i
=+2whenthephaseshiftiswrittenintermsof
theidleroordinates. ThisisinagreementwithEq. 6.
Figure4. Coinideneprolesfor=+ 1
2
. DetetorAand
Baresannedsimultaneouslywithdierentveloities.
For = -1
2
,detetor Aisstilldisplaedtwotimes
slowerthan detetor B, but now the negativesign
in-diates that the sense of one of the displaements is
hanged. Detetor A movestowardsthe pump beam
whiledetetorBmovebakwardsthepumpbeam. The
resultsareshownin Fig. 5. InFig. 5atheoinidene
ountsareplottedasafuntionofthedetetorA
posi-tion whilein Fig. 5b theyareplotted asafuntionof
thedetetor Bposition. Fromthepointofviewof
de-tetorA= -1
2
andthewavevetorisk 1
2
(idlerside)
= 1
2 k
0
. FromthepointofviewofdetetorB
s =-2
andk 1(signalside)=k
0 .
Figure5. Coinideneprolesfor=+ 1
2
. DetetorAand
Baresannedsimultaneouslywithdierentveloities.
V Disussion
FromEq. 6and experimental resultspresentedin
pre-vioussetion,itislearthatthewavelengthassoiated
totheoinidenepatternsanbeontinuouslyvaried
andthat itanassumeanyvalue,evenafration ora
multipleofthesinglephotonone. Thisfat anbe
ex-plainedbythephaseentanglementbetweensignaland
idlerphotons. Italsoseemsthatithasnoimpliations
ontheenergyoftheindividual photonsneitheronthe
totalenergyofthebiphotons. Itislearfromall
experi-mentsutilizingtwinphotonsfromtheparametri
down-onversion,thattheproessbehindallquantumeets
isentanglement. Theentanglementisaonsequeneof
aproessthat weouldalltransferof spetrumfrom
the pump beam to the biphotons. A partiular ase
ofthat isthetransferofangularspetrumdesribedin
Ref. [11℄ dealing with the transverse degrees of
free-domoftheeld. Intheexperimentwepresenthere,it
is nie to be able to understand the main features in
onnetionwith entanglementand transfer of angular
spetrum for a simple ase utilizing a monomode
ap-proah. Thisisoneofthevirtuesoftheinterferometer
presented.
It is interesting however, to analyze some possible
interpretations. When=0,onedetetorisxedand
anbeviewed asa singlephoton wavepaket
interfer-enebeausethephasedierenedependsonlyon
tra-jetoriesforthesamephoton. When=+1or=-2,
bothdetetorsaremovedsimultaneouslywiththesame
veloity. Inthisase,theexperimentorrespondstoan
unfolded versionof atwo-photon wavepaket
interfer-ene. Note that inoneof thepreviousexperiments[4℄,
it was neessary to prepare the state of the eld by
manipulating thepumpbeam,in orderto avoidsingle
photoninterferene. Inthepresentonguration,eah
rystal plays the role of one slit in the analogy with
a double slit experiment. However, eah twin photon
pairisalways(the probabilityismuh bigger)emitted
bythesamerystalandnever(theprobabilityismuh
smaller)bydierentones. Thisorrespondsto having
bothphotonspassingthroughoneoftheslitsandnever
onephotonthrougheahslit. Forthisreason,itisnot
neessarytohange thepump laserbeamprole. But
theanalogyisomplete.
When=+ 1
2
,forexample,itisnotpossibleto
as-signsomephysialmeaningtothewavelengthobserved
by one of the detetors anymore. For the same set
of measurements, the wavevetoris three times larger
than the singlephoton one(orresponding to a
wave-lengththreetimessmallerthanthesinglephotonone)
fromthepointofviewoftheonjugateddetetor. This
resultshowsthat assigning awavelengthto something
we all biphoton an be dangerous. The energy of
eahindividualphotonisnothangedduringthe
inter-ferene proess, neither during the detetion proess,
andthedisussionaboutthisapparentwavelengthmay
turn into speulation. However, diration properties
are atuallyhangedand that mayhaveonsequenes
in imagingand other appliations. Some possible
ap-pliations an be envisaged. For example, the
oin-idene patterns ould be used for measuring the
re-fration index forsomematerial plaedin the pathof
the beams. Dierential measurements ould be
per-formed in a ollinear onguration with both beams
passing through the sample. We ould also think of
twinbeamswithrossedpolarizationsusedfor
birefrin-genemeasurements. Theinterferometerisquiteuseful
forfundamental researh onthe prodution,detetion
andmanipulationof entangledstates. Reently,itwas
used in our laboratory for produing position
entan-gledstateswihwereoupledtopolarization
entangle-ment sothat wewere ableto measure theplarization
entanglementthorughpositioninterferene[14℄.
Exper-imentsurrentlyinprogressshowthatitisalsosuitable
for studying quantum distilation, puriation and
de-oherene.
VI Conlusion
A two-photon interferometer without double-slits and
withoutbeam-splitters ispresented. The frequenyof
isvaried. Thesefrequeniesareassoiatedtothesingle
and to the two-photon wavepakets. Frequenies that
arefrationsandmultiplesofthesinglephotononeare
observed,inagreementwiththeory. Thesevariable
fre-quenyosillationsarewellunderstood intermsofthe
quantum theoryandtwopartileentanglement. Some
appliations in the new eld of quantum imaging and
measurementsof refrationindexesarepromising
pos-sibilities. Theinterferomenter is being usedin
experi-mentsforproduingandmanipulatingentangledstates.
Theentangledstatesareessentialin theeld of
quan-tum information.
Aknowledgment
We thank Drs. Carlos Monken and Sebasti~ao de
Padua for lending one of the detetors and also for
many helpfull disussions. Finanial support was
pro-videdbyBrazilianageniesCNPq,PRONEX,FAPERJ
andFUJB.
Referenes
[*℄ Correspondingauthor.
E-mailaddress: phsrif.ufrj.br
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