Cascaded cold atomic ensembles in a diamond configuration as a spectrally entangled multiphoton source

0.63 MB·

by  H. H. Jen

Cascadedcoldatomicensembles ina diamondconfiguration as a spectrally entangled multiphoton
source
H. H. Jen1,∗
1Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
(Dated:March27,2017)
Wetheoreticallyinvestigatethespectralentanglement of amultiphotonsource generatedfromthecascade
emissionsinthecascadedcoldatomicensembles.Thisphotonsourceishighlydirectional, guaranteedunder
thefour-wave mixing condition, and isalsohighly frequency-correlated due tofinitedriving pulse durations
and superradiant decay constants. We utilize Schmidt decomposition to study the bipartite entanglement of
the biphoton states projected from the multiphoton ones. This entropy of entanglement can be manipulated
7
bycontrollingthedrivingparametersandsuperradiant decayrates.Moreovertheprojectedbiphotonstatesin
1
the cascaded scheme can have larger entanglement than the one produced from only one atomic ensemble,
0
2 whichresultsfromlargercapacityinmultipartiteentanglement.Thiscascadedschemeenablesamultiphoton
sourceusefulinquantuminformationprocessing.Italsoallowsforpotentialapplicationsinmultimodequantum
r
communicationandspectralshapingofhigh-dimensionalcontinuousfrequencyentanglement.
a
M
I. INTRODUCTION capacityincontinuousvariables[23]isalsopresentinvarious
4
degrees of freedom, for example, the transverse momentum
2
Quantum computation and quantum information process- [24, 25], space [26], time [27–29], and orbital angular mo-

] ing [1, 2] promise to outperform classical implementations mentaoflight[30–37].Intheperspectiveofspectralshaping,
h
p for efficient algorithm, secure communication [3], and gen- in Ref. [21], we have investigated the frequency-entangled
- uine teleportation of quantum states [4]. This superiority of two-photon state in the scheme of multiplexed cold atomic
nt quantumnesseven envisionsa quantumnetwork or quantum ensembles. This two-photonstate is generated from the cas-
a internet [5] which links variousquantum systems to process cadeemissionsinthediamond-typeatomicconfiguration.Its
u tasksthatareintractableinclassicalregimes.Thesequantum upperandlowertransitions(alsodenotedassignalandidler)
q systemsinclude,tonameafew,photonicqubits,trappedions, are respectively in the telecom and infrared bandwidths. In
[ atomic ensembles, and solid state systems [4, 6], in which themultiplexedscheme,amultiplecommonexcitationpulses
2 unfortunatelybothstrengthsandweaknessescoexist[6].The areappliedsimultaneouslytotherespectiveatomicensembles
v factthatthereisnoprefectquantumsystem (highefficiency, along with individually controlled frequency shifters for the
7 longcoherencetime,strongcoupling,andscalability,forex- cascade emissions. Under the condition of weak excitations
6 ample)demandsan integrabilityof the well-controlledinter- asinthequantumrepeaterprotocol[8],weeffectivelycreate
5 facesbetweenthesequantumsystems. a biphoton state with additive spectral functionswhere indi-
7
0 A good quantum interface [7] involves an efficient gener- vidualfrequencyshiftsforthesignalandidlerphotonscanbe
. ation, distribution, and storageof quantuminformation.One manipulated.Thereforethespectralpropertyofthisbiphoton
1 manifestation of these functionalities is long-distance quan- state can be shaped in either modifying the central frequen-
0
tum communication [8] using cold atomic ensembles for a ciesorcontrollingthephasesofthephotons[22].Wefurther
7
1 quantumrepeater[9, 10]. The buildingblocksfor this quan- analyzetheentropyof entanglementof themultiplexedtwo-
: tumrepeaterprotocolthereforerelyona generationoflight- photonsourcebySchmidtdecomposition,whichgrowsasthe
v matterentanglement[11,12] andstorageofit[13–16]using numberofmultiplexedensemblesincreases.
i
X Λ-type atomic configurations, making the atomic ensemble
Here in contrast we propose to generate a multiphoton
r (mostlyalkalimetals)agoodcandidateforquantumnetwork.
a However one drawback for the fiber-based quantum infor- source out of the cascade emissions from the cascaded cold
atomicensembles.Wetakeatwo-photonsourceofquantum-
mation transmission is the attenuation loss for D line tran-
correlatedsignal and idler photonsemitted from a diamond-
sitionsinalkalimetals.To reacha minimal-lossopticalfiber
type atomic configuration, as an initial seed for multiphoton
transmission, telecommunication (telecom) bandwidth from
generation in the cascaded atomic ensembles. This is moti-
diamond-type atomic configurations [17–19] serves the pur-
vatedby the multiphotongenerationfromspontaneouspara-
pose and allows for an optimal operation in long-distance
metric down conversion (SPDC) where multiple cascaded
quantumcommunication.
nonlinear crystals are pumped sequentially [38]. Similar to
In addition to the advantage of the telecom bandwidth in
the sequential pumping scheme in SPDC, we can take the
fibertransmission,continuousfrequencyentanglementofcas-
infrared (telecom) photonof the biphotonsource along with
cadeemissionsfromdiamond-typeatomictransitionscanbe
a telecom(infrared)drivingfield to generateanothersponta-
generated [20] and spectrally shaped [21, 22] to create a
neously emitted cascade emissions satisfying the four-wave
highly entangled biphoton source. This high communication
mixing condition. In this way we enable a k-photon source
withminfraredandntelecomphotonsfulfillingk=m+n.
Sincethelastcascadedatomicensemblealwaysproducetwo
∗[email protected] photonsintheinfraredandtelecombandwidthsrespectively,
2
(a) actionpictureusingthesamenotationof[21],
(cid:507)
2 2
(cid:525)a (cid:537)a AE (cid:537)i AE' (cid:525)b (cid:507)1 (cid:525)b 1 aˆ†s 3 VI =− ∆m N |miµhm|− (cid:18)Ω2mPˆm† +h.c.(cid:19)
(cid:537)s (cid:537)b(cid:525) (cid:525)a aˆ†i mX=1,2 µX=1 mX=a,b
b 0
(cid:524) (cid:118)aˆ†aˆ†aˆ† 0 −i (cid:26) gmaˆkm,λmQˆ†me−i∆ωmt−h.c.(cid:27), (1)
(b) s s' i' mX=s,i kmX,λm
AE'' wherewe set~=1forsimplicity,anddenoteλm asthepo-
larizationsofphotons.Thecollectivedipoleoperatorsarede-
(cid:525)
(cid:525)(cid:58)aa AE (cid:525) AE' (cid:525)b (cid:524) (cid:118)aˆb†saˆ†s'aˆ†s''aˆ†i'' 0 fiQRˆna†sebd≡ifarPseqPµˆuae†|2n≡icµiehPs3|Ωeµia|k(1sbi·)rµµfho,0ra|etnwidkoaQ·ˆpr†iµu,m≡PpˆbP†fie≡µld|3Ps.iµCµhe|02n|ietrµiakhli1·f|rreµei,qkwub·erinµth-,
b
cies and wavevectors of these four fields are ω and
a(b),s(i)
k respectively.Signalandidler photoncouplingcon-
a(b),s(i)
cFcisoaIcdGnrefi.egae1utm.erdia(stCbsioiyoonlnto.wsrIonionptnhutlheimneiepnc)safiePsetcrlooaddpfseo(Ωdase)asd,tcbhhmweeuismtilhgteinspwiahnliogtahtlnoeadnandisddioaluetmwrrcopoenh-dpfor-htotooymntpoepnthaadietreotcaˆmua†snsi,ci-- sgretsac(nit)itosfnoarorefctoghsne(ciqi)suewahnexetirpzereedwssbeioohsnaosvn.eiǫckafimbes,λlodmrsbeaˆidskmt(hǫ,λekmmp,,oλalmanrdi·zddˆaˆm∗mtio)insintdhtioe-
unit direction of the dipole operators. The detuningsare ∆
ings ∆1,2, driving theatomic ensembles (AEs). For demonstration 1
we plot (a) three and (b) four-photon generations in the cascaded = ωa ω1 and ∆2 = ωa + ωb ω2, and for later conve-
− −
scheme. In (a) aˆ†i is sent to AE’ along with a pump field Ωb, and niencewedefine∆ωs ≡ωs −ω2+ω3 −∆2 and∆ωi ≡ωi
thenisconverted intoapairofphotons aˆ†s′ andaˆ†i′.Theexcitation − ω3 with the atomic level energies ω1,2,3. The upper level
and photon emission angles are denoted as θa,b,s,i with respect to |2iallowsforatelecomwavelengthwithin1.3-1.5µm[17]if
thelong-axisof AE.In(b) aˆ†i and aˆ†i′ aresent sequentiallytoAE’ 6S1/2,7S1/2,or4D3/2(5/2)levelsareconsidered.
and AE” interacting with pump fields Ωb, and then effectively are With the Hamiltonian of Eq. (1), we can construct self-
convertedintothreephotonsaˆ†s′,aˆ†s′′,andaˆ†i′′.|Ψiaretheeffective consistentSchro¨dingerequationsassuming there is only one
multiphotonstatesinthecascadedscheme. atomicexcitation[20,21].Thisisvalidwhenweakandlarge
detunedexcitationpulsesareconsidered,satisfying ∆
1,2
| |≫
Ω . After adiabatically eliminating the atomic levels 1
a,b
| | | i
atleastoneinfraredortelecomphotonisgenerated.However and 2 in the coupled Schro¨dingerequations, we derive the
| i
iftheinfraredandtelecomphotonscanbeconvertedbackand probabilityamplitudeofthebiphotonstate 1k ,1k [20],
| s ii
forth with each other [18, 19], we can have arbitrary m and
n. Furthermore, we investigate its entanglement property in N t t′
continuousfrequencyspaces. Our studies allow for an alter- Ds,i(t)=gi∗gs∗ ei∆k·rµZ Z dt′′dt′ei∆ωit′ei∆ωst′′
native setting for multiphoton generation from cold atomic µX=1 −∞ −∞
ensembles, and demonstrate spectral shaping to control the b(t′′)e(−Γ2N3 +iδωi)(t′−t′′), (2)
entanglementof this multiphotonsource[39], advancingthe ×
developmentofmultimodequantumcommunication.
where b(t) = Ω (t)Ω (t)/(4∆ ∆ ), resulting from the re-
a b 1 2
In this paper, we first discuss the Hamiltonian and four- quiredadiabaticconditionsfordrivingpulses.Four-wavemix-
wave mixing condition in Sec. II, and propose to generate ing (FWM) condition N ei∆k·rµ represents the phase-
a multiphoton source in the scheme of cascaded atomic en- matchingconditionwhePn∆µ=k1=ka +kb ks ki 0in
sembles in Sec. III. Then in Sec. IV we study the bipartite − − →
thelimitoflargenumberofatomsN,andguaranteestocreate
entanglement properties of the multiphoton source, and we
ahighlycorrelatedphotonpair.Furthermoretheidlerphoton
concludeinSec. V. issuperradiant[17,41,42]thatΓN =(Nµ¯+1)Γ quantifies
3 3
a superradiantdecayrate in the levelof 3 with an intrinsic
| i
decayrateΓ .Thegeometricalconstantµ¯ [43]relatestothe
3
shapeoftheAE,andtheassociatedcollectivefrequencyshift
II. CASCADEEMISSIONSANDFOUR-WAVEMIXING
CONDITION [44–46] is denotedas δωi. This reflects the nature of collec-
tive radiation [47] due to induced dipole-dipole interactions
in the dissipation [48]. Note that a complete description of
We considera Rb atomicensemble(AE)with a diamond-
thecollectivefrequencyshiftrequiresnon-RWAtermsinthe
type configuration shown in the inset of Fig. 1(a). The cor-
Hamiltonian[46,48].
related cascade emissions of signal and idler photons (aˆ†
s(i) This biphoton state 1k ,1k has two main features. One
asshorthandsforaˆ† below)arecreatedbypumpingthe is the strong directiona|litys ofithie emitted photons, which is
km,λm
atomicsystemwithtwoclassicalfields.Withdipoleapproxi- determinedbyFWM.InthermodynamiclimitofAE,wehave
mationoflight-matterinteractionsandrotating-waveapprox- ∆k=0.Ifcounter-propagatingexcitations[17]areusedasin
imation(RWA)[40],weexpresstheHamiltonianintheinter- Fig. 1(a)withexcitationandphotonemissionanglesdenoted
3
asθ , wecanderivetheconditionfortheemittedangles ∆ω ∆ω withinthespectralrangeof1/τ .Moreen-
a,b,s,i s i eff
≈−
as tangledbiphotonstatecanbemadewithanincreasingoptical
densityoftheAEorlongerpulses[20],makingf(ω ,ω )less
λ s i
b(cosθ cosθ )=cosθ cosθ , factorizablealigningontheaxisof∆ω = ∆ω [21].Inthe
a i b s s i
λ − − −
a next section we propose to use this two-photon source as a
λ
b(sinθ +sinθ )=sinθ +sinθ , (3) seed to create multiphoton states in the scheme of cascaded
a i b s
λa AEs.
where λ are the wavelengths of the excitation fields, and
a,b
we haveassumed k = k and k = k . Sinceθ are
i a s b a,b
given when the ex|cit|ation|s a|re ap|plie|d, θ| |can be decided III. MULTIPHOTONSTATESFROMTHECASCADE
s,i
EMISSIONSINTHECASCADEDSCHEME
fromEq. (3).Tohavesomeestimatefortheangles,wehave
(θ ,θ ) = (4.9◦,9.9◦) when we set (θ ,θ ) = (5◦,10◦) and
i s a b
λ /λ = 2. For the same ratio of the wavelengths, we have Before we investigate the multiphoton states generated
b a
(θ ,θ )=(7.9◦,13.9◦)whenweset(θ ,θ )=(4◦,10◦).Ifin from the cascaded atomic ensembles, we note that through-
i s a b
radiansthatθ 1andsettingθ =2θ ,wederiveθ ( outthepaper,wefocusonlyontheeffectivepurestatesgen-
a,b,s,i b a s
≪ ≈
θ ) 2θ ,whichindicatesthatthesignalandidlerphotonsare eration. In general for an open quantum system, the system
b i
≈
emittedwithcorrespondingexcitationanglesofθ andθ re- becomes mixed states inevitably [49] due to the interactions
b a
spectively,andtheyfollowthedirectionsalmosttangenttothe with the reservoir or the imperfections in experiments. The
longaxisofAE.Thisdirectionispreferentialinexperiments, mixedstatescanbeinterpretedasstatisticalmixturesofpure
whichallowsforstronglight-mattercouplings. states, which can be expressed as a density operator, ρˆ =
Theotherfeatureisthatthebiphotonstateisprobabilistic. p ψ ψ with the constraints of p = 1 and 0 <
k k| kih k| k k
Accordingto Eq. (2), we can expressthe completeand nor- pPk 1. ρˆ becomes a pure state when Pone of pk’s is unity,
≤
malizedstateas thereforethisdensityoperatorshowsageneralrepresentation
ofanyquantumsystem.Forexampleofthespontaneousemis-
1
|Ψi= 1+ Ds,i 2|0⊗Ni[|vaci+Ds,i|1ks,1kii], (4) asimonplpitruodceedssaminpaintgw[o1-]letovethleataotmom(i|c0istaanteds.|1Ini)a, diteascctrsipltiikoenaonf
| |
p generalizedamplitude damping, the atom evolvesto the sta-
whichinvolvesN atomic groundwith vacuumandbiphoton
tionarymixedstateρ =p0 0 +(1 p)1 1 withsome
nstoattegserneesrpaetectainvyelpy.hSoitnocnes.|DThse,ir|e≪for1e,emxpoesrtiomfethnetatlilmyeitAreEqudioreess probabilityp.Similar∞ly,ρ∞|cainhd|escribe−alo|ssihof|photondue
toattenuationordecoherencefromenvironmentif 0 and 1
repeatedexcitationsuntilthebiphotonstateiscreated,which | i | i
aredenotedasvacuumandone-photonstatesrespectively.
can be confirmed via photon detections. The degree of cor-
In general, a spontaneous emission is a random process
relationofthephotons(second-ordercorrelationfunctionfor
[1, 40] in time and space, where the emitted direction has
example)canbemeasuredaswellbyaconditionaldetection
[17, 42] of the idler photon after the signal one is detected. a uniformly dΩˆ/(4π) distribution in a solid angle of dΩˆ =
Below and throughout the paper, we focus on the effective sinθdθdφinsphericalcoordinates.Incontrasttothisrandom
process in space, the biphoton state from the cascade emis-
biphotonandmultiphotonstates, thereforethe normalization
sions in a diamond configuration under FWM condition is
fortheseeffectivestatesisneglected.Duetotheirprobabilis-
highlydirectionalandcorrelated,whichthusmakestheeffec-
tic feature, these effective states can be confirmed only via
conditionaldetectionsorpostselections. tive state 1k ,1k valid if photonloss or other noisy chan-
| s ii
nelsthatdeteriorateitsfidelitycanbeneglected.Althoughthis
Specifically we use normalized Gaussian pulses where
Ωa(b)(t) = [√πτa(b)]−1Ω˜a(b)e−t2/τa2(b) with the pulse areas ltiamngitlsemouernitndviessttiilglaattiioonnotortphuerpifiucraetisotantepsr,owceednuorete[t1h,at2a3n, 5en0]-
Ω˜ and pulse widths τ . Consideringthe long time limit
a,b a(b) can be applied to the mixed entangled states by local oper-
that D (t ), we derive the probabilityamplitude D
s,i si ationsand classicalcommunication.Thereforethe purestate
→ ∞
afterinsertingtheGaussianformsintoEq. (2)andredefining
picture we focus here can still give insights to quantum in-
∆ω as∆ω δω and∆ω +δω respectively,whichreads
s(i) s i i i formationprocessingwiththeeffectivemultiphotonstatepre-
−
sented in this work. Other discussions of inseparability cri-
D (∆ω ,∆ω )= Ω˜aΩ˜bgs∗gi∗ Nµ=1ei∆k·rµf(ω ,ω ).(5) terion,boundentanglement,ormultipartiteentanglementfor
si s i 4∆ ∆ √2Pπ τ2+τ2 s i mixedstatescanbereferredtoRefs. [23,39].Formeasuring
1 2 a b
p entanglement, it does not imply Bell nonlocality except for
Thespectralfunctionofthebiphotonstateis purestates,andquantifyingentanglementforbipartitemixed
state involves various measures not agreeable to the partial
f(ω ,ω )= e−(∆ωs+∆ωi)2τe2ff/8, (6) vonNeumannentropy[23].Thatbeingthe case, ourinvesti-
s i Γ2N3 −i∆ωi ggaletimonenstuosfinbgipSacrhtimteidpturdeecstoamtepsoisnitSioenc.toIVquparnotvifiydethaeneunptpaner-
whereτ √2τ τ / τ2+τ2. Thisspectrallycorrelated boundtotheentanglementmeasure.
eff ≡ a b a b
biphoton state involvespa Gaussian weighting modulated by To generate a multiphoton state from the cascade emis-
a Lorentzian. The spectral function is most significant when sions, we propose to couple one of the two photons with
4
other AE along with a corresponding pump field, such that field,suchthatwecansimplifyEq. (9)as
(k+1)-photonsourcecanbe createdfromak-photonstate.
As demonstrated in Fig. 1, three and four-photonstates are f3,B1 e−(∆ωs+∆ωs′+∆ωi′)2τe2ff/8e−(∆ωs′+∆ωi′)2τb2/4
= .
cartoeamteicdeinnstehmebclaesscaudseedd stochgeemnee.raAteE’coarnrdelaAteEd”pahreottownosoaˆt†her Γ23 Γ2N3 −i(∆ωs′ +∆ωi′) Γ2N3 −i∆ωi′
s′,i′ (10)
and aˆ† respectivelyunder the FWM condition.Since the
s′′,i′′
initial seed of two-photon source is highly entangled in fre- Similarly,ifweannihilatesignalphotonoftheinitialtwo-
quency space, the generated multiphoton source is expected photonseed,wehavealternativelytheeffectivethree-photon
tobealsoentangledincontinuousfrequencyspaces. stateas(B2todenotethesecondrouteforthree-photonstate
generation,involvingtwoidlerandonesignalphotons)
The generated multiphoton spectral functions can be de-
rivedbyproductsof thetwo-photonspectralonesofEq. (6) Ψ =f aˆ†aˆ† aˆ† 0 , (11)
and invoke the conditionfor centralfrequenciesof the anni- | i3,B2 3,B2 i s′ i′| i
where its dimensionless spectral function is [after removing
hilated photon, that is ω = ω′ + ω′ ω for example in
i s i − b theextrafrequencyshiftasinderivingEq. (10)]
thecaseofAE’inFig. 1(a).Thisconditionalso satisfiesthe
energyconservationas if four fields are plane waves. Below f3,B2 e−(∆ωi+∆ωs′+∆ωi′)2τe2ff/8e−(∆ωs′+∆ωi′)2τa2/4
wedemonstratethreeandfour-photonstatespectralfunctions, = .
Γ2 ΓN ΓN
andtheirentanglementpropertieswillbediscussedinthenext 3 ( 23 −i∆ωi)( 23 −i∆ωi′)
section. (12)
Note that the overall constants in spectral functions do not
make an effect on spectral distributions, thus we regularize
them in dimensionlessforms.The overallconstantshowever
A. Three-photonstate determine the generation rates which are small in general
sinceweakandlargedetunedexcitationsareused.
Eqs. (10) and (12) are two of the main results in this
AsinFig. 1(a)whichwedenotea routeB1forthethree-
subsection.Obviouslythesethree-photonstatesareentangled
photon generation, the effective three-photonstate involving
in frequency spaces, meaning there is no possible ways to
twosignalandoneidlerphotonscanbeexpressedas
factorizethesespectralfunctions.Meanwhile Ψ differs
3,B1
| i
from Ψ specificallyinamodulatedLorentzianfunction
3,B2
| i
Ψ =f aˆ†aˆ† aˆ† 0 . (7) on signal distribution ∆ωs′. This results from the annihila-
| i3,B1 3,B1 s s′ i′| i tion of the idler photon in the route B1, which replaces the
Lorentzian with the correlated signal and idler photon pair.
We derive the dimensionless and effective spectral function Also different timescales τ in routes B1(2) respectively
b(a)
by multiplying a typical biphoton state spectral function of for a joint Gaussian profile of the photonpair aˆ† aˆ† suggest
s′ i′
Eq. (6)withtheonegeneratedbyaplane-waveidlerphoton anindependentcontrolovertheir spectralfunctionsbyvary-
aˆ†i andadrivingfieldΩbinAE’, ingpulsedurations.
Below we derive the spectral functionfor the four-photon
stateofFig. 1(b)usingthethree-photonstatesinthecascaded
f /Γ2 = e−(∆ωs+∆ωi)2τe2ff/8e−(∆ωs′+∆ωi′)2τb2/4,(8) schemeofFig. 1(a),andalsotheotherpossiblespectralfunc-
3,B1 3 ΓN ΓN tionsinalternativeroutes.
23 −i∆ωi 23 −i∆ωi′
where τ appears when we let τ in τ . The above B. Four-photonstate
b a eff
becomes,aftersettingω =ω′ +ω→′ ∞ω andusing∆ =ω
i s i− b 2 a
+ωb ω2, Hereusingthesamefashiontogeneratethree-photonstates,
−
the four-photon ones can be also created in the cascaded
e−(∆ωs+∆ωs′+∆ωi′+∆a3+δωi)2τe2ff/8 scheme. As in Fig. 1(b), the idler photon aˆ†i′ emitted from
f /Γ2 = AE’ is annihilatedwith an extracouplingfield Ω in AE” to
3,B1 3 Γ2N3 −i(∆ωs′ +∆ωi′ +∆a3+δωi) generateanewlycorrelatedpairofphotonsaˆ†s′′aˆ†i′′b.Theeffec-
e−(∆ωs′+∆ωi′)2τb2/4 tivefour-photonstatewiththreesignalandoneidlerphotons
. (9) becomes
× Γ2N3 −i∆ωi′ Ψ =f aˆ†aˆ† aˆ† aˆ† 0 , (13)
| i4,C1 4,C1 s s′ s′′ i′′| i
The extra frequency shift ∆ ω ω in the above can where again the dimensionless spectral function can be de-
a3 a 3
≡ −
be removedalong with δω by applyingan externalZeeman rivedas
i
5
f4,C1 = e−(∆ωs+∆ωs′+∆ωs′′+∆ωi′′)2τe2ff/8e−(∆ωs′+∆ωs′′+∆ωi′′)2τb2/4e−(∆ωs′′+∆ωi′′)2τb2/4. (14)
Γ3 ΓN ΓN ΓN
3 23 −i(∆ωs′ +∆ωs′′ +∆ωi′′) 23 −i(∆ωs′′ +∆ωi′′) ( 23 −i∆ωi′′)
(cid:2) (cid:3)(cid:2) (cid:3)
Other four possible routes to generate four-photon states byannihilatingaˆ† fromtheroutesB1(2)respectivelyalong
s(i)
aredemonstratedinAppendixA.Inprinciplethereshouldbe withthecouplingfieldsΩ .Thesymmetryissatisfiedwhen
a(b)
six differentspectralfunctionswherethreeof themare from aˆs′′ aˆs′ andaˆi′′ aˆi′,andthiseffectivefour-photonstate
routes B1 and B2 respectively. Since there is one spectral with↔twosignaland↔twoidlerphotonsis
functionwhichissymmetrictoeachotherinrespectiveroutes,
making a total of five possible spectral functionsin our cas- Ψ =f aˆ† aˆ†aˆ† aˆ† 0 , (15)
| i4,C3 4,C3 s′ i′ s′′ i′′| i
cadedscheme.Thissymmetricfour-photonstateisgenerated
withthespectralfunction
f4,C3 = e−(∆ωs′+∆ωi′+∆ωs′′+∆ωi′′)2τe2ff/8e−(∆ωs′+∆ωi′)2τa2/4e−(∆ωs′′+∆ωi′′)2τb2/4. (16)
Γ3 ΓN ΓN ΓN
3 23 −i(∆ωs′′ +∆ωi′′) ( 23 −i∆ωi′)( 23 −i∆ωi′′)
(cid:2) (cid:3)
These spectral functions have a common weighting of 29],
Gaussian envelope involving four photon frequencies, indi-
catingto possessa genuinek-partyentanglement[23] fork- K1(ω,ω′)ψn(ω′)dω′ =λnψn(ω), (17)
Z
photon source in our proposed cascaded scheme. This gen-
uine multiphoton entanglement means to exclude any pos- K (ω,ω′)φ (ω′)dω′ =λ φ (ω), (18)
2 n n n
sible bipartite splittings or groupings [23]. For example of Z
Eq. (16) from the route C3, two groups of photons aˆ† where
s′,i′
andaˆ†s′′,i′′ respectivelywouldbe able tobe factorizedif this K1(ω,ω′) f(ω,ω1)f∗(ω′,ω1)dω1, (19)
common weighting of e−(∆ωs′+∆ωi′+∆ωs′′+∆ωi′′)2τe2ff/8 is ≡Z
absent.Multipartiteentanglementisstillanongoingresearch K (ω,ω′) f(ω ,ω)f∗(ω ,ω′)dω . (20)
2 2 2 2
evenforpurestatesweconsiderhere,thereforetogiveanin- ≡Z
tuitive study of entanglementpropertyof our proposed mul- The spectral function f comes from the effective biphoton
tiphoton sources, in the next section we introduce Schmidt state(apairofaˆ† andaˆ†forexample)
s i
decompositionto investigate their bipartite entanglementsin
continuousfrequencyspaces. Ψ = f(ω ,ω )aˆ†(ω )aˆ†(ω )0 dω dω , (21)
| i Z s i s s i i | i s i
= λ ˆb†cˆ† 0 , (22)
n n n| i
Xn p
with two mode functions in the Schmidt bases, determining
IV. ENTROPYOFENTANGLEMENT theeffectivephotonoperators,
ˆb† ψ (ω )aˆ†(ω )dω , (23)
To gaininsightsof the spectralentanglementin themulti- n ≡Z n s s s s
photonsources from the cascaded scheme of AEs, we study
cˆ† φ (ω )aˆ†(ω )dω . (24)
the bipartite entropy of entanglement in these sources. The n ≡Z n i i i i
multiphoton states can be projected or collapsed to the ef-
The above decomposition has been used to quantify the
fectivebiphotonstatebyconditionalmeasurements,suchthat
spectralentanglementof two-photonsourcefromparametric
we can quantify and analyze their spectral entanglement by
downconversion[28,29]andcascadeemissions[20],andin
Schmidtdecomposition[28]. The bipartite entropyof entan-
spectral shaping of biphoton state in a multiplexed scheme
glement is S = ∞ λ log λ [51], intended for pure
− n=1 n 2 n [21,22].
states.SchmidteigePnvaluesλn determinetheprobabilitiesof
nthmodefunctions,withwhichthestatecanbeexpressedas
n√λnˆb†ncˆ†n|0i, for two effective photon operatorsˆbn and A. Sinthree-photonstates
cPˆnassociatedwithmodefunctionsψnandφnrespectively.
Schmidtdecompositionisdonenumericallyassolvingthe Forbipartiteentanglementpropertyofathree-photonstate
eigenvalue problems of the one-photon spectral kernels [28, with the spectral function f , three possible projected
3,B1
6
1 functionf .Theinsetsof(a)and(b)demonstratethespec-
3,B2
20 (a) 20 (b) tralfunctionsoftheprojectedbiphotonstateswithannihilated
0.8
Γ3 Γ3 photonsaˆi andaˆs′ respectively.Theentropyofentanglement
0.6 ω/s’ 0 ω/s 0 S is 1.79 and 0.09 respectively, again indicating a more en-
∆ ∆
n tangled source in the former case. Fig. 3(b) is less entan-
λ
0.4 −20−20 0 20 −2−010 0 10 gledduetoaremovalofthesignalphotondependence∆ωs′
0.2 ∆ωi’/Γ3 ∆ωi’/Γ3 intheentanglingfunctione−(∆ωs′+∆ωi′)2τa2/4,similartothe
cases in Fig. 2. The reason why Fig. 3(a) is less entangled
0 thanFig. 2(a)isthatanextraentanglingLorentzianfunction
1 2 3 4 5 6 7 8 9 10
n [ΓN3 /2−i(∆ωs′ +∆ωi′)]−1 present in Eq. (10). Fig. 3(b)
showsa relativelysmaller entangledbiphotonsource, result-
FIG.2. (Coloronline)FirsttenSchmidteigenvaluesλn ofthepro- ingfromaspectralfunctionwithmorealigneddistributionson
njeaclteadndspiedclterralpdhiosttorinbsutaiorensp|ofs3t,-Bse1l|e.cTtehdestope±ctr2a0l0raΓngetshrfoourgbhoothutsiagl-l ∆ωi(i′) = 0 comparedto Fig. 2(b), in a somewhatdistorted
3 fashionofFig.2(b).ThisisduetotwofactorizableLorentzian
filogsusroefsgwehneerrealwitey.aTlshoesientsΓetN3s=(a)5aΓn3da(nbd)aτrae=prτobje=cte0d.2s5pΓec−3t1rawlditihsotrui-t functionsofidlerphotonsaˆiandaˆi′ infrequencies∆ωi(i′)of
butionswith∆ωs(s′) =0respectivelywithcorrespondingSchmidt Eq. (12)whenprojectingoutthesignalphotonaˆs′.
eigenvalues((cid:3))and(◦). Sincethedecayconstantsoftwoatomicensembles,AEand
AE’, are notnecessarily the same, we furtherinvestigatethe
spectralpropertywith differentΓN andΓN′. Asan example
3 3
biphotonstatescanbederivedbyeitherannihilatingphotons fromEq. (10)andprojectingoutaˆs,wehave
saˆhso,waˆst′h,eosrpaˆeci′t.raIlndtihsteribinusteiotsnsopfr(oaj)ecatnindg(obu)toaˆfsFaingd. aˆ2s,′ wree- f3→2,B1 = e−(∆ωs′+∆ωi′)2τe2ff/8 e−(∆ωs′+∆ωi′)2τb2/4.
smpeencttiSveilsy2b.3y7saenttdin0g.1∆5ωress(sp′e)c=tive0l.yT,ihnediecnattrionpgyaomfoerneteanntgalne-- Γ23 Γ2N3 −i(∆ωs′ +∆ωi′) ΓN32′ −i∆ωi′
(25)
gled source in the former case. The huge differencein these
twoprojectionsresultsfromaremovalofanentanglingfunc- In Fig. 4 we show the results for different decay constants
tione−(∆ωs′+∆ωi′)2τb2/4 ofEq. (10)whenprojectingoutaˆs′. comparedtoFig2(a).TheentropyofentanglementS forthe
ThisGaussianfunctionsupposestoentanglephotonsaˆs′ and insets(a)and(b)are3.9and0.89respectively.ThesmallΓN3
aˆi′ ininfinitesignalandidlerfrequencyspaces. Amodulat- in(a)providesasharpdistributionalongthehighlyentangled
ing Lorentzianfunctionof idler photon howeverconfinesits axis ∆ωs′ = ∆ωi′, therefore making this projected spec-
spectraldistributionintoafinitebandwidthof∼ΓN3 .Project- tralfunctionm−oreentangled.IncontrastthesmallΓN3 ′ in(b)
ingoutasignalphotonaˆs′ thuscollapsestheentanglingfunc- limitsthefactorizableLorentzianidlerdistributionaˆi′,allow-
tionintoe−(∆ωi′)2τb2/4,renderinganotherfinitebandwidthof ing for a squeezed distribution in ∆ωi′ and a less entangled
the idler photon with FWHM (full-width at half-maximum) biphotonsource.
4√ln2/τ . In essence these factorizable idler functions, In this subsection we have demonstrated a rich spectral
b
∼
Gaussian and Lorentzian, rotate the spectral distribution to- propertyoftheprojectedbiphotonstatefromthethree-photon
ward the axis ∆ωi′ = 0, as we show in Fig. 2(b). The third source in the cascaded scheme. The entanglement property
possible projectedbiphotonstate is to projectout aˆi′, which canbeverydifferentdependingonhowthecounterpartofthe
hasasimilarspectralpropertytotheoneannihilatingaˆs′. source is collapsed. In the perspective of generating a more
Likewise in Fig. 3, we show the results for the spectral
1
20 (a) 10 (b)
1
0.8 Γ3 20 (a) Γ3 5 (b) 00..68 ∆ωΓ/s’3 0 ∆ωΓ/s’3 0
λn00..46 ∆ω/s’−200−20 0 20 ∆ω/i’−−1050 −10 0 10 λn00..24 −20−20 ∆ω0i’/Γ3 20 −10−10 ∆ωi0’/Γ3 10
∆ω/Γ ∆ω/Γ
0.2 i’ 3 i 3
0
1 2 3 4 5 6 7 8 9 10
0 n
1 2 3 4 5 6 7 8 9 10
n
FIG.4. (Coloronline)FirsttenSchmidteigenvaluesλn ofthepro-
jFeIcGte.d3.sp(eCcotrlaolrdoinstlrinibeu)tFioinrsst|tfe3n,BS2c|h.mTihdetpeaigraemnveatleuressaλrenthoeftshaemperaos- jΓeN3cteadndspΓecN3tr′a.lTdhiestrpiabruatmioentser|sf3a,rBe1t|hweistahmdeifafesreinntFdige.ca2ywcohnilsetatnhtes
inFig. 2whiletheinsets(a)and(b)areprojectedspectraldistribu- insets(a) and (b) areprojected spectral distributionsas in Fig2(a)
tionswith∆ωi(s′) =0andcorrespondingSchmidteigenvalues((cid:3)) withΓN3 /Γ3 =1(5)andΓN3′/Γ3 =5(1)respectively.Correspond-
and(◦)respectively. ingSchmidteigenvaluesare((cid:3))and(◦).
7
(a)100 (b) 0.2
5
Γ3 0.1
10−5 ω/i’ 0
∆ −5 0−10 0 10
n ∆ω/Γ
λ 10−10 −5 ∆ω0/Γ 5 0.3 i’ 3
i’’ 3
0.2
10−15 0.1
0
−10 0 10
2 4 6 8 10 ∆ω /Γ
n i’’ 3
FIG.5. (Color online) Three-dimensional isosurface plots for pro-
at(jrebnac)dlteaˆdndiios′′srt.mprieTbacuhltitreziaoelidnsdos|isfsiunt4rr,i(Cfbaa1u)c|teai=onpndl0so(.t|b3sf)4war,Crehes1ep|rceehwcottihistvheeeynalynsf.hnooirhwrileatsitpleteedcdptiahvnoedtloyanxp(iraaol)jesaˆpcsteeocd-r maFFniIogGnd.i.he2i6.lf.a(uatne)(cdCλtiopnolhnososrhtoooownfnslstihaˆnaesen′)paaFbrnoridrujesptacˆttstede′′den.csTrSpehceaehcsmetpraaiindrlatldmoeigsiegtatreeriirbntsuvhatmailrouiencestsshc|eλaflns4ea,B,mai3nne|ddawictswiatitohn-
inganextremelylowentropyofentanglement. (b)Twoidlermode
probability densities (solid and dashed for the first and the second
entangledphotonsource,thespectralfunctionfromrouteB1 modes)in∆ωi′(i′′)respectively.
serves better than B2, meanwhile a less(more) superradiant
decayconstantΓN3 (ΓN3 ′)ofAE(AE’)intherouteB1isfavor- ofthe modes(upto 99.98%)in thisbiphotonsource.In Fig.
ableforthispurpose. 6(b), we show their mode probabilitydensities. As expected
theFWHMofthemodeprobabilitydensityfollowsthespec-
traldistributionatthecutof∆ωi′(i′′) =0.Thefirstmodeof
B. Sinfour-photonstates theidleraˆi′′ hasashortenedlinewidththantheoneofaˆi′ due
to a squaredLorentzian functionof [ΓN3 /2−i(∆ωi′′)]−2 in
Forfour-photonstates,thereareingeneralsixpossiblepro- f4,C3with∆ωs′′ =0.
jections to biphotonones. Thoughof plenty of possible pro- We can also manipulateS by modifyingthe drivingpulse
jected biphoton states for a total of five spectral functions durations τ and τ . When we increase them to τ = τ =
a b a b
demonstrated in Sec. III.B and Appendix A, there are only 1Γ−1 in the projected f as in Fig. 6, we find S be-
3 4,C3
a few of qualitatively different spectral functions. As an ex- comes0.13, which allowsfora moreentangledsource since
ample, we choose the spectral function f4,C1 of Eq. (14). the entangling Gaussian function e−(∆ωi′+∆ωi′′)2τe2ff/8 has
Based on the observation from projected three-photonstates a tighter photon correlation on the distribution axis ∆ωi′ =
oinraˆthse′′pinrefv4i,oCu1spsruobvsiedcetsiolens,sweenteaxnpgelcetdtmhautltaipphrootjoecntsiotanteosf.Faˆoi′r′ −=∆1Γω−3i′′1.iOnnththeeasoythmemrheatrnidcswehtteinngw,ewseefitnτad=S b0e.2c5oΓm−3e1s0an.0d2τ3b,
comparisons, in Fig. 5 we show two projectionsof f4,C1 in whichisevensmallerthanthesymmetriccaseinFig. 6.This
three-dimensionalisosurfaceplotswhichprovideaqualitative reflectsthecompetitionofthisentanglingandtwo otherdis-
distinctionintheprojectedspectralfunctions.InFig.5(a)and entanglingGaussianfunctions.Thoughτ stillincreasesin
eff
(b)weset∆ωs(i′′) =0respectively,whichshowatiltedand the asymmetricsetting, a moreconfineddistributionfromτb
anaxialdistribution.Thetilteddistributionpotentiallyallows limitstheoverallprojectedspectralfunctiondistribution,thus
foramoreentangledmultiphotonstatewhilewenotethatfur- decreasingS.
therprojectingoutphotonsaˆi′′ andaˆs respectivelyin(a)and
(b)givesthesamespectralfunctionoftheprojectedbiphoton
states. V. DISCUSSIONANDCONCLUSION
Less entangled biphoton states from f can be derived
4,C1
byprojectingoutapairofphotons(aˆs′′(i′′),aˆs′)or(aˆs′′,aˆi′′). The multiphoton states generated from the cascade emis-
Theseprojectionsbasicallycollapsef4,C1intoasingleGaus- sionsin the schemeof cascadedAE are advantageousin the
sian function of e−[∆ωs′′(i′′)]2τb2/4 or e−(∆ωs′)2τb2/4, which perspective of well-controlled AE preparations. Large-scale
againconfinesitsspectraldistributionwithoutentanglingwith implementation of such cascaded scheme is feasible when
otherphotons.Asanotherexampleofdisentanglingphotons, preparingthe atoms either in free-space, multiplexed setting
wenotethatthespectralfunctionf ofEq.(16)allowsfor [21, 52], or opticallattices[53], butwill sufferfromthe low
4,C3
theleastentangledbiphotonsource.Byannihilatingapairof generationrate.Lowgenerationrateresultsfromtheweakex-
photonsofeitheraˆs′(i′)oraˆs′′(i′′),wecollapsef4,C3intotwo citationsandfew-photonlevelofemissions.Alsolowlightis
singleGaussianfunctions.Choosing∆ωs′,s′′ =0inf4,C3,in moresubjecttopropagationattenuation,makingthedetection
Fig. 6weshowthecollapsedspectraldistributionoftwoidler of photons difficult. However this can be overcome by rais-
photons, which has extremely low entropy of entanglement ingtherepetitionratesinexcitations.Finitespectralwindows
S = 0.028. The eigenvalues λ are plotted in a logarithmic forcollapsingthemultiphotonspectralfunctionsarenotcon-
n
scale, showing an abruptdecrease of Schmidtnumbers. The sideredhere.Butweexpectofnosignificantmodificationsin
first two eigenvalues are 0.997 and 0.0028, occupying most theirspectralpropertiesqualitatively,whereessentiallythefi-
8
nite spectral window of projection averages out the spectral schemesafullcontrolofS .Inadditiontothespectralshap-
M
distribution.Agatetime ofseveralhundredsofnanoseconds ingofthecascadeemissionsbymodifyingdrivingconditions
inthephotoncountingdevicewouldbeenoughtoneglectthe [20]ormultiplexingAEs[21,22],thecascadedschemehere
effectfromfinitedetectionwindows. providesanalternativeroutetospectrallyshapeanevenmore
Theothermeritofthe multiphotonsourceinthe cascaded entangled biphoton source, thus overcomes the limitation of
scheme is flexibility to entangle the photons either in signal S inthemultiplexedscheme.
oridlerfrequencies,makingarepertoireofmanypossibleen- Inconclusion,weproposeacascadedschemetogeneratea
tangledmultiphotonsources.Thesignalfrequencyisbestfor multiphotonsourcefromthecascadeemissionsoftheatomic
optical fiber transmission while the idler one is preferential ensembles.Highlyspectrallyentangled(k+1)-photonsource
inquantumstorage.Inourschemeahighlyentangledphoton can be created using k-photon state as the seed along with
sourcecanbegeneratedbyincreasingtheexcitationpulsedu- anappropriatedrivingfieldeitherintheloweroruppertran-
rationsin thesymmetricsettingortightenthespectraldistri- sition of the diamondconfiguration.Under the FWM condi-
butionontheaxis∆ω +∆ω =0thatconservesthebiphoton tion, this highly directionaland frequency-correlatedphoton
s i
energy. By appropriately projecting out the photon counter- sourceareusefulforquantuminformationprocessingandap-
partsin themultiphotonspectralfunction,theentropyofen- plicabletomultimodequantumcommunication.Furthermore
tanglementand the spectralmode functionscan be modified such entangled multiphoton source can be spectrally shaped
and manipulated to serve different purposes requiring either withcontrollabledrivingconditionsandensembleproperties
pureorentangledstates. (forexampleatomicdensityandgeometry),whichcouldpo-
Formoreentangledbiphotonsource,wehavedemonstrated tentiallybeimplementedinquantumspectroscopy[54].
a larger entropy of entanglementS = 2.37 or 1.79 from the ACKNOWLEDGMENTS
projectedthree-photonstatescomparedto1.33[20]fromjust
one AE under the same driving conditions. This shows that
This work is supported by the Ministry of Science and
a more entangled biphoton source can be generated from
Technology(MOST), Taiwan, under Grant No. MOST-103-
a multiphoton source with even larger capacity in the gen-
2112-M-001-011.
uine multipartite entanglement. Our cascaded scheme here
can also combine with the multiplexed one [21]. The multi-
plexedschememanipulatesthespectralpropertyofthebipho-
AppendixA:Otherroutesforfour-photonstategeneration
tonstatebymodifyingtheircentralfrequenciesorphases[22].
Its maximal entropy of entanglement S can be described
M InSec. III.B,wehaveinvestigatedfour-photonstategener-
by S + S . S is the entropy of entanglement from one AE
d ationinthecascadedscheme,anddemonstratedthefirstroute
while S log (N ) with N , the number of multi-
d ≡ 2 MP MP C1.Otherfourpossibleroutestogeneratefour-photonstates
plexed AEs. Therefore the multiplexed scheme increases s
d canusethree-photonstates Ψ and Ψ asseeds.The
asNMPincreases.Meanwhilethecascadedschemeenablesa route C2 below is to annih|ilaite3,Baˆ†1 of |Ψi3,B2with an extra
multiphotonsourcefromsequentially-coupledAEsusingdi- s′ | i3,B1
couplingfieldΩ inthethirdAE”togenerateanewlycorre-
amond configurations. Its bipartite entanglement can be ex- a
tractedfromthebiphotonstatescollapsedfromthemultipho- latedpairofphotonsaˆ†s′′aˆ†i′′.Theeffectivestateis|Ψi4,C2 =
ton ones. This way the cascaded scheme modifies and ma- f aˆ†aˆ†aˆ† aˆ† 0 involvingtwosignalandtwo idlerpho-
4,C2 s i′ s′′ i′′| i
nipulatesS effectively,makingthecombinationofthesetwo tonswiththespectralfunction
f4,C2 = e−(∆ωs+∆ωi′+∆ωs′′+∆ωi′′)2τe2ff/8e−(∆ωi′+∆ωs′′+∆ωi′′)2τb2/4e−(∆ωs′′+∆ωi′′)2τa2/4. (A1)
Γ3 ΓN ΓN ΓN
3 23 −i(∆ωi′ +∆ωs′′ +∆ωi′′) ( 23 −i∆ωi′)( 23 −i∆ωi′′)
(cid:2) (cid:3)
TherouteC3involvestwosignalandtwoidlerphotons,generatedfromasymmetriccouplingbetweenaˆ† andpumpfields
s(i)
Ω inthethirdAE”usingthree-photonstatesfromroutesB1(2)respectively.Itsspectralfunctionhasbeenshowninthemain
a(b)
paper. For route C4, which annihilates aˆ† of Ψ with an extra coupling field Ω in the third AE”. The effective state is
i′ | i3,B2 b
Ψ =f aˆ†aˆ† aˆ† aˆ† 0 involvingtwosignalandtwoidlerphotonswiththespectralfunction
| i4,C4 4,C4 i s′ s′′ i′′| i
f4,C4 = e−(∆ωs′+∆ωi+∆ωs′′+∆ωi′′)2τe2ff/8e−(∆ωs′+∆ωs′′+∆ωi′′)2τa2/4e−(∆ωs′′+∆ωi′′)2τb2/4. (A2)
Γ3 ΓN ΓN ΓN
3 ( 23 −i∆ωi) 23 −i(∆ωs′′ +∆ωi′′) ( 23 −i∆ωi′′)
(cid:2) (cid:3)
ThelastrouteC5annihilatesaˆ† of Ψ withanextracouplingfieldΩ inthethirdAE”.Theeffectivestateis Ψ =
s′ | i3,B2 a | i4,C5
f aˆ†aˆ†aˆ† aˆ† 0 involvingonesignalandthreeidlerphotonswiththespectralfunction
4,C5 i i′ s′′ i′′| i
f4,C5 = e−(∆ωi+∆ωi′+∆ωs′′+∆ωi′′)2τe2ff/8e−(∆ωi′+∆ωs′′+∆ωi′′)2τa2/4e−(∆ωs′′+∆ωi′′)2τa2/4. (A3)
Γ3 ΓN ΓN ΓN
3 ( 23 −i∆ωi)( 23 −i∆ωi′)( 23 −i∆ωi′′)
9
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