Non-Markovian dynamics and steady-state entanglement of cavity arrays in finite-bandwidth squeezed reservoirs

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by  Stefano Zippilli

Non-Markoviandynamics andsteady-state entanglement ofcavityarrays infinite-bandwidth
squeezed reservoirs
S. Zippilli1 and F. Illuminati1,2,3
1DipartimentodiIngegneriaIndustriale,Universita` degliStudidiSalerno,ViaGiovanniPaoloII132,I-84084Fisciano(SA),Italy
2INFN,SezionediNapoli,GruppoCollegatodiSalerno,I-84084Fisciano(SA),Italy
3CNISMUnita` diSalerno,I-84084Fisciano(SA),Italy
(Dated:February16,2014)
Whentwochainsof quantumsystemsaredriven attheirendsbyatwo-modesqueezed reservoir, theyap-
proachasteadystatecharacterizedbytheformationofmanyentangledpairs.Eachpairismadeofoneelement
ofthefirstandoneofthesecondchain. Thiseffecthasbeenalreadypredictedundertheassumptionofbroad-
bandsqueezing. Hereweinvestigatethesituationoffinite-bandwidthreservoirs. Thisisdonebymodelingthe
4
drivingbathastheoutputfieldofanon-degenerateparametricoscillator.Theresultingnon-Markoviandynam-
1
icsisstudiedwithinthetheoreticalframeworkofcascadeopenquantumsystems.Itisshownthattheformation
0
of pair-entangledstructuresoccurs aslong asthenormal-mode splittingof thearraysdoes not overcome the
2
squeezingbandwidthofthereservoir.

b
e PACSnumbers: 42.50.Dv,03.67.Bg,42.50.Ex03.65.Yz
F

6
1 I. INTRODUCTION lator, essentially the strengths of the nonlinearity and of the
dissipation. We show that the pair-entanglement replication

] mechanismpreviouslyfoundinthecaseofabroadbandreser-
The ability to manipulate, engineer and control entangle-
h
voir continues to hold also in the more realistic scenario of
p ment is of primary importance for the development of new
finitebandwidth. Thisfindingisthusquitepromisingforthe
- quantumtechnologies,whichareexpectedto outperformthe
t actualrealizabilityofthemechanisminconcreteexperimental
n correspondingclassicalimplementationsinmanyfieldsofap-
a plication, including quantum computation, communication, setups.
u Thepaperisorganizedasfollows. InSec.IIweintroduce
metrology, and sensing. In particular, the manipulation of
q themodelintermsofquantumLangevinequations.InSec.III
entanglementover large quantum ensembles [1], and the ef-
[ weanalyzethebasicdynamicsoftheoutputfieldofthenon-
ficienttransfer and distributionof quantumresourcesamong
2 different systems [2] are fundamental tasks for the develop- degenerate parametric oscillator (Sec. IIIA), and of the two
v ment of scalable devices for quantum computation and in- arrays(SecIIIB).Arelevantresultofthisanalysisistheiden-
1 tificationoftheregimesinwhichonerecoversthebroadband
formation processing [3]. In this context, a scheme was re-
4 centlyproposed[4] forthe efficienttransferofquantumcor- Markovian dynamics considered previously [4]. In Sec IV,
2
wereportnumericalresultsforthesteady-stateentanglement
relations from a broadbandtwo-mode entangled reservoir to
8
betweenthe two arrays. InSec.V we analyzethefeasibility
. non-directlyinteractingchainsofquantumsystems. Thisphe-
1 oftheschemewithrealisticexperimentalparameters.Finally,
nomenonisobservedatthesteadystateoftheopenincoherent
0 inSec.VIwedrawourconclusionsanddiscusssomepossible
dynamics in the framework of quantum reservoir engineer-
4
outlooks.
1 ing [5, 6]. Specifically, it was shown [4], that at the steady
: state,manypairsofentangledsubsystemsreplicatetheorigi-
v
nalentanglementofthereservoir,asillustratedinFig.1. This
Xi mechanismappearsto be general, holdingbothfor arraysof II. THEMODEL
r linearlycoupledharmonicoscillatorsandtwo-levelsystems.
a In the present work we generalize the previous investiga- Weanalyzethedynamicsoftwoindependent,non-directly
tionbyrelaxingthebroadbandconditionofthereservoir. In- interactingone-dimensionalarrays(chains)ofNlinearlycou-
deed,forfinitebandwidththeresultingdynamicsofthearrays pled harmonic oscillators. In Fig. 1 we illustrate a specific
is in generalnon-Markovian. In order to study this problem realization in terms of electromagnetic field modes confined
we consider in detail the specific physicalsystem producing insidecavities. Oneoftheendsofeacharrayisincoherently
theentangledreservoir. Inthisway,thefullcoupleddynam- coupled,with rate ζa, to the outputfield ofa non-degenerate
ics of the arrays and of such system is Markovian and can parametricoscillator. Thecouplingisnon-reciprocalandre-
be efficiently analyzed in the standard approach of cascade alizesacascadeconfiguration[7–10,16],inwhichthearrays
open quantum systems [7–10]. We will study the dynamics have no effect on the dynamics of the parametric oscillator.
oftwodistinctarraysoflinearlycoupledharmonicoscillators This situation can be realized using non-reciprocal devices
interacting locally with the output field of a non-degenerate suchas,forinstance,standardopticalcirculatorsbasedonthe
parametric oscillator. The latter is one of the most efficient Faraday effect [16, 17]. The creation and annihilation oper-
sources of entangled fields and it is routinely used in many ators for the two arrays are aξ,j and a†ξ,j, where the index ξ
applicationsofquantumtechnologies[11–15]. Itsoutputisa takesthevaluesI orII anddistinguishesbetweenthetwoar-
continuoussqueezedfieldwhosestatistical propertiescanbe rays, while j indicates the element in each array, as shown
controlledthroughthe characteristic parametersof the oscil- in Fig. 1. The oscillators in each chain are resonant, but the
2
fieldoperators[7,8]. Then,theequationsofmotionforeach
arrayread
a˙ (t) = (κ +ζ ) a (t) iη a (t) (3)
ξ,1 1 a ξ,1 1 ξ,2
− −
2κ a(in)(t) 2ζ r(out)(t),
− 1 ξ,1 − a ξ
a˙ (t) = κpa (t) iη pa (t) iη a (t),
ξ,j j ξ,j j 1 ξ,j 1 j ξ,j
− − − − −
2κ a(in)(t) for j 2...N 1
−q j ξ,1 ∈{ − }
a˙ (t) = κ a (t) iη a (t) 2κ a(in)(t),
ξ,N − N ξ,N − N−1 ξ,N−1 − N ξ,N
p
withξ I,II . Herea(in)arethezero-averageinputnoiseop-
∈{ } ξ,j
erators for each cavity, whose only nonvanishingcorrelation
function is (cid:28)a(ξi,nj)(t)a(ξi′n,)j′†(t′)(cid:29) = δξ,ξ′δj,j′δ(t−t′). We remark
thatin Eq.(3) the operatorforthe outputfield of the nonde-
generateparametricoscillator, r(out), acts as a source term in
ξ
theequationforthefirstcavityofeacharray.Theequationsof
FIG.1: Twoarraysofcavitiesaredrivenattheirendsbytwo-mode motionforthefieldoperatorsoftheparametricoscillatorare
squeezed light. Arrowsdenote theinter-arraypairsof cavitiesthat
bsuefficocmieentelnytalanrgglee.d atsteady statewhenthesqueezing bandwidth is b˙I(t) = −(κ0+ζb)bI(t)+αb†II(t)
2κ b(in)(t) 2ζ r(in)(t),
− 0 I − b I
frequency of the two chains can be different. In each array, b˙II(t) = −(pκ0+ζb)bII(t)+pαb†I(t)
cavitiesarecoupledwithnearest-neighborstrengthηjandlose − 2κ0b(IiIn)(t)− 2ζbrI(iIn)(t), (4)
excitationsatarateκ .Thetwoarraysareassumedtobesym- p p
j
metric, i.e. they have the same interaction strengths η and where b(in) and r(in) are the corresponding input noise oper-
j ξ ξ
lossratesκ . Deviationsfromthissymmetricconditionhave
j ators, whose correlationfunctionssatisfy b(in)(t)b(in)†(t ) =
been previouslyinvestigated[4] showing that the replication (cid:28) ξ ξ′ ′ (cid:29)
mechanism is significantly resilient to random asymmetries. r(in)(t)r(in)†(t ) = δ δ(t t). The noise operator r(in) ac-
The total Hamiltonian for the two arrays, in interaction pic- (cid:28) ξ ξ′ ′ (cid:29) ξ,ξ′ − ′ ξ
counts for the modes of the reservoir which are involved in
tureis
thedrivingprocess. Theoperatorb(in) accountsforthe other
ξ
N−1 possibledissipationchannels,withdecayrateκ0, whoseout-
Ha = ηj aξ,j†aξ,j+1+aξ,j+1†aξ,j . (1) put field is lost and cannot be used to drive the two arrays.
ξXI,II Xj=1 (cid:16) (cid:17) The outputfield operatorscorrespondingto each inputoper-
∈{ }
atorarerelatedtothesystemoperators(parametricoscillator
Thetwomodesofthenondegenerateparametricoscillatorare andarrays)throughthestandardrelations[7,8]
bξ andb†ξ,whereξ ∈ {I,II}indicates,respectively,themodes
whoseoutputdrivesthearrayI and II,andtheyareresonant r(out)(t) = r(in)(t)+ 2ζ b (t), (5)
ξ ξ b ξ
atthefrequenciesofthecavitiesofthecorrespondingarrays. p
Moreover, they mutually interact with coupling strength α. aξ(o,ujt)(t) = a(ξi,nj)(t)+ 2κjaξ,j(t), (6)
Throughoutthepresentworkthenonlinearcouplingstrength q
b(out)(t) = b(in)(t)+ 2κ b (t). (7)
αis assumed, withoutloss ofgenerality,tobe realandposi- ξ ξ 0 ξ
tive. A differentassumption wouldcorrespondto a different p
InparticularEq.(5)canbeusedtoeliminatetheoutputfield
squeezingphase, buttheoverallentanglementdynamicswill
in the equationsfor the arraysand to make explicittheir de-
remain unaffected. The corresponding parametric Hamilto-
pendenceontheoperatorsoftheparametricoscillatorandon
nianreads
thecorrespondinginputnoisefield.
H = iα b b b b . (2)
b I† II† I II
−
(cid:16) (cid:17)
III. DYNAMICS
Thetwomodesarefurthercoupledtotheoutputmodeswhich
drivethearrayswithrateζ ,seeFig.1. Theoutputfieldoper-
b
atorsarer(out) andr(out)† andtheyfulfillthestandardbosonic The dynamics described by the linear equations (3), (4)
ξ ξ and(5)isGaussian,henceitiscompletelydeterminedbythe
fieldcommutationrelations[rξ(out)(t),rξ(o′ut)(t′)†]=δξ,ξ′δ(t−t′). equationsfortheaverageandthecorrelationsofthefieldop-
The cascade quantum dynamicsis convenientlydescribed erators (first and second statistical moments). In particular,
in terms of Heisenberg quantum Langevin equations for the assuming that the initial fields averages are zero, they will
3
remain zero at all times. This is the situation that we as- fieldoperatorsr = r(out),r(out),r(out)†,r(out)† ,as
out I II I II
sume through the rest of this article. Hence, in the follow- (cid:18) (cid:19)
ing,wewillfocusourinvestigationontheanalysisofthecor-
(out)(t,t ) = r (t) r (t ) . (11)
relation matrix of the array operators a(t), whose elements Cr ′ j,k { out }j out ′ k
are defined, in terms of the vector ofCthe 4N field operators n o D (cid:8) (cid:9) E
a= aI,1···aI,N,aII,1···aII,N,a†I,1···a†I,N,a†II,1···a†II,N ,as dTehpeecnodrsreosnploynodninthgestdeiaffdeyr-esntacteeocfotrhreeltaimtioenamrgautmrixenCtsr(o,,sutat)n(td−cat′n)
(cid:16) (cid:17)
beexpressedas
(t) = a(t) a(t) . (8)
{Ca }j,k { }j { }k
ThismatrixcontainsalltheDinformationaEboutthestateofthe Cr(o,sutt)(τ) = δ(τ)Y+ α2ζb e−αα¯¯ι |τ|Wι (12)
arraysandcanbeused,forexample,toanalyzetheentangle- Xι= ι
±
mentpropertiesof the steady state. In particular, we will be
wherethe parametersα¯ are the decayrates ofthe field cor-
interestedinthebipartiteentanglementofpairsofinter-array
relation functions. In t±urn, as discussed below, they deter-
cavitymodes,quantifiedbythelogarithmicnegativity.
mine the bandwidth over which the statistical properties of
The logarithmic negativity, , for two generic bosonic
EN the reservoir are of the same order of magnitude. They are
modes can be computed as follows. Given a generic corre-
definedas
lationmatrix fortwomodesthecorrespondinglogarithmic
C
negativity canbeeasilyevaluatedas[18]:
N α¯ =ζ +κ α. (13)
E b 0
± ±
N =max 0, logν , (9) InEq.(12),the4 4matrix hasonlytwonon-zeroelements:
E − − × Y
(cid:8) (cid:9)
where ν denotes the smallest symplectic eigenvalue of the = =1. (14)
matrix 1− ( + T) T with {Y}1,3 {Y}2,4
2T C C T
It corresponds to the correlation matrix for two modes in
1 0 1 0 vacuum, and it is asymmetric and nonvanishing because of
i 0 i 0 the non-commutativity of the corresponding field operators.
= −  . (10)
T  00 −1i 00 1i  Moreover,W±isthe4×04m1atri∓x1 0
In the following sections we will use this definition to com- = 1 0 0 ∓1  , (15)
pEanu(Ncdtaevs)[ebcjoIo,thnjdIIt]ha,rewrahlyoegrreaersjiptIheacmntidivcejIlnIye,agaraentdtihvteihtycealvooifgtyatwriniotdhimccaeicvsiontyfegthmaetoifivdiretsyst W± ∓01 ∓01 01 10 
andthepartofEq.(12)thatcontainsthesematricesaccounts
ofthetwomodesofthereservoir (PO).
EN for a finite number of excitations and for the correlations of
the two modes. The corresponding spectral density matrix
A. Spectralpropertiesoftheoutputfieldofthenondegenerate Cr(out)(ω)= ∞ dτeiωτCr(o,sutt)(τ)isgivenby
R−∞
parametricoscillator e αζ
(out)(ω) = + a . (16)
Cr Y α¯2+ω2Wι
Beforeconsideringindetailtheresultsforthesteadystate Xι= ι
e ±
of the open dynamics for the two arrays, in this subsection
It is the sum of two Lorentzian whose bandwidths are
we describe the mainfeaturesof the steady state of the non-
α¯ and α¯ respectively, and it is related to the correla-
+
degenerate parametric oscillator which are useful in the fol- tion functi−ons of the output field in frequency space and,
lowing. The open dynamics of a nondegenerate parametric
hence, to the spectrum of the squeezed reservoir as fol-
oscillator(see[19–22]foradetaileddiscussion)isdescribed
lows. Let as define the vector of operators in frequency
bythequantumLangevinequation(4)whichadmitsasteady
space r˜ (ω) = r˜(out)(ω),r˜(out)(ω),r˜(out) (ω),r˜(out) (ω) , with
statesolutiononlyiftheratesforthedecayprocessesaresuf- out I II I † II †
ficientlylargetocounterbalancetheeffectsoftheparametric r˜(out)(ω) = 1(cid:16) ∞ dτ eiωτ r(out)(τ), which implie(cid:17)s that
ξ √2 ξ
amplification. Inthiscasetheparametricoscillatorissaidto r˜(out)(ω) † = r˜(oRu−t)∞( ω). Then r˜ (ω) r˜ (ω) =
workbelowthreshold.Theconditionforthisregimeis ξ ξ † − { out }j { out ′ }k
nδ(ω+ ωo) (out)(ω). We note that,Dsince the system HaEmil-
′ r
C
α<ζ +κ , tonianisdefinedininteractionpicture,thefrequencyωisnot
b 0 e
anabsolutefrequency,butitisdefinedrelativetothefrequen-
whereζ +κ accountsforthetotaldissipationrate.Thiscon- ciesofthemodesofthenondegenerateparametricoscillator,
b 0
ditionwillalwaysbeassumedinthefollowing. whichinturnareequaltothefrequenciesofthecavitymodes.
The output field is completely characterized by the two- Being relative, it can take both positive and negativevalues.
time correlation matrix (out)(t,t ) whose elements are de- Inparticularthemodeatω=0ininteractionpicture,r˜(out)(0),
Cr ′ ξ
fined,intermsoftheelementsofthevectorofthefouroutput corresponds to the spectral componentof the output field at
4
thefrequencyofthecavitiesofthearrayξintheoriginalpic- B. Reducednon-Markoviandynamicsforthetwoarrays
ture.
The spectral density matrix in Eq. (16) can be used to
To gain insight into the system dynamics and in order
construct the squeezing spectrum, that is the noise spec-
to draw a clear comparison with the case of a broadband
tral density associated to the collective quadrature X =
squeezed reservoir [4], in this subsection we discuss the re-
(cid:18)(1rI(,out1),+1r,I(ou1t))†T.−IrtI(ioIsutg)i−vernI(oIbuty)†(cid:19)/√2 ≡ uT rout, where u = dtauinceeddnaoftne-rMtraarckionvgiaonutdythneamdeicgsreoefsthoef tfwreoedaorrmayosfththateipsaorba--
− − metricoscillator. Weanticipatethatnoapproximationisper-
formedinthederivationofthereduceddynamicalequations,
S(ω) = uT (out)(ω)u
Cr hence the corresponding dynamics of the arrays is equal to
= 1 e4αζb . (17) thatobtaineddirectlyfromthefullmodelinEqs.(3),(4)and
− α¯2 +ω2 (5).
+
The equations (4) and (5) for the nondegenerateparamet-
According to our definitions, the output field is two-mode ricoscillatordonotdependonthedegreesoffreedomofthe
squeezed when S(ω) < 1. Specifically, Eq. (17) indicates arrays. Therefore, they can be solved and used to obtain a
thatthespectralcomponentsoftheoutputfieldaresqueezed set of closed equationsforthe dynamicsof the arraysalone.
over a bandwidth α¯+, with maximum squeezing attained at Theresultingequationforthecorrelationmatrixofthearrays
thecentralfrequencyω = 0. Ontheotherhand,theorthogo- (t), whose definition was given in Eq. (8), is linear with a
a
nalquadratureY = vT rout, withv = i(1, 1, 1,1)T, isanti- tCime-dependentsourcetermoftheform
− −
squeezed,namelyitsvarianceislargerthenone. Indeed,the
correspondingnoisespectrumisgivenby
∂
(t)= (t)+ (t) T + (t), (20)
4αζ ∂tCa MaCa Ca Ma Na
T(ω)=1+ b , (18)
α¯2 +ω2
− where isthe4N 4Nmatrixofcoefficientscorresponding
a
M ×
which exhibitsanti-squeezingover a bandwidthα¯ . It is in- tothehomogeneouspartofthesystemofequations(3);itisa
terestingtopointoutthatsqueezedandanti-squeez−edquadra- tridiagonalmatrixwhose elements, alongthe maindiagonal,
are
tures are characterized by non-equalbandwidths, which can
be even quite different. In particular, this is true when the
squeezing is maximum. Indeed, maximum squeezing is ob-
= (ζ +κ ) forℓ 0,N,2N,3N ,
tainedinthelimitα¯ 0,namelyatthethresholdofthepara- {Ma}1+ℓ,1+ℓ − a 1 ∈{ }
metricoscillator. In−th→iscase,perfectsuppressionofthefluc- {Ma}j+ℓ,j+ℓ =−κj forℓ∈{0,N,2N,3N}, and j∈{2···N},
tuationsofthesqueezedquadratureisobservedatthecentral (21)
frequencyω = 0,thatisS(0) 0;then,asalreadyremarked,
∼
thesqueezingextendsoverafinitebandwidthα¯ .Ontheother
+
while the non-zero elements along the first upper and lower
hand, in order to satisfy the Heisenberguncertaintyrelation,
diagonalsare
the fluctuations of the orthogonal anti-squeezed quadrature
must diverge at the central frequencyand the corresponding
bandwidthvanishes.
= =
Two-modesqueezingisastrongsufficientconditionforen- {Ma}j+ℓ,j+ℓ+1 {Ma}j+ℓ+1,j+ℓ −{Ma}j+ℓ+2N,j+ℓ+2N+1
= = iη
tanglement in continuous-variablesystems [14, 15]. In par- −{Ma}j+ℓ+2N+1,j+ℓ+2N − j
forℓ 0,N , and j 1 N 1 .
ticular, the squeezing spectrum at a given frequencyω mea-
∈{ } ∈{ ··· − }
surestheentanglementbetweenthespectralcomponentatfre- (22)
quencyωofthefirstmode,andthespectralcomponentatfre-
quency ωofthesecondone.Thecorrespondinglogarithmic
negativi−ty (PO)(ω)canbeevaluatedapplyingthedefinitionin The time-dependent inhomogeneous source term Na(t) in
EN Eq.(20)isa4N 4N matrixdefinedas
Eq.(9)tothecorrelationmatrix (out)(ω)definedinEq.(16). ×
r
C
Inourcase, thesmallestsymplecticeigenvalueν (ω),evalu-
atedforeachspectralcomponente,isindeedequal−tothevalue t
ofthecorrespondingsqueezingspectrum Na(t) = Qa+2ζaZ dτ eMa(t−τ)ZCr(out)(τ,t)ZT
0 h
ν (ω)=S(ω). (19) +ZCr(out)(t,τ)ZTeMaT(t−τ), (23)
i
−
Correspondingly, maximum entanglement is found for the where a is a matrix whose non-zero elements are
Q
spectralcomponentsatthe centralfrequencyω = 0, and the {Qa}j+ℓ,2N+j+ℓ = 2κj for ℓ ∈ {0,N} and j ∈ {1···N}. The
entanglementextendsoverabandwidthα¯ . quantity (out)(t,t )isthe4 4correlationmatrixalreadyde-
+ r ′
C ×
5
finedinEq.(11). Finally, isthe4N 4matrixdefinedas where H is the Hamiltonianof the two arrays, Eq. (1), and
a
Z ×
thetime-dependentKossakowskimatrix (t),responsiblefor
a
1 thenon-MarkoviancharacterofthedynaKmics,isdefinedas
.
Z=0.. 10... 10... 10...  , (24) wwblihotehcrkeJs1atr2heNen4iusNllth×mea42tJNKrNicas=1(ey×.ts)m .2−pB=Nlr1eoc2ia21dNtdieJbcnam1tnNidt2ayNatlr(imimt!x)iaJ,ttri,x, and the mis(s(32in09g))
wherethemissingentriesareallzeros. Eqs.(20)and(28)allowforadirectanalysisofthelimitof
Inparticular,ifweassumethattheparametricoscillatoris, infinitebandwidth.Thiscorrespondstothesituationinwhich
attheinitialtimeofthearraysdynamics,initsstationarystate, the parametersα¯ , defined in Eq. (13), are the largestin the
thentheresultinEq.(12)canbeusedtoexpress a(t)as systemdynamics±. Since,bydefinition,α¯+ > α¯ , thislimitis
N obtainedwhenα¯ ismuchlargerthentheeigen−valuesof ,
a
1 thatis,inparticu−lar,when M
(t)= +2ζ T αζ ζ (25)
Na Qa aZYZ − b a α¯
ι=X+, ι α¯ ζ ,κ ,η . (31)
×1−e(Maα−¯α¯ι)t ZWιZT +ZWι{Z−T} 1−eT(MTaα−¯α¯ι)t, In this limit, the steady−s≫tateacorjreljation matrix of the reser-
 Ma− ι Ma − ι  voir,Eq.(12),takestheform
where and are defined in Eqs. (14) and (15) respec-
ι
tively. YThe firsWt two termsof this expressionaccountfor the (out)(τ) = δ(τ) +αζ Wι (32)
Cr,st Y b α¯2 
deffisesciptaotfiothnes eonftathneglaerdraryesse,rwvohiirl.eItthdeepleanstdsteormnbiosthdutheetporothpe-  Xι=± ι 
andEq.(25)reducesto
ertiesoftheoutputfieldoftheparametricoscillator(through
theparametersα¯ )andofthetwoarrays(throughthematrix T
Ma). ± Na(t)=Qa+2ζaZYZT +2αζbζa ZWα¯2ιZ .
In the following, we are interested in the steady state of ι=X+, ι
{ −}
the arrays. The corresponding correlation matrix (st) can (33)
a
C
be computedfromEq. (20). Defining the linear operator
a
which operates on a generic correlation matrix as L= Hencethecorrespondingmasterequationforthereducedden-
a
+ T, the steady state can then be exCpressLedCfor- sitymatrixofthearraysisequaltotheoneintroducedin[4]:
Ma C C Ma
mallyas
ρ˙ = i H ,ρ
a a a
−
(st) = 1 , (26) (cid:2)N (cid:3)
Ca −L−a N0 + κj 2aξ,jρaa†ξ,j−a†ξ,jaξ,jρa−ρaa†ξ,jaξ,j
where isthetime-independentpartofEq.(25),namely Xj=1 ξX=I,II(cid:16) (cid:17)
0
N
1 +ζa(n¯ +1) 2aξ,1ρaa†ξ,1−a†ξ,1aξ,1ρa−ρaa†ξ,1aξ,1
N0 =Qa+2ζaZYZT −αζbζa α¯ (27) ξX=I,II(cid:16) (cid:17)
1 T ι+=X{+,−} ι T 1 . +ζan¯ξX=I,II(cid:16)2a†ξ,1ρaaξ,1−aξ,1a†ξ,1ρa−ρaaξ,1a†ξ,1(cid:17)
×"Ma−α¯ι ZWιZ ZWιZ MTa −α¯ι# −2ζam¯ aI,1ρaaII,1+aII,1ρaaI,1
a a (cid:0) ρ ρ a a +h.c. , (34)
WefinallynotethatEq.(20)isequivalenttothefollowing − I,1 II,1 a− a I,1 II,1
non-Markovianmaster equationin Lindbladformforthe re- (cid:1)
where the parameters n¯ and m¯ account, respectively, for the
duceddensitymatrixρ ofthetwoarrays:
a numberofexcitationsandforthecorrelationsofthereservoir,
andaregivenby
ρ˙ = i H , ρ (28)
a a a
−
+ (cid:2) (t(cid:3)) 2a ρ a a a ρ ρ a a , 1 1 1 1
Xj,k {Ka }j,kh k a j− j k a− a j ki n¯ =αζb α¯2 − α¯2+!, m¯ =αζb α¯2 + α¯2+! . (35)
− −
6
Specifically,accordingtoEq.(32),theyfulfilltherelations IV. RESULTS
(cid:28)rξ(out)†(t)rξ(out)(t′)(cid:29)st = n¯ δ(t−t′), In this section we analyze numerically the entanglement
properties of the steady state of the two arrays. We char-
rI(out)(t)rI(oIut)(t′) st = m¯ δ(t−t′), (36) acterize the steady state in terms of the logarithmic negativ-
D E ity (cav)[j , j ] for the modes j and j of first and second
where the label st indicates that the averages are performed arraEyNrespeIctiIvIelly, which is evaIluatedIaIpplying the general
overthesteadystate. definition, Eq. (9), to the reducedcorrelation matrix of each
Therefore,inthislimitwerecoveralltheresultsof[4]and, pair of cavity modes which, in turn, is obtained form the
in particular, when κj = 0, the exactsteady state solution of full steady state correlation matrix defined in Eq. (26). In
Eq.(34)takestheform particular, we analyze the normalized logarithmic negativity
E(cav)[j , j ] = (cav)[j , j ]/( (cav)[j , j ]+1), whichtakes
N N I II EN I II EN I II
valuesbetween0and1.
ρ(ast) = Uj̺I,j⊗̺II,j U†j (37) Weareinterestedinidentifyingparameterregimesinwhich
Oj=1
the entanglement replication mechanism described in [4] is
where̺ isthethermalstate still visible even if the assumptions assumed in [4] are not
ξ,j
strictly satisfied. According to the mechanism of entangle-
̺ = ∞ 1 n¯T n n n , mtieesnwtritehpleiqcuatailoinnddiecsecsriableodnginth[e4]t,waotathrreasytseaardeyesntatatengthleedcwavitih-
ξ,j 1+n¯ 1+n¯ ! | ih |
Xn=0 T T their logarithmic negativity equal to that of the driving two-
modesqueezedfield. Thisresultisexactifthedrivingentan-
with n¯ = 1 (2n¯+1)2 4m¯2 1 averagephotons, and it gledreservoirisbroadband. AsdiscussedinSec. IIIB1this
is the sTame2fo(cid:20)rpall the cav−ities. −Mo(cid:21)reover U is a squeezing limitisexpressedbytheconditionEq.(31).
j
The behavior of the entanglementfor the pairs of cavities
transformationfor the two modes a and a , and it is de-
I,j II,j
withequalindicesasafunctionofthebandwidthofthedriv-
finedas
ing reservoir is reported in Fig. 2, in the ideal situation in
Uj =e(−1)js0(cid:16)aI,jaII,j−a†I,ja†II,j(cid:17), wanhdicahlloontlhyerthceavfiirtsitescaavrietyloisnsleeascsh(κarr=ay0c,anj)d.isHsiepraeteth(eζam,ax0i)-
j
∀
mumoftheentanglementofthereservoir,namelythevalueof
wherethesqueezingparameters canbeexpressedas
0
(PO)(0)(seeSec.IIIA),isfixedbykeepingconstanttheratio
EN
n¯ n¯ α/ζ . Fig. 2 shows how the amount of replicated entangle-
tanh(s )= − T . (38) b
0 m¯ mentreduceswith decreasingbandwidth. At largevaluesof
ζ thebandwidthislargeandtheentanglementofthepairsof
b
Correspondingly,inthesteadystate,thecorrelationfunctions cavitieswithequalindicesapproachesthevalueoftheentan-
forthearraysare glementofthereservoir. Ontheotherhand,asζ islowered,
b
theinter-arraypair-entanglementisreducedcorrespondingly.
a†ξ,jaξ′,j′ st =n¯ δj,j′ δξ,ξ′ ThearrowsontheupperpartofFig.2correspondtothere-
D E sultofFig.3,4and5. Fig.3correspondstotherightmostar-
Daξ,ja†ξ′,j′Est =(n¯+1)δj,j′ δξ,ξ′ row.Herebothα¯+andα¯ arelargerthentheparametersofthe
DaI,jaII,j′Est =Da†I,ja†II,j′Est =(−1)j+1m¯ δj,j′ awreraoybss(eηrjv,eκjaansudbζsat)a,ntthiae−lberonatadnbgalnedmceonntdrietipolniciastisoanti.sfiSepde,cainfid-
aξ,jaξ,j′ st = a†ξ,ja†ξ,j′ st =0, (39) cally, the plot in Fig. 3 (a) reports the steady-state pairwise
D E D E entanglementofalltheinter-arraypairsofcavities(including
and, hence, the logarithmic negativity for each pair of cav- those with differentindices) for two equal arraysof N = 10
ities with equal indices along the two arrays is given by cavities each. We observe that only the cavities with equal
(cav)[j, j] = log(2n¯+1 2m¯) = log(1 4αζ /α¯ ), indicesarestronglyentangled,inagreementwiththepredic-
EN − − − − b +
which is equal to the logarithmic negativity between the tionsofthebroadbandmodel[4].Thefactthatinthiscasethe
modesofthereservoir. squeezingbandwidthofthereservoirislargecanbeseenlook-
In the nextsection we analyze numericallythe validity of ingatFig.3(c).Here,thevaluesofthelogarithmicnegativity
thislimit,andhowtheresultsof[4]aremodifiedwhencondi- E(PO)(ω)= (PO)(ω)/( (PO)(ω)+1)andofthesqueezingspec-
N EN EN
tion(31)isnotfulfilled. trumS(ω)of theoutputfieldof theparametricoscillatorare
A final remark is in order. Eq. (31) implies that if the comparedwiththerangeoffrequencies,indicatedbythever-
parametric oscillator is very close to the threshold condition tical lines, that correspondsto the frequenciesof the normal
(α¯ 0), that is, as discussed in Sec. IIIA, when the entan- modesofthearrays.Thelatterareevaluatedastheimaginary
gle−m∼ent is maximum, then the broadband assumption is no partoftheeigenvaluesofthematrix definedinEqs.(21)
a
M
longervalidevenifthesqueezingspectrumisrelativelybroad and(22). Indetail, thisplotshowsthatthebandwidthofthe
(itsbandwidthis,infact,givenbyα¯ ). squeezingissignificantlylargecomparedtothenormalmode
+
7
1
(a) 1 (b)
(cav)E [j,j]N 0000....2468 ijjjjn====15910 (cav)EN0.50112345jI6I78910 1234j5I678910 (normal modes)EN0.15023k4I5I678190 1234k5I 678910
0 1
10−2 10−1 100 101 102
ζ (units of ζ ) ω)
b a S( (c)
ω) , 0.5
FIG.2: Logarithmicnegativityforinter-arraypairsofcavitieswith O)(
equal indicesalongtwoequal arraysof N = 10cavitieseach, asa (PEN
function of the emission rate ζ of the parametric oscillator, when
b 0
theratio α/ζb = 0.648 iskept constant inorder tofixthe value of −5 −2 ω (unit0s of ζ ) 2 5
theentanglementatthecentralfrequencyoftheoutputfield,i.e. the a
reservoirthatdrivesthetwoarrays. Theinsetspecifiestheindexof
thecavitiescorresponding toeach curve. Thegraysolidthickline FIG.4: AsinFig.3 withζ = ζ and α = 0.648ζ . Theseresults
b a a
indicates the value of the entanglement at the central frequency of correspondtothecentralarrowinFig.2.
thereservoir. Theotherparametersare,η = ζ , j 1,...N and
j a ∀ ∈ { }
κ =0, j 0,...N . Thearrowsonthetoppartoftheplotindicate
j ∀ ∈{ }
theparameterscorresponding,respectively,totheresultsofFigs.3,
4and5. (a) 0.4 (b)
(a) 1 (b) (cav)EN00..12 (normal modes)EN0.02
(cav)EN0.015 (normal modes)EN0.05 012345jI6I78910 1234j5I678910 1234k5II678190 1234kI5678910
12345jI6I78910 1234j5I678910 1234k5II678910 1234k5I 678910 ωωS()) , 0.51 (c)
1 (c) O)(
ωS() (PEN
ω) , 0.5 −05 −2 0 2 5
(PO)E(N ω (units of ζa )
0 FIG.5:AsinFig.3withζ =0.1ζ andα=0.0648ζ .Theseresults
−10 −5 0 5 10 b a a
ω (units of ζ ) correspondtotheleftmostarrowinFig.2.
a
FIG.3: (a)Logarithmicnegativityforpairsofcavitymodesintwo
equal arrays of N = 10 cavities: jI and jII are the indices of the correspondingentanglementforpairsofnormalmodesofthe
cavitiesofthefirstandthesecondarrayrespectively.(b)Logarithmic arrays is reported in Fig. 3 (b). Here, the indices k , with
ξ
negativity for the pairs of normal modes: kI and kII are indices of ξ I,II , indicate the normal modes of the arrays, where
thenormalmodesofthefirstandthesecondarrayrespectively,with ∈ { }
k = 1 andand k = 10 arethe lowest andhigherfrequency
increasingorderineachindexcorrespondingtomodesofincreasing ξ ξ
modesrespectively. Weobservethatamodeofthefirstarray
frequency. (c) Entanglement E(PO)(ω) (solid black line) and two-
N atfrequency,say,ω relativetothefrequencyofthecavities
modesqueezingspectrumS(ω)(dashedblueline)oftheoutputfield kI
inarrayI isentangledwiththereciprocalmodeofthesecond
of the parametric oscillator. The system parameters are η = ζ ,
j a
1∀6j.4∈8{ζ1,,..α¯.N},=κj3=.520ζ,∀).j∈Th{0e,se...Nre}s,uζltbs=co1r0rζeaspaonnddαto=t6h.e48riζgah(tαm¯+os=t barerauyndweirtshtooopdpboysiotebsfreerqvuinegncthyaωt,kaIIs=disωcuNs−skeId=in−SωekcI..ITIIhAis,ctahne
a a
arrowinFi−g.2. driving two-mode field exhibits a similar structure of entan-
glementbetweenthedifferentspectralcomponents. Namely,
thespectralcomponentofthefirstmodeofthesqueezedreser-
splitting of the arrays and that all the normal modes feel a voiratfrequencyωrelativetothefrequencyofthecavitiesis
reservoirwithroughlythesamelevelofentanglement.Inthis entangledwiththecomponentofthesecondmodeatrelative
case, the Markoviandescriptionof Ref. [4] is accurate. The frequency ω, and the normal modes of the cavities simply
−
8
1 1
0.8 0.8
in
(cav)E [j,j]N 00..46 ijn=1 (cav)E [j,j]N 00..46 jjj===159
j=5 j=10
j=9
0.2 j=10 0.2
0 0
10−2 α10 0(units of ζ ) 102 10−1 100ζ (units of ζ )101 102
+ a a b
FIG. 6: Logarithmic negativity for the pairs of cavities withequal FIG. 8: Logarithmic negativity for the pairs of cavities with equal
indicesalongtwoequalarraysofN =10cavities,asafunctionofthe indicesalongtwoequalarraysofN =10cavities,asafunctionofthe
parameterα¯+,whenα¯ =0.01ζaiskeptconstant.Theinsetspecifies rateζa atwhichthearrayexchangephotonswiththereservoir. The
theindexofthecaviti−escorrespondingtoeachcurve. Thegrayline insetspecifiestheindexofthecavitiescorrespondingtoeachcurve.
indicates the value of the entanglement at the central frequency of Hereallthelineswith j,1arealmostsuperimposed. Thegrayline
thereservoir. Theother parameters areη = ζ , j 1,...N and indicatesthevalueoftheentanglementatthecentralfrequencyofthe
κj =0,∀j∈{0,...N}. j a ∀ ∈ { } rejservo0i,r....TNhe,aontdheαrp=ar0a.m64e8teζrs. areηj =0.1ζb,∀j∈{1,...N},κj =0,
∀ ∈{ } b
1
1
0.8
0.8
]
(cav)E [j,jN 00..46 ijn=1 (cav)E [j,j]N 00..46 ijjjn===159
0.2 j=2 j=10
j=5 0.2
0 j=9
10−2 10−1 100 101 j=11002 0
α+ (units of ζa ) 100 101η (units of ζ )102 103
a
FIG.7: AsinFig.6withκ =0.5ζ .
0 b
FIG. 9: Logarithmic negativity for the pairs of cavities with equal
indices along two equal arrays of N = 10 cavities, as a function
of thecoupling strengthη η , j 1,...N . Theinset specifies
reproducethissamefeature. ≡ j ∀ ∈ { }
theindexofthecavitiescorrespondingtoeachcurve. Thegrayline
When the bandwidth of the correlations of the squeezed indicates the value of the entanglement at the central frequency of
reservoirisreducedasinFig.4,correspondingtothecentral thereservoir. Theotherparametersare j 1,...N ,κ = 0, j
∀ ∈ { } j ∀ ∈
arrowin Fig.2, thereplicationmechanismisnolongeropti- {0,...N},ζb =10ζa,andα=0.648ζb.
mal and the entanglementof the pairs of cavities with equal
indices is reduced. In terms of the normal modes it corre-
spondstomaximumentanglementbetweenthenormalmodes terα¯ determinesthebandwidthoftheanti-squeezedquadra-
whosefrequencyisclosertothecentralfrequencyofthereser- ture.−Fig.6showstheentanglementofpairsofcavitymodes
voir, asillustrated in Fig. 4 (b). Whenthe bandwidthis fur- asafunctionofα¯ whenα¯ isfixedtoasmallvalue. Inthis
+
ther reduced,as in Fig. 5, which correspondsto the leftmost situation, the broadbandco−ndition, Eq. (31), is not satisfied.
arrows in Fig. 2, only few normal modesclose to resonance Hence,theMarkovianmasterequationusedin[4]isnotvalid
with the driving squeezed reservoir (the central modes) be- inthe entirerangeofparametersofFig. 6. Nevertheless,we
comeentangled(Fig.5(b)). Asthenormalmodescanbeex- observethatentanglementreplicationisstill inorderaslong
pressedintermsofthemodesofallthecavities,correspond- asthesqueezingbandwidthα¯ issufficientlylarge.Thisresult
+
inglytheoriginalentanglementofthereservoirisredistributed showsthattheentanglementreplicationisnotstrictlybasedon
amongallpairsofcavities,atthepriceofitsmagnitudebeing theMarkoviannatureofthedynamics.
stronglyreduced,asillustratedinFig.5(a). Asimilarsituation,withanadditionaldecaychannelatrate
InSec.IIIA,wehaveshownthatthecorrelationfunctions κ affecting the dynamics of the parametric oscillator, is an-
0
of the reservoir are characterized by two time scales associ- alyzed in Fig. 7. It corresponds, for example, to the case in
atedtotheparametersα¯ ,whereα¯ determinesthebandwidth which both mirrorsof the Fabry-Pe´rotcavity used to realize
+
ofthesqueezedquadratu±re(andhenceoftheentanglement)of theparametricoscillator,haveafinitetransmissivitybutonly
thefieldemittedbytheparametricoscillator,andtheparame- the output from one mirror is efficiently controlled to drive
9
1
(a) (b) (a)
0.4 1
(cav)EN0.2012345jII6789 1234jI56789 (normal modes)EN0.501234k5II6789 1234kI56789 (cav)E [j,j]N0000....2468 ijjjjjn=====125910
1 100 −2 10−1 100 101 102
ω) ζ (units of ζ )
S( (c) b a
ω) , 0.5 1
O)( in
(PEN 0.8 j=1
−0600 −400 −200ω (unit0s of ζa 2)00 400 600 (cav)E [j,j]N00..46 jjjj====25910
FIG.10: AsinFig.3withN =9,η =300ζ , j 1,...N ,κ =0,
j 0,...N ,ζ =10ζ ,andα=6.4j8ζ . a ∀ ∈{ } j 0.2 (b)
∀ ∈{ } b a a
0
10−2 10−1 100 101 102
ζ (units of ζ )
thearrays.Inthiscasepartofthequantumcorrelationswhich a b
are built up by the parametric oscillator are lost in the un- 1
(c)
controlledoutputandtheactualreservoirexperiencedbythe
0.8
arraysisnotinapuresqueezedstatebutinamixedsqueezed
in
]
troherpedrlemircaaotlriosltnaartaegs.elrIontnhFgaingas.t7htheweceshqoaubreaseceztreivnreigstthbiceanofdrnewsqeiudtetohnfcieisenostafontfhgeltehsmeameanre-t (cav)E [j,jN00..46 jjj===125
j=9
rays.Specifically,althoughthemaximumentanglementofthe
j=10
reservoir(thethick,graylineinFig.7)issignificantlyreduced 0.2
with respectto the situation illustrated in Fig. 6, the qualita-
0
tive behaviorof the entanglementis similar and, in the limit 10−2 10−1 100 101 102
ζ (units of ζ )
of large bandwidth (large α¯+), the entanglementof the cavi- b a
tiesreplicatesthe entanglementofthe reservoir. Also inthis
case we have verifiedthat, whenthe bandwidthof the reser- FIG.11: (a)AsinFig.2withκ =ζ .(b)AsinFig.8withκ =ζ
N a N a
voirissufficientlylarge,onlythemodeswithequalindicesare (κN isvariedtogetherwiththeparameter ζa). (c)AsinFig.2with
entangled. κj = 0.1ζa for j ∈ {1,...N}. The arrow on the top part of plot (c)
indicatestheparameterscorrespondingtotheresultsinFig.12.
So far we have analyzed the steady-state entanglementas
a functionoftheparametersofthe parametricoscillator. We
willnowstudythesteady-stateentanglementasafunctionof
mentdecreasesforallpairs.
theparametersofthe arrayswhenthe propertiesofthe para-
metricoscillatorarekeptfixed. Inparticular,weconsiderthe A particular situation is achieved when ζa is sufficiently
oscillator’s parameters given in Fig. 3, correspondingto the small and ηj is sufficiently large. In this case it is possible
situation of good entanglement replication, and we increase to have just a single normal mode resonant with the driving
either the exchange rate of photons ζ between the driving squeezed field. In Fig. 10 this is realized using arrays with
a
reservoirandthearraysorthecavity-cavitycouplingstrengths anoddnumberofcavitiessuchthatthereisalwaysanormal
ηjinsideeacharray.Whenζaisincreased,asshowninFig.8, modeat zero frequency. Hence when ηj is large, the central
onlythe entanglementof the first pair of cavities is affected, normal modes of the two arrays are the only modes that in-
and it decays to zero. Indeed, a large value of ζ means a teractwiththereservoirandtheygetefficientlyentangled,as
a
large line-width of the first cavity in each array. Hence, the shown in Fig. 10 (b). Due to the particular symmetric situ-
system behaveseffectively as if the externalreservoir would ationconsidered,thecentralnormalmodescanbeexpressed
interact directly with the second element in each array. All asalinearcombinationwithequalweightofthecavitymodes
other pairs of inter-array cavities with equal indices remain withoddindices. Correspondingly,allthecavitymodeswith
unaffected and continue to replicate the entanglementof the odd indices are entangled in pairs, as shown in Fig. 10 (a),
reservoir. On the other hand, when ζ is fixed and the cou- againatthepriceofarathersmallvalueofthereplicateden-
a
plings η are increased, as shown in Fig. 9, then all pairs of tanglement.
j
cavitiesareaffected;correspondingly,thereplicatedentangle- Ingeneral,whenfurthersourcesofdissipationarepresent,
10
15
(a) 1 (b) DdB 10
(cav)EN0.01512345jI6I78910 1234j5I678910 (normal modes)EN0.501234k5II678190 1234kI5678910 @NoiseSpectrum --105500.0 0.2 0.4 Ω@0G.6HzD 0.8 1.0 1.2
1 FIG. 13: Spectrum of the squeezed (solid, blue line) and anti-
ω) (c) squeezed (dashed, red line) quadratures, measured indB and eval-
S(
ω) , 0.5 luoagted[,Tu(ωsi)n]greEspqes.ct(iv1e7l)y.anTdhe(1p8a)r,amaset1e0rs×arelogα10=[S(0ω.8)]GaHnzd,1ζ0 ×=
(PO)E(N 1ex.1p1eG0riHmze,natnadlfiκ0nd=in0g.s05ofGRHezf..[T2h3i]s.behaviorisinagreementwithbthe
0
−10 −5 0 5 10
ω (units of ζ )
a
(a) 0.4 (b)
TFIhGes.e1r2e:suAltssicnoFrriegs.p3onwditthoζthbe=arζroawanidnκFjig=.101.1(ζca)for j ∈ {1,...N}. (cav)EN00..12 (normal modes)EN0.02
0 1
1 2
2 3
3 4 5 4
jII 4 5 1 2 jI3 kII 5 1 2 kI3 4 5
entanglementreplicationisfurtherdegraded,andthepairwise 1
steady-state entanglement can never attain the value of the ω)
minaFxiimg.1u1m. eNnteavnegrltehmeleensts,ofthtehequsqaulietaetzievderbeeshearvvoioirr,daessschriobwend ωS() , 0.5 (c)
O)(
previouslyisstill observed. InFig.11(a)and(b), wereport (PEN
the situation in which the two arrays are open at both ends
0
with the decay rate of the last cavity of each array equal to −5 0 5
ω (units of ζ )
the rate at which the first cavity exchangesphotonswith the
driving reservoir: κ = ζ . In this case E(cav)[j, j] saturates
N a N FIG.14: AsinFig.3withN = 5,η = 1GHz, j 1,...N ,ζ =
tovaluessmallerthanthevalueofthedrivingfieldwhenthe j ∀ ∈ { } a
0.1GHzκ = 0.1GHz, j 1,...N ,α = 0.8GHz,ζ = 1.1GHz,
squeezingbandwidthofthereservoirincreases. Inparticular, j ∀ ∈ { } b
andκ =0.05GHz.
allbutthefirstandthelastpairssaturatetothesamevalueof 0
entanglement,seeFig.11(a).Itisalsointerestingtonote,see
Fig.11(b),thatinthissituationonlytheentanglementofthe
V. EXPERIMENTALIMPLEMENTATION
firstandofthelastpairsdependsonthedissipationratesofthe
arrays,whiletheentanglementforalltheotherpairsremains
constant. The behavior of the first pair at large ζ is similar Investigation of the effects of a finite bandwidth is im-
a
to that discussed in Fig. 8, while the decay of E(cav)[j, j] for portant because the experimental realization of broadband
N
the last pair is due to the fact that at large κ the last pair is squeezing is extremely challenging. Recently, squeezing of
N
effectivelydecoupledfromtherestofthetwoarrays. light at telecommunication wavelength with a bandwidth of
several GHz and with noise reduction of 3dB ( 10dB
∼ ∼
when corrected for detector inefficiency) has been reported
Finally, in Fig. 11 (c) we consider the situation in which in[23]. Eqs.(17)and(18)yieldresultsconsistentwiththose
allthecavitiesarelossy. Inthiscasethemaximumreplicated of Ref. [23], as shown in Fig. 13, when α = 0.8 GHz,
entanglementissmallerthanthatofthedrivingfieldanditis ζb =1.1GHz,andκ0 =0.05GHz.
different for each pair. We note that even if all the cavities Arraysofcavitiesintherangeoftelecomwavelengthsare
dissipate and thus the replicatedentanglementis reduced, in activelyinvestigatedinvariousphysicalimplementations.Re-
any case, as long as the squeezing bandwidth is sufficiently alization of cavity arrays looks particularly promising using
large, only the inter-arraypairs of cavities with the same in- photoniccrystals. Exploitingthelatter,inRef.[24–26]cavity
dicescangetentangled,regardlessoftheeffectofadditional arraysofhighqualityfactorQ(Q 106)havebeenstudiedex-
∼
decaychannelsaffectingthedynamicsofthecavities. Anex- perimentally: thereportedcavitydecayratesareoftheorder
ampleisshowinFig.12whichcorrespondstotheparameters of 1 GHz, while the cavity-cavitycouplingstrengthsrange
∼
indicatedbythearrowintheupperpartofFig.11(c). from60to2000GHz. Theyarecontrolledbyrealizingdiffer-