Quantum coherent control of highly multipartite continuous-variable entangled states by tailoring parametric interactions

1.6 MB·English

by  Giuseppe Patera

Quantum coherent control of highly multipartite continuous-variable entangled states
by tailoring parametric interactions
Giuseppe Patera,1,∗ Carlos Navarrete-Benlloch,2,3 Germ´an J. de Valca´rcel,2 and Claude Fabre4
1Laboratoire de Physique des Lasers, Atomes et Mol´ecules,
Universit´e Lille 1, 59655 Villeneuve d’Ascq Cedex, France, EU
2Departament d’O`ptica, Universitat de Val`encia, Dr. Moliner 50, 46100 Burjassot, Spain, EU
3Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany, EU
4Laboratoire Kastler Brossel, Universit´e Pierre et Marie Curie-Paris6,
ENS, CNRS, 4 place Jussieu CC74, 75252 Paris cedex 05, France, EU
(Dated: January 17, 2012)
Thegenerationofcontinuous-variablemultipartiteentangledstatesisimportantforseveralproto-
2
colsofquantuminformationprocessingandcommunication,suchasone-wayquantumcomputation
1
orcontrolled densecoding. In thisarticle wetheoretically show that multimodeoptical parametric
0
oscillators can produce a great variety of such states by an appropriate control of the parametric
2
interaction,whatweaccomplishbytailoringeitherthespatio-temporalshapeofthepump,orthege-

n ometryofthenonlinearmedium. Specificexamplesinvolvingcurrentlyavailableopticalparametric
a oscillators are given, henceshowing that ourideas are within reach of present technology.
J

5 PACSnumbers: 42.50.Dv–Quantum stateengineeringandmeasurement,42.65.Yj–Opticalparametricoscil-
1 latorsandamplifiers,42.65.Re–Ultrafastprocesses,opticalpulsegenerationandpulsecompression.

]
h
I. INTRODUCTION researchfor the implementation of reliable and practical
p
quantum protocols for quantum communication and in-
-
t formation processing. Several of the promised benefits
n Squeezed states of light were introduced several
requirethoughaquantum“substrate”thatis createdby
a decadesagoasstateswhichcouldhelpbeatingthelimits
u set by quantum mechanics on the precision of measure- distributingquantumcorrelations(entanglement)among
q a number of degrees of freedom (modes) increasing with
ments performed with coherent light [1, 2]. These are
[ the complexity of the task to achieve. The protocols of
states in which one of the quadratures of light (equiva-
quantumtelecloning[14,15]andcontrolleddensecoding
1 lent to the position and momentum of a mechanical os-
v cillator)hasaquantumuncertaintybelowthevacuumor [16,17]constituteparadigmaticexamplesofthisscenario
5 forsmallnumberofmodes,whileone-wayquantumcom-
“shotnoise”level,atthe expenseofincreasingthe quan-
0 putation [18–21], in which the computation is achieved
tum fluctuations in the orthogonal quadrature. These
1 by applying local measurements to a set of modes ini-
states found their way into the new century thanks to
3
tially in a cluster state, is a most promising example in
. their widely proved applications in ultra-precise metrol-
1 the large number of modes regime.
ogy (such as gravitational wave detection [3, 4] or beam
0
displacements [5, 6]), as well as in the field of quantum However,thegenerationofsuchmultipartiteentangled
2
1 information with continuous variables [7, 8], where the states by means of optical devices requires experimental
: highest-quality entangled states (the basic ingredient of configurationswhosecomplexityincreaseswiththenum-
v
manyquantuminformationprotocols)knowntodateare ber of modes involved [10, 17, 22–25]. In contrast, a
i
X currentlyobtainedbymixingsqueezedbeamswithlinear practicalsourceshouldbe compact,scalable,andpermit
r optics [9, 10]. tomasterthequantumpropertiesofthegeneratedstates
a Squeezed light can be obtained via the parametric even when the number of modes is very large.
down-conversion process that takes place inside a sec- A continuous-variable cluster state source with these
ond order nonlinear medium pumped by a laser beam properties was proposed in [26]—and refined in subse-
[1]. In order to increase the nonlinear interaction, it is quent papers [27, 28] (see also [20])—, which consisted
customary to insert the nonlinear crystal in an optical in an OPO driven by a multifrequency pump field in
cavity—dealing then with a so-called “optical paramet- thepresenceofconcurrentnonlinearities. Recently,these
ricoscillator”(OPO)—,andlargelevelsofsqueezingare ideashavebeenbroughttothelaboratory,andtherecord
obtained in the down-converted field when the OPO is of 15 quadripartite cluster states have been generated
operated close to threshold, 93% of noise reduction be- along60consecutivelongitudinalmodesoftheOPOcav-
ing the current benchmark [11] (see also [12, 13]). ity [29]. This interesting approach is however limited by
Ontheotherhand,quantuminformationhasreacheda the number of frequencies that can be coupled within
stage where real-world applications stimulate an intense the phase-matching bandwidth, the need of frequency-
sensitive measurements, and the fact that a nonlinear
crystal with different phase-matching properties has to
be engineered for each particular cluster state that one
∗Electronicaddress: [email protected] desires to create. An even more simple and promising
2
approach based on the synchronously operation of only nel function or matrix (depending on the continuum or
one single mode vacuum squeezer and a quantum non- discrete nature of the modes involved) which describes
demolition gate was proposed very recently in [30]. the coupling among the different relevant signal-idler
Inthispaperweproposeanalternativeapproachbased modes. As shown in previous studies, the diagonaliza-
on the use of a naturally multimode optical parametric tion of such kernel is instrumental for determining the
oscillator,either inthe spatial[31–42]orinthe temporal objectswithwell-definedquantumproperties,whichturn
domain [43, 44]. Below threshold, they produce multi- out to be linear combinations of the signal-idler modes.
mode squeezed states, which has been observed experi- Such linear combinations were termed “supermodes” in
mentally [42, 45]. When operated above threshold, such [43] and the next subsections are devoted to their intro-
multimode devices have been recently proved to be ca- duction in both single-pass and cavity devices. We will
pable of producing non-critically squeezed states of light consider collinear type I degenerate phase matching for
via the phenomena of spontaneous symmetry breaking definiteness but our treatment can be generalized easily
[32, 33, 35, 41, 46] and pump clamping [37, 42]. to other types of phase matching.
Theadvantageoftheschemeintroducedinthecurrent
article for the generation of cluster states over previous
proposals[20,26–28]isthatonedoesnotneedtoengineer
A. Single-pass parametric interactions
the couplings between each mode throughcrystal design
(which is specific of each coupling scheme): choosing in-
Inthesingle-passconfigurationthepumpbeamampli-
stead the temporal or spatial shape of the pump (which
fiesparametricallythequantumnoiseimpingingthecrys-
is a flexible technique) is enough to directly control the
tal around the subharmonic frequencies and the down-
characteristics of the generated quantum state, even for
converted field is measured at the exit of that crystal.
a very large number of modes. We will show that it is
In the undepleted pump approximation, holding when
possible to master in this way the number of entangled
parametricgainislow,the “output”bosonoperatorsare
modes,aswellasthedistributionoftheirquantumcorre-
genericallyrelatedtothe“input”onesviathewellknown
lations, thus permitting the generationof arbitrary mul-
Bogoliubov transformation
timode Gaussian quantum states of any dimension. The
technique thatwe proposeis thereforereminiscentofthe
widelyused“coherentcontrol”ofevolutionofatomsand aˆ (ξ)= dξ′ C(ξ,ξ′)aˆ (ξ′)+S(ξ,ξ′)aˆ† (ξ′) ,
out in in
molecules that is obtained by appropriately shaping the
Z h i(1)
pulses with which they interact [47, 48].
where ξ is a general continuous index standing for fre-
The article is organized as follows. In Section II
quency or/and transverse position1, and C and S are
we show how both the light generated via single-pass
Green functions that solve the propagation equations
spontaneousparametricdownconversionin the low-gain
(see [49] for details). In the very weak conversion limit
regime,aswellasthatleavingamultimode opticalpara-
a perturbative treatment can be applied to first order
metric oscillator below threshold, can be described as
in the crystal length, which physically means consider-
a combination of independent, squeezed modes (termed
ing only generation of single photon pairs. In such case
“supermodes” later on). After considering a particular
C(ξ,ξ′)=δ(ξ ξ′)andS(ξ,ξ′)=gK(ξ,ξ′),whereg is
(butphysicallyrelevant)exampleoftheparametricinter- −
acouplingconstantproportionaltothelengthl andnon-
actionto showhowthese supermodes arise(SectionIII), c
linear susceptibility χ(2) of the crystaland to the square
we demonstrate that the squeezing and spatio-temporal
root of the total pump irradiance P, and K(ξ,ξ′) is a
shapeofsuchmodescanbecontrolledeitherbytailoring
kernel. Hence in this regime (1) becomes
the shape of the pumping field, or the geometry of the
nonlinearmedium;weconsiderseparatelythecasesofan
OPOwitheithermanylongitudinalmodes(SectionIVA) aˆ (ξ) aˆ (ξ)+g dξ′K(ξ,ξ′)aˆ† (ξ′). (2)
out ≈ in in
or just a few transverse modes (Section IVB) available Z
for parametric down-conversion. We then explain how
As we will see immediately, the form of K is universal,
our results can be used for the generation of arbitrary
and is given by the product of the pump amplitude and
cluster states of copropagating modes in Section V, and
somefunctiondescribingtheoverlapbetweenmodesover
give our conclusions in Section VI.
the crystal. However, the special form of K depends on
theconsideredproblemandinthefollowingwetreatsuc-
cessively the temporal and spatial aspects of single-pass
II. GENERALIZED SUPERMODE
parametric interaction in the single photon pair approx-
DESCRIPTION OF WEAK PARAMETRIC
imation (2).
INTERACTIONS
The dynamics of parametric interactions in the low-
gain regime (for single-pass devices) or below its oscilla- 1 Transverse means orthogonal with respect to a propagation di-
tion threshold (for cavity devices) is controlledby a ker- rection,inwhichcasetheparaxialapproximationisassumed.
3
Letusbeginbythespectral/temporalaspectsinwhich B. Intracavity parametric interactions
caseξ =ωcorrespondstothefrequencyofthemonochro-
matic modes (we call this the temporal case). Assuming
Whenanopticalcavityisusedtoenhancetheefficiency
thatallthe parametricallycoupledmodeshavethe same
ofthenonlinearprocess,thepreviousnonlinearcouplings
transverse dependence, and neglecting diffraction inside
represented by kernel K are projected onto the cavity
the crystal, the coupling kernel K(ω,ω′) turns out to be
modes, and the multi-dimensional spectral and spatial
[49]
properties of the correlated photons can be lost due to
such filtering. However for a given length and geometry
K(ω,ω′)=α (ω+ω′)D(ω,ω′), (3)
p ofthecavity,whichweassumetohaveacylindricalsym-
metry around the optical axis, a great number of modes
αp(ω) being proportional to the spectral pump ampli- can be simultaneously sustained: in the frequency do-
tude at frequency ω, and D the usual phase-matching mainthey are the series of longitudinalmodes separated
function2 by the free spectralrangeofthe cavity;inthe transverse
spatialdomaintheyarethesetofLaguerre-Gaussmodes
D(ω,ω′)= l1c Z−llcc//22dzei∆k(ω,ω′)z =sinc[Φ(ω,ω′)], {TIEnMthple}plt∈∈eZmN.poral case we assume the pump consisting
(4) ofanunlimitedseriesofpulsesatagivenrepetitionrate,
where ∆k(ω,ω′) = k(ω+ω′) k(ω) k(ω′), k(ω) is equal to the free spectral range of the cavity: this is
− −
theopticalwavenumberatfrequencyωinsidethecrystal, what is called a Synchronously Pumped OPO (SPOPO)
sinc(x)=sin(x)/x is the sinus cardinal function, and [43, 44]. The pump spectrum is thus a frequency comb,
consisting of a large number of frequency components,
Φ(ω,ω′)=∆k(ω,ω′)lc/2, (5) eachofwhichgivesriseto signal-idlerphotons belonging
to different longitudinal modes through the parametric
is a phase mismatch. downconversionprocess,andselectedbyenergyandlin-
Letusnowturntothe spatialcasewithintheparaxial ear momentum conservation.
approximation, in which case ξ = r is the transverse In the spatial case, on the contrary, the pump is as-
spatialvariable(dξ =d2r). Wedefinethis“spatialcase” sumed monochromatic with a given spatial profile. In
in the sense that pump is assumed monochromatic and this case, owing to the linearity of the interaction in the
focus is put on the signal/idler (multimode) field at just below threshold regime, one can focus on, say, the sub-
the subharmonic frequency, in which case the kernel is harmonicsignalphotons. Howeverthesephotonsbelong,
given by [40]: in general, to different transverse modes TEM (degen-
pl
erate in frequency), characterizedby a constant value of
r+r′ the sum f = 2p+ l , known as “family index” [37]. As
K(r,r′)=αp 2 ∆(r−r′), (6) well, even several f|a|milies of transverse modes can be-
(cid:18) (cid:19)
come relevant in the case of a degenerate cavity such as
α (r) being the normalized pump amplitude at trans- the the confocal [31, 34, 51] or the self-imaging cavity
p
verse point r, and ∆(r) the diffraction function [40, 52], in which many different families (with different
familyindicesf)resonateatthesamefrequency. Wewill
ik +lc/2 dz ik use a single generic index to label this discrete series of
∆(r) = s exp s r2 (7) modes.
4πl z 4z| |
c Z−lc/2 (cid:18) (cid:19) It is instructive in this intracavity interaction prob-
1 π r 2 lem to write down the interaction Hamiltonian in the
= Si ,
πlc2oh "2 − (cid:12)lcoh(cid:12) !# undepleted pump approximation, which can be written
(cid:12) (cid:12) in general as
(cid:12) (cid:12)
ks being the phase-matched sign(cid:12)al wa(cid:12)venumber inside ~g
the crystal, l = 2l /k the “coherence length”, and Hˆ =i dξdξ′K(ξ,ξ′)aˆ†(ξ)aˆ†(ξ′)+H.c., (8)
z coh c s I 2
Si(z)= sinc(u)du the sine integral function. Z
0 p
In the general case, the spatial and spectral aspects where the continuous boson operators aˆ(ξ) satisfy the
R
ofthe parametricinteractionaresimultaneouslypresent, standard commutation relations
giving rise to new interesting features [50] which we will
not consider here. [aˆ(ξ),aˆ(ξ′)] = [aˆ†(ξ),aˆ†(ξ′)]=0, (9a)
[aˆ(ξ),aˆ†(ξ′)] = δ(ξ ξ′). (9b)
−
and the kernel K(ξ,ξ′) plays a role analogous to that in
2 In[49]afactor exp{i[k(ω)−k(ω′)]lc/2} isincludedintheex- single-pass devices.
pression(4). Hereitisabsentbecauseweareimplicitlyworking
In the temporal/spectral case (SPOPO) the Hamilto-
intheinteractionpicturewithrespecttotimeandspace,analo-
goustoEq. (17)in[49],see[43,44]fordetails. nianof the nonlinearinteractioncanbe written interms
4
ofdiscretecreationandannihilationoperatorsofthecav- is formally expressed by the Bloch-Messiah decomposi-
ity longitudinal modes: tion [55].
Letusintroducetheeigenmodesofthecontinuousker-
Hˆ =i~g K aˆ†aˆ†+H.c. (10) nel K(ξ,ξ′), which are the solutions sn(ξ) of the Fred-
I 2 ij i j holm integral equation:
i,j
X
where K are the specialization of the continuous- F [s (ξ)] dξ′K(ξ,ξ′)s (ξ′)=Λ s (ξ). (13)
ij n n n n
≡
variable kernel K (3–5) to the relevant longitudinal Z
modes around the subharmonic, and aˆi is the boson op- Starting from the eigenmodes sn, we can define the
erator for the signal-idler mode i verifying [aˆi,aˆ†j]=δi,j. associated supermode annihilation operator Sˆn as
The dynamics of the intracavity modes is then given by
the following quantum Langevin equations Sˆ = dξs (ξ)aˆ(ξ), (14)
n n
Z
daˆ
dti =−γaˆi+ 2γaˆi,in+γσ Kijaˆ†j (11) where we assume sn(ξ) to be normalized so that
j [Sˆ ,Sˆ† ] = δ . These operators annihilate a photon
p X n m n,m
inamode whichis acombinationofeither spatialorfre-
where γ is the cavity loss rate (assumed identical for all
quency modes, and we call it “supermode” [43] because
signal/idlermodes–which,bytheway,cannotbedistin-
it is the combination of many cavity modes and has a
guished from one another in this collinear type I degen-
nontrivialspatialorspectral/temporalshapeingeneral.
erate phase matching case), and σ2 =(g/γ)2 is a dimen-
Using (14) and (13) into Eq. (2) we get
sionless pumping parameter proportional to the actual
pump power (see, e.g. [44] for more details). Sˆ =Sˆ +gΛ Sˆ† . (15)
out,n in,n n in,n
Inthespatialcase(asdefinedinthepreviousSection),
undercircumstancesinwhichonlyafewsignalmodesare Whenthecrystalisinsertedinanopticalcavity—OPO
relevant [37], Hamiltonian (10) holds by identifying the case—, Ref. [43] shows that the supermodes can be de-
indices (i,j) with the available TEM modes at the sig- fined as a discrete combination of the cavity eigenmodes
pl
nalfrequency,whosedynamicsareruledbythequantum insteadofthecontinuouscombinationwrittenabove,cor-
Langevin equations (11). In degenerate cavities, such as responding to the diagonalization of matrix K in Eq.
confocal or self-imaging cavities, a discrete representa- (11). These two possible expressions are identical in the
tion is still possible, but a continuous one (8) is more limitwherethecavitymodesaretightenoughsothatthe
helpful, in which case the quantum Langevin equations sum can be assimilated to an integral. In the following
become [40] weassumethatthisconditionisfulfilled. Inthiscase,the
evolution of these supermode operators is governedby a
∂aˆ(r) set of uncoupled quantum Langevin equations obtained
= γaˆ(r)+ 2γaˆ (r)+γσ d2r′K(r,r′)aˆ†(r′).
∂t − in by introducing the discrete versions of (14) and (13) in
Z
p (12) Eq. (11):
For self-imaging cavities [40] the kernel is given by (6)
and (7), while for the confocal case it has a slightly dif- dSˆn = γSˆ + 2γSˆ +γσΛ Sˆ†. (16)
ferentexpression[31]owedtothe factthatevenandodd dt − n in,n n n
transverse families resonate at different frequencies. p
This equation shows that according to this definition,
any multimode OPOcanbe in fact seenas a set ofinde-
pendentsingle-modeOPOs,andthereforecanproduce a
C. Supermodes
set of copropagating squeezed supermodes. Hence each
supermode is independently squeezed, and the squeezed
We have seen in the previous Section that the output quadrature can be shown to have a noise level at zero
fromparametricdevicesinthelow-gain(orbelowthresh- frequencyequalto(Λ Λ )2/(Λ + Λ )2 (1setting
1 n 1 n
old) regime is governed by a kernel K, which is equal, theshot-noiselevel)|wh|e−n|the|syste|m|iso|per|atedcloseto
under equivalent conditions, in the single-pass and in-
threshold, where Λ is the eigenvalue of largest absolute
1
tracavity cases. As both the input-output relations in value.
single-pass configurations and quantum Langevin equa-
tionsincavitydevicesarelinearintheconsideredregime,
diagonalization of the kernel allows a considerable sim- III. A GENERAL METHOD FOR TAILORING
plification in the description of the problem as well as THE SUPERMODE SPECTRUM
a clear physical picture of the entities in which clean
quantum properties are concentrated. This approach
The solution of Eq. (13) is simple when the kernel
[40, 43, 44, 49] is the single-partite and continuous-
factorizes,
variableversion[53]oftheSchmidtdecompositionforthe
bi-partite bi-photon wave function [54], which in general K(ξ,ξ′)=f(ξ)f(ξ′), (17)
5
in which case the unique eigenmode is, within a mul- therefore shown that a kernel having the form (19) with
tiplicative factor, s = f, with corresponding eigen- (21)willhaveexactly(2N +1)(2N +1)eigenmodes,or
1 + −
value Λ1 = dξ′f(ξ′)2. This result can be extended to 4N+N− modes if the constant terms b±0 = 0. As shown
the case where there is a set of N orthogonal functions in Appendix A a simple way to diagonalize the kernel
f N sucRh that defined by (21) consists in using auxiliary functions
{ n}n=1
N zt1,t2 (x)=e−σ2x2t β+x t β−x , (22)
K(ξ,ξ′)= f (ξ)f (ξ′), (18) n1,n2 1 n1 2 n2
n n
nX=1 where ti=1,2 stands for any o(cid:0)f the(cid:1)trig(cid:0)onom(cid:1)etric func-
tions cos or sin, in terms of which actual eigenvectors
in which case Eq. (13) has N solutions s = f (save
n n
multiplicative constant), Λ = dξ′f (ξ′)2. When K and eigenvalues can be found by linear combinations of
n n
them. The expressions for eigenvalues and eigenvectors
has the same type of decomposition as (18), but with
R areawfullycumbersomebutonecanshowthatwhenever
non orthogonal functions, just linearly independent, one
can show that (13) has still N solutions [56], which are
now different from the functions f . β+ 2, β− 2 8σ2, (23)
n n1 n2 ≫
TheanalyticaldescriptionofthekernelK intermsofa
(cid:0) (cid:1) (cid:0) (cid:1)
basisoflinearlyindependentfunctions(supermodes)like the functions (22) are approximate eigenvectors indeed.
inEq. (18)hasaverysimplesolutionandphysicalinter- For each couple (n ,n ) in (22) two doubly degenerate
1 2
pretation in the case where the kernel can be factorized eigenvalues are found, of equal magnitude but opposite
as sign: π/32σ2b+ b− . The positive one is associated
± n1 n2
to t = cos and the negative one to t = sin. As for the
1 1
K(x,x′)=K+(x+x′)K−(x−x′). (19) cases(np1 =0,n2)and(n1,n2 =0),inwhichβn+1 =0and
β− =0 respectively, the eigenvalues read π/8σ2b+b−
In the spatial case this is the form encountered [40], see n2 0 n2
and π/8σ2b+b− ,respectively. Finally,theeigenvalue
(6): K+ relates to the pump, while K− has to do with − 0 n2 p
associatedtothefundamentalGaussianeigenvector,that
the crystal. In the temporal case, while this is not the is, to tphe case (n =0,n =0), reads π/2σ2b+b−.
most general kernel one can find in actual applications, 1 2 0 0
it has been shownin [44, 49] that it is a sensible approx- We have therefore shown that onepis able to master
imation to many real cases. The function K in (19) the number of supermodes and the magnitude and sign
+
can be manipulated by tailoring the pump shape, while oftheir eigenvaluesbya properchoiceofthe modulation
in order to tune the function K one needs to play with amplitudes ofthe parametric multimode interactionker-
−
the geometry of the nonlinear crystal. nel (21). In Figure 1 we give two examples of kernels,
Wenowexhibitaseriesof“simple”kernelswhichallow designed so as to lead to eigenvalues respectively pro-
a high degree of control over their spectra and, as we portional to 1,1, 1, 1 and 4,1,1,1,1 . We note
{ − − } {− }
will show in the next Section, find applications in actual that the results shown in the figures have been obtained
systems. by numerically diagonalizing the kernels, what gives ad-
Letusstartbythesimplestcase,thesymmetricGaus- ditional support to our previous approximate analytical
siankernel,definedbyK+(x)=K−(x)=e−12σ2x2,read- treatment.
ing Our analysis has been restricted so far to the case of
kernels with symmetric Gaussians. However, it can be
K(x,x′)=e−σ2x2e−σ2x′2, (20) readily extended to the more physical case of a kernel
factorizing in two Gaussian functions of variables x+x′
as in (17), whose only supermode is the Gaussian func- and x x′ having unequal widths [44, 49]. In fact this
−
tions (x)=e−σ2x2,withcorrespondingeigenvalueΛ = is the most general situation as it is not always possible
1 1
to configure the OPO/OPAin sucha symmetric way (in
π/2σ2.
SPOPOs, for instance, for typical situations one has a
Let us now consider the previous symmetric kernel
p difference of one order of magnitude between Gaussian
(20), now multiplied by cosine functions of x+x′ and
widths). Inthis casethekernelweproposehasagainthe
x x′, i.e.:
− factorized form (19), now with
N±
K±(x)=e−21σ2x2n=0 b±n cos βn±x , (21) K±(x)=e−21σ±2x2 N± b±n cos βn±x , (24)
X (cid:0) (cid:1)
n=0
whichisaquasiperiodicmodulationoftheGaussianker- X (cid:0) (cid:1)
nel. In (21), b±,β± N are constants and β± = 0 by instead of (21), where we allowed for different widths,
definition. Th{isnleands}nt=o0a kernel K (19) whic0h can be σ−1, alongthe directionsx+x′ andx x′. InAppendix
± −
written like in Eq. (18)—with non-orthogonal functions Bwegivetheeigenvaluesandeigenvectorsofthisgeneral
f —using well known trigonometric formulae. We have case.
n
6
FIG. 1: (a) Density plot of the kernel—see (19) and (21)—particularized to the following parameters: σ = 0.005, b+ = 1,
1
β+ = 3πσ, b+ = 1, b− = 1 β+ = 6πσ, the rest of b’s being zero; in (b) we show its associated eigenvectors with the
1 2 0 2
− −
corresponding eigenvalues. (c) Density plot of thekernelfor a different choice of theparameters: σ=0.005, b = 2, b =1,
b+ =1,β− =3πσ,b− =1,β− =6πσ,therestofb’sbeingzero;in(d)weshowthecorrespondingeigenvectors0ande−igenv1alues.
0 1 2 2
IV. CONTROLLING THE NUMBER OF howto implementin realdevices the quasiperiodicmod-
SUPERMODES AND THE EIGENVALUES IN ulations we introduced in the previous Section.
ACTUAL PARAMETRIC DEVICES
We will divide our presentation in two Subsections.
Thefirstonedealswithwhatwecallthecontinuouscase,
The method we sketched out above can be practically which includes both problems in which a continuous bo-
applied to spatial or temporal modes of the OPO/OPA son representation is used (single-pass devices) or prob-
since in both cases a factorizedform(19) of the kernelis lemsinwhich,beingthatrepresentationdiscrete,theker-
a very good approximation(it is even exact in the “spa- nelcanbetreatedasacontinuousfunction(SPOPO[44]
tial case”),and both K admit as well an approximated and the self-imaging OPO [40]). The second Subsection
±
Gaussian form, see e.g. [44, 49, 53]. The point is then deals with what we call the discrete case, in which a dis-
7
crete boson representation is used and a discrete treat- diffraction gratings perform the inverse Fourier trans-
mentoftheproblemissimpler,giventherelativelysmall form so that the outgoing field has the following tem-
numberofmodesinvolved,likeanOPOtunedtoasingle poral shape: α (t) = M(t) α (t), i.e. the usual
p,out p,in
⊗
transverse mode family [37]. convolution, which is used to pump the SPOPO. It is
then straightforward to see that, by using α (ω) in
p,out
Eq. (3), one obtains a modulation of the kernel K(x,x′)
as in Eq. (24), with b+ =b , b− =1 and b− =0. As a
A. The continuous case n n 0 n>0
consequence,the eigenvaluesΛ areproportionalto b
n n
±
accordingto the analyticalsolutionsgiveninSectionIII.
AsfortheGaussianformofthesubkernelsK wenote
±
that both the phase-matching function D (4,5) of the
b. Using pulse shapers leading to special waveforms:
temporal case and the diffraction function ∆ (7) of the
Inthis case weassume thata pulse shaperhas beenpro-
spatial case can be well approximated by the Gaussian
grammed so as to convert the frequency comb coming
e−21τ12(ω+ω′)2−12σ−2(ω−ω′)2. In the temporal case [44, 49] from the pumping laser (whatever its temporal wave-
form could be) into into a sequence of equal rectangu-
(k′ k′)l
τ = p− s c, (25) lar pulses at the frequency comb repetition rate. Hence
1
2 the pump spectrum is proportional to sinc(τ ω), τ be-
p p
ing the pulse duration. Then, if τ−1 is much smaller
k′ being the derivatives of the pump/signal wavenum- p
p,s thanthewidthofthecrystalresponsefunctionalongthe
ber with respect to frequency at phase matching, while direction ω +ω′—equal to τ−1 defined in Eq. (25)—,
1
in the spatial case τ = 0. Hence if the pump spec-
1 i.e. for“long”pulses,the kernelcanbe approximatedby
btreailngamthpelitinuddieviαdpu(aωl)puhlasseaduGraatuisosnia,nthsehatopteale−ke21rτnp2eωl2,caτnp e−12σ+2(ω+ω′)2−21σ−2(ω−ω′)2sinc[τp(ω+ω′)], see Eq. (3),
where σ τ [44]. As a consequence, as sinc(τ ω) can
beapproximatedbyK =e−12σ+2(ω+ω′)2−21σ−2(ω−ω′)2 with be appro+x∼ima1ted by a finite (Fourier) sum of fupnctions
σ2 =τ2+τ2. Ifthe pumphasnotaGaussianshapebut cos n2πω , with L large as compared to τ−1, the ker-
+ 1 p L 1
can be expressed as a sum over cos functions, τp = 0 nel reads as in (24). As the coefficients of the Fourier
(hence σ2 = τ2) and the Gaussian kernel will be multi- seri(cid:0)es, b+,(cid:1)are all very similar (because of the function
+ 1 n
plied by those modulations (see below). sinc(τω),whoseFouriertransformisrectangular)weex-
As the pump amplitude αp is a factor of K+, see Eq. pect a large degree of degeneracy between eigenmodes,
(3), a simple and practical way to tailor the kernel is by asisactuallyevidencedbynumericallydiagonalizingthe
shaping the pump, as we will see more precisely now. corresponding kernel.
In Figure 2a we trace the kernel corresponding to a
1. Tailoring the pump realisticsituationwhereaSOPOcavity,withthenonlin-
earityofa 100µm-thickBIBOcrystal(τ 20fs;see [44]
1
∼
for details), is pumped by a train of square pulses. The
We focus on the temporal case [43, 44, 49] and hence
eigenvaluesΛ obtainedbythenumericaldiagonalization
deal with pumps consisting of pulses. However the ideas n
of K for increasing values of τ are shown in Figure 2b
put forward below can be applied equally to the spatial p
andarecomparedwith the correspondingcases ofGaus-
case [40] by substituting the pulse shapers we consider
sian pump pulses: as expected, the degree of degeneracy
by amplitude masks or simply by superposing different
increases dramatically with the duration of the square
plane waves.
pump pulses. For instance, for 1ps pulses the first hun-
a. Using pulse shapers with harmonic spectral re-
dred eigenvalues differ by less than 1%, see the inset in
sponse: Afirstpossiblemethodconsistsofusingapulse
Figure 2b, what allows the generationof very high qual-
shaper [57] with severalwell separatedamplitude modu-
ity, highly multi-dimensional entanglement.
lationsontheSpatialLightModulator(SLM).Theeffect
ofsuchpulseshaperinthe frequencydomaincanberep-
c. Using delay lines A third method consists in us-
resented in the general case by the modulating function
ing delay lines that, starting from one comb, allow the
superposition of several combs so that each “tooth” in
N
M(ω)= b cos(β ω), (26) the temporal domain is made of a series of pulses de-
n n
layed/advanced by t with respect to the first one; one
n=0 n
X
thengetsasmanyeigenvaluesastherearedifferentsuper-
where β = 0 by definition. The incoming pump field, posed combs, the eigenvalues being proportional to the
0
of temporal envelope α (t), is sent onto a diffraction amplitude of each comb as we show next. For example,
p,in
grating and the image α (ω) is created in the Fourier α(t) being the (normalized) envelope of the pump field
p,in
plane of a lens in correspondence of the SLM mask. Im- associated to a train of pulses incoming the OPA or the
mediately after the mask, the outgoing field has a spec- SPOPOcavity,theenvelopeofthepumpfieldassociated
trum α (ω) = M(ω)α (ω). Then another lens and to a generic superposition of delayed/advanced trains of
p,out p,in
8
FIG. 2: (a) kernel evaluated in the realistic case of a 100µm thick BIBO crystal inside a 4m ring OPO cavity pumped by a
c.w. train of 500fs rectangular pump pulses for a degenerate type I critically phase matching operation at 0.4µm pumping. q
andmdenotetheindicesofthecavitylongitudinalmodes. (b)Eigenvaluesobtainedbynumericaldiagonalization ofthekernel
shown in(a)when differenttrainsof pumppulseswithGaussian (grey-dashedlines)andrectangular (black-solid lines) shapes
areconsidered;thedurationofthepulsesarespecifiedinthefigure. Λ1 referstotheeigenvaluewiththelargestabsolutevalue.
note that thecaption is a zoom of theupper-left corner of thefigure.
pulses reads whereΦ(ω,ω′)isgivenin(5). Ifthecrystalsarearranged
symmetrically by couples at distances d between mid-
N n
planes, then the above expression becomes
α (t)=b α(t)+ b [α(t t )+α(t+t )], (27)
p 0 n n n
−
n=1
X
where b and b are real coefficients controllable in the N/2
0 n d
experiment. In the Fourier domain Eq. (27) reads D(ω,ω′)=2 cos nΦ(ω,ω′) sinc[Φ(ω,ω′)], (30)
l
n=1 (cid:20) c (cid:21)
N X
α (ω)=α(ω) b + b cos(t ω) , (28)
p 0 n n
" n=1 # where now the sum extends over couples of crystals3.
X
When the pump spectrum is narrow as compared with
α(ω) being the Fourier transform of α(t). Since we have
the width of D along the direction ω+ω′—the quantity
thesameeigenproblemasinthepreviouscases,theeigen-
τ−1 defined in Eq. (25)—a safe approximation consists
values Λn will be again proportional to bn. 1
±
in setting Φ(ω,ω′) τ (ω+ω′) in cos dnΦ(ω,ω′) .
→ 1 lc
Hence, h i
2. Tailoring the crystal response
d d
Another interesting possibility in the temporal case cos nΦ(ω,ω′) cos nτ (ω+ω′) ,
1
is to modify not the pump temporal shape but the ef- (cid:20) lc (cid:21)≈ (cid:20)lc (cid:21)
fect of the nonlinear medium, which allows us to con-
trol the function K+. It can be accomplished by using acts as a harmonic modulation (equivalent to b+ = 2,
several identical non-linear crystals which are not per- β+ = dnτ , b− = 1, and b− = 0 in our previonus no-
fectly phase-matched and separated by fixed distances. n lc 1 0 n>0
tation) of the single-crystalkernel, and the latter can be
In the general case where N crystals of thickness l are
c approximated by a factorized Gaussian form as already
used,centeredatplanesz =z ,thecorrespondingphase-
n discussed, so the full kernel takes the form (24).
matching function D, see (4), reads
D(ω,ω′) = 1 zn+lc/2dzei∆k(ω,ω′)z (29)
l
c Xn Zzn−lc/2 3 If the number of crystals is odd, say N = 2M +1, then Eq.
N 2z (30)ismodifiedbysubstitutingthesumupperlimitbyM,and
= exp i nΦ(ω,ω′) sinc[Φ(ω,ω′)], adding a term equal to sinc[Φ(ω,ω′)] to the result, as can be
l
n=1 (cid:20) c (cid:21) easilychecked.
X
9
B. Discrete case Thisexpressionisclosetotheoneassumedin(19). Itcan
actuallybebroughttothatexactformbyintroducingthe
Hybrid Laguerre-Gauss modes
As commented in Section IIB, the discrete represen-
tation is useful when dealing a low number of signal
C (r) = 1 [L (r)+L (r)], (36a)
modes. Atypicalconfigurationcorrespondingtothiscase l √2(1+δl,0) l −l
is givenby OPO with monochromaticpump, whose cav- S (r) = 1 [L (r) L (r)], (36b)
ityistunedatthesubharmonictosometransversemode l i√2(1+δl,0) l − −l
family represented by the family index f [37]. Recall
which are respectively proportional to coslφ and sinlφ.
that family f contains the f +1 Laguerre-Gauss modes
L (r) with l =l ,l +2,...,f,being l equalto 0 The kernel can then be written as
(f−l)/2,±l 0 0 0
for even families and 1 for odd families. K(r,r′)= χ H (r)H (r′). (37)
l l l
In order to simplify the upcoming discussion, we as-
H=C,S l
sume that the the pump beam has a coaxial cylindri- X X
calsymmetry,inwhichcaseOrbitalAngularMomentum In other words, we have just shown that the Hybrid
(OAM) conservation ensures that the down-converted Laguerre-Gaussmodes C andS are the supermodes for
l l
photons must haveopposite OAMs. This implies that, if the present OPO configuration. In addition, both the
one uses the basis of Laguerre-Gauss modes TEM , the “sine” and “cosine” type modes have the same eigen-
pl
parametric down-conversionHamiltonian takes the form value Λ = χ (H = C,S). Note that nothing pre-
H,l l
[37] vents χ from being complex, say χ = χ exp(iψ ),
l l l l
| |
what means that the supermodes with different l can
χ
Hˆ =i~g l aˆ†aˆ† +H.c., (31) be squeezed along different directions of phase space, di-
I Xl 1+δ0,l l −l Aresctfioornt(hπe+nuψml)b/e2rinofpaavratiilcaubllaersfuopremrmodoedses{HNl,(rit)}iHs=giCv,eSn.
with by the number of modes contained in family f, that is,
N =f +1.
+∞ 2 Let us now consider the situation in which the pump
χ =2π rdrα (r) l (r) , (32)
l Z0 p hR(f−l)/2 i has a Gaussian shape, that is, αp(r)=wsGp(r), where
α (r) and l (r) being the transverse profiles of 1 2 r2
thpe (normaRliz(efd−)l)/p2ump field and the Laguerre-Gauss Gp(r)= wprπ exp(cid:18)−wp2(cid:19), (38)
modes 4 at the cavitywaist, respectively,andaˆ† the cre-
l is a TEM mode with spot size w =ρw , w being the
ation operator associated to the Laguerre-Gauss mode 00 p s s
spot size of the transversemodes at the signal frequency
L (r), whichweabbreviateasL (r)fromnowon
(f−|l|)/2,l l andρ ascalingfactor(the factorw inα is includedfor
becausea fixedvalue ofthe familyindex f is assumedas s p
dimensionalreasons,aswedefinedα asadimensionless,
explained. p
normalized pump amplitude). It is obvious from (32)
The continuous boson operators aˆ(r) defined on the
that χ > χ > ... > χ ; in other words, the lower
transverse plane can be expressed in terms of Laguerre- l0 l0+2 f
the OAM, the better the signal modes overlap with the
Gauss modal operators as
pump profile. However, for a large enough ρ, the pump
profile is somehow seen as plane by all the signalmodes,
1
aˆ(r)= [Ll(r)aˆl+L−l(r)aˆ−l]. (34) and the couplings χl become independent of l:
1+δ
l,0
l
X 2 w +∞ 2 2
fTohlleowkienrgneelxKpr(ers,srio′)no:f Hamiltonian (8) has therefore the χl −ρ≫→12πrπwps Z0 rdrhRl(f−l)/2(r)i =rπρ−(31.9)
Hence, by pumping with a wide Gaussian, one can bring
χ
K(r,r′)= l [L−l(r)Ll(r′)+L−l(r′)Ll(r)]. all the couplings χl to a common value, at the expense
1+δ0,l ofdecreasingthem,henceincreasingtheOPOthreshold.
l
X
(35) Notehoweverthat,asdiscussednext,thedesiredeffectis
obtainedevenfor moderatevaluesofρ,sothatthe OPO
threshold stays within reasonable limits.
In Fig. 3 we show the ratios r =χ /χ as a function
l l l0
4 Explicitly,wehave of ρ for the first families. Note that the ρ axis starts at
1/√2, which is the value one has in the doubly resonant
2p! 1 √2r l 2r2 r2
l(r)= Ll exp , (33) configuration (cavity resonant both for the signal and
Rp sπ(p+l)!w w ! p(cid:18)w2(cid:19) (cid:18)−w2(cid:19) the pump). Note also how rl goes very quickly above
whereLl(x)arethemodifiedLaguerrepolynomialsandwisthe 0.5, which is the value above which one finds more than
p
spotsizeofthebeam atthecavitywaist. 90% squeezing for the corresponding supermodes [37].
10
FIG.3: Ratiobetweenthecouplingsofthelmodesandthel0 modesasafunctionofthethicknessoftheGaussianpump. The
insets show the corresponding change in the threshold pump power (normalized to the threshold of the ρ = 1/√2 situation),
that is, thepower needed to make thesignal field oscillate inside the cavity.
In the insets, we show R (ρ) = χ2(ρ = 1/√2)/χ2(ρ), one gets χ = χ . Note that, in the generalcase where
th l0 l0 1 − 3
whichgivestheratiobetweenthe pumppowerneededto theresonatoristunedtofamilyf,onecanhavecomplete
make the signal field oscillate for a given ρ and that for controlofallthe ratiosbetween the coupling parameters
ρ = 1/√2; note that the threshold is not dramatically by using 1+(f l )/2 TEM beams with adjustable
0 00
−
increased for reasonable values of ρ (it is even decreased amplitudes as pumping beams.
for small ρ). In the second example we show the very interesting
One can even tune the actual values of the couplings case where some of the couplings become zero by using
χ tospecificvaluesbypumpingnotwithasingleTEM an appropriate pump shape. To show that this extreme
l 00
mode,butwithacombinationofTEM modesofdiffer- caseisindeedpossible,weconsiderthepreviousexample
00
ent widths. To show this, let us consider two examples withthefollowingchoiceofthemixinganglebetweenthe
within the third family of transverse modes (f = 3, and Gaussians:
thenl =1,3). First, wewanttomakeχ = χ ,sothat
1 − 3 ρ 3 1+2ρ2 4 1+2ρ4
the eigenvalues of the f +1 = 4 supermodes have the tanθ = a b a, (42)
same magnitude, but opposite sign. In this case, we can (cid:18)ρb(cid:19) (cid:18)1+2ρ2a(cid:19) 1+2ρ4b
use the following pump profile
in which case it is simple to show that χ = 0, while χ
1 3
canbelargeenoughwithaproperelectionofρ . Itcan
a,b
α (r)=w [G (r)cosθ G (r)sinθ], (40)
p s a − b be interesting, for example, to choose the mixing angle
between the Gaussians in order to cancel the coupling
that is, a superposition of two Gaussians with opposite
with the TEM modes in a generalOPO, so as to favor
00
phase, and spot sizes w = ρ w and w = ρ w . Note
a a s b b s the coupling to the TEM modes, thus forcing it to
10/01
thatthistypeofpumpshapecanbeeasilyobtainedinthe
emitthesignalandidlermodeswithoppositeOAM.This
labbymixingthe Gaussianbeamsonabeamsplitter. It
way,onecaninduceaspontaneousbreakingoftheradial
isthenstraightforwardtoshowthatchoosingthe mixing
symmetry, what has been predicted to give rise to some
angle such that
remarkable quantum properties [35, 41].
The methods discussed so far allow us to tune at will
tanθ = ρa 3 1+2ρ2b 4 1+4ρ4a, (41) the eigenvalues of the supermodes, but not the spatial
ρ 1+2ρ2 1+4ρ4 profile of the supermodes themselves (they are the Hy-
(cid:18) b(cid:19) (cid:18) a(cid:19) b