Emission of correlated photon pairs from superluminal perturbations in dispersive media

0.41 MB·English

by  Francesco Dalla Piazza

Emission of correlated photon pairs from superluminal perturbations in dispersive
media.
F. Dalla Piazza,1 F. Belgiorno,3 S. L. Cacciatori,1,2 and D. Faccio1,4
1 Dipartimento di Scienza e Alta Tecnologia, Universita` dellInsubria, Via Valleggio 11, IT-22100 Como, Italy,
2 I.N.F.N., Sezione di Milano, Via Celoria 16, 20133, Milano, Italy
3 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo 32, 20133 Milano, Italy
4 School of Engineering and Physical Sciences, SUPA,
Heriot-Watt University, Edinburgh EH14 4AS, UK
(Dated: January 12, 2012)
We develop a perturbative theory that describes a superluminal refractive perturbation propa-
gating in a dispersive medium and the subsequent excitation of the quantum vacuum zero-point
2 fluctuations. WefindaprocesssimilartotheanomalousDopplereffect: photonsareemittedincor-
1 relatedpairsandmainlywithinaCerenkov-likecone,oneontheforwardandtheotherinbackward
0 directions. The number of photon pairs emitted from the perturbation increases strongly with the
2 degree of superluminality and under realizable experimental conditions, it can reach up to ∼10−2

n photons per pulse. Moreover, it is in principle possible to engineer the host medium so as to mod-
a ify the effective group refractive index. In the presence of “fast light” media, e.g. a with group
J index smaller than unity, a further ∼10× enhancement may be achieved and the photon emission

1 spectrumischaracterizedbytwosharppeaksthat,infutureexperimentswould clearlyidentifythe
1 correlated emission of photon pairs.

]
h I. INTRODUCTION a changein the internalstate, e.g. a transitioninenergy
p
fromoneleveltoanother. InthenormalDopplerregime,
-
t a positive frequency photon will thus be emitted with a
n SincetheanalysisofSchwinger[1]agreatamountofre-
spontaneous transition from an upper to a lower energy
a searchhasbeendevotedtothestudyofparticlesproduc-
u tionfromthevacuum. Intheseminalpaper[2],Hawking state. In much the same way, in the anomalous Doppler
q regime a negative frequency photon (positive frequency
predictedthatthe presenceofanhorizonaroundablack
[ in the laboratory reference frame) will be spontaneously
hole induces the particle production. Actually, the exis-
emitted in combination with a transition from a lower
1 tence of the Hawking radiation does not require a gravi-
to a higher state. The energy required for this process
v tationalcollapse,butratherthekeyelementsareaquan-
4 tum field and an event horizon associated with a curved is providedby the superluminal translationalmovement.
5 Thesuperluminalsourcewillthereforecontinuouslyemit
space time metric [3–7]. Another mechanism providing
3 pairs of photons, one in the anomalous and the other in
particle production is the dynamical Casimir effect [8].
2 the normal Doppler regions [12]. Anomalous Doppler
. Here, a two dimensional quantum theory of a massless
1 emission has never actually been observed before due to
scalarfieldisconsideredthatisinfluencedbythe motion
0 the obvious difficulty in actually realising an experimen-
of a perfectly reflecting boundary (mirror). The vacuum
2 tal system in which to observe these effects.
expectationvalueoftheenergy-momentumtensorforan
1
In this work we extend the same concept proposed to
: arbitrary mirror trajectory shows a non vanishing radia-
v generateeventhorizonsinopticalmedia: anintenselaser
tion flux.
i pulsefocusedintothemediumwillinducealocalincrease
X Hereweconsiderafurthereffectbywhichparticlepro-
in the refractive index and this create a perturbation in
r duction is induced by the superluminal motion of a per-
the dielectric medium (DP) that travels locked with the
a
turbation of the refractive index in a dielectric medium
laser pulse itself [5, 7]. Laser pulses will typically travel
withdispersion. Differently fromthe Hawkingeffect,the
atawell-definedgroupvelocitysothatthiskindofDPis
production of particles is not induced by the presence
of an horizon but by the fast motion of the perturba-
tion. Therefore, this effect can be better recognized in
v
the context of the anomalous Doppler effect [9–12]. Fol- normal Doppler cn(ω)cosθ=1
lowing the description given by Ginzburg, light emitted
region
by a generic moving source will be Doppler shifted ac-
v
cording to the formula ω(θ) = ω′/γ|1 − v/cn(ω)cosθ|
[12], where ω′ is the comoving reference frame value of
anomalous Doppler
the emitted frequency, θ is the observation angle and
region
γ = 1/ 1−v2/c2. For v/cn(ω)cosθ < 1 we have the
normal Doppler effect but for v/cn(ω)cosθ > 1, i.e. for
p
a superluminal source we have the so-called anomalous
Doppler effect (see Fig. 1). Emission of a photon on be- FIG.1: Schematicalrepresentationofnormalandanomalous
half of the moving source must in general correspond to Doppleremission from a “source” moving with velocity v.
2
idealforcreatingtheanalogueofaneventhorizon. How- Thisperturbationtravelsacrossthemediumatthesame
ever, by properly choosing how the laser pulse overlaps speed of the light pulse. The medium was supposed to
with the dielectric medium it is possible to generate a be nondispersive. The authors find that there is no pair
DPthattravelswithanyarbitraryvelocity. Forexample, production unless the pulse velocity v satisfies the con-
recent experimental results clearly show that it is possi- dition v ≥ c , where c is the velocity of light in vacuum
n0
ble to generateisolatedpeakswithin the laserpulse that endn istheuniformandconstantbackgroundrefraction
0
travel faster than the vacuum speed of light, c [14, 15]. index. The model analyzed is based on the perturbative
Withinthecontextofthesefindingswethereforeconsider approachintroducedbySchu¨tzholdet al. in[18]andthe
theproblemofhowasuperluminalDPinteractswiththe interaction representation for the electromagnetic field
zero-point fluctuations of the surrounding quantum vac- in the presence of a dielectric constant ǫ(~x,t) depending
uum. This same problem was first tackled in Ref. [13] on space and time is considered. Moreover, a uniform
where it was shown that, in a dispersion-less medium a background and a constant value ǫ = n2 for the di-
b 0
superluminal DP will spontaneously emit photons that electric constant are assumed. Thus, the disturbance is
are excited from the vacuum state. The photons appear ǫ(~x,t)−ǫ . Following [18], the interaction Hamiltonian
b
as correlated pairs and their wavelength decreases from is H = ξΠ2 with ξ := 1 1 − 1 and the number
the far-infrared to the visible region as the DP velocity I 2 ǫ(~x,t) ǫb
is increased from ∼ c to many times c. Here we extend ofparticlescreated,labeled(cid:16)bythe mo(cid:17)mentum~k andthe
these results to a dispersive medium the results and we polarization µ, is:
consider in more detail the nature of the correlatedpho-
ω ω
tons andpossible experimentalimplementation. We also N = 1 2|ξ˜(k +k )|2[1−(~e ·~e )2], (1)
consider how the correlated photon pair emission may ~k1µ1 V2 1 2 ~k2 ~k1µ1
X~k2
be enhanced by properly engineering the vacuum states,
i.e. by using media in which the effective group velocity
where k = (~k,ω ) = (~k,ck/n ), with k = |~k|, ~e =~k/k,
at the emitted frequencies is also superluminal, so-called ~k 0 ~k
the subscripts 1 and 2 label the particles of the emit-
“fast light media” [16].
ted couple, and the particular shape of the perturbation
is contained in the function ξ. Considering a Gaussian
behavior for the refraction index n2 =ǫ(x,t,y,z):
II. NONDISPERSIVE CASE
We start by briefly summarising the main results and n2 =n20+2n0ηe−[(x−vt)2+y2+z2]/2σ2 (2)
equationsforthenon-dispersivecasetreatedinRef.[13].
Anintenselightpulseinducestheperturbationofthere- the expression for the number of created particles
fractive index through the optical nonlinear Kerr effect. becomes[28]:
N =22σ6π2 η2 d~k′ω1ω2e−σ2|~k1+~k2|2 δ((~k +~k ) − c (k +k )) 2[1−(~e ·~e )2]. (3)
~k1,µ1 v2 n6 V 1 2 x vn 1 2 ~k2 ~k1µ1
0 Z (cid:20) 0 (cid:21)
The deltafunction givesaconstraintonthe possibleval- as
ues of the momenta for the emitted particles.
1
S = E~ ·εE~dx4, (4)
2
III. DISPERSIVE CASE Z
where εE~ is
A. Perturbation
When considering actual experiments, the previous εE~(~r,t)=ε(~r,i∂t)E~(~r,t). (5)
model should only be consideredas an approximationof
arealmediumandonehastotakeintoaccountthepres-
We will not specify the operator ε now, but we will as-
ence of the dispersion. In this section we generalize the
sumeforittobeasymmetricoperator. Theunperturbed
model of [13] for dispersive media. To introduce disper-
conjugate momentum is then
sioninthemodelwehavetorefinethetheoreticapproach
leading to the expressionfor the number of emitted par-
ticles. Let us write the dispersive electromagneticaction Π~ =εE~. (6)
3
If we set IV. THE MODEL
E~ (~r)= e−iωtE~(~r,t)dt, (7)
ω
R Consideraperturbationoftherefractiveindexmoving
Z
inthexdirectionwithvelocityv,i.e. n(ω)(ω)+δn(x−vt)
Π~ (~r)= e−iωtΠ~(~r,t)dt, (8)
ω and assume that the ω dependence of the term δn is
R
Z negligible. Thus, at the first order in δn the dielectric
we can write
constant is ǫ(ω)2 +2n(ω)δn(x−vt). The perturbation
1 1 1
E~ = Π~ = eiωt Π~ (~r)dω. (9) term(19)maybetakenasanoperatorthatwecanwrite
ω
ε 2π R ε(~r,ω) formally as
Z
Weintroduceaperturbationε(~r,i∂ )7→ε(~r,i∂ )+δε(~r−
t t
1
~vt,i∂t). To proceed, we note that now the momentum is H =− E~δǫE~ =−E~ (n(i∂ )δn(x−vt))SE~, (20)
I t
2
Π~ =εE~ +δεE~. (10)
where the apex S implies the Weyl prescription for the
To describe the Hamiltonian we have to invert this op-
maximally symmetrised operator [19]. The perturbative
erator in order to express the electric field in terms of
approach used here may be justified by evaluating the
the momentum. This canbe done perturbatively. Letus
main parameters under investigation. We expect to ob-
introduce an order parameter η so that
serve a similar behaviour to the non-dispersive case for
δε(~r−~vt,i∂t)=ηδǫ(~r−~vt,i∂t). (11) which there is a peak of emission for λmax = 3σ [13],
Next we write where σ is the radius of the gaussian-shaped perturba-
tion. It is reasonable to expect that the perturbative
E~ =E~ +ηE~ +... (12)
0 1 approach works if the measured wavelength λ satisfies
where the dots are higher orders in η. Then, we have σ/λ>>η. Forexample,ifηisoftheorderof0.01−0.001
the perturbative approach is justified for emitted wave-
Π~ =εE~ +η(δǫE~ +εE~ ), (13)
0 0 1 lengths well below 100 nm if σ ∼ 1 µm. In this sit-
which gives uation the operator appearing in equation (20) can be
Π~ =εE~ , (14) approximated by the more tractable one −12E~δǫE~ =
0 −E~(n(i∂ )δn(x−vt))SE~ ≃ −E~δn(x − vt)n(i∂ )E~, in
εE~ =−δǫE~ . (15) t t
1 0 whichtheoperatorn(i∂ )actsjustontheelectricfieldE
t
We may therefore quantize the theoryperturbatively, by andnotonthevariationoftherefractiveindexδn(x−vt).
using the unperturbed momentum operator Π~ and the This perturbation has the same form as the interaction
perturbation Hamiltonian HI considered in [18], cf. equation (59). In
the present case δn(x−vt) plays the role of ξ(~r,t) and
1 1 1
δE~ =ηE~1 =−ε(δεE~0)=−ε(δεεΠ~). (16) the electric field of Π~ˆ. These considerations allows us to
apply the perturbative scheme of [18]. The expressionof
The action is then
the electric field E is (see [17], equation (32), coherently
1
S = Π~ ·E~dx4 adapted to the notation of [18]):
2
Z
1 1 1 1 1
= Π~ · Π~dx4− Π~ · (δε Π~)dx4+... ω 1/2 1
2 ε 2 ε ε E~(~r,t)=i a (t)ei~k·~r
1Z 1 1Z 1 1 2V n(ω)ng(ω) ~kµ
= Π~ · Π~dx4− Π~ ·δε Π~dx4+...(17) X~kµ (cid:16) (cid:17) h
2 ε 2 ε ε
Z Z −a† (t)e−i~k·~r ~e , (21)
where the dots are higher order terms and we have used ~kµ ~kµ
the symmetry of ε−1 (and dropped total time deriva- i
tives). where n (ω) = d(n(ω)ω) (ω)+ωdn(ω) = c and k =
Thus, the Hamiltonian density is g dωk g dω vg(ω)
(~k,ω) = (~k,ck/(n(ω)), with k = |~k| and ~e =~k/k. The
~k
H=H0+HI, (18) action of the operator n(i∂ ) on the electric field E~ is:
t
where H is the unperturbed hamiltonian (with disper-
0
sion) and
n(i∂ )E~(i∂ ) e−iωktE dk
t t ω
11 1
H :=− Π~ ·δε Π~ (19) Z
I
2ε ε
= n(ω)E dk, (22)
ω
is the first order perturbation, which reduces to the
Z
hamiltonian used in Refs. [13, 18] in absence of disper-
sion. where Eω are the Fourier modes in the expansion (21)
The perturbative computation is then carried forward of the electric field. Thus, the number of particles of
starting from this expression. momentum ~k and polarization µ per unit of volume
1 1
4
generated by the disturbance HI turns out to be: where Nˆk1µ1 is the number operator. Expanding the
product of the fields and the number operator in the
<Nˆk1µ1 >=<ψ(t→∞)|Nˆk1µ1|ψ(t→∞)> bracket with the vacuum state and defining f(k) :=
= d4x1 d4x2 <0|HI(x1)Nˆk1µ1HI(x2)|0>(,23) 2ωV 1/2 n(ω)n1g(ω) we obtain the expression:
Z Z (cid:0) (cid:1)
Nkµ = d4x1 d4x2 <0| f(k1)f(k2)f(k3)f(k4)aˆ~k1µ1~e~k1µ1aˆ~k2µ2~e~k2µ2aˆ~†kµaˆ~kµaˆ~†k3µ3~e~k3µ3aˆ~†k4µ4~e~k4µ4
Z Z ~k1µX1,~k2µ2,
~k3µ3,~k4µ4
δn(x )δn(x )n(ω )n(ω )e−i(k2+k1)xei(k3+k4)x|0>, (24)
1 2 2 4
where the argument x of δn is a four-vector and in our inside the Cerenkovconeθ =arccos(c/vn )(anomalous
0 0
particular case it depends just on x and t as x−vt. We Dopplerregion,seeFig.1),andthe otheris emittedout-
indicate with a tilde the four-dimensional Fourier trans- side the cone, in the normal Doppler region.
form g˜(~k, ck) = dtdxdydzg(~x,t)ei~k·~x−inck0t and using In the dispersive case we cannot always identify distinct
n0
the commutation rules for the operators a and a† we cones of emission for the photons as before. The delta
R k k function gives the condition:
obtain:
c c
Nk1µ1 =f(k1)2 δ˜n(k1+k2)2f(k2)2[n(ω1)+n(ω2)]2 k1x− vn(ω )k1 + k2x− vn(ω )k2 =0. (28)
~kX2µ2 (cid:18) 1 (cid:19) (cid:18) 2 (cid:19)
(25) Similarly to the non-dispersive case, this equation im-
(e ·e )2. plies that whenever k1x/k1 > c/[vn(ω1)], the sec-
~k1µ1 ~k2µ2 ond photon must satisfy k /k < c/[vn(ω )]. In
2x 2 2
terms of emission angles for the two photons, we find
The space time dependence of the perturbation dn is of
θ < arccos{c/[vn(ω )]} and θ > arccos{c/[vn(ω )]}.
theformδn(t,x,y,z)≡δn(x−vt,y,z). Hence,usingthe 1 1 2 2
Therefore, if arccos{c/[vn(ω )]} > arccos{c/[vn(ω )]},
variables u = x−vt, w = x+vt, the Fourier transform 1 2
of δn is
δ˜n(k1+k2)= 2vπδ k1x+k2x− vc n(kω1 ) + n(kω2 ) normal Doppler 2 (a)
(cid:18) (cid:18) 1 2 (cid:19)(cid:19) region
(26) !2 !1 v
1
dudydzδn(u,y,z)ei(k1x+k2x)u+i(k1y+k2y)y+i(k1z+k2z)z. anomalous Doppler
Z region
(b)
A. Superluminality emission cones normal Doppler
As for the non dispersive case this last result is very region !2 1 2 !1 v
anomalous Doppler
meaningful from a physical point of view. The support
region
of the delta distribution gives precise conditions for the
possible states of the emitted particles. First we ana-
lyze the condition given by the delta function in the non
dispersive case. Its argument gives the constraint: FIG.2: Schematical representation of correlated photon pair
emission from a superluminal perturbation moving with ve-
c c locity v in the presence of dispersion: the Cerenkov cone is
k − k + k − k =0. (27)
1x vn 1 2x vn 2 split in two and depending on the medium dispersion and
(cid:18) 0 (cid:19) (cid:18) 0 (cid:19)
frequencyofemittedphotons,regionsinwhichnophotonsor
This equation implies that whenever k /k > c/(vn ), bothphotonsareemittedmayappear. Forthecaseofnormal
1x 1 0
thesecondphotonmustsatisfyk /k <c/(vn ),i.e. out dispersion(refractiveincreaseswithincreasingfrequency),the
2x 2 0
of every pair of emitted photons, one photon is emitted figureshows thesituation for ω1 <ω2 (a) and ω1 >ω2 (b).
5
the two cones overlap and if arccos{c/[vn(ω )]} < condensed media or ∼1 ns in gasseous media [20]. This
1
arccos{c/[vn(ω )]} there is a gap between them. In the situationmaythereforebeadequatelyapproximatedbya
2
first case there is a region in which both photons can be tanh-like function. We alsonote thata plasma frontwill
emitted(differently fromthe non-dispersivecase),whilst locally reduce the refractive index (δn < 0) as opposed
inthesecondcasethereisaregioninwhichnophotonat to the Kerr effect that will usually increase the refrac-
allcanbeemitted. ThetwocasesareshowninFigs.2(a) tive index (δn > 0). Yet the result of the calculations
and(b)respectively. Thenon-dispersivecaseisobtained depends not onthe signof δn but only on its amplitude,
in the limiting situation in which arccos{c/[vn(ω )]} = velocity and physical dimensions.
1
arccos{c/[vn(ω )]}. The shape of the perturbation therefore is taken as:
2
δn=ηtanhx−vte−2yσ2y2e−2zσ2z2.
Gaussian shape of the perturbation σ
x
Again we use the variables u = x−vt, w = x+vt, and
We now focus attention on the problem of determin- y and z unchanged. Repeating the same computation as
ing the actual number of emitted photons, starting from above we obtain:
the assumption of a Gaussian shape for the refractive
index perturbation δn = ηe−2σ12[(x−vt)2+y2+z2]. This Nk1 = σx2σy2vσ2Vz2η2π2π2n2(ω ω)n12(ω ) d3~k2 (30)
form for the perturbation is what one may expect when 1 g 1 Z
a local refractive index variation is induced by a laser ω2 csch2 πσx(k +k )
pulse through the nonlinear Kerr effect. Indeed, laser n2(ω2)n2g(ω2) 2 1x 2x
h i
pfiulelsaeslotnygpibcoatllhylhoanvgeituGdaiunsasliaann-dshtarpaendsvienrtseensciotyo,rdIi,nparteos- e−σy2(k1y+k2y)2e−σz2(k1z+k2z)2[n(ω1)+n(ω2)]2
so that δn2I, where n2 is the nonlinear Kerr index, will k12k22+(~k1·~k2)2 δ k +k − 1(ω +ω ) 2,
havethesameform. Atypicalvalueforη,forexamplein k2k2 1x 2x v 1 2
fusedsilicaglass,is∼0.001wheren ∼3·10−16 cm2/W 1 2 !(cid:20) (cid:18) (cid:19)(cid:21)
2
andwetakeI ∼3·1012 W/cm2. Insertingthisδnprofile Form these relations for the total emitted photon num-
in(25),using →V d3~k 1 andsumming overthe bers it is immediately apparent that it is advantageous
~k (2π)3
polarisation states µ and µ we obtain: to keep a large a perturbation as possible with each
P 1 R 2 of the longitudinal and transverse dimensions σ
x,y,z
Nk1 = σ6vη22Vπ2n2(ω ω)n12(ω ) d3~k2 (29) cshonowtrnibubteilnogw,eqiutatlluyrnins tohuetmthualttiptlhiceattiavnehprper-ofaficletoer.mAitss
1 g 1 Z
e−σ2|~k1+~k2|2 ω2 [n(ω )+n(ω )]2 roughly half the number of photons of a dimensionally
n2(ω )n2(ω ) 1 2 similar Gaussian perturbation, indicating that the main
2 g 2
role is played by the transient switch-on and off of the
k12k22+(~k1·~k2)2 δ(k +k − 1(ω +ω )) 2. perturbation. We alsonotethedependence onthegroup
k2k2 1x 2x v 1 2 refractive index at the emission frequency. As we shall
1 2 !(cid:20) (cid:21)
discussbelow,thisallowsanadditionaldegreeoffreedom
Note that in the nondispersive limit, i.e. dn(ω) = 0, the forenhancingorcontrollingthephotonemissionprocess.
dω
expression for the number of emitted particles N re-
k1
duces to equation (3).
B. Numerical analysis - Gaussian shape
Hyperbolic tangent shape of the perturbation
The non trivial dispersion relation appearing in equa-
We canconsideranothershapefor the perturbationof tion (29) makes the dispersive case more intricate to an-
the refractive index. In particular, let us focus on a pro- alyze,evennumerically,thanthe nondispersiveone. We
file given by a hyperbolic tangent function. This allows perform the numerical analysis in fused silica where, for
us to investigatehow the particulargeometryofthe per- completenessweshowthefulldispersionlawgivenbythe
turbation affects the maximum of the wavelengths and Sellmeier relation [21]:
thenumberofemittedparticles. Moreover,atanh-shape
a λ2 a λ2 a λ2 1/2
may arise if for example the dominant refractive varia- 1 2 3
n(λ)=Re 1+ + + , (31)
tion is related to plasma generatedthroughmultiphoton λ2−l2 λ2−l2 λ2−l2
(cid:20) 1 2 3(cid:21)
ortunnellingionisationbyanintenselaserpulse. Intense
where
laser pulses may indeed efficiently ionise the medium
within the first few optical cycles and thus create a very a =0.473115591 l =0.0129957170
1 1
steep moving plasma front followed by a nearly constant
a =0.631038719 l =4.12809220·10−3
plasma density that will decay (through electron recom- 2 2
a =0.906404498 l =98.7685322.
bination) over time scales of the order of ∼ 100 fs in 3 3
6
~k =(k ,k ,0)=(k cosθ ,k sinθ ,0), then:
(cid:7)(cid:21)(cid:4) (cid:7)(cid:18)(cid:18) (cid:7)(cid:6) (cid:7)(cid:2)(cid:3) (cid:7)(cid:5) (cid:7)(cid:1) 2 2x 2y 2 2 2 2
(cid:1)(cid:19)(cid:4) (cid:12)(cid:23)(cid:14) (cid:1)(cid:3) (cid:12)(cid:17)(cid:14) |~k1+~k2|2 =k12+k22+2k1k2cos(θ2−θ1)≥k12+k22
(cid:11)(cid:14) −2k k
(cid:15)(cid:20)(cid:12)(cid:10)(cid:1)(cid:2)(cid:19)(cid:6) (cid:22)(cid:2)(cid:3) (~k·~k′)2 =[k k cos(θ −θ )]2 ≤k21k22
1 2 2 1 1 2
(cid:2)(cid:19)(cid:1) (cid:1)
(cid:1) (cid:18) (cid:1) (cid:4) (cid:5) (cid:6) The two conditions are both satisfied for θ2 −θ1 = π,
(cid:15)(cid:2)(cid:20)(cid:12)(cid:10)(cid:11)(cid:14) (cid:15)(cid:12)(cid:10)(cid:11)(cid:14) i.e. the two vectors point in opposite directions. The
(cid:7)(cid:2)(cid:3) (cid:7)(cid:5) (cid:7)(cid:1) condition given by the delta function in equation (29) is
(cid:1)(cid:3) (cid:12)(cid:16)(cid:14) (cid:6) (cid:8)(cid:8)(cid:9)(cid:9)(cid:2)(cid:1)(cid:10)(cid:10)(cid:11)(cid:11) (cid:12)(cid:13)(cid:14) k1x +k2x − vc(n(kω11) + n(kω22)) = 0 that can be rewritten
(cid:22) (cid:11)(cid:14)(cid:5) ask1cosθ1−k2cosθ1−vcn(kω11)−vcn(kω22) =0,whereθ1 is
(cid:2)(cid:3) (cid:12)(cid:10) (cid:4) the angle between ~k and the x axis and, consequently,
(cid:15)(cid:1) 1
(cid:1) π+θ the angle between k and the x axis. As observed
2 2
(cid:1) (cid:3) before, the maximum of equation (29) depends only on
(cid:1) (cid:4) (cid:5) (cid:6) (cid:3) (cid:1) (cid:4)
(cid:15)(cid:12)(cid:10)(cid:11)(cid:14) (cid:15)(cid:2)(cid:12)(cid:10)(cid:11)(cid:14) k1, k2 and~k1·~k2, Thus, at the maxima we have:
d
N (k ,k ,~k ·~k )+λg(k ,k ,θ )
FIG. 3: (a) log10Nλ1,λ2 for k1x corresponding to the maxi- dθ1 k1,k2 1 2 1 2 1 2 1
mum. (b) Maxima of log10Nλ1,λ2 for θ1 = 0, θ2 = π, and = hd λg(k ,k ,θ )=λ(k −k )sinθ =i0,
σ = 1µm. (c) Maxima of log10Nλ1,λ2 for θ1 = 0, θ2 = π, dθ1 1 2 1 1 2 1
and σ = 2µm. (d) Relation between the wavelengths of the
emitted couples for β increasing from 2, in the right part, to where λ is a Lagrange multiplier. Being k and k in
1 2
20, in the left part of thepicture. general not equal and λ 6= 0, we therefore have θ = 0.
1
Wealsoverifiednumericallythatindeedthemaximumof
N isatθ =0: WevisualizethisinFigure3(a),that
k1,k2 1
showsanexample(σ =2andβ =5)ofthedistributionof
photonnumbersinLogscale. The twostraightlines cor-
respond to the condition k2 =k2 and k2 =k2 and the
1 1x 2 2x
As in the non dispersive case, the delta distribution in curverepresentstherelationbetweenλ andλ givenby
1 2
equation (29) gives a constraintthat furnishes a relation the delta function in Eq. (29). As discussed, these three
betweenthemomentaoftheproducedparticles. Inprin- curvesintersectatthe maximumofN implying that
k1,k2
ciple, a component of the momentum, say k2x, could be indeed maximum emission occurs along the propagation
expressed as a function of the other ones, so that by in- direction θ =0.
1
tegrating over k2y and k2z one could find explicitly the Weperformthenumericalanalysisforperturbationswith
functionN~k1 givingthedistributionoftheproducedpar- radius σ = 1 µm and σ = 2µm and for increasing
ticles as a function of the momentum. However, for a values of β. As explained, to give an estimation of
generic dispersion law, the constraint cannot be analyti- the number of emitted couples for every single DP we
callysolved. Wethereforestudythecriticalpointsofthe
integrand of equation (29) that we will call N . This
k1,k2 (a)σ=1µm (b)σ=2µm
is a function of five variables, because the six compo-
nents of the momenta are notallindependent due to the β λ1max λ2max Nλ1,λ2 β λ1max λ2max Nλ1,λ2
constraintgivenbythedelta. Thesymmetryoftheprob- m m m m
lem allows to restrict the analysis to the planes k1z = 0 2 2.51e-6 4.98e-6 6.13e-7 2 3.93e-6 7.02e-6 4.14e-8
and k = 0 reducing the number of variables to three.
2z 3 1.93e-6 3.06e-6 6.68e-6 3 3.50e-6 5.38e-6 2.14e-6
This function describes the distribution of emitted cou-
4 1.54e-6 2.17e-6 3.01e-5 4 2.95e-6 4.11e-6 1.27e-5
ples as function of the momenta k and k . Its max-
1 2
5 1.26e-6 1.66e-6 9.35e-5 5 2.49e-6 3.26e-6 4.28e-5
imashowthemomentaofthe coupleofparticlesemitted
6 1.08e-6 1.36e-6 2.33e-4 6 2.13e-6 2.68e-6 1.11e-4
withhighestprobability. Therefore,theeffectivenumber
of emitted particles can be found integrating this dis- 7 0.94e-6 1.15e-6 5.02e-4 7 1.87e-6 2.69e-6 2.44e-4
tribution. We note that the maximum of the function 8 0.83e-6 0.99e-6 8.73e-4 8 1.65e-6 1.96e-6 4.80e-4
Nk1,k2 is located around θ1 = 0 and θ2 = π. Indeed, in 9 0.75e-6 0.87e-6 1.74-3 9 1.48e-6 1.73e-6 8.69e-4
equation (29) all factors depend on ~k and ~k by their 10 0.68e-6 0.78e-6 2.91e-3 10 1.35e-6 1.54e-6 1.47e-3
1 2
modulus except for the terms |~k +~k |2 and (~k ·~k )2 15 0.47e-6 0.52e-6 2.09e-2 15 0.92e-6 1.00e-6 1.11e-2
1 2 1 2
depending on the relative orientation between the two 20 0.36e-6 0.39e-6 8.19e-2 20 0.70e-6 0.75e-6 4.63e-2
wave vectors. For fixed k and k the function N
1 2 k1,k2
is maximum for small |~k1 +~k2|2 and large (~k1 ·~k2)2. TABLE I:Maxima of Nλ1,λ2 for increasing β and two values
Let be ~k = (k ,k ,0) = (k cosθ ,k sinθ ,0) and of σ.
1 1x 1y 1 1 1 1
7
have to evaluate the integrand of the equation (29), i.e. (cid:17)(cid:4)(cid:16) (cid:17)(cid:14) (cid:17)(cid:12)
vNakl1u,ek2of:=δ(0d)Nh/(adsΩbdeΩe′ndkadpkp′r)o.xiImnattehde wcoitmhpLut/a(t2iπon) [tsheee (cid:12)(cid:16) (cid:10)(cid:20)(cid:11) (cid:13)(cid:5)(cid:19) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:4)(cid:6)(cid:7)(cid:8) (cid:10)(cid:21)(cid:11)
Eq.(26)]. In the followingwe reportallphoton numbers (cid:8)(cid:11)
in function of wavelength rather than wave-vector (thus (cid:18)(cid:4)(cid:16) (cid:10)(cid:7)(cid:12)(cid:5)(cid:19)
indicated as N ) and integrated over the azimuthal (cid:9)(cid:12)
λ1,λ2
angles ϕ and ϕ .
1 2
Figures 3(b) and (c) show the calculatedphoton num- (cid:12) (cid:16) (cid:4) (cid:12)
(cid:12) (cid:13) (cid:14) (cid:15)
bers Nλ1,λ2 in Log scale for the value of λ2 at which (cid:9)(cid:6)(cid:10)(cid:7)(cid:8)(cid:11) (cid:9)(cid:4)(cid:6)(cid:10)(cid:7)(cid:8)(cid:11)
emission is maximum and for η = 0.001, two different
values of the perturbation size, σ = 1 and 2 µm respec-
FIG. 4: (a) log10Nλ1,λ2 for θ1 = 0, θ2 = π and for a hy-
tively. Figure 3(d) shows both wavelengths λ1 = λ1max perbolic tangent shape of the perturbation. (b) Correlated
and λ2 = λ2max at which emission is maximum and for photon wavelengths for β increasing from 2 (top right hand
2<β <20. At lowestβ, λ1 and λ2 have the largestval- point) to 20 (bottom left point).
ues and monotonically decreases and β increases. These
points are also shown for the two values of σ consid-
ered, 1 µm (blue dots) and 2 µm (red dots). We observe
velocitiesexceedingthespeedoflighthavebeenobserved
thatthephotonemissionispeakedatwavelengthsλ
1max inseveralexperiments[22–26]wherebythemediumisei-
that increases with the perturbation diameter. We also
ther chosen so as to have an absorption resonance close
note that, as in the non dispersive case [13], λ and
1max to the frequency of interest or is structurally modified,
λ decreasefromthemid-infraredregionintothevis-
2max e.g. into Bragg grating structures so that the dispersion
ible region as β, i.e. the perturbation velocity increases.
curve is strongly modified (without absorption). This
However,differentlyfromthenon-dispersivecase,weun-
property can therefore be exploited to effectively engi-
derlinethatingeneralλ andλ aredifferent,i.e.
1max 2max neerthe (dispersionpropertiesofthe)vacuumstates,in-
the photons are correlated but at different wavelengths.
creasing the number of emitted photons and, moreover,
However,forincreasingβ thisdifferencebecomessmaller
providing an additional tool for investigating the corre-
with λ →λ as β →∞.
1max 2max lation properties of the produced couples. We emulate
Some numerical values of N from these graphs are
λ1,λ2 suchsuperluminalgroupvelocitiesby introducinga sim-
given in Table I(a) and (b). An estimation of the actual
ple Lorentzian correction on top a backgroundSellmeier
numberofemittedparticlescanbeobtainedbyintegrat-
relation. Wethereforeobtainadispersionlawwithanar-
ing the function N overλ, θ and θ : this furnishes,
λ1,λ2 1 2 row peak whose maximum slope is placed in correspon-
for η =0.001, σ =1µm, β =20 and a 5 cm propagation
dence to the point of the highest photon emission for a
distance, a number of emitted photons for every DP in
mediumcharacterisedonlybythebackgrounddispersion.
an angle of 30 degree of ∼3·10−4.
Wegiveanexampleofsuchamodifieddispersionlawfor
the fused silica in Fig. 5(a) and the correspondinggroup
index n (ω)(λ(ω)) − λ(ω)dn(λ) in Fig. 5(b). We then
C. Numerical analysis - Hyperbolic tangent shape g dλ
perform the analysis for a DP of Gaussian shape with
η = 0.001, σ = 1µm and traveling at β = 20 in such a
We perform a similar numerical analysis for a hyper-
medium. In Figure 5(c) we show the distribution of the
bolic tangent shape of the perturbation of the refractive
emittedcouplesinamediumwiththebackgrounddisper-
indexbystudyingtheintegrandofequation(30). Adapt-
sionlawandinFigure5(d) the distributionproducedby
ingthepreviousargumenttothiscasewesearchitsmax-
the same perturbation in a medium with the Lorentzian
ima in the direction of the propagation of the DP, i.e.
correction to the dispersion. As a first comment, we ob-
for θ = 0. We assume a perturbation with dimensions
serve that the number of emitted particles increases by
σ =1.1µm and σ =σ =1µm and η =0.001. An es-
x y z a factor ten due to the dependence of Eq. (29) on the
timation similar to the gaussiancase, for β =20 and a 5
groupindicesn (ω )andn (ω ). Moreover,theemission
g 1 g 2
cmpropagationdistance,givesanumberofemittedpho-
spectrumisstronglydistortedbutmostimportantly,two
tonsforeveryDPinanangleof30degreeof∼1.5·10−4.
distinct maxima appear. These two peaks are related to
enhanced emission separately at λ or at λ . In virtue
1 2
of the nature of the two emitted photons, enhancement
D. Fast light at one wavelength will necessarily lead to enhancement
atthecorrelatedphotonwavelength. Therefore,observa-
Wenowreturntoafeatureoutlinedearlieron,i.e. the tion of these two emission peaks in an experiment could
dependence of the photon emissivity on the group index be considered as evidence of correlated photon emission
at the emitted frequencies [see e.g. Eq. (29)]. From our fromthesuperluminalperturbationbymeasuringonlyin
equations it is clear that by a reduction in the groupve- the forward direction. We performed a similar analysis
locity at ω and/or ω may greatly influence the actual for the case of silicon as this is a widely used material
1 2
number of emitted photons at both frequencies. Group inoptics andwaveguidetechnologywhere bothslowand
8
(cid:6)(cid:7)(cid:8)(cid:9)
(cid:2)(cid:16)(cid:5) (cid:2)(cid:18)(cid:5) pump pulse
(cid:14)(cid:7)(cid:8)
(cid:6)(cid:7)(cid:10)(cid:11) waveguide
(cid:15) (cid:15)(cid:17) v=c/cos
(cid:6)(cid:7)(cid:8)
(cid:6)(cid:7)(cid:10)(cid:14)
(cid:6)(cid:7)(cid:12)(cid:13)(cid:9) (cid:6) (cid:14) (cid:12) (cid:10) (cid:9)(cid:7)(cid:8)(cid:9) (cid:14) (cid:10) (cid:1) low-index substrate
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
(cid:9)(cid:7)(cid:10) (cid:2)(cid:19)(cid:5) (cid:8) FIG.7: Schematicalrepresentationofanexperimentallayout
(cid:2)(cid:20)(cid:5)
forcreatingasuperluminalrefractiveindexperturbationina
(cid:22)(cid:23)(cid:17)(cid:24)(cid:6)(cid:9)(cid:1)(cid:1)(cid:6)(cid:14)(cid:9)(cid:7)(cid:14) (cid:22)(cid:23)(cid:17)(cid:24)(cid:6)(cid:9)(cid:1)(cid:1)(cid:6)(cid:14)(cid:14)(cid:7)(cid:8) waveguide.
(cid:9) (cid:9)
(cid:9) (cid:6) (cid:14) (cid:9)(cid:7)(cid:6) (cid:9)(cid:7)(cid:12) (cid:9)(cid:7)(cid:11) (cid:9)(cid:7)(cid:21)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
only if v > c/n(ω). This effect bears a strong resem-
blance to the anomalous Doppler effect and in this sense
FIG.5: Dispersionlaw(a)andgroupindex(b)forthemodi- represent the first detailed analysis of a setting in which
fieddispersion lawofthefused silica. (c)log10Nλ1,λ2 forthe theeffectmayactuallybeobserved. Thenumberofemit-
standard dispersion law of the fused silica. (d) log10Nλ1,λ2 tedphotonpairsisrelativelysmallclosetothe threshold
for the modified dispersion law of thefused silica.
v =c/n(ω) but increases significantly if v is much larger
(e.g. 10−20×)largerthanc. Ouranalysisfullyaccounts
for material dispersion and, doing so, introduces an in-
(cid:14)(cid:15)(cid:19) (cid:20)
(cid:2)(cid:17)(cid:5) (cid:2)(cid:8)(cid:5) triguingdependenceonthegroupindexorgroupvelocity
ofthe emittedphotons. Thisdependence andcarefulen-
(cid:6)(cid:14)(cid:15)(cid:16)(cid:16) (cid:6)(cid:7) (cid:18) gineering of the host medium may be used to enhance
photon emission by at least an order of magnitude and
alsoprovidesdirectevidenceofcorrelatedemissioninthe
(cid:14)(cid:15)(cid:18) form of two correlated peaks in the output spectra.
(cid:12) (cid:13) (cid:14) (cid:12) (cid:13) (cid:14)
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5) Fromouranalysisitwouldseemthatexperimentsarein-
(cid:21)(cid:12)(cid:11)(cid:22)(cid:14) (cid:21)(cid:12)(cid:11)(cid:22)(cid:13) deedfeasible. Refractiveindexperturbationsoftheorder
(cid:2)(cid:9)(cid:5) (cid:14)(cid:15)(cid:16) (cid:2)(cid:10)(cid:5) of those used in this work, η ∼ 0.001, may be obtained
(cid:13)
by focusing laser pulses in fused silica or other nonlinear
(cid:25)(cid:1)(cid:1)(cid:12)(cid:13) (cid:25)(cid:1)(cid:1)(cid:12)(cid:13) media and higher values have been observed. According
(cid:7)(cid:12)(cid:11) (cid:12) (cid:7)(cid:12)(cid:11)(cid:13) toourresultsthis wouldresultina photonpairemission
(cid:24) (cid:24)
(cid:23) (cid:23) rate of ∼ 10−6 pairs/pulse at β = 20 in a 5 mm long
(cid:11) (cid:11) waveguide, i.e. ∼ 1 photon/second if a MHz repetition-
(cid:12) (cid:13) (cid:14) (cid:12) (cid:13) (cid:14) ratelaserisused. Thesenumbersmaybeincreasedbyan
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
orderofmagnitudeormoree.g. byusingfastlightmedia
or higher DP amplitudes. The high velocities used here
FIG. 6: Dispersion law (a) and group index (b) for the mod- for the perturbation may be obtained experimentally by
ified dispersion law of the silicon. (c) log10Nλ1,λ2 for the sending an extended(approximately “plane wave”)laser
standard dispersion law of the silicon. (d) log10Nλ1,λ2 for pulse at an angle onto the host medium that could be,
themodified dispersion law. for example, a waveguide (see Fig. 7). The pulse would
intersect the waveguide with an angle θ and the Kerr
effect would create a DP only at the intersection point
fast light may be engineered. We obtain a similar in- that travels with speed c/cosθ (if we approximate the
crease of the emitted particles as shown in Fig. 6 for a refractive index of air ∼ 1). Therefore, an incidence an-
perturbation of dimension σ =1µm and velocity β =4. gle of ∼ 80−85 degrees (i.e. close to normal incidence
on the waveguide)would guaranteeβ ∼5−20. We note
that a similar approach has actually been used in pre-
vious pioneering experiments in which a DC field was
E. Conclusions convertedtoTeraHertzradiationbyasuperluminallaser
pulse excitation of an array of biased capacitors [27]. In
Perturbative analysis of quantum fluctuation excita- ourcaseweareexcitingasuperluminalDPinadielectric
tion from a travelling refractive index perturbation in- medium andcorrelatedphotonpairs wouldbe generated
dicates that emission of correlated photon pairs occurs and collected from the two ends of the waveguide.
9
Acknowledgments from l’Universit´a dell’Insubria and Fondazione Cariplo,
Univercomo and Banca del Monte di Lombardia.
TheauthorsthankT.KraussandG.Ortenziforfruit-
ful discussions. F. D.P. acknowledges financial support
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