Spectrum of the parametric down converted radiation calculated in the Wigner function formalism

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Springer Nature in The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics, Vol. 13 on January 2001, available at: https://doi.org/10.1007/s100530170292 © 2001 EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag Spectrum of the parametric down converted radiation calculated in the Wigner function formalism Alberto Casado1, Trevor Marshall2, Ram´on Risco-Delgado1 and Emilio Santos3. 1Departamento de F´ısica Aplicada, Universidad de Sevilla, 41092 Sevilla, Spain. 2Department of Mathematics, University of Manchester, Manchester M13 9PL, U. K. 3Departamento de F´ısica Moderna, Universidad de Cantabria, 39005 Santander, Spain. Abstract We continue the study of parametric down conversion within the framework of the Wigner representation, by using a Maxwellian approach developed in a recent paper [A. Casado et al., Eur. Phys. J. D 11, 465 (2000)]. This gives a mechanism, inside the crystal, for the production of the downconverted radiation. We obtain the electric field to second order in the coupling constant by using the Green’s function method, and compare our treatment with the standard hamiltonian approach. The spectrum of the down-converted radiation is calculated as a function of the parameters of the nonlinear crystal (in particular the length) and the radius of the pumpimg beam. PACS. 42.50.Ar Photon statistics and coherence theory – 03.65.Sq Semiclassical theories and applications – 42.50.Lc Quantum fluctuations, quantum noise, and quantum jumps 1 1 INTRODUCTION For many years the state of the radiated field corresponding to quantum parametric down-conversion (PDC from now on) has been studied. A single monochromatic laser converts into pairs of highly correlated photons fulfilling the frequency matching conditions [1, 2]. PDC is usually considered as a typical quantum phenomena, not only because there is no solution of the classical Maxwell equations which represents the so called “spontaneous splitting” of the laser, but also because there is a very short correlation time between the conjugate beams [3, 4] and a high visibility of interference patterns in joint detection experiments [5]. These properties have been used in order to test Bell’s inequalities [6] and to show other nonclassical aspects of the down-converted light [7]. The theory of PDC in the Wigner function formalism of quantum optics was treated in an earlier series of papers by using a standard hamiltonian approach [8, 9, 10, 11]. We described how the radiated field is produced via the coupling between the laser beam and the zeropoint radiation inside the crystal. We also studied the process of light detection, stressing the fact that all detectors integrate the light intensity over a large time window [11]. We should recall that the hamiltonian formalism was originally developed, during the 1960s, in parallel [13] with a treatment, rather similar to ours, which also took account of the zeropoint field. Indeed the name by which PDC was known during that period was spontaneous parametric fluorescence, and such ways of describing the interaction persist up to the present day. More recently we have developed the theory of PDC by starting from the Maxwell equations inside the crystal, in place of the usual hamiltonian standard model [12]. We showed that the production and propagation of PDC light is entirely equivalent to classical electromagnetic field theory, provided that we consider the zeropoint field entering the crystal along with the laser beam. An explicit expression for the first order electric field amplitude in the far field approximation was obtained by using the Green’s function method, in order to calculate the cross-correlation of photon counts. Here we shall use the same formulation in order to study the spectral properties of the PDC radiation. In Sec.II we summarize the main results of the Wigner representation of second-order nonlinear optical phenomena in the Maxwellian approach [12], and in Sec. III we calculate the electric field to second order in the coupling parameter. In addition to the part of the field that corresponds to the splitting of the laser (PDC), there appear some new 2 contributions coming from the up and down conversion of the zeropoint field. In Section IV we analyze the relation between the different contributions of the field to the autocorrelation, and a comparison is made with the results in the hamiltonian approach. Finally, in Sec. V we obtain the PDC spectrum, i.e. the intensity of the down-converted radiation, at each point, as a function of the frequency, and we consider the long and short crystal approximations. 2 WIGNER REPRESENTATION OF PDC The study of PDC in the Wigner representation within a Maxwellian approach starts with the evolution equation for the electric field operator in the Heisenberg picture. By considering a second-order nonlinear isotropic medium, the corresponding equation is [12] ∇2ˆ E−1 c2 f ∂2ˆ E ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)ˆ E(t′)dt′=β∂2ˆ E2 ∂t2,(1) cfbeing the speed of light in free space. χis the linear susceptibility of the medium, and βis a coupling constant which is defined by β≡2µ0d, (2) where dis the bilinear susceptibility and µ0the magnetic permeability of free space. Eq. (1) is an inhomogeneous wave equation in which the source of radiation is a quadratic function of ˆ E. In order to solve it an adiabatic switch on of the interaction is considered by substituting λ(t)βfor β,λ(t) being a slowly varying function of time, so that λ(t) = 0 at t→ −∞, and λ(t)=1 at t≥0. The state of the radiation at t→ −∞ is that corresponding to a laser, i.e. a coherent state |ϕ⟩, fulfilling ˆ E(+)|ϕ⟩=E(+) laser|ϕ⟩,(3) ˆ E(+) being the part of the electric field operator that only contains destruction operators: ˆ E(+) =iX k ¯hωk L3 0!1 2 ˆake−iωkt+ik·r,(4) 3 L3 0being the normalization volume, and ωk=|k|cf. It is possible to use the vacuum field as the initial state if we perform the following change of variables: ˆ E(+) =ˆ E′(+) +E(+) laser,(5) By substituting (5) into Eq. (1), we have ∇2(ˆ E′+Elaser)−1 c2 f ∂2(ˆ E′+Elaser) ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)[ ˆ E′(t′) + Elaser(t′)]dt′ =λ(t)β∂2(ˆ E′+Elaser)2 ∂t2.(6) Let us restrict our attention to Eq. (6) for t≥0. If there were no laser beam incoming to the nonlinear medium, i.e. if we made Elaser = 0 in Eq. (6), then this equation would represent the evolution of the vacuum due to the presence of the crystal, and would give rise to just a modified vacuum. If we take into account that the laser is very intense, it seems reasonable to discard the term β∂2ˆ E′2/∂t2from (6), because its contribution to the radiated field is very small compared with the others. Hence Eq. (6) reduces to the following equation, which is linear in the field operators: ∇2(ˆ E′+Elaser)−1 c2 f ∂2(ˆ E′+Elaser) ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)[ ˆ E′(t′) + Elaser(t′)]dt′ =β∂2(2Elaser ˆ E′+E2 laser) ∂t2.(7) Let us now pass to the Wigner representation. As is well known, the evolution equations of the Wigner field amplitudes are the same as the Heisenberg equations of motion of the quantum field amplitudes, whenever these are linear. Then, in order to go to the Wigner representation we simply remove the hats in Eq. (7) (we shall remove also the prime in order to simplify the notation), so that ∇2(E+Elaser)−1 c2 f ∂2(E+Elaser) ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)[E(t′) + Elaser(t′)]dt′ 4 =β∂2(2ElaserE+E2 laser) ∂t2.(8) Because we are in the Heisenberg picture the state does not change with time and we shall use the Wigner function corresponding to the initial state, i.e. the Wigner function of the vacuum state Wvacuum({αk},{α∗ k}) = Y k 2 πe−2αkα∗ k,(9) αkbeing the complex amplitude corresponding to the mode kof the zeropoint radiation EZP =E(+) ZP +E(−) ZP , where E(+) ZP =iX k ¯hωk L3 0!1 2 αke−iωkt+ik·r;E(−) ZP = [E(+) ZP ]∗.(10) E(+) ZP is the positive frequency part of the vacuum field. On the other hand, from (9) it follows trivially that ⟨αk⟩= 0 ; ⟨αkαk′⟩= 0 ; ⟨αkα∗ k′⟩=1 2δk,k′.(11) 3 SECOND-ORDER PERTURBATION THEORY FOR THE CALCULATION OF THE RADIATED FIELD The electric field can be expressed as an expansion in powers of the small parameter β[9]: E(r, t) = E0(r, t) + βE1(r, t) + β2E2(r, t) + ... (12) Now a set of coupled equations is obtainted by substituting (12) into (8): ∇2(E0+Elaser)−1 c2 f ∂2(E0+Elaser) ∂t2 −∂2 ∂t2Zt −∞ χ(t−t′)[E0(t′) + Elaser(t′)]dt′= 0,(13) 5 ∇2E1−1 c2 f ∂2E1 ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)E1(t′)dt′ =∂2(2E0Elaser +E2 laser) ∂t2,(14) ∇2E2−1 c2 f ∂2E2 ∂t2−∂2 ∂t2Zt −∞ χ(t−t′)E2(t′)dt′= 2∂2(E1Elaser) ∂t2,(15) and so on. To zeroth order in β,E0is given by Eq.(10), i.e. E0≡EZP . On the other hand, from now on we shall consider Elaser as a quasimonochromatic beam of frequency ω0, wave vector k0, and radius R: Elaser =E(+) laser +E(−) laser, where E(+) laser(r, t) = V(r)e−iω0t+ik0·r;V(r) = V0e−x2+y2 2R2;k0=ω0 c0 uz,(16) c0being the velocity of light corresponding to the frequency ω0. We are taking a coordinate system OXY Z,Obeing the center of the crystal and uz an unitary vector in the direction of the pumping. Strictly speaking (16) is a solution of the homogeneous wave equation only in the limit case R→ ∞ [1], but we follow the usual approximation, Eq.(16), for the lateral size of the beam. In experimental practice the laser is often focussed in a point several centimeters beyond the crystal. As a result the shape of the laser beam within the crystal is conical, rather than cylindrical. The net effect is that the light emitted from differents points of the crystal is focussed in a point placed at a finite distance of the crystal, where the detector is placed. In our calculation we will ignore this complication and consequently the emitted signal and idler beams are taken as consisting of parallel rays. By using the well known retarded solution of the inhomogeneous wave equation, the radiated field to first order can be obtained from (14) in the following way: E1(r, t) = −1 4πZΩd3r′S1(r′, t −|r−r′| c′) |r−r′|,(17) where the integration is carried over the volume Ω of the crystal. c′is the speed of light (as a function of the frequency) in the non linear medium and S1≡∂2(2EZP Elaser +E2 laser) ∂t2(18) 6 is the source of the field. It, and therefore E1, contains terms of frequencies ω0−ωk,ωk−ω0,ω0+ωk, 2ω0and 0. let us consider them briefly. (i) The terms of frequency 2ω0(second harmonic generation) and frequency 0 (rectification of the input laser field) come from E2 laser. As is usual in PDC experiments, we shall deal only with the part of the spectrum which contains oscillatory terms of frequency lower than ω0, so we can ignore them. The rest of the terms come from 2EZP Elaser. They are the following: (ii) A term of frequency ω0−ωk(ω0> ωk) that we shall call EP DC 1; (iii) a term of frequency ωk−ω0(ωk> ω0). This term is a down-conversion of the zero point field (ZPF) and hence we shall call it EZDC 1. In order to keep its frequencies lower than ω0,ωkmust also fulfill 2ω0> ωk. These two terms can be written together as EDC 1(r, t)≡EPDC 1(r, t) + EZDC 1(r, t) =i 2πX k ¯hω L3 0!1 2 αk(ω0−ω)2e−i(ω−ω0)t ×ZΩd3r′V(r′)ei(k−k0)·r′ei(ω−ω0)|r−r′| cωo−ω |r−r′|+ c.c ; 0 < ω < 2ω0,(19) where, for 0 < ω < ω0it represents EP DC 1, and for ω0< ω < 2ω0,EZDC 1. We have simplified the notation writing ωk≡ω. On the other hand, we have put cωo−ω≡c(ω0−ω), because the components of the radiated field travel with different velocities inside the crystal. Here we have made the customary assumption of considering the crystal embedded in a linear medium with the same dispersion [14]. (iv) Finally, the term of frequency ω0+ωk. This is an up-conversion of the ZPF – we shall call it EZUC 1–; its frequencies are always greater than ω0and therefore we ignore its contribution to E1. Nevertheless it plays an important role in E2and we must write its expression: EZUC 1(r, t) = i 2πX k ¯hω L3 0!1 2 αk(ω0+ω)2e−i(ω0+ω)t ×ZΩd3r′V(r′)ei(k0+k)·r′ei(ω+ω0)|r−r′| cωo+ω |r−r′|+ c.c,(20) where cωo+ω≡c(ω0+ω). 7 In the far field approximation 1 |r−r′|≈1 r;|r−r′| ≈ r(1 −r·r′ r2),(21) Eqs. (19) and (20) read EDC 1(r, t) = i 2πr X k ¯hω L3 0!1 2 αk(ω0−ω)2e−i(ω−ω0)tei(ω−ω0)r cωo−ω ×ZΩd3r′V(r′)ei(k−k0)·r′e−i(ω−ω0)r·r′ rcωo−ω+ c.c ; 0 < ω < 2ω0,(22) EZUC 1(r, t) = i 2πr X k ¯hω L3 0!1 2 αk(ω0+ω)2e−i(ω0+ω)tei(ω0+ω)r cωo+ω ×ZΩd3r′V(r′)ei(k0+k)·r′e−i(ω+ω0)r·r′ rcωo+ω+ c.c.(23) We will now calculate the second-order field. From (15), we have E2(r, t) = −1 4πZΩd3r′′ S2(r′′, t −|r−r′′| c′) |r−r′′|,(24) where S2≡∂2(2E1Elaser) ∂t2.(25) Therefore the frequencies that appear are in principle those of E1plus or minus ω0. Let us analyse the contributions of E1(EP DC 1, EZDC 1and EZUC 1) that induce frequencies lower than ω0in E2: (i) From EPDC 1, the frequencies will be ω0−ω±ω0. Remembering that ω < ω0−see discussion above Eq. (19)−the “+” sign can never give a contribution of frequency lower than ω0. Then only the contribution with “−” sign is relevant. (ii) EZDC 1gives frequencies ω−ω0±ω0, where we had ω > ω0. Therefore the “+” sign does not contribute, and the “−” sign contributes only if ω < 3ω0. (iii) Finally, EZUC 1gives ω+ω0±ω0without restriction, in principle, for ω. Hence, only the contribution with “−” sign must be considered in case that ω < ω0. 8 IPDC(ν, x, z) = β2F(r) 8π2(1 + c2 ω0−ν c2 ν ) ×Z2π 0dϕ Z1 −1d(cos θ)Zω0 0dωω3 c3 ω (ω0−ω)4δ(ω0−ω−ν) ×exp (−R2[(ω0−ω)2x2 r2c2 ω0−ω +ω2 c2 ω sin2θ+ 2(ω0−ω)x rcω0−ω ω cω sin θcos ϕ]) ×sinc2(L 2[(ω0−ω)z rcω0−ω +ω cω cos θ−k0]).(45) The integration in ϕis straightforward: Zπ −πdϕ e−mcos ϕ≃(2π m)1/2em,(46) if m≫1. In our case m= 2R2(ω0−ω)x rcω0−ω ω cω sin θ, (47) which is, roughly speaking, the squared ratio between the radius of the laser and the wavelength of the radiation, much greater than 1, and therefore IPDC(ν, x, z) = β2F(r) 8π3/2(1+c2 ω0−ν c2 ν )Z1 −1d(cos θ)Zω0 0ω3dωδ(ω0−ω−ν)(ω0−ω)4 × exp −R2[(ω0−ω)2x2 r2c2 ω0−ω+ω2 c2 ωsin2θ−2(ω0−ω)x rcω0−ω ω cωsin θ] c3 ωRr(ω0−ω)xω sin θ rcωcω0−ω ×sinc2(L 2[(ω0−ω)z rcω0−ω +ω cω cos θ−k0]).(48) By performing the integration in ω IPDC(ν, x, z) = β2F(r)ν4(ω0−ν)3 8π3/2(1 + c2 ω0−ν c2 ν )Z1 −1d(cos θ) 15 × exp −R2[νx rcν−(ω0−ν) cω0−νsin θ]2 c3 ω0−νRrνx(ω0−ν) sin θ rcνcω0−ν ×sinc2(L 2[νz rcν +ω0−ν cω0−ν cos θ−ω0 c0 ]).(49) In the former expression there is one gaussian factor and one sinc squared. These functions are important only for a small range of frequencies. This means that in the case of large values of Ror large values of Lone of these functions can be approximated by a delta. Let us study these two limiting cases, which correspond to Rbeing much greater or much smaller than Lx/r. CASE I: Long crystal (L→ ∞) If we multiply and divide by L/2 and take the limit L→ ∞ in the appropriate place we get IPDC(ν, x, z) = β2F(r)ν4(ω0−ν)3 4Lπ1/2c3 ω0−ν (1 + c2 ω0−ν c2 ν )Z1 −1d(cos θ) × exp −R2[νx rcν−(ω0−ν) cω0−νsin θ]2 Rrνx(ω0−ν) sin θ rcνcω0−ν ×δ νz rcν +ω0−ν cω0−ν cos θ−ω0 c0!.(50) Using Zdx δ[f(x)]g(x) = X i g(xi) |f′(xi)|;f(xi) = 0, and making x/r = sin α,z/r = cos α, we obtain IPDC(ν, α, r) = β2F(r)ν4 4Lπ1/2 (ω0−ν)2 c2 ω0−ν (1 + c2 ω0−ν c2 ν ) 16 × exp   −R2"ν cνsin α−(ω0−ν) cω0−νr1−c2 ω0−ν (ω0−ν)2[ω0 c0−ν cνcos α]2#2   Rsνsin α(ω0−ν) cνcω0−νr1−c2 ω0−ν (ω0−ν)2[ω0 c0−ν cνcos α]2 .(51) By considering typical values of the experimental parameters (R≈10−3m, ω0≈ 1015rad.s−1) we have Rω0 c2 ≈108,(52) and then, for a given value of α,the relevant contributions to the intensity are those coming from frequencies nearly equal to να,the frequency that maximizes IPDC(ν, α, r). It is the one that makes the exponent of the first term equals to zero. By taking into account that x2+z2=r2we have the following relation between ναand αfor the optimun position of the detector: cos α=z r= [ ν2 α c2 να−(ω0−να)2 c2 ω0−να +ω2 0 c2 0 ]c0cνα 2ναω0 .(53) By taking into account (52) we shall express the argument of the exponential factor of (51) to second order in ν−ναand to zeroth order elsewhere. For simplicity only two velocities of light will be considered, namely the corresponding to the laser c0and cν≃cω0−ν≡c. After some easy calculations in which we use Eq. (44), we arrive at the following expression for the PDC spectrum in the long crystal case: IPDC(ν, α, r) = ¯hR3LV 2 0β2ν3 α(ω0−να)2 2π1/2r2sin α×e−(ν−να)2 2∆2 α,(54) where ∆α=cναsin α √2Rω01−c c0cos α,(55) and, in the case c≡cν≃cω0−ν, the relation between ναand αis given by να=ω0 2×1−c2 c2 0 1−c c0cos α.(56) 17 For instance, if we consider the degenerate case ν=ω0/2, αis given by the relation cos α=cω0/2/c0, and then ∆α=c 2√2Rsin α,(57) expression that coincides with the inverse of the correlation time between signal and idler photons (see Eq. (50) of Ref. [12]). This agreement between the coherence time of the signal (or the idler) beam and the cross-correlation time is remarkable and it is the basis of the most relevant properties of PDC (in common language it is expressed by saying that the two partner photons are emitted at the same time). CASE II: Short crystal (R→ ∞) Now let us explore the consequences of assuming a very large radius of the pumping: R→ ∞. Taking into account that lim R→∞ Re−R2x2=√πδ(x),(58) the spectrum is, from (49) IPDC 1(ν, x, z) = β2F(r)ν4(ω0−ν)3 8πRc3 ω0−νZ1 −1d(cos θ) ×δ(νx rcν−(ω0−ν) cω0−νsin θ) Rrνx(ω0−ν) sin θ rcνcω0−ν ×sinc2(L 2[νz rcν +ω0−ν cω0−ν cos θ−ω0 c0 ]). Therefore IPDC(ν, α, r) = β2F(r)ν3cν(ω0−ν)2 8πR2c2 ω0−νsin α(1 + c2 ω0−ν c2 ν ) × sinc2(L 2[ν cνcos α+ω0−ν cω0−νr1−ν2c2 ω0−ν c2 ν(ω0−ν)2sin2α−ω0 c0]) r1−ν2c2 ω0−ν c2 ν(ω0−ν)2sin2α .(59) By considering typical values (L≈10−2m, ω0≈1015rad.s−1) we have 18 Lω0 c≈105, and then, for a given value of α,the relevant contributions to the intensity are those coming from frequencies nearly equal to να,the frequency that maximizes IPDC(ν, α, r). It is the one that makes the argument of the sinc2equals to zero, and coincides with (53). Now, by expanding to first order in of ν−ναthe argument of the sinc and to zeroth order elsewhere, and by using Eq. (44) we arrive at the following result: IPDC(ν, α, r) = ¯hL2R2V2 0β2ν3 α(ω0−να)2 4πr2sin αr1−ν2 α (ω0−να)2sin2α×sinc2ν−να ∆α, (60) where ∆α=2cναcos α−c c0ω0 Lω01−c c0cos α.(61) Again, if we consider the degenerate case, it can be easily proved that the width of the sinc2function is of the same order of the quantity ccos α/L sin2α, corresponding to the inverse of the correlation time between signal and idler photons (see Eq. (41) of Ref. [12]). 6 DISCUSSION: PDC EXPERIMENTS AND LOCAL REALISM This is the sixth of a series of articles devoted to the study of parametric down conversion in the Wigner-function formalism. In the first four [8]-[11] we started from a hamiltonian approach and showed that all PDC experiments performed until the end of 1997 may be interpreted within the formalism. It is also possible to interpret more recent experiments; for instance we may predict all the correlations which are actually measured in reported quantum teleportation experiments [18], [19]. This simply confirms that the Wigner representation is a valid formulation of quantum optics, fully equivalent to the more common Hilbert space formalism. But the Wigner function 19 provides an intuitive picture of PDC in terms of classical (Maxwell) waves propagating causally in space and time. The only difference with standard classical electrodynamics is the presence of a random electromagnetic radiation (zeropoint field, ZPF) filling the whole space. That radiation may be expanded in plane waves and the probability distribution for the amplitudes is given by Eq. (9). A crucial point for the possibility of a classical picture is the fact that the Wigner distribution is, in the case of PDC, positive definite. In our last two papers on PDC ([12] and the present one) we use an approach more fundamental than the model hamiltonian; in fact we start from the quantized electromagnetic field in the nonlinear crystal. As a result we have shown that the production of PDC radiation is formally equivalent to the classical polarization of the crystal by the combined action of the laser pump and the zeropoint field, which causes the crystal to reemit radiation. We have studied the stochastic properties of that radiation by calculating the cross-correlation of signal and idler [12] and their autocorrelations (in the present paper). In both cases we have used a standard perturbative expansion of the retarded (causal) Green’s function. It is interesting that the intensity of the crosscorrelation (calculated in Ref. [12]) may be obtained to second order by just calculating the field to first order, whilst we need the field to second order for the autocorrelation, so that the latter is substantially more difficult than the former. From these correlations we have been able to derive the correlation time of what in standard quantum language are called “signal and idler photons”, and also the spectrum of the emitted radiation, all in terms of the parameters of the crystal and the pumping beam. It is remarkable that in the Wigner-function formalism there is no trace of “photons”; we have just correlated waves. Effects like the strong correlation between colour and direction of emission, which in the standard (Hilbert space) approach appears as derived from the conservation of energy and momentum in the process of splitting of a laser photon, are, in the Wignerfunction approach, a consequence of energy and momentum conservation of the field. And the directionality appears, as is typical in wave optics, due to the interference (constructive in a specific direction, destructive otherwise) of the radiation emitted from all points of the crystal. Then it is not surprising that the bigger the crystal the better the directionality of the emitted radiation, as shown in [12] and the present paper, a relation which is less clear in the standard (Hilbert space) approach. Even effects so typically quantal as the “photon entanglement”, appear in the Wigner-function formalism just as correlations which involve both the zeropoint and the superimposed radi20 ation, whilst normal correlation involves only the radiation above the ZPF (see section 5 of Ref. [11]). The above discussion leads us to an apparently paradoxical situation. We have arrived at a purely wave (classical) picture of PDC and, nevertheless, PDC is the phenomenon most frequently claimed to exhibit non-classical aspects of light, like teleportation or violation of the Bell inequality. Clearly this situation requires a deeper study, which we now make. We begin by substituting a better expression for the somewhat ambiguous word “classical”. We propose “local realist” or “local hidden variables” (LHV) model in the well defined sense given by Bell [20]. We shall call local realist any theory where the probability of a coincidence photocount may be obtained from the expression P12 =Zρ(λ)P1(λ, ϕ1)P2(λ, ϕ2)dλ, ρ(λ)≥0, Zρ(λ)dλ = 1; 0 ≤P1(λ, ϕ1), P2(λ, ϕ2)≤1,(62) and similar expressions for single counts, triple coincidence counts, etc. Here λrepresent the hidden variables and ϕ1,ϕ2are controllable parameters (e.g. angles of polarization of the polarizers) . As is well known, from (62) it is possible to derive the Bell inequalities which are, therefore, necessary conditions for local realism. Many of these inequalities have been reported to be violated in experiments, but in all of them there exist loopholes for the refutation of LHV models ([21] and [22]). Actually the inequalities which have been violated in experiments are derived from (62) plus additional hypotheses which cannot be tested. Typical of these is “no-enhancement”[6]. We shall see in the following that the additional hypotheses are naturally violated in a LHV model derived from the Wigner-function formalism of PDC. In the analysis of the experiments, which we have made using the Wignerfunction formalism [8]-[11], the joint detection probability appears in the form P12 = constant ×Zdt1Zdt2⟨(I1(r1, t1)−I01)(I2(r1, t2)−I02)⟩,(63) where I1(I2) is the light intensity arriving at the first (second) detector, placed at r1(r2), at time t1(t2). The integrals extend over appropriate detection time-windows, and I01, I02 are parameters corresponding to the average intensity of the zeropoint. Actually those modes of the radiation which contain only zeropoint, without additional radiation, contribute equally to the 21 averages of I1and I01, and therefore cancel out in the difference, and similarly for I2−I02. Consequently only a few modes are needed in practice, but there is no problem if we include more modes than those strictly needed. In any case Eq. (63) was derived including only modes corresponding to a beam of almost parallel wave vectors. If this is not the case we should write Eq. (63) using the Poynting vector rather than the intensity (see below). The proportionality constant in Eq. (63) is irrelevant for many purposes, including the test of those “Bell inequalities” which are derived using additional hypotheses. But that constant is very relevant for the test of genuine Bell inequalities, derived from local realism alone [23]. Consequently we shall write Eq. (63) in a more complete form, which also shows the precise meaning of the average represented by ⟨⟩. We have P12 =ZW({αk},{α∗ k})Q1({αk},{α∗ k}, ϕ1)Q2({αk},{α∗ k}, ϕ2)dNαkdNα∗ k, (64) Qj({αk},{α∗ k}, ϕj) = ηj(hνj)−1Zdt1Zd2rj[Ij({αk},{α∗ k}, ϕj,rj, tj)−I0j], (65) where j= 1,2, Wis the “vacuum” Wigner function (Eq. (9)), Nis the number of modes (in practice we should take the limit N→ ∞ at some appropriate moment), klabels the wave vector and polarization of one mode. I1and I2are complicated functions of the amplitudes {αk}and {α∗ k}which take account of the evolution, including the effect of the nonlinear crystal and the various optical devices present in the experiment (we refer to our articles [8]-[11] for details). These devices may contain controllable parameters which we have labeled ϕ1and ϕ2. In addition to the time integration we have included an integration over the surface aperture of the detector. We have divided by the typical energy of one “photon” so that Qjbecomes dimensionless. Finally ηjis the quantum efficiency of the detector. The relevant question is whether (64) may be considered a particular case of (62). If the answer is affirmative (negative) the formalism provides (does not provide) an explicit LHV model for the experiment, which therefore is (is not) compatible with local realism. We see that (64) looks precisely like (62) with the amplitudes {αk}and {α∗ k}playing the role of the hidden variables λ. Indeed, the Wigner function W, playing the role of ρ(λ), is positive definite (see (9)) and normalized. The problem appears with the positivity of Qj. (The requirement Qj≤1, certainly holds for the low quantum efficiencies of the experiments). Now Qjmay be negative because the difference I1−I01 22 (or I2-I02)is not always positive. The problem is not the huge value of the zeropoint energy (about 105w/cm2for the ZPF in the visible range), because the threshold intensity I0cancels precisely that intensity. The problem lies in the fluctuation of the intensity. For the weak light signals of the experiments Ijmay have fluctuations such that Ij< I0j. The problem of the non-positivity of Ij-I0jis alleviated by the time and space integrations in (65). Indeed, the fluctuations of the intensity are strongly reduced by those integrations as the Heisenberg (uncertainty) relations show. But we can guarantee the positivity of Qjonly in the limit of infinitely wide time-windows and infinitely large apertures, which is non-physical. Consequently we conclude that it is not possible to interpret directly the Wigner-function formalism as a LHV model for the PDC experiments. In spite of the above conclusion, it is not without interest to study whether some modification of Qjmight give a LHV theory compatible with the experiments, though not precisely with the quantum predictions (which are given by the unmodified Eq. (64)). The question of a modification of quantum theory has been rejected by most, because of the spectacular success of that theory. However, we are not proposing any modification in the foundations of the quantum theory, but rather a change in the description of that complicated macroscopic system which is a photon counter. Eq. (65) is the quantum prediction (in the Wigner representation) for the behaviour of an ideal detector. Every experimentalist knows that a real device is quite different from an ideal one in many respects. So what we propose is a realist theory of detection, in place of the model of instantaneous collapse which has been used hitherto. An objection to any attempt at interpreting Eq. (65), or any small modification of it, as a “classical” (LHV) model is that any classical detector should be sensitive to the total radiation intensity, Ij, rather than to the difference Ij−I0j. We do not agree with that. In fact, it is natural to assume that the detection should depend on the total flux of energy crossing the aperture during the time-window and this flux is given by an appropriate integral of the Poynting vector, rather than the intensity. Therefore we should write, instead of Eq. (65) the following Qj({αk},{α∗ k}, ϕj) = ηj(hνj)−1Zdt1Zd2rjSj({αk},{α∗ k}, ϕj,rj, tj),(66) where Sjis the component of the Poynting vector perpendicular to the entrance area of the detector. Eq. (66) is not equal to Eq. (65 ), but the 23 average shown in (64) is the same in both cases. The reason is that the average of the Poynting vector of the ZPF alone is zero, and the average of the contribution of the signal is the same whether we use the component of the Poynting vector or the intensity. Therefore we have not modified the quantum prediction up to this point. Still Eq. (66) is not positive definite, but now it is not difficult to imagine that a modification might be possible by making it positive without departing too much from the ideal quantum prediction Eq. (62). A model of such a detector has been presented elsewhere (for other purposes) which suggests that it is possible [24]. Finally we comment on the no-enhancement assumption. In a beam splitter (semitransparent mirror or polarizer) we should take into account both the signal and the “vacuum” zeropoint fields, so that at the outgoing channels (1 and 2) of this device, the electromagnetic fields E(+) 1and E(+) 2, are given in terms of the incoming signal, E(+) S, and zeropoint, E(+) ZP , fields by E(+) 1(r, t) = TE(+) S(r, t) + iRE(+) ZP (r, t), E(+) 2(r, t) = TE(+) ZP (r, t) + iRE(+) S(r, t),(67) where T(R) is the transmission (reflection) coefficient. This fact is essential in order to preserve the conmutation relations in the beam-splitter. The “no-enhancement assumption” essentially means that the intensity of a signal outgoing from a polarizer can never be greater than the incoming signal, and it is considered plausible because the beam-splitter divides the intensity. Eq. (67) clearly shows that this assumption is naturally violated in any theory where the zeropoint field is real. The intensity at the outgoing channel may be greater than the intensity at the incoming channel for some realizations of the fields, although it will be certainly smaller on the average. A local realist theory, obtained by modifying the Wigner representation of quantum optics, along the lines discussed above, will violate both the “noenhancement” assumption and the Bell-type inequalities derived from it. In conclusion, parametric down conversion is an experimental arena for testing some important conceptual features of quantum mechanics. The Maxwellian approach introduced here offers a new perspective on the meaning of these experiments. On the other hand PDC is frequently used as a way to implement the techniques of quantum information, and our treatment can help by giving a better understanding of the correlation properties of this kind of light. Special attention has been paid to the spectrum of PDC, because this is one of the main features of the phenomenon. Two limit 24