Abstract

We continue the analysis of our previous papers that were devoted to type I parametric downconversion, the extension to type II being straightforward. We show that entanglement, in the Wigner representation, is merely a correlation that involves both signals and vacuum fluctuations. An analysis of the detection process opens the way to a complete description of parametric downconversion in terms of pure Maxwell electromagnetic waves.

© 1998 Optical Society of America

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References

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  1. A. Casado, T. W. Marshall, and E. Santos, J. Opt. Soc. Am. B 14, 494–502 (1997).
    [Crossref]
  2. A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
    [Crossref]
  3. A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
    [Crossref]
  4. L. de la Peña and A. M. Cetto, The Quantum Dice. An Introduction to Stochastic Electrodynamics (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  5. T. W. Marshall and E. Santos, in The Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier, S. Jeffers, S. Roy, J. P. Vigier, and G. Hunter, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1997), pp. 67–77.
  6. A. V. Belinsky and D. N. Klyshko, Phys. Lett. A 166, 303 (1992).
    [Crossref]
  7. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
    [Crossref] [PubMed]
  8. Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29–34 (1994);Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, Phys. Rev. A 50, 23–50 (1994);Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564–2568 (1994);Phys. Lett. A 191, 201–207 (1994).
    [Crossref] [PubMed]
  9. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
    [Crossref] [PubMed]
  10. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chapter 17; G. H. Rieke, Detection of Light: from the Ultraviolet to the Submillimeter (Cambridge U. Press, Cambridge, UK, 1994).
  12. J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978).
    [Crossref]
  13. J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526–535 (1974).
    [Crossref]
  14. T. W. Marshall and E. Santos, Found. Phys. 18, 185–201 (1988);Phys. Rev. A 39, 6271–6283 (1989);M. Ferrero, T. W. Marshall, and E. Santos, Am. J. Phys. 58, 683–688 (1990);T. W. Marshall, Found. Phys. 21, 209–219 (1991);E. Santos, Phys. Rev. A 46, 3646–3656 (1992).
    [Crossref] [PubMed]

1997 (3)

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

A. Casado, T. W. Marshall, and E. Santos, J. Opt. Soc. Am. B 14, 494–502 (1997).
[Crossref]

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

1994 (2)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29–34 (1994);Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, Phys. Rev. A 50, 23–50 (1994);Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564–2568 (1994);Phys. Lett. A 191, 201–207 (1994).
[Crossref] [PubMed]

1992 (1)

A. V. Belinsky and D. N. Klyshko, Phys. Lett. A 166, 303 (1992).
[Crossref]

1988 (1)

T. W. Marshall and E. Santos, Found. Phys. 18, 185–201 (1988);Phys. Rev. A 39, 6271–6283 (1989);M. Ferrero, T. W. Marshall, and E. Santos, Am. J. Phys. 58, 683–688 (1990);T. W. Marshall, Found. Phys. 21, 209–219 (1991);E. Santos, Phys. Rev. A 46, 3646–3656 (1992).
[Crossref] [PubMed]

1978 (1)

J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978).
[Crossref]

1974 (1)

J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526–535 (1974).
[Crossref]

Belinsky, A. V.

A. V. Belinsky and D. N. Klyshko, Phys. Lett. A 166, 303 (1992).
[Crossref]

Casado, A.

A. Casado, T. W. Marshall, and E. Santos, J. Opt. Soc. Am. B 14, 494–502 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

Cetto, A. M.

L. de la Peña and A. M. Cetto, The Quantum Dice. An Introduction to Stochastic Electrodynamics (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Clauser, J. F.

J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978).
[Crossref]

J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526–535 (1974).
[Crossref]

de la Peña, L.

L. de la Peña and A. M. Cetto, The Quantum Dice. An Introduction to Stochastic Electrodynamics (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Fernández-Rueda, A.

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

Horne, M. A.

J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526–535 (1974).
[Crossref]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

A. V. Belinsky and D. N. Klyshko, Phys. Lett. A 166, 303 (1992).
[Crossref]

Kwiat, P. G.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Marshall, T. W.

A. Casado, T. W. Marshall, and E. Santos, J. Opt. Soc. Am. B 14, 494–502 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

T. W. Marshall and E. Santos, Found. Phys. 18, 185–201 (1988);Phys. Rev. A 39, 6271–6283 (1989);M. Ferrero, T. W. Marshall, and E. Santos, Am. J. Phys. 58, 683–688 (1990);T. W. Marshall, Found. Phys. 21, 209–219 (1991);E. Santos, Phys. Rev. A 46, 3646–3656 (1992).
[Crossref] [PubMed]

T. W. Marshall and E. Santos, in The Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier, S. Jeffers, S. Roy, J. P. Vigier, and G. Hunter, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1997), pp. 67–77.

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Risco-Delgado, R.

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

Rubin, M. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chapter 17; G. H. Rieke, Detection of Light: from the Ultraviolet to the Submillimeter (Cambridge U. Press, Cambridge, UK, 1994).

Santos, E.

A. Casado, T. W. Marshall, and E. Santos, J. Opt. Soc. Am. B 14, 494–502 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

T. W. Marshall and E. Santos, Found. Phys. 18, 185–201 (1988);Phys. Rev. A 39, 6271–6283 (1989);M. Ferrero, T. W. Marshall, and E. Santos, Am. J. Phys. 58, 683–688 (1990);T. W. Marshall, Found. Phys. 21, 209–219 (1991);E. Santos, Phys. Rev. A 46, 3646–3656 (1992).
[Crossref] [PubMed]

T. W. Marshall and E. Santos, in The Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier, S. Jeffers, S. Roy, J. P. Vigier, and G. Hunter, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1997), pp. 67–77.

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29–34 (1994);Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, Phys. Rev. A 50, 23–50 (1994);Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564–2568 (1994);Phys. Lett. A 191, 201–207 (1994).
[Crossref] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

Shih, Y.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Shih, Y. H.

Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29–34 (1994);Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, Phys. Rev. A 50, 23–50 (1994);Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564–2568 (1994);Phys. Lett. A 191, 201–207 (1994).
[Crossref] [PubMed]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

Shimony, A.

J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chapter 17; G. H. Rieke, Detection of Light: from the Ultraviolet to the Submillimeter (Cambridge U. Press, Cambridge, UK, 1994).

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Zeilinger, A.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Found. Phys. (1)

T. W. Marshall and E. Santos, Found. Phys. 18, 185–201 (1988);Phys. Rev. A 39, 6271–6283 (1989);M. Ferrero, T. W. Marshall, and E. Santos, Am. J. Phys. 58, 683–688 (1990);T. W. Marshall, Found. Phys. 21, 209–219 (1991);E. Santos, Phys. Rev. A 46, 3646–3656 (1992).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

Phys. Lett. A (2)

Y. H. Shih and A. V. Sergienko, Phys. Lett. A 186, 29–34 (1994);Y. H. Shih, A. V. Sergienko, M. H. Rubin, T. E. Kiess, and C. O. Alley, Phys. Rev. A 50, 23–50 (1994);Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564–2568 (1994);Phys. Lett. A 191, 201–207 (1994).
[Crossref] [PubMed]

A. V. Belinsky and D. N. Klyshko, Phys. Lett. A 166, 303 (1992).
[Crossref]

Phys. Rev. A (3)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122–5133 (1994).
[Crossref] [PubMed]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 55, 3879 (1997).
[Crossref]

A. Casado, A. Fernández-Rueda, T. W. Marshall, R. Risco-Delgado, and E. Santos, Phys. Rev. A 56, 2477–2480 (1997).
[Crossref]

Phys. Rev. D (1)

J. F. Clauser and M. A. Horne, Phys. Rev. D 10, 526–535 (1974).
[Crossref]

Phys. Rev. Lett. (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337–4341 (1995).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881–1927 (1978).
[Crossref]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chapter 17; G. H. Rieke, Detection of Light: from the Ultraviolet to the Submillimeter (Cambridge U. Press, Cambridge, UK, 1994).

L. de la Peña and A. M. Cetto, The Quantum Dice. An Introduction to Stochastic Electrodynamics (Kluwer Academic, Dordrecht, The Netherlands, 1996).

T. W. Marshall and E. Santos, in The Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier, S. Jeffers, S. Roy, J. P. Vigier, and G. Hunter, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1997), pp. 67–77.

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Figures (3)

Fig. 1
Fig. 1

Process of parametric downconversion.

Fig. 2
Fig. 2

Polarization entanglement in noncollinear type II down conversion.

Fig. 3
Fig. 3

Tests of Bell’s inequalities by use of noncollinear type II downconversion.

Equations (45)

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H=j=o,ekωjkαjk*αjk+igVk,kf(k, k)exp(-iωpt)αok*αek*+c.c.,
V(r, t)={V(t)exp[i(kpr-ωpt)]+c.c.}u,
kpk+k.
ωp=ωk+ωk.
Ej(r, t)=Ej(+)(r, t)+Ej(-)(r, t),
Ej(+)(r, t)=ikωjk2L31/2jkαjk(t)exp(ikr),
j=o, e,
αjk(t)=αjk(0)exp(-iωjkt),
α˙ek=-iωekαek+gVkf(k, k)exp(-iωpt)αok*,
αek(0)=αek(vac)(0)+gVkf(k, k)×uΔt2 (ωp-ωek-ωok)αok*(vac)(0)+g2|V|2kkf(k, k)f*(k, k)×uΔt2 (ωok+ωek-ωp)×uΔt2 (ωek-ωek)αek(vac)(0),g|V|1,
u(x)=(sin x/x)exp(ix).
αek(vac)(0)=αek(vac)(-Δt)exp(-iωekΔt),
αek*(vac)(0)=αek*(vac)(-Δt)exp(iωekΔt).
αek(t)=αek(0)exp(-iωekt).
ωo+ωe=ωp,ko+ke=kp.
ωj min<ωjk<ωj max,|ktr|ωj min/c,
eke,oko;eo=0.
Fj(+)(r, t)exp(iωjt)Ej(+)(r, t)=i[k]j(ωk/2L3)1/2αk(0)exp(ikr)×exp[i(ωj-ωjk)t]j,(j=o, e),
Fj(+)(rB, t)=Fj(+)(rA, t-rAB/c)exp[iωj(rAB/c)],
(j=e, o),
Fe(+)(r, t)=Fe(+)(r, t)e={[1+g2|V|2J]Fe(+)(vac)(r, t)+gVGFo(-)(vac)(r, t)}e,
Fo(+)(r, t)=Fo(+)(r, t)o={[1+g2|V|2J]Fo(+)(vac)(r, t)+gVGFe(-)(vac)(r, t)}o.
Fe(+)(vac)(r, t)=i[k]eωek2L31/2 exp(ikr)×exp[i(ωe-ωek)t]αek(vac)(0)
GFo(-)(vac)(r, t)=i[k]eωek2L31/2 exp(ikr)×exp[i(ωe-ωek)t]βk,
βk=[k]of(k, k)uΔt2 (ωp-ωek-ωok)αok*(vac)(0),
JFe(+)(vac)(r, t)
=i[k]eωek2L31/2 exp(ikr)exp[i(ωe-ωek)t]γk,
γk=[k]o[k]ef(k, k)f*(k, k)×uΔt2 (ωok+ωek-ωp)×uΔt2 (ωek-ωek)αek(vac)(0).
Fe(+)(r, t)Fe(-)(r, t)-Fe(+)(vac)(r, t)Fe(-)(vac)(r, t)=2g2|V|2GFo(-)(vac)(r, t)G*Fo(+)(vac)(r, t)g2|V|2μe(t-t),Fe(+)(r, t)Fe(+)(r, t)=0.
Fe(+)(0, t)Fo(+)(0, t)=gVν(t-t),
Fe(+)(0, t)Fo(-)(0, t)=Fe(-)(0, t)Fo(+)(0, t)=0.
P1(r1, t)I(r1, t)-I0(r1),
P12(r1, t; r2, t+τ)[I(r1, t)-I0(r1)]×[I(r2, t+τ)-I0(r2)].
P12(r1, t; r2, t+τ)λλ|Fλ(+)(r1, t)×Fλ(+)(r2, t+τ)|2,
F1(+)(0, t)=Fe(+)(0, t)i+Fo(+)(0, t)j,
F2(+)(0, t)=Fe(+)(0, t)i+Fo(+)(0, t)j,
F(+)(r1, t)=[F1(+)(r1, t)(cos ϕ1i+sin ϕ1 j)]×(cos ϕ1i+sin ϕ1j)=exp[iω(d/c)][Fe(+)(0,t-d/c)cos ϕ1+Fo(+)(0, t-d/c)sin ϕ1]×(cos ϕ1i+sinϕ1j).
F(+)(r2, t+τ)=[F2(+)(r2, t+τ)×(cos ϕ2i+sin ϕ2j)]×(cos ϕ2i+sin ϕ2j)=exp[iω(d/c)]×[Fe(+)(0, t+τ-d/c)cos ϕ2+Fo(+)(0, t+τ-d/c)sin ϕ2]×(cos ϕ2i+sin ϕ2j).
P12=K sin2(ϕ2+ϕ1),
P0ΔtE^(-)(t)E^(+)(t)dt,
P0ΔtI(t)-I0dt,
PI(t)-I0=limT1T 0T[I(t)-I0]dt.
1Δt 0Δt[I(t)-I0]dt
P0Δt[I(t)-I0]dt+,
P12=P1(λ, ϕ1)P2(λ, ϕ2)=ρ(λ)P1(λ, ϕ1)P2(λ, ϕ2)dλ,

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