Abstract

A microscopic multimode theory of collinear type-I spontaneous parametric downconversion in a cavity is presented. Single-mode and multimode correlation functions have been derived using fully quantized atom and electromagnetic field variables. From a first principles calculation the FWHM of the single-mode correlation function and the cavity enhancement factor have been obtained in terms of mirror reflectivities and the first-order crystal dispersion coefficient. The values obtained are in good agreement with recent experimental results [Phys. Rev. A 62 , 033804 (2000)].

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. Z. Y. Ou and L. Mandel, �??Violation of Bell�??s inequality and classical probability in a two-photon correlation experiment,�?? Phys. Rev. Lett. 61, 50-53 (1988).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. S. L. Braunstein and H. J. Kimble, �??Teleportation of continuous quantum variables,�?? Phys. Rev. Lett. 80, 869-872 (1998).
    [CrossRef]
  8. L. Vaidman, �??Teleportation of quantum states,�?? Phys. Rev. A 49, 1473-1476 (1994).
    [CrossRef] [PubMed]
  9. D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, �??Experimental quantum teleportation,�?? Nature 390, 575-579 (1997).
    [CrossRef]
  10. J-W Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger, �??Experimental entanglement swapping : Entangling photons that never interacted,�?? Phys. Rev. Lett. 80, 3891-3894 (1998).
    [CrossRef]
  11. D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 82, 1345-1349 (1999).
    [CrossRef]
  12. J. G. Rarity and P. R. Tapster, �??Two-color photons and nonlocality in fourth-order interference,�?? Phys. Rev. A 41, 5139-5146 (1990).
    [CrossRef] [PubMed]
  13. X. Y. Zou, L. J. Wang and L. Mandel, �??Induced coherence and indistinguishability in optical interference,�?? Phys. Rev. Lett. 67, 318-321 (1991).
    [CrossRef] [PubMed]
  14. C. K. Hong, Z. Y. Ou and L. Mandel, �??Measurement of subpicosecond time intervals between two photons by interference,�?? Phys. Rev. Lett. 59, 2044-2046 (1987).
    [CrossRef] [PubMed]
  15. Z. Y. Ou and Y. J. Lu, �??Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,�?? Phys. Rev. Lett. 83, 2556-2559 (1999).
    [CrossRef]
  16. Y. J. Lu and Z. Y. Ou, �??Optical parametric oscillator far below threshold: Experiment vs theory,�?? Phys. Rev. A 62, 033804-033804-11 (2000).
    [CrossRef]
  17. M. J. Collett and C. W. Gardiner, �??Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,�?? Phys. Rev. A 30, 1386-1391 (1984).
    [CrossRef]
  18. R. Andrews, E. R. Pike and S. Sarkar, �??�??The role of second-order nonlinearities in the generation of localized photons,�??�?? Pure Appl. Opt. 7, 293-299 (1998).
    [CrossRef]

Nature

D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, �??Experimental quantum teleportation,�?? Nature 390, 575-579 (1997).
[CrossRef]

Phys. Rev. A

L. Vaidman, �??Teleportation of quantum states,�?? Phys. Rev. A 49, 1473-1476 (1994).
[CrossRef] [PubMed]

Z. Y. Ou, X. Y. Zou, L. J. Wang and L. Mandel, �??Experiment on nonclassical fourth-order interference,�?? Phys. Rev. A 42, 2957-2965 (1990).
[CrossRef] [PubMed]

C. K. Hong and L. Mandel, �??�??Theory of parametric frequency down-conversion of light,�??�?? Phys. Rev. A 31, 2409-2418 (1985).
[CrossRef] [PubMed]

J. G. Rarity and P. R. Tapster, �??Two-color photons and nonlocality in fourth-order interference,�?? Phys. Rev. A 41, 5139-5146 (1990).
[CrossRef] [PubMed]

Y. J. Lu and Z. Y. Ou, �??Optical parametric oscillator far below threshold: Experiment vs theory,�?? Phys. Rev. A 62, 033804-033804-11 (2000).
[CrossRef]

M. J. Collett and C. W. Gardiner, �??Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,�?? Phys. Rev. A 30, 1386-1391 (1984).
[CrossRef]

Phys. Rev. Lett.

X. Y. Zou, L. J. Wang and L. Mandel, �??Induced coherence and indistinguishability in optical interference,�?? Phys. Rev. Lett. 67, 318-321 (1991).
[CrossRef] [PubMed]

C. K. Hong, Z. Y. Ou and L. Mandel, �??Measurement of subpicosecond time intervals between two photons by interference,�?? Phys. Rev. Lett. 59, 2044-2046 (1987).
[CrossRef] [PubMed]

Z. Y. Ou and Y. J. Lu, �??Cavity enhanced spontaneous parametric down-conversion for the prolongation of correlation time between conjugate photons,�?? Phys. Rev. Lett. 83, 2556-2559 (1999).
[CrossRef]

Z. Y. Ou and L. Mandel, �??Violation of Bell�??s inequality and classical probability in a two-photon correlation experiment,�?? Phys. Rev. Lett. 61, 50-53 (1988).
[CrossRef] [PubMed]

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

C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, andW. K. Wootters, �??Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,�?? Phys. Rev. Lett. 70, 1895-1899 (1993).
[CrossRef] [PubMed]

S. L. Braunstein and H. J. Kimble, �??Teleportation of continuous quantum variables,�?? Phys. Rev. Lett. 80, 869-872 (1998).
[CrossRef]

D. C. Burnham and D. L. Weinberg, �??Observation of simultaneity in parametric production of optical photon pairs,�?? Phys. Rev. Lett. 25, 84-87 (1970).
[CrossRef]

J-W Pan, D. Bouwmeester, H. Weinfurter and A. Zeilinger, �??Experimental entanglement swapping : Entangling photons that never interacted,�?? Phys. Rev. Lett. 80, 3891-3894 (1998).
[CrossRef]

D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter and A. Zeilinger, Phys. Rev. Lett. 82, 1345-1349 (1999).
[CrossRef]

Pure Appl. Opt.

R. Andrews, E. R. Pike and S. Sarkar, �??�??The role of second-order nonlinearities in the generation of localized photons,�??�?? Pure Appl. Opt. 7, 293-299 (1998).
[CrossRef]

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Figures (1)

Figure 1:
Figure 1:

Schematic showing a cavity of length d bounded by an input mirror M1 and an exit mirror M2. The shaded area represents the nonlinear crystal which fills the cavity.

Equations (22)

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A ( 2 ) ( r 1 , r 2 , r 3 , t ) = { ( i h ) 5 t d t 1 t 1 d t 2 t 2 d t 3 t 3 d t 4 t 4 d t 5
× < a 1 , a 2 , s 3 μ a , ( 1 ) ( t 1 ) μ b , ( 2 ) ( t 2 ) μ c , ( 3 ) ( t 3 ) μ d , ( 3 ) ( t 4 ) μ e , ( 3 ) ( t 5 ) g 1 , g 2 , g 3 >
× < 0 , a k 0 λ 0 E a ( r 1 , t 1 ) E b ( r 2 , t 2 ) E c ( r 3 , t 3 ) E d ( r 3 , t 4 ) E e ( r 3 , t 5 ) 0 , α k 0 λ 0 > }
+ ( r 1 r 2 )
E ( r , t ) = i∫ d 3 k j = 1,2 ( ħ kc 16 π 3 ε 0 ) 1 2 ε j ( k ) U k j ( r ) a k j exp ( iωt )
U in , k j ( r ) = t 2 o exp ( i k ( ) · r ) D k t 2 o exp ( i k ( + ) · r + ikd ) D k
D k = 1 + r 2 o exp ( 2 ikd )
k ( ± ) = k ( sin θ cos ϕ , sin θ sin ϕ , ± cos θ )
d 3 k = 0 k 2 dk 0 π 2 sin θdθ 0 2 π d ϕ
U out , k j ( r ) = exp ( i k ( ) · r ) + R k j exp ( i k ( + ) · r )
R k j exp ( ikd ) + r 2 j exp ( ikd ) D k
G cav ( 2 ) ( z 1 , t 1 ; z 2 , t 2 ) = π ε 2 d ( t 2 o 2 t 1 p ) ω k 0 2 ω k 0 2 dx exp ( x ) sin ( dv x 2 2 ) ( dv x 2 2 ) 1 1 + r 2 o exp ( i 2 dvx ) 2
k i = k i * + k i ω k i ω k i = ω k i * ( ω k i ω k i * ) + 1 2 2 k i ω k i 2 ω k i = ω k i * ( ω k i ω k i * ) 2 +
ω k 1 * + ω k 2 * = ω k 0 * ; k 1 * + k 2 * = k 0 *
1 1 + r 2 o exp ( i 2 dvx ) 2 1 ( 1 + r 2 o ) 2 l = N l = N 1 [ n 1 2 ( vdx ) 2 + 1 ]
n 1 = 2 r 2 o 1 + r 2 o
G cav ( 2 ) = ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 sin [ ( 2 N + 1 2 ) π τ ˜ ] sin [ π τ ˜ 2 ] exp ( τ ˜ n 1 )
A SM = ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 exp ( τ ˜ n 1 )
[ sin [ ( 2 N + 1 2 ) π τ ˜ ] sin [ π τ ˜ 2 ] ] 2 = ( 2 N + 1 ) + 2 N ( 2 cos ( 2 π τ ˜ ) ) + ( 2 N 1 ) ( 2 cos ( 2 [ 2 π τ ˜ ] ) )
+ + 2 cos ( 2 N [ 2 π τ ˜ ] )
A MM ( 2 ) ~ 2 N + 1 ( t 2 o 2 t 1 p ) n 1 ( 1 + r 2 o ) 2 exp ( τ ˜ n 1 )
γ = ( count rate bandwidth ) cavity ( count rate bandwidth ) no cavity

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