## Photon correlations of a sub-threshold optical parametric oscillator

Optics Express, Vol. 10, Issue 11, pp. 461-468 (2002)

http://dx.doi.org/10.1364/OE.10.000461

Acrobat PDF (138 KB)

### 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

## 1. Introduction

1. 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]

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]

5. 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]

6. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. **70**, 1895–1899 (1993). [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]

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]

2. 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]

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]

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]

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]

## 2. Spontaneous down-conversion amplitude for a crystal in a cavity

**r**⃗

_{3}in free space and for one-atom detectors located at

**r**⃗

_{1}and

**r**⃗

_{2}, the generalized amplitude for detecting pairs of down-converted photons from a single crystal atom is given by [18

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]

*E*

^{a,…,e}are components of the electric field vector and it should be noted that in (1) repeated superscripts are being summed over. The initial state of the electromagnetic field, |0,

*α*

^{λ0}

_{k0}〉, consists of a coherent state with wave-vector

*k*

_{0}and polarization index

*λ*

_{0}(the monochromatic pump beam) with other modes in the vacuum state |0〉 . |

*g*

_{1},

*g*

_{2},

*g*

_{3}> is the wave-function describing the ground state of the detector atoms and the crystal atom and |

*a*

_{1},

*a*

_{2},

*g*

_{1}〉 describes the detector atoms in excited states |

*a*

_{1},

*a*

_{2}〉 and a source atom finally in the ground state |

*g*

_{3}〉 after the two-photon emission process.

*μ*

^{a,…,e}(

*t*) denotes components of the interaction-picture electric dipole moment operator for the multi-level atom at position

**r**

_{j}.

*d*, all embedded in a linear medium of the same refractive index [3

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

19. F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A **43**, 2480–2497 (1991). [CrossRef] [PubMed]

_{1}from inside the cavity is zero, the quantized electric field is given as

*U*

_{k⃗j}(

*r*⃗) are denoted

*a*

_{k⃗j}where

*ε*

_{j}(

*k*⃗) (

*j*=1,2) denotes the mode polarization vector. The spatial function

*U*

_{k⃗j}(

*r*⃗) for

*U*

_{in,k⃗j}(

*r*⃗) and is defined by the following

*t*

_{2o}and

*r*

_{2o}being the amplitude transmission and reflection coefficients of M

_{2}respectively for the signal and idler photons; we have taken the reflection coefficient of M

_{1}as

*r*

_{1o}= - 1 consistent with a perfectly reflecting and infinitely thin mirror. The wave vectors

*k*⃗

^{(-)}and

*k*⃗

^{(+)}describe backward and forward propagating photons and are defined in terms of polar (

*θ*) and azimuthal (

*ϕ*) angles by

*θ*is measured from the direction of the incident pump beam and therefore

*θ*= 0 corresponds to collinear propagation. In (2) the

*k*integral is defined as

*r*⃗ =

*r*⃗

_{3}and (7) when

*r*⃗ =

*r*⃗

_{1,2}for the spatial mode function, we obtain, on performing a simple integration over the irradiated volume of the crystal

*t*

_{1p}is the amplitude transmission coefficient of M

_{1}at the pump frequency. Crystal dispersion has been taken into account with the following wave-vector expansion:

*τ*′ = (

*t*

_{1}-

*t*

_{2}) +

*v*[

*z*

_{2}-

*z*

_{1}) and the integration variable

*x*=

*ω*

_{k⃗i}

^{*};

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]

*z*

_{1},

*z*

_{2}are the positions of the detectors along the axis of the cavity. For simplicity, we can assume that the detectors are situated at equal distances from the cavity. For a high-Q cavity we use the following approximation [19

19. F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A **43**, 2480–2497 (1991). [CrossRef] [PubMed]

*l*= 0 describes the detection of the degenerate mode. After substituting Eq. (12) in Eq. (9) we obtain to a good approximation the following amplitude

*N*= 0. We therefore obtain the single-mode amplitude

*A*

_{SM}as

## 3. Multimode pair-photon count rate

^{-12}s. Since detectors cannot measure such rapid oscillations in time, only an average is recorded, and therefore the cosine terms make a vanishing contribution to the count rate. Hence the multimode amplitude is given by

*N*+1) compared to the single-mode count rate, but shows the same time dependence as the single-mode case.

## 5. Enhancement-factor per mode

*γ*, is defined by the following:

*l*= 0) with respect to the correlation time

*τ*′. The FWHM, (Δ

*ω*)

_{cav}, of this function is a reasonable estimate of the bandwidth and is approximately

*k*is a constant. In the absence of the cavity we use the right-hand-side of Eq. (32) in [21

21. R. Andrews, E. R. Pike, and S. Sarkar, “Photon correlations and interference intype-I optical parametric down-conversion,” J. Opt. B: Quantum Semiclass. Opt. **1**, 588–597 (1999). [CrossRef]

_{nocav}=

*k*. The enhancement factor,

*γ*, is then given by,

## 4. Results

_{2}transmission coefficient of 1.5% and

*v*= 8×C10

^{-9}m

^{-1}s [20

20. B. Zysset, I. Biaggio, and P. Gunter, “Refractive indices of orthorhombic KNbO_{3} : Dispersion and temperature dependence,” J. Opt. Soc. Am. B **9**, 380–386 (1992). [CrossRef]

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]

^{4}which compares well with the results of Ou et. al. who obtained 5.5 × 10

^{4}. Any differences between our predictions and that of Ou et. al. is most probably due to the fact that we assumed that M

_{1}was a perfectly reflecting mirror.

## 5. Conclusion

## References and links

1. | D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. |

2. | Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, “Experiment on nonclassical fourth-order interference,” Phys. Rev. A |

3. | C. K. Hong and L. Mandel, “Theory of parametric frequency down-conversion of light,” Phys. Rev. A |

4. | Z. Y. Ou and L. Mandel, “Violation of Bell’s inequality and classical probability in a two-photon correlation experiment,” Phys. Rev. Lett. |

5. | P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. |

6. | C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. |

7. | S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett. |

8. | L. Vaidman, “Teleportation of quantum states,” Phys. Rev. A |

9. | D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature |

10. | J-W Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, “Experimental entanglement swapping : Entangling photons that never interacted,” Phys. Rev. Lett. |

11. | D. Bouwmeester, J-W Pan, M. Daniell, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. |

12. | J. G. Rarity and P. R. Tapster, “Two-color photons and nonlocality in fourth-order interference,” Phys. Rev. A |

13. | X. Y. Zou, L. J. Wang, and L. Mandel, “Induced coherence and indistinguishability in optical interference,” Phys. Rev. Lett |

14. | C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

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. |

16. | Y. J. Lu and Z. Y. Ou, “Optical parametric oscillator far below threshold: Experiment vs theory,” Phys. Rev. A |

17. | M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and travelling-wave light fields produced in parametric amplification,” Phys. Rev. A |

18. | R. Andrews, E. R. Pike, and S. Sarkar, “The role of second-order nonlinearities in the generation of localized photons,” Pure Appl. Opt. |

19. | F. De Martini, M. Marrocco, P. Mataloni, L. Crescentini, and R. Loudon, “Spontaneous emission in the optical microscopic cavity,” Phys. Rev. A |

20. | B. Zysset, I. Biaggio, and P. Gunter, “Refractive indices of orthorhombic KNbO |

21. | R. Andrews, E. R. Pike, and S. Sarkar, “Photon correlations and interference intype-I optical parametric down-conversion,” J. Opt. B: Quantum Semiclass. Opt. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(270.4180) Quantum optics : Multiphoton processes

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 28, 2002

Revised Manuscript: May 21, 2002

Published: June 3, 2002

**Citation**

Roger Andrews, E. Pike, and Sarben Sarkar, "Photon correlations of a sub-threshold optical parametric oscillator," Opt. Express **10**, 461-468 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-11-461

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### References

- 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]
- 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]
- 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]
- L. Vaidman, �??Teleportation of quantum states,�?? Phys. Rev. A 49, 1473-1476 (1994). [CrossRef] [PubMed]
- D. Bouwmeester, J-W Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, �??Experimental quantum teleportation,�?? Nature 390, 575-579 (1997). [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]
- 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]
- 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]
- 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]
- 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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