## Theory of photon statistics and squeezing in quantum interference of a sub-threshold parametric oscillator

Optics Express, Vol. 11, Issue 1, pp. 7-13 (2003)

http://dx.doi.org/10.1364/OE.11.000007

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### Abstract

A multimode theory describing quantum interference of a sub-threshold optical parametric oscillator (OPO) with a coherent local oscillator (LO) in a homodyne detection scheme is presented. Analytic expressions for the count rates in terms of the correlation time and relative phase difference between the LO and OPO have been derived. The spectrum of squeezing is also derived and the threshold for squeezing obtained in terms of the crystal nonlinearity and LO and OPO beam intensities.

© 2003 Optical Society of America

## 1. Introduction

## 2. Amplitude for pair detection in homodyne

*r*⃗

_{1}and

*r*⃗

_{2}respectively, the amplitude for pair detection in the Heisenberg representation is described by

*E*⃗

_{H}(

*r*⃗

_{i},

*t*

_{i}) is the Heisenberg electric field operator at the detectors. We assume that the input pump laser to the OPO and the local oscillator are in single-mode coherent states, |α

*,α*

_{LO}*〉, with the LO emitting at the degenerate frequency of the OPO. We also assume single-pass operation of a one-sided OPO as in previous work [7*

_{pump}7. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express **10**, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

*E*⃗

_{I}(

*r*⃗

_{i},

*t*) are the interaction-picture electric field operators and ε

*,φ*

_{p}*are the amplitude and phase of the pump respectively;*

_{p}7. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express **10**, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

9. 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]

*k*

_{1},

*k*

_{2}as the wave vectors of the signal and idler photons and

*k*

_{0}the wave vector of the input pump photon. In obtaining (2) we employed beam splitter transformations for the electric field at the detectors:

*E*⃗

_{out}and

*E*⃗

_{LO}are the electric fields of the OPO and LO respectively. Equation (2) describes the phenomenon of quantum interference in the detection of pairs from the LO and OPO. In order to simplify (2) we follow similar calculations in previous work [7

7. R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express **10**, 461–468 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461. [CrossRef] [PubMed]

*,φ*

_{l}*are the amplitude and phase of the LO, τ is the correlation time,*

_{l}*d*is the length of the OPO cavity and

*v*is the first-order dispersion coefficient of the nonlinear crystal;

*S*and

*n*

_{1}are defined as

*t*

_{2o},

*r*

_{2o}are the amplitude transmission, reflection coefficients of the output mirror of the OPO at the signal and idler frequencies and

*V*the volume of the irradiated crystal. We obtain the count rate which is proportional to

## 3. Spectrum of squeezing

*r*⃗

_{1}and

*r*⃗

_{2}. The difference in powers of the two beams is obtained from

*P*(

*t*) defined as

*P*(

*t*)〉 is described by

*Q*(ω), is defined as

*is the frequency of the signal photon detected at D1. Taking the Fourier transform of the R.H.S of (10) we obtain the normalized spectrum of squeezing at the degenerate frequency*

_{k}## 4. Results

*q*= 2, Δ = 2φ

*- φ*

_{l}*= 0;*

_{p}*q*= 2 and Δφ = π and

*q*= 1, Δφ = π respectively. These results agree qualitatively with the experimental results of Lu and Ou in Fig. 3 of their paper [6]. We confirm with our model photon antibunching with

*q*= 1, Δφ = π at zero time delay (Fig. 2c) and at non-zero time delays with

*q*= 2, Δφ = π (Fig. 2b).

^{-9}mm

^{3}, squeezing is obtained when

## 5. Conclusion

## References and links

1. | G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. |

2. | A. Kiraz, S. Falth, C. Bechner, B. Gayral, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu, and A. Imamoglu, “Photon correlation spectroscopy of a single quantum dot,” Phys. Rev. B |

3. | C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. |

4. | R. Brouri, A. Beveratos, J. Poizat, and P. Grangier, “Photon antibunching in the fluorescence of individual color centers in diamond,” Opt. Lett. |

5. | L. Fleury, J. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. |

6. | Y. J. Lu and Z. Y. Ou, “Observation of nonclassical photon statistics due to quantum interference,” Phys. Rev. Lett. |

7. | R. Andrews, E. R. Pike, and Sarben Sarkar, “Photon correlations of a sub-threshold optical parametric oscillator,” Opt. Express |

8. | A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator,” Phys. Rev. A |

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

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 12, 2002

Revised Manuscript: December 11, 2002

Published: January 13, 2003

**Citation**

Roger Andrews, E. Pike, and Sarben Sarkar, "Theory of photon statistics and squeezing in quantum interference of a sub-threshold parametric oscillator," Opt. Express **11**, 7-13 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-1-7

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### References

- G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, �??Limitations on practical quantum cryptography,�?? Phys. Rev. Lett. 85, 1330-1333 (2000). [CrossRef] [PubMed]
- A. Kiraz, S. Falth, C. Bechner , B. Gayral, W. V. Schoenfeld, P. M. Petroff, Lidong Zhang, E. Hu and A. Imamoglu, �??Photon correlation spectroscopy of a single quantum dot,�?? Phys. Rev. B 65, 161303-161304 (2002). [CrossRef]
- C. Santori, M. Pelton, G. Solomon, Y. Dale, and Y. Yamamoto, �??Triggered single photons from a quantum dot,�?? Phys. Rev. Lett. 86, 1502-1505 (2001). [CrossRef] [PubMed]
- R. Brouri, A. Beveratos, J. Poizat, P. Grangier, �??Photon antibunching in the fluorescence of individual color centers in diamond,�?? Opt. Lett. 25, 1294-1296 (2000). [CrossRef]
- L. Fleury, J. Segura, G. Zumofen, B. Hecht and U. P. Wild, �??Nonclassical photon statistics in single-molecule fluorescence at room temperature,�?? Phys. Rev. Lett. 84, 1148-1151 (2000). [CrossRef] [PubMed]
- Y. J. Lu and Z. Y. Ou, �??Observation of nonclassical photon statistics due to quantum interference,�?? Phys. Rev. Lett. 88, 023601-1-023601-4 (2002).
- R. Andrews, E. R. Pike and Sarben Sarkar, �??Photon correlations of a sub-threshold optical parametric oscillator,�?? Opt. Express 10, 461-468 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-11-461</a>. [CrossRef] [PubMed]
- A. Gatti and L. Lugiato, �??Quantum images and critical fluctuations in the optical parametric oscillator,�?? Phys. Rev. A 52, 1675-1690 (1995). [CrossRef] [PubMed]
- 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]

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