## Pulsed Sagnac source of polarization entangled photon pairs |

Optics Express, Vol. 20, Issue 22, pp. 25022-25029 (2012)

http://dx.doi.org/10.1364/OE.20.025022

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### Abstract

Photonic quantum information experiments demand bright and highly entangled photon pair sources. The combination of periodic poling and collinear excitation geometry allows the use of considerably longer crystals for parametric down-conversion. We demonstrate a picosecond-pulsed laser pumped source of high quality polarization entangled photon pairs. The phase of the output biphoton state is affected by the relative phase of the two-color interferometer and the phase of the nonlinearly interacting Gaussian beams. We measure the influence of these onto the phase of the output state. The presented source is a promising candidate for a compact, semiconductor laser driven source of entangled photon pairs.

© 2012 OSA

## 1. Introduction

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

3. J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature **423**, 417–422 (2003). [CrossRef] [PubMed]

4. J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. **86**, 4435–4438 (2001). [CrossRef] [PubMed]

5. M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. **92**, 077901 (2004). [CrossRef] [PubMed]

6. M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. **32**(16,) 2339–2341 (2007). [CrossRef] [PubMed]

8. R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express **17**, 18920–18933 (2009). [CrossRef]

9. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A **70**, 043818 (2004). [CrossRef]

## 2. Experimental set-up and source characterisation

_{4}(PPKTP) crystal embedded in a triangular two-color Sagnac interferometer [10

10. B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A **69**, 013803 (2004). [CrossRef]

*φ*can be adjusted by various means as detailed below. The phase settings

*φ*= 0,

*π*correspond to the Bell states Ψ

^{+}and Ψ

^{−}, respectively. The Ψ

^{+}-state we created in the experiment yields a correlation visibility of 98.70(9)% in the A/D (±45°) basis and 99.88(3)% in the H/V (horizontal/vertical) basis, where accidental coincidence counts of 2 counts per second have been subtracted. Figure 2 shows the results of the visibility measurements. In addition, we performed state tomography of the output state [15

15. D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A **64**, 052312 (2001). [CrossRef]

*μ*W and each measurement point was averaged over 10 s. The density matrix was reconstructed using the maximum-likelihood estimation method. In order to obtain the measurement errors we performed a 100 run Monte Carlo simulation of the data with a Poissonian noise model applied to the measured values. We note that the measurements of the achieved tangle and state fidelity were performed via long single mode optical fibers which could have led to some averaging of the measurement results over different bases due to polarization fluctuations in the fibers. We measured the total brightness per pump power of our source to be 39 700 pairs/s per mW. Spectral widths (FWHM) of the produced protons were measured to be 1.79 nm and 2.67 nm.

## 3. Phase of the output state

^{+}〉 and |Ψ

^{−}〉, the phase between the state components |

*HV*〉 and |

*VH*〉, and tangle as shown in Fig. 3(a, b and c). At the nominal center position we set the pump phase to produce the |Ψ

^{+}〉 state. Figure 3(b) shows the real and imaginary part of an off-diagonal element (i.e. the coherence between |

*HV*〉 and |

*VH*〉) of the reconstructed density matrix. The tangle of the state maintained its high value (around 90%) throughout the measurement, but decreased when moving away from the central position, as expected due to the imbalance between the clockwise and counter-clockwise amplitudes.

^{−6}. This amounts to a phase shift between the clockwise and counter-clockwise beams of

*φ*

_{air}(

*x*) = 0.07

*πx*where x is the shift of the crystal away from the loop center in mm. Consequently, this shift is transferred to the phase of the entangled state. However, it does not yet fully account for the observed effect and we further consider the exact phase acquired in the nonlinear interaction.

*ϕ*

_{Gouy}= − arctan

*τ*, where

*τ*=

*z/z*

_{0}is the scaled position

*z*in the propagation direction and

*z*

_{0}is the Rayleigh range of the beam. In particular, passing through a focus a Gaussian beam receives a

*π*phase shift.

17. G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. **39**, 3597–3640 (1968). [CrossRef]

*z*

_{0}and where there is no birefringent walk-off, which is appropriate for the crystal we use. For a nonlinear crystal of geometrical length

*L*or, equivalently, focusing parameter

*ξ*=

*L*/(2

*z*

_{0}), the converted complex on-axis field amplitude for a beam centered at position

*f*within the crystal is then proportional to

*f*=

*L*/2 the focus is centered in the crystal. The highest conversion efficiency is obtained at a nominal zero phase difference between the pump and down-conversion beams. The phase mismatch

*σ*=

*z*

_{0}Δ

*k*is set by the linear dispersive properties of the crystal (Δ

*k*=

*k*

_{pump}−

*k*

_{signal}−

*k*

_{idler}) and the denominator in Eq. (2) comes from the Gouy phases of the three interacting waves [18

18. R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A **81**, 053805 (2010). [CrossRef]

20. N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express **15**, 7211–7217 (2007). [CrossRef] [PubMed]

*f*in the clockwise direction, we will have

*L*−

*f*in the counter-clockwise position. The resultant phase shift is then

*φ*

_{Gouy}(

*σ*,

*f*) = arg{

*H*(

*σ*,

*ξ*,

*f*)} − arg{

*H*(

*σ*,

*ξ*,

*L*−

*f*)} = 2arg{

*H*(

*σ*,

*ξ*,

*f*)}.

*ξ*and focus position

*f*there is an optimum phase mismatch

*σ*

_{opt}(

*ξ*,

*f*), which best compensates the Gouy phase and achieves the highest conversion efficiency. Here we have to point out that we investigate this phase shift in spontaneous parametric down-conversion where the produced photon pair may be non-degenerate and

*σ*depends very sensitively on all three wavelengths and the operating temperature. Because the Sagnac source works equally well for the degenerate and non-degenerate cases and we do not apply any narrow-band filtering we assume that at the chosen temperature the spectra of the two photons will only be determined by the function H and that the phase mismatch at the centers of these spectra is

*σ*

_{opt}.

*w*

_{0}= 25(3)

*μ*m corresponding to

*z*

_{0}= 8.9(21) mm (inside the crystal) or

*ξ*= 0.84(20). Using the values for atmospheric pressure, temperature, and humidity during the measurement, we modelled the air refractive index using Ciddor’s approach [21

21. Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp.

22. P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. **35**, 1566–1573 (1996). [CrossRef] [PubMed]

^{−6}for the wavelengths of 404 nm and 808 nm. Subtracting the resulting air phase shift (blue, dashed line in Fig 4(a)) from the measured total phase data yields the net (Gouy) phase.

*φ*

_{Gouy}(

*f*(

*x*),

*σ*

_{opt}(

*f*(

*x*))) calculated for this value of

*ξ*and the shift obtained in the measurement are similar in size. Nevertheless, the theoretical curve predicts a more rapid phase change when the focus is shifted through the end parts of the crystal than through the center. Our data shows a small reduction of the phase shift close to the center of the crystal but the shift is more monotonous. In order to have the two waists co-located with the crystal in place they need to be positioned off-center before the crystal is inserted. Since the exact waist locations are difficult to measure precisely and the refractive indices of the crystal are only known approximately, offsets of several millimeters could occur. Such an offset has the effect of displacing the nominal zero Gouy phase for the clock- and counter-clockwise directions in opposite directions and tends to flatten the dependence of their difference on the crystal position. On the other hand, the model to which we compare our experimental results is optimal for second harmonic generation and neither takes into account the finite spectral widths, nor the differences in the Rayleigh ranges of the three interacting beams. Further deviations could stem from higher order modal contributions. A tighter quantitative comparison between theory and experiment would require higher spatial density data as well as data-sets for a few different values of the focusing parameter

*ξ*.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett. |

3. | J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature |

4. | J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett. |

5. | M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett. |

6. | M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett. |

7. | P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. |

8. | R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express |

9. | P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A |

10. | B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A |

11. | T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A |

12. | A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A |

13. | O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express |

14. | M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express |

15. | D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A |

16. | D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010). |

17. | G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. |

18. | R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A |

19. | H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun. |

20. | N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express |

21. | Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp. |

22. | P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. |

**OCIS Codes**

(000.1600) General : Classical and quantum physics

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: September 11, 2012

Revised Manuscript: October 4, 2012

Manuscript Accepted: October 4, 2012

Published: October 17, 2012

**Citation**

Ana Predojević, Stephanie Grabher, and Gregor Weihs, "Pulsed Sagnac source of polarization entangled photon pairs," Opt. Express **20**, 25022-25029 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-22-25022

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### References

- D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature390, 575–579 (1997). [CrossRef]
- T. Jennewein, G. Weihs, J.-W. Pan, and A. Zeilinger “Experimental nonlocality proof of quantum teleportation and entanglement swapping,” Phys. Rev. Lett.88, 017903 (2002). [CrossRef] [PubMed]
- J.-W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, “Experimental entanglement purification of arbitrary unknown states,” Nature423, 417–422 (2003). [CrossRef] [PubMed]
- J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger “Experimental demonstration of four-photon entanglement and high-fidelity teleportation,” Phys. Rev. Lett.86, 4435–4438 (2001). [CrossRef] [PubMed]
- M. Eibl, N. Kiesel, M. Bourennane, C. Kurtsiefer, and H. Weinfurter “Experimental realization of a three-qubit entangled W state,” Phys. Rev. Lett.92, 077901 (2004). [CrossRef] [PubMed]
- M. Hendrych, M. Mičuda, and J. P. Torres “Tunable control of the frequency correlations of entangled photons,” Opt. Lett.32(16,) 2339–2341 (2007). [CrossRef] [PubMed]
- P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett.100, 133601 (2008). [CrossRef] [PubMed]
- R. Rangarajan, M. Goggin, and P. Kwiat “Optimizing type-I polarization-entangleg photons,” Opt. Express17, 18920–18933 (2009). [CrossRef]
- P. S. K. Lee, M. P. van Exter, and J. P. Woerdman“Increased polarization-entangled photon flux via thinner crystals,” Phys. Rev. A70, 043818 (2004). [CrossRef]
- B.-S. Shi and A. Tomita “Generation of a pulsed polarization entangled photon pair using a Sagnac interferometer,” Phys. Rev. A69, 013803 (2004). [CrossRef]
- T. Kim, M. Fiorentino, and F. N. C. Wong “Phase-stable source of polarization-entangled photons using a polarization Sagnac interferometer,” Phys. Rev. A73, 012316 (2006). [CrossRef]
- A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Phys. Rev. A77, 032314 (2008).
- O. Kuzucu and F. N. C. Wong “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
- M. Hentschel, H. Hübel, A. Poppe, and A. Zeilinger “Pulsed Sagnac source of narrow-band polarization entangled photons,” Opt. Express15, 15377–15386 (2007).
- D. F. V. James, P. Kwiat, W. Munro, and A. White“Measurement of qubits,” Phys. Rev. A64, 052312 (2001). [CrossRef]
- D. R. Hamel “Realization of novel entangled photon sources using periodically poled materials,” Master’s thesis, University of Waterloo (2010).
- G. D. Boyd and D. A. Kleiman “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39, 3597–3640 (1968). [CrossRef]
- R. S. Bennink“Optimal collinear Gaussian beams for spontaneous parametric down-conversion,” Phys. Rev. A81, 053805 (2010). [CrossRef]
- H. E. Major, C. B. E. Gawith, and P. G. R. Smith “Gouy phase compensation in quasi phase matching,” Opt. Commun.281, 5036–5040 (2008). [CrossRef]
- N. Lastzka and R. Schnabel “The Gouy phase shift in nonlinear interactions of waves,” Opt. Express15, 7211–7217 (2007). [CrossRef] [PubMed]
- Engineering metrology toolbox, http://emtoolbox.nist.gov/Wavelength/Ciddor.asp .
- P. E. Ciddor “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt.35, 1566–1573 (1996). [CrossRef] [PubMed]

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