## Generation of paired photons in a quantum separable state in Bragg reflection waveguides |

Optics Express, Vol. 19, Issue 4, pp. 3115-3123 (2011)

http://dx.doi.org/10.1364/OE.19.003115

Acrobat PDF (802 KB)

### Abstract

This work proposes and analyses a novel approach for the generation of separable (quantum uncorrelated) photon pairs based on spontaneous parametric down-conversion in Bragg reflection waveguides composed of semiconductor AlGaN layers. This platform allows the removal of any spectral correlation between paired photons that propagate in different spatial modes. The photons can be designed to show equal or different spectra by tuning the structural parameters and hence the dispersion of the waveguide.

© 2011 Optical Society of America

## 1. Introduction

*χ*

^{(2)}nonlinear media is a well-known parametric process that allows the generation of photon pairs. In this process, the interaction of an intense pump beam with the atoms of a nonlinear medium mediates the generation of pairs of photons with lower frequency. The spatio-temporal properties of the down-converted photons depend on the specific SPDC configuration considered as well as on the spatial and temporal characteristics of the pump beam.

1. P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A **72**, 032306 (2005). [CrossRef]

3. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. **79**, 135 (2007). [CrossRef]

4. L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. **12**, 093027 (2010). [CrossRef]

*Ga*

_{x}_{1−}

*N slabs can be tailored to generate photon pairs in a quantum separable state. Quasi-phase-matching (QPM) of the core slab is used to satisfy the phase-matching condition, while the tailoring of the dispersive properties of the waveguide, enhanced by using different types of modes for each of the interacting waves, allows us to control the frequency correlations between the down-converted photons.*

_{x}## 2. Description of the quantum state of the down-converted photons

*designate the frequency deviations from the corresponding central frequencies. The signal and idler photons are generated in specific spatial modes of the waveguide as will be described later.*

_{s,i}*ω*,

_{s}*ω*) is given by [19

_{i}19. Notice that here *E _{p}*(

*ω*+

_{s}*ω*) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators

_{i}*is*/2). Different definitions of

_{k}L*E*and

_{p}*a*and

_{s}*a*lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where

_{i}*E*refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators

_{p}*a*and

_{s}*a*is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−

_{i}*i*Δ

*/2). Indeed, when*

_{k}L*E*and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett.

_{p}**90**, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case. [CrossRef] [PubMed]

*=*

_{k}*k*–

_{p}*k*–

_{s}*k*and

_{i}*s*=

_{k}*k*+

_{p}*k*+

_{s}*k*.

_{i}*k*are the longitudinal (

_{p,s,i}*z*) components of the wavevector of all the interacting photons.

*E*is the spectral amplitude of the pump beam of central frequency

_{p}*d*Ω

*Ω*

_{s}d*|Φ(Ω*

_{i}*, Ω*

_{s}*)|*

_{i}^{2}= 1.

*j*=

*p, s, i*.

*N*are the inverse group velocities. Under these conditions, the biphoton amplitude can be written as Upon inspecting Eq. (3), one can show that if the inverse group velocities of the signal (idler) and pump are equal

_{j}*N*=

_{p}*N*(

_{s}*N*=

_{p}*N*), then increasing the bandwidth of the pump beam bandwidth such that Δ

_{i}*ω*≫ 1/|

_{p}*N*–

_{p}*N*|

_{s,i}*L*allows us to erase all the frequency correlations between the signal and idler photons. Notice that in this case, even though there is no entanglement between the signal and idler photons, the bandwidth of one the photons is larger than the bandwidth of the other photon. The quantum state is separable but the photons are distinguishable by their spectra.

20. A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. **63**, 415 (1995). [CrossRef]

21. J. H. Eberly, “Schmidt analysis of pure-state entanglement,” Laser Phys. **16**, 921 (2006). [CrossRef]

*λ*are the Schmidt eigenvalues and

_{n}*U*and

_{n}*V*are the corresponding Schmidt modes. The degree of entanglement of the two-photon state can be quantified by means of the purity of either of the subsystems (signal or idler photons) that make up the whole system. The purity of either subsystem is given by

_{n}*P*=

*K*

^{−1}, where

*K*= 1 corresponds to a separable two-photon state, while an increasing value of

*K*corresponds to an increase in the degree of entanglement.

## 3. Design of BRW structures to generate uncorrelated photon pairs

23. G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the Al_{x}Ga_{1−x}N and In_{y}Ga_{1−y}N alloys,” J. Appl. Phys. **89**, 1108 (2001). [CrossRef]

24. S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. **87**, 062106 (2005). [CrossRef]

*– 2*

_{k}*π*/Λ = 0, where the phase-mismatch function Δ

*is taken at the central frequencies of all the interacting waves.*

_{k}*u*(

_{j}*x*),

*j*=

*p,s, i*, are the mode functions describing the transverse distribution of the electric field in the waveguide. The overlap reaches 40.5% for Structure 1 and 19.4% for Structure 2. The combination of the high effective nonlinear coefficient and the overlap results in an efficiency that is still much higher than with other phase-matching platforms in waveguides or in bulk media. Although the thickness of the core of both structures is sufficiently large so that higher-order modes (both TIR and Bragg modes) could exist, they lack phase-matching and their overlap is very small.

## 4. Conclusion

*Ga*

_{x}_{1−}

*. Quasi-phase-matching of the waveguide core is used to achieve phase-matching at the desired wavelength. The control of waveguide dispersion is used to control the frequency correlation between the generated photons.*

_{x}N## Acknowledgments

## References and links

1. | P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, “Quantum gate characterization in an extended Hilbert space,” Phys. Rev. A |

2. | I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science |

3. | P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing,” Rev. Mod. Phys. |

4. | L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, “Design of bright, fiber-coupled and fully factorable photon pair sources,” N. J. Phys. |

5. | T. Aichele, A. I. Lvovsky, and S. Schiller, “Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state,” Eur. Phys. J. D |

6. | W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlation in multiphoton states,” Phys. Rev. A |

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. | J. P. Torres, F. Macià, S. Carrasco, and L. Torner, “Engineering the frequency correlations of entangled two-photon states by achromatic phase matching,” Opt. Lett. |

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

10. | J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. |

11. | Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion,” Phys. Rev. A |

12. | A. B. U’Ren, K. Banaszek, and I. A. Walmsley, “Photon engineering for quantum information processing,” Quantum Inf. Comput. |

13. | S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, “Spatial-to-spectral mapping in spontaneous parametric down-conversion,” Phys. Rev. A |

14. | A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. |

15. | X. Shi, A. Valencia, M. Hendrych, and J. Torres, “Generation of indistinguishable and pure heralded single photons with tunable bandwidth,” Opt. Lett. |

16. | A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, “Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices,” Phys. Rev. Lett. |

17. | B. R. West and A. S. Helmy, “Dispersion tailoring of the quarter-wave Bragg reflection waveguide,” Opt. Express |

18. | P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, “Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides,” Opt. Lett. |

19. | Notice that here ω + _{s}ω) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators _{i}is/2). Different definitions of _{k}LE and _{p}a and _{s}a lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where _{i}E refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators _{p}a and _{s}a is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−_{i}iΔ/2). Indeed, when _{k}LE and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. _{p}90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case. [CrossRef] [PubMed] |

20. | A. Ekert and P. L. Knight, “Entangled quantum systems and the Schmidt decomposition,” Am. J. Phys. |

21. | J. H. Eberly, “Schmidt analysis of pure-state entanglement,” Laser Phys. |

22. | J. Jin, |

23. | G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, “Improved refractive index formulas for the Al |

24. | S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, “Submicron periodic poling and chemical patterning of GaN,” Appl. Phys. Lett. |

25. | M. A. Nielsen and I. L. Chuang, |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 17, 2010

Revised Manuscript: January 19, 2011

Manuscript Accepted: January 26, 2011

Published: February 2, 2011

**Citation**

Jirí Svozilík, Martin Hendrych, Amr S. Helmy, and Juan P. Torres, "Generation of paired photons in a quantum separable state in Bragg reflection waveguides," Opt. Express **19**, 3115-3123 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3115

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### References

- P. P. Rohde, G. J. Pryde, J. L. O’Brien, and T. C. Ralph, "Quantum gate characterization in an extended Hilbert space," Phys. Rev. A 72, 032306 (2005). [CrossRef]
- I. A. Walmsley, and M. G. Raymer, "Toward quantum-information processing with photons," Science 307, 1733 (2005). [CrossRef] [PubMed]
- P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, "Linear optical quantum computing," Rev. Mod. Phys. 79, 135 (2007). [CrossRef]
- L. E. Vicent, A. B. U’Ren, R. Rangarajan, C. I. Osorio, J. P. Torres, L. Zhang, and I. A. Walmsley, "Design of bright, fiber-coupled and fully factorable photon pair sources," N. J. Phys. 12, 093027 (2010). [CrossRef]
- T. Aichele, A. I. Lvovsky, and S. Schiller, "Optical mode characterization of single photons prepared by means of conditional measurements on a biphoton state," Eur. Phys. J. D 18, 237 (2002). [CrossRef]
- W. P. Grice, A. B. U’Ren, and I. A. Walmsley, "Eliminating frequency and space-time correlation in multiphoton states," Phys. Rev. A 64, 063815 (2001). [CrossRef]
- 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]
- J. P. Torres, F. Macià, S. Carrasco, and L. Torner, "Engineering the frequency correlations of entangled two photon states by achromatic phase matching," Opt. Lett. 30, 314 (2005). [CrossRef] [PubMed]
- M. Hendrych, M. Mičuda, and J. P. Torres, "Tunable control of the frequency correlations of entangled photons," Opt. Lett. 32, 2339 (2007). [CrossRef]
- J. P. Torres, M. Hendrych, and A. Valencia, "Angular dispersion: an enabling tool in nonlinear and quantum optics," Adv. Opt. Photon. 2, 319 (2010). [CrossRef]
- Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Controllable frequency entanglement via auto-phase-matched spontaneous parametric down-conversion," Phys. Rev. A 67, 053810 (2003). [CrossRef]
- A. B. U’Ren, K. Banaszek, and I. A. Walmsley, "Photon engineering for quantum information processing," Quantum Inf. Comput. 3, 480 (2003).
- S. Carrasco, J. P. Torres, L. Torner, A. V. Sergienko, B. E. Saleh, and M. C. Teich, "Spatial-to-spectral mapping in spontaneous parametric down-conversion," Phys. Rev. A 70, 043817 (2004). [CrossRef]
- A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, "Shaping the waveform of entangled photons," Phys. Rev. Lett. 99, 243601 (2007). [CrossRef]
- X. Shi, A. Valencia, M. Hendrych, and J. Torres, "Generation of indistinguishable and pure heralded single photons with tunable bandwidth," Opt. Lett. 33, 875 (2008). [CrossRef] [PubMed]
- A. B. U’Ren, R. K. Erdmann, M. de la Cruz-Gutierrez, and I. A. Walmsley, "Generation of Two-Photon States with an Arbitrary Degree of Entanglement Via Nonlinear Crystal Superlattices," Phys. Rev. Lett. 97, 223602 (2006). [CrossRef]
- B. R. West, and A. S. Helmy, "Dispersion tailoring of the quarter-wave Bragg reflection waveguide," Opt. Express 14, 4073 (2006). [CrossRef] [PubMed]
- P. Abolghasem, M. Hendrych, X. Shi, J. P. Torres, and A. Helmy, "Bandwidth control of paired photons generated in monolithic Bragg reflection waveguides," Opt. Lett. 34, 2000 (2009). [CrossRef] [PubMed]
- Notice that here Ep(ωs + ωi) refers to the spectral amplitude of the pump beam at the input face of the waveguide, while the operators as(ω0s + Ωs) and ai(ω0i + Ωi) refer to the quantum state at the output face of the waveguide. Under these conditions (see [8]), the exponential factor contained in the biphoton amplitude is of the form exp(iskL/2). Different definitions of Ep and as and ai lead to slightly different expressions for the exponential factor of the biphoton amplitude. This is the case, for instance, in [6], where Ep refers to the spectral amplitude of the pump beam at the output face of the nonlinear crystal, while the same definition of the quantum operators as and ai is used. Now the exponential factor contained in the biphoton amplitude is of the form exp(−iΔkL/2). Indeed, when Ep and the quantum operators are referred to the center of the nonlinear crystal, there is no exponential factor at all [see S. P. Walborn, A. N. de Oliveira, S. Padua, and C. H. Monken, "Multimode Hong-Ou-Mandel interference," Phys. Rev. Lett. 90, 143601 (2003)]. Of course, all of these expressions are related and should yield the same result when calculating correlation functions, since the electric field operators should also be correspondingly modified in each case. [CrossRef] [PubMed]
- A. Ekert, and P. L. Knight, "Entangled quantum systems and the Schmidt decomposition," Am. J. Phys. 63, 415 (1995). [CrossRef]
- J. H. Eberly, "Schmidt analysis of pure-state entanglement," Laser Phys. 16, 921 (2006). [CrossRef]
- J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE Press, 2002).
- G. M. Laws, E. C. Larkins, I. Harrison, C. Molloy, and D. Somerford, "Improved refractive index formulas for the AlxGa1−xN and InyGa1−yN alloys," J. Appl. Phys. 89, 1108 (2001). [CrossRef]
- S. Pezzagna, P. Vennéguès, N. Grandjean, A. D. Wieck, and J. Massies, "Submicron periodic poling and chemical patterning of GaN," Appl. Phys. Lett. 87, 062106 (2005). [CrossRef]
- M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, 2000).

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