## Photon-pair generation in arrays of cubic nonlinear waveguides |

Optics Express, Vol. 20, Issue 24, pp. 27441-27446 (2012)

http://dx.doi.org/10.1364/OE.20.027441

Acrobat PDF (2141 KB)

### Abstract

We study photon-pair generation in arrays of cubic nonlinear waveguides through spontaneous four-wave mixing. We analyze numerically the quantum statistics of photon pairs at the array output as a function of waveguide dispersion and pump beam power. We show flexible spatial quantum state control such as pump-power-controlled transition between bunching and anti-bunching correlations due to nonlinear self-focusing.

© 2012 OSA

## 1. Introduction

1. J. C. F. Matthews, A. Politi, A. Stefanov, and J.
L. O’Brien, “Manipulation of multiphoton entanglement in
waveguide quantum circuits,” Nature Photonics **3**, 346–350 (2009). [CrossRef]

1. J. C. F. Matthews, A. Politi, A. Stefanov, and J.
L. O’Brien, “Manipulation of multiphoton entanglement in
waveguide quantum circuits,” Nature Photonics **3**, 346–350 (2009). [CrossRef]

2. A. Politi, J. C. F. Matthews, and J.
L. O’Brien, “Shor’s quantum factoring algorithm on a
photonic chip,” Science **325**, 1221–1221 (2009). [CrossRef] [PubMed]

3. L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame,
“Polarization entangled state measurement on a chip,” Phys.
Rev. Lett. **105**, 200503 (2010). [CrossRef]

4. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J.
L. O’Brien, “Quantum walks of correlated
photons,” Science **329**, 1500–1503 (2010). [CrossRef] [PubMed]

*nonlinear waveguide arrays*further opens the possibility for enhanced spatial quantum state control and improved clarity of spatial correlations [5

5. A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y.
S. Kivshar, “Spontaneous parametric down-conversion and
quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev.
Lett. **108**, 023601 (2012). [CrossRef] [PubMed]

5. A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y.
S. Kivshar, “Spontaneous parametric down-conversion and
quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev.
Lett. **108**, 023601 (2012). [CrossRef] [PubMed]

6. A. Rai and D.
G. Angelakis, “Dynamics of nonclassical light in integrated
nonlinear waveguide arrays and generation of robust continuous-variable
entanglement,” Phys. Rev. A **85**, 052330 (2012). [CrossRef]

7. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar,
“Generation of correlated photons in nanoscale silicon waveguides,”
Opt. Express **14**, 12388–12393 (2006). [CrossRef] [PubMed]

8. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi
Itabashi, “Entanglement
generation using silicon wire waveguide,” Appl. Phys. Lett. **91**, 201108 (2007). [CrossRef]

## 2. SFWM in WGAs at low pump powers

7. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar,
“Generation of correlated photons in nanoscale silicon waveguides,”
Opt. Express **14**, 12388–12393 (2006). [CrossRef] [PubMed]

8. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi
Itabashi, “Entanglement
generation using silicon wire waveguide,” Appl. Phys. Lett. **91**, 201108 (2007). [CrossRef]

*β*= 2

*β*−

_{p}*β*−

_{s}*β*. Here

_{i}*β*are the propagation constants for pump, signal and idler. In a WGA they depend on normalized transverse momenta

_{p,s,i}*k*as

_{p,s,i}*C*is the coupling coefficient between the waveguides [9

9. D. N. Christodoulides, F. Lederer, and Y. Silberberg,
“Discretizing light behaviour in linear and nonlinear waveguide
lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

5. A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y.
S. Kivshar, “Spontaneous parametric down-conversion and
quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev.
Lett. **108**, 023601 (2012). [CrossRef] [PubMed]

9. D. N. Christodoulides, F. Lederer, and Y. Silberberg,
“Discretizing light behaviour in linear and nonlinear waveguide
lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

*n*is a waveguide number. Since

*N*∈ , then the transverse momenta

9. D. N. Christodoulides, F. Lederer, and Y. Silberberg,
“Discretizing light behaviour in linear and nonlinear waveguide
lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

4. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J.
L. O’Brien, “Quantum walks of correlated
photons,” Science **329**, 1500–1503 (2010). [CrossRef] [PubMed]

10. L. G. Helt, M. Liscidini, and J. E. Sipe,
“How does it scale? comparing quantum and classical nonlinear optical processes in
integrated devices,” J. Opt. Soc. Am. B **29**, 2199–2212 (2012). [CrossRef]

*Ĥ*=

*Ĥ*

^{(lin)}+

*Ĥ*

^{(nonlin)},

*n*and

_{s}*n*are the waveguide numbers describing the positions of the signal, and the idler photons, and

_{i}*n*. Δ

_{p}*β*

^{(0)}is the linear four-wave mixing phase-mismatch in a single waveguide,

*γ*is a nonlinear coefficient. The normalized pump field profile evolution along the propagation distance

*z*is defined through the classical coupled-mode equations [9

**424**, 817–823 (2003). [CrossRef] [PubMed]

*ψ*

_{ns,ni}(

*z*) in a Schrodinger-type equation. The equation is obtained from the Hamiltonian, and it has a form ¨ similar to that of quadratic media [11

11. M. Grafe, A. S. Solntsev, R. Keil, A. A. Sukhorukov, M. Heinrich, A. Tunnermann, S. Nolte, A. Szameit, and Y.
S. Kivshar, “Biphoton generation in quadratic waveguide
arrays: A classical optical simulation,” Sci. Rep. **2**, 562 (2012). [CrossRef] [PubMed]

_{ns,ni}= |

*ψ*

_{ns,ni}(

*L*)|

^{2}, where

*L*is the propagation length. In order to find correlations for the signal and idler photons in k-space we apply the two-dimensional Fourier-transform,

*L*= 1 and nonlinearity

*γ*= 1. The physical value of the nonlinear coefficient can be determined following the approach of Ref. [10

10. L. G. Helt, M. Liscidini, and J. E. Sipe,
“How does it scale? comparing quantum and classical nonlinear optical processes in
integrated devices,” J. Opt. Soc. Am. B **29**, 2199–2212 (2012). [CrossRef]

*C*= 5, and consider a pump beam coupled only to the central waveguide (

*n*= 0).

**424**, 817–823 (2003). [CrossRef] [PubMed]

*β*

^{(0)}= −18, zero Δ

*β*

^{(0)}= 0 and normal Δ

*β*

^{(0)}= 18. In the case of anomalous GVD, photons in a pair tend to end up mostly away from the central waveguide, with higher probability to be at either the same or the opposite waveguides, see photon-pair probability correlation in Fig. 1(c). This behavior corresponds to weakly pronounced simultaneous spatial bunching and antibunching. The quantum statistics in this case is quasi-anyonic, which can be interesting for quantum simulations [12

12. J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wrhoff, M. G. Thompson, and J. L. O’Brien, “Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions,” http://arxiv.org/abs/1106.1166 (2011).

*β*= 0. For zero GVD the signal and idler photons mostly leave the structure from the same waveguides, thus demonstrating strong spatial bunching behavior [Fig. 1(e)]. Figure 1(f) shows that the transverse wavenumbers for photon pairs satisfy the relations

13. J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P.
M. Fauchet, “Optical solitons in a silicon
waveguide,” Opt. Express **15**, 7682–7688 (2007). [CrossRef] [PubMed]

## 3. SFWM and pump self-focusing at high pump powers

14. A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R.
M. De La Rue, “Spatiotemporal nonlinear optics in arrays of
subwavelength waveguides,” Phys. Rev. A **82**, 041802 (2010). [CrossRef]

*P*for photon-pair generation is much smaller: in a 10 mm long Si waveguide

*P*≈ 0.1 – 0.2 W [7

7. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar,
“Generation of correlated photons in nanoscale silicon waveguides,”
Opt. Express **14**, 12388–12393 (2006). [CrossRef] [PubMed]

8. H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi
Itabashi, “Entanglement
generation using silicon wire waveguide,” Appl. Phys. Lett. **91**, 201108 (2007). [CrossRef]

*spatially filtered from the pump beam*and show strongly pronounced anti-bunching. This demonstrates that by tuning the pump wavelength and power we can get a great degree of control on the spatial photon-pair correlations, which is useful for applications in quantum information.

*R*= (∑

_{ni=ns}Γ

_{ns,ni})/(∑

_{ni=−ns}Γ

_{ns,ni}) and study the dynamics of this ratio with respect to pump power [Fig. 4(a) ( Media 1)], while also tracking the spatial photon-pair correlations for anomalous [Fig. 4(b) ( Media 1)] and normal GVD [Fig. 4(c) ( Media 1)] and the pump propagation [Fig. 4(d) ( Media 1)]. We observe that the pump power tuning provides access to a wide range of photon-pair quantum statistics. We also see that in general, normal dispersion can lead to stronger spatial antibunching, as in this case only angled phase-matching can be satisfied.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. C. F. Matthews, A. Politi, A. Stefanov, and J.
L. O’Brien, “Manipulation of multiphoton entanglement in
waveguide quantum circuits,” Nature Photonics |

2. | A. Politi, J. C. F. Matthews, and J.
L. O’Brien, “Shor’s quantum factoring algorithm on a
photonic chip,” Science |

3. | L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame,
“Polarization entangled state measurement on a chip,” Phys.
Rev. Lett. |

4. | A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J.
L. O’Brien, “Quantum walks of correlated
photons,” Science |

5. | A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y.
S. Kivshar, “Spontaneous parametric down-conversion and
quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev.
Lett. |

6. | A. Rai and D.
G. Angelakis, “Dynamics of nonclassical light in integrated
nonlinear waveguide arrays and generation of robust continuous-variable
entanglement,” Phys. Rev. A |

7. | J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar,
“Generation of correlated photons in nanoscale silicon waveguides,”
Opt. Express |

8. | H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi
Itabashi, “Entanglement
generation using silicon wire waveguide,” Appl. Phys. Lett. |

9. | D. N. Christodoulides, F. Lederer, and Y. Silberberg,
“Discretizing light behaviour in linear and nonlinear waveguide
lattices,” Nature |

10. | L. G. Helt, M. Liscidini, and J. E. Sipe,
“How does it scale? comparing quantum and classical nonlinear optical processes in
integrated devices,” J. Opt. Soc. Am. B |

11. | M. Grafe, A. S. Solntsev, R. Keil, A. A. Sukhorukov, M. Heinrich, A. Tunnermann, S. Nolte, A. Szameit, and Y.
S. Kivshar, “Biphoton generation in quadratic waveguide
arrays: A classical optical simulation,” Sci. Rep. |

12. | J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wrhoff, M. G. Thompson, and J. L. O’Brien, “Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions,” http://arxiv.org/abs/1106.1166 (2011). |

13. | J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P.
M. Fauchet, “Optical solitons in a silicon
waveguide,” Opt. Express |

14. | A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R.
M. De La Rue, “Spatiotemporal nonlinear optics in arrays of
subwavelength waveguides,” Phys. Rev. A |

15. | C. J. Benton and D.
V. Skryabin, “Coupling induced anomalous group velocity
dispersion in nonlinear arrays of silicon photonic wires,” Opt.
Express |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(270.0270) Quantum optics : Quantum optics

(080.1238) Geometric optics : Array waveguide devices

**ToC Category:**

Four-Wave Mixing in Waveguides and Fibers

**History**

Original Manuscript: September 5, 2012

Revised Manuscript: October 19, 2012

Manuscript Accepted: October 20, 2012

Published: November 19, 2012

**Virtual Issues**

Nonlinear Photonics (2012) *Optics Express*

**Citation**

Alexander S. Solntsev, Andrey A. Sukhorukov, Dragomir N. Neshev, and Yuri S. Kivshar, "Photon-pair generation in arrays of cubic nonlinear waveguides," Opt. Express **20**, 27441-27446 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27441

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### References

- J. C. F. Matthews, A. Politi, A. Stefanov, and J. L. O’Brien, “Manipulation of multiphoton entanglement in waveguide quantum circuits,” Nature Photonics3, 346–350 (2009). [CrossRef]
- A. Politi, J. C. F. Matthews, and J. L. O’Brien, “Shor’s quantum factoring algorithm on a photonic chip,” Science325, 1221–1221 (2009). [CrossRef] [PubMed]
- L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett.105, 200503 (2010). [CrossRef]
- A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. O’Brien, “Quantum walks of correlated photons,” Science329, 1500–1503 (2010). [CrossRef] [PubMed]
- A. S. Solntsev, A. A. Sukhorukov, D. N. Neshev, and Y. S. Kivshar, “Spontaneous parametric down-conversion and quantum walks in arrays of quadratic nonlinear waveguides,” Phys. Rev. Lett.108, 023601 (2012). [CrossRef] [PubMed]
- A. Rai and D. G. Angelakis, “Dynamics of nonclassical light in integrated nonlinear waveguide arrays and generation of robust continuous-variable entanglement,” Phys. Rev. A85, 052330 (2012). [CrossRef]
- J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, “Generation of correlated photons in nanoscale silicon waveguides,” Opt. Express14, 12388–12393 (2006). [CrossRef] [PubMed]
- H. Takesue, Y. Tokura, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, and S. ichi Itabashi, “Entanglement generation using silicon wire waveguide,” Appl. Phys. Lett.91, 201108 (2007). [CrossRef]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature424, 817–823 (2003). [CrossRef] [PubMed]
- L. G. Helt, M. Liscidini, and J. E. Sipe, “How does it scale? comparing quantum and classical nonlinear optical processes in integrated devices,” J. Opt. Soc. Am. B29, 2199–2212 (2012). [CrossRef]
- M. Grafe, A. S. Solntsev, R. Keil, A. A. Sukhorukov, M. Heinrich, A. Tunnermann, S. Nolte, A. Szameit, and Y. S. Kivshar, “Biphoton generation in quadratic waveguide arrays: A classical optical simulation,” Sci. Rep.2, 562 (2012). [CrossRef] [PubMed]
- J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wrhoff, M. G. Thompson, and J. L. O’Brien, “Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions,” http://arxiv.org/abs/1106.1166 (2011).
- J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express15, 7682–7688 (2007). [CrossRef] [PubMed]
- A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A82, 041802 (2010). [CrossRef]
- C. J. Benton and D. V. Skryabin, “Coupling induced anomalous group velocity dispersion in nonlinear arrays of silicon photonic wires,” Opt. Express17, 5879–5884 (2009). [CrossRef] [PubMed]

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