## Azimuthal distinguishability of entangled photons generated in spontaneous parametric down-conversion

Optics Express, Vol. 15, Issue 22, pp. 14636-14643 (2007)

http://dx.doi.org/10.1364/OE.15.014636

Acrobat PDF (150 KB)

### Abstract

We experimentally demonstrate that paired photons generated in different sections of a down-conversion cone, when some of the interacting waves show Poynting vector walk-off, carry different spatial correlations, and therefore a different degree of spatial entanglement. This is shown to be in agreement with theoretical results. We also discuss how this *azimuthal distinguishing information* of the down-conversion cone is relevant for the implementation of quantum sources aimed at the generation of entanglement in other degrees of freedom, such as polarization.

© 2007 Optical Society of America

## 1. Introduction

1. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. **3**, 305 – 310 (2007). [CrossRef]

2. A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum,” Phys. Rev. Lett. **91**, 227902 (2003). [CrossRef] [PubMed]

*quantum coin tossing*protocol with qutrits [3

3. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “ Experimental Quantum Coin Tossing,” Phys. Rev. Lett. **94**, 040501 (2005). [CrossRef] [PubMed]

4. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat , “Generation of Hyperentangled Photon Pairs,” Phys. Rev. Lett. **95**, 260501 (2005). [CrossRef]

5. J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A **68**050301(R) (2003). [CrossRef]

6. C. K. Law and J. H. Eberly, “Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion,” Phys. Rev. Lett. **92**, 127903 (2004). [CrossRef] [PubMed]

7. H. H. Arnaut and G. A. Barbosa, “Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion,” Phys. Rev. Lett. **85**, 286 (2000). [CrossRef] [PubMed]

8. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**, 313–316 (2001). [CrossRef] [PubMed]

*Are the entangled spatial properties of the photons modified depending of the location in the down-conversion cone where they are detected?*

9. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A **72**, 033803 (2005). [CrossRef]

10. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, R773 (1999). [CrossRef]

11. J. Altepeter, E. Jeffrey, and P. Kwiat, “Phase-compensated ultra-bright source of entangled photons,” Opt. Express , **13**, 8951–8959(2005). [CrossRef] [PubMed]

12. J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. **7**, 235–239 (2005). [CrossRef]

## 2. Experimental set-up and results

*λ*= 405

_{p}*nm*, is appropriately spatially filtered to obtain a beam with a gaussian profile, while a half wave plate (HWP) is used to control the polarization. The pump beam is focalized to

*w*

_{0}= 136

*μm*beam waist on the input face of a

*L*= 5mm thick lithium iodate crystal, cut at 42° for Type I degenerate collinear phase matching. The generated photons, signal and idler, are ordinary polarized, in opposition to the extraordinary polarized pump beam. The crystal is tilted to generate paired photons which propagate inside the nonlinear crystal with a non-collinear angle of

*φ*= 4°. Due to the crystal birefringence, the pump beam exhibits Pointing vector walk-off with angle

*ρ*

_{0}= 4.9°, while the generated photons do not exhibit spatial walk-off. Fig. 1(b) represents the transversal section of the down-conversion cone. The directions of propagation of the signal and the idler photons over this ring are determined by the azimuthal angle

*α*, which is the angle between the plane of propagation of the down-converted photons and the

*YZ*plane. To determine experimentally the position of the crystal optics axis, and the origin of

*α*, we measure the relative position of the pump beam in the plane

*XY*at the input and output faces of the nonlinear crystal using a CCD camera.

*f*system with focal length f=50

*cm*. Low-pass filters are used to remove the remaining pump beam radiation. After the filters, the photons are coupled into multimode fibers. In order to increase our spatial resolution, we use small pinholes of 300

*μ*m of diameter. We keep the idler pinhole fixed and measure the coincidence rate while scanning the signal photon transverse spatial shape with a motorized

*XY*translation stage. Finally, as we are interested in the different spatial correlations at different azimuthal positions of the downconversion ring, instead of rotating the whole detection system, the nonlinear crystal and the polarization of the pump beam are rotated around the propagation direction. Due to slight misalignments of the rotation axis of the crystal, after every rotation it is necessary to adjust the tilt of the crystal to achieve generation of photons at the same non-collinear angle in all the cases.

*α*= 0°, 90°, 180° and 270°. The movie shows the experimental and theoretical spatial shape of the signal photon corresponding to other values of the angle

*α*. Each point of these images corresponds to the recording of a 10

*s*measurement. The typical number of coincidences at the maximum is around 10 photons per second. The resolution of the experimental images is 50 × 50 pixels. The different spatial shapes measured of the mode function of the signal photons clearly show that the down-conversion cone does not posses azimuthal symmetry. This agrees with the theoretical predictions presented in the lower row of Fig. 2. Note that no fitting parameter has been used whatsoever. Slight discrepancies between experimental data and theoretical predictions might be due to the small, but not negligible, bandwidth of the pump beam and to the fact that the resolution of our system is limited by the detection pinholes size.

*α*= 90° presents a nearly gaussian shape. We will show below that this effect happens whenever φ ≃

*ρ*

_{0}, which corresponds to our experimental conditions. On the other hand the mode function shown in Figs. 2 (a) and (c) are highly elliptical.

## 3. Azimuthal distinguishability of paired photons generated in different sections of the the down-conversion cone

**z**

_{1}(see Fig. 1) with longitudinal wavevector

*k*= [(

_{s}*ω*/

_{s}n_{s}*c*)

^{2}- |

**p**|

^{2}]

^{1/2}and transverse wavevector

**p**= (

*p*,

_{x}*p*). Similarly, the idler photon propagates along the

_{y}**z**

_{2}direction with longitudinal wavevector

*k*, and transverse wavevector

_{i}**q**. Here we consider the signal and idler photons as purely monochromatic, due to the use of a narrow pinhole in the idler side, which selects a very small bandwidth of frequencies of the down-converted ring. Although photons detected in different parts of the down-conversion cone might present slightly different polarizations [13

13. A. Migdall, “Polarization directions of noncollinear phase-matched optical parametric downconversion output,” J. Opt. Soc. Am. B , **14**1093–1098 (1997). [CrossRef]

*d*

**p**

*d*

**q**Φ(

**p**,

**q**)

*a*

_{s}^{†}(

**p**)

*a*

_{i}^{†}(

**q**)|0,0〉, where the mode function writes [14

14. M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A **54**, 5349 (1996). [CrossRef] [PubMed]

12. J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. **7**, 235–239 (2005). [CrossRef]

15. G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P. Torres, “Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion,” Phys. Rev. A **72**, 065802 (2005). [CrossRef]

*α*) due to the presence of spatial walk off.

*The latter is the cause of the azimuthal symmetry breaking of the down-conversion cone*. Both effects turn out to be important when the length of the crystal

*L*is larger than the non-collinear length

*L*=

_{nc}*w*

_{0}/ sin

*φ*and the walk-off length

*L*=

_{w}*w*

_{0}/tan

*ρ*

_{0}. Our experimental configuration is fully in this regime. We should notice that in a collinear SPDC configuration, Poynting vector walk-off also introduces ellipticity of the mode function [16

16. M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion,” Phys. Rev. Lett. **99**, 063901 (2007). [CrossRef] [PubMed]

*β*in the (

*p*,

_{x}*P*) plane of the loci of perfect phase matching transverse momentum, which writes tan

_{y}*β*= (sin

*φ*- tan

*ρ*

_{0}cos

*φ*sin

*α*)/(tan

*ρ*

_{0}cos

*α*). If

*φ*≃

*ρ*

_{0}and

*α*= 90°, the spatial mode function of the signal photons shows a nearly gaussian shape, due to the compensation of the non-collinear and walk-off effects. All these results are in agreement with experimental data in Fig. 2.

_{s}(

**p**) = Φ(

**p**,

**q**= 0) in terms of orbital angular momentum modes. The mode function can be described by superposition of spiral harmonics [17

17. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. **88**, 013601 (2002). [CrossRef] [PubMed]

_{s}(

*ρ*,

*φ*) = (2

*π*) Σ

*(*

_{m}a_{m}*ρ*)exp(

*imφ*), where

*a*(

_{m}*ρ*) = 1/(2

*π*)

^{1/2}∫

*dφ*Φ

_{s}(

*ρ*,

*φ*)exp(-

*imφ*), and ρ and

*φ*are cylindrical coordinates in the transverse wave-number space. The weight of the

*m*-harmonic is given by

*C*= ∫

_{m}*ρdρ*|

*a*(

_{m}*ρ*) |

^{2}.

*l*= 0, while the idler photon is projected into

_{p}**q**= 0, which corresponds to projection into a large area gaussian mode (

*l*= 0). Figures 3(a) and (b) show the weight of the mode

_{i}*l*= 0, and the weight of all other OAM modes, as a function of the angle

_{s}*α*for two different pump beam widths. We observe that the OAM correlations of the two-photon state change along the down-conversion cone due to the azimuthal symmetry breaking induced by the spatial walk-off. This implies that the correlations between OAM modes do not follow the relationship

*l*=

_{p}*l*+

_{s}*l*. From Fig. 3 it is clearly observed that for larger pump beams the azimuthal changes are smoothed out, since in this case the non-collinear and walk-off lengths are much larger than the crystal length.

_{i}*w*

_{0}= 100

*μ*m, and Figs. 3(e)and3(f) for

*w*

_{0}= 600

*μ*m, for

*α*= 0,90°. Notice that the weight of the

*l*= 0 mode is maximum for

_{s}*α*= 90°, which therefore is the optimum angle for the generation of heralded single photons with a gaussian-like shape. This effect can be clearly observed in Figs. 2(b), 3(d) and 3(f). On the contrary, for

*α*= 270°, the combined effects of the noncollinear and walk off effects make the weight of the

*l*= 0 mode to obtain its minimum value. This is of relevance in any quantum information protocol where the generated photons, no matter the degree of freedom where the quantum information is encoded, are to be coupled into single mode fibers.

_{s}*K*= 1/

*Trρ*

^{2}

_{s}, where

*ρ*=

_{s}*Tr*|Ψ〉 〈Ψ|, is the density matrix that describe the quantum state of the signal photon, after tracing out the spatial variables corresponding to the idler photon. The Schmidt number [6

_{i}6. C. K. Law and J. H. Eberly, “Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion,” Phys. Rev. Lett. **92**, 127903 (2004). [CrossRef] [PubMed]

*K*= 1 corresponding to a product state, while larger values of

*K*corresponds to increasingly larger values of the degree of entanglement. The degree of entanglement is maximum for

*α*= 0, and minimum for

*α*= 90 °, as shown in Fig. 4(a). The degree of entanglement is known to decrease with increasing filtering [18

18. M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, “Effect of spatial filtering on the Schmidt decomposition of entangled photons,” Phys. Rev. A **74**, 012309 (2006). [CrossRef]

*w*, as shown in Fig. 4(b), and to increase for larger values of the pump beam width (

_{s}*w*

_{0}).

## 4. Effects on the generation of polarization entanglement

10. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A **60**, R773 (1999). [CrossRef]

4. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat , “Generation of Hyperentangled Photon Pairs,” Phys. Rev. Lett. **95**, 260501 (2005). [CrossRef]

_{1}(

**p**,

**q**) = Φ(

*α*= 0,

**p**,

**q**) exp (

*ip*tan

_{y}*ρ*+

_{s}L*iq*tan

_{y}*ρ*) describes the spatial shape of the photons generated in the first nonlinear crystal,

_{i}L*ρ*are the spatial walk-off angles of the down-converted photons traversing the second nonlinear crystal, and Φ

_{s,i}_{2}(

**p**,

**q**) = Φ(

*α*= 90°,

**p**,

**q**) corresponds to the photons generated in the second nonlinear crystal. The quantum state in the polarization space is obtained tracing out the spatial variables, i.e.,

*ρ*=

_{p}*Tr*|Ψ〉 〈Ψ|, which gives

_{s}*ξ*= ∫

*d*

**p**

*d*

**q**Φ

_{1}(

**p**,

**q**) Φ

_{2}

^{*}(

**p**,

**q**).

*P*) of the quantum state given by Eq. (3), which writes

*P*= 1/2 (1 + |

*ξ*|

^{2}). The concurrence of the polarization-entangled state, which writes writes

*C*= |

*ξ*|, quantifies the quality of the polarization entangled state. Figure 5 shows the concurrence of the quantum state for two different crystal lengths. If spatial walk-off effects are negligible, |

*ξ*| = 1 and spatial and polarization variables can be separated. Therefore, both the purity and the concurrence are equal to 1. This is the case shown in Fig. 5 for a crystal length of

*L*= 0.5 mm. Notwithstanding, this is not generally he case, as demonstrated above.

## 5. Conclusions

## References and links

1. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. |

2. | A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum,” Phys. Rev. Lett. |

3. | G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, “ Experimental Quantum Coin Tossing,” Phys. Rev. Lett. |

4. | J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat , “Generation of Hyperentangled Photon Pairs,” Phys. Rev. Lett. |

5. | J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A |

6. | C. K. Law and J. H. Eberly, “Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion,” Phys. Rev. Lett. |

7. | H. H. Arnaut and G. A. Barbosa, “Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion,” Phys. Rev. Lett. |

8. | A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature |

9. | P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, “How focused pumping affects type-II spontaneous parametric down-conversion,” Phys. Rev. A |

10. | P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, “Ultrabright source of polarization-entangled photons,” Phys. Rev. A |

11. | J. Altepeter, E. Jeffrey, and P. Kwiat, “Phase-compensated ultra-bright source of entangled photons,” Opt. Express , |

12. | J. P. Torres, G. Molina-Terriza, and L. Torner, “The spatial shape of entangled photon states generated in non-collinear, walking parametric downconversion,” J. Opt. B: Quantum Semiclass. Opt. |

13. | A. Migdall, “Polarization directions of noncollinear phase-matched optical parametric downconversion output,” J. Opt. Soc. Am. B , |

14. | M. H. Rubin, “Transverse correlation in optical spontaneous parametric down-conversion,” Phys. Rev. A |

15. | G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych, and J. P. Torres, “Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion,” Phys. Rev. A |

16. | M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe, and S. P. Kulik, “Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion,” Phys. Rev. Lett. |

17. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum,” Phys. Rev. Lett. |

18. | M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, “Effect of spatial filtering on the Schmidt decomposition of entangled photons,” Phys. Rev. A |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(270.0270) Quantum optics : Quantum optics

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 17, 2007

Revised Manuscript: October 12, 2007

Manuscript Accepted: October 12, 2007

Published: October 22, 2007

**Citation**

Clara I. Osorio, Gabriel Molina-Terriza, Blanca G. Font, and Juan P. Torres, "Azimuthal distinguishability of entangled photons generated in spontaneous parametric down-conversion," Opt. Express **15**, 14636-14643 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14636

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### References

- G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007). [CrossRef]
- A. Vaziri, J. Pan, T. Jennewein, G. Weihs and A. Zeilinger, "Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum," Phys. Rev. Lett. 91, 227902 (2003). [CrossRef] [PubMed]
- G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, " Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005). [CrossRef] [PubMed]
- J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005). [CrossRef]
- J. P. Torres, A. Alexandrescu and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68050301(R) (2003). [CrossRef]
- C. K. Law and J. H. Eberly, "Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion," Phys. Rev. Lett. 92, 127903 (2004). [CrossRef] [PubMed]
- H. H. Arnaut and G. A. Barbosa, "Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion," Phys. Rev. Lett. 85, 286 (2000). [CrossRef] [PubMed]
- A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001). [CrossRef] [PubMed]
- P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, "How focused pumping affects type-II spontaneous parametric down-conversion," Phys. Rev. A 72, 033803 (2005). [CrossRef]
- P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999). [CrossRef]
- J. Altepeter, E. Jeffrey, and P. Kwiat, "Phase-compensated ultra-bright source of entangled photons," Opt. Express, 13, 8951-8959(2005). [CrossRef] [PubMed]
- J. P. Torres, G. Molina-Terriza and L. Torner, "The spatial shape of entangled photon states generated in noncollinear, walking parametric downconversion," J. Opt. B: Quantum Semiclass. Opt. 7, 235-239 (2005). [CrossRef]
- A. Migdall, "Polarization directions of noncollinear phase-matched optical parametric downconversion output," J. Opt. Soc. Am. B, 141093-1098 (1997). [CrossRef]
- M. H. Rubin, "Transverse correlation in optical spontaneous parametric down-conversion," Phys. Rev. A 54, 5349 (1996). [CrossRef] [PubMed]
- G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych and J. P. Torres, "Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion," Phys. Rev. A 72, 065802 (2005). [CrossRef]
- M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe and S. P. Kulik, "Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 99, 063901 (2007). [CrossRef] [PubMed]
- G. Molina-Terriza, J. P. Torres and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]
- M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, "Effect of spatial filtering on the Schmidt decomposition of entangled photons," Phys. Rev. A 74, 012309 (2006). [CrossRef]

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