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Violation of a Bell Inequality Using Fractional Fourier Transforms

Violation of a Bell Inequality Using Fractional Fourier Transforms. Paulo H S Ribeiro Instituto de Física - UFRJ. Paraty August-2007. Quantum Optics Group at IF - UFRJ. Author list. IF/UFRJ DS Tasca MP Almeida SP Walborn PH Souto Ribeiro. Univ. Brétagne-Sud P Pellat-Finett.

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Violation of a Bell Inequality Using Fractional Fourier Transforms

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  1. Violation of a Bell Inequality Using Fractional Fourier Transforms Paulo H S Ribeiro Instituto de Física - UFRJ Paraty August-2007

  2. Quantum Optics Group at IF - UFRJ

  3. Author list IF/UFRJDS Tasca MP Almeida SP Walborn PH Souto Ribeiro Univ. Brétagne-SudP Pellat-Finett DF/UFMGCH Monken

  4. Twin Photon Generator

  5. Transverse Momentum Correlations

  6. Motivation

  7. Motivation

  8. Typical Bell-inequality experiment

  9. Bell inequality:Dichotomic degree of freedom Dichotomic degree of freedom: 2 orthogonal axis Exemple: polarization of light Probabilities

  10. Bell inequalitiesBell States CHSH inequalities: Binary base Polarization

  11. Non-separability

  12. Non-separability

  13. Non-separability

  14. EPR correlations between position and momentum of the photons S. Mancini, V. Giovannetti, D. Vitali, and P. TombesiPhys. Rev. Lett. 88, 120401 (2002). Lu-Ming Duan, G. Giedke, J. I. Cirac, and P. Zoller Phys. Rev. Lett. 84, 2722 (2002).

  15. Mancini´s inequality

  16. Mancini´s inequality

  17. Duan´s inequality

  18. Dichotomization of transverse momentum

  19. The fractional Fourier transform Integral form When a = p/2  Ordinary Fourier transform

  20. Optical implementation Optical implementation of a Fourier transform. D e ZF are conjugate planes. Imaging of the field in plane D onto plane ZI with unit magnification. Optical implementation of a Fractional Fourier transform of arbitrary order f.

  21. The Fractional Fourier Transform First works related: N. Wiener. Hermitian polynomials and fourier analysis. J. Math. Phys. MIT, 8:70–73, 1929. E. U. Condon. Immersion of the fourier transform in a continuous of functional transformations. Proc. Nat. Acad. Sc. USA, 23:158–164, 1937. A. L. Patterson. Zeits. Kristal, 112:22–32, 1959. First application: Method for solving partial diferential equations 1980; V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," J. Inst. Appl. Math.25, 241–265 (1980).

  22. The Fractional Fourier Transform Application in physics and engeneering: R. S. Khare. Fractional fourier analysis of defocused images. Opt. Comm.,12:386–388, 1974. A. W. Lohmann, "Image rotation, Wigner rotation and the fractional Fourier transform," J. Opt. Soc. Am.A 10, 2181–2186 (1993). Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Sig. Processing42 (11), 3084–3091 (1994). Haldun M. Ozaktas, Zeev Zalevsky and M. Alper Kutay. "The Fractional Fourier Transform with Applications in Optics and Signal Processing". John Wiley & Sons (2001). Series in Pure and Applied Optics.

  23. The Fractional Fourier Transform Application in physics and engeneering: Soo-Chang Pei and Jian-Jiun Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Sig. Processing49 (8), 1638–1655 (2001). D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review33, 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)

  24. The Fractional Fourier Transform Applications in quantum optics; Yangjian Cai, Qiang Lin, and Shi-Yao Zhu; Coincidence fractional Fourier transform with entangled photon pairs and incoherent light;Appl. Phys. Lett. 86, 021112 (2005) Fei Wang, Yangjian Cai and Sailing He; Experimental observation of coincidence fractional Fourier transform with a partially coherent beam;Opt. Exp., 16, 6999(2006) Violation of Bell inequalities using the fractional momentum of the photon.

  25. Fractional momentum analyzer FFT analyzer

  26. Fractional momentum analyzer

  27. Rotations in momentum space Angular spectrum Transverse amplitude

  28. Clauser-Horne-Shimony-Holt(CHSH) inequality C(,) is the coincidence counting rate: Maximal violation:

  29. Analogy with Polarization

  30. Experimental set-up DS Tasca et al. arXiv:quant-ph/0605061- Experiment SP Walborn, et al. arXiv:quant-ph/0612141- Theory

  31. Experimental set-up FFT orders for the maximal violation Implemented FFT orders

  32. Results

  33. Discussion: positivity of the Wigner function Expectation value for the observable F:

  34. Discussion: positivity of the Wigner function

  35. Conclusion • We need to show that this negativity is actually responsible for the violation! • Anyway a qubit can be buildt with dichotomized transverse • momentum of the photon - A. F. Abouraddy et al., Phys. Rev. A • 75, 052114 (2007), arXiv:0708.0653v1 [quant-ph] .

  36. Aditivity of the FFT

  37. Advanced wave picture position-position (correlation) Max. Count rate: Min. Count rate: position-momentum (no correlation) momentum-momentum (anti-correlation) Min. Count rate: Equal: Max. Count rate:

  38. FFT – Ray optics Matrix representation of a paraxial ray: Paraxial aproximation: small angles with the paraxial direction z. Paraxial free propagation: Thin lens refraction:

  39. FFT – Ray optics • Optical system: PROPAGATION + LENS + PROPAGATION

  40. FFT – Ray optics Fourier plane • Fourier Transform (z = f): Fourier ray: Fourier ray scaled coordinate:

  41. FFT – Ray optics Image plane • Imaging system Image ray: Aditivity: Scaled coordinate of the image ray:

  42. FFT – Ray optics • Arbitrary rotations: Fractional Fourier Transform Fracional focal length:

  43. FFT – Ray optics In the paraxial approximation: Mapping the transverse momentum Fractional momentum of order f Image

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