Violation of a Bell Inequality Using Fractional Fourier

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 DF/UFMG CH Monken

Twin Photon Generator

Transverse Momentum Correlations

Motivation

Typical Bell-inequality experiment

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

Bell inequalities Bell States CHSH inequalities: Binary base Polarization

Non-separability

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

Mancini´s inequality

Duan´s inequality

Dichotomization of transverse momentum

$The fractional Fourier transform Integral form When = p/2 Ordinary Fourier transform$

The fractional Fourier transform Integral form When = p/2 Ordinary Fourier transform

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.

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).

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 timefrequency representations, " IEEE Trans. Sig. Processing 42 (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.

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. Processing 49 (8), 1638– 1655 (2001). D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications, " SIAM Review 33, 389 -404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT. )

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.

Fractional momentum analyzer FFT analyzer

Fractional momentum analyzer

Rotations in momentum space Angular spectrum Transverse amplitude

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

Analogy with Polarization

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

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

Results

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

Discussion: positivity of the Wigner function

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), ar. Xiv: 0708. 0653 v 1 [quant-ph].

Aditivity of the FFT

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:

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

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

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

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

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

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