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Implementation of quantum and classical discrete fractional Fourier transforms
Abstract
Fourier transforms, integer and fractional, are ubiquitous mathematical tools in basic and applied science. Certainly, since the ordinary Fourier transform is merely a particular case of a continuous set of fractional Fourier domains, every property and application of the ordinary Fourier transform becomes a special case of the fractional Fourier transform. Despite the great practical importance of the discrete Fourier transform, implementation of fractional orders of the corresponding discrete operation has been elusive. Here we report classical and quantum optical realizations of the discrete fractional Fourier transform. In the context of classical optics, we implement discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem. Moreover, we apply this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions. The proposed approach is versatile and could find applications in various fields where Fourier transforms are essential tools.
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Such bi-partite quantum walks give rise to interesting quantum correlation patterns.
“that gives rise to interesting quantum correlation patterns [157,158]”
From Two-photon interference: the Hong–Ou–Mandel effect (2021) · cites this paper as [157]
The computational complexity of BosonSampling is rooted in the hardness of evaluating the permanent in the scattering amplitudes.
“The computational complexity of the problem is rooted in the hardness [562] of evaluating the permanent in the scattering amplitudes [563]”
From Photonic quantum information processing: a review (2019) · cites this paper as [562]
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An algorithm for the machine calculation of complex Fourier series
Cooley, J. W., & Tukey, J. W. (1965) · Mathematics of Computation · doi:10.1090/s0025-5718-1965-0178586-1 · Cited by 13,154
Measurement of subpicosecond time intervals between two photons by interference
Hong, C. K., Ou, Z. Y., & Mandel, L. (1987) · Physical Review Letters · doi:10.1103/physrevlett.59.2044 · Cited by 4,369
Discretizing light behaviour in linear and nonlinear waveguide lattices
Christodoulides, D. N., Lederer, F., & Silberberg, Y. (2003) · Nature · doi:10.1038/nature01936 · Cited by 1,692
Self-citationDiscrete optics in femtosecond-laser-written photonic structures
Szameit, A., & Nolte, S. (2010) · Journal of Physics B: Atomic, Molecular and Optical Physics · doi:10.1088/0953-4075/43/16/163001 · Cited by 610 · cited 2×
“1⁄4 i ð Þ u q ð Þ p : ð21Þ Equation (21) shows that at Z1⁄4p/2 the point-spread function becomes proportional to the amplitude of the corresponding eigenstates depending on the excited site.”
Self-citationQuantum and Classical Correlations in Waveguide Lattices
Bromberg, Y., Lahini, Y., Morandotti, R., & Silberberg, Y. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.253904 · Cited by 318 · cited 1×
“27).”
Laser written waveguide photonic quantum circuits
Marshall, G. D., Politi, A., Matthews, J. C. F., Dekker, P., Ams, M., Withford, M. J., & O'Brien, J. L. (2009) · Optics Express · doi:10.1364/oe.17.012546 · Cited by 308
Experimental observation of N00N state Bloch oscillations
Lebugle, M., Gräfe, M., Heilmann, R., Perez-Leija, A., Nolte, S., & Szameit, A. (2015) · Nature Communications · doi:10.1038/ncomms9273 · Cited by 60
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Research works printed but not linked: 30
- Namias, V. The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Maths. Applics 25, 241–265 (1980). · DOI · Paper page
- Almeida, L. B. The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084–3091 (1994). · DOI · Paper page
- Ozaktas, H. M. & Mendlovic, D. Fractional Fourier Optics. J. Opt. Soc. Am. A 12, 743–751 (1995). · DOI
- Ozaktas, H. M., Zalevsky, Z. & Kutay, M. A. The Fractional Fourier Transform with Applications in Optics and Signal Processing Wiley (2001). · DOI · Paper page
- Lohmann, A. W. & Soffer, B. H. Relationship between the Radon–Wigner and fractional Fourier transforms. J. Opt. Soc. Am. A 11, 1798–1801 (1994). · DOI · Paper page
- Man’ko, M. A. Fractional Fourier transform in information processing, tomography of optical signal, and green function of harmonic oscillator. J. of Russ. Laser Research 20, 226–228 (1999). · DOI
- Dorsch, R. G., Lohmann, A. W., Bitran, Y., Mendlovic, D. & Ozaktas, H. M. Chirp filtering in the fractional Fourier domain. Appl. Opt. 33, 7599–7602 (1994). · DOI
- Lohmann, A. W. & Mendlovic, D. Fractional Fourier transform: photonic implementation. Appl. Opt. 33, 7661–7664 (1994). · DOI
- Lohmann, A. W. Image rotation, Wigner rotation, and the Fractional Fourier transform. J. Opt. Soc. Am. A 10, 2181–2186 (1993). · DOI · Paper page
- Mendlovic, D. & Ozaktas, H. Fractional Fourier transforms and their optical implementation: I. J. Opt. Soc. Am. A 10, 1875–1881 (1993). · DOI · Paper page
- Atakishiyev, N. M. & Wolf, K. B. Fractional Fourier-Kravchuk transform. J. Opt. Soc. Am. A 14, 1467–1477 (1997). · DOI
- Tichy, M. C., Mayer, K., Buchleitner, A. & Molmer, K. Stringent and efficient assessment of Boson Sampling Devices. Phys. Rev Lett. 113, 020502–020506 (2014). · DOI · Paper page
- Perez-Leija, A., Keil, R. & Moya-Cessa, H. Perfect transfer of path-entangled photons in J x-photonic lattices, A. Szameit, and D. N. Christodoulides. Phys. Rev. A 87, 022303 (2013). · DOI
- Humphreys, P. C., Barbieri, M., Datta, A. & Walmsley, I. A. Quantum enhanced multiple phase estimation. Phys. Rev. Lett. 111, 070403 (2013). · DOI · Paper page
- Perez-Leija, A. et al. Eigenstate-assisted longitudinal quantum state transfer and qubit-storage in photonic and spin lattices, 45-th annual meeting of the APS Division of Atomic. Molecular and optical Physics 59, J4.010 (2014).
- Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated Quantum computation of molecular energies. Science 309, 1704–1707 (2005). · DOI · Paper page
- Wolf, K. B. Integral transforms in science and engineering Vol. 11, Mathematical Concepts and Methods for Science and Engineering (1979). · DOI
- Narducci, L. M. & Orzag, M. Eigenvalues and Eigenvectors of angular momentum operator J x without theory of rotations. Am. J. of Phys 40, 1811–1814 (1972). · DOI
- Lieb, E., Schultz, T. & Matts, D. Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961). · DOI
- Cormick, C., Bermudez, A., Huelga, S. F. & Plenio, M. Preparation of the ground state of a spin chain by dissipation in a structured environment. New J. Phys. 15, 073027 (2013). · DOI
- Bobrov, I. B., Kalashnikov, D. A. & Krivitsky, L. A. Imaging of spatial correlations of two-photon states. Phys. Rev. A 89, 043814 (2014). · DOI
- Peeters, W. H., Renema, J. J. & van Exter, M. P. Engineering of two-photon spatial quantum correlations behind a double slit. Phys. Rev. A 79, 043817 (2009). · DOI
- Poem, E., Gilead, Y., Lahini, Y. & Silberberg, Y. Fourier processing of quantum light. Phys. Rev. A 86, 023836 (2012). · DOI
- Wiseman, H. M. & Milburn, G. J. Quantum Measurement and control Cambridge University Press (2010). · DOI
- Orlowski, A. & Paul, H. Phase retrieval in quantum mechanics. Phys. Rev. A 50, R921–R924 (1994). · DOI
- Wood, J. & Barry, D. T. Radon transformation of time-frequency distributions for analysis of multicomponent signals. IEEE Transac. Signal Process. 42, 3166–3177 (1994). · DOI · Paper page
- Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847 (1989). · DOI
- Raymer, M. G., Beck, M. & McAlister, D. F. Quantum Optics VI eds Harvey J. D., Wall D. F. Springer-Verlag (1994).
- Ozaktas, H. M. & Mendlovic, D. Fractional Fourier transforms and their optical implementations. J. Opt. Soc. Am. A 10, 2522–2531 (1993). · DOI · Paper page
- Marhic, M. E. Roots of the identity operator and optics. J. Opt. Soc. Am. A 12, 1448 (1995). · DOI
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Bouchard, F., Sit, A., Zhang, Y., Fickler, R., Miatto, F. M., Yao, Y., Sciarrino, F., & Karimi, E. (2021) · Reports on Progress in Physics · doi:10.1088/1361-6633/abcd7a · Cited by 232
Tapering of femtosecond laser-written waveguides
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Self-citationTunable Generation of Spatial Entanglement in Nonlinear Waveguide Arrays
Raymond, A., Zecchetto, A., Palomo, J., Morassi, M., Lemaître, A., Raineri, F., Amanti, M. I., Ducci, S., & Baboux, F. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.233602 · Cited by 11
Tailoring quantum walks in integrated photonic lattices
Raymond, A., Cathala, P., Morassi, M., Lemaître, A., Raineri, F., Ducci, S., & Baboux, F. (2025) · Optics Express · doi:10.1364/oe.571522 · Cited by 1
“Indeed, discrete fractional Fourier transforms can be realized in waveguide arrays through engineered coupling profiles that mimic the J x operator algebra [48–50].”
Metadata
- Authors
- Weimann, S., Perez-Leija, A., Lebugle, M., Keil, R., Tichy, M., Gräfe, M., Heilmann, R., Nolte, S., Moya-Cessa, H., Weihs, G., Christodoulides, D. N., & Szameit, A.
- Venue
- Nature Communications
- Published
- 2016-03-23
- Volume
- 7
- Issue
- 1
- ISSN
- 2041-1723
- PMC
- PMC4814576
- Cited by
- 114
- References
- 7
- Language
- en
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