Implementation of quantum and classical discrete fractional Fourier transforms

Nature Communications

Volumn 7, Issue , 2016, Pages

  Weimann, Steffen ^a   Perez Leija, Armando ^a   Lebugle, Maxime ^a   Keil, Robert ^b   Tichy, Malte ^c   Gräfe, Markus ^a   Heilmann, René ^a   Nolte, Stefan ^a   Moya Cessa, Hector ^d   Weihs, Gregor ^b   Christodoulides, Demetrios N ^e   Szameit, Alexander ^a  

^a FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

^b UNIVERSITY OF INNSBRUCK   (Austria)

^c AARHUS UNIVERSITY   (Denmark)

^d INSTITUTO NACIONAL DE ASTROFÍSICA   (Mexico)

^e UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIOPHYSICS; FOURIER TRANSFORM; NUMERICAL METHOD; PHYSICAL ACTIVITY; QUANTUM MECHANICS; WAVE ACTION;

FOURIER TRANSFORMATION; OPTICS;

EID: 84962261006     PISSN: None     EISSN: 20411723     Source Type: Journal    
DOI: 10.1038/ncomms11027     Document Type: Article

References (37)

1. Cooley, J. & Tukey, J. W. An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297-301 (1965).
2. Namias, V. The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Maths. Applics 25, 241-265 (1980).
3. Almeida, L. B. The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42, 3084-3091 (1994).
4. Ozaktas, H. M. & Mendlovic, D. Fractional Fourier Optics. J. Opt. Soc. Am. A 12, 743-751 (1995).
5. Ozaktas, H. M., Zalevsky, Z. & Kutay, M. A. The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
6. Lohmann, A. W. & Soffer, B. H. Relationship between the Radon-Wigner and fractional Fourier transforms. J. Opt. Soc. Am. A 11, 1798-1801 (1994).
7. 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).
8. 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).
9. Lohmann, A. W. & Mendlovic, D. Fractional Fourier transform: photonic implementation. Appl. Opt. 33, 7661-7664 (1994).
10. Lohmann, A. W. Image rotation, Wigner rotation, and the Fractional Fourier transform. J. Opt. Soc. Am. A 10, 2181-2186 (1993).
11. Mendlovic, D. & Ozaktas, H. Fractional Fourier transforms and their optical implementation: I. J. Opt. Soc. Am. A 10, 1875-1881 (1993).
12. Atakishiyev, N. M. & Wolf, K. B. Fractional Fourier-Kravchuk transform. J. Opt. Soc. Am. A 14, 1467-1477 (1997).
13. Tichy, M. C., Mayer, K., Buchleitner, A. & Molmer, K. Stringent and efficient assessment of Boson Sampling Devices. Phys. Rev Lett. 113, 020502-020506 (2014).
14. Perez-Leija, A., Keil, R. & Moya-Cessa, H. Perfect transfer of path-entangled photons in Jx-photonic lattices, A. Szameit, and D. N. Christodoulides. Phys. Rev. A 87, 022303 (2013).
15. Humphreys, P. C., Barbieri, M., Datta, A. & Walmsley, I. A. Quantum enhanced multiple phase estimation. Phys. Rev. Lett. 111, 070403 (2013).
16. 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).
17. Aspuru-Guzik, A., Dutoi, A. D., Love, P. J. & Head-Gordon, M. Simulated Quantum computation of molecular energies. Science 309, 1704-1707 (2005).
18. Wolf, K. B. Integral transforms in science and engineering Vol. 11 (Mathematical Concepts and Methods for Science and Engineering, 1979).
19. Narducci, L. M. & Orzag, M. Eigenvalues and Eigenvectors of angular momentum operator Jx without theory of rotations. Am. J. of Phys 40, 1811-1814 (1972).
20. Christodoulides, D. N., Lederer, F. & Silberberg, Y. Discretizing light behavior in linear and nonlinear waveguide lattices. Nature 424, 817-823 (2003).
21. Szameit, A. & Nolte, S. Discrete optics in femtosecond-laser-written photonic structures. J. Phys. B 43, 163001 (2010).
22. Lieb, E., Schultz, T. & Matts, D. Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407-466 (1961).
23. 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).
24. Bobrov, I. B., Kalashnikov, D. A. & Krivitsky, L. A. Imaging of spatial correlations of two-photon states. Phys. Rev. A 89, 043814 (2014).
25. 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).
26. Poem, E., Gilead, Y., Lahini, Y. & Silberberg, Y. Fourier processing of quantum light. Phys. Rev. A 86, 023836 (2012).
27. Bromberg, Y., Lahini, Y., Morandotti, R. & Silberberg, Y. Quantum and classical correlations in waveguide lattices. Phys. Rev. Lett. 102, 253904 (2009).
28. Marshall, G. D. et al. Laser written waveguide photonic quantum circuits. Opt. Exp. 17, 12546-12554 (2009).
29. Lebugle, M. et al. Experimental observation of N00N state Bloch oscillations. Nat. Commun. 6, 8273 (2015).
30. Hong, C. K., Ou, Z. Y. & Mandel, L. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59, 2044 (1987).
31. Wiseman, H. M. & Milburn, G. J. Quantum Measurement and control (Cambridge University Press, 2010).
32. Orlowski, A. & Paul, H. Phase retrieval in quantum mechanics. Phys. Rev. A 50, R921-R924 (1994).
33. Wood, J. & Barry, D. T. Radon transformation of time-frequency distributions for analysis of multicomponent signals. IEEE Transac. Signal Process. 42, 3166-3177 (1994).
34. Vogel, K. & Risken, H. Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847 (1989).
35. Raymer, M. G., Beck, M. & McAlister, D. F. Quantum Optics VI. (eds Harvey, J. D. & Wall, D. F. ) (Springer-Verlag, 1994).
36. Ozaktas, H. M. & Mendlovic, D. Fractional Fourier transforms and their optical implementations. J. Opt. Soc. Am. A 10, 2522-2531 (1993).
37. Marhic, M. E. Roots of the identity operator and optics. J. Opt. Soc. Am. A 12, 1448 (1995).