Abstract

We report on prime number decomposition by use of the Talbot effect, a well-known phenomenon in classical near field optics whose description is closely related to Gauss sums. The latter are a mathematical tool from number theory used to analyze the properties of prime numbers as well as to decompose composite numbers into their prime factors. We employ the well-established connection between the Talbot effect and Gauss sums to implement prime number decompositions with a novel approach, making use of the longitudinal intensity profile of the Talbot carpet. The new algorithm is experimentally verified and the limits of the approach are discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  6. O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
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    [Crossref] [PubMed]
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    [Crossref]
  9. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
    [Crossref]
  10. S. Nowak, C. Kurtsiefer, T. Pfau, and C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett. 22, 1430–1432 (1997).
    [Crossref]
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    [Crossref] [PubMed]
  12. S. Cherukulappurath, D. Heinis, J. Cesario, N. F. van Hulst, S. Enoch, and R. Quidant, “Local observation of plasmon focusing in Talbot carpets,” Opt. Express 17, 23772–23784 (2009).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  19. V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
    [Crossref]
  20. V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
    [Crossref]
  21. M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
    [Crossref] [PubMed]
  22. M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
    [Crossref] [PubMed]
  23. M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
    [Crossref] [PubMed]
  24. T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
    [Crossref]
  25. X. Peng and D. Suter, “NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors,” Europhys. Lett. 84, 40006 (2008).
    [Crossref]
  26. M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
    [Crossref]
  27. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17, 20966–20974 (2009).
    [Crossref] [PubMed]
  28. M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
    [Crossref]
  29. M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
    [Crossref]
  30. V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
    [Crossref]
  31. W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
    [Crossref]

2012 (1)

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

2011 (4)

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

2009 (4)

S. Cherukulappurath, D. Heinis, J. Cesario, N. F. van Hulst, S. Enoch, and R. Quidant, “Local observation of plasmon focusing in Talbot carpets,” Opt. Express 17, 23772–23784 (2009).
[Crossref]

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express 17, 20966–20974 (2009).
[Crossref] [PubMed]

2008 (5)

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

X. Peng and D. Suter, “NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors,” Europhys. Lett. 84, 40006 (2008).
[Crossref]

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
[Crossref] [PubMed]

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

2007 (4)

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

M. R. Dennis, N. I. Zheludev, and F. J. G. de Abajo, “The plasmon Talbot effect,” Opt. Express 15, 9692–9700 (2007).
[Crossref] [PubMed]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

2001 (1)

M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
[Crossref]

2000 (1)

O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
[Crossref]

1998 (1)

1997 (3)

1996 (2)

J. F. Clauser and J. P. Dowling, “Factoring integers with Young’s N-slit interferometer,” Phys. Rev. A 53, 4587–4590 (1996).
[Crossref] [PubMed]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

1995 (1)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

1989 (1)

1967 (1)

1881 (1)

L. Rayleigh, “On copying diffraction–gratings, and some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Alley, C. O.

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

Arndt, M.

Averbukh, I. Sh.

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

Banaszek, K.

Baruchel, J.

Berry, M.

M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
[Crossref]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

Bigourd, D.

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

Case, W. B.

Cesario, J.

Chapman, M. S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Chatel, B.

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

Cherukulappurath, S.

Clauser, J. F.

J. F. Clauser and J. P. Dowling, “Factoring integers with Young’s N-slit interferometer,” Phys. Rev. A 53, 4587–4590 (1996).
[Crossref] [PubMed]

Cloetens, P.

Cronin, A. D.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
[Crossref]

David, C.

de Abajo, F. J. G.

De Martino, C.

Deachapunya, S.

Dennis, M. R.

Dowling, J. P.

J. F. Clauser and J. P. Dowling, “Factoring integers with Young’s N-slit interferometer,” Phys. Rev. A 53, 4587–4590 (1996).
[Crossref] [PubMed]

Ekstrom, C. R.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Enoch, S.

Ertmer, W.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Friesch, O.

O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
[Crossref]

Garuccio, A.

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

Gilowski, M.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Girard, B.

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

Guigay, J. P.

Haase, D.

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

Hammond, T. D.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

He, X.

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

Heinis, D.

Jentsch, Ch.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

Kumar, S.

M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
[Crossref] [PubMed]

Kurtsiefer, C.

Liu, L.

Mahesh, T. S.

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

Maier, H.

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

Marzoli, I.

M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
[Crossref]

O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
[Crossref]

McMorran, B. J.

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
[Crossref]

Mehring, M.

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

Merkel, W.

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

Montgomery, W. D.

Müller, K.

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

Müller, T.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Nakagawa, K.

M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
[Crossref] [PubMed]

Nowak, S.

Paulus, G. G.

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

Peng, X.

X. Peng and D. Suter, “NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors,” Europhys. Lett. 84, 40006 (2008).
[Crossref]

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

Pfau, T.

Pritchard, D. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Quidant, R.

Rajendran, N.

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

Rasel, E. M.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Rayleigh, L.

L. Rayleigh, “On copying diffraction–gratings, and some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[Crossref]

Sadgrove, M.

M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
[Crossref] [PubMed]

Schleich, W. P.

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
[Crossref]

O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
[Crossref]

K. Banaszek, K. Wódkiewicz, and W. P. Schleich, “Fractional Talbot effect in phase space: A compact summation formula,” Opt. Express 2, 169–172 (1998).
[Crossref] [PubMed]

Schlenker, M.

Schmiedmayer, J.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Shih, Y.

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

Shih, Y. H.

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

Shor, P. W.

P. W. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,” SIAM Journal on Computing 26, 1484–1509 (1997).
[Crossref]

Štefanák, M.

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

Suter, D.

X. Peng and D. Suter, “NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors,” Europhys. Lett. 84, 40006 (2008).
[Crossref]

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Tamma, V.

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

Tannian, B. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Tomandl, M.

van Hulst, N. F.

Wehinger, S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

Wendrich, T.

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

Wódkiewicz, K.

Wölk, S.

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

Zhang, H.

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

Zheludev, N. I.

Zubairy, M. S.

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

Appl. Opt. (1)

Europhys. Lett. (1)

X. Peng and D. Suter, “NMR implementation of factoring large numbers with Gauß sums: Suppression of ghost factors,” Europhys. Lett. 84, 40006 (2008).
[Crossref]

Found. Phys. (1)

V. Tamma, C. O. Alley, W. P. Schleich, and Y. H. Shih, “Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers,” Found. Phys. 42, 111–121 (2012).
[Crossref]

Int. J. Quantum Inf. (1)

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “Factorization of integers with multi-path optical interference,” Int. J. Quantum Inf. 9, 423–430 (2011).
[Crossref]

J. Mod. Opt. (2)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, and Y. Shih, “New factorization algorithm based on a continuous representation of truncated Gauss sums,” J. Mod. Opt. 56, 2125–2132 (2009).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

M. Štefaňák, D. Haase, W. Merkel, M. S. Zubairy, and W. P. Schleich, “Factorization with exponential sums,” J. Phys. A 41, 304024 (2008).
[Crossref]

New J. Phys. (5)

W. Merkel, S. Wölk, W. P. Schleich, I. Sh. Averbukh, B. Girard, and G. G. Paulus, “Factorization of numbers with Gauss sums: II. Suggestions for implementation with chirped laser pulses,” New J. Phys. 13, 103008 (2011).
[Crossref]

M. Štefaňák, W. Merkel, W. P. Schleich, D. Haase, and H. Maier, “Factorization with Gauss sums: scaling properties of ghost factors,” New J. Phys. 9, 370 (2007).
[Crossref]

S. Wölk, W. Merkel, W. P. Schleich, I. Sh. Averbukh, and B. Girard, “Factorization of numbers with Gauss sums: I. Mathematical background,” New J. Phys. 13, 103007 (2011).
[Crossref]

O. Friesch, I. Marzoli, and W. P. Schleich, “Quantum carpets woven by Wigner functions,” New J. Phys. 2, 4 (2000).
[Crossref]

B. J. McMorran and A. D. Cronin, “An electron Talbot interferometer,” New J. Phys. 11, 033021 (2009).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

L. Rayleigh, “On copying diffraction–gratings, and some phenomena connected therewith,” Philos. Mag. 11, 196–205 (1881).
[Crossref]

Phys. Rev. A (4)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near–field imaging of atom diffraction gratings: The atomic Talbot effect,” Phys. Rev. A 51, R14–R17 (1995).
[Crossref]

V. Tamma, H. Zhang, X. He, A. Garuccio, W. P. Schleich, and Y. Shih, “Factoring numbers with a single interferogram,” Phys. Rev. A 83, 020304 (2011).
[Crossref]

J. F. Clauser and J. P. Dowling, “Factoring integers with Young’s N-slit interferometer,” Phys. Rev. A 53, 4587–4590 (1996).
[Crossref] [PubMed]

T. S. Mahesh, N. Rajendran, X. Peng, and D. Suter, “Factorizing numbers with the Gauss sum technique: NMR implementations,” Phys. Rev. A 75, 062303 (2007).
[Crossref]

Phys. Rev. Lett. (4)

M. Gilowski, T. Wendrich, T. Müller, Ch. Jentsch, W. Ertmer, E. M. Rasel, and W. P. Schleich, “Gauss Sum Factorization with Cold Atoms,” Phys. Rev. Lett. 100, 030201 (2008).
[Crossref] [PubMed]

M. Sadgrove, S. Kumar, and K. Nakagawa, “Enhanced Factoring with a Bose-Einstein Condensate,” Phys. Rev. Lett. 101, 180502 (2008).
[Crossref] [PubMed]

M. Mehring, K. Müller, I. Sh. Averbukh, W. Merkel, and W. P. Schleich, “NMR Experiment Factors Numbers with Gauss Sums,” Phys. Rev. Lett. 98, 120502 (2007).
[Crossref] [PubMed]

D. Bigourd, B. Chatel, W. P. Schleich, and B. Girard, “Factorization of Numbers with the Temporal Talbot Effect: Optical Implementation by a Sequence of Shaped Ultrashort Pulses,” Phys. Rev. Lett. 100, 030202 (2008).
[Crossref] [PubMed]

Phys. World (1)

M. Berry, I. Marzoli, and W. P. Schleich, “Quantum carpets, carpets of light,” Phys. World 14, 39 (2001).
[Crossref]

SIAM Journal on Computing (1)

P. W. Shor, “Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer,” SIAM Journal on Computing 26, 1484–1509 (1997).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1 (a) Evaluation of |S15(l)|2. For l = 3 and l = k · 3, |S15(l)|2 evaluates to 3 15. For l = 5 and l = k · 5, |S15(l)|2 is equal to 5 15. (b) Evaluation of |S13(l)|2. Here |S13(l)|2 evaluates to 1 13 for every l since 13 is a prime number. Thus, all prime factors of a number N intersect the line connecting |SN (1)|2 and |SN (N)|2 giving rise to the name “line of factors”.
Fig. 2
Fig. 2 Experimental setup: Light of a Nd:Yag laser (λ = 532 nm) is coupled into a single mode fiber SMF. A collimation lens CF is placed at its focal length f behind the fiber, such that the beam waist of the resulting Gaussian beam is located at f behind the lens where the grating is located. A CMOS camera with 1280 × 1024 pixels of side length 5.2 μm and a 10× magnifying microscope objective is placed at a distance z behind the grating. The camera was mounted on a translation stage which can travel 150 mm along the z axis.
Fig. 3
Fig. 3 (a) Measured Talbot carpet for a grating with period d = 200 μm and a slit width w = 10 μm. The yellow line indicates the line along which the intensity is evaluated for the factorization algorithm. (b) Intensity along the yellow line as measured in the experiment (green line) and as evaluated theoretically (black dashed line) obtained by evaluation of the Fresnel diffraction integral with 250 slits and a Gaussian beam width of σ = 5100 μm measured by the knife-edge method. (c) Result of the proposed factorization algorithm using the experimental data. We see that it finds the correct prime factors 3 and 9 of N = 27.

Equations (11)

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S N ( l ) = m = w m exp ( i 2 π m 2 l N ) ,
S N ( l ) = 1 N m = 0 N 1 exp ( i 2 π m 2 l N ) = 1 N G ( l , N ) ,
T ( ξ ) = n = rect ( ξ n d w ) with rect ( x ) = { 1 for 1 / 2 < x < 1 / 2 , 0 else .
E ( x , z ) = exp ( i k z i π / 4 ) λ z T ( ξ ) E 0 exp { i k 2 z [ ( x ξ ) 2 ] } d ξ .
T ( ξ ) = m = A m exp ( i m 2 π d ξ ) with A m = w d sinc ( m w d ) .
E ( x , z ) = E 0 exp ( i k z ) m = A m exp [ i 2 π ( m x d + m 2 z L T ) ] ,
I ( x , z ) = I 0 | m = A m exp [ i 2 π ( m x d + m 2 z L T ) ] | 2 , with I 0 = E 0 2 .
I ( q d , z ) = I 0 | n = A n exp ( i 2 π n 2 z L T ) | 2 = I 0 | S N = L T ( l = z ) | 2 .
I ( q d , z ) = I 0 | S N = L T ( l = z ) | 2 = I 0 N 2 | G ( l = z , N = L T ) | 2 for N < 2 d w .
E ( x , y , z = 0 ) = E 0 exp ( x 2 + y 2 2 σ 2 ) .
I ( x , z ) = I 0 λ z | T ( ξ ) exp ( ξ 2 2 σ 2 ) exp { i π λ z [ ( x ξ ) 2 ] } d ξ | 2 .

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