Abstract
In this paper, we connect our approach of factoring numbers using the continuous truncated Gauss sum (Wölk et al., J. Mod. Optic, 2009) with the phenomenon of quantum carpets. In particular, we demonstrate that the degree of degeneracy of the ratio ℓ ∕ N translates into a crossing of the canals and ridges contained in the design of quantum carpets. In this way, quantum carpets represent an experimental implementation of our idea of factorization with degeneracies.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bartlett J, Kaplan J (2002) Bartlett’s familiar quotations. Little, Brown and Company, Boston
Middleton D (1960) An introduction to statistical communication theory. McGraw-Hill, New York
Berry M, Marzoli I, Schleich WP (2001) Quantum carpets, carpets of light. Phys World 14:39
Kaplan AE, Marzoli I, Lamb WE Jr., Schleich WP (2000) Multimode interference: highly regular pattern formation in quantum wave packet evolution. Phys Rev A 61:032101
Marzoli I, Saif F, Bialynicki-Birula I, Friesch OM, Kaplan AE, Schleich WP (1998) Quantum Carpets made simple. Acta Physica Slovaca 48:323
For a recent overview see for example Marzoli I, Kaplan AE, Saif F, Schleich WP (2008) Quantum carpets of a slightly relativistic particle. Fortschr. d. Physik 56:967
Nowak S, Kurtsiefer Ch., Pfau T, David C (1997) High-order Talbot fringes for atomic matter waves. Opt Lett 22:1430
Ahn J, Hutchinson DN, Rangan C, Bucksbaum PH (2001) Quantum phase retrivial of a Rydberg wave packet using a half-cycle pulse. Phys Rev Lett 86:1179
Katsuki H, Chiba H, Meier Ch., Girard B, Ohmori K (2009) Actively tailored spatiotemporal images of quantum interference on the picometer and femtosecond scales. Phys Rev Lett 102:103602; see also the excellent review by Ohmori K (2009); Wave-packet and coherent dynamics. Annu Rev Phys Chem 60:487
Deng L, Hagley EW, Denschlag J, Simsarian JE, Edwards M, Clark CW, Helmerson K, Roston SL, Philipps WD (1999) Temporal, matter-wave-dispersion Talbot effect. Phys Rev Lett 83:5407
Gustavsson M, Haller E, Mark MJ, Danzl JG, Hart R, Daley AJ, Nägerl H-C (2010) Interference of interacting matter waves. New J Phys 12:065029
Berry MV, Klein S (1996) Integer, fractional and fractal Talbot effect. J Mod Opt 43:2139
Case WB, Tomandl M, Deachapunya S, Arndt M (2009) Realization of optica carpets in the Talbot and Talbot-Lau configuration. Opt Express 17:20966
See for example Maier H, Schleich WP (2012) Prime numbers 101: A primer on number theory. Wiley-VCH, New York
Shor P (1994) In: Goldwasser S (ed) Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, IEEE Computer Society Press, New York, pp. 124–134, ; for an elementary introduction to the Shor algorithm see for example Mack R, Schleich WP, Haase D, Maier H (2009) In: Arendt W, Schleich WP (eds.) Mathematical Analysis of Evolution, Information and Complexity. Wiley-VCH, Berlin
Mack H, Bienert M, Haug F, Straub FS, Freyberger M, Schleich WP (2002) In: Mataloni P, De Martini F (eds.) Experimental quantum computation and information. Elsevier, Amsterdam
Wölk S, Merkel W, Schleich WP, Averbukh ISh, Girard B (2011) Factorization of numbers with Gauss sums: I. mathematical background. New J Phys 13:103007
See for example Mehring M, Müller K, Averbukh ISh, Merkel W, Schleich WP (2007) NMR experiment factors numbers with Gauss sums. Phys Rev Lett 98:120502; Mahesh T, Rajendran N, Peng X, Suter D (2007) Factoring numbers with the Gauss sum technique: NMR implementations. Phys Rev A 75:062303; Gilowski M, Wendrich T, Müller T, Jentsch C, Ertmer W, Rasel EM, Schleich WP (2008) Gauss sum Factorization with cold atoms. Phys Rev Lett 100:030201; Sadgrove M, Kumar S, Nakagawa K (2008) Enhanced factoring with a Bose-Einstein condensate. Phys Rev Lett 101:180502; Tamma V, Zhang H, He X, Garruccio A, Schleich WP, Shih Y (2011) Factoring numbers with a single interferogram. Phys Rev A 83:020304
Wölk S, Feiler C, Schleich WP (2009) Factorization of numbers with truncated Gauss sums at rational arguments. J Mod Opt 56:2118
Schleich WP (2001) Quantum optics in phase space. Wiley-VCH, Weinheim
Harter W (2001) Quantum-fractal revival structure in CN quadratic spectra: Base-N quantum computer registers. Phys Rev A 64:012312
Acknowledgment
We are grateful to M. Jakob, K.A.H. van Leuwwen, M. Štefańǎk, and M.S. Zubairy for many fruitful discussions on this topic. In this context, one of us (WPS) appreciates the inspiring discussions at the University of Vienna in the summer of 2009 with W. Case and M. Tomandl. This research was partially supported by the Max Planck Prize of WPS awarded by the Humboldt Foundation and the Max Planck Society. Moreover, WPS expresses his sincere thanks to the organizers L. Cohen and M.O. Scully of the Middleton Festival in Princeton 2007 for a most stimulating conference.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Wölk, S., Schleich, W.P. (2012). Quantum Carpets: Factorization with Degeneracies. In: Cohen, L., Poor, H., Scully, M. (eds) Classical, Semi-classical and Quantum Noise. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6624-7_18
Download citation
DOI: https://doi.org/10.1007/978-1-4419-6624-7_18
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-6623-0
Online ISBN: 978-1-4419-6624-7
eBook Packages: EngineeringEngineering (R0)