We examine the free time evolution of a rectangular one dimensional Schrödinger wave packet of constant phase during the early stage which in the paraxial wave approximation is identical to the diffraction of a scalar field from a single slit. Our analysis, based on numerics and the Cornu spiral reveals considerable intricate detail behavior in the density and phase of the wave. We also point out a concentration of the intensity that occurs on axis and propose a new measure of width that expresses this concentration.

© 2012 OSA

1. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
2. G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959). [CrossRef]
3. C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974). [CrossRef]
4. J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969). [CrossRef]
5. A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988). [CrossRef]
6. O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003). [CrossRef]
7. In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952). [CrossRef]
8. J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).
9. M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).
10. M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003). [CrossRef]
11. W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009). [CrossRef] [PubMed]
12. T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992). [CrossRef]
13. W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009). [CrossRef] [PubMed]
14. A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005). [CrossRef]
15. See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002). [CrossRef] [PubMed]
16. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
17. Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010). [CrossRef]
18. I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002). [CrossRef] [PubMed]
19. M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003). [CrossRef]
20. K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010). [CrossRef]
21. R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010). [CrossRef]
22. T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009). [CrossRef]
23. L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184. [CrossRef]
24. D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003). [CrossRef]
25. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001). [CrossRef]
26. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
27. The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
28. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
29. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
30. R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).
31. M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004). [CrossRef]
32. E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
33. M. Gonçalves (personal communication, 2011).
34. A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010). [CrossRef]
35. T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004). [CrossRef]

Am. J. Phys.

• C. Jönsson, “Electron diffraction at multiple slits,” Am. J. Phys.42, 4–11 (1974). [CrossRef]
• J. A. Leavitt and F. A. Bills, “Single-slit diffraction pattern of a thermal atomic potassium beam,” Am. J. Phys.37, 905–912 (1969). [CrossRef]
• O. Nairz, M. Arndt, and A. Zeilinger, “Quantum interference experiments with large molecules,” Am. J. Phys.71, 319–325 (2003). [CrossRef]

App. Phys. B

• T. Kraemer, J. Herbig, M. Mark, T. Weber, C. Chin, H. C. Nägerl, and R. Grimm, “Optimized production of a cesium Bose-Einstein condensate,” App. Phys. B79, 1013–1019 (2004). [CrossRef]

Appl. Phys. B.

• T. Sleator, T. Pfau, V. Balykin, and J. Mlynek, “Imaging and focusing of an atomic beam with a large period standing light wave,” Appl. Phys. B.54, 375–379 (1992). [CrossRef]

Chemical Physics

• K. Vogel, F. Gleisberg, N. L. Harshman, P. Kazemi, R. Mack, L. Plimak, and W. P. Schleich, “Optimally focusing wave packets,” Chemical Physics375, 133–143 (2010). [CrossRef]

J. Mod. Opt.

• R. Mack, V. P. Yakovlev, and W. P. Schleich, “Correlations in phase space and the creation of focusing wave packets,” J. Mod. Opt.57, 1437–1444 (2010). [CrossRef]

J. Phys. A: Math. Gen.

• M. Andreata and D. Dodonov, “On shrinking and expansion of radial wave packets,” J. Phys. A: Math. Gen.36, 7113–7128 (2003). [CrossRef]
• M. J. W. Hall, “Incompleteness of trajectory-based interpretations of quantum mechanics,” J. Phys. A: Math. Gen.37, 9549–9556 (2004). [CrossRef]

J. Phys.: Condens. Matter

• M. Oberthaler and T. Pfau, “One-, two-and three-dimensional nanostructures with atom lithography,” J. Phys.: Condens. Matter15, R233–R255 (2003). [CrossRef]

Opt. Express

• W. B. Case, M. Tomandl, S. Deachapunya, and M. Arndt, “Realization of optical carpets in the Talbot and Talbot-Lau configurations,” Opt. Express17(23), 20966–20974 (2009). [CrossRef] [PubMed]

Phys. Rev.

• In a seminal paper, Marcos Moshinsky studied the propagation of a matter wave suddenly released from a shutter. For this reason these functions are sometimes called Moshinsky functions. See M. Moshinsky, “Diffraction in time,” Phys. Rev.88, 625–631 (1952). [CrossRef]

Phys. Rev. A

• A. Turlapov, A. Tonyushkin, and T. Sleator, “Talbot-Lau effect for atomic de Broglie waves manipulated with light,” Phys. Rev. A71, 043612 (2005). [CrossRef]
• T. Reisinger, A. Patel, H. Reingruber, K. Fladischer, W. E. Ernst, G. Bracco, H. I. Smith, and B. Holst, “Poisson’s spot with molecules,” Phys. Rev. A79, 053823 (2009). [CrossRef]
• A. Jaouadi, N. Gaaloul, B. Viaris de Lesegno, M. Telmini, L. Pruvost, and E. Charron, “Bose-Einstein condensation in dark power-law laser traps,” Phys. Rev. A82, 023613 (2010). [CrossRef]

Phys. Rev. E

• T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001). [CrossRef]

Phys. Rev. Lett.

• J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987). [CrossRef] [PubMed]
• W. Schnitzler, N. M. Linke, R. Fickler, J. Meijer, F. Schmidt-Kaler, and K. Singer, “Deterministic ultracold ion source targeting the Heisenberg limit,” Phys. Rev. Lett.102, 070501 (2009). [CrossRef] [PubMed]
• See for example: M. Mützel, S. Tandler, D. Haubrich, D. Meschede, K. Peithmann, M. Flaspöhler, and K. Buse, “Atom lithography with a holographic light mask,” Phys. Rev. Lett.88, 083601 (2002). [CrossRef] [PubMed]
• A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett.85, 2733–2736 (2000). [CrossRef] [PubMed]
• Z. Liao, M. Al-Amri, and M.S. Zubairy, “Quantum lithography beyond the diffraction limit via Rabi oscillations,” Phys. Rev. Lett.105, 183601 (2010). [CrossRef]
• I. Bialynicki-Birula, M. A. Cirone, J. P. Dahl, M. Fedorov, and W. P. Schleich, “In- and outbound spreading of a free-particle s-wave,” Phys. Rev. Lett.89, 060404 (2002). [CrossRef] [PubMed]

Rev. Mex. Fis.

• J. Goldemberg and H. M. Nussenzveig, “On the possibility of the experimental observation of diffraction in time effects,” Rev. Mex. Fis.VI.3, 105–115 (1957).

Rev. Mod. Phys.

• A. Zeilinger, R. Gähler, C. G. Shull, W. Treimer, and W. Mampe “Single- and double-slit diffraction of neutrons,” Rev. Mod. Phys.60, 1067–1073 (1988). [CrossRef]

Z. f. Phys.

• G. Möllenstedt and C. Jönsson, “Elektronenmehrfachinterferenz an regelmäßig hergestellen Feinspalten,” Z. f. Phys.155, 427–474 (1959). [CrossRef]

Z. Naturforsch.

• M. Bozic, D. Arsenovic, and L. Vuskovic, “Transverse momentum distribution of atoms in an interferometer,” Z. Naturforsch.56a, 173–177 (2001).

Other

• M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon Press, London, 1964).
• L. Novotny, “The history of near-field optics” in Progress in Optics vol. 50, E. Wolf, ed. (Elsevier, Amsterdam, 2007) pp. 137–184. [CrossRef]
• D. Courjon, Near-field Microscopy and Near-Field Optics (World Scientific Publishing, Singapore, 2003). [CrossRef]
• E. Sadurní, W. B. Case, and W. P. Schleich, in preparation.
• M. Gonçalves (personal communication, 2011).
• The paraxial approximation is expected to hold as long as the wavelength is much less than the slit width.
• R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
• M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
• R. W. Wood, Physical Optics (Optical Society of America, Washington, 1988).

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