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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 23 — Nov. 12, 2007
  • pp: 15387–15401

The von Neumann picture: a new representation for ultrashort laser pulses

Susanne Fechner, Frank Dimler, Tobias Brixner, Gustav Gerber, and David J. Tannor  »View Author Affiliations

Optics Express, Vol. 15, Issue 23, pp. 15387-15401 (2007)

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In recent years, the use of joint time-frequency representations to characterize and interpret shaped femtosecond laser pulses has proven to be very useful. However, the number of points in a joint time-frequency representation is daunting as compared with those in either the frequency or time representation. In this article we introduce the use of the von Neumann representation, in which a femtosecond pulse is represented on a discrete lattice of evenly spaced time-frequency points using a non-orthogonal Gaussian basis. We show that the information content in the von Neumann representation using a lattice of N points in time and √N points in frequency is exactly the same as in a frequency (or time) array of N points. Explicit formulas are given for the forward and reverse transformation between an N-point frequency signal and the von Neumann representation. We provide numerical examples of the forward and reverse transformation between the two representations for a variety of different pulse shapes; in all cases the original pulse is reconstructed with excellent precision. The von Neumann representation has the interpretational advantages of the Husimi representation but requires a bare minimum number of points and is stably and conveniently inverted; moreover, it avoids the periodic boundary conditions of the Fourier representation.

© 2007 Optical Society of America

OCIS Codes
(320.5550) Ultrafast optics : Pulses
(070.7425) Fourier optics and signal processing : Quasi-probability distribution functions

ToC Category:
Ultrafast Optics

Original Manuscript: August 7, 2007
Revised Manuscript: September 12, 2007
Manuscript Accepted: September 14, 2007
Published: November 5, 2007

Susanne Fechner, Frank Dimler, Tobias Brixner, Gustav Gerber, and David J. Tannor, "The von Neumann picture: a new representation for ultrashort laser pulses," Opt. Express 15, 15387-15401 (2007)

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  1. E. Wigner, "On the Quantum Correction For Thermodynamic Equilibrium," Phys. Rev. 40, 749-759 (1932). [CrossRef]
  2. K. Husimi, "Some Formal Properties of the Density Matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).
  3. R. Kosloff, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, "Wavepacket Dancing - Achieving Chemical Selectivity By Shaping Light-Pulses," Chem. Phys. 139(1), 201-220 (1989). [CrossRef]
  4. D. J. Tannor and Y. Jin, Mode selective Chemistry, chap. Design of Femtosecond Pulse Sequences to Control Photochemical Products, pp. 333-345 (Kluwer Academic Publishers, 1991). [CrossRef]
  5. J. Paye, "The Chronocyclic Representation of Ultrashort Light Pulses," IEEE J. Quantum Electronics 28, 2262-2272 (1992). [CrossRef]
  6. J. Paye, "Space-Time Wigner functions and their application to the analysis of a pulse shaper," J. Opt. Soc. Am. B 12, 1480-1490 (1995). [CrossRef]
  7. G. Vogt, P. Nuernberger, R. Selle, F. Dimler, T. Brixner, and G. Gerber, "Analysis of femtosecond quantum control mechanisms with colored double pulses," Phys. Rev. A 74(3), 033413 (2006). [CrossRef]
  8. B. J. Pearson, J. L. White, T. C. Weinacht, and P. H. Bucksbaum, "Coherent control using adaptive learning algorithms," Phys. Rev. A 6306(6), 063412 (2001). [CrossRef]
  9. T. C. Weinacht and P. H. Bucksbaum, "Using feedback for coherent control of quantum systems," J. Opt. B: Quantum Semiclass. Opt. 4(3), R35-R52 (2002). [CrossRef]
  10. B. Amstrup, G. J. Tóth, G. Szab, H. Rabitz, and A. Lörincz, "Genetic Algorithm With Migration On Topology Conserving Maps For Optimal-Control of Quantum-Systems," J. Phys. Chem. 99(14), 5206-5213 (1995). [CrossRef]
  11. T. Brixner, N. H. Damrauer, B. Kiefer, and G. Gerber, "Liquid-phase adaptive femtosecond quantum control: Removing intrinsic intensity dependencies," J. Chem. Phys. 118(8), 3692-3701 (2003). [CrossRef]
  12. S. Mukamel, C. Ciordas-Ciurdariu, and V. Khidekel, "Wigner spectrograms for femtosecond pulse-shaped heterodyne and autocorrelation measurements," IEEE J. Quantum Electron. 32(8), 1278-1288 (1996). [CrossRef]
  13. B. Schäfer-Bung, R. Mitrić, V. Bonačić-Koutecký, A. Bartelt, C. Lupulescu, A. Lindinger, V. Vajda, S.M. Weber, and L. Wöste, "Optimal control of ionization processes in NaK: Comparison between theory and experiment," J. Phys. Chem. A 108(19), 4175-4179 (2004). [CrossRef]
  14. K.-H. Hong, J.-H. Kim, Y. Kang, and C. Nam, "Time-frequency analysis of chirped femtosecond pulses using Wigner distribution function," Appl. Phys. B 74, 231-236 (2002). [CrossRef]
  15. D. Lalović, D. M. Davidović, and N. Bijedić, "Quantum mechanics in terms of non negative smoothed Wigner functions," Phys. Rev. A 46, 1206-1212 (1992). [CrossRef] [PubMed]
  16. H.-W. Lee, "Generalized antinormal ordered quantum phase-space distribution functions," Phys. Rev. A 50, 2746-2749 (1994). [CrossRef] [PubMed]
  17. J. von Neumann, "Die Eindeutigkeit der Schr dingerschen Operatoren," Math. Ann. 104, 570 (1931). [CrossRef]
  18. M. Boon and J. Zak, "Discrete coherent states on the von Neumann lattice," Phys. Rev. B 18, 6744-6751 (1978). [CrossRef]
  19. M. J. Davis and E. J. Heller, "Semiclassical Gaussian basis set method for molecular vibrational wave functions," J. Chem. Phys. 71, 3383 (1979). [CrossRef]
  20. D. J. Tannor, Introduction to Quantum Mechanics A Time-Dependent Perspective (Palgrave Macmillan, 2007).
  21. R. Kosloff, "Time-Dependent Quantum-MechanicalMethods forMolecular Dynamics," J. Phys. Chem. 92, 2087 (1988). [CrossRef]
  22. R. Kosloff, Numerical Grid Methods and Their Application to Schrodinger’s Equation, chap. The Fourier Method, 1st ed.(Springer-Verlag GmbH, 1993-09-00 1993-09) pp. 175-194,.
  23. A. M. Perelomov, "On the Completeness of a System of Coherent States," Theor. Math. Phys. 11, 156 (1971). [CrossRef]
  24. W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001). [CrossRef]
  25. S. S. Mizrahi, "Quantum mechanics in the Gaussian wave-packet phase space representation," Physica A 127, 241-264 (1984). [CrossRef]

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