Abstract

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

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    [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).
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  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).
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  5. J. Paye, "The Chronocyclic Representation of Ultrashort Light Pulses," IEEE J. Quantum Electronics 28, 2262-2272 (1992).
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  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).
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  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).
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  25. S. S. Mizrahi, "Quantum mechanics in the Gaussian wave-packet phase space representation," Physica A 127, 241-264 (1984).
    [CrossRef]

2006 (1)

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]

2004 (1)

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]

2003 (1)

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]

2002 (2)

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]

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]

2001 (1)

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]

1996 (1)

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]

1995 (2)

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]

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]

1994 (1)

H.-W. Lee, "Generalized antinormal ordered quantum phase-space distribution functions," Phys. Rev. A 50, 2746-2749 (1994).
[CrossRef] [PubMed]

1992 (2)

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]

J. Paye, "The Chronocyclic Representation of Ultrashort Light Pulses," IEEE J. Quantum Electronics 28, 2262-2272 (1992).
[CrossRef]

1989 (1)

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]

1988 (1)

R. Kosloff, "Time-Dependent Quantum-MechanicalMethods forMolecular Dynamics," J. Phys. Chem. 92, 2087 (1988).
[CrossRef]

1984 (1)

S. S. Mizrahi, "Quantum mechanics in the Gaussian wave-packet phase space representation," Physica A 127, 241-264 (1984).
[CrossRef]

1979 (1)

M. J. Davis and E. J. Heller, "Semiclassical Gaussian basis set method for molecular vibrational wave functions," J. Chem. Phys. 71, 3383 (1979).
[CrossRef]

1978 (1)

M. Boon and J. Zak, "Discrete coherent states on the von Neumann lattice," Phys. Rev. B 18, 6744-6751 (1978).
[CrossRef]

1971 (1)

A. M. Perelomov, "On the Completeness of a System of Coherent States," Theor. Math. Phys. 11, 156 (1971).
[CrossRef]

1940 (1)

K. Husimi, "Some Formal Properties of the Density Matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

1932 (1)

E. Wigner, "On the Quantum Correction For Thermodynamic Equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

1931 (1)

J. von Neumann, "Die Eindeutigkeit der Schr dingerschen Operatoren," Math. Ann. 104, 570 (1931).
[CrossRef]

Amstrup, B.

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]

Bartelt, A.

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]

Bijedic, N.

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]

Bonacic-Koutecký, V.

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]

Boon, M.

M. Boon and J. Zak, "Discrete coherent states on the von Neumann lattice," Phys. Rev. B 18, 6744-6751 (1978).
[CrossRef]

Brixner, T.

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]

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]

Bucksbaum, P. H.

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]

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]

Ciordas-Ciurdariu, C.

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]

Damrauer, N. H.

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]

Davidovic, D. M.

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]

Davis, M. J.

M. J. Davis and E. J. Heller, "Semiclassical Gaussian basis set method for molecular vibrational wave functions," J. Chem. Phys. 71, 3383 (1979).
[CrossRef]

Dimler, F.

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]

Gaspard, P.

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]

Gerber, G.

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]

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]

Heller, E. J.

M. J. Davis and E. J. Heller, "Semiclassical Gaussian basis set method for molecular vibrational wave functions," J. Chem. Phys. 71, 3383 (1979).
[CrossRef]

Hong, K.-H.

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]

Husimi, K.

K. Husimi, "Some Formal Properties of the Density Matrix," Proc. Phys. Math. Soc. Jpn. 22, 264-314 (1940).

Kang, Y.

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]

Khidekel, V.

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]

Kiefer, B.

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]

Kim, J.-H.

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]

Kosloff, R.

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]

R. Kosloff, "Time-Dependent Quantum-MechanicalMethods forMolecular Dynamics," J. Phys. Chem. 92, 2087 (1988).
[CrossRef]

Lalovic, D.

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]

Lee, H.-W.

H.-W. Lee, "Generalized antinormal ordered quantum phase-space distribution functions," Phys. Rev. A 50, 2746-2749 (1994).
[CrossRef] [PubMed]

Lindinger, A.

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]

Lörincz, A.

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]

Lupulescu, C.

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]

Mitric, R.

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]

Mizrahi, S. S.

S. S. Mizrahi, "Quantum mechanics in the Gaussian wave-packet phase space representation," Physica A 127, 241-264 (1984).
[CrossRef]

Mukamel, S.

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]

Nam, C.

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]

Nuernberger, P.

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]

Paye, J.

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]

J. Paye, "The Chronocyclic Representation of Ultrashort Light Pulses," IEEE J. Quantum Electronics 28, 2262-2272 (1992).
[CrossRef]

Pearson, B. J.

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]

Perelomov, A. M.

A. M. Perelomov, "On the Completeness of a System of Coherent States," Theor. Math. Phys. 11, 156 (1971).
[CrossRef]

Rabitz, H.

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]

Rice, S. A.

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]

Schäfer-Bung, B.

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]

Selle, R.

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]

Szab, G.

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]

Tannor, D. J.

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]

Tersigni, S.

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]

Tóth, G. J.

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]

Vajda, V.

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]

Vogt, G.

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]

von Neumann, J.

J. von Neumann, "Die Eindeutigkeit der Schr dingerschen Operatoren," Math. Ann. 104, 570 (1931).
[CrossRef]

Weber, S.M.

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]

Weinacht, T. C.

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]

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]

White, J. L.

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]

Wigner, E.

E. Wigner, "On the Quantum Correction For Thermodynamic Equilibrium," Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wöste, L.

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]

Zak, J.

M. Boon and J. Zak, "Discrete coherent states on the von Neumann lattice," Phys. Rev. B 18, 6744-6751 (1978).
[CrossRef]

Appl. Phys. B (1)

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]

Chem. Phys. (1)

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Figures (5)

Fig. 1.
Fig. 1.

Definition of the von Neumann parameters. A discrete signal in the frequency or in the time domain can be mapped one-to-one on the von Neumann joint time-frequency grid. This grid covers the complete spectral and temporal ranges, Ω and T, defined by the Fourier relation, but the spectral and temporal resolutions (δt and δω in time and frequency domain and Δt and Δω on the von Neumann grid) are different such that the total number of sample points, N, remains the same in all three cases. The FWHM of the Gaussian basis function of the von Neumann representation σ t and σ ω are illustrated with respect to the grid spacing by the circle.

Fig. 2.
Fig. 2.

Two examples of the von Neumann transformation. The first example is a chirped laser pulse with a quadratic spectral phase. Its spectral intensity (a) and phase (b) is sampled at 121 frequency points and shown as black lines. The second example is an ultrashort laser pulse with the first half of the spectrum shifted forward and the second half backward in time. The corresponding spectral intensity (d) and phase (e) are again plotted in black. These electric fields were transformed onto a von Neumann grid of 11 × 11 points and subsequently transformed back to the frequency domain. The von Neumann intensity of the chirped pulse (c) shows a characteristic shape and the von Neumann intensity of the double pulse (f) displays two subpulses of different central frequency. The reconstructed spectral intensity and phase is added in (a), (b), (c) and (d) as red circles. Apart from small discrepancies, the reconstruction quality is very good and the original signal is recovered.

Fig. 3.
Fig. 3.

Example for the reconstruction of von Neumann representations. A von Neumann representation was defined on a 11 × 11 von Neumann grid. It is shown on the left side in intensity (a) and phase (b). In order to test the quality of the von Neumann transformation it was transformed both to the frequency (c) and the time domain (d) representation using 121 sample points with red lines indicating the intensity and black dashed lines the corresponding phases. The von Neumann picture after transforming back from the spectral domain to the von Neumann grid is displayed as intensity (e) and phase (f). Since the phase for positions with negligible intensity has no meaning, we used “phase blanking” for simplification of the plot, i.e. setting all those von Neumann phase values to zero for which the corresponding intensities are less than 5% of the maximal intensity. The agreement between the original and the reconstructed von Neumann representation is excellent.

Fig. 4.
Fig. 4.

The relation between the von Neumann phase and the phase of the electric field. A double pulse in the von Neumann plane is shown in intensity (a) and phase (b). The two parts of the double pulse have the same von Neumann intensity, which leads to equally intense pulses in the frequency and time domains. The von Neumann phase at these points is set to 0.75 π rad at ω 1 = 2.33 fs-1and to -0.25 π rad at ω 2 =2.38 fs-1. The corresponding signal in frequency domain (c) is represented as intensity (red line) and phase (black dashed line). The arrows connecting (b) and (c) show that the signal phase is identical with the von Neumann phase at positions ω 1 and ω 2.

Fig. 5.
Fig. 5.

Shaped ultrashort laser pulse in different representations. This example is based on a Gaussian spectrum and piecewise linear spectral phase, creating “colored double pulses” with temporally shifted “red” and “blue” components. (a) Electric field (Fourier) representation in the frequency domain (upper panel) and in the time domain (lower panel) with intensity (red) and phase (dashed black). The two representations are equivalent and connected via Fourier transformation. (b) Real-valued and oscillating temporal electric field. (c) Wigner representation with positive and negative valued interference structures in the middle. (d) Non-negative Husimi representation obtained by smoothing the Wigner representation with a Gaussian. For these two jtf representations a resolution of 121 × 121 was applied in order to obtain time and frequency marginals with the same resolution as the electric field representations of 121 sample points. The von Neumann representation on a 11 × 11 grid with intensity (e) and phase (f) contains equal information as the other representations.

Tables (1)

Tables Icon

Table 1. Comparison between different representations for ultrashort laser pulses. The spectral, as well as the temporal representation, with and without SVEA of ultrashort laser pulses are compared with the most common jtf-representations (Wigner and Husimi). The von Neumann representation has favorable properties in all of the examined criteria.

Equations (27)

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α ˜ ω n t m ( ω ) = ( 2 α π ) 1 4 exp [ α ( ω ω n ) 2 i t m ( ω ω n ) ]
α ω n t m ( t ) = ( 1 2 απ ) 1 4 exp [ 1 4 α ( t t m ) 2 i t ω n ] .
σ ω = 4 ln 2 α .
σ t σ ω = 8 ln 2 .
α = σ t 2 σ ω .
σ ω Ω = σ t T .
α = T 2 Ω .
1 = ( n , m ) , ( i , j ) α ω n t m S ( n , m ) , ( i , j ) 1 α ω i t j ,
S ( n , m ) , ( i , j ) = α ω n t m α ω i t j
S ( n , m ) , ( i , j ) = 2 α π exp [ α 2 ( ω n ω i ) 2 1 8 α ( t j t m ) 2
+ i 2 ( ω i ω n ) ( t j + t m ) ] .
ε = n , m Q ˜ ω n t m α ω n t m ,
Q ˜ ω n t m = i , j S ( n , m ) , ( i , j ) 1 α ω i t j ε
= i , j S ( n , m ) , ( i , j ) 1 α ω i t j ω ω ε = i , j S ( n , m ) , ( i , j ) 1 α ˜ ω i t j * ( ω ) ε ˜ ( ω )
= i , j S ( n , m ) , ( i , j ) 1 α ω i t j t t ε d t = i , j S ( n , m ) , ( i , j ) 1 α ˜ ω i t j * ( t ) ε ( t ) d t .
Ω = 2 π δt .
T = 2 π δ ω .
V = T × Ω = T × 2 π δ t = 2 π N .
Δ t × Δ ω = 2 π
k 2 = T Δ t × Ω Δ ω = T Ω ( 2 π ) = 2 π N ( 2 π ) = N ,
Q = Ψ α p 0 q 0 2 .
α p 0 q 0 ( q ) = q α p 0 q 0 = ( 1 π σ 2 ) 1 / 4 exp [ ( i p 0 q h ¯ ) ( q q 0 ) 2 2 σ 2 ] ,
Q ( ω n , t m ) = α ͂ ω n t m ( ω ) * ε ͂ ( ω ) d ω 2 = α ω n t m ε 2 .
ε ͂ ( ω ) = ε ͂ ( ω ) exp [ i Φ ε ( ω ) ] ,
ε ͂ ( ω ) = ( 2 α π ) 1 / 4 n , m Q ͂ ω n t m exp [ α ( ω ω n ) 2 ] exp { i [ t m ( ω ω n ) + Φ ω n t m ] } ,
ε ͂ ( ω ) = Â 1 ( ω ) exp { i [ t 1 ( ω ω 1 ) + Φ ω 1 t 1 ] } + A ̂ 2 ( ω ) exp { i [ t 2 ( ω ω 2 ) + Φ ω 2 t 2 ] }
Φ ε ( t i ) = t i ω i + Φ ω i t i

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