Abstract

We theoretically study the carrier-envelope phase-dependent inversion generated in a two-level system by excitation with a few-cycle pulse. On the basis of the invariance of the inversion under time reversal of the exciting field, parameters are introduced to characterize the phase sensitivity of the induced inversion. Linear and nonlinear phase effects are numerically studied for rectangular and sinc-shaped pulses. Furthermore, analytical results are obtained in the limits of weak fields, as well as strong dephasing, and by nearly degenerate perturbation theory for sinusoidal excitation. The results show that the phase-sensitive inversion in the ideal two-level system is a promising route for constructing carrier-envelope phase detectors.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).
  2. S. Hughes, "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses," Phys. Rev. Lett. 81, 3363-3366 (1998).
    [CrossRef]
  3. J. H. Shirley, "Solution of the Schrödinger equation with a Hamiltonian periodic in time," Phys. Rev. 138, B979-B987 (1965).
    [CrossRef]
  4. R. Gush and H. P. Gush, "Scattering of intense light by a two-level system," Phys. Rev. A 6, 129-140 (1972).
    [CrossRef]
  5. F. Ahmad, "Semiclassical electrodynamics of a two-level system," Phys. Rev. A 12, 1539-1545 (1975).
    [CrossRef]
  6. J. V. Moloney and W. J. Meath, "Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field," Mol. Phys. 31, 1537-1548 (1976).
    [CrossRef]
  7. J. V. Moloney and W. J. Meath, "Induced transition probabilities and energies for the strongly coupled two-level system," Phys. Rev. A 17, 1550-1554 (1978).
    [CrossRef]
  8. W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
    [CrossRef]
  9. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996).
    [CrossRef] [PubMed]
  10. A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
    [CrossRef] [PubMed]
  11. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
    [CrossRef] [PubMed]
  12. U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
    [CrossRef] [PubMed]
  13. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, "Determining the carrier-envelope offset frequency of 5-fs pulses with extreme nonlinear optics in ZnO," Opt. Lett. 27, 2127-2129 (2002).
    [CrossRef]
  14. T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
    [CrossRef] [PubMed]
  15. A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
    [CrossRef] [PubMed]
  16. G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
    [CrossRef]
  17. M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).
  18. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, "Role of the carrier-envelope offset phase of few-cycle pulses in nonperturbative resonant nonlinear optics," Phys. Rev. Lett. 89, 127401 (2002).
    [CrossRef] [PubMed]
  19. O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, F. X. Kärtner, G. Khitrova, and H. M. Gibbs, "Carrier-wave Rabi flopping: role of the carrier-envelope phase," Opt. Lett. 29, 2160-2162 (2004).
    [CrossRef] [PubMed]
  20. G. F. Thomas, "Dipole interaction of a multilevel system with a continuous-wave or Gaussian-pulsed laser," Phys. Rev. A 32, 1515-1525 (1985).
    [CrossRef] [PubMed]
  21. A. Brown and W. J. Meath, "On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules," J. Chem. Phys. 109, 9351-9365 (1998).
    [CrossRef]
  22. V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer-Verlag, 1992).
    [CrossRef]
  23. A. J. Jerri, Introduction to Integral Equations with Applications (Wiley, 1999).
  24. O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
    [CrossRef]
  25. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
    [CrossRef]
  26. P. R. Certain and J. O. Hirschfelder, "New partitioning perturbation theory. I. General formalism," J. Chem. Phys. 52, 5977-5987 (1970).
    [CrossRef]
  27. J. O. Hirschfelder, "Almost degenerate perturbation theory," Chem. Phys. Lett. 54, 1-3 (1978).
    [CrossRef]
  28. P. K. Aravind and J. O. Hirschfelder, "Two-state systems in semiclassical and quantized fields," J. Phys. Chem. 88, 4788-4801 (1984).
    [CrossRef]
  29. B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys. 18, 756-763 (1977).
    [CrossRef]

2004 (3)

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, F. X. Kärtner, G. Khitrova, and H. M. Gibbs, "Carrier-wave Rabi flopping: role of the carrier-envelope phase," Opt. Lett. 29, 2160-2162 (2004).
[CrossRef] [PubMed]

2003 (1)

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

2002 (3)

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, "Role of the carrier-envelope offset phase of few-cycle pulses in nonperturbative resonant nonlinear optics," Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef] [PubMed]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, "Determining the carrier-envelope offset frequency of 5-fs pulses with extreme nonlinear optics in ZnO," Opt. Lett. 27, 2127-2129 (2002).
[CrossRef]

O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
[CrossRef]

2001 (1)

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

2000 (2)

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (3)

A. Brown and W. J. Meath, "On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules," J. Chem. Phys. 109, 9351-9365 (1998).
[CrossRef]

S. Hughes, "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses," Phys. Rev. Lett. 81, 3363-3366 (1998).
[CrossRef]

W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
[CrossRef]

1996 (1)

1993 (1)

M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).

1985 (1)

G. F. Thomas, "Dipole interaction of a multilevel system with a continuous-wave or Gaussian-pulsed laser," Phys. Rev. A 32, 1515-1525 (1985).
[CrossRef] [PubMed]

1984 (1)

P. K. Aravind and J. O. Hirschfelder, "Two-state systems in semiclassical and quantized fields," J. Phys. Chem. 88, 4788-4801 (1984).
[CrossRef]

1978 (2)

J. O. Hirschfelder, "Almost degenerate perturbation theory," Chem. Phys. Lett. 54, 1-3 (1978).
[CrossRef]

J. V. Moloney and W. J. Meath, "Induced transition probabilities and energies for the strongly coupled two-level system," Phys. Rev. A 17, 1550-1554 (1978).
[CrossRef]

1977 (1)

B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys. 18, 756-763 (1977).
[CrossRef]

1976 (1)

J. V. Moloney and W. J. Meath, "Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field," Mol. Phys. 31, 1537-1548 (1976).
[CrossRef]

1975 (1)

F. Ahmad, "Semiclassical electrodynamics of a two-level system," Phys. Rev. A 12, 1539-1545 (1975).
[CrossRef]

1972 (1)

R. Gush and H. P. Gush, "Scattering of intense light by a two-level system," Phys. Rev. A 6, 129-140 (1972).
[CrossRef]

1970 (1)

P. R. Certain and J. O. Hirschfelder, "New partitioning perturbation theory. I. General formalism," J. Chem. Phys. 52, 5977-5987 (1970).
[CrossRef]

1965 (1)

J. H. Shirley, "Solution of the Schrödinger equation with a Hamiltonian periodic in time," Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Ahmad, F.

F. Ahmad, "Semiclassical electrodynamics of a two-level system," Phys. Rev. A 12, 1539-1545 (1975).
[CrossRef]

Akulin, V. M.

V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer-Verlag, 1992).
[CrossRef]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Angelow, G.

Apolonski, A.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Aravind, P. K.

P. K. Aravind and J. O. Hirschfelder, "Two-state systems in semiclassical and quantized fields," J. Phys. Chem. 88, 4788-4801 (1984).
[CrossRef]

Baltuska, A.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Bhat, R. D.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Brabec, T.

Brown, A.

A. Brown and W. J. Meath, "On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules," J. Chem. Phys. 109, 9351-9365 (1998).
[CrossRef]

Burgdörfer, J.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Certain, P. R.

P. R. Certain and J. O. Hirschfelder, "New partitioning perturbation theory. I. General formalism," J. Chem. Phys. 52, 5977-5987 (1970).
[CrossRef]

Chen, Y.

Cho, S. H.

Corkum, P. B.

M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).

Cundiff, S. T.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Dietrich, P.

M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).

Dombi, P.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

Ell, R.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

Fortier, T. M.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Fujimoto, J. G.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
[CrossRef]

Gallagher, T. F.

W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
[CrossRef]

Gibbs, H. M.

Goulielmakis, E.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Griffith, W. M.

W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
[CrossRef]

Gush, H. P.

R. Gush and H. P. Gush, "Scattering of intense light by a two-level system," Phys. Rev. A 6, 129-140 (1972).
[CrossRef]

Gush, R.

R. Gush and H. P. Gush, "Scattering of intense light by a two-level system," Phys. Rev. A 6, 129-140 (1972).
[CrossRef]

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Hänsch, T. W.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996).
[CrossRef] [PubMed]

Haus, H. A.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
[CrossRef]

Hirschfelder, J. O.

P. K. Aravind and J. O. Hirschfelder, "Two-state systems in semiclassical and quantized fields," J. Phys. Chem. 88, 4788-4801 (1984).
[CrossRef]

J. O. Hirschfelder, "Almost degenerate perturbation theory," Chem. Phys. Lett. 54, 1-3 (1978).
[CrossRef]

P. R. Certain and J. O. Hirschfelder, "New partitioning perturbation theory. I. General formalism," J. Chem. Phys. 52, 5977-5987 (1970).
[CrossRef]

Holzwarth, R.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Hughes, S.

S. Hughes, "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses," Phys. Rev. Lett. 81, 3363-3366 (1998).
[CrossRef]

Ippen, E. P.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, "Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser," Opt. Lett. 24, 411-413 (1999).
[CrossRef]

Ivanov, M. Y.

M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).

Jerri, A. J.

A. J. Jerri, Introduction to Integral Equations with Applications (Wiley, 1999).

Jones, D. J.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Kakehata, M.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Karlov, N. V.

V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer-Verlag, 1992).
[CrossRef]

Kärtner, F. X.

Khitrova, G.

Krausz, F.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996).
[CrossRef] [PubMed]

Lemell, C.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Lezius, M.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Lindner, F.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Meath, W. J.

A. Brown and W. J. Meath, "On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules," J. Chem. Phys. 109, 9351-9365 (1998).
[CrossRef]

J. V. Moloney and W. J. Meath, "Induced transition probabilities and energies for the strongly coupled two-level system," Phys. Rev. A 17, 1550-1554 (1978).
[CrossRef]

J. V. Moloney and W. J. Meath, "Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field," Mol. Phys. 31, 1537-1548 (1976).
[CrossRef]

Metzler, G.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

Misra, B.

B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys. 18, 756-763 (1977).
[CrossRef]

Mitrofanov, O.

O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
[CrossRef]

Moloney, J. V.

J. V. Moloney and W. J. Meath, "Induced transition probabilities and energies for the strongly coupled two-level system," Phys. Rev. A 17, 1550-1554 (1978).
[CrossRef]

J. V. Moloney and W. J. Meath, "Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field," Mol. Phys. 31, 1537-1548 (1976).
[CrossRef]

Morgner, U.

Mücke, O. D.

Noel, M. W.

W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
[CrossRef]

Paulus, G. G.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Pfeiffer, L. N.

O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
[CrossRef]

Poppe, A.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996).
[CrossRef] [PubMed]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Roos, P. A.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Scheuer, V.

Schibli, T. R.

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

Shirley, J. H.

J. H. Shirley, "Solution of the Schrödinger equation with a Hamiltonian periodic in time," Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Sipe, J. E.

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

Spielmann, C.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996).
[CrossRef] [PubMed]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Sudarshan, E. C.

B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys. 18, 756-763 (1977).
[CrossRef]

Tempea, G.

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Thomas, G. F.

G. F. Thomas, "Dipole interaction of a multilevel system with a continuous-wave or Gaussian-pulsed laser," Phys. Rev. A 32, 1515-1525 (1985).
[CrossRef] [PubMed]

Torizuka, K.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

Tritschler, T.

Tschudi, T.

Udem, T.

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

Walther, H.

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Wegener, M.

West, K. W.

O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
[CrossRef]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Xu, L.

Appl. Phys. Lett. (1)

O. Mitrofanov, L. N. Pfeiffer, and K. W. West, "Generation of low-frequency components due to phase-amplitude modulation of subcycle far-infrared pulses in near-field diffraction," Appl. Phys. Lett. 81, 1579-1581 (2002).
[CrossRef]

Chem. Phys. Lett. (1)

J. O. Hirschfelder, "Almost degenerate perturbation theory," Chem. Phys. Lett. 54, 1-3 (1978).
[CrossRef]

J. Chem. Phys. (2)

P. R. Certain and J. O. Hirschfelder, "New partitioning perturbation theory. I. General formalism," J. Chem. Phys. 52, 5977-5987 (1970).
[CrossRef]

A. Brown and W. J. Meath, "On the effects of absolute laser phase on the interaction of a pulsed laser with polar versus nonpolar molecules," J. Chem. Phys. 109, 9351-9365 (1998).
[CrossRef]

J. Math. Phys. (1)

B. Misra and E. C. G. Sudarshan, "The Zeno's paradox in quantum theory," J. Math. Phys. 18, 756-763 (1977).
[CrossRef]

J. Phys. Chem. (1)

P. K. Aravind and J. O. Hirschfelder, "Two-state systems in semiclassical and quantized fields," J. Phys. Chem. 88, 4788-4801 (1984).
[CrossRef]

Laser Phys. (1)

M. Y. Ivanov, P. B. Corkum, and P. Dietrich, "Coherent control and collapse of symmetry in a two-level system in an intense laser field," Laser Phys. 3, 375-380 (1993).

Mol. Phys. (1)

J. V. Moloney and W. J. Meath, "Phase and temporal average transition probabilities for a multi-level system in a sinusoidal field," Mol. Phys. 31, 1537-1548 (1976).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (1)

J. H. Shirley, "Solution of the Schrödinger equation with a Hamiltonian periodic in time," Phys. Rev. 138, B979-B987 (1965).
[CrossRef]

Phys. Rev. A (5)

R. Gush and H. P. Gush, "Scattering of intense light by a two-level system," Phys. Rev. A 6, 129-140 (1972).
[CrossRef]

F. Ahmad, "Semiclassical electrodynamics of a two-level system," Phys. Rev. A 12, 1539-1545 (1975).
[CrossRef]

J. V. Moloney and W. J. Meath, "Induced transition probabilities and energies for the strongly coupled two-level system," Phys. Rev. A 17, 1550-1554 (1978).
[CrossRef]

W. M. Griffith, M. W. Noel, and T. F. Gallagher, "Phase and rise-time dependence using rf pulses in multiphoton processes," Phys. Rev. A 57, 3698-3704 (1998).
[CrossRef]

G. F. Thomas, "Dipole interaction of a multilevel system with a continuous-wave or Gaussian-pulsed laser," Phys. Rev. A 32, 1515-1525 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett. (7)

A. Apolonski, A. Poppe, G. Tempea, C. Spielmann, T. Udem, R. Holzwarth, T. W. Hänsch, and F. Krausz, "Controlling the phase evolution of few-cycle light pulses," Phys. Rev. Lett. 85, 740-743 (2000).
[CrossRef] [PubMed]

S. Hughes, "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses," Phys. Rev. Lett. 81, 3363-3366 (1998).
[CrossRef]

O. D. Mücke, T. Tritschler, M. Wegener, U. Morgner, and F. X. Kärtner, "Role of the carrier-envelope offset phase of few-cycle pulses in nonperturbative resonant nonlinear optics," Phys. Rev. Lett. 89, 127401 (2002).
[CrossRef] [PubMed]

U. Morgner, R. Ell, G. Metzler, T. R. Schibli, F. X. Kärtner, J. G. Fujimoto, H. A. Haus, and E. P. Ippen, "Nonlinear optics with phase-controlled pulses in the sub-two-cycle regime," Phys. Rev. Lett. 86, 5462-5465 (2001).
[CrossRef] [PubMed]

T. M. Fortier, P. A. Roos, D. J. Jones, S. T. Cundiff, R. D. R. Bhat, and J. E. Sipe, "Carrier-envelope phase-controlled quantum interference of injected photocurrents in semiconductors," Phys. Rev. Lett. 92, 147403 (2004).
[CrossRef] [PubMed]

A. Apolonski, P. Dombi, G. G. Paulus, M. Kakehata, R. Holzwarth, T. Udem, C. Lemell, K. Torizuka, J. Burgdörfer, T. W. Hänsch, and F. Krausz, "Observation of light-phase-sensitive photoemission from a metal," Phys. Rev. Lett. 92, 073902 (2004).
[CrossRef] [PubMed]

G. G. Paulus, F. Lindner, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, "Measurement of the phase of few-cycle laser pulses," Phys. Rev. Lett. 91, 253004 (2003).
[CrossRef]

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Other (3)

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).

V. M. Akulin and N. V. Karlov, Intense Resonant Interactions in Quantum Electronics (Springer-Verlag, 1992).
[CrossRef]

A. J. Jerri, Introduction to Integral Equations with Applications (Wiley, 1999).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Dynamics of the two-level system for excitation with a field Ω ( t ) (black curve) and the corresponding time-reversed field Ω ( t ) (gray curve). (a) Time-dependent electric field, parametrized in terms of the normalized Rabi frequency. (b) Evolution of the inversion for a Rabi frequency Ω R = 1.3 ω c , a transition frequency ω b a = 1.7 ω c , and a dephasing rate γ 2 = f c 10 .

Fig. 2
Fig. 2

Bloch vector trajectories for excitation of a two-level system with a driving field (black curve) and the time-reversed field (gray curve). The fields and system parameters are the same as for Fig. 1.

Fig. 3
Fig. 3

Phase-dependent steady-state inversion w s after interaction with a two-cycle sinc pulse, for Rabi frequencies Ω R = 1.5 ω c (solid curve) and Ω R = 2.3 ω c (dashed curve). The transition frequency is ω b a = 1.5 ω c , and the dephasing time is T 2 = 10 f c .

Fig. 4
Fig. 4

Rectangular two-cycle pulse: power spectrum for CE phases 0 (solid curve) and π 2 (dashed curve).

Fig. 5
Fig. 5

Average inversion w ¯ s and modulation amplitude Δ as a function of the Rabi frequency Ω R and the transition frequency ω b a in units of ω c . Displayed are w ¯ s after interaction with rectangular (a) single-cycle and (b) two-cycle pulses and Δ after interaction with rectangular (c) single-cycle and (d) two-cycle pulses.

Fig. 6
Fig. 6

Modulation depth δ as a function of the Rabi frequency Ω R and the transition frequency ω b a in units of ω c : perturbative (dashed curves) and exact numerical result (solid curves).

Fig. 7
Fig. 7

Average inversion w ¯ s and modulation amplitude Δ as a function of the Rabi frequency Ω R and the transition frequency ω b a in units of ω c . Displayed are w ¯ s after interaction with sinc-shaped (a) single-cycle and (b) two-cycle pulses and Δ after interaction with sinc-shaped (c) single-cycle and (d) two-cycle pulses.

Fig. 8
Fig. 8

Excitation with a rectangular two-cycle pulse: exact numerical result (solid curves) and strong dephasing approximation (dashed curves) for the (a) average inversion w ¯ s and (b) modulation amplitude Δ as a function of the normalized dephasing rate γ 2 ω c . The Rabi frequency is Ω R = 0.25 ω c , and the transition frequency is ω b a = 1.5 ω c .

Fig. 9
Fig. 9

Excitation with a sinc-shaped two-cycle pulse: exact numerical result (solid curve) and strong dephasing approximation (dashed curve) for the average inversion w ¯ s as a function of the normalized dephasing rate γ 2 ω c . The Rabi frequency is Ω R = 0.25 ω c , and the transition frequency is ω b a = 1.5 ω c .

Fig. 10
Fig. 10

Excitation with a rectangular two-cycle pulse: exact numerical result (solid curves) and strong dephasing approximation (dashed curves) for the (a) average inversion w ¯ s and (b) modulation amplitude Δ as a function of the normalized dephasing rate γ 2 ω c . The Rabi frequency is Ω R = ω c , and the transition frequency is ω b a = 1.5 ω c . The inset shows Δ at an increased scale.

Fig. 11
Fig. 11

Excitation with a sinc-shaped two-cycle pulse: exact numerical result (solid curves) and strong dephasing approximation (dashed curve) for the (a) average inversion w ¯ s and (b) modulation amplitude Δ as a function of the normalized dephasing rate γ 2 ω c . The Rabi frequency is Ω R = ω c , and the transition frequency is ω b a = 1.5 ω c . For Δ, only the numerical result is shown because the approximation is not CE phase sensitive for sinc pulses.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

s ̇ 1 = ω b a s 2 γ 2 s 1 ,
s ̇ 2 = ω b a s 1 + 2 Ω s 3 γ 2 s 2 ,
s ̇ 3 = 2 Ω s 2 .
Ω ( t ) = Ω R [ ϵ ( t ) exp ( i ω c t + i ϕ CE ) + ϵ * ( t ) exp ( i ω c t i ϕ CE ) ] 2 ,
A ( t ) = [ cos ( ω b a t ) sin ( ω b a t ) 0 sin ( ω b a t ) cos ( ω b a t ) 0 0 0 1 ] .
u ̇ = 2 Ω sin ( ω b a t ) w γ 2 u ,
v ̇ = 2 Ω cos ( ω b a t ) w γ 2 v ,
w ̇ = 2 Ω sin ( ω b a t ) u 2 Ω cos ( ω b a t ) v .
w s ( ϕ CE ) = w ¯ s + Δ cos ( 2 ϕ CE ) .
w ¯ s = [ w s ( 0 ) + w s ( π 2 ) ] 2 ,
Δ = [ w s ( 0 ) w s ( π 2 ) ] 2 ,
w ( t ) = w 0 4 t d t 0 d τ cos ( ω b a τ ) exp ( γ 2 τ ) Ω ( t ) Ω ( t τ ) w ( t τ ) .
w ( 1 ) ( t ) = w 0 [ 1 4 0 d τ cos ( ω b a τ ) exp ( γ 2 τ ) t d t Ω ( t ) Ω ( t τ ) ] .
Ω ( ω ) = d t Ω ( t ) exp ( i ω t ) ,
w s ( 1 ) = w 0 [ 1 0 d ω Ω ( ω ) 2 H ( ω ) ] .
H ( ω ) = 4 π γ 2 ( ω 2 + γ 2 2 + ω b a 2 ) ( ω 2 + γ 2 2 + ω b a 2 ) 2 4 ω b a 2 ω 2 .
H ( ω ) = 4 π γ 2 2 ( ω ω b a ) 2 + γ 2 2 ,
H ( ω ) = 2 δ ( ω ω b a ) .
Ω ( ω ) = Ω R [ ϵ ( ω ω c ) exp ( i ϕ CE ) + ϵ * ( ω ω c ) exp ( i ϕ CE ) ] 2 .
δ = Δ ( w ¯ s w 0 ) ,
δ p = ( ω b a 2 ω c 2 ) ( ω b a + ω c ) + ω c Ω R 2 ( ω b a 2 + ω c 2 ) ( ω b a + ω c ) ω c Ω R 2 ;
δ p = ( ω b a 2 ω c 2 ) ( ω b a 2 + ω c 2 )
w ̇ ( t ) = 4 w ( t ) 0 d τ cos ( ω b a τ ) exp ( γ 2 τ ) Ω ( t ) Ω ( t τ ) ,
w ( t ) = w 0 exp [ 4 0 d τ cos ( ω b a τ ) exp ( γ 2 τ ) t d t Ω ( t ) Ω ( t τ ) ] .
w s = w 0 exp [ 0 d ω Ω ( ω ) 2 H ( ω ) ] ,
w s = w 0 exp [ 4 π γ 2 0 d ω Ω ( ω ) 2 ] = w 0 exp [ 4 γ 2 d t Ω ( t ) 2 ] ,
u ̇ = 2 Ω M u γ 2 ( u 1 , u 2 , 0 ) T ,
M ( t ) = [ 0 0 sin ( ω b a t ) 0 0 cos ( ω b a t ) sin ( ω b a t ) cos ( ω b a t ) 0 ]
u ( t 0 ) = T ̂ exp [ t 0 t 0 2 Ω ( t ) M ( t ) d t ] u 0 = lim N n = N N exp [ 2 Ω ( t n ) M n t 0 N ] u 0 = lim N n = N N [ I + 2 Ω ( t n ) M n t 0 N ] u 0 ,
M n 1 M n 2 M n u 0 = M n M n 2 M n 1 u 0 ,
M n M n 2 M n 1 u 0 = w 0 ( 1 ) 2 m = 1 2 cos [ ω b a ( t n 2 m 1 t n 2 m ) ] ( 0 , 0 , 1 ) T .
M n M n 2 M n 1 u 0 = w 0 ( 1 ) ( 1 ) 2 m = 1 ( 1 ) 2 cos [ ω b a ( t n 2 m 1 t n 2 m ) ] [ sin ( ω b a t ) , cos ( ω b a t ) , 0 ] T .
M ̃ ( t ) = [ 0 0 sin ( ω b a t ) exp ( γ 2 t ) 0 0 cos ( ω b a t ) exp ( γ 2 t ) sin ( ω b a t ) exp ( γ 2 t ) cos ( ω b a t ) exp ( γ 2 t ) 0 ] .
M ̃ n M ̃ n 2 M ̃ n 1 u 0 = w 0 ( 1 ) 2 m = 1 2 { exp [ γ 2 ( t n 2 m 1 t n 2 m ) ] cos [ ω b a ( t n 2 m 1 t n 2 m ) ] } ( 0 , 0 , 1 ) T .
M ̃ n M ̃ n 2 M ̃ n 1 u 0 = w 0 ( 1 ) ( 1 ) 2 m = 1 ( 1 ) 2 { exp [ γ 2 ( t n 2 m 1 t n 2 m ) ] cos [ ω b a ( t n 2 m 1 t n 2 m ) ] } exp ( γ 2 t ) [ sin ( ω b a t ) , cos ( ω b a t ) , 0 ] T .
c n = U n c 0 ,
c n j = exp ( i n W j T 0 ) c 0 j .
c b ( n T 0 ) 2 = [ exp ( i n W b T 0 ) exp ( i n W a T 0 ) ] T a a T b a T a a T b b T a b T b a 2 ,
ρ β β ( t ) = m β , m χ + exp ( i ω m t ) 2 m α , m χ + exp ( i ω m t 0 ) 2 + m β , m χ exp ( i ω m t ) 2 m α , m χ exp ( i ω m t 0 ) 2 + 2 R { exp [ 2 i q ( t t 0 ) ] [ m α , m χ + exp ( i ω m t 0 ) ] [ m χ α , m exp ( i ω m t 0 ) ] [ m χ + β , m exp ( i ω m t ) ] [ m β , m χ exp ( i ω m t ) ] } .
χ + = N [ cos ( θ ) ( α , 0 η β , 1 ) + sin ( θ ) ( β , 1 + η α , 2 ) ] ,
χ = N [ sin ( θ ) ( α , 0 η β , 1 ) + cos ( θ ) ( β , 1 + η α , 2 ) ] ,
ϵ j j = ω 0 2 η 2 ( ω 0 + ω ) ,
ϵ k k = ω 0 2 ω + η 2 ( ω 0 + ω ) ,
ϵ j k = ϵ k j = η ( ω 0 + ω ) η 3 ( ω 0 + ω ) .
ρ β β ( T 2 ) = 2 N 4 [ sin ( θ ) η cos ( θ ) ] 2 [ cos ( θ ) + η sin ( θ ) ] 2 [ 1 cos ( 2 q T ) ] .
ρ β β ( T 2 T 0 4 ) = 2 N 4 [ sin ( θ ) + η cos ( θ ) ] 2 [ cos ( θ ) η sin ( θ ) ] 2 [ 1 cos ( 2 q T ) ] .
δ = 2 η sin ( θ ) cos ( θ ) [ sin ( θ ) 2 cos ( θ ) 2 ] ( 1 η 2 ) sin ( θ ) 2 cos ( θ ) 2 ( 1 6 η 2 + η 4 ) + η 2 .

Metrics