Abstract

We examine the effect of a frequency-chirped external force field on the final energy that has been absorbed by two classical mechanical oscillators, by quantum mechanical two- and three-level systems, and by electron-positron pairs that were created from the quantum field theoretical Dirac vacuum. By comparing the final dynamical responses to the original force field with that associated with the corresponding time-reversed field, we can test the sensitivity of each of these five systems to the temporal phase information contained in the field. We predict that the linear oscillator, the two-level atom, and the pair-creation process triggered by a spatially homogeneous field are remarkably immune to this phase, whereas the quartic oscillator, the three-level atom, or the pair-creation process caused by a space-time field absorb the provided energy differently depending on the temporal details of the external field.

© 2020 Optical Society of America

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References

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  1. S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
    [Crossref]
  2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
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  4. R. L. McCrory, Proceedings of Frontiers in Optics (Optical Society of America, 2006).
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).
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    [Crossref]
  7. Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
    [Crossref]
  8. H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
    [Crossref]
  9. M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
    [Crossref]
  10. A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
    [Crossref]
  11. A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
    [Crossref]
  12. J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82, 664–679 (1951).
    [Crossref]
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    [Crossref]
  14. F. Sauter, “Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69, 742–764 (1931).
    [Crossref]
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    [Crossref]
  16. B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
    [Crossref]
  17. S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
    [Crossref]
  18. A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
    [Crossref]
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    [Crossref]
  21. F. Hebenstreit and F. Fillion-Gourdeau, “Optimization of Schwinger pair production in colliding laser pulses,” Phys. Lett. B 739, 189–195 (2014).
    [Crossref]
  22. S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
    [Crossref]
  23. J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
    [Crossref]
  24. J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
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  28. C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
    [Crossref]
  29. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975).
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    [Crossref]
  33. Q. Su and R. Grobe, “Dirac vacuum as a transport medium for information,” Phys. Rev. Lett. 122, 023603 (2019).
    [Crossref]
  34. A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
    [Crossref]
  35. T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
    [Crossref]
  36. B. Thaller, The Dirac Equation (Springer, 1992).
  37. W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, 1985).
  38. P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
    [Crossref]
  39. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [Crossref]
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2020 (1)

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

2019 (3)

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

Q. Su and R. Grobe, “Dirac vacuum as a transport medium for information,” Phys. Rev. Lett. 122, 023603 (2019).
[Crossref]

2018 (2)

Q. Z. Lv, Q. Su, and R. Grobe, “Manipulation of the vacuum to control its field-induced decay,” Phys. Rev. Lett. 121, 183606 (2018).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

2017 (2)

A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
[Crossref]

B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
[Crossref]

2014 (1)

F. Hebenstreit and F. Fillion-Gourdeau, “Optimization of Schwinger pair production in colliding laser pulses,” Phys. Lett. B 739, 189–195 (2014).
[Crossref]

2013 (6)

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
[Crossref]

H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
[Crossref]

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

2012 (1)

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

2010 (3)

T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
[Crossref]

C. K. Dumlu, “Schwinger vacuum pair production in chirped laser pulses,” Phys. Rev. D 82, 045007 (2010).
[Crossref]

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

2009 (1)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

2008 (1)

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

2004 (1)

P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
[Crossref]

1985 (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219 (1985).
[Crossref]

1951 (1)

J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82, 664–679 (1951).
[Crossref]

1946 (1)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

1936 (1)

W. Heisenberg and H. Euler, “Consequences of Dirac theory of the positron,” Z. Phys. 98, 714–732 (1936).
[Crossref]

1934 (1)

G. Breit and J. A. Wheeler, “Collision of two light quanta,” Phys. Rev. 46, 1087–1091 (1934).
[Crossref]

1931 (1)

F. Sauter, “Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69, 742–764 (1931).
[Crossref]

Alkofer, R.

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Allen, L.

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

Bergeman, T.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

Bloch, F.

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

Breit, G.

G. Breit and J. A. Wheeler, “Collision of two light quanta,” Phys. Rev. 46, 1087–1091 (1934).
[Crossref]

Bulanov, S. S.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

Cheng, T.

T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
[Crossref]

Clow, S. D.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

Di Piazza, A.

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

Dong, S. S.

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

Duffing, G.

G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (F. Vieweg u. Sohn, 1918).

Dumlu, C. K.

C. K. Dumlu, “Schwinger vacuum pair production in chirped laser pulses,” Phys. Rev. D 82, 045007 (2010).
[Crossref]

Eberly, J. H.

P. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

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

Euler, H.

W. Heisenberg and H. Euler, “Consequences of Dirac theory of the positron,” Z. Phys. 98, 714–732 (1936).
[Crossref]

Fillion-Gourdeau, F.

F. Hebenstreit and F. Fillion-Gourdeau, “Optimization of Schwinger pair production in colliding laser pulses,” Phys. Lett. B 739, 189–195 (2014).
[Crossref]

Flores, R.

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

Gonoskov, A.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Gonoskov, I.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Greiner, W.

W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, 1985).

Grobe, R.

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

Q. Su and R. Grobe, “Dirac vacuum as a transport medium for information,” Phys. Rev. Lett. 122, 023603 (2019).
[Crossref]

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

Q. Z. Lv, Q. Su, and R. Grobe, “Manipulation of the vacuum to control its field-induced decay,” Phys. Rev. Lett. 121, 183606 (2018).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
[Crossref]

P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
[Crossref]

Guckenheimer, J.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1983).

Harvey, C.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Hatsagortsyan, K. Z.

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

Hebenstreit, F.

F. Hebenstreit and F. Fillion-Gourdeau, “Optimization of Schwinger pair production in colliding laser pulses,” Phys. Lett. B 739, 189–195 (2014).
[Crossref]

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Heisenberg, W.

W. Heisenberg and H. Euler, “Consequences of Dirac theory of the positron,” Z. Phys. 98, 714–732 (1936).
[Crossref]

Holland, D.

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

Holmes, P.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1983).

Ilderton, A.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Jiang, M.

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
[Crossref]

Keitel, C. H.

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

Kim, A.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

Kohlfürst, C.

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

C. Kohlfürst, “Electron-positron pair production in structured pulses of electric fields,” Master’s thesis (Graz University, 2012).

Krausz, F.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Krekora, P.

P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
[Crossref]

Li, Z. L.

B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
[Crossref]

A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
[Crossref]

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
[Crossref]

Lv, Q. Z.

Q. Z. Lv, Q. Su, and R. Grobe, “Manipulation of the vacuum to control its field-induced decay,” Phys. Rev. Lett. 121, 183606 (2018).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Marklund, M.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

McCrory, R. L.

R. L. McCrory, Proceedings of Frontiers in Optics (Optical Society of America, 2006).

McGinnes, C.

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

Milonni, P.

P. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

Mitter, M.

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Mourou, G.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219 (1985).
[Crossref]

Müller, B.

W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, 1985).

Müller, C.

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

Mur, V. D.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

Narozhny, N. B.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

Nees, J.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

Nurriman, A.

A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
[Crossref]

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

Popov, V. S.

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

Rafelski, J.

W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, 1985).

Sang, H. B.

H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
[Crossref]

Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
[Crossref]

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

Sauter, F.

F. Sauter, “Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69, 742–764 (1931).
[Crossref]

Sayipjamal, D.

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

Schwinger, J. S.

J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82, 664–679 (1951).
[Crossref]

Sergeev, A.

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

Strickland, D.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219 (1985).
[Crossref]

Su, Q.

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

Q. Su and R. Grobe, “Dirac vacuum as a transport medium for information,” Phys. Rev. Lett. 122, 023603 (2019).
[Crossref]

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

Q. Z. Lv, Q. Su, and R. Grobe, “Manipulation of the vacuum to control its field-induced decay,” Phys. Rev. Lett. 121, 183606 (2018).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
[Crossref]

P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
[Crossref]

Tang, S.

B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
[Crossref]

Thaller, B.

B. Thaller, The Dirac Equation (Springer, 1992).

Trallero-Herrero, C.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

Unger, J.

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

von Winckel, G.

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Weinacht, T.

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

Wheeler, J. A.

G. Breit and J. A. Wheeler, “Collision of two light quanta,” Phys. Rev. 46, 1087–1091 (1934).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

Xie, B. S.

B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
[Crossref]

A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
[Crossref]

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
[Crossref]

Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
[Crossref]

Chin. Phys. B (2)

M. Jiang, B. S. Xie, H. B. Sang, and Z. L. Li, “Enhanced electron-positron pair creation by the frequency chirped laser pulse,” Chin. Phys. B 22, 100307 (2013).
[Crossref]

A. Nurriman, Z. L. Li, and B. S. Xie, “Enhanced electron-positron pair production by frequency chirping in one- and two-color laser pulse fields,” Chin. Phys. B 26, 020301 (2017).
[Crossref]

Chin. Phys. Lett. (2)

Z. L. Li, H. B. Sang, and B. S. Xie, “Enhanced electron-positron pair production of a vacuum in a strong laser pulse field by frequency variation,” Chin. Phys. Lett. 30, 071201 (2013).
[Crossref]

H. B. Sang, M. Jiang, and B. S. Xie, “Electron-positron pair creation from vacuum by using negative frequency chirping laser pulses,” Chin. Phys. Lett. 30, 111201 (2013).
[Crossref]

Comm. Theor. Phys. (1)

A. Nurriman, B. S. Xie, Z. L. Li, and D. Sayipjamal, “Electron-positron pair production in a strong laser field enhanced by an assisted high frequency weak field,” Comm. Theor. Phys. 59, 331–334 (2013).
[Crossref]

Contemp. Phys. (1)

T. Cheng, Q. Su, and R. Grobe, “Introductory review on quantum field theory with space–time resolution,” Contemp. Phys. 51, 315–330 (2010).
[Crossref]

Matter Radiat. Extremes (1)

B. S. Xie, Z. L. Li, and S. Tang, “Electron-positron pair production in ultrastrong laser fields,” Matter Radiat. Extremes 2, 225–242 (2017).
[Crossref]

Opt. Commun. (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219 (1985).
[Crossref]

Phys. Lett. B (1)

F. Hebenstreit and F. Fillion-Gourdeau, “Optimization of Schwinger pair production in colliding laser pulses,” Phys. Lett. B 739, 189–195 (2014).
[Crossref]

Phys. Rev. (3)

J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. 82, 664–679 (1951).
[Crossref]

G. Breit and J. A. Wheeler, “Collision of two light quanta,” Phys. Rev. 46, 1087–1091 (1934).
[Crossref]

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[Crossref]

Phys. Rev. A (2)

J. Unger, S. S. Dong, Q. Su, and R. Grobe, “Optimal supercritical potentials for the electron-positron pair-creation rate,” Phys. Rev. A 100, 012518 (2019).
[Crossref]

J. Unger, S. S. Dong, R. Flores, Q. Su, and R. Grobe, “Infinite-dimensional optimization applied to pair creation from the vacuum,” Phys. Rev. A 99, 022128 (2019).
[Crossref]

Phys. Rev. D (2)

C. K. Dumlu, “Schwinger vacuum pair production in chirped laser pulses,” Phys. Rev. D 82, 045007 (2010).
[Crossref]

C. Kohlfürst, M. Mitter, G. von Winckel, F. Hebenstreit, and R. Alkofer, “Optimizing the pulse shape for Schwinger pair production,” Phys. Rev. D 88, 045028 (2013).
[Crossref]

Phys. Rev. E (2)

S. S. Dong, R. Flores, J. Unger, Q. Su, and R. Grobe, “Superposition principle for the simultaneous optimization of collective responses,” Phys. Rev. E 98, 012221 (2018).
[Crossref]

C. McGinnes, D. Holland, Q. Su, and R. Grobe, “Universal scaling laws for optimally excited nonlinear oscillators,” Phys. Rev. E 101, 032202 (2020).
[Crossref]

Phys. Rev. Lett. (6)

Q. Z. Lv, Q. Su, and R. Grobe, “Manipulation of the vacuum to control its field-induced decay,” Phys. Rev. Lett. 121, 183606 (2018).
[Crossref]

Q. Su and R. Grobe, “Dirac vacuum as a transport medium for information,” Phys. Rev. Lett. 122, 023603 (2019).
[Crossref]

S. S. Bulanov, N. B. Narozhny, V. D. Mur, J. Nees, and V. S. Popov, “Multiple colliding electromagnetic pulses: a way to lower the threshold of e+ e- pair production from vacuum,” Phys. Rev. Lett. 104, 220404 (2010).
[Crossref]

A. Gonoskov, I. Gonoskov, C. Harvey, A. Ilderton, A. Kim, M. Marklund, G. Mourou, and A. Sergeev, “Probing nonperturbative QED with optimally focused laser pulses,” Phys. Rev. Lett. 111, 060404 (2013).
[Crossref]

S. D. Clow, C. Trallero-Herrero, T. Bergeman, and T. Weinacht, “Strong field multiphoton inversion of a three-level system using shaped ultrafast laser pulses,” Phys. Rev. Lett. 100, 233603 (2008).
[Crossref]

P. Krekora, Q. Su, and R. Grobe, “Klein paradox in spatial and temporal resolution,” Phys. Rev. Lett. 92, 040406 (2004).
[Crossref]

Rev. Mod. Phys. (2)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. 84, 1177–1228 (2012).
[Crossref]

Z. Phys. (2)

F. Sauter, “Über das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69, 742–764 (1931).
[Crossref]

W. Heisenberg and H. Euler, “Consequences of Dirac theory of the positron,” Z. Phys. 98, 714–732 (1936).
[Crossref]

Other (12)

C. Kohlfürst, “Electron-positron pair production in structured pulses of electric fields,” Master’s thesis (Graz University, 2012).

R. L. McCrory, Proceedings of Frontiers in Optics (Optical Society of America, 2006).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, 1995).

B. Thaller, The Dirac Equation (Springer, 1992).

W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, 1985).

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

P. Milonni and J. H. Eberly, Lasers (Wiley, 1988).

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung (F. Vieweg u. Sohn, 1918).

I. Kovacic and M. J. Brennan, eds., The Duffing Equation: Nonlinear Oscillators and Their Behavior (Wiley, 2011).

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, 1983).

G. W. F. Drake, ed., Handbook of Atomic, Molecular, and Optical Physics (Springer, 2005).

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Figures (7)

Fig. 1.
Fig. 1. Temporal profiles of the chirped electric force fields ${ F }({t})$ and ${{ F }_{ R }}({t})$ used to excite the classical oscillators and quantum systems discussed in Section 2. We also show their energy spectrum $|{ F }(\omega ){|^2} = |{{ F }_{ R }}(\omega ){|^2}$. In all of our numerical illustrations, we use the atomic unit system and ${ F }({t}) = {{2}^{1/2}}\;{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]\;{\rm a}.{\rm u}$.
Fig. 2.
Fig. 2. Time dependence of the energy ${ e }({t}) = {({dx}/{\rm dt})^2}/{2}\;{ + }\;{\Omega ^2}{{x}^2}/{2}$ of a harmonic oscillator driven by the chirped force field ${ F }({t})$ and its time-reversed form ${ F_R}({t})$, presented in Fig. 1, with parameters $\Omega = {1}\;{\rm a.}{\rm u.}$ and ${ F }({t}) = {{2}^{1/2}}\;{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]\;{\rm a}.{\rm u}$. Here and in all of the following figures the energies ${ e }({t})$ and ${{ e }_{ R }}({t})$ are also in atomic units.
Fig. 3.
Fig. 3. Time dependence of the energy ${ e }({t}) = {({dx}/{dt})^2}/{2}\;{ + }\;{{x}^4}/{4}$ of a quartic oscillator driven by the chirped force field ${ F }({t})$ and its time-reversed form ${{ F }_{ R }}({t})$, as presented in Fig. 1, ${ F }({t}) = {{2}^{1/2}}$ ${\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$.
Fig. 4.
Fig. 4. Time dependence of the total energy ${ e }({t}) \equiv {\omega _1}|{{ C }_1}({t}){|^2} + {\omega _2}$ $|{{ C }_2}({t}){|^2}$ of a two-level atom excited by the chirped force ${ F }({t}) = {{2}^{1/2\:}}{\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$ and by its temporally reversed form ${{ F }_{ R }}({t})$. [The energies are ${\omega _1} = {0}\;{\rm a}.{\rm u}.$ and ${\omega _2} = {3}\;{\rm a}.{\rm u}.$; ${{d}_{12}} = {{d}_{21\:}} = {0}.{5}\;{\rm a}.{\rm u}.$]
Fig. 5.
Fig. 5. Time dependence of the energy ${\omega _1}|{{ C }_1}({t}){|^2} + \;{\omega _2}|{{ C }_2}({t}){|^2} + \;{\omega _3}|{{ C }_3}({t}){|^2}$ of a three-level atom when pumped by the chirped force ${ F }({t}) = {{2}^{1/2}}$ ${\sin}[({1}.{4}\;{ + }\;{0}.{065}\;{t}){t}]$ and by its temporally reversed force ${{ F }_{ R }}({t})$. Left figure: ${{d}_{12}} = {{d}_{23}} = {1}$ a.u. Right figure: ${{d}_{12}} = {1}\;{\rm a}.{\rm u}.$ but ${{d}_{23}} = {2}\;{\rm a}.{\rm u}$. The energies were ${\omega _1} = {0}\;{\rm a}.{\rm u}.$, ${\omega _2} = {3}\;{\rm a}.{\rm u}.$, and ${\omega _3} = {6}\;{\rm a}.{\rm u}$.
Fig. 6.
Fig. 6. Growth of the energy of the created electrons during the interaction with a chirped external electric field. [The numerical box of length ${L} = {2}.{4}$ a.u. was discretized into ${{N}_x} = {512}$ spatial grid points. The temporal evolution was discretized into 5000 grid points. The electric field was given by ${ F }({t})\; =\; {{ F }_0}\;{\exp}[\; - \;{({t} \; - \; {{t}_1})^2}/({2}{\tau ^2})]\;{ \cos}[(\omega \;+ \;{ b }({t} \;-\; {{t}_1}))({t} \;-\; {{t}_1})]$, with ${ b } = {5}.{6} \times\;{{10}^6}\;{\rm a}.{\rm u}.$, $\omega = {2}.{8}\;{{ C }^2}$, ${\tau} = \;{5}.{325}\; \times \;{{10}^{ - 4}}\;{\rm a}.{\rm u}.$, ${{t}_{1\:}} =$ $\;{0}.{004}\;{\rm a}.{\rm u}.$, and ${{ F }_0} = {5}\;{{ C }^3}$.
Fig. 7.
Fig. 7. Growth of the total energy of the created electrons during the interaction with a chirped external field. The numerical box of length ${L}= {2.4}$ a.u. was discretized into ${{ N}_x} = {512}$ spatial grid points. The temporal evolution was discretized into 5000 grid points. The scalar potential was given by ${ V}({x},{t})= {F}({t},{ x})$ ${V}({ x})$, where ${V}({ x}) = {{V}_0}\{{\tanh}[({ x} - {D})/{W}] - {\tanh}[({x} + {D})/{W}]\}/{2}$, with ${{V}_{0\:}} = {5}\;{{c}^2}$, ${W} = {0}.{ 5/c}$, ${D} = {0}.{6}\;{\rm a}.{\rm u}.$, and ${F}({t},{ x}) = {\rm exp}[\; - \;{({t} - {{ t}_1})^2}/({2}{\tau ^2})]{\cos}[(\omega + {b}({t} - {{t}_1}))({t} - {{t}_1})\; - \;{x}\omega /{ C }]$, with ${b} = {5}.{6} \times {{10}^6}\;{\rm a}.{\rm u}.$, $\omega = {2}.{8}$ ${{c}^2}$, $\tau = {5}.{325}\; \times \;{{10}^{ - 4}}\;{\rm a}.{\rm u}.$, and ${{ t}_1} = {0}.{004}\;{\rm a}.{\rm u}$.

Equations (36)

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H ( t ) = ω 1 | 1 1 | + ω 2 | 2 2 | + F ( t ) d 12 { | 1 2 | + | 2 1 | } ,
i d C 1 ( t ) / d t = ω 1 C 1 ( t ) + F ( t ) d 12 C 2 ( t ) ,
i d C 2 ( t ) / d t = ω 2 C 2 ( t ) + F ( t ) d 21 C 1 ( t ) .
H ( t ) = ω 1 | 1 1 | + ω 2 | 2 2 | + ω 3 | 3 3 | + F ( t ) d 12 { | 1 2 | + | 2 1 | } + F ( t ) d 23 { | 2 3 | + | 3 2 | } ,
i d C 1 ( t ) / d t = ω 1 C 1 ( t ) + F ( t ) d 12 C 2 ( t ) ,
i d C 2 ( t ) / d t = ω 2 C 2 ( t ) + F ( t ) d 12 C 1 ( t ) + F ( t ) d 23 C 3 ( t ) ,
i d C 3 ( t ) / d t = ω 3 C 3 ( t ) + F ( t ) d 23 C 2 ( t ) .
F ( t ) = F 0 exp [ ( t t 1 ) 2 / ( 2 τ 2 ) ] cos [ ( ω + b ( t t 1 ) ) ( t t 1 ) ] ,
N inc ( t ) k ( 1 | k ; d | k ; d ( t ) | 2 ) ,
N c o h ( t ) k k k | k ; d | k ; d ( t ) | 2 ,
| K ; d ( t ) = Σ P C P ( K , t ) | P ; u + Σ k D k ( K , t ) | k ; d ,
| k ; d ( t ) = k D k ( k , t ) | k ; d ,
V ( x ) = V 0 { tanh [ ( x D ) / W ] tanh [ ( x + D ) / W ] ) } / 2 ,
F ( t , x ) = exp [ t 2 / ( 2 τ 2 ) ] cos [ ( ω + b t ) t ω x / c ] ,
d S 1 ( t ) / d t = ω 0 S 2 ( t ) ,
d S 2 ( t ) / d t = ω 0 S 1 ( t ) f ( t ) S 3 ( t ) ,
d S 3 ( t ) / d t = f ( t ) S 2 ( t ) ,
G [ ω 0 , f ( t ) ] { { 0 , ω 0 , 0 } , { ω 0 , 0 , f ( t ) } , { 0 , f ( t ) , 0 } } .
P = { { P 11 , P 12 , P 13 } , { P 12 , P 22 , P 23 } , { P 13 , P 23 , P 33 } }
P 11 = [ f ( t n ) 2 + ω 0 2 cos ( Ω n Δ t ) ] Ω n 2 ,
P 12 = ω 0 Ω n 1 sin ( Ω n Δ t ) ] ,
P 13 = ω 0 f ( t n ) Ω n 2 [ 1 cos ( Ω n Δ t ) ] ,
P 22 = cos ( Ω n Δ t ) ,
P 23 = f ( t n ) Ω n 1 sin ( Ω n Δ t ) ,
P 33 = [ ω 0 2 + f ( t n ) 2 cos ( Ω n Δ t ) ] Ω n 2 .
S ( T ) = P t o t { 0 , 0 , 1 } ,
S R ( T ) = P R , t o t { 0 , 0 , 1 } ,
e ( T ) = S 3 ( T ) ( ω 2 / 2 ) + ω 2 / 2 = { 0 , 0 , 1 } P t o t { 0 , 0 , 1 } ( ω 2 / 2 ) + ω 2 / 2 ,
e R ( T ) = S R , 3 ( T ) ( ω 2 / 2 ) + ω 2 / 2 = { 0 , 0 , 1 } P R , t o t { 0 , 0 , 1 } ( ω 2 / 2 ) + ω 2 / 2.
P R , t o t T = [ P ( t 1 ) P ( t 2 ) P ( t N 1 ) P ( t N ) ] T = [ P ( t N ) P ( t N 1 ) P ( t 2 ) P ( t 1 ) ] T = M P ( t N ) P ( t N 1 ) P ( t 2 ) P ( t 1 ) M = M P t o t M ,
Ψ | U | Ψ = Ψ | U R | Ψ ,
j | U ( n ) | k = k | U ( n ) | j .
j | U | k = k | U R | j .
Ψ | U | Ψ = ϕ | U R | ϕ
l Ψ | U | Ψ = Σ k C k k | U Σ j C j | j = Σ k Σ j C k C j k | U | j = Σ k Σ j C k C j j | U R | k = ϕ | U R | ϕ
ϕ | U | Ψ = ϕ | U R | Ψ .

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