Abstract

We demonstrate numerically that the long-wavelength nonlinear dipole moment and ionization rate versus electric field strength $F$ for different noble gases can be scaled onto each other, revealing universal functions that characterize the form of the nonlinear response. We elucidate the physical origin of the universality by using a metastable state analysis of the light-atom interaction in combination with a scaling analysis. Our results also provide a powerful new means of characterizing the nonlinear response in the mid-infrared and long-wave infrared for optical filamentation studies.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. L. Kadanoff, “Scaling and universality in statistical physics,” Phys. A 163(1), 1–14 (1990).
    [Crossref]
  2. E. Najafi and A. H. Darooneh, “A new universality class in corpus of texts; a statistical physics study,” Phys. Lett. A 382(17), 1140–1148 (2018).
    [Crossref]
  3. P. Naidon and S. Endo, “Efimov physics: a review,” Rep. Prog. Phys. 80(5), 056001 (2017).
    [Crossref]
  4. X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At., Mol. Opt. Phys. 38(15), 2593–2600 (2005).
    [Crossref]
  5. F. Cloux, B. Fabre, and B. Pons, “Semiclassical description of high-order-harmonic spectroscopy of the cooper minimum in krypton,” Phys. Rev. A 91(2), 023415 (2015).
    [Crossref]
  6. S.-F. Zhao, L. Liu, and X.-X. Zhou, “Multiphoton and tunneling ionization probability of atoms and molecules in an intense laser field,” Opt. Commun. 313, 74–79 (2014).
    [Crossref]
  7. P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
    [Crossref]
  8. K. Schuh, P. Panagiotopoulos, M. Kolesik, S. W. Koch, and J. V. Moloney, “Multi-terawatt 10 micron pulse atmospheric delivery over multiple rayleigh ranges,” Opt. Lett. 42(19), 3722–3725 (2017).
    [Crossref]
  9. L. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Phys. JETP 20, 1307–1314 (1965).
  10. M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1(5), 323–331 (2014).
    [Crossref]
  11. A. Bahl, J. M. Brown, E. M. Wright, and M. Kolesik, “Assessment of the Metastable Electronic State Approach as a microscopically self-consistent description for the nonlinear response of atoms,” Opt. Lett. 40(21), 4987–4990 (2015).
    [Crossref]
  12. A. Bahl, E. M. Wright, and M. Kolesik, “Nonlinear optical response of noble gases via the metastable electronic state approach,” Phys. Rev. A 94(2), 023850 (2016).
    [Crossref]
  13. G. Gamow, “Zur Quantentheorie des Atomkernes,” Eur. Phys. J. A 51(3-4), 204–212 (1928).
    [Crossref]
  14. A. J. F. Siegert, “On the derivation of the dispersion formula for nuclear reactions,” Phys. Rev. 56(8), 750–752 (1939).
    [Crossref]
  15. L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
    [Crossref]
  16. N. Moiseyev, Non-Hermitian quantum mechanics (Cambridge University Press, 2011).
  17. D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A: Math. Theor. 47(3), 035305 (2014).
    [Crossref]
  18. J. M. Brown and M. Kolesik, “Properties of stark resonant states in exactly solvable systems,” Adv. Math. Phys. 2015, 1–11 (2015).
    [Crossref]
  19. P. Lindl, “Completeness relations and resonant state expansions,” Phys. Rev. C: Nucl. Phys. 47(5), 1903–1920 (1993).
    [Crossref]
  20. O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep. 396(2), 41–113 (2004).
    [Crossref]
  21. O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
    [Crossref]
  22. A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
    [Crossref]
  23. O. Smirnova, “Validity of the Kramers-Henneberger approximation,” J. Exp. Theor. Phys. 90(4), 609–616 (2000).
    [Crossref]
  24. S. Chu and W. Reinhardt, “Intense field multiphoton ionization via complex dressed states: Application to the H atom,” Phys. Rev. Lett. 39(19), 1195–1198 (1977).
    [Crossref]
  25. M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
    [Crossref]
  26. G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64(1), 013409 (2001).
    [Crossref]
  27. A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
    [Crossref]
  28. A. Bohm, “Resonances/decaying states and the mathematics of quantum physics,” Rep. Math. Phys. 67(3), 279–303 (2011).
    [Crossref]
  29. G. García-Calderón, A. Máttar, and J. Villavicencio, “Hermitian and non-hermitian formulations of the time evolution of quantum decay,” Phys. Scr. T151, 014076 (2012).
    [Crossref]
  30. H. L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, 1987).
  31. N. Moiseyev, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Phys. Rep. 302(5-6), 212–293 (1998).
    [Crossref]
  32. J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
    [Crossref]
  33. R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. 36(48), 12065–12080 (2003).
    [Crossref]
  34. A. Emmanouilidou and N. Moiseyev, “Stark and field-born resonances of an open square well in a static external electric field,” J. Chem. Phys. 122(19), 194101 (2005).
    [Crossref]
  35. J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
    [Crossref]
  36. B. Gyarmati and T. Vertse, “On the normalization of Gamow functions,” Nucl. Phys. A 160(3), 523–528 (1971).
    [Crossref]
  37. Y. B. Zel’dovich, “On the theory of unstable states,” J. Exp. Theor. Phys. (U.S.S.R.) 39, 776–780 (1960).
  38. J. Julve and F. J. de Urries, “Inner products of resonance solutions in 1d quantum barriers,” J. Phys. A: Math. Theor. 43(17), 175301 (2010).
    [Crossref]
  39. W. J. Romo, “Inner product for resonant states and shell-model applications,” Nucl. Phys. A 116(3), 617–636 (1968).
    [Crossref]

2018 (1)

E. Najafi and A. H. Darooneh, “A new universality class in corpus of texts; a statistical physics study,” Phys. Lett. A 382(17), 1140–1148 (2018).
[Crossref]

2017 (3)

P. Naidon and S. Endo, “Efimov physics: a review,” Rep. Prog. Phys. 80(5), 056001 (2017).
[Crossref]

K. Schuh, P. Panagiotopoulos, M. Kolesik, S. W. Koch, and J. V. Moloney, “Multi-terawatt 10 micron pulse atmospheric delivery over multiple rayleigh ranges,” Opt. Lett. 42(19), 3722–3725 (2017).
[Crossref]

A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
[Crossref]

2016 (2)

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

A. Bahl, E. M. Wright, and M. Kolesik, “Nonlinear optical response of noble gases via the metastable electronic state approach,” Phys. Rev. A 94(2), 023850 (2016).
[Crossref]

2015 (4)

A. Bahl, J. M. Brown, E. M. Wright, and M. Kolesik, “Assessment of the Metastable Electronic State Approach as a microscopically self-consistent description for the nonlinear response of atoms,” Opt. Lett. 40(21), 4987–4990 (2015).
[Crossref]

J. M. Brown and M. Kolesik, “Properties of stark resonant states in exactly solvable systems,” Adv. Math. Phys. 2015, 1–11 (2015).
[Crossref]

P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
[Crossref]

F. Cloux, B. Fabre, and B. Pons, “Semiclassical description of high-order-harmonic spectroscopy of the cooper minimum in krypton,” Phys. Rev. A 91(2), 023415 (2015).
[Crossref]

2014 (4)

S.-F. Zhao, L. Liu, and X.-X. Zhou, “Multiphoton and tunneling ionization probability of atoms and molecules in an intense laser field,” Opt. Commun. 313, 74–79 (2014).
[Crossref]

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1(5), 323–331 (2014).
[Crossref]

D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A: Math. Theor. 47(3), 035305 (2014).
[Crossref]

A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
[Crossref]

2012 (2)

G. García-Calderón, A. Máttar, and J. Villavicencio, “Hermitian and non-hermitian formulations of the time evolution of quantum decay,” Phys. Scr. T151, 014076 (2012).
[Crossref]

L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
[Crossref]

2011 (2)

J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
[Crossref]

A. Bohm, “Resonances/decaying states and the mathematics of quantum physics,” Rep. Math. Phys. 67(3), 279–303 (2011).
[Crossref]

2010 (1)

J. Julve and F. J. de Urries, “Inner products of resonance solutions in 1d quantum barriers,” J. Phys. A: Math. Theor. 43(17), 175301 (2010).
[Crossref]

2008 (1)

2005 (2)

A. Emmanouilidou and N. Moiseyev, “Stark and field-born resonances of an open square well in a static external electric field,” J. Chem. Phys. 122(19), 194101 (2005).
[Crossref]

X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At., Mol. Opt. Phys. 38(15), 2593–2600 (2005).
[Crossref]

2004 (1)

O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep. 396(2), 41–113 (2004).
[Crossref]

2003 (1)

R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. 36(48), 12065–12080 (2003).
[Crossref]

2001 (1)

G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64(1), 013409 (2001).
[Crossref]

2000 (1)

O. Smirnova, “Validity of the Kramers-Henneberger approximation,” J. Exp. Theor. Phys. 90(4), 609–616 (2000).
[Crossref]

1998 (1)

N. Moiseyev, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Phys. Rep. 302(5-6), 212–293 (1998).
[Crossref]

1997 (1)

O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
[Crossref]

1993 (1)

P. Lindl, “Completeness relations and resonant state expansions,” Phys. Rev. C: Nucl. Phys. 47(5), 1903–1920 (1993).
[Crossref]

1990 (1)

L. Kadanoff, “Scaling and universality in statistical physics,” Phys. A 163(1), 1–14 (1990).
[Crossref]

1977 (1)

S. Chu and W. Reinhardt, “Intense field multiphoton ionization via complex dressed states: Application to the H atom,” Phys. Rev. Lett. 39(19), 1195–1198 (1977).
[Crossref]

1971 (1)

B. Gyarmati and T. Vertse, “On the normalization of Gamow functions,” Nucl. Phys. A 160(3), 523–528 (1971).
[Crossref]

1968 (1)

W. J. Romo, “Inner product for resonant states and shell-model applications,” Nucl. Phys. A 116(3), 617–636 (1968).
[Crossref]

1965 (1)

L. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Phys. JETP 20, 1307–1314 (1965).

1960 (1)

Y. B. Zel’dovich, “On the theory of unstable states,” J. Exp. Theor. Phys. (U.S.S.R.) 39, 776–780 (1960).

1939 (1)

A. J. F. Siegert, “On the derivation of the dispersion formula for nuclear reactions,” Phys. Rev. 56(8), 750–752 (1939).
[Crossref]

1928 (1)

G. Gamow, “Zur Quantentheorie des Atomkernes,” Eur. Phys. J. A 51(3-4), 204–212 (1928).
[Crossref]

Bahl, A.

A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
[Crossref]

A. Bahl, E. M. Wright, and M. Kolesik, “Nonlinear optical response of noble gases via the metastable electronic state approach,” Phys. Rev. A 94(2), 023850 (2016).
[Crossref]

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

A. Bahl, J. M. Brown, E. M. Wright, and M. Kolesik, “Assessment of the Metastable Electronic State Approach as a microscopically self-consistent description for the nonlinear response of atoms,” Opt. Lett. 40(21), 4987–4990 (2015).
[Crossref]

A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
[Crossref]

Bohm, A.

A. Bohm, “Resonances/decaying states and the mathematics of quantum physics,” Rep. Math. Phys. 67(3), 279–303 (2011).
[Crossref]

Brody, D. C.

D. C. Brody, “Biorthogonal quantum mechanics,” J. Phys. A: Math. Theor. 47(3), 035305 (2014).
[Crossref]

Brown, J. M.

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

J. M. Brown and M. Kolesik, “Properties of stark resonant states in exactly solvable systems,” Adv. Math. Phys. 2015, 1–11 (2015).
[Crossref]

A. Bahl, J. M. Brown, E. M. Wright, and M. Kolesik, “Assessment of the Metastable Electronic State Approach as a microscopically self-consistent description for the nonlinear response of atoms,” Opt. Lett. 40(21), 4987–4990 (2015).
[Crossref]

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1(5), 323–331 (2014).
[Crossref]

J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
[Crossref]

Cavalcanti, R. M.

R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. 36(48), 12065–12080 (2003).
[Crossref]

Chu, S.

S. Chu and W. Reinhardt, “Intense field multiphoton ionization via complex dressed states: Application to the H atom,” Phys. Rev. Lett. 39(19), 1195–1198 (1977).
[Crossref]

Civitarese, O.

O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep. 396(2), 41–113 (2004).
[Crossref]

Cloux, F.

F. Cloux, B. Fabre, and B. Pons, “Semiclassical description of high-order-harmonic spectroscopy of the cooper minimum in krypton,” Phys. Rev. A 91(2), 023415 (2015).
[Crossref]

Cycon, H. L.

H. L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, 1987).

Darooneh, A. H.

E. Najafi and A. H. Darooneh, “A new universality class in corpus of texts; a statistical physics study,” Phys. Lett. A 382(17), 1140–1148 (2018).
[Crossref]

de Urries, F. J.

J. Julve and F. J. de Urries, “Inner products of resonance solutions in 1d quantum barriers,” J. Phys. A: Math. Theor. 43(17), 175301 (2010).
[Crossref]

Emmanouilidou, A.

A. Emmanouilidou and N. Moiseyev, “Stark and field-born resonances of an open square well in a static external electric field,” J. Chem. Phys. 122(19), 194101 (2005).
[Crossref]

Endo, S.

P. Naidon and S. Endo, “Efimov physics: a review,” Rep. Prog. Phys. 80(5), 056001 (2017).
[Crossref]

Fabre, B.

F. Cloux, B. Fabre, and B. Pons, “Semiclassical description of high-order-harmonic spectroscopy of the cooper minimum in krypton,” Phys. Rev. A 91(2), 023415 (2015).
[Crossref]

Froese, R.

H. L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, 1987).

Gadella, M.

O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep. 396(2), 41–113 (2004).
[Crossref]

Gamow, G.

G. Gamow, “Zur Quantentheorie des Atomkernes,” Eur. Phys. J. A 51(3-4), 204–212 (1928).
[Crossref]

García-Calderón, G.

G. García-Calderón, A. Máttar, and J. Villavicencio, “Hermitian and non-hermitian formulations of the time evolution of quantum decay,” Phys. Scr. T151, 014076 (2012).
[Crossref]

Giacconi, P.

R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. 36(48), 12065–12080 (2003).
[Crossref]

Gyarmati, B.

B. Gyarmati and T. Vertse, “On the normalization of Gamow functions,” Nucl. Phys. A 160(3), 523–528 (1971).
[Crossref]

Hamonou, L.

L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
[Crossref]

Ivanov, M. Y.

G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64(1), 013409 (2001).
[Crossref]

Jakobsen, P.

Jakobsen, P. K.

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
[Crossref]

Julve, J.

J. Julve and F. J. de Urries, “Inner products of resonance solutions in 1d quantum barriers,” J. Phys. A: Math. Theor. 43(17), 175301 (2010).
[Crossref]

Kadanoff, L.

L. Kadanoff, “Scaling and universality in statistical physics,” Phys. A 163(1), 1–14 (1990).
[Crossref]

Keldysh, L.

L. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Soviet Phys. JETP 20, 1307–1314 (1965).

Kirsch, W.

H. L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, 1987).

Koch, S. W.

Kolesik, M.

K. Schuh, P. Panagiotopoulos, M. Kolesik, S. W. Koch, and J. V. Moloney, “Multi-terawatt 10 micron pulse atmospheric delivery over multiple rayleigh ranges,” Opt. Lett. 42(19), 3722–3725 (2017).
[Crossref]

A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
[Crossref]

A. Bahl, E. M. Wright, and M. Kolesik, “Nonlinear optical response of noble gases via the metastable electronic state approach,” Phys. Rev. A 94(2), 023850 (2016).
[Crossref]

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

J. M. Brown and M. Kolesik, “Properties of stark resonant states in exactly solvable systems,” Adv. Math. Phys. 2015, 1–11 (2015).
[Crossref]

A. Bahl, J. M. Brown, E. M. Wright, and M. Kolesik, “Assessment of the Metastable Electronic State Approach as a microscopically self-consistent description for the nonlinear response of atoms,” Opt. Lett. 40(21), 4987–4990 (2015).
[Crossref]

P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
[Crossref]

M. Kolesik, J. M. Brown, A. Teleki, P. Jakobsen, J. V. Moloney, and E. M. Wright, “Metastable electronic states and nonlinear response for high-intensity optical pulses,” Optica 1(5), 323–331 (2014).
[Crossref]

A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
[Crossref]

J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
[Crossref]

M. Kolesik and J. V. Moloney, “Perturbative and non-perturbative aspects of optical filamentation in bulk dielectric media,” Opt. Express 16(5), 2971–2988 (2008).
[Crossref]

Lin, C. D.

X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At., Mol. Opt. Phys. 38(15), 2593–2600 (2005).
[Crossref]

Lindl, P.

P. Lindl, “Completeness relations and resonant state expansions,” Phys. Rev. C: Nucl. Phys. 47(5), 1903–1920 (1993).
[Crossref]

Liu, L.

S.-F. Zhao, L. Liu, and X.-X. Zhou, “Multiphoton and tunneling ionization probability of atoms and molecules in an intense laser field,” Opt. Commun. 313, 74–79 (2014).
[Crossref]

Lotti, A.

J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
[Crossref]

Máttar, A.

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[Crossref]

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A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
[Crossref]

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P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
[Crossref]

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P. Naidon and S. Endo, “Efimov physics: a review,” Rep. Prog. Phys. 80(5), 056001 (2017).
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O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
[Crossref]

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O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
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[Crossref]

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[Crossref]

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[Crossref]

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[Crossref]

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J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
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L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
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O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
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G. García-Calderón, A. Máttar, and J. Villavicencio, “Hermitian and non-hermitian formulations of the time evolution of quantum decay,” Phys. Scr. T151, 014076 (2012).
[Crossref]

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A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
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L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
[Crossref]

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P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
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[Crossref]

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G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64(1), 013409 (2001).
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A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
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[Crossref]

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S.-F. Zhao, L. Liu, and X.-X. Zhou, “Multiphoton and tunneling ionization probability of atoms and molecules in an intense laser field,” Opt. Commun. 313, 74–79 (2014).
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A. Emmanouilidou and N. Moiseyev, “Stark and field-born resonances of an open square well in a static external electric field,” J. Chem. Phys. 122(19), 194101 (2005).
[Crossref]

J. Exp. Theor. Phys. (1)

O. Smirnova, “Validity of the Kramers-Henneberger approximation,” J. Exp. Theor. Phys. 90(4), 609–616 (2000).
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J. Exp. Theor. Phys. (U.S.S.R.) (1)

Y. B. Zel’dovich, “On the theory of unstable states,” J. Exp. Theor. Phys. (U.S.S.R.) 39, 776–780 (1960).

J. Lightwave Technol. (1)

A. Bahl, A. Teleki, P. K. Jakobsen, E. M. Wright, and M. Kolesik, “Reflectionless beam propagation on a piecewise linear complex domain,” J. Lightwave Technol. 32(22), 4272–4278 (2014).
[Crossref]

J. Math. Phys. (1)

J. M. Brown, P. K. Jakobsen, A. Bahl, J. V. Moloney, and M. Kolesik, “Stark and field-born resonances of an open square well in a static external electric field,” J. Math. Phys. 57(3), 032105 (2016).
[Crossref]

J. Phys. A: Math. Gen. (1)

R. M. Cavalcanti, P. Giacconi, and R. Soldati, “Decay in a uniform field: an exactly solvable model,” J. Phys. A: Math. Gen. 36(48), 12065–12080 (2003).
[Crossref]

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J. Julve and F. J. de Urries, “Inner products of resonance solutions in 1d quantum barriers,” J. Phys. A: Math. Theor. 43(17), 175301 (2010).
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X. M. Tong and C. D. Lin, “Empirical formula for static field ionization rates of atoms and molecules by lasers in the barrier-suppression regime,” J. Phys. B: At., Mol. Opt. Phys. 38(15), 2593–2600 (2005).
[Crossref]

J. Phys.: Conf. Ser. (1)

L. Hamonou, T. Morishita, O. I. Tolstikhin, and S. Watanabe, “Siegert-state method for ionization of molecules in strong field,” J. Phys.: Conf. Ser. 388(3), 032030 (2012).
[Crossref]

Nat. Photonics (1)

P. Panagiotopoulos, P. Whalen, M. Kolesik, and J. Moloney, “Super high power mid-infrared femtosecond light bullet,” Nat. Photonics 9(8), 543–548 (2015).
[Crossref]

Nucl. Phys. A (2)

B. Gyarmati and T. Vertse, “On the normalization of Gamow functions,” Nucl. Phys. A 160(3), 523–528 (1971).
[Crossref]

W. J. Romo, “Inner product for resonant states and shell-model applications,” Nucl. Phys. A 116(3), 617–636 (1968).
[Crossref]

Opt. Commun. (1)

S.-F. Zhao, L. Liu, and X.-X. Zhou, “Multiphoton and tunneling ionization probability of atoms and molecules in an intense laser field,” Opt. Commun. 313, 74–79 (2014).
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Opt. Express (1)

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Phys. Lett. A (1)

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[Crossref]

Phys. Rep. (2)

O. Civitarese and M. Gadella, “Physical and mathematical aspects of gamow states,” Phys. Rep. 396(2), 41–113 (2004).
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N. Moiseyev, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Phys. Rep. 302(5-6), 212–293 (1998).
[Crossref]

Phys. Rev. (1)

A. J. F. Siegert, “On the derivation of the dispersion formula for nuclear reactions,” Phys. Rev. 56(8), 750–752 (1939).
[Crossref]

Phys. Rev. A (5)

A. Bahl, E. M. Wright, and M. Kolesik, “Nonlinear optical response of noble gases via the metastable electronic state approach,” Phys. Rev. A 94(2), 023850 (2016).
[Crossref]

F. Cloux, B. Fabre, and B. Pons, “Semiclassical description of high-order-harmonic spectroscopy of the cooper minimum in krypton,” Phys. Rev. A 91(2), 023415 (2015).
[Crossref]

J. M. Brown, A. Lotti, A. Teleki, and M. Kolesik, “Exactly solvable model for non-linear light-matter interaction in an arbitrary time-dependent field,” Phys. Rev. A 84(6), 063424 (2011).
[Crossref]

A. Bahl, J. K. Wahlstrand, S. Zahedpour, H. M. Milchberg, and M. Kolesik, “Nonlinear optical polarization response and plasma generation in noble gases: Comparison of metastable-electronic-state-approach models to experiments,” Phys. Rev. A 96(4), 043867 (2017).
[Crossref]

G. L. Yudin and M. Y. Ivanov, “Nonadiabatic tunnel ionization: Looking inside a laser cycle,” Phys. Rev. A 64(1), 013409 (2001).
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S. Chu and W. Reinhardt, “Intense field multiphoton ionization via complex dressed states: Application to the H atom,” Phys. Rev. Lett. 39(19), 1195–1198 (1977).
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O. I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, “Siegert pseudo-states as a universal tool: Resonances, $\mathit {S}$S matrix, green function,” Phys. Rev. Lett. 79(11), 2026–2029 (1997).
[Crossref]

Phys. Scr. (1)

G. García-Calderón, A. Máttar, and J. Villavicencio, “Hermitian and non-hermitian formulations of the time evolution of quantum decay,” Phys. Scr. T151, 014076 (2012).
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N. Moiseyev, Non-Hermitian quantum mechanics (Cambridge University Press, 2011).

H. L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger Operators (Springer-Verlag, 1987).

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Figures (7)

Fig. 1.
Fig. 1. Nonlinear dipole moment $p_\textrm {nl}(F)$ versus field strength $F$ for different noble gas atoms. The symbols represent the ssMESA numerical results, while the continuous lines are generated using the universal form $M({\cal F})$ in (8) along with the tabulated values for the scale parameters $\alpha _a$ and $\beta _a$ for each species given in the Appendix.
Fig. 2.
Fig. 2. Collapse of the nonlinear dipole moment curves in noble gases onto a single universal form $M({\cal F})/M(0)$ versus scaled field strength ${\cal F}$ shown as the thick solid line, where ${\cal F}=\beta _a F$. The numerical data based on ssMESA for each atomic species is shown by the various symbols.
Fig. 3.
Fig. 3. Ionization rate $\Gamma (F)$ versus field strength $F$ for the different noble gases. The symbols represent the ssMESA numerical results, while the continuous lines are generated using the universal form $G({\cal F})$ in (8) along with the tabulated values for the scale parameters $\alpha _a$ and $\beta _a$ for each species given in the Appendix.
Fig. 4.
Fig. 4. Scaled total composite potentials versus scaled coordinate $X$ for the four noble gas species in the vicinity of the saddle point of the potential.
Fig. 5.
Fig. 5. Scaled composite potentials for four noble gas species for small distances from the nucleus. Here, the different species can not be scaled onto each other.
Fig. 6.
Fig. 6. Nonlinear dipole moment (left vertical axes) and ionization rates (right vertical axes) obtained for two different SAE potentials of Krypton are shown in panel a). Panel b) depicts the same curves after suitable horizontal and vertical scaling, demonstrating that the curves share the same shape.
Fig. 7.
Fig. 7. Complex-valued spatial axis: Contour C follows the real axis except at very large distance from the origin, when it starts to deviate into upper complex plane. Outgoing Stark resonance states are normalizable and mutually orthogonal when integrated along such a contour.

Tables (1)

Tables Icon

Table 1. Nonlinear Response Scaling Parameters

Equations (16)

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

Γ ( F ) e 2 ( 2 U a ) 3 / 2 / 3 F ,
i ψ t = 1 2 r 2 ψ + V a ( r ) ψ x F ( t ) ψ ,
H [ ψ ] = 1 2 d 3 r   ψ [ 1 2 r 2 + V a ( r ) x F ] ψ ,
E j ( F ) u j ( r , F ) = [ 1 2 r 2 + V a ( r ) x F ] u j ( r , F ) .
ψ ( r , t ) u 0 ( r , F ( t ) ) exp ( i 0 t d t E 0 ( F ( t ) ) ) ,
p ( F ) = d 3 r   u 0 ( r , F ) x u 0 ( r , F ) ,
Γ ( F ) = 2 { E 0 ( F ) } .
p nl ( a ) ( F ) = α a 3 F 3 M ( F ) , Γ ( a ) ( F ) = α a G ( F ) ,
E 0 u 0 = [ 1 2 r 2 + V a ( r ) F x ] u 0   .
E 0 u 0 = [ 1 2 R 2 + s a V a ( s a R ) F X ] u 0   .
Γ ( a ) ( F ) = α a G ( β a F ) p nl ( a ) ( F ) = α a 3 ( β a F ) 3 M ( β a F )
G ( F ) = exp [ 0.1692048194155632 0.810669873391612 / F + 56.391127621143774 F 823.1689378085457 F 2 + 4152.098445656342 F 3 7390.473021873485 F 4 0.7555077122737133 log F ]
M ( F ) = exp [ 5.049542998716102 + 63.07083179334633 F 2 + 50607.81357765684 F 4 6632204.58068692 F 6 1019206807.5646534 F 8 + 184485603638.703 F 10 7509132994894.033 F 12 ]
1 2 r 2 ψ + V a ( r ) ψ x F ψ = E ψ   ,
d y d z C d x   ψ E 1 ( x , y , z ) ψ E 2 ( x , y , z ) = δ E 1 , E 2   ,
p ( F ) = d y d z C d x   ψ ( x , y , z , F )   x   ψ ( x , y , z , F )   ,

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