Abstract

Ultrafast pump-high-harmonic-generation-probe spectroscopy aims to provide a unique observation window into electronic dynamics while using the infrared or visible light sources. While it is widely accepted that the role of excited bound states in high-harmonic generation is negligible, its dynamics play a significant role in time-resolved pump-probe measurements. Here we show that the time-resolved pump-high-harmonic-generation-probe measurement may reveal a significant (up to 20%) contribution of the quantum interference in electron ionization and recombination with atomic system, with the initial or the final state being an excited bound state. Interplay of two dephasing mechanisms of electron-ion and electron-atom collisions yields decay and recovery of the time-resolved signal, respectively, signifying the role of the quantum interference involving excited bound states in recovery mode. Our theory, based on the density matrix Liouville space formalism, is supported by experimental measurements in argon gas.

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

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    [Crossref]

2018 (1)

2017 (2)

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

D. Kartashov and M. N. Shneider, “Femtosecond filament initiated, microwave heated cavity-free nitrogen laser in air,” J. Appl. Phys. 121, 113303 (2017).
[Crossref]

2016 (1)

J. Marangos, “Development of high harmonic generation spectroscopy of organic molecules and biomolecules,” J. Phys. B: At. Mol. Opt. Phys. 49, 132001 (2016).
[Crossref]

2015 (4)

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

X. Yuan, P. Wei, C. Liu, Z. Zeng, Y. Zheng, J. Jiang, X. Ge, and R. Li, “Enhanced high-order harmonic generation from excited argon,” Appl. Phys. Lett. 107, 041110 (2015).
[Crossref]

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
[Crossref]

2013 (2)

K. E. Dorfman, B. P. Fingerhut, and S. Mukamel, “Time-resolved broadband raman spectroscopies: A unified six-wave-mixing representation,” The J. Chem. Phys. 139, 124113 (2013).
[Crossref] [PubMed]

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
[Crossref]

2012 (1)

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

2011 (1)

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

2010 (1)

H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

2009 (3)

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

N. Rohringer and R. Santra, “Multichannel coherence in strong-field ionization,” Phys. Rev. A 79, 053402 (2009).
[Crossref]

2008 (2)

M. Spanner and P. Brumer, “Probing electron transfer within alkali-metal halides via high-order harmonic generation,” Phys. Rev. A 78, 033840 (2008).
[Crossref]

U. Harbola and S. Mukamel, “Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions,” Phys. Rep. 465, 191–222 (2008).
[Crossref]

2006 (1)

S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006).
[Crossref] [PubMed]

2005 (1)

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005).
[Crossref]

1997 (1)

F. I. Gauthey, B. M. Garraway, and P. L. Knight, “High harmonic generation and periodic level crossings,” Phys. Rev. A 56, 3093–3096 (1997).
[Crossref]

1996 (1)

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

1994 (2)

S. Mori and O. Sueoka, “Excitation and ionization cross sections of he, ne and ar by positron impact,” J. Phys. B: At. Mol. Opt. Phys. 27, 4349–4364 (1994).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

1992 (1)

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[Crossref] [PubMed]

1982 (1)

M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982).
[Crossref]

1965 (1)

J. H. Shirley, “Solution of the schrödinger equation with a hamiltonian periodic in time,” Phys. Rev. 138, B979–B987 (1965).
[Crossref]

1963 (1)

R. Kubo, “Stochastic liouville equations,” J. Math. Phys. 4, 174–183 (1963).
[Crossref]

Ališauskas, S.

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
[Crossref]

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

Baltuška, A.

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
[Crossref]

Bandrauk, A.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Baykusheva, D.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

Bennett, K.

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

Bertrand, J.

H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

Brabec, T.

S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006).
[Crossref] [PubMed]

Brumer, P.

M. Spanner and P. Brumer, “Probing electron transfer within alkali-metal halides via high-order harmonic generation,” Phys. Rev. A 78, 033840 (2008).
[Crossref]

Bruner, B. D.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

Cireasa, R.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Constant, E.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Corkum, P.

H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

Cormier, E.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Dagan, M.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

Descamps, D.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Ditmire, T.

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Donnelly, T. D.

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Dorfman, K. E.

P. Wei, M. Qin, K. E. Dorfman, X. Yuan, C. Liu, Z. Zeng, X. Ge, X. Zhu, Q. Liang, B. Yao, Q. J. Wang, H. Li, J. Liu, Y. Zhang, S. Y. Jeong, G. S. Yun, D. E. Kim, P. Lu, and R. Li, “Probing electron-atom collision dynamics in gas plasma by high-order harmonic spectroscopy,” Opt. Lett. 43, 1970–1973 (2018).
[Crossref] [PubMed]

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

K. E. Dorfman, B. P. Fingerhut, and S. Mukamel, “Time-resolved broadband raman spectroscopies: A unified six-wave-mixing representation,” The J. Chem. Phys. 139, 124113 (2013).
[Crossref] [PubMed]

Dudovich, N.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

Fabre, B.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Falcone, R.

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Feit, M.

M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982).
[Crossref]

Fingerhut, B. P.

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

K. E. Dorfman, B. P. Fingerhut, and S. Mukamel, “Time-resolved broadband raman spectroscopies: A unified six-wave-mixing representation,” The J. Chem. Phys. 139, 124113 (2013).
[Crossref] [PubMed]

Fleck, J.

M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982).
[Crossref]

Garraway, B. M.

F. I. Gauthey, B. M. Garraway, and P. L. Knight, “High harmonic generation and periodic level crossings,” Phys. Rev. A 56, 3093–3096 (1997).
[Crossref]

Gauthey, F. I.

F. I. Gauthey, B. M. Garraway, and P. L. Knight, “High harmonic generation and periodic level crossings,” Phys. Rev. A 56, 3093–3096 (1997).
[Crossref]

Ge, X.

Gijsbertsen, A.

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
[Crossref]

Grassi, G.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Harbola, U.

U. Harbola and S. Mukamel, “Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions,” Phys. Rep. 465, 191–222 (2008).
[Crossref]

Higuet, J.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Horn?, L.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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O. Smirnova and M. Ivanov, Multielectron High Harmonic Generation: Simple Man on a Complex Plane (John Wiley & Sons, Ltd, 2014), chap. 7, pp. 201–256.

Ivanov, M. Y.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005).
[Crossref]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

Jensen, F.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Jeong, S. Y.

Jiang, J.

X. Yuan, P. Wei, C. Liu, Z. Zeng, Y. Zheng, J. Jiang, X. Ge, and R. Li, “Enhanced high-order harmonic generation from excited argon,” Appl. Phys. Lett. 107, 041110 (2015).
[Crossref]

Kartashov, D.

D. Kartashov and M. N. Shneider, “Femtosecond filament initiated, microwave heated cavity-free nitrogen laser in air,” J. Appl. Phys. 121, 113303 (2017).
[Crossref]

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
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H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
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Knight, P. L.

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M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

Kraus, P. M.

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
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Krause, J. L.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
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Kubo, R.

R. Kubo, “Stochastic liouville equations,” J. Math. Phys. 4, 174–183 (1963).
[Crossref]

Kulander, K. C.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
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M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
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Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
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Li, H.

Li, R.

Liang, Q.

Liu, C.

Liu, J.

Lu, P.

Lucchese, R.

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
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Madsen, L.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Mairesse, Y.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
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J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
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J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
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Mignolet, B.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Mori, S.

S. Mori and O. Sueoka, “Excitation and ionization cross sections of he, ne and ar by positron impact,” J. Phys. B: At. Mol. Opt. Phys. 27, 4349–4364 (1994).
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Mukamel, S.

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
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K. E. Dorfman, B. P. Fingerhut, and S. Mukamel, “Time-resolved broadband raman spectroscopies: A unified six-wave-mixing representation,” The J. Chem. Phys. 139, 124113 (2013).
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U. Harbola and S. Mukamel, “Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions,” Phys. Rep. 465, 191–222 (2008).
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S. Mukamel and S. Rahav, “Chapter 6 - ultrafast nonlinear optical signals viewed from the molecule’s perspective: Kramers-heisenberg transition-amplitudes versus susceptibilities,” in “Advances in Atomic, Molecular, and Optical Physics,” vol. 59 of Advances In Atomic, Molecular, and Optical Physics, P. B. E. Arimondo and C. Lin, eds. (Academic Press, 2010), pp. 223 – 263.

S. Mukamel, Principles of nonlinear optical spectroscopy, 6 (Oxford University, 1999).

Neuman, K.

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
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W. Parson, Modern Optical Spectroscopy: With Exercises and Examples from Biophysics and Biochemistry (SpringerBerlin Heidelberg, 2007).
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D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006).
[Crossref] [PubMed]

Penka, E.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Perry, M.

T. D. Donnelly, T. Ditmire, K. Neuman, M. Perry, and R. Falcone, “High-order harmonic generation in atom clusters,” Phys. Rev. Lett. 76, 2472–2475 (1996).
[Crossref] [PubMed]

Petit, S.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Pons, B.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Pugžlys, A.

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
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Qin, M.

Rahav, S.

S. Mukamel and S. Rahav, “Chapter 6 - ultrafast nonlinear optical signals viewed from the molecule’s perspective: Kramers-heisenberg transition-amplitudes versus susceptibilities,” in “Advances in Atomic, Molecular, and Optical Physics,” vol. 59 of Advances In Atomic, Molecular, and Optical Physics, P. B. E. Arimondo and C. Lin, eds. (Academic Press, 2010), pp. 223 – 263.

Remacle, F.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Rohringer, N.

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
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N. Rohringer and R. Santra, “Multichannel coherence in strong-field ionization,” Phys. Rev. A 79, 053402 (2009).
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Ruf, H.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Rupenyan, A.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Santra, R.

N. Rohringer and R. Santra, “Multichannel coherence in strong-field ionization,” Phys. Rev. A 79, 053402 (2009).
[Crossref]

Schafer, K. J.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[Crossref] [PubMed]

Schneider, J.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
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Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University, 1997).
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Shafir, D.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
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Shirley, J. H.

J. H. Shirley, “Solution of the schrödinger equation with a hamiltonian periodic in time,” Phys. Rev. 138, B979–B987 (1965).
[Crossref]

Shneider, M. N.

D. Kartashov and M. N. Shneider, “Femtosecond filament initiated, microwave heated cavity-free nitrogen laser in air,” J. Appl. Phys. 121, 113303 (2017).
[Crossref]

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
[Crossref]

Smirnova, O.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
[Crossref] [PubMed]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

O. Smirnova, S. Patchkovskii, Y. Mairesse, N. Dudovich, and M. Y. Ivanov, “Strong-field control and spectroscopy of attosecond electron-hole dynamics in molecules,” Proceedings of the National Academy of Sciences 106, 16556–16561 (2009).
[Crossref]

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005).
[Crossref]

O. Smirnova and M. Ivanov, Multielectron High Harmonic Generation: Simple Man on a Complex Plane (John Wiley & Sons, Ltd, 2014), chap. 7, pp. 201–256.

Soifer, H.

D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Y. Ivanov, O. Smirnova, and N. Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature 485, 343–346 (2012).
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M. Spanner and P. Brumer, “Probing electron transfer within alkali-metal halides via high-order harmonic generation,” Phys. Rev. A 78, 033840 (2008).
[Crossref]

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005).
[Crossref]

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M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982).
[Crossref]

Sueoka, O.

S. Mori and O. Sueoka, “Excitation and ionization cross sections of he, ne and ar by positron impact,” J. Phys. B: At. Mol. Opt. Phys. 27, 4349–4364 (1994).
[Crossref]

Thiré, N.

J. Higuet, H. Ruf, N. Thiré, R. Cireasa, E. Constant, E. Cormier, D. Descamps, E. Mével, S. Petit, B. Pons, Y. Mairesse, and B. Fabre, “High-order harmonic spectroscopy of the cooper minimum in argon: Experimental and theoretical study,” Phys. Rev. A 83, 053401 (2011).
[Crossref]

Tolstikhin, O. I.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Villeneuve, D.

H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

O. Smirnova, Y. Mairesse, S. Patchkovskii, N. Dudovich, D. Villeneuve, P. Corkum, and M. Y. Ivanov, “High harmonic interferometry of multi-electron dynamics in molecules,” Nature 460, 972–977 (2009).
[Crossref] [PubMed]

Villeneuve, D. M.

S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006).
[Crossref] [PubMed]

Wang, Q. J.

Wei, P.

Wörner, H.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Hornỳ, E. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. Madsen, A. Bandrauk, F. Remacle, and H. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
[Crossref]

H. Wörner, J. Bertrand, D. Kartashov, P. Corkum, and D. Villeneuve, “Following a chemical reaction using high-harmonic interferometry,” Nature 466, 604–607 (2010).
[Crossref] [PubMed]

Wörner, H. J.

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

Yao, B.

Yuan, X.

Yun, G. S.

Zeng, Z.

Zhang, S. B.

S. B. Zhang, D. Baykusheva, P. M. Kraus, H. J. Wörner, and N. Rohringer, “Theoretical study of molecular electronic and rotational coherences by high-order-harmonic generation,” Phys. Rev. A 91, 023421 (2015).
[Crossref]

P. M. Kraus, S. B. Zhang, A. Gijsbertsen, R. Lucchese, N. Rohringer, and H. Wörner, “High-harmonic probing of electronic coherence in dynamically aligned molecules,” Phys. Rev. Lett. 111, 243005 (2013).
[Crossref]

Zhang, Y.

Zhao, Z.

S. Patchkovskii, Z. Zhao, T. Brabec, and D. M. Villeneuve, “High harmonic generation and molecular orbital tomography in multielectron systems: Beyond the single active electron approximation,” Phys. Rev. Lett. 97, 123003 (2006).
[Crossref] [PubMed]

Zheng, Y.

X. Yuan, P. Wei, C. Liu, Z. Zeng, Y. Zheng, J. Jiang, X. Ge, and R. Li, “Enhanced high-order harmonic generation from excited argon,” Appl. Phys. Lett. 107, 041110 (2015).
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Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum optics (Cambridge University, 1997).
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Appl. Phys. Lett. (1)

X. Yuan, P. Wei, C. Liu, Z. Zeng, Y. Zheng, J. Jiang, X. Ge, and R. Li, “Enhanced high-order harmonic generation from excited argon,” Appl. Phys. Lett. 107, 041110 (2015).
[Crossref]

Chem. Rev. (1)

M. Kowalewski, B. P. Fingerhut, K. E. Dorfman, K. Bennett, and S. Mukamel, “Theory of high-harmonic generation by low-frequency laser fields,” Chem. Rev. 117, 12165–12226 (2017).
[Crossref] [PubMed]

J. Appl. Phys. (1)

D. Kartashov and M. N. Shneider, “Femtosecond filament initiated, microwave heated cavity-free nitrogen laser in air,” J. Appl. Phys. 121, 113303 (2017).
[Crossref]

J. Comput. Phys. (1)

M. Feit, J. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Comput. Phys. 47, 412 – 433 (1982).
[Crossref]

J. Math. Phys. (1)

R. Kubo, “Stochastic liouville equations,” J. Math. Phys. 4, 174–183 (1963).
[Crossref]

J. Mod. Opt. (1)

M. Y. Ivanov, M. Spanner, and O. Smirnova, “Anatomy of strong field ionization,” J. Mod. Opt. 52, 165–184 (2005).
[Crossref]

J. Phys. B: At. Mol. Opt. Phy. (1)

D. Kartashov, S. Ališauskas, A. Pugžlys, M. N. Shneider, and A. Baltuška, “Theory of a filament initiated nitrogen laser,” J. Phys. B: At. Mol. Opt. Phy. 48, 094016 (2015).
[Crossref]

J. Phys. B: At. Mol. Opt. Phys. (2)

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Figures (4)

Fig. 1
Fig. 1 Schematic of the PHHGP measurement. Strong pump pulse creates superposition of electronic states in the atom, which consists of the ground g, excited state e and continuum of state i. This dynamics undergoes dephasing due to collisions between neutral atoms and electrons governed by rate γ b b , ions and electrons determined by γ b c . After a delay T a probe laser pulse generates HHG in three-step process from the atom in superposition of electronic states which involvesionization and recombination pathways from bound-to bound d m m b b and bound-to-continuum d m m b c , m , m = e , g electronic transitions. The resulting HHG spectra is recorded in the series of snapshots.
Fig. 2
Fig. 2 Time-profile of the excited bound state population ρ e e ( t ) obtained from numerical solution of Eq. (29).
Fig. 3
Fig. 3 Simulation of the snapshot HHG spectra of Argon for three-step model (top two rows) and including excited bound state (bottom two rows). Different panels correspond to interpulse delay T = 100 fs, 10 ps, 80 ps, and 200 ps.
Fig. 4
Fig. 4 PHHGP signal vs interpulse delay T, experiment [21] - red, theoretical model in Eq. (33) - blue. Inset - same but for a model with the single bound (ground) state.

Equations (34)

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H = 1 2 v 2 + V ( r ) E ( t ) r ,
E ( t ) = E 2 π σ e t 2 2 σ 2 cos  ( ω 0 t ) ,
| ψ ( z , t ) = m e i ω m t c m ( t ) | ψ m ( z ) + d v c i ( v , t ) e i ω v t | v ,
c ˙ m ( t ) = i n μ m n ( b ) E ( t ) c n ( t ) e i ω m n t + i d v μ m i ( v ) E ( t ) c i ( v , t ) e i ( ω m ω v ) t
c ˙ i ( v , t ) = i ( v 2 2 ω v ) c i ( v , t ) E ( t ) c i ( v , t ) v + i μ i m ( v ) E ( t ) e i ( ω m ω v ) t c m ( t ) ,
c i ( v , t ) = i m 0 t d t E ( t ) μ i m ( v + A ( t ) A ( t ) ) × e i t t d t [ ( v + A ( t ) A ( t ) ) 2 / 2 ω v ] c m ( t ) e i ( ω m ω v ) t
c ˙ m ( t ) = i n [ μ m n ( b ) + μ m n ( c ) ( t ) ] E ( t ) c n ( t ) e i ω m n t
ρ ˙ = i [ H , ρ ]
ρ ˙ e g ( s ) = i ( ω e g + δ ω e g ( t ) ) ρ e g ( s ) ,
ρ e g ( s ) ( t ) = exp  [ i ω e g t i 0 t d t δ ω e g ( t ) ] ρ e g ( s ) ( 0 ) .
δ ω e g ( t ) δ ω e g ( t ) = 2 γ δ ( t t ) .
exp   [ i 0 t d t δ ω e g ( t ) ] = exp   [ γ t ] ,
ρ ˙ e g ( s ) = ( i ω e g + γ ) ρ e g ( s ) .
k B T e 2 U p ξ 2 = 13 e V ,
n e = ρ ¯ i i P k B T 0 ,
Λ = 4 π n e λ D e 3 ,
Γ = E C o u l o m b k B T e = e 2 4 π ϵ 0 k B T e ( 4 π n e 3 ) 1 / 3
ω p = ( 4 π e 2 n e m e ) 1 / 2 1.45 × 10 6 s 1 ,
2 π ν a , a ( g ) = 2 π n 0 σ 0 ( k B T 0 m 0 ) 1 / 2 9.6 × 10 8  s 1 ,
2 π ν e l , a ( g ) = 2 π n 0 σ 0 ( k B T e m e ) 1 / 2 1.3 × 10 11  s 1 .
2 π ν e l , a ( e ) = 2 π ρ ¯ e e n 0 σ 0 ( k B T e m e ) 1 / 2 1.2 × 10 9  s 1 .
ν = 4 π N i Z i 2 e 2 μ 1 / 2 ( k B T ) 3 / 2 log Λ
2 π ν e l , i = 2 π 2.91 10 6 n e log   ( Λ e l , i ) T e 3 / 2 1.7 × 10 11   s 1 ,
log  ( Λ e l , i ) = 23 log  ( N e 1 / 2 Z i T e 3 / 2 ) ,
z ( t ) = ψ ( z , t ) | z | ψ ( z , t ) = m n μ m n ( b ) c m * ( t ) c n ( t ) + n d v μ i n ( v ) c i * ( v , t ) c n ( t ) + c . c .
z ( t ) = ψ ( z , t ) | z | ψ ( z , t ) = m n μ m n ( b ) ρ n m ( t ) e γ b b t + n d v μ i n ( v ) ρ n i ( v , t ) e γ b c t + c . c .
γ b c = 2 π ( ν e l , i + 1 2 ( ν e l , a ( g ) + ν e l , a ( e ) ) ) = 2.3 × 10 11   s 1 ,
γ b b = 2 π ν e l , a ( e ) = 1.2 × 10 9   s 1 ,
ρ ˙ m n = γ m n ρ m n i k ( μ m k ( t ) ρ k n μ k n ( t ) ρ m k E ) ( t ) ,
μ m n ( t ) = μ m n ( b ) ( 1 δ m n ) e [ i ω m n + γ b b ] t + 1 2 ( μ m n ( c ) * ( t ) + μ n m ( c ) ( t ) ) e [ i ω m n + γ b c ] t .
μ m n ( c ) ( t ) = i 0 d τ ( π ϵ + i τ / 2 ) 1 / 2 E ( t τ ) × μ i m * ( p s t A ( t ) ) μ i n ( p s t A ( t τ ) ) e i S s t ( t , τ ) ,
S ( ω ) = | d t z ¨ ( t ) e i ω t | 2 .
S ( T ) = 0 d t | z ( t ) + z ( t + T ) | 2 ,
z ( t ) = 2 Re [ m , n μ m n ( t ) ρ n m ] .

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