Abstract

We report experimental observations of rotated echoes of alignment induced by a pair of time-delayed and polarization-skewed femtosecond laser pulses interacting with an ensemble of molecular rotors. Rotated fractional echoes, rotated high order echoes and rotated imaginary echoes are directly visualized by using the technique of coincident Coulomb explosion imaging. We show that the echo phenomenon not only exhibits temporal recurrences but also spatial rotations determined by the polarization of the time-delayed second pulse. The dynamics of echo formation is well described by the laser-induced filamentation in rotational phase space. The quantum-mechanical simulation shows good agreements with the experimental results.

© 2017 Optical Society of America

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References

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  4. Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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    [Crossref]
  24. K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
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  26. I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A 139, 449–453 (1989).
    [Crossref]
  27. R.W. Robinett, “Quantum wave packet revivals,” Physics Reports 392, pp. 1–119 (2004).
    [Crossref]
  28. R.W. Boyd, “Nonlinear Optics,” (Academic, 1992).
  29. B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623 (1995); “Polarization of Molecules Induced by Intense Nonresonant Laser Fields,” J. Phys. Chem. 99, 15 686 (1995).
    [Crossref] [PubMed]
  30. D. Lynden-Bell, “Statistical mechanics of violent relaxation in stellar systems,” Mon. Not. R. Astr. Soc. 136, 101 (1967).
    [Crossref]
  31. A. J. Lichtenberg, Phase-space Dynamics of Particles (Wiley, 1969).
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  33. B. Ya. Dubetskii and V. P. Chebotaev, “Imaginary echo in a gas in a Doppler expanded transition,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya,  50, 1530 (1986) [Bull. Acad. Sci. USSR, Phys. Ser. (English Transl.) 50, 70 (1986)].
  34. R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
    [Crossref]
  35. H. Y. Jiang, C. Y. Wu, H. Zhang, H. B. Jiang, H. Yang, and Q. H. Gong, “Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses,” Opt. Express 18, 8990 (2010).
    [Crossref] [PubMed]
  36. E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
    [Crossref]

2016 (3)

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
[Crossref]

2015 (3)

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
[Crossref]

K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
[Crossref] [PubMed]

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
[Crossref]

2014 (1)

E. Hemsing, G. Stupakov, D. Xiang, and A. Zholents, “Beam by design: Laser manipulation of electrons in modern accelerators,” Rev. Mod. Phys. 86, 897 (2014).
[Crossref]

2013 (1)

M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais, “Manipulation of molecules with electromagnetic fields,” Mol. Phys. 111, 1648 (2013).
[Crossref]

2012 (3)

S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion,”. Isr. J. Chem. 52, 414 (2012).
[Crossref]

Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
[Crossref]

M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, ‘Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86, 023613 (2012).
[Crossref]

2010 (3)

Y. Ohshima and H. Hasegawa, “Coherent rotational excitation by intense nonresonant laser fields,” Int. Rev.Phys. Chem. 29, 619 (2010).

D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
[Crossref] [PubMed]

H. Y. Jiang, C. Y. Wu, H. Zhang, H. B. Jiang, H. Yang, and Q. H. Gong, “Alignment structures of rotational wavepacket created by two strong femtosecond laser pulses,” Opt. Express 18, 8990 (2010).
[Crossref] [PubMed]

2009 (1)

G. Stupakov, “Using the Beam-Echo Effect for Generation of Short-Wavelength Radiation,” Phys. Rev. Lett. 102, 074801 (2009).
[Crossref] [PubMed]

2005 (1)

T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
[Crossref] [PubMed]

2004 (1)

R.W. Robinett, “Quantum wave packet revivals,” Physics Reports 392, pp. 1–119 (2004).
[Crossref]

2003 (1)

H. Stapelfeldt and T. Seideman, “Aligning molecules with strong laser pulses,” Rev. Mod. Phys. 75, 543 (2003).
[Crossref]

2000 (1)

R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
[Crossref]

1998 (1)

A. Bulatov, A. Kuklov, B. E. Vugmeister, and H. Rabitz, “Echo in optical lattices: Stimulated revival of breathing oscillations,” Phys. Rev. A 57, 3788 (1998).
[Crossref]

1996 (1)

L. K. Spentzouris, J. -F. Ostiguy, and P. L. Colestock, “Direct Measurement of Diffusion Rates in High Energy Synchrotrons Using Longitudinal Beam Echoes,” Phys. Rev. Lett. 76, 620 (1996).
[Crossref] [PubMed]

1995 (1)

B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623 (1995); “Polarization of Molecules Induced by Intense Nonresonant Laser Fields,” J. Phys. Chem. 99, 15 686 (1995).
[Crossref] [PubMed]

1989 (1)

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A 139, 449–453 (1989).
[Crossref]

1986 (2)

J. Parker and C. R. Stroud, “Coherence and decay of Rydberg wave packets,” Phys. Rev. Lett. 56, 716 (1986).
[Crossref] [PubMed]

B. Ya. Dubetskii and V. P. Chebotaev, “Imaginary echo in a gas in a Doppler expanded transition,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya,  50, 1530 (1986) [Bull. Acad. Sci. USSR, Phys. Ser. (English Transl.) 50, 70 (1986)].

1985 (1)

B.Ya. Dubetskii and V. P. Chebotaev, “Echoes in classical and quantum ensembles with determinate frequencies,” Pis’ma Zh.Eksp.Teor. Fiz. 41, 267 (1985) [JETP Lett. 41, 328 (1985)].

1967 (2)

D. Lynden-Bell, “Statistical mechanics of violent relaxation in stellar systems,” Mon. Not. R. Astr. Soc. 136, 101 (1967).
[Crossref]

R. W. Gould, T. M. O’Neil, and J. H. Malmberg, “Plasma Wave Echo,” Phys. Rev. Lett. 19, 219 (1967).
[Crossref]

1965 (1)

R. M. Hill and D .E. Kaplan, “Cyclotron Resonance Echo,” Phys. Rev. Lett. 14, 1062 (1965).
[Crossref]

1964 (1)

N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a Photon Echo,” Phys. Rev. Lett. 13, 567 (1964).
[Crossref]

1953 (1)

E. L. Hahn, “Free nuclear induction,” Physics Today 6(11), 4 (1953).
[Crossref]

1950 (1)

E. L. Hahn, “Spin Echoes,” Phys. Rev. 80, 589 (1950).
[Crossref]

Abella, I. D.

N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a Photon Echo,” Phys. Rev. Lett. 13, 567 (1964).
[Crossref]

Antonsen, T. M.

M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, ‘Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86, 023613 (2012).
[Crossref]

Auffeves, A.

T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
[Crossref] [PubMed]

Averbukh, I. Sh.

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
[Crossref]

S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion,”. Isr. J. Chem. 52, 414 (2012).
[Crossref]

I. Sh. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. A 139, 449–453 (1989).
[Crossref]

Billard, F.

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
[Crossref]

Boyd, R.W.

R.W. Boyd, “Nonlinear Optics,” (Academic, 1992).

Brown, R. W.

R. W. Brown, Y.-C. N. Cheng, E. M. Haacke, M. R. Thompson, and R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design (Wiley-Blackwell, 2014).
[Crossref]

Brune, M.

T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
[Crossref] [PubMed]

Bulatov, A.

A. Bulatov, A. Kuklov, B. E. Vugmeister, and H. Rabitz, “Echo in optical lattices: Stimulated revival of breathing oscillations,” Phys. Rev. A 57, 3788 (1998).
[Crossref]

Chebotaev, V. P.

B. Ya. Dubetskii and V. P. Chebotaev, “Imaginary echo in a gas in a Doppler expanded transition,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya,  50, 1530 (1986) [Bull. Acad. Sci. USSR, Phys. Ser. (English Transl.) 50, 70 (1986)].

B.Ya. Dubetskii and V. P. Chebotaev, “Echoes in classical and quantum ensembles with determinate frequencies,” Pis’ma Zh.Eksp.Teor. Fiz. 41, 267 (1985) [JETP Lett. 41, 328 (1985)].

Chen, J. H.

Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
[Crossref]

Chen, Z. H.

Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
[Crossref]

Cheng, Y.-C. N.

R. W. Brown, Y.-C. N. Cheng, E. M. Haacke, M. R. Thompson, and R. Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design (Wiley-Blackwell, 2014).
[Crossref]

Colby, E.

D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
[Crossref] [PubMed]

Colestock, P. L.

L. K. Spentzouris, J. -F. Ostiguy, and P. L. Colestock, “Direct Measurement of Diffusion Rates in High Energy Synchrotrons Using Longitudinal Beam Echoes,” Phys. Rev. Lett. 76, 620 (1996).
[Crossref] [PubMed]

Deng, H. X.

Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
[Crossref]

Ding, J. G.

Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
[Crossref]

Ding, J. X.

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
[Crossref]

Dörner, R.

R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
[Crossref]

Doyle, J. M.

M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais, “Manipulation of molecules with electromagnetic fields,” Mol. Phys. 111, 1648 (2013).
[Crossref]

Dubetskii, B. Ya.

B. Ya. Dubetskii and V. P. Chebotaev, “Imaginary echo in a gas in a Doppler expanded transition,” Izvestiya Akademii Nauk SSSR, Seriya Fizicheskaya,  50, 1530 (1986) [Bull. Acad. Sci. USSR, Phys. Ser. (English Transl.) 50, 70 (1986)].

Dubetskii, B.Ya.

B.Ya. Dubetskii and V. P. Chebotaev, “Echoes in classical and quantum ensembles with determinate frequencies,” Pis’ma Zh.Eksp.Teor. Fiz. 41, 267 (1985) [JETP Lett. 41, 328 (1985)].

Dunning, M.

E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
[Crossref]

D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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Faucher, O.

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
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M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, ‘Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86, 023613 (2012).
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S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion,”. Isr. J. Chem. 52, 414 (2012).
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E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
[Crossref]

Gershnabel, E.

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
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S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion,”. Isr. J. Chem. 52, 414 (2012).
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D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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Gong, X. C.

K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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R. W. Gould, T. M. O’Neil, and J. H. Malmberg, “Plasma Wave Echo,” Phys. Rev. Lett. 19, 219 (1967).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
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N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a Photon Echo,” Phys. Rev. Lett. 13, 567 (1964).
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K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
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Y. Ohshima and H. Hasegawa, “Coherent rotational excitation by intense nonresonant laser fields,” Int. Rev.Phys. Chem. 29, 619 (2010).

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E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
[Crossref]

D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
[Crossref] [PubMed]

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E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, “Echo-enabled harmonics up to the 75th order from precisely tailored electron beams,” Nat. Photonics 10, 512 (2016).
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M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, ‘Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86, 023613 (2012).
[Crossref]

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B. Friedrich and D. Herschbach, “Alignment and Trapping of Molecules in Intense Laser Fields,” Phys. Rev. Lett. 74, 4623 (1995); “Polarization of Molecules Induced by Intense Nonresonant Laser Fields,” J. Phys. Chem. 99, 15 686 (1995).
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G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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Jiang, H. Y.

Jobe, K.

D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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R. M. Hill and D .E. Kaplan, “Cyclotron Resonance Echo,” Phys. Rev. Lett. 14, 1062 (1965).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
[Crossref]

G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
[Crossref]

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S. Fleischer, Y. Khodorkovsky, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Molecular Alignment Induced by Ultrashort Laser Pulses and Its Impact on Molecular Motion,”. Isr. J. Chem. 52, 414 (2012).
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K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
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M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais, “Manipulation of molecules with electromagnetic fields,” Mol. Phys. 111, 1648 (2013).
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N. A. Kurnit, I. D. Abella, and S. R. Hartmann, “Observation of a Photon Echo,” Phys. Rev. Lett. 13, 567 (1964).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
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G. Karras, E. Hertz, F. Billard, B. Lavorel, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Orientation and Alignment Echoes,” Phys. Rev. Lett. 114, 15361 (2015).
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M. Lemeshko, R. V. Krems, J. M. Doyle, and S. Kais, “Manipulation of molecules with electromagnetic fields,” Mol. Phys. 111, 1648 (2013).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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Z. T. Zhao, D. Wang, J. H. Chen, Z. H. Chen, H. X. Deng, J. G. Ding, C. Feng, Q. Gu, M. M. Huang, T. H. Lan, Y. B. Leng, D. G. Li, G. Q. Lin, B. Liu, E. Prat, X. T. Wang, Z. S. Wang, K. R. Ye, L. Y. Yu, H. O. Zhang, J. Q. Zhang, Me. Zhang, Mi. Zhang, T. Zhang, S. P. Zhong, and Q. G. Zhou, “First lasing of an echo-enabled harmonic generation free-electron laser,” Nature Photonics 6, 360 (2012).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

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T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
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R. W. Gould, T. M. O’Neil, and J. H. Malmberg, “Plasma Wave Echo,” Phys. Rev. Lett. 19, 219 (1967).
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D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
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T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
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K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
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R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, “Cold Target Recoil Ion Momentum Spectroscopy: a “momentum microscope” to view atomic collision dynamics,” Physics Reports 330, 95–192 (2000).
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D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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T. Meunier, S. Gleyzes, P. Maioli, A. Auffeves, G. Nogues, M. Brune, J. M. Raimond, and S. Haroche, “Rabi Oscillations Revival Induced by Time Reversal: A Test of Mesoscopic Quantum Coherence,” Phys. Rev. Lett. 94, 010401 (2005).
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K. Mizuse, K. Kitano, H. Hasegawa, and Y. Ohshima, “Quantum unidirectional rotation directly imaged with molecules,” Sci. Adv. 1, e1400185 (2015).
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M. Herrera, T. M. Antonsen, E. Ott, and S. Fishman, ‘Echoes and revival echoes in systems of anharmonically confined atoms,” Phys. Rev. A 86, 023613 (2012).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

G. Karras, E. Hertz, F. Billard, B. Lavorel, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Experimental observation of fractional echoes,” Phys. Rev. A 94, 033404 (2016).
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K. Lin, P. F. Lu, J. Y. Ma, X. C. Gong, Q. Y. Song, Q. Y. Ji, W. B. Zhang, H. P. Zeng, J. Wu, G. Karras, G. Siour, J.-M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. Sh. Averbukh, “Echoes in Space and Time,” Phys. Rev. X 6, 041056 (2016).

K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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D. Xiang, E. Colby, M. Dunning, S. Gilevich, C. Hast, K. Jobe, D. McCormick, J. Nelson, T. O. Raubenheimer, K. Soong, G. Stupakov, Z. Szalata, D. Walz, S. Weathersby, and M. Woodley, “Demonstration of the Echo-Enabled Harmonic Generation Technique for Short-Wavelength Seeded Free Electron Lasers,” Phys. Rev. Lett. 105, 114801 (2010).
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K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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K. Lin, Q. Y. Song, X. C. Gong, Q. Y. Ji, H. F. Pan, J. X. Ding, H. P. Zeng, and J. Wu, “Visualizing molecular unidirectional rotation,” Phys. Rev. A 92, 013410 (2015).
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Opt. Express (1)

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Figures (4)

Fig. 1
Fig. 1

Rotated echo formation. The first pulse is applied at t = 0, ϕ = 0, with Ω1/σ = 1. The second pulse is applied at t = T (such that σT = 5), at crossing angle β = −π/12 with respect to the first pulse, (a)–(c) Laser-induced filamentation of the phase-space density distributions. (a) Initial phase-space density distribution. (b) Folding and alignment shortly after the first pulse at t = 0.1T. (c) Filamentation at t = T after the first pulse. (d)–(f) Rotated echo formation after the second pulse, Ω21 = 0.333. The polarization angles of the first and the second pulses, as well as the rotation angles of echoes at different times are indicated by the white dashed lines. (d) Rotated 1/2 fractional echo at (t, ϕ) = (1.46T, 1.5β). (e) Rotated second order echo at (t, ϕ) = (2.82T, 3β). (f) Rotated imaginary echo at (t, ϕ) = (−1.2T, −β). (g)–(i) Rotated echo formation after the second pulse, Ω21 = 0.16 (close to the value used in our experiments). (g) Rotated 1/2 fractional echo at (t, ϕ) = (1.533T, 1.5β). (h) Rotated second order echo at (t, ϕ) = (3.13T, 3β). (i) Rotated imaginary echo at (t, ϕ) = (−0.865T, −β). (j)–(l) are the corresponding polar plots of the angular distributions for (g)–(i), respectively. A constant angular-independent background was subtracted in these polar plots. The theoretically expected directions of the rotated echoes are indicated by the gray dashed lines.

Fig. 2
Fig. 2

Experimental setup. Two linearly polarized pump pulses P1 and P2 are employed to stimulate molecular alignment echoes of N2O. An intense circularly polarized probe pulse is used to trace the molecular angular distribution at various times. The meaurement is performed in a reaction microscope of COLTRIMS. BS, beam splitter; HWP, zero-order half wave plate; QWP, zero-order quarter wave plate; DL, delay line.

Fig. 3
Fig. 3

Experimentally measured time-dependent angular distribution of (a) the 1/2 fractional, the full and the 2 nd order echoes, and (d) the rotated 1/2 fractional, the rotated full, and the rotated 2 nd order echoes of N2O. In both cases the echo region is highlighted between the vertical dashed white lines. The corresponding polar plots of the angular distributions are shown in (b)–(c) and (e)–(f). The angular-independent negative background was subtracted in these polar plots. (b) and (c) for the 1/2 and 2 nd order echoes, and (e) and (f) for the rotated 1/2 and 2 nd order echoes. The red solid lines in (b)–(c) and (e)–(f) are the results of quantum simulations, which agree well with the experiments. The theoretically expected directions of echoes and the rotated ones are indicated by the gray dashed lines.

Fig. 4
Fig. 4

(a) Time-dependent angular distribution of rotated imaginary echo. (b) The measured alignment factor as a function of probe time delay. The red and blue curves show the full revivals of the alignment induced by P1 and P2 respectively. The rotated imaginary echo is presented by the magenta curve. (c) and (d) are the polar plots of the angular distributions of the rotated imaginary echoes at alignment and anti-alignment times respectively, where the angular-independent negative background was subtracted. The red curves in (c) and (d) are quantum simulations. The theoretically expected directions of the rotated echoes are indicated by the gray dashed lines.

Equations (2)

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f ( ω , ϕ , t ) = 1 2 π 1 2 π σ exp [ - [ ω - Ω sin ( 2 ω t - 2 ϕ ) ] 2 2 σ 2 ] .
ϕ = β + [ ( m π / T + β / T ) ] ( t - T ) ( mod π ) .

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