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Based on the laser-induced-phase model, periodic quantum phase modulation of the dipole response in atomic He is studied theoretically. The two-level system of the transition 1s2→1s2p with a delay width of 1.8 × 109 s-1 and an energy difference of 21.2 eV between the excited state and the ground state is used in the calculation. The system is excited by attosecond laser pulse from high harmonic generator, and the spectral response of the system is of single isolated symmetric Lorentzian absorption line. After the excitation, near infrared (NIR) femtosecond laser pulse train with a repetition rate of 5 GHz, central frequency 780 nm, and pulse duration of 100 fs, is utilized to periodically modify the spontaneous decay of the excited 1s2p level. The incremental phase step Δφ depends on the intensity of the NIR laser pulse, while the initial offset phase φ can be controlled independently by partially overlapping the first NIR pulse with the excitation. Simulated results show that the Lorentzian absorption line is transformed into comb-like spectral structure with equal gap depending on the repetition rate of the NIR pulse train. The line shape of each comb tooth is symmetric Lorentzian line by setting φ = Δφ/2 = π/2, while it is Fano line by setting φ = Δφ = π. The location of the comb structure is mainly dependent on the energy difference between the excited state and the ground state, while it can be slightly tuned by controlling the incremental phase step Δφ. We develop an analytic description of the comb-like spectral structure by Fourier analysis, depending on both the atomic and the phase-control properties. The analytical expressions can be readily used to estimate the exact experimental parameters. The universality of this mechanism allows the spectral modulation in arbitrary atomic system at arbitrary frequency, including the hard X-ray regime, by using reference transitions in highly charged ions. The generalization of this approach should thus not only enable relative frequency measurement and relevant applications at extremely high frequencies, but also open the way for pulse shaping at arbitrary frequencies.
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Keywords:
- absorption line /
- dipole response /
- periodic quantum phase modulation
[1] Fraunhofer J 1817 Annalen der Physik 56 264
[2] Kirchhoff G, Bunsen R 1860 Annalen der Physik 186 161
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[10] Lin C D, Chu W C 2013 Science 340 694
[11] Liu Z Y, Cavaletto S M, Ott C, Meyer K, Mi Y, Harman Z, Keitel C H, Pfeifer T 2015 Phys. Rev. Lett. 115 033003
[12] Chini M, Zhao B, Wang H, Cheng Y, Hu S X, Chang Z 2012 Phys. Rev. Lett. 109 073601
[13] Mashiko H, Yamaguchi T, Oguri K, Suda A, Gotoh H 2014 Nat. Commun. 5 6599
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[18] Chen S, Wu M, Gaarde M B, Schafer K J 2013 Phys. Rev. A 88 033409
[19] Fano U, Cooper J W 1968 Rev. Mod. Phys. 40 441
[20] Pollard W T, Mathies R A 1992 Annu. Rev. Phys. Chem. 43 497
[21] Tannoudji C C, Diu B, Lalöe F 1977 Quantum Mechanics (Vol. II) (New York: Wiley-Interscience Press) pp1095-1108
[22] Rosen N Zener C 1932 Phys. Rev. 40 502
[23] Lewenstein M, Zakrzewski J, Rzązewski K 1986 J. Opt. Soc. Am. B 3 22
[24] Budker D, Kimball D F, DeMille D P 2004 Atomic Physics (Oxford: Oxford University Press) pp13-18
[25] Drake G W F 2006 High Precision Calculations for Helium (Springer Handbook of Atomic Molecular and Optical Physics) (New York: Springer) pp107-217
[26] Pekarek S, Klenner A, Sdmeyer T, Fiebig C, Paschke K, Erbert G, Keller U 2012 Opt. Express 20 4248
[27] Lim J, Chen H, Xu S, Yang Z, Chang G, Kärtner F X 2014 Opt. Lett. 39 2060
[28] Klenner A, Golling M, Keller U 2014 Opt. Express 22 11884
[29] Bernitt S, Brown G V, Rudolph J K, et al. 2012 Nature 492 225
[30] Derevianko A, Johnson W R 1997 Phys. Rev. A 56 1228
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[1] Fraunhofer J 1817 Annalen der Physik 56 264
[2] Kirchhoff G, Bunsen R 1860 Annalen der Physik 186 161
[3] Liu Z Y, Shi Y C, Hu B T 2014 Acta Phys. Sin. 63 184206 (in Chinese) [刘作业, 史彦超, 胡碧涛 2014 物理学报 63 184206]
[4] Liu Z Y, Sun S H, Shi Y C, Ding P J, Liu Q C, Liu X L, Ding B W, Hu B T 2013 Chin. Phys. B 22 075204
[5] Sun S H, Liu X, Liu Z Y, Wang X, Ding P, Liu Q, Guo Z, Hu B T 2013 Chin. Phys. Lett. 30 045202
[6] Zhang Z X, Xu R J, Song L W, Wang D, Liu P, Leng Y X 2012 Acta Phys. Sin. 61 184209 (in Chinese) [张宗昕, 许荣杰, 宋立伟, 王丁, 刘鹏, 冷雨欣 2012 物理学报 61 184209]
[7] Dai X C, Richter M, Li H B, Bristow A D, Falvo C, Mukamel S, Cundiff S T 2012 Phys. Rev. Lett. 108 193201
[8] Ott C, Kaldun A, Raith P, Meyer K, Laux M, Evers J, Keitel C H, Greene C H, Pfeifer T 2013 Science 340 716
[9] Ott C, Kaldun A, Argenti L, Raith P, Meyer K, Laux M, Zhang Y, Blättermann A, Hagstotz S, Ding T, Heck R, Madroñero J, Martín F, Pfeifer T 2014 Nature 516 374
[10] Lin C D, Chu W C 2013 Science 340 694
[11] Liu Z Y, Cavaletto S M, Ott C, Meyer K, Mi Y, Harman Z, Keitel C H, Pfeifer T 2015 Phys. Rev. Lett. 115 033003
[12] Chini M, Zhao B, Wang H, Cheng Y, Hu S X, Chang Z 2012 Phys. Rev. Lett. 109 073601
[13] Mashiko H, Yamaguchi T, Oguri K, Suda A, Gotoh H 2014 Nat. Commun. 5 6599
[14] Zhao J, Lein M 2012 New J. Phys. 14 065003
[15] Lucchini M, Herrmann J, Ludwig A, Locher R, Sabbar M, Gallmann L, Keller U 2013 New J. Phys. 15 103010
[16] Chen S, Bell M J, Beck A R, Mashiko H, Wu M, Pfeiffer A N, Gaarde M B, Neumark D M, Leone S R, Schafer K J 2012 Phys. Rev. A 86 063408
[17] Chen S, Wu M, Gaarde M B, Schafer K J 2013 Phys. Rev. A 87 033408
[18] Chen S, Wu M, Gaarde M B, Schafer K J 2013 Phys. Rev. A 88 033409
[19] Fano U, Cooper J W 1968 Rev. Mod. Phys. 40 441
[20] Pollard W T, Mathies R A 1992 Annu. Rev. Phys. Chem. 43 497
[21] Tannoudji C C, Diu B, Lalöe F 1977 Quantum Mechanics (Vol. II) (New York: Wiley-Interscience Press) pp1095-1108
[22] Rosen N Zener C 1932 Phys. Rev. 40 502
[23] Lewenstein M, Zakrzewski J, Rzązewski K 1986 J. Opt. Soc. Am. B 3 22
[24] Budker D, Kimball D F, DeMille D P 2004 Atomic Physics (Oxford: Oxford University Press) pp13-18
[25] Drake G W F 2006 High Precision Calculations for Helium (Springer Handbook of Atomic Molecular and Optical Physics) (New York: Springer) pp107-217
[26] Pekarek S, Klenner A, Sdmeyer T, Fiebig C, Paschke K, Erbert G, Keller U 2012 Opt. Express 20 4248
[27] Lim J, Chen H, Xu S, Yang Z, Chang G, Kärtner F X 2014 Opt. Lett. 39 2060
[28] Klenner A, Golling M, Keller U 2014 Opt. Express 22 11884
[29] Bernitt S, Brown G V, Rudolph J K, et al. 2012 Nature 492 225
[30] Derevianko A, Johnson W R 1997 Phys. Rev. A 56 1228
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