Abstract

We theoretically investigate high-order harmonic generation (HHG) from solids in two-color fields. It is found that under the premise of maintaining the same amplitude, the intensity of the second plateau can be enhanced by two to three orders in a proper two-color field compared with the result in the monochromatic field with the same frequency as the driving pulse of the two-color field. This can be attributed to the fact that most excited electrons can be driven to the top of the first conduction band due to the larger vector potential of the two-color fields, which leads to the higher electron population of upper conduction bands. Moreover, we also find that isolated attosecond pulses can be generated from solids by choosing a proper two-color field that allows the electrons to reach the top of the first conduction band only once. This work provides a promising method for extending the range of solid HHG spectra in experiments.

© 2017 Optical Society of America

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  5. T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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2017 (4)

2016 (9)

Z. Guan, X. X. Zhou, and X. B. Bian, “High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses,” Phys. Rev. A 93(3), 033852 (2016).
[Crossref]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016).
[Crossref]

G. Li, Y. Zheng, X. Ge, Z. Zeng, and R. Li, “Frequency modulation of high-order harmonic generation in an orthogonally polarized two-color laser field,” Opt. Express 24(16), 18685–18694 (2016).
[Crossref] [PubMed]

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Physics 13(3), 262–265 (2016).
[Crossref]

T. Y. Du, Z. Guan, X. X. Zhou, and X. B. Bian, “Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields,” Phys. Rev. A 94(2), 023419 (2016).
[Crossref]

S. Han, H. Kim, Y. W. Kim, Y. J. Kim, S. Kim, I. Y. Park, and S. W. Kim, “High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanostructure,” Nat. Commun. 7, 13105 (2016).
[Crossref] [PubMed]

C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016).
[Crossref]

C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016).
[Crossref] [PubMed]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

2015 (9)

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015).
[Crossref]

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

S. Haessler, T. Balciunas, G. Fan, L. E. Chipperfield, and A. Baltuska, “Enhanced multi-colour gating for the generation of high-power isolated attosecond pulses,” Sci. Rep. 5, 10084 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015).
[Crossref]

2014 (4)

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
[Crossref] [PubMed]

2010 (6)

P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82(5), 053413 (2010).
[Crossref]

J. G. Chen, S. L. Zeng, and Y. J. Yang, “Generation of isolated sub-50-as pulses by quantum path control in the multicycle regime,” Phys. Rev. A 82(4), 043401 (2010).
[Crossref]

H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81(6), 063813 (2010).
[Crossref]

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010).
[Crossref]

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

2009 (2)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

L. E. Chipperfield, J. S. Robinson, J. W. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102(6), 063003 (2009).
[Crossref] [PubMed]

2007 (2)

Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007).
[Crossref] [PubMed]

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Physics 3(6), 381–387 (2007).
[Crossref]

2005 (1)

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005).
[Crossref]

2000 (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

1997 (1)

M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389 (1997).
[Crossref]

1986 (1)

M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atoms ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

1952 (1)

J. C. Slater and J. C. Slater, “A soluble problem in energy bands,” Phys. Rev. 87(5), 807–835 (1952).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010).
[Crossref]

Ammosov, M.

M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atoms ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).

Anh-Thu, L.

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Arbeiter, M.

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

Arpin, P.

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Balciunas, T.

S. Haessler, T. Balciunas, G. Fan, L. E. Chipperfield, and A. Baltuska, “Enhanced multi-colour gating for the generation of high-power isolated attosecond pulses,” Sci. Rep. 5, 10084 (2015).
[Crossref] [PubMed]

Baltuska, A.

S. Haessler, T. Balciunas, G. Fan, L. E. Chipperfield, and A. Baltuska, “Enhanced multi-colour gating for the generation of high-power isolated attosecond pulses,” Sci. Rep. 5, 10084 (2015).
[Crossref] [PubMed]

Bauer, D.

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

Bian, X. B.

T. Y. Du and X. B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25(1), 151–158 (2017).
[Crossref] [PubMed]

Z. Guan, X. X. Zhou, and X. B. Bian, “High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses,” Phys. Rev. A 93(3), 033852 (2016).
[Crossref]

T. Y. Du, Z. Guan, X. X. Zhou, and X. B. Bian, “Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields,” Phys. Rev. A 94(2), 023419 (2016).
[Crossref]

T. Y. Du and X. B. Bian, “Mechanism of high-order harmonic generation from periodic potentials,” arXiv:1606.06433 (2016).

Brabec, T.

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

Browne, D. A.

Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017).
[Crossref] [PubMed]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

Chaloupka, J. L.

C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016).
[Crossref] [PubMed]

Chang, Z.

Chen, J. G.

J. G. Chen, S. L. Zeng, and Y. J. Yang, “Generation of isolated sub-50-as pulses by quantum path control in the multicycle regime,” Phys. Rev. A 82(4), 043401 (2010).
[Crossref]

Chen, M. C.

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Cheng, Y.

Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007).
[Crossref] [PubMed]

Chew, A.

Chini, M.

Chipperfield, L. E.

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F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
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C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
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[Crossref]

Lin, C. D.

C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
[Crossref] [PubMed]

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Liu, C.

C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016).
[Crossref]

Liu, H.

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Physics 13(3), 262–265 (2016).
[Crossref]

Luu, T. T.

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref] [PubMed]

Mancuso, C. A.

C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016).
[Crossref] [PubMed]

Marangos, J. P.

L. E. Chipperfield, J. S. Robinson, J. W. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102(6), 063003 (2009).
[Crossref] [PubMed]

McDonald, C. R.

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Meier, T.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Meiwes-Broer, K. H.

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

Midorikawa, K.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82(5), 053413 (2010).
[Crossref]

Moulet, A.

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref] [PubMed]

Murnane, M. M.

C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016).
[Crossref] [PubMed]

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Nam, C. H.

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005).
[Crossref]

Ndabashimiye, G.

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

Orlando, G.

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Park, I. Y.

S. Han, H. Kim, Y. W. Kim, Y. J. Kim, S. Kim, I. Y. Park, and S. W. Kim, “High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanostructure,” Nat. Commun. 7, 13105 (2016).
[Crossref] [PubMed]

Park, J. Y.

I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005).
[Crossref]

Popmintchev, T.

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

Protopapas, M.

M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389 (1997).
[Crossref]

Reis, D. A.

Y. Y. S. You, D. A. Reis, and S. Ghimire, “Anisotropic high-harmonic generation in bulk crystals,” Nat. Physics 13(4), 345–349 (2017).
[Crossref]

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Physics 13(3), 262–265 (2016).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015).
[Crossref]

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010).
[Crossref]

Ren, X.

Robert, L.

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Robinson, J. S.

L. E. Chipperfield, J. S. Robinson, J. W. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102(6), 063003 (2009).
[Crossref] [PubMed]

Schafer, K. J.

Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017).
[Crossref] [PubMed]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015).
[Crossref]

Schmidt, B. E.

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

Schubert, O.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Sistrunk, E.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010).
[Crossref]

Skruszewicz, S.

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

Slater, J. C.

J. C. Slater and J. C. Slater, “A soluble problem in energy bands,” Phys. Rev. 87(5), 807–835 (1952).
[Crossref]

J. C. Slater and J. C. Slater, “A soluble problem in energy bands,” Phys. Rev. 87(5), 807–835 (1952).
[Crossref]

Song, X.

Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007).
[Crossref] [PubMed]

Takahashi, E. J.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82(5), 053413 (2010).
[Crossref]

Thire, N.

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

Tiggesbaumker, J.

S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015).
[Crossref] [PubMed]

Tisch, J. W.

L. E. Chipperfield, J. S. Robinson, J. W. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102(6), 063003 (2009).
[Crossref] [PubMed]

Toru, M.

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Urbanek, B.

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Vampa, G.

K. F. Lee, X. Ding, T. J. Hammond, M. E. Fermann, G. Vampa, and P. B. Corkum, “Harmonic generation in solids with direct fiber laser pumping,” Opt. Lett. 42(6), 1113–1116 (2017).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Wang, G.

C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
[Crossref] [PubMed]

Wang, H.

H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015).
[Crossref]

H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81(6), 063813 (2010).
[Crossref]

Wei, H.

C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
[Crossref] [PubMed]

Wen, Y.

H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015).
[Crossref]

Wu, M.

Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017).
[Crossref] [PubMed]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016).
[Crossref]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015).
[Crossref]

Xu, Z.

Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007).
[Crossref] [PubMed]

Xue, S.

H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015).
[Crossref]

Yakovlev, V. S.

P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015).
[Crossref]

Yang, Y. J.

J. G. Chen, S. L. Zeng, and Y. J. Yang, “Generation of isolated sub-50-as pulses by quantum path control in the multicycle regime,” Phys. Rev. A 82(4), 043401 (2010).
[Crossref]

Yin, Y.

You, Y. S.

You, Y. Y. S.

Y. Y. S. You, D. A. Reis, and S. Ghimire, “Anisotropic high-harmonic generation in bulk crystals,” Nat. Physics 13(4), 345–349 (2017).
[Crossref]

Zeng, S. L.

J. G. Chen, S. L. Zeng, and Y. J. Yang, “Generation of isolated sub-50-as pulses by quantum path control in the multicycle regime,” Phys. Rev. A 82(4), 043401 (2010).
[Crossref]

Zeng, Z.

C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016).
[Crossref]

G. Li, Y. Zheng, X. Ge, Z. Zeng, and R. Li, “Frequency modulation of high-order harmonic generation in an orthogonally polarized two-color laser field,” Opt. Express 24(16), 18685–18694 (2016).
[Crossref] [PubMed]

Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007).
[Crossref] [PubMed]

Zhangjin, C.

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Zhao, K.

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

Zheng, Y.

C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016).
[Crossref]

G. Li, Y. Zheng, X. Ge, Z. Zeng, and R. Li, “Frequency modulation of high-order harmonic generation in an orthogonally polarized two-color laser field,” Opt. Express 24(16), 18685–18694 (2016).
[Crossref] [PubMed]

Zhou, X. X.

Z. Guan, X. X. Zhou, and X. B. Bian, “High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses,” Phys. Rev. A 93(3), 033852 (2016).
[Crossref]

T. Y. Du, Z. Guan, X. X. Zhou, and X. B. Bian, “Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields,” Phys. Rev. A 94(2), 023419 (2016).
[Crossref]

Zusin, D.

C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016).
[Crossref] [PubMed]

J. Phys. B (1)

C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010).
[Crossref]

Nat. Commun. (2)

S. Han, H. Kim, Y. W. Kim, Y. J. Kim, S. Kim, I. Y. Park, and S. W. Kim, “High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanostructure,” Nat. Commun. 7, 13105 (2016).
[Crossref] [PubMed]

C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014).
[Crossref] [PubMed]

Nat. Photonics (3)

T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010).
[Crossref]

M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014).
[Crossref]

Nat. Physics (4)

Y. Y. S. You, D. A. Reis, and S. Ghimire, “Anisotropic high-harmonic generation in bulk crystals,” Nat. Physics 13(4), 345–349 (2017).
[Crossref]

H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Physics 13(3), 262–265 (2016).
[Crossref]

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010).
[Crossref]

P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Physics 3(6), 381–387 (2007).
[Crossref]

Nature (3)

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015).
[Crossref] [PubMed]

G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015).
[Crossref] [PubMed]

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016).
[Crossref] [PubMed]

Opt. Commun. (1)

H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

J. C. Slater and J. C. Slater, “A soluble problem in energy bands,” Phys. Rev. 87(5), 807–835 (1952).
[Crossref]

Phys. Rev. A (9)

C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016).
[Crossref]

M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016).
[Crossref]

P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015).
[Crossref]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The schematic plot of the motion of a Bloch electron. Three different transition paths from C1 to C2 are shown. The band structure is obtained from the Bloch theory.
Fig. 2
Fig. 2 The HHG spectra in the two-color fields with different relative phases. The blue line and the red line indicate the dependence of the peak electric field and the peak vector potential on the relative phase, respectively.
Fig. 3
Fig. 3 The electric fields, the vector potentials of the laser pulses and the corresponding HHG spectra. The black solid lines and the red doted lines represent the monochromatic field and the two-color field, respectively. (a) The electric fields of the monochromatic and the two-color field. (b) Corresponding vector potentials. (c) The HHG spectra in the two laser fields.
Fig. 4
Fig. 4 Electron populations of the solid system in (a) the monochromatic field and (b) the two-color field. Their partial enlargements are shown on (c) and (d), respectively. Blue areas are band gaps. Time-frequency analysis of the time-dependent current in (e) the monochromatic field and (f) the two-color field. The white solid lines represent the semi-classical trajectories of the electrons on C1. The white doted lines and the white dashed line represent the trajectories of the electrons transited from C1 to C2 through the path three and other paths, respectively. The white solid lines, the white doted lines and the white dashed lines are the semi-classical trajectories calculated by εC1[A(t)], εC2[π/a0 + A(t)] and εC2[A(t)], respectively.
Fig. 5
Fig. 5 The total electron population on each conduction band. (a) and (c) are the population on C1 in the monochromatic field and the two-color field, respectively. (b) and (d) are the population on high-lying conduction band in the monochromatic field and the two-color field, respectively. We do not show the population on C4 in (b), because it cannot be seen in such a scale.
Fig. 6
Fig. 6 (a) The HHG spectrum in the two-color field. (b) Time-frequency analysis of the time-dependent current. (c) The isolated attosecond pulse produced by filtering the 64th–84th harmonics. (d) The time-dependent population of C1 to C3.

Equations (6)

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ε n ( t ) = ε n [ k ( t ) ] = ε n [ k 0 + A ( t ) ] .
H ^ ( t , x ) = H ^ 0 x E ( t ) ,
E ( t ) = { E 1 cos ( ω 1 t ) + E 2 cos [ ω 2 ( t τ delay ) ] } f ( t ) ,
H ^ 0 ψ n ( x ) = ε n ψ n ( x ) ï i j Ň
ψ ( x , t + Δ t ) = 1 i H ^ Δ t / 2 1 + i H ^ Δ t / 2 ψ ( x , t ) .
j ( t ) = ψ ( t ) | p ^ | ψ ( t ) .

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