Abstract

A compact numerical method for simulating ultrafast pulse interaction with inhomogeneously broadened multi-level media is reported. We use a low-dispersion pseudospectral scheme with fourth order time stepping for Maxwell’s equations, and a weakly coupled operator splitting method for the Bloch equations where inhomogeneous broadening and relaxations are also taken into account. The underlying physics is briefly discussed with emphasis on the formalism used.

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References

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  1. T. Bartel, P. Gaal, K. Reimann, M. Woerner, and T. Elsaesser, “Generation of single-cycle THz transients with high electric-field amplitudes,” Opt. Lett. 30, 2805–2807 (2005).
    [CrossRef] [PubMed]
  2. R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
    [CrossRef] [PubMed]
  3. V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
    [CrossRef] [PubMed]
  4. J. N. Sweetser and I. A. Walmsley, “Linear pulse propagation in stationary and nonstationary multilevel media in the transient regime,” J. Opt. Soc. Am. B 13, 601–612 (1996).
    [CrossRef]
  5. L. E. E. Araujo, “Ultrashort pulse propagation in multilevel systems,” Phys. Rev. A 72, 053802 (2005).
    [CrossRef]
  6. S. Hughes, “Breakdown of the Area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. A 81, 3363–3366 (1998).
  7. R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
    [CrossRef] [PubMed]
  8. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
    [CrossRef]
  9. J. A. Gruetzmacher and N. F. Scherer, “Finite-difference time-domain simulations of ultrashort pulse propagation incorporating quantum-mechanical response functions,” Opt. Lett. 28, 573–575 (2003).
    [CrossRef] [PubMed]
  10. S. Hughes, “Subfemtosecond soft-x-ray generation from a two-level atom: extreme carrier-wave Rabi flopping,” Phys. Rev. A 62, 055401 (2000).
    [CrossRef]
  11. V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1998).
    [CrossRef]
  12. Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
    [CrossRef]
  13. F. Schlottau, M. Piket-May, and K. Wagner, “Modeling of femtosecond pulse interaction with inhomogeneously broadened media using an iterative predictor-corrector FDTD method,” Opt. Express 13, 182–194 (2005).
    [CrossRef] [PubMed]
  14. B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
    [CrossRef]
  15. B. Bidegaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Methods Partial Differ. Equ. 19, 284–300 (2003).
    [CrossRef]
  16. A. Bourgeade and O. Saut, “Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals,” J. Comp. Phys. 213, 823–843 (2006).
    [CrossRef]
  17. S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).
  18. Q. H. Liu, “The pseudospectral time-domain (PSTD) method: A new algorithm for solutions of Maxwell’s equations,” in Antennas and Propagation Society International Symposium Digest (IEEE, 1997), Vol. 1, pp. 122–125.
    [CrossRef]
  19. T.-W. Lee and S. C. Hagness, “Pseudospectral time-domain methods for modeling optical wave propagation in second-order nonlinear materials,” J. Opt. Soc. Am. B 21, 330–342 (2004).
    [CrossRef]
  20. S. Rosseland, “On the transmission of radiation through an absorbing medium in motion, with applications to the theory of sun-spots and solar rotation,” Astrophys. J. 63, 342–367 (1926).
    [CrossRef]
  21. S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
    [CrossRef]
  22. R. B. Sidje, “Expokit: a software package for computing matrix exponentials,” ACM Trans. Math. Softw. 24, 130–156 (1998).
    [CrossRef]
  23. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

2010

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

2008

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

2006

A. Bourgeade and O. Saut, “Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals,” J. Comp. Phys. 213, 823–843 (2006).
[CrossRef]

2005

2004

2003

2000

S. Hughes, “Subfemtosecond soft-x-ray generation from a two-level atom: extreme carrier-wave Rabi flopping,” Phys. Rev. A 62, 055401 (2000).
[CrossRef]

B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
[CrossRef]

1998

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1998).
[CrossRef]

R. B. Sidje, “Expokit: a software package for computing matrix exponentials,” ACM Trans. Math. Softw. 24, 130–156 (1998).
[CrossRef]

S. Hughes, “Breakdown of the Area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. A 81, 3363–3366 (1998).

1996

1995

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef] [PubMed]

1993

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
[CrossRef] [PubMed]

1966

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1926

S. Rosseland, “On the transmission of radiation through an absorbing medium in motion, with applications to the theory of sun-spots and solar rotation,” Astrophys. J. 63, 342–367 (1926).
[CrossRef]

Allegrini, M.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Araujo, L. E. E.

L. E. E. Araujo, “Ultrashort pulse propagation in multilevel systems,” Phys. Rev. A 72, 053802 (2005).
[CrossRef]

Arnold, J. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef] [PubMed]

Bartel, T.

Bidegaray, B.

B. Bidegaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Methods Partial Differ. Equ. 19, 284–300 (2003).
[CrossRef]

B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
[CrossRef]

Blanes, S.

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

Bouloufa, N.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Bourgeade, A.

A. Bourgeade and O. Saut, “Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals,” J. Comp. Phys. 213, 823–843 (2006).
[CrossRef]

B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
[CrossRef]

Bucksbaum, P. H.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
[CrossRef] [PubMed]

Casas, F.

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

Chotia, A.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Comparat, D.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Cui, N.

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Dulieu, O.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

Elsaesser, T.

Gaal, P.

Gogny, D. M.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef] [PubMed]

Gong, S.

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Gruetzmacher, J. A.

Hagness, S. C.

Herrmann, J.

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1998).
[CrossRef]

Hughes, S.

S. Hughes, “Subfemtosecond soft-x-ray generation from a two-level atom: extreme carrier-wave Rabi flopping,” Phys. Rev. A 62, 055401 (2000).
[CrossRef]

S. Hughes, “Breakdown of the Area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. A 81, 3363–3366 (1998).

Jones, R. R.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
[CrossRef] [PubMed]

Kalosha, V. P.

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1998).
[CrossRef]

Lee, T.-W.

Li, R.

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Liu, Q. H.

Q. H. Liu, “The pseudospectral time-domain (PSTD) method: A new algorithm for solutions of Maxwell’s equations,” in Antennas and Propagation Society International Symposium Digest (IEEE, 1997), Vol. 1, pp. 122–125.
[CrossRef]

Mukamel, S.

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

Niu, Y.

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Oteo, J. A.

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

Piket-May, M.

Pillet, P.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Reignier, D.

B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
[CrossRef]

Reimann, K.

Ros, J.

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

Rosseland, S.

S. Rosseland, “On the transmission of radiation through an absorbing medium in motion, with applications to the theory of sun-spots and solar rotation,” Astrophys. J. 63, 342–367 (1926).
[CrossRef]

Saut, O.

A. Bourgeade and O. Saut, “Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals,” J. Comp. Phys. 213, 823–843 (2006).
[CrossRef]

Scherer, N. F.

Schlottau, F.

Sidje, R. B.

R. B. Sidje, “Expokit: a software package for computing matrix exponentials,” ACM Trans. Math. Softw. 24, 130–156 (1998).
[CrossRef]

Sweetser, J. N.

Viteau, V.

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Wagner, K.

Walmsley, I. A.

Woerner, M.

Xia, K.

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

You, D.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef] [PubMed]

ACM Trans. Math. Softw.

R. B. Sidje, “Expokit: a software package for computing matrix exponentials,” ACM Trans. Math. Softw. 24, 130–156 (1998).
[CrossRef]

Astrophys. J.

S. Rosseland, “On the transmission of radiation through an absorbing medium in motion, with applications to the theory of sun-spots and solar rotation,” Astrophys. J. 63, 342–367 (1926).
[CrossRef]

Eur. J. Phys.

S. Blanes, F. Casas, J. A. Oteo, and J. Ros, “A pedagogical approach to the Magnus expansion,” Eur. J. Phys. 31, 907–918 (2010).
[CrossRef]

IEEE Trans. Antennas Propag.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

J. Comp. Phys.

B. Bidegaray, A. Bourgeade, and D. Reignier, “Introducing physical relaxations terms in Bloch equations,” J. Comp. Phys. 170, 603–613 (2000).
[CrossRef]

A. Bourgeade and O. Saut, “Numerical methods for the bidimensional Maxwell-Bloch equations in nonlinear crystals,” J. Comp. Phys. 213, 823–843 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Numer. Methods Partial Differ. Equ.

B. Bidegaray, “Time discretizations for Maxwell-Bloch equations,” Numer. Methods Partial Differ. Equ. 19, 284–300 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. Hughes, “Subfemtosecond soft-x-ray generation from a two-level atom: extreme carrier-wave Rabi flopping,” Phys. Rev. A 62, 055401 (2000).
[CrossRef]

L. E. E. Araujo, “Ultrashort pulse propagation in multilevel systems,” Phys. Rev. A 72, 053802 (2005).
[CrossRef]

S. Hughes, “Breakdown of the Area theorem: carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. A 81, 3363–3366 (1998).

R. W. Ziolkowski, J. M. Arnold, and D. M. Gogny, “Ultrafast pulse interactions with two-level atoms,” Phys. Rev. A 52, 3082–3094 (1995).
[CrossRef] [PubMed]

Y. Niu, K. Xia, N. Cui, S. Gong, and R. Li, “Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium,” Phys. Rev. A 78, 063835 (2008).
[CrossRef]

Phys. Rev. Lett.

R. R. Jones, D. You, and P. H. Bucksbaum, “Ionization of Rydberg atoms by subpicosecond half-cycle electromagnetic pulses,” Phys. Rev. Lett. 70, 1236–1239 (1993).
[CrossRef] [PubMed]

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell-Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1998).
[CrossRef]

Science

V. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, and P. Pillet, “Optical pumping and vibrational cooling of molecules,” Science 321, 232–234 (2008).
[CrossRef] [PubMed]

Other

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, 1995).

Q. H. Liu, “The pseudospectral time-domain (PSTD) method: A new algorithm for solutions of Maxwell’s equations,” in Antennas and Propagation Society International Symposium Digest (IEEE, 1997), Vol. 1, pp. 122–125.
[CrossRef]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, 1987).

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Figures (4)

Fig. 1
Fig. 1

Propagation of a 2π hyperbolic secant soliton shown in a two-level absorber of length L = 37.5 μ m. The left figure shows the envelope |(t, z)| of the electric field E(t,z)= (t, z)cosϕ (t,z). The envelope has been extracted from the full-field data by use of a Hilbert transform. The right figure shows the traveling inversion bump ρ 22(t,z) – ρ 11(t,z) of the medium. The “flattenings” on the inversion bump are manifestations of counter-rotating terms in the matter-field interaction [7].

Fig. 2
Fig. 2

Photon echo(es) shown in a two-level absorber for a simulation with Δt = 0.318fs for a total duration of 11ps. The top figure shows the injected pulse envelopes obtained through a Hilbert transform of the full-field, while the inset shows the population inversion 〈ρ 22ρ 11Σ during the passage of the first pulse. The second figure (below) shows the envelope(s) of the polarization(s) P(t) = ��(t)cosϕ(t)obtained through a Hilbert transform for vp /c = 1/(50π)(dotted), vp /c = 1/(100π)(dashed) and vp /c = 1/(200π)(solid).

Fig. 3
Fig. 3

Few-cycle pulse propagation shown in a 6-level ladder configuration. The left figure shows the rapid absorption of the pulse in the medium. Even after only one pulselength the peak amplitude has been reduced to half its initial amplitude. The right figure shows the pulse spectrum during propagation.

Fig. 4
Fig. 4

The figure on the left shows the normalized energy during the passage of the pulse at the entrance of the medium, while the figure on the right shows the level populations.

Equations (34)

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

H ^ = ω i ( v ) | i i | E ( r , t ) μ i j ( θ , ϕ ) | i j | ,
Σ = n ( v , θ , ϕ ) d v d Ω ,
d t | ρ = L ˇ | ρ .
| O = O j k | j k ,
O 1 | O 2 = Tr ( O ^ 1 O ^ 2 ) .
L ˇ = i H ^ ( H ^ ) .
L ˇ L ˇ + R ˇ .
d t ρ ^ = ( i ) 1 [ H ^ , ρ ^ ] + i , j ( γ ^ i j ρ ^ γ ^ i j 1 2 ρ ^ γ ^ i j γ ^ i j 1 2 γ ^ i j γ ^ i j ρ ^ ) ,
d t ρ m n = ( i ) 1 [ H ^ , ρ ^ ] m n 1 2 j ( γ j m + γ j n ) ρ m n , m n
d t ρ m m = ( i ) 1 [ H ^ , ρ ^ ] m n + j m γ m j ρ j j j m γ j m ρ m m
R ˇ = i , j [ γ ^ i j γ ^ i j 1 2 ( γ ^ i j γ ^ i j ) ( γ ^ i j γ ^ i j ) ] ,
P = N a μ | ρ Σ ,
d t P = N a μ | L ˇ | ρ Σ .
| ρ ( t ) = U ˇ 0 ( t , t 0 ) exp [ n = 1 Ω ˇ ( t , t 0 ) ] | ρ ( t 0 ) ,
Ω ˇ 1 ( t , t 0 ) = t 0 t d τ L ˇ I ( τ ) ,
t H = × E ,
t E = × H ζ μ | L ˇ 0 + E L ˇ 1 ) | ρ Σ ,
d t | ρ = ( L ˇ 0 + E L ˇ 1 ) | ρ ,
L ˇ 0 = i H ^ 0 ( H ^ 0 ) + R ˇ ,
L ˇ 1 = i μ ( μ ^ ) .
| ρ n + 1 / 2 = U ˇ 0 U ˇ I n | ρ n 1 / 2 ,
U ˇ 0 = exp ( L ˇ 0 Δ t ) ,
U ˇ I n = exp ( L ˇ I n Δ t ) ,
U ˇ I n = exp ( i E n μ ^ Δ t ) * exp ( i E n μ ^ Δ t ) ,
| ρ n + 1 / 2 = U ˇ 0 [ exp ( i E n μ ^ Δ t ) * exp ( i E n μ ^ Δ t ) ] | ρ n 1 / 2 .
| ρ n + 1 / 2 = U ˇ 0 1 / 2 U ˇ I n U ˇ 0 1 / 2 | ρ n 1 / 2 .
d F = d 1 ( 2 π i k d d ( F ) )
( 𝔻 t H x ) i , j , k n = ( y E z z E y ) i , j , k n ,
( 𝔻 t H y ) i , j , k n = ( z E x x E z ) i , j , k n ,
( 𝔻 t H z ) i , j , k n = ( x E y y E x ) i , j , k n ,
( 𝔻 t E x ) i , j , k n + 1 / 2 = ( y H z z H y ) i , j , k n + 1 / 2 ( ζ μ x | ( L ˇ 0 + E L ˇ 1 | ρ Σ ) i , j , k n + 1 / 2 ,
( 𝔻 t E y ) i , j , k n + 1 / 2 = ( z H x x H z ) i , j , k n + 1 / 2 ( ζ μ y | ( L ˇ 0 + E L ˇ 1 | ρ Σ ) i , j , k n + 1 / 2 ,
( 𝔻 t E z ) i , j , k n + 1 / 2 = ( x H y y H x ) i , j , k n + 1 / 2 ( ζ μ z | ( L ˇ 0 + E L ˇ 1 | ρ Σ ) i , j , k n + 1 / 2 ,
g ( v ) = 1 π v p 2 exp ( v 2 v p 2 ) .

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