Abstract

A new graphical perturbation technique, based on the adiabatic solution of the density matrix, is developed to calculate nonlinear optical polarization. In each step of the perturbation, the adiabatic solution is expressed by a pair of conjugate diagrams describing the propagation of eigenstates. Using simple diagrammatic rules defined here, we calculate various third- and higher-order nonlinear optical polarizations including that for the seventh-harmonic generation. Inspection of higher-order calculation shows a consistent graphical pattern that can be used to directly write the nonlinear optical polarization of any desired order.

© 2006 Optical Society of America

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  1. N. Mukherjee, "Constructive interference and efficient vacuum-ultraviolet generation in resonant six-wave mixing," Phys. Rev. A 51, 3221-3226 (1995).
    [CrossRef] [PubMed]
  2. L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
    [CrossRef] [PubMed]
  3. J. F. Ward, "Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory," Rev. Mod. Phys. 37, 1-18 (1965).
    [CrossRef]
  4. A. Yariv, "The application of time evolution operators and Feynman diagrams to nonlinear optics," QE-13, 943-950 (1977).
  5. T. K. Yee and T. K. Gustafson, "Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and CW responses," Phys. Rev. A 18, 1597-1617 (1978).
    [CrossRef]
  6. D. E. Nikonov and M. O. Scully "Diagrammatic representation of quantum interference in lasting without inversion," in Coherent Phenomena and Amplification without Inversion, A. L. Andreev, O. A. Kocharovskaya, and P. Mandel, eds., Proc. SPIE 2798, 198-204 (1996).
    [CrossRef]
  7. P. W. Milonni, and J. H. Eberly, "Temporal coherence in multiphoton absorption. Far off-resonance intermediate states," J. Chem. Phys. 68, 1602-1613 (1978).
    [CrossRef]
  8. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  9. D. Marcuse, Principles of Quantum Electronics (Academic, 1980).

1996 (2)

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

D. E. Nikonov and M. O. Scully "Diagrammatic representation of quantum interference in lasting without inversion," in Coherent Phenomena and Amplification without Inversion, A. L. Andreev, O. A. Kocharovskaya, and P. Mandel, eds., Proc. SPIE 2798, 198-204 (1996).
[CrossRef]

1995 (1)

N. Mukherjee, "Constructive interference and efficient vacuum-ultraviolet generation in resonant six-wave mixing," Phys. Rev. A 51, 3221-3226 (1995).
[CrossRef] [PubMed]

1978 (2)

P. W. Milonni, and J. H. Eberly, "Temporal coherence in multiphoton absorption. Far off-resonance intermediate states," J. Chem. Phys. 68, 1602-1613 (1978).
[CrossRef]

T. K. Yee and T. K. Gustafson, "Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and CW responses," Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

1977 (1)

A. Yariv, "The application of time evolution operators and Feynman diagrams to nonlinear optics," QE-13, 943-950 (1977).

1965 (1)

J. F. Ward, "Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory," Rev. Mod. Phys. 37, 1-18 (1965).
[CrossRef]

Deng, L.

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Milonni, and J. H. Eberly, "Temporal coherence in multiphoton absorption. Far off-resonance intermediate states," J. Chem. Phys. 68, 1602-1613 (1978).
[CrossRef]

Garrett, W. R.

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

Gustafson, T. K.

T. K. Yee and T. K. Gustafson, "Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and CW responses," Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

Lee, D. Z.

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

Marcuse, D.

D. Marcuse, Principles of Quantum Electronics (Academic, 1980).

Milonni, P. W.

P. W. Milonni, and J. H. Eberly, "Temporal coherence in multiphoton absorption. Far off-resonance intermediate states," J. Chem. Phys. 68, 1602-1613 (1978).
[CrossRef]

Mukherjee, N.

N. Mukherjee, "Constructive interference and efficient vacuum-ultraviolet generation in resonant six-wave mixing," Phys. Rev. A 51, 3221-3226 (1995).
[CrossRef] [PubMed]

Nikonov, D. E.

D. E. Nikonov and M. O. Scully "Diagrammatic representation of quantum interference in lasting without inversion," in Coherent Phenomena and Amplification without Inversion, A. L. Andreev, O. A. Kocharovskaya, and P. Mandel, eds., Proc. SPIE 2798, 198-204 (1996).
[CrossRef]

Payne, M. G.

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

Scully, M. O.

D. E. Nikonov and M. O. Scully "Diagrammatic representation of quantum interference in lasting without inversion," in Coherent Phenomena and Amplification without Inversion, A. L. Andreev, O. A. Kocharovskaya, and P. Mandel, eds., Proc. SPIE 2798, 198-204 (1996).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Ward, J. F.

J. F. Ward, "Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory," Rev. Mod. Phys. 37, 1-18 (1965).
[CrossRef]

Yariv, A.

A. Yariv, "The application of time evolution operators and Feynman diagrams to nonlinear optics," QE-13, 943-950 (1977).

Yee, T. K.

T. K. Yee and T. K. Gustafson, "Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and CW responses," Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

J. Chem. Phys. (1)

P. W. Milonni, and J. H. Eberly, "Temporal coherence in multiphoton absorption. Far off-resonance intermediate states," J. Chem. Phys. 68, 1602-1613 (1978).
[CrossRef]

Phys. Rev. A (3)

T. K. Yee and T. K. Gustafson, "Diagrammatic analysis of the density operator for nonlinear optical calculations: pulsed and CW responses," Phys. Rev. A 18, 1597-1617 (1978).
[CrossRef]

N. Mukherjee, "Constructive interference and efficient vacuum-ultraviolet generation in resonant six-wave mixing," Phys. Rev. A 51, 3221-3226 (1995).
[CrossRef] [PubMed]

L. Deng, W. R. Garrett, M. G. Payne, and D. Z. Lee, "Effect of the odd-photon destructive interference on laser-induced transparency and multiphoton excitation and ionization in rubidium," Phys. Rev. A 54, 4218-4225 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

D. E. Nikonov and M. O. Scully "Diagrammatic representation of quantum interference in lasting without inversion," in Coherent Phenomena and Amplification without Inversion, A. L. Andreev, O. A. Kocharovskaya, and P. Mandel, eds., Proc. SPIE 2798, 198-204 (1996).
[CrossRef]

Rev. Mod. Phys. (1)

J. F. Ward, "Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory," Rev. Mod. Phys. 37, 1-18 (1965).
[CrossRef]

Other (3)

A. Yariv, "The application of time evolution operators and Feynman diagrams to nonlinear optics," QE-13, 943-950 (1977).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

D. Marcuse, Principles of Quantum Electronics (Academic, 1980).

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Figures (8)

Fig. 1
Fig. 1

Stepwise generation of σ n m ( ω j ) via single-photon coupling to an intermediate level ν. (a) and (b) are the conjugate pair of diagrams representing the propagation of the ket eigenstate n and the bra state m, respectively. The other conjugate pair of diagrams (c) and (d) describe a similar propagation via single-photon emission.

Fig. 2
Fig. 2

Creation of third-harmonic polarization σ n m ( 3 ω ) . The conjugate pair (a) and (b) show the first-order expansion of σ n m ( 3 ω ) in the electric field. (a1), (a2), (b1), and (b2) extend the expansion to the next higher order. In each step of expansion the interaction is expressed by a pair of conjugate diagrams.

Fig. 3
Fig. 3

Generation of Stokes polarization σ n m ( ω r ) through vibronic resonances in the presence of pump and Stokes fields E P and E r . Conjugate diagrams (a) and (b) represent the first-order expansion of σ n m ( ω r ) the interacting electric fields. (a1)–(a4) and (b1)–(b4) extend the expansion to the next higher order in the electric fields. For each field component the interaction is expressed by a pair of conjugate diagrams describing the propagation of the bra and ket eigenstates.

Fig. 4
Fig. 4

Generation of third-order polarization in two-photon resonant four-wave mixing. The conjugate diagrams (a) and (b) represent the generation of σ n m ( ω ) via two-photon resonant interactions described by the resonant density-matrix elements σ μ m ( ω 1 + ω 2 ) and σ n ν ( ω 1 + ω 2 ) . For a specific pair of resonant levels a and b with ω b a ω 1 + ω 2 , the nonlinear polarization is expressed by the conjugate pair of diagrams (c) and (d).

Fig. 5
Fig. 5

Expansion of two-photon resonant σ a b ( ω 1 + ω 2 ) by the set of conjugate diagrams (a)–(d).

Fig. 6
Fig. 6

Graphical expansion of the seventh-order density matrix σ m n ( 7 ω ) . The conjugate diagrams 6.1(a) and 6.1(b) represent the first-order expansion of σ m n ( 7 ω ) . 6.2(a) and 6.2(b) show the perturbation expansion to the next higher order. The conjugate pair of diagrams in 6.3–6.8 show steps of the graphical perturbation up to the seventh order.

Fig. 7
Fig. 7

(a) Coherent mixing of eight energy levels to generate the seventh-harmonic polarization σ m n ( 7 ω ) . (b) Sequential breaking of σ m n ( 7 ω ) in terms of the lower-order density matrices. In the final step, σ m n ( 7 ω ) is expressed as a linear combination of the level population ρ m m , ρ a a , etc.

Fig. 8
Fig. 8

Eleventh-harmonic generation: coherent mixing of 12 atomic levels to generate σ m n ( 7 ω ) .

Equations (27)

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d ρ n m d t = ( i ω n m γ n m ) ρ n m + i E ν [ μ n ν ρ ν m ρ n ν μ ν m ] ,
d d t σ n m ( ω j ) + i Δ n m ( ω j ) σ n m ( ω j ) = i i , ν E i [ μ n ν σ ν m ( ω j ω i ) σ n ν ( ω j ω i ) μ ν m ] + i i , ν E i * [ μ n ν σ ν m ( ω j + ω i ) σ n ν ( ω j + ω i ) μ ν m ] ,
σ n m ( ω j , t ) = i , υ E i μ n ν Δ n m ( ω j ) σ ν m ( ω j ω i ) ( a ) i , υ σ n ν ( ω j ω i ) E i μ ν m Δ n m ( ω j ) ( b ) + i , υ E i * μ n ν Δ n m ( ω j ) σ ν m ( ω j + ω i ) ( c ) i , υ σ n ν ( ω j + ω i ) E i * μ ν m Δ n m ( ω j ) ( d ) .
σ μ m ( 2 ω ) = ν μ μ ν E σ ν m ( ω ) Δ μ m ( 2 ω ) ν σ μ ν ( ω ) Δ μ m ( 2 ω ) μ ν m E ,
σ μ m ( 2 ω ) = ν μ μ ν μ ν m Δ μ m ( 2 ω ) ( E ) 2 [ ρ m m ρ ν ν Δ ν m ( ω ) ρ ν ν ρ μ μ Δ μ ν ( ω ) ] .
σ n m ( ω 3 ) = ( E ) 3 1 Δ n m ( ω 3 ) μ ν μ n μ μ μ ν μ ν m [ ρ m m ρ ν ν Δ μ m ( 2 ω ) Δ ν m ( ω ) ρ ν ν ρ μ μ Δ μ m ( 2 ω ) Δ μ ν ( ω ) ρ ν ν ρ μ μ Δ n ν ( 2 ω ) Δ μ ν ( ω ) + ρ μ μ ρ n n Δ n ν ( 2 ω ) Δ n μ ( ω ) ] .
P ( ω 3 ) = N ( E ) 3 n μ ν m μ n μ μ μ ν μ ν m μ m n ρ n n [ 1 ( ω μ n + 3 ω ) ( ω ν n + 2 ω ) ( ω m n + ω ) + 1 ( ω μ n ω ) ( ω ν n + 2 ω ) ( ω m n + ω ) + 1 ( ω μ n ω ) ( ω ν n 2 ω ) ( ω m n + ω ) + 1 ( ω m n 3 ω ) ( ω ν n 2 ω ) ( ω μ n ω ) ] .
σ n m ( ω r ) = 1 Δ n m ( ω r ) μ [ μ n μ E P σ μ m ( ω r ω P ) σ n μ ( ω r ω P ) μ μ m E P ] ,
Fig . 3 ( a 1 ) μ μ ν E r σ ν m ( ω P ) 1 Δ μ m ( ω r ω P ) ,
Fig . 3 ( a 2 ) σ μ ν ( ω P ) μ ν m E r 1 Δ μ m ( ω r ω P ) ,
Fig . 3 ( a 3 ) μ μ ν E P * σ ν m ( ω r ) 1 Δ μ m ( ω r ω P ) ,
Fig . 3 ( a 4 ) σ μ ν ( ω r ) μ ν m E P * 1 Δ μ m ( ω r ω P ) ,
σ μ m ( ω r ω P ) = Fig. 3 ( a 1 ) + Fig. 3 ( a 2 ) + Fig. 3 ( a 3 ) + Fig . 3 ( a 4 ) .
σ μ m ( ω r ω P ) = ν ( μ μ ν E r ) ( μ ν m E P * ) 2 Δ μ m ( ω r ω P ) [ ( ρ m m ρ ν ν ) Δ ν m ( ω P ) ( ρ ν ν ρ μ μ ) Δ μ ν ( ω r ) ] + ν ( μ μ ν E P * ) ( μ ν m E r ) 2 Δ μ m ( ω r ω P ) [ ( ρ ν ν ρ μ μ ) Δ μ ν ( ω P ) + ( ρ m m ρ ν ν ) Δ ν m ( ω r ) ] .
P ( ω r ) = N n , m σ n m ( ω r ) μ m n ,
P ( ω r ) = N n , m , μ , ν μ m n { ( μ n μ E P ) ( μ μ ν E r ) ( μ ν m E P * ) 3 Δ n m ( ω r ) Δ μ m ( ω r ω P ) [ ( ρ m m ρ ν ν ) Δ ν m ( ω P ) + ( ρ μ μ ρ ν ν ) Δ μ ν ( ω r ) ] + ( μ n μ E P ) ( μ μ ν E P * ) ( μ ν m E r ) 3 Δ n m ( ω r ) Δ μ m ( ω r ω P ) [ ( ρ m m ρ ν ν ) Δ ν m ( ω r ) + ( ρ μ μ ρ ν ν ) Δ μ ν ( ω P ) ] + ( μ n ν E r ) ( μ ν μ E P * ) ( μ μ m E P ) 3 Δ n m ( ω r ) Δ n μ ( ω r ω P ) [ ( ρ ν ν ρ n n ) Δ n ν ( ω r ) + ( ρ ν ν ρ μ μ ) Δ ν μ ( ω P ) ] + ( μ n ν E P * ) ( μ ν μ E r ) ( μ μ m E P ) 3 Δ n m ( ω r ) Δ n μ ( ω r ω P ) [ ( ρ ν ν ρ n n ) Δ n ν ( ω P ) + ( ρ ν ν ρ μ μ ) Δ ν μ ( ω r ) ] } .
P ( ω ) = N n [ μ n a E 3 ( ω ω b n ) μ b n μ b n E 3 ( ω ω n a ) μ n a ] σ a b ( ω 1 + ω 2 ) .
σ a b ( ω 1 + ω 2 ) = Fig . 5 ( a ) + Fig . 5 ( b ) + Fig . 5 ( c ) + Fig . 5 ( d ) = 1 Δ a b ( ω 1 + ω 2 ) m [ μ a m E 1 σ m b ( ω 2 ) σ a m ( ω 2 ) μ m b E 1 + μ a m E 2 σ m b ( ω 1 ) σ a m ( ω 1 ) μ m b E 2 ] ,
P ( ω ) = N 3 n , m [ ( μ a m E 1 ) ( μ m b E 2 ) μ b n ( μ n a E 3 ) ( ω n b + ω ) ( ω b a + i γ a b ω 1 ω 2 ) ( ω m a ω 1 ) + ( μ a m E 2 ) ( μ m b E 1 ) μ b n ( μ n a E 3 ) ( ω n b + ω ) ( ω b a + i γ a b ω 1 ω 2 ) ( ω m a ω 2 ) + ( μ a m E 1 ) ( μ m b E 2 ) ( μ b n E 3 ) μ n a ( ω n a ω ) ( ω b a + i γ a b ω 1 ω 2 ) ( ω m a ω 1 ) + ( μ a m E 2 ) ( μ m b E 1 ) ( μ b n E 3 ) μ n a ( ω n a ω ) ( ω b a + i γ a b ω 1 ω 2 ) ( ω m a ω 2 ) ] ( ρ a a ρ b b ) .
σ m n ( ω 7 ) = μ m a E σ a n ( 6 ω ) 1 Δ m n ( 7 ) σ m f ( 6 ω ) μ f n E 1 Δ m n ( 7 ) ,
σ a n ( 6 ω ) = μ a b E σ b n ( 5 ω ) 1 Δ a n ( 6 ) σ a f ( 5 ω ) μ f n E 1 Δ a n ( 6 ) .
σ a n ( 6 ω ) = μ a b μ b c σ c n ( 4 ω ) ( E ) 2 1 Δ a n ( 6 ) Δ b n ( 5 ) μ a b σ b f ( 4 ω ) μ f n ( E ) 2 1 Δ a n ( 6 ) Δ b n ( 5 ) μ a b σ b f ( 4 ω ) μ f n ( E ) 2 1 Δ a n ( 6 ) Δ a f ( 5 ) + σ a e ( 4 ω ) μ e f μ f n ( E ) 2 1 Δ a n ( 6 ) Δ a f ( 5 ) .
σ a n ( 6 ω ) = μ a b μ b c μ c d μ d e μ e f μ f n ( E ) 6 × [ ρ n n Δ a n ( 6 ) Δ b n ( 5 ) Δ c n ( 4 ) Δ d n ( 3 ) Δ e n ( 2 ) Δ f n ( 1 ) + ρ f f Δ a f ( 5 ) Δ b f ( 4 ) Δ c f ( 3 ) Δ d f ( 2 ) Δ e f ( 1 ) Δ n f ( 1 ) + ρ e e Δ a e ( 4 ) Δ b e ( 3 ) Δ c e ( 2 ) Δ d e ( 1 ) Δ f e ( 1 ) Δ n e ( 2 ) + ρ d d Δ a d ( 3 ) Δ b d ( 2 ) Δ c d ( 1 ) Δ e d ( 1 ) Δ f d ( 2 ) Δ n d ( 3 ) + ρ c c Δ a c ( 2 ) Δ b c ( 1 ) Δ d c ( 1 ) Δ e c ( 2 ) Δ f c ( 3 ) Δ n c ( 4 ) + ρ b b Δ a b ( 1 ) Δ c b ( 1 ) Δ d b ( 2 ) Δ e b ( 3 ) Δ f b ( 4 ) Δ n b ( 5 ) + ρ a a Δ b a ( 1 ) Δ c a ( 2 ) Δ d a ( 3 ) Δ e a ( 4 ) Δ f a ( 5 ) Δ n a ( 6 ) ] ,
σ m n ( 7 ω ) = μ m a μ a b μ b c μ c d μ d e μ e f μ f n ( E ) 7 × [ ρ n n Δ m n ( 7 ) Δ a n ( 6 ) Δ b n ( 5 ) Δ c n ( 4 ) Δ d n ( 3 ) Δ e n ( 2 ) Δ f n ( 1 ) + ρ f f Δ m f ( 6 ) Δ a f ( 5 ) Δ b f ( 4 ) Δ c f ( 3 ) Δ d f ( 2 ) Δ e f ( 1 ) Δ n f ( 1 ) + ρ e e Δ m e ( 5 ) Δ a e ( 4 ) Δ b e ( 3 ) Δ c e ( 2 ) Δ d e ( 1 ) Δ f e ( 1 ) Δ n e ( 2 ) + ρ d d Δ m d ( 4 ) Δ a d ( 3 ) Δ b d ( 2 ) Δ c d ( 1 ) Δ e d ( 1 ) Δ f d ( 2 ) Δ n d ( 3 ) + ρ c c Δ m c ( 3 ) Δ a c ( 2 ) Δ b c ( 1 ) Δ d c ( 1 ) Δ e c ( 2 ) Δ f c ( 3 ) Δ n c ( 4 ) + ρ b b Δ m b ( 2 ) Δ a b ( 1 ) Δ c b ( 1 ) Δ d b ( 2 ) Δ e b ( 3 ) Δ f b ( 4 ) Δ n b ( 5 ) + ρ a a Δ m a ( 1 ) Δ b a ( 1 ) Δ c a ( 2 ) Δ d a ( 3 ) Δ e a ( 4 ) Δ f a ( 5 ) Δ n a ( 6 ) + ρ m m Δ a m ( 1 ) Δ b m ( 2 ) Δ c m ( 3 ) Δ d m ( 4 ) Δ e m ( 5 ) Δ f m ( 6 ) Δ n m ( 7 ) ] .
ρ b b 1 Δ m b ( 2 ) Δ a b ( 1 ) Δ c b ( 1 ) Δ d b ( 2 ) Δ e b ( 3 ) Δ f b ( 4 ) Δ n b ( 5 ) .
σ m n ( 11 ω ) = μ m a μ a b μ b c μ c d μ h i μ i j μ j n ( E ) 11 × [ ρ n n Δ m n ( 11 ) Δ a n ( 10 ) Δ b n ( 9 ) Δ e n ( 6 ) Δ h n ( 3 ) Δ i n ( 2 ) Δ j n ( 1 ) + ρ j j Δ m j ( 10 ) Δ a j ( 9 ) Δ b j ( 8 ) Δ e j ( 5 ) Δ h j ( 2 ) Δ i j ( 1 ) Δ n j ( 1 ) + + ρ e e Δ m e ( 5 ) Δ a e ( 4 ) Δ b e ( 3 ) Δ f e ( 1 ) Δ i e ( 4 ) Δ j e ( 5 ) Δ n e ( 6 ) + + ρ a a Δ m a ( 1 ) Δ b a ( 1 ) Δ c a ( 2 ) Δ e a ( 4 ) Δ i a ( 8 ) Δ j a ( 9 ) Δ n a ( 10 ) + ρ m m Δ a m ( 1 ) Δ b m ( 2 ) Δ c m ( 3 ) Δ e m ( 5 ) Δ i m ( 9 ) Δ j m ( 10 ) Δ n m ( 11 ) ] .
σ m n ( 11 ω ) = μ m a μ a b μ b c μ h i μ i j μ j n ( E ) 11 q = 0 11 [ ρ q q p = 0 p q 11 1 Δ p q ( q p ) ] ,

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