Abstract

We present a general formulation for quantum macroscopic nonlinear optics, which can be considered as a fully quantized version of its semiclassical counterpart in which the electric field is treated as the classical variable. Our formulation begins with the fundamental minimal-coupling Lagrangian, which is then transformed into the multipolar Lagrangian. After a quantum preservative expansion by using a Hamiltonian decomposition proposed in this paper, we have found the formal relations between the Heisenberg operators of macroscopic polarization density and macroscopic electric field. Finally, the linearized quantum effects for nonlinear optics are also discussed.

© 2009 Optical Society of America

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  1. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467-491 (1991).
    [Crossref] [PubMed]
  2. B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306-4322 (1992).
    [Crossref] [PubMed]
  3. R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
    [Crossref] [PubMed]
  4. H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
    [Crossref]
  5. L. Knöll, S. Scheel, and D.-G. Welsch, “QED in dispersing and absorbing media,” in Coherence and Statistics of Photons and Atoms, J.Peřina, ed. (Wiley, 2001).
  6. L. G. Suttorp and M. Wubs, “Field quantization in inhomogeneous absorptive dielectrics,” Phys. Rev. A 70, 013816 (2004).
    [Crossref]
  7. M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860-1865 (1984).
    [Crossref]
  8. P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845-6857 (1990).
    [Crossref] [PubMed]
  9. L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925-930 (1997).
    [Crossref]
  10. E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
    [Crossref]
  11. S. Scheel and D.-G. Welsch, “Quantum theory of light and noise polarization in nonlinear optics,” Phys. Rev. Lett. 96, 073601 (2006).
    [Crossref] [PubMed]
  12. E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. London, Ser. A 372, 265-273 (1980).
    [Crossref]
  13. E. A. Power and T. Thirunamachandran, “Further remarks on the Hamiltonian of quantum optics,” J. Opt. Soc. Am. B 2, 1100-1105 (1985).
    [Crossref]
  14. W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
    [Crossref]
  15. C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801-853 (1996).
    [Crossref]
  16. R. W. Boyd, Nolinear Optics, 2nd ed. (Academic, New York, 2003).
  17. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84-87 (1970).
    [Crossref]
  18. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  19. R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255-284 (1966).
    [Crossref]

2006 (1)

S. Scheel and D.-G. Welsch, “Quantum theory of light and noise polarization in nonlinear optics,” Phys. Rev. Lett. 96, 073601 (2006).
[Crossref] [PubMed]

2004 (1)

L. G. Suttorp and M. Wubs, “Field quantization in inhomogeneous absorptive dielectrics,” Phys. Rev. A 70, 013816 (2004).
[Crossref]

1998 (2)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
[Crossref]

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

1997 (1)

L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925-930 (1997).
[Crossref]

1996 (1)

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801-853 (1996).
[Crossref]

1995 (1)

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

1992 (1)

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306-4322 (1992).
[Crossref] [PubMed]

1991 (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467-491 (1991).
[Crossref] [PubMed]

1990 (1)

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845-6857 (1990).
[Crossref] [PubMed]

1985 (1)

1984 (1)

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860-1865 (1984).
[Crossref]

1980 (1)

E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. London, Ser. A 372, 265-273 (1980).
[Crossref]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84-87 (1970).
[Crossref]

1966 (1)

R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255-284 (1966).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

Barnett, S. M.

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306-4322 (1992).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nolinear Optics, 2nd ed. (Academic, New York, 2003).

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84-87 (1970).
[Crossref]

Drummond, P. D.

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845-6857 (1990).
[Crossref] [PubMed]

Duan, L. -M.

L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925-930 (1997).
[Crossref]

Dung, H. T.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
[Crossref]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467-491 (1991).
[Crossref] [PubMed]

Guo, G. -C.

L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925-930 (1997).
[Crossref]

Henry, C. H.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801-853 (1996).
[Crossref]

Hillery, M.

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860-1865 (1984).
[Crossref]

Huttner, B.

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306-4322 (1992).
[Crossref] [PubMed]

Jeffers, J.

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

Kazarinov, R. F.

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801-853 (1996).
[Crossref]

Knöll, L.

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
[Crossref]

L. Knöll, S. Scheel, and D.-G. Welsch, “QED in dispersing and absorbing media,” in Coherence and Statistics of Photons and Atoms, J.Peřina, ed. (Wiley, 2001).

Kubo, R.

R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255-284 (1966).
[Crossref]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467-491 (1991).
[Crossref] [PubMed]

Loudon, R.

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

Matloob, R.

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

Mlodinow, L. D.

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860-1865 (1984).
[Crossref]

Power, E. A.

E. A. Power and T. Thirunamachandran, “Further remarks on the Hamiltonian of quantum optics,” J. Opt. Soc. Am. B 2, 1100-1105 (1985).
[Crossref]

E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. London, Ser. A 372, 265-273 (1980).
[Crossref]

Scheel, S.

S. Scheel and D.-G. Welsch, “Quantum theory of light and noise polarization in nonlinear optics,” Phys. Rev. Lett. 96, 073601 (2006).
[Crossref] [PubMed]

L. Knöll, S. Scheel, and D.-G. Welsch, “QED in dispersing and absorbing media,” in Coherence and Statistics of Photons and Atoms, J.Peřina, ed. (Wiley, 2001).

Schmidt, E.

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

Suttorp, L. G.

L. G. Suttorp and M. Wubs, “Field quantization in inhomogeneous absorptive dielectrics,” Phys. Rev. A 70, 013816 (2004).
[Crossref]

Thirunamachandran, T.

E. A. Power and T. Thirunamachandran, “Further remarks on the Hamiltonian of quantum optics,” J. Opt. Soc. Am. B 2, 1100-1105 (1985).
[Crossref]

E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. London, Ser. A 372, 265-273 (1980).
[Crossref]

Vogel, W.

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

Weinberg, D. L.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84-87 (1970).
[Crossref]

Welsch, D. -G.

S. Scheel and D.-G. Welsch, “Quantum theory of light and noise polarization in nonlinear optics,” Phys. Rev. Lett. 96, 073601 (2006).
[Crossref] [PubMed]

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
[Crossref]

L. Knöll, S. Scheel, and D.-G. Welsch, “QED in dispersing and absorbing media,” in Coherence and Statistics of Photons and Atoms, J.Peřina, ed. (Wiley, 2001).

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

Wubs, M.

L. G. Suttorp and M. Wubs, “Field quantization in inhomogeneous absorptive dielectrics,” Phys. Rev. A 70, 013816 (2004).
[Crossref]

J. Mod. Opt. (1)

E. Schmidt, J. Jeffers, S. M. Barnett, L. Knöll, and D.-G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377-402 (1998).
[Crossref]

J. Opt. Soc. Am. B (1)

Phys. Rev. A (8)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467-491 (1991).
[Crossref] [PubMed]

B. Huttner and S. M. Barnett, “Quantization of electromagnetic field in dielectrics,” Phys. Rev. A 46, 4306-4322 (1992).
[Crossref] [PubMed]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, “Electromagnetic field quantization in absorbing dielectrics,” Phys. Rev. A 52, 4823-4838 (1995).
[Crossref] [PubMed]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics,” Phys. Rev. A 57, 3931-3942 (1998).
[Crossref]

L. G. Suttorp and M. Wubs, “Field quantization in inhomogeneous absorptive dielectrics,” Phys. Rev. A 70, 013816 (2004).
[Crossref]

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860-1865 (1984).
[Crossref]

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845-6857 (1990).
[Crossref] [PubMed]

L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925-930 (1997).
[Crossref]

Phys. Rev. Lett. (2)

S. Scheel and D.-G. Welsch, “Quantum theory of light and noise polarization in nonlinear optics,” Phys. Rev. Lett. 96, 073601 (2006).
[Crossref] [PubMed]

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84-87 (1970).
[Crossref]

Proc. R. Soc. London, Ser. A (1)

E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. London, Ser. A 372, 265-273 (1980).
[Crossref]

Rep. Prog. Phys. (1)

R. Kubo, “The fluctuation-dissipation theorem,” Rep. Prog. Phys. 29, 255-284 (1966).
[Crossref]

Rev. Mod. Phys. (1)

C. H. Henry and R. F. Kazarinov, “Quantum noise in photonics,” Rev. Mod. Phys. 68, 801-853 (1996).
[Crossref]

Other (4)

R. W. Boyd, Nolinear Optics, 2nd ed. (Academic, New York, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

W. Vogel and D.-G. Welsch, Quantum Optics (Wiley-VCH, 2006).
[Crossref]

L. Knöll, S. Scheel, and D.-G. Welsch, “QED in dispersing and absorbing media,” in Coherence and Statistics of Photons and Atoms, J.Peřina, ed. (Wiley, 2001).

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Equations (87)

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L 0 = 1 2 ( ε 0 | A ̇ + φ | 2 μ 0 1 | × A | 2 ) d 3 r + 1 2 s , α m s α x ̇ s α 2 + ( A J ρ φ ) d 3 r ,
J = s , α e s α x ̇ s α δ ( r x s α ) ,     ρ = s , α e s α δ ( r x s α ) ,
B = × A ,     E = φ A ̇ .
B = 0 ,     × E + B t = 0.
E = ρ / ε 0 ,     × B ε 0 μ 0 E t = μ 0 J
R s = M s 1 α m s α x s α ,     M s = α m s α ,     Q s = α e s α .
P s ( r ) = α e s α ( x s α R s ) 0 1 δ ( r R s μ ( x s α R s ) ) d μ + Q s R s 0 1 δ ( r μ R s ) d μ ,
M s = α x ̇ s α × N s α
N s α ( r ) = e s α ( x s α R s ) 0 1 m s α + ( M s m s α ) μ M s δ [ ( r R s ) μ ( x s α R s ) ] d μ Q s N s R s 0 1 μ δ ( r μ R m ) d μ ,
P ( r ) = s P s ( r ) ,     M ( r ) = s M s ( r ) .
P ( r ) = ρ ( r ) ,     P ̇ ( r ) + × M ( r ) = J ( r ) ,
D = 0 ,     × H D t = 0 ,
B = μ 0 ( H + M ) ,     D = ε 0 E + P .
L R = ( ε 0 2 | A ̇ | 2 1 2 μ 0 | × A | 2 + M × A A ̇ P ) d 3 r + 1 2 s , α m s α x ̇ s α 2 V Coulomb ,
V Coulomb = 1 2 ε 0 | P | 2 d 3 r
Π = δ L R δ A ̇ = ε 0 A ̇ P = ε 0 E P = D = D ,
p s α = L R x ̇ s α = m s α x ̇ s α + N s α × B d 3 r .
H = Π A ̇ d 3 r + s , α p s α x ̇ s α L R = 1 2 ( ε 0 1 | Π | 2 + μ 0 1 | B | 2 ) d 3 r + s , α p s α 2 2 m s α + | P | 2 2 ε 0 d 3 r + ( Π P ε 0 + B s , α p s α m s α × N s α ) d 3 r + s , α 1 2 m s α | N s α × B d 3 r | 2 ,
[ x ̂ r α , j , p ̂ s β , k ] = i δ r α , s β δ j k ,     [ A ̂ j ( r ) , Π ̂ k ( r ) ] = i δ j k ( r r ) ,
δ j k ( r r ) = ( j r k r δ j k Δ r ) 1 4 π | r r | ,
t A ̂ = ε 0 1 ( Π ̂ + P ̂ ) ,
t Π ̂ = μ 0 1 × × A ̂ .
× × E ̂ + ε 0 μ 0 t 2 E ̂ = μ 0 t 2 P ̂ .
P ̂ I ( t ) = exp ( i H ̂ 0 t ) P ̂   exp ( i H ̂ 0 t ) ,
P ̂ ( t ) = U ̂ ( t 0 , t ) P ̂ I ( t ) U ̂ ( t , t 0 )
U ̂ ( t , t 0 ) = T ( exp [ i t 0 t d s H ̂ int I ( s ) ] ) = 1 + 1 i t 0 t d s 1 H ̂ i n t I ( s 1 ) + ( 1 i ) 2 t 0 t d s 1 t 0 s 1 d s 2 H ̂ int I ( s 1 ) H ̂ int I ( s 2 ) + ( 1 i ) 3 t 0 t d s 1 t 0 s 1 d s 2 t 0 s 2 d s 3 H ̂ int I ( s 1 ) H ̂ int I ( s 2 ) H ̂ int I ( s 3 ) + ,
H ̂ int I ( t ) = exp ( i H ̂ 0 t ) H ̂ int   exp ( i H ̂ 0 t ) .
Π ̂ ( r ) P ̂ ( r ) d 3 r = Π ̱ ̂ ( k ) P ̱ ̂ ( k ) d 3 k .
d 3 r Π ̂ ( r ) P ̂ ( r ) = mac d 3 r Π ̂ ( r ) P ̂ ( r ) + mic d 3 r Π ̂ ( r ) P ̂ ( r ) ,
mac d 3 r Π ̂ ( r ) P ̂ ( r ) = | k | < 2 π Λ c d 3 k Π ̱ ̂ ( k ) P ̱ ̂ ( k )
mic d 3 r Π ̂ ( r ) P ̂ ( r ) = | k | 2 π Λ c d 3 k Π ̱ ̂ ( k ) P ̱ ̂ ( k ) ,
H ̂ = H ̂ 0 + H ̂ int ,
H ̂ mic = mic ε 0 1 | Π ̂ | 2 + μ 0 1 | B ̂ | 2 2 d 3 r + s , α p ̂ s α 2 2 m s α + mic | P ̂ | 2 2 ε 0 d 3 r + mic ( Π ̂ P ̂ ε 0 + B ̂ s , α p ̂ s α m s α × N ̂ s α ) d 3 r + s , α 1 2 m s α | mic N ̂ s α × B ̂ d 3 r | 2
H ̂ em mac = mac ε 0 1 | Π ̂ | 2 + μ 0 1 | B ̂ | 2 2 d 3 r
H ̂ int = 1 ε 0 mac Π ̂ P ̂ d 3 r + 1 2 ε 0 mac | P ̂ | 2 d 3 r
[ A ̱ ̂ m ( k ) , Π ̂ ̱ n ( k ) ] = i ( δ m n k m k n k 2 ) δ ( k + k ) ,
[ H ̂ em mac , H ̂ m i c ] = 0.
U ̂ ( t 0 , t ) O ̂ I ( t ) U ̂ ( t , t 0 ) = O ̂ I ( t ) + t 0 t d t 1 t 1 [ U ̂ ( t 0 , t 1 ) O ̂ I ( t ) U ̂ ( t 1 , t 0 ) ] = O ̂ I ( t ) + t 0 t d t 1 [ t 1 U ̂ ( t 0 , t 1 ) O ̂ I ( t ) U ̂ ( t 1 , t 0 ) + U ̂ ( t 0 , t 1 ) O ̂ I ( t ) t 1 U ̂ ( t 1 , t 0 ) ] = O ̂ I ( t ) + i t 0 t d t 1 U ̂ ( t 0 , t 1 [ H ̂ int I ( t 1 ) , O ̂ I ( t ) ] ) U ̂ ( t 1 , t 0 ) ,
U ̂ t ( t , t 0 ) = i H ̂ int I ( t ) U ̂ ( t , t 0 ) ,
U ̂ t ( t 0 , t ) = i U ̂ ( t 0 , t ) H ̂ int I ( t ) .
H ̂ int I ( t ) = 1 ε 0 mac Π ̂ I ( r , t ) P ̂ I ( r , t ) d 3 r + 1 2 ε 0 mac | P ̂ I ( r , t ) | 2 d 3 r .
H ̂ int I ( t ) H ̂ int I ( t ) exp ( ϵ | t | ) ,   with   ϵ 0 + .
P ̂ μ ( r , t ) = P ̂ μ I ( r , t ) + i t d t 1 U ̂ ( , t 1 ) [ H ̂ int I ( t 1 ) , P ̂ μ I ( r , t ) ] U ̂ ( t 1 , ) = P ̂ μ I ( r , t ) + 1 i α 1 d 3 r 1 t d t 1 1 2 E ̂ α 1 ( r 1 , t 1 ) U ̂ ( , t 1 ) [ P ̂ α 1 I ( r 1 , t 1 ) , P ̂ μ I ( r , t ) ] U ̂ ( t 1 , ) + 1 i α 1 d 3 r 1 t d t 1 1 2 U ̂ ( , t 1 ) [ P ̂ α 1 I ( r 1 , t 1 ) , P ̂ μ I ( r , t ) ] U ̂ ( t 1 , ) E ̂ α 1 ( r 1 , t 1 ) ,
U ̂ ( , t 1 ) E ̂ α 1 I ( r 1 , t 1 ) = E ̂ α 1 ( r 1 , t 1 ) U ̂ ( , t 1 ) ,
E ̂ α 1 I ( r 1 , t 1 ) U ̂ ( t 1 , ) = U ̂ ( t 1 , ) E ̂ α 1 ( r 1 , t 1 ) .
C ̂ + O ̂ = 1 2 ( C ̂ O ̂ + O ̂ C ̂ ) .
P ̂ μ ( r , t ) = P ̂ μ I ( r , t ) + 1 i α 1 d 3 r 1 t d t 1 E ̂ α 1 , + ( r 1 , t 1 ) U ̂ ( , t 1 ) [ P ̂ α 1 I ( r 1 , t 1 ) , P ̂ μ I ( r , t ) ] U ̂ ( t 1 , ) .
P ̂ μ ( r , t ) = P ̂ μ ( 0 ) ( r , t ) + P ̂ μ ( 1 ) ( r , t ) + P ̂ μ ( 2 ) ( r , t ) + P ̂ μ ( 3 ) ( r , t ) + ,
P ̂ μ ( n ) ( r , t ) = d 3 r 1 d 3 r n + d τ 1 + d τ n E ̂ α 1 , + ( r 1 , τ 1 ) E ̂ α 2 , + ( r 2 , τ 2 ) E ̂ α n , + ( r n , τ n ) G ̂ α n α n 1 α 1 μ ( n ) ( r n , r n 1 , , r 1 ; τ n , τ n 1 , , τ 1 ; r , t ) m ,
G ̂ α n α n 1 α 1 μ ( n ) ( r n , r n 1 , , r 1 ; τ n , τ n 1 , , τ 1 ; r , t ) = ( 1 i ) n θ ( t τ 1 ) θ ( τ 1 τ 2 ) θ ( τ n 1 τ n ) [ P ̂ α n I ( r n , τ n ) , [ , [ P ̂ α 2 I ( r 2 , τ 2 ) , [ P ̂ α 1 I ( r 1 , τ 1 ) , P ̂ μ I ( r , t ) ] ] ] ]
θ ( t ) = { 1 , t 0 0 , t < 0. }
P ̂ μ ( n ) ( r , t ) = ε 0 ( 2 π ) n d 3 r 1 d 3 r n + d ω 1 + d ω n E ̱ ̂ α 1 , + ( r 1 , ω 1 ) E ̱ ̂ α 2 , + ( r 2 , ω 2 ) E ̱ ̂ α n , + ( r n , ω n ) χ ̂ α n α n 1 α 1 μ ( n ) ( r n , r n 1 , , r 1 ; ω n , ω n 1 , , ω 1 ; r , t ) exp ( i l = 1 n ω l t ) ,
P ̱ ̂ μ ( n ) ( r , t ) = ε 0 ( 2 π ) n d 4 n σ E ̱ ̂ α 1 , + E ̱ ̂ α 2 , + E ̱ ̂ α n , + χ ̂ α n α n 1 α 1 μ ( n ) ( r , t ) exp ( i l = 1 n 1 ω l t ) ,
χ ̂ α n α 1 μ ( n ) ( r n , , r 1 ; ω n , , ω 1 ; r , t ) = ε 0 1 + d τ 1 + d τ n G ̂ α n α 1 μ ( n ) ( r n , , r 1 ; τ n , , τ 1 ; r , t ) exp ( i l = 1 n ω l ( t τ l ) ) .
E ̂ ( r , t ) = 1 2 π 0 + E ̱ ̂ ( r , ω ) exp ( i ω t ) d ω + H .c .
H ̂ int I ( t ) = d 3 r E α ( r , t ) P ̂ α I ( r , t ) ,
P ̂ μ ( n ) ( r , t ) = d 4 n σ E α 1 E α 2 E α n G ̂ α n α n 1 α 1 μ ( n ) ( r , t ) ,
ρ ̂ th = exp ( H ̂ mic / K B T ) tr [ exp ( H ̂ mic / K B T ) ] ,
G ̂ α n α 1 μ ( n ) ( τ n , , τ 1 ; r , t ) = tr [ ρ ̂ th G ̂ α n α 1 μ ( n ) ( τ n , , τ 1 ; r , t ) ] = tr [ ρ ̂ th G ̂ α n α 1 μ ( n ) ( τ n t , , τ 1 t ; r , 0 ) ] = G α n α 1 μ ( n ) ( τ n t , , τ 1 t ; r )
χ ̂ α n α 1 μ ( n ) ( r , t ) = χ α n α 1 μ ( n ) ( r ) .
χ ̂ α n α 1 μ ( n ) ( r , t ) = χ α n α 1 μ ( n ) ( r ) + Δ χ ̂ α n α 1 μ ( n ) ( r , t ) ,
P ̂ μ ( n ) ( r , t ) = P ̂ μ ( n ) ( r , t ) + Δ P ̂ μ ( n ) ( r , t ) ,
P ̂ μ ( n ) ( r , t ) = ε 0 ( 2 π ) n d 4 n σ χ α n α n 1 α 1 μ ( n ) ( r ) E ̱ ̂ α 1 , + E ̱ ̂ α n 1 , + E ̱ ̂ α n   exp ( i l = 1 n ω l t ) ,
Δ P ̂ μ ( n ) ( r , t ) = ε 0 ( 2 π ) n d 4 n σ E ̱ ̂ α 1 , + E ̱ ̂ α n 1 , + E ̱ ̂ α n Δ χ ̂ α n α n 1 α 1 μ ( n ) ( r , t ) exp ( i l = 1 n ω l t ) ,
Δ P ̂ μ ( n ) ( r , t ) = n d 4 n σ Δ E ̂ α 1 E α 2 E α n 1 E α n G α n α n 1 α 1 μ ( n ) ( r )
Δ P ̂ μ ( n ) ( r , t ) = d 4 n σ E α 1 E α n 1 E α n Δ G ̂ α n α n 1 α 1 μ ( n ) ( r , t ) ,
G α n α p α q α 1 μ ( n ) ( r n , , r p , , r q , , r 1 ; t n , , t p , , t q , , t 1 ; r ) = G α n α q α p α 1 μ ( n ) ( r n , , r q , , r p , , r 1 ; t n , , t q , , t p , , t 1 ; r ) .
Δ P ̂ μ ( r , t ) = n Δ P ̂ μ ( n ) ( r , t ) = ε 0 d 3 r 1 + d t 1 Δ E ̂ α 1 ( r 1 , t 1 ) R α 1 μ ( r 1 , t 1 , r , t ) ,
R α 1 μ ( r 1 , t 1 , r , t ) = n R α 1 μ ( n ) ( r 1 , t 1 , r , t ) = θ ( t t 1 ) n n ε 0 d 4 ( n 1 ) σ G α n α 2 α 1 μ ( n ) E α 2 E α n
R ͇ α 1 μ ( r 1 , ω 1 , r , ω ) = 2 π δ ( r r 1 ) χ α 1 μ ( r , ω 1 , ω ) ,
χ α β ( r , ω 1 , ω ) = χ α β ( 1 ) ( ω ) δ ( ω ω 1 ) + 1 π E ̱ α 2 ( ω ω 1 ) χ α 2 α β ( 2 ) ( ω ω 1 , ω 1 ) + 3 ( 2 π ) 2 + d ω 2 E ̱ α 2 ( ω ω 2 ) E ̱ α 3 ( ω 2 ω 1 ) χ α 3 α 2 α β ( 3 ) ( ω 2 ω 1 , ω ω 2 , ω 1 ) +
Δ P ̱ ̂ μ ( r , ω ) = ε 0 + d ω 1 Δ E ̱ ̂ α 1 ( r , ω 1 ) χ α 1 μ ( r , ω 1 , ω ) .
× × Δ E ̱ ̂ ( r , ω ) ω 2 c 2 + d ω 1 ε ( r , ω , ω 1 ) Δ E ̱ ̂ ( r , ω 1 ) = ω 2 μ 0 Δ P ̱ ̂ ( r , ω ) ,
Δ E ̱ ̂ ( r , ω ) = μ 0 ω ω 1 G ( r , ω , r 1 , ω 1 ) Δ P ̱ ̂ ( r 1 , ω 1 ) d 3 r 1 d ω 1 ,
× × G ( r , ω , r 1 , ω 1 ) d ω 2 ω c ε ( r , ω , ω 2 ) ω 2 c G ( r , ω 2 , r 1 , ω 1 ) = I δ ( r r 1 ) δ ( ω ω 1 ) .
G ( r , ω , r 1 , ω 1 ) = G T ( r 1 , ω 1 , r , ω ) ,
G ( r , ω , r 1 , ω 1 ) = G ( r 1 , ω 1 , r , ω ) ,
Im   G ( r 2 , ω 2 , r 1 , ω 1 ) = 1 2 i [ G ( r 2 , ω 2 , r 1 , ω 1 ) G ( r 1 , ω 1 , r 2 , ω 2 ) ] = G ( r , ω , r 2 , ω 2 ) ω c ε A ( r , ω , ω 3 ) ω 3 c G ( r , ω 3 , r 1 , ω 1 ) d 3 r d ω d ω 3 ,
R α 1 μ ( n ) ( r 1 , t 1 , r , t ) = θ ( t t 1 ) i ε 0 [ Δ P ̂ α 1 ( r 1 , t 1 ) , Δ P ̂ μ ( r , t ) ] ( n 1 ) = θ ( t t 1 ) i ε 0 m = 0 n 1 [ Δ P ̂ α 1 ( m ) ( r 1 , t 1 ) , Δ P ̂ μ ( n m 1 ) ( r , t ) ] .
Δ P ̱ ̂ α ( r , ω ) , Δ P ̱ ̂ β ( r 1 , ω 1 ) = 4 π ε 0 ε A , α β ( r , ω , ω 1 ) δ ( r r 1 ) ,
[ Δ E ̱ ̂ α ( r 2 , ω 2 ) , Δ E ̱ ̂ β ( r 1 , ω 1 ) ] = ω 1 ω 2 c 2 ε 0 4 π   Im   G α β ( r 2 , ω 2 , r 1 , ω 1 ) .
U ̂ ( t , t 0 ; λ ) = T ( exp i λ t 0 t d t 1 H ̂ int I ( t 1 ) ) ,
P ̂ μ ( r , t ; λ ) = U ̂ ( , t ; λ ) P ̂ μ I ( r , t ) U ̂ ( t , ; λ ) .
P ̂ μ ( n ) ( r , t ) = | 1 n ! n P ̂ μ ( r , t ; λ ) λ n | λ = 0 .
U ̂ ( t , ; λ ) λ = U ̂ ( t , ; λ ) i t d t 1 d 3 r 1 E α 1 ( r 1 , t 1 ) P ̂ α 1 ( r 1 , t 1 ; λ ) .
P ̂ μ ( n ) ( r , t ) = 1 n ! i t d t 1 d 3 r 1 E α 1 ( r 1 , t 1 ) | n 1 [ P ̂ α 1 ( r 1 , t 1 ; λ ) , P ̂ μ ( r , t ; λ ) ] λ n 1 | λ = 0 = 1 i n t d t 1 d 3 r 1 E α 1 ( r 1 , t 1 ) [ P ̂ α 1 ( r 1 , t 1 ) , P ̂ μ ( r , t ) ] ( n 1 ) = 1 i n t d t 1 d 3 r 1 E α 1 ( r 1 , t 1 ) m = 0 n 1 [ P ̂ α 1 ( m ) ( r 1 , t 1 ) , P ̂ μ ( n 1 m ) ( r , t ) ] .
d 4 ( n 1 ) σ E α 2 E α n G ̂ α n α n 1 α 1 μ ( n ) ( r , t ) = θ ( t t 1 ) i n [ Δ P ̂ α 1 ( r 1 , t 1 ) , Δ P ̂ μ ( r , t ) ] ( n 1 ) .

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