Abstract

A system of coupled quantum harmonic oscillators (QHOs) whose Hamiltonian conserves photon number begets a one-photon correspondence principle (OPCoP), which allows solutions to the classical linear Maxwell equations for propagation in matter to be reinterpreted as a precise description of a one-photon state. We state, prove, and explore the OPCoP, whereby the vast extant body of linear, sourceless optical waveguide theory [Optical Waveguide Theory (Chapman & Hall, 1983)] can be applied to the exact analysis of one-photon propagation in optical fibers.

© 2007 Optical Society of America

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References

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  1. D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).
  2. I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).
  3. I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.
  4. I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
  5. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  6. P. M. Delaney, M. R. Harris, and R. G. King, "Fiber-optic laser scanning confocal microscope suitable for fluorescence imaging," Appl. Opt. 33, 573-577 (1994).
    [CrossRef] [PubMed]
  7. C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, Bangalore, India, 1984), pp. 175-179.
  8. R. W. C. Vance and F. Ladouceur, "One-photon electrodynamics in optical fiber with fluorophore systems. II. One-polariton propagation in fluorophore-driven fibers from the one-photon correspondence principle," J. Opt. Soc. Am. B 24, 942-958 (2007).
  9. L. Schiff, Quantum Mechanics (McGraw-Hill, 1968).
  10. R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).
  11. Wikipedia entry, "Quantum harmonic oscillator," http://en.wikipedia.org/wiki/Quantumlowbarharmoniclowbaroscillator.
  12. Wolfram Scienceworld entry, "Simple harmonic oscillator--quantum mechanical," http://scienceworld.wolfram.com/physics/SimpleHarmonicOscillatorQuantumMechanical.html.
  13. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford U. Press, 1982), pp. 136-140.
  14. J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
    [CrossRef]
  15. B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
    [CrossRef] [PubMed]
  16. G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins U. Press, 1996).
  17. E. Schrödinger, "The constant crossover of micro- to macro- mechanics," Naturwiss. 40, 664-666 (1926).
  18. W. Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford U. Press, 2003).
  19. R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
    [CrossRef]
  20. M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).
  21. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press, 2005).

2007

1996

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).

1994

1992

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

1963

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[CrossRef]

1958

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

1926

E. Schrödinger, "The constant crossover of micro- to macro- mechanics," Naturwiss. 40, 664-666 (1926).

Barnett, S. M.

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

Bennett, C. H.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, Bangalore, India, 1984), pp. 175-179.

Bialynicki-Birula, I.

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

Brassard, G.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, Bangalore, India, 1984), pp. 175-179.

Delaney, P. M.

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford U. Press, 1982), pp. 136-140.

Gerry, C.

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press, 2005).

Glauber, R. J.

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins U. Press, 1996).

Harris, M. R.

Hopfield, J. J.

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

Huttner, B.

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

King, R. G.

Knight, P. L.

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press, 2005).

Ladouceur, F.

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Marcuse, D.

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).

Rossmann, W.

W. Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford U. Press, 2003).

Schiff, L.

L. Schiff, Quantum Mechanics (McGraw-Hill, 1968).

Schrödinger, E.

E. Schrödinger, "The constant crossover of micro- to macro- mechanics," Naturwiss. 40, 664-666 (1926).

Scully, M. O.

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Suhail Zubiary, M.

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

van Loan, C. F.

G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins U. Press, 1996).

Vance, R. W. C.

Acta Phys. Pol. A

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

Appl. Opt.

J. Opt. Soc. Am. B

Naturwiss.

E. Schrödinger, "The constant crossover of micro- to macro- mechanics," Naturwiss. 40, 664-666 (1926).

Phys. Rev.

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

R. J. Glauber, "Coherent and incoherent states of the radiation field," Phys. Rev. 131, 2766-2788 (1963).
[CrossRef]

Phys. Rev. A

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

Prog. Opt.

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).

Other

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, Bangalore, India, 1984), pp. 175-179.

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press, 2005).

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).

G. H. Golub and C. F. van Loan, Matrix Computations (John Hopkins U. Press, 1996).

W. Rossmann, Lie Groups: An Introduction Through Linear Groups (Oxford U. Press, 2003).

L. Schiff, Quantum Mechanics (McGraw-Hill, 1968).

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

Wikipedia entry, "Quantum harmonic oscillator," http://en.wikipedia.org/wiki/Quantumlowbarharmoniclowbaroscillator.

Wolfram Scienceworld entry, "Simple harmonic oscillator--quantum mechanical," http://scienceworld.wolfram.com/physics/SimpleHarmonicOscillatorQuantumMechanical.html.

P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford U. Press, 1982), pp. 136-140.

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Figures (4)

Fig. 1
Fig. 1

Ladder conceptualization of the QHO.

Fig. 2
Fig. 2

Another factorization of the QHO Hamiltonian.

Fig. 3
Fig. 3

Simple occupancy calculation.

Fig. 4
Fig. 4

Two-level atom.

Equations (85)

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x ̂ = 2 m Δ E ( a exp ( i ξ ) + a exp ( i ξ ) ) ,
p ̂ = i m Δ E 2 ( a exp ( i ξ ) a exp ( i ξ ) ) ,
H ̂ = Δ E a a + ( E 0 Δ E ) I ,
[ a , a ] = I .
[ x ̂ , p ̂ ] = i I .
H ̂ = E 0 Δ E + Δ E diag ( 1 2 3 ) ,
a = [ 0 0 0 0 1 0 0 0 0 2 0 0 0 0 3 0 ] ,
a = [ 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 ] .
a j , k = { j exp ( i α j ) , k = j + m 0 , otherwise ,
H ̂ = ω ( a a + 1 2 ) , [ a , a ] = I ,
x ̂ ψ = def ψ x ̂ ψ = α ( ψ ) cos ( ω t + arg ( α ( ψ ) ) ) , α C ,
p ̂ ψ = def ψ p ̂ ψ = m d x ̂ ψ d t .
i d ψ d t = H ̂ ψ
[ ψ ( t ) ] k = ψ k exp ( i ( k + 1 2 ) ω t ) ,
α ( ψ ) = 2 m ω ( 1 ψ 0 ψ 1 * + 2 ψ 1 ψ 2 * + 3 ψ 2 ψ 3 * + ) exp ( i ξ ) .
x ̂ ψ = ψ x ̂ ψ = j = 0 x ̂ j , j ψ j 2 + 2 Re ( j = 0 k = j + 1 ( x ̂ j , k ψ j * ψ k exp ( i ( ω j ω k ) t ) ) ) ,
ω j = ω 0 + j ω j 0 , x ̂ = b + b , p ̂ = i m ω ( b b ) , b j , k = { b j , k = j + j 0 0 , otherwise
W 1 = { { ψ k } k = 0 : ψ k C k = 0 ψ k 2 = 1 } .
f ( x ) 2 d x = 1 .
ψ = { ψ k } k = 0 ψ ( x , t ) = k = 0 ψ k exp ( i ( k + 1 2 ) ω t ) ϕ k ( x ) ,
ψ k = ϕ k ( x ) ψ ( x ) d x ,
ϕ n ( x ) = exp ( i n ξ ) m ω h n ( m ω x ) ,
h n ( y ) = exp ( y 2 2 ) H n ( y ) 2 n n ! π ,
H 0 ( y ) = 1 , H 1 ( y ) = 2 y , H n + 1 ( y ) = 2 y H n ( y ) 2 n H n 1 ( y ) .
x ̂ f ( x ) = x f ( x ) , p ̂ f ( x ) = i d f ( x ) d x ,
a f ( x ) = exp ( i ξ ) 2 ( m ω x ̂ f ( x ) i p ̂ f ( x ) m ω ) = exp ( i ξ ) 1 2 ( m ω x f ( x ) m ω d f ( x ) d x ) ,
a f ( x ) = exp ( i ξ ) 2 ( m ω x ̂ f ( x ) i p ̂ f ( x ) m ω ) = exp ( i ξ ) 1 2 ( m ω x f ( x ) + m ω d f ( x ) d x ) ,
H ̂ f ( x ) = ( p ̂ 2 2 m + 1 2 m ω 2 x ̂ 2 ) f ( x ) = 2 2 m d 2 f ( x ) d x 2 + 1 2 m ω 2 x 2 f ( x ) ,
i ψ ( x , t ) t = 2 2 m 2 ψ ( x , t ) x 2 + 1 2 m ω 2 x 2 ψ ( x , t ) ,
L ( W 1 , W 1 ) = { x : W 1 W 1 ; x ( α 1 ψ 1 ( u ) + α 2 ψ 2 ( u ) ) = α 1 x ( ψ 1 ( u ) ) + α 2 x ( ψ 2 ( u ) ) ; ψ 1 ( u ) , ψ 2 ( u ) W 1 } .
ad x ̂ : L ( W 1 , W 1 ) L ( W 1 , W 1 ) , ad x ̂ p = [ x ̂ , p ] .
p ̂ = i d d x + g ( x ) , p ̂ f ( x ) = i d d x f ( x ) + g ( x ) f ( x ) ,
a ψ 0 ( x ) = m ω ( x + i g ( x ) m ω ) ψ 0 ( x ) + m ω d ψ 0 ( x ) d x = 0 ψ 0 ( x ) = m ω π exp ( m ω x 2 2 ) exp ( i 0 x g ( u ) d u ) ,
ψ n ( x ) a n ψ 0 ( x ) ( m ω ( x i g ( x ) m ω ) m ω d d x ) n ψ 0 ( x ) ψ n ( x ) = m ω h n ( m ω x ) exp ( i 0 x g ( u ) d u ) .
ψ ( x , t ) = m ω n ( α n h n ( m ω x ) exp ( i ( n + 1 2 ) ω t ) exp ( i 0 x g ( u ) d u ) ) = m ω exp ( i ω t 2 ) exp ( i 0 x g ( u ) d u ) n ( α n h n ( m ω x ) exp ( i n ω t ) ) ,
ψ ( x , t ) 2 = m ω n ( α n h n ( m ω x ) exp ( i n ω t ) ) 2 .
F ( θ ) = ψ exp ( i θ ( a exp ( i ξ ) + a exp ( i ξ ) ) ) ψ .
F coh ( θ , α , t ) = exp ( i θ α cos ( ω t + arg ( α ) ) ) 0 exp ( i θ ( a e i ξ + a exp ( i ξ ) ) ) 0 ( 0 ) ,
F coh ( θ , α , t ) = ψ coh ( α , t ) exp ( i θ ( a e i ξ + a exp ( i ξ ) ) ) ψ coh ( α , t ) ,
ψ coh ( α , t ) = D coh ( α exp ( i ( ω t + ξ ) ) ) 0 ,
D coh ( α exp ( i ( ω t + ξ ) ) ) = exp ( 1 2 ( a α * exp ( i ( ω t + ξ ) ) a α exp ( i ( ω t + ξ ) ) ) ) ,
P x ( x ) = ψ ( x ) 2 , P p ( p ) = 1 2 π exp ( i p x ) ψ ( x ) d x 2 .
exp ( i p x ) exp ( i ϕ ( x ) ) ψ ( x ) d x exp ( i ρ ( p ) ) exp ( i p x ) ψ ( x ) d x = exp ( i p x ) ( exp ( i ϕ ( x ) ) exp ( i ρ ( p ) ) ) ψ ( x ) d x = 0 ( exp ( i ϕ ( x ) ) exp ( i ρ ( p ) ) ) ψ ( x ) = 0 almost everywhere ,
D coh ( u ) D coh ( v ) = D coh ( v ) D coh ( u ) = D coh ( u + v ) , u , v C ,
D coh ( u ) 0 = exp ( 1 2 ( a u * a u ) ) 0 = exp ( u 2 8 ) exp ( a u * 2 ) exp ( a u 2 ) 0 = exp ( u 2 8 ) exp ( a u * 2 ) 0 ,
H ̂ = ω j [ a j a j + 1 2 ] ,
[ a j , a k ] = δ j k I ,
H ̂ = diag [ ω 1 , ω 2 , ω 3 , ] + 2 ω j I ,
H ̂ = K 0 I + K j , k , m , n a j m a k n ,
H ̂ = K 0 I + K j , k a j a k ,
H ̂ = [ K 11 K 12 K 13 K 12 * K 22 K 23 K 13 * K 23 * K 33 ] + K 0 I ,
K = Φ Ω Φ = Φ Ω Φ 1 , Ω = diag [ ω 1 , ω 2 , ω 3 , ] .
a ̃ j = k ( Φ 1 ) j k a k = k Φ k j * a k a j = k Φ j k a ̃ k ,
a ̃ j = k Φ k j a k a j = k Φ j k * a ̃ k ,
[ a ̃ j , a ̃ k ] = ( m Φ m , j * a m ) n Φ n , k * a n n Φ n , k * a n ( m Φ m , j * a m ) = m n ( Φ m , j Φ n , k * ( a m a n a n a m ) ) = m n ( Φ m , j Φ n , k * [ a m , a n ] ) = m n ( Φ m , j Φ n , k * δ m , n ) I = m ( Φ m , j Φ m , k * ) I = m ( Φ j , m T Φ m , k * ) I = ( m Φ j , m 1 Φ m , k ) * I = ( Φ Φ 1 ) j , k * I = δ j , k I .
H ̂ = K 0 I + K j k a j a k = K 0 I + j , k ( a j a k ( m Φ j , m ( Ω Φ 1 ) m , k ) ) = K 0 I + m ( ω m a ̃ m a ̃ m ) ,
n 1 , n 2 , n 3 , ( t ) = ( a ̃ 1 ) n 1 exp ( i ω 1 t ) n 1 ! ( a ̃ 2 ) n 2 exp ( i ω 2 t ) n 2 ! ( a ̃ 3 ) n 3 exp ( i ω 3 t ) n 3 ! 0 , 0 , , ( 0 ) ,
n 1 , n 2 , n 3 , ( t ) = ( a 1 ) n 1 exp ( i ω 1 t ) n 1 ! ( a 2 ) n 2 exp ( i ω 2 t ) n 2 ! ( a 3 ) n 3 exp ( i ω 3 t ) n 3 ! 0 , 0 , , ( 0 ) ,
H ̂ n 1 , n 2 , n 3 , ( t ) = ( K 0 + j n j ω j ) n 1 , n 2 , n 3 , ( t ) .
ψ 1 ( t ) ψ 2 ( t ) ψ 3 ( t ) = def ψ 1 , ψ 2 , ψ 3 , = def ( j ( k ψ j , k ( a ̃ j ) k exp ( i k ω j t ) k ! ) ) 0 , 0 , 0 , ( 0 ) ,
ψ j ( t ) = k ψ j , k ( a ̃ j ) k exp ( i k ω j t ) k ! 0 ( 0 ) .
ψ Φ 1 ψ = Φ ψ = ψ ̃ ,
( ( 28 + 2 × 16 ) × 10 3 kg mol 1 2600 kg m 3 × 6.022 × 10 23 entities mol 1 ) 3 = 3.84 × 10 29 m 3 3 = 3.4 × 10 10 m .
( 1 p ) l c + p τ = n l c p τ ( n 1 ) l c ; p 1 .
ζ IA ω p τ = ( n 1 ) IA ω l c = ( n 1 ) k V I ω 2 = ( n 1 ) V I ν 2 λ = ( n 1 ) V I λ c 2 .
ζ ( ψ coh ( α , t ) ) = ζ ( exp ( α 2 8 ) [ 0 ( t ) + α * 2 1 ! 1 ( t ) + α * 2 2 2 2 ! 2 ( t ) + α * 3 2 3 3 ! 3 ( t ) ] ) = Pr ( in state 1 ) + Pr ( in state 2 ) + Pr ( in state 3 ) + = 1 Pr ( in state 0 ) = 1 exp ( α 2 8 ) ,
α = 8 log ( 1 ζ ) 2 2 ζ .
exp ( α 2 8 ) exp ( exp ( i ω t ) α * 2 a ) 0 1 + exp ( i ω t ) α * 2 a 1 + α 2 8 0 ,
1 1 , 0 2 , 0 3 , , 0 N , 0 1 , 1 2 , 0 3 , , 0 N , 0 1 , 0 2 , 1 3 , , 0 N , , 0 1 , 0 2 , 0 3 , , 1 N ,
exp ( α 2 8 ) exp ( α * 2 exp ( i ω t ) ( j = 1 N a ̃ j ) ) 0 = j = 1 N exp ( α 2 8 N ) exp ( α * 2 N exp ( i ω t ) a ̃ j ) 0 j = 1 N exp ( α 2 8 N ) exp ( 1 + α * 2 N exp ( i ω t ) a ̃ j ) 0 .
ψ ̃ coh , j ( α * 2 exp ( i ω j t ) ) = exp ( α 2 8 ) exp ( α * 2 exp ( i ω j t ) a ̃ j ) 0 ,
ψ ̃ coh , j ( α * 2 e i ω j t ) = exp ( α 2 8 ) exp ( α * 2 exp ( i ω j t ) a ̃ j ) 0 = exp ( α 2 8 ) exp ( α * 2 exp ( i ω j t ) k Φ k j a k ) 0 = k exp ( α Φ k j 2 8 ) exp ( α * 2 Φ k j exp ( i ω j t ) a k ) 0 = ψ coh , 1 ( α * 2 Φ 1 j exp ( i ω j t ) ) ψ coh , 2 ( α * 2 Φ 2 j exp ( i ω j t ) ) ψ coh , 3 ( α * 2 Φ 3 j exp ( i ω j t ) ) = ψ coh , 1 ( α * 2 Φ 1 j exp ( i ω j t ) ) , ψ coh , 2 ( α * 2 Φ 2 j exp ( i ω j t ) ) , ψ coh , 3 ( α * 2 Φ 3 j exp ( i ω j t ) ) .
x ̂ k j = Re ( α * Φ k j exp ( i ξ j i ω j t ) ) = α * Φ k j cos ( ω j t ξ j arg ( α * Φ k j ) ) .
ψ ̃ coh , 1 ( α 1 * 2 exp ( i ω 1 t ) ) ψ ̃ coh , 2 ( α 2 * 2 exp ( i ω 2 t ) ) ψ ̃ coh , 3 ( α 3 * 2 exp ( i ω 3 t ) ) = j ( k exp ( α j * Φ k j 2 8 ) exp ( α j * 2 Φ k j exp ( i ω j t ) a k ) ) 0 = k ( j exp ( α j * Φ k j 2 8 ) exp ( α j * 2 Φ k j exp ( i ω j t ) a k ) ) 0 = k ( exp ( j α j * Φ k j 2 8 ) exp ( ( j α j * 2 Φ k j exp ( i ω j t ) ) a k ) ) 0 .
x ̂ k ( t ) = Re ( j α j * Φ k j exp ( i ξ j i ω j t ) ) = j α j * Φ k j cos ( ω j t ξ j arg ( α j * Φ k j ) ) .
j α j * Φ k j exp ( i ξ j i ω j t ) = j ( Φ exp ( i Ω t ) ) k j α j * = j ( ( Φ exp ( i Ξ i Ω t ) ) k j ( m ( Φ 1 ) j m β m ) ) = m ( ( j ( Φ exp ( i Ξ i Ω t ) ) k j ( Φ 1 ) j m ) β m ) = m ( ( ( Φ exp ( i Ξ i Ω ) Φ 1 ) k m ) β m ) = m ( ( exp ( i Φ Ξ Φ 1 i Φ Ω Φ 1 t ) k m ) β m ) = m ( ( exp ( i K K t ) ) k m β m ) = m ( ( exp ( i K t ) exp ( i K ) ) k m β m ) = j ( ( exp ( i K t ) ) k j ( m ( exp ( i K ) ) j m β m ) ) = j ( ( exp ( i K t ) ) k j γ j ) ,
Ξ = diag [ ξ 1 , ξ 2 , ξ 3 , ] , Ω = diag [ ω 1 , ω 2 , ω 3 , ] , K = Φ Ω Φ 1 , K = Φ Ξ Φ 1 ,
α * = Φ 1 β , γ = exp ( i K ) β = exp ( i K ) Φ α * ,
[ K , K ] = 0 ,
x ̂ k ( t ) = Re ( m ( ( exp ( i K K t ) ) k m β m ) ) = Re ( m ( ( exp ( i K t ) ) k m γ m ) ) = 1 2 m ( ( exp ( i K K t ) ) k m β m ) + 1 2 m ( ( exp ( i K * K * t ) ) k m β m * ) = 1 2 m ( ( exp ( i K t ) ) k m γ m ) + 1 2 m ( ( exp ( + i K * t ) ) k m γ m * )
ψ k ( t ) = exp ( i K 0 t ) j α j Φ k j exp ( i ω j t ) = exp ( i K 0 t ) m ( ( exp ( i K t ) ) k m β m ) .
ψ k ( t ) = exp ( i K 0 t ) m ( ( exp ( i K ) exp ( i K t ) ) k m β m ) = exp ( i K 0 t ) m ( ( exp ( i K t ) exp ( i K ) ) k m β m ) = exp ( i K 0 t ) m ( ( exp ( i K K t ) ) k m β m ) = exp ( i K 0 t ) m ( ( exp ( i K t ) ) k m γ m ) ,
ψ ( t ) = exp ( i K t ) ψ ( 0 ) i d ψ d t = K ψ = H ̂ ψ .
m [ B m l 1 ( C l 1 ψ ̃ coh , 1 ( κ 1 , l 1 , m * 2 exp ( i ω 1 t ) ) ) l 2 ( C l 2 ψ ̃ coh , 2 ( κ 2 , l 2 , m * 2 exp ( i ω 2 t ) ) ) ] = m [ A m ψ ̃ coh , 1 ( η 1 , m * 2 exp ( i ω 1 t ) ) ψ ̃ coh , 2 ( η 2 , m * 2 exp ( i ω 1 t ) ) ] ,
α j * = m A m η j , m * .

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