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Quantum noise properties of parametric processes

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic

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Abstract

In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homodyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.

©2005 Optical Society of America

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Figures (11)
Fig. 1.
Fig. 1. Illustration of the constituent two-mode processes in a four-mode parametric interaction driven by two pump waves

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Fig. 2.
Fig. 2. Illustration of the architecture of one stage in a communication link with phase-insensitive amplification. The attenuator ◁ is followed by a parametric amplifier ▷. The signal, idler and scattered modes are labeled 1, 2r and 2r+1, respectively.

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Fig. 3.
Fig. 3. Illustration of the constituent two-mode processes in a phase-sensitive parametric interaction driven by two pump waves.

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Fig. 4.
Fig. 4. Illustration of the architecture of one stage in a communication link with phase-sensitive amplification. The attenuator ◁ is followed by a parametric amplifier▷. The signal and scattered modes are labeled 1 and r+1, respectively.

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Fig. 5.
Fig. 5. Noise figures [Eqs. (51) and (52)] plotted as functions of the transmittance T. The solid line represents the signal, whereas the dashed curve represents the idler.

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Fig. 6.
Fig. 6. Noise figures [Eqs. (45), (46), (53) and (54)] plotted as functions of the relative phase x. The solid line represents the signal, whereas the dashed curve represents the idler.

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Fig. 7.
Fig. 7. Noise figures [Eqs. (61) and (62)] plotted as functions of the gain G. The solid curve represents the signal, whereas the dashed curve represents the idler.

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Fig. 8.
Fig. 8. Homodyne noise-figure [Eq. (86)] plotted as a function of the input phase ξ and the local-oscillator phase η. Dark shadings denote low noise figures, whereas light shadings denote high noise figures. The contour spacing is 4 dB.

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Fig. 9.
Fig. 9. Homodyne noise-figure [Eq. (86)] plotted as a function of (a) the input phase ξ, for the case in which the local-oscillator phase η=0, and (b) η, for the case in which ξ=0.

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Fig. 10.
Fig. 10. Direct noise-figure plotted as a function of the input phase ξ. The solid curve represents the exact noise figure [Eqs. (46) and (87)], whereas the dashed curve represents the approximate noise figure [Eq. (88)].

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Fig. 11.
Fig. 11. Direct noise-figure plotted as a function of the relative phase x. The solid curve represents the exact noise figure [Eqs. (46) and (93)], whereas the dashed curve represents the approximate noise figure [Eq. (94)].

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Equations (163)

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a j ( z ) = μ ̄ ( z ) a j ( 0 ) + ν ̄ ( z ) a k ( 0 ) ,
a k ( z ) = − ν ̄ * ( z ) a j ( 0 ) + μ ̄ * ( z ) a k ( 0 ) .
d jk = ( ∣ μ ̄ ∣ 2 − ∣ ν ̄ ∣ 2 ) ( a j † a j − a k † a k ) + 2 ( μ ̄ ν ̄ * a j a k † + μ ̄ * ν ̄ a j † a k ) .
d j = a j a l † e − i ϕ + a j † a l e i ϕ ,
〈 d j ( ϕ ) 〉 = 2 ∣ α l ∣ 〈 q j ( ϕ ) 〉 ,
〈 δ d j 2 ( ϕ ) 〉 = 4 ∣ α l ∣ 2 〈 δ q j 2 ( ϕ ) 〉 + 1 l 〈 n j 〉 .
a 1 + ( z ) = ν ( z ) a 1 − † ( 0 ) + μ ( z ) a 1 + ( 0 ) .
a 1 + ( z ) = μ ( z ) a 1 + ( 0 ) + ν ( z ) a 2 − † ( 0 ) ,
a 1 + ( z ) = μ ̄ ( z ) a 1 + ( 0 ) + ν ̄ ( z ) a 2 + ( 0 ) .
a 2 ( z ) = v 21 ( z ) a 1 † ( 0 ) + μ 22 ( z ) a 2 ( 0 ) + ν 23 ( z ) a 3 † ( 0 ) + μ 24 ( z ) a 4 ( 0 ) ,
a 1 ( z r ′ ) = μ ̄ ( z r ′ − z ″ r − 1 ) a 1 ( z ″ r − 1 ) + ν ̄ ( z r ′ − z ″ r − 1 ) a 2 r + 1 ( z ″ r − 1 ) ,
a 1 ( z r ″ ) = μ ( z r ″ − z r ′ ) a 1 ( z r ′ ) + ν ( z r ″ − z r ′ ) a 2 r † ( z r ′ ) ,
a 1 ( z r ″ ) = μ 11 ( z r ″ , z ″ r − 1 ) a 1 ( z r − 1 ″ ) + ν 12 r ( z r ″ , z r ) a 2 r † ( z r ′ )
+ μ 12 r + 1 ( z r ″ , z ″ r − 1 ) a 2 r + 1 ( z ″ r − 1 ) ,
a 1 ( z s ) = μ 11 ( z s ) a 1 ( 0 ) + ∑ r = 1 s [ ν 12 r ( z s ) a 2 r † ( 0 ) + μ 12 r + 1 ( z s ) a 2 r + 1 ( 0 ) ] .
a 1 + ( z ) = μ ( z ) a 1 + ( 0 ) + ν ( z ) a 1 + † ( 0 ) ,
a 2 ( z ) = μ 21 ( z ) a 1 ( 0 ) + ν 21 ( z ) a 1 † ( 0 ) + μ 22 ( z ) a 2 ( 0 )
+ ν 22 ( z ) a 2 † ( 0 ) + μ 23 ( z ) a 3 ( 0 ) + ν 23 ( 0 ) a 3 † ( 0 ) ,
a 1 ( z r ′ ) = μ ̄ ( z r ′ − z ″ r − 1 ) a 1 ( z ″ r − 1 ) + ν ̄ ( z r ′ − z ″ r − 1 ) a r + 1 ( z ″ r − 1 ) ,
a 1 ( z r ″ ) = μ ( z r ″ − z r ′ ) a 1 ( z r ′ ) + ν ( z r ″ − z r ′ ) a 1 † ( z r ′ ) ,
a 1 ( z r ″ ) = μ 11 ( z r ″ , z ″ r − 1 ) a 1 ( z ″ r − 1 ) + ν 11 ( z r ″ , z ″ r − 1 ) a 1 † ( z ″ r − 1 )
+ μ 1 r + 1 ( z ″ r , z ″ r − 1 ) a r + 1 ( z ″ r − 1 ) + ν 1 r + 1 ( z r ″ , z ″ r − 1 ) a r + 1 † ( z ″ r − 1 ) ,
a 1 ( z s ) = μ 11 ( z s ) a 1 ( 0 ) + ν 11 ( z s ) a 1 † ( 0 )
+ ∑ r = 1 s [ μ 1 r + 1 ( z s ) a r + 1 ( 0 ) + ν 1 r + 1 ( z s ) a r + 1 † ( 0 ) ] ,
a j ( z ) = ∑ k [ μ j k ( z ) a k ( 0 ) + ν j k ( z ) a k † ( 0 ) ] ,
∑ l [ μ j l ( z ) ν k l ( z ) − μ k l ( z ) ν j l ( z ) ] = 0 ,
∑ l [ μ j l ( z ) μ k l * ( z ) − ν j l ( z ) ν k l * ( z ) ] = δ j k .
∑ l [ ∣ μ j l ( z ) ∣ 2 − ∣ ν j l ( z ) ∣ 2 ] = 1 .
a j ( z ) = α j ( z ) + v j ( z ) ,
α j ( z ) = μ j i ( z ) α i ( 0 ) + v j i ( z ) α i * ( 0 )
v j ( z ) = ∑ k [ μ j k ( z ) a k ( 0 ) + v j k ( z ) a k † ( 0 ) ]
a j 2 = α j 2 + 2 α j v j + v j 2 ,
a j a j † = ∣ α j ∣ 2 + α j v j † + α j * v j + v j v j † ,
a j † a j = ∣ α j ∣ 2 + α j v j † + α j * v j + v j † v j ,
( a j † a j ) 2 = ∣ α j ∣ 4 + [ α j 2 ( v j † ) 2 + ∣ α j ∣ 2 ( v j † v j + v j v j † ) + ( α j * ) 2 v j 2 ]
+ ( v j † v j ) 2 + 2 ∣ α j ∣ 2 ( α j v j † + α j * v j ) + 2 ∣ α j ∣ 2 v j † v j
+ ( α j v j † + α j * v j ) v j † v j + v j † v j ( α j v j † + α j * v j ) .
v j ∣ 0 〉 = ∑ k v j k ∣ 1 k 〉 ,
v j † ∣ 0 〉 = ∑ k μ j k * ∣ 1 k 〉 ,
v j 2 ∣ 0 〉 = ∑ k [ μ j k ν j k ∣ 0 〉 + 2 1 2 ν j k 2 ∣ 2 k 〉 + ∑ l ≠ k ν j k ν j l ∣ 1 k 1 l 〉 ] ,
v j v j † ∣ 0 〉 = ∑ k [ ∣ μ j k ∣ 2 ∣ 0 + 2 1 2 μ j k * ν j k ∣ 2 k 〉 + ∑ l ≠ k μ j k * ν j l ∣ 1 k 1 l 〉 ] ,
v j † v j ∣ 0 〉 = ∑ k [ 2 1 2 μ j k * ν j k ∣ 2 k 〉 + ∣ ν j k ∣ 2 ∣ 0 〉 + ∑ l ≠ k μ j l * ν j k ∣ 1 k 1 l 〉 ] ,
( v j † ) 2 ∣ 0 〉 = ∑ k [ 2 1 2 ( μ j k * ) 2 ∣ 2 k 〉 + μ j k * ν j k * ∣ 0 〉 + ∑ l ≠ k μ j k * μ j l * ∣ 1 k 1 l 〉 ] ,
〈 q j 〉 = ( α j e − i ϕ + α j * e i ϕ ) ⁄ 2 ,
〈 δ q j 2 〉 = [ ( ∑ k μ j k ν j k ) e − i 2 ϕ + ∑ k ( ∣ μ j k ∣ 2 + ∣ ν j k ∣ 2 ) + ( ∑ k μ j k ν j k ) * e i 2 ϕ ] ⁄ 4 ,
〈 δ q j 2 〉 = ∑ k ∣ λ j k ∣ 2 ⁄ 4 .
〈 n j 〉 = ∣ α j ∣ 2 + ∑ k ∣ ν j k ∣ 2 ,
〈 δ n j 2 〉 = α j 2 ∑ k ( μ j k ν j k ) * + ∣ α j ∣ 2 ∑ k ( ∣ μ j k ∣ 2 + ∣ ν j k ∣ 2 ) + ( α j 2 ) * ∑ k μ j k ν j k
+ 2 ∑ k ∣ μ j k ν j k ∣ 2 + ∑ k ∑ l > k ∣ μ j k * ν j l + μ j l * ν j k ∣ 2 ,
〈 δ n j 2 〉 = ∣ α j ∣ 2 ∑ k ∣ λ j k ' ∣ 2 + 2 ∑ k ∣ μ j k ν j k ∣ 2 + ∑ k ∑ l > k ∣ U j k * ν j l + μ j l * ν j k ∣ 2 .
α j ( z ) = ∑ i [ μ j i ( z ) α i ( 0 ) + ν j i ( z ) α i * ( 0 ) ] ,
S i = 4 ∣ α i ∣ 2 ,
S i = ∣ α i ∣ 2 .
S j = 4 ∣ α j ∣ 2 cos ⁡ 2 ( ϕ j − ϕ ) ⁄ ∑ k ∣ λ j k ∣ 2 ,
S j = 4 ∣ α j ∣ 2 ⁄ ∑ k ∣ λ j k ∣ 2 .
S j = ( ∣ α j ∣ 2 + ∑ k ∣ ν j k ∣ 2 ) 2 ∣ α j ∣ 2 ∑ k ∣ λ j k ' ∣ 2 + 2 ∑ k ∣ μ j k ν j k ∣ 2 + ∑ k ∑ l > k ∣ μ j k * ν j l + μ j l * ν j k ∣ 2 ,
S j ≈ ∣ α j ∣ 2 ⁄ ∑ k ∣ λ j k ' ∣ 2 .
F 1 ( z ) = 1 ⁄ T ,
F 2 ( z ) = 1 ⁄ ( 1 − T ) ,
∣ α 1 ( z ) ∣ 2 = T ∣ α 1 ∣ 2 + ( 1 − T ) ∣ α 2 ∣ 2 + 2 [ T ( 1 − T ) ] 1 2 ∣ α 1 α 2 ∣ cos ⁡ ξ ,
∣ α 2 ( z ) ∣ 2 = ( 1 − T ) ∣ α 1 ∣ 2 + T ∣ α 2 ∣ 2 − 2 [ T ( 1 − T ) ] 1 2 ∣ α 1 α 2 ∣ cos ⁡ ξ ,
S 1 ( z ) = 4 G 〈 n 1 〉 ⁄ ( 2 G − 1 ) ,
S 2 ( z ) = 4 ( G − 1 ) 〈 n 1 〉 ⁄ ( 2 G − 1 ) ,
F 1 ( z ) = 1 + ( G − 1 ) ⁄ G ,
F 2 ( z ) = 1 + G ⁄ ( G − 1 ) .
S 1 ( z ) = [ G 〈 n 1 〉 + G − 1 ] 2 ⁄ [ G ( 2 G − 1 ) 〈 n 1 〉 + G ( G − 1 ) ] ,
S 2 ( z ) = [ ( G − 1 ) 〈 n 1 〉 + G − 1 ] 2 ⁄ [ ( G − 1 ) ( 2 G − 1 ) 〈 n 1 〉 + G ( G − 1 ) ] .
F 1 ( z ) ≈ 1 + ( G − 1 ) ⁄ G ,
F 2 ( z ) = 1 + G ⁄ ( G − 1 ) .
S j ( z ) = 4 ∣ κ j i ∣ 2 〈 n i 〉 ⁄ ∑ k ∣ κ j k ∣ 2 ,
F j ( z ) = ∑ k ∣ κ j k ∣ 2 ⁄ ∣ κ j i ∣ 2 ,
S j ( z ) = [ ∣ κ j i ∣ 2 〈 n i 〉 + ∑ k ∣ κ j k ∣ 2 σ j k ] 2 ⁄ [ ∣ κ j i ∣ 2 ∑ k ∣ κ j k ∣ 2 〈 n i 〉 + ∑ k ∑ l > k ∣ κ j k κ j l ∣ 2 σ k l ] .
F j ( z ) ≈ ∑ k ∣ κ j k ∣ 2 ⁄ ∣ κ j i ∣ 2 .
S 1 ( z ) = 4 G T 〈 n 1 〉 ⁄ ( 2 G − 1 ) ,
F 1 ( z ) = ( 2 G − 1 ) ⁄ G T ,
S 1 ( z ) = [ G T 〈 n 1 〉 + G − 1 ] 2 ⁄ [ G ( 2 G − 1 ) T 〈 n 1 〉 + G ( G − 1 ) ] .
F 1 ( z ) ≈ ( 2 G − 1 ) ⁄ G T .
S 1 ( z ) = 4 〈 n 1 〉 ⁄ [ 1 + 2 s ( G − 1 ) ] ,
F 1 ( z ) = 1 + 2 s ( G − 1 ) .
S 1 ( z ) = [ 〈 n 1 〉 + s ( G − 1 ) ] 2 ⁄ { [ 1 + 2 s ( G − 1 ) ] 〈 n 1 〉 + s ( G − 1 ) [ 1 + s ( G − 1 ) ] } .
F 1 = 1 + 2 s ( G − 1 ) .
S 1 ( z ) = 4 T f G T i 〈 n 1 〉 ⁄ [ 1 + 2 T f ( G − 1 ) ] ,
F 1 ( z ) = [ 1 + 2 T f ( G − 1 ) ] ⁄ ( T f G T i ) ,
S 1 ( z ) = [ T f G T i 〈 n 1 〉 + T f ( G − 1 ) ] 2 ⁄ { ( T f G T i ) [ 1 + 2 T f ( G − 1 ) ] 〈 n 1 〉
+ T f ( G − 1 ) [ 1 + T f ( G − 1 ) ] } .
F 1 ( z ) ≈ [ 1 + 2 T f ( G − 1 ) ] ⁄ ( T f G T i ) .
∣ α 1 ( z ) ∣ 2 = ∣ α 1 ( 0 ) ∣ 2 { 2 G − 1 + 2 [ G ( G − 1 ) ] 1 2 cos ⁡ ξ } ,
ϕ 1 ( z ) = ϕ 1 ( 0 ) + ϕ μ + tan ⁡ − 1 [ ( G − 1 ) 1 2 sin ⁡ ξ G 1 2 + ( G − 1 ) 1 2 cos ⁡ ξ ] ,
〈 n 1 ( z ) 〉 = H ( ξ ) 〈 n 1 〉 + G − 1 ,
〈 δ q 1 2 ( z ) 〉 = H ( η ) ⁄ 4 ,
〈 δ n 1 2 ( z ) 〉 = H ( ξ ) H ( ζ ) 〈 n 1 〉 + 2 G ( G − 1 ) ,
ζ = ξ − tan ⁡ − 1 [ ( G − 1 ) 1 2 sin ⁡ ξ G 1 2 + ( G − 1 ) 1 2 cos ⁡ ξ ] .
S 1 ( z ) = 4 〈 n 1 〉 H ( ξ ) cos ⁡ 2 [ ( ζ − η ) ⁄ 2 ] ⁄ H ( η ) ,
F 1 ( z ) = H ( η ) ⁄ { H ( ξ ) cos ⁡ 2 [ ( ζ − η ) ⁄ 2 ] } ,
S 1 ( z ) = [ H ( ξ ) 〈 n 1 〉 + G − 1 ] 2 ⁄ [ H ( ξ ) H ( ζ ) 〈 n 1 〉 + 2 G ( G − 1 ) ] .
F 1 ( z ) ≈ H ( ζ ) ⁄ H ( ξ ) .
∣ α 1 ( z ) ∣ 2 = G ∣ α 1 ∣ 2 + ( G − 1 ) ∣ α 2 ∣ 2 + 2 [ G ( G − 1 ) ] 1 2 ∣ α 1 α 2 ∣ cos ⁡ ξ ,
∣ α 2 ( z ) ∣ 2 = ( G − 1 ) ∣ α 1 ∣ 2 + G ∣ α 2 ∣ 2 + 2 [ G ( G − 1 ) ] 1 2 ∣ α 1 α 2 ∣ cos ⁡ ξ ,
S j ( z ) = 4 ∣ α j ( z ) ∣ 2 ⁄ ( 2 G − 1 ) ,
F j ( 2 ) = ( 2 G − 1 ) ∣ α j ( 0 ) ⁄ α j ( z ) ∣ 2 ,
S j ( z ) = [ [ α j ( z ) ] 2 + G − 1 ] 2 ⁄ [ ( 2 G − 1 ) ∣ α j ( z ) ∣ 2 + G ( G − 1 ) ] .
F j ( z ) ≈ ( 2 G − 1 ) ∣ α j ( 0 ) ⁄ α j ( z ) ∣ 2 .
∣ α 1 ( z ) ∣ 2 = T ∣ α 1 ( 0 ) ∣ 2 { 2 G − 1 + 2 [ G ( G − 1 ) ] 1 2 cos ⁡ ξ } ,
ϕ 1 ( z ) = ϕ 1 ( 0 ) + ϕ μ + ϕ μ ̄ + tan ⁡ − 1 [ ( G − 1 ) 1 2 sin ⁡ ξ G 1 2 + ( G − 1 ) 1 2 cos ⁡ ξ ] ,
〈 n 1 ( z ) 〉 = H ( ξ ) T 〈 n 1 〉 + G − 1 ,
〈 δ q 1 2 ( z ) 〉 = H ( η ) ⁄ 4 ,
〈 δ n 1 2 ( z ) 〉 = H ( ξ ) H ( ζ ) T 〈 n 1 〉 + 2 G ( G − 1 ) ,
S 1 ( z ) = 4 T 〈 n 1 〉 H ( ξ ) cos ⁡ 2 [ ( ζ − η ) ⁄ 2 ] ⁄ H ( η ) ,
F 1 ( z ) = H ( η ) ⁄ { H ( ξ ) T cos ⁡ 2 [ ( ζ − η ) ⁄ 2 ] } ,
S 1 ( z ) = [ H ( ξ ) T 〈 n 1 〉 + G − 1 ] 2 ⁄ [ H ( ξ ) H ( ζ ) T 〈 n 1 〉 + 2 G ( G − 1 ) ] .
F 1 ( z ) ≈ H ( ζ ) ⁄ [ H ( ξ ) T ] .
S 1 ( z ) = 4 ∣ α 1 ( 0 ) ∣ 2 ⁄ [ 1 + s ( L − 1 ) ] ,
F 1 ( z ) = 1 + s ( L − 1 ) .
S 1 ( z ) ≈ { ∣ α 1 ( 0 ) ∣ 2 + [ s ( L − 1 ) − 1 ] ⁄ 4 } 2 [ 1 + s ( L − 1 ) ] ∣ α 1 ( 0 ) ∣ 2 + { [ 1 + s ( L − 1 ) ] 2 − 2 } ⁄ 8 .
F 1 ( z ) ≈ 1 + s ( L − 1 ) .
a j ( z ) = u j ( z ) + v j ( z ) ,
u j ( z ) = μ j 1 ( z ) a 1 ( 0 ) + ν j 1 ( z ) a 1 † ( 0 )
v j ( z ) = ∑ k > 1 [ μ j k ( z ) a k ( 0 ) + ν j k ( z ) a k † ( 0 ) ]
a j 2 = u j 2 + 2 u j v j + v j 2 ,
a j a j † = u j u j † + u j v j † + u j † v j + v j v j † ,
a j † a j = u j † u j + u j † v j + u j v j † + v j † v j ,
( a j † a j ) 2 = ( u j † u j ) 2 + [ u j 2 ( v j † ) 2 + u j u j † v j † v j + u j † u j v j v j † + ( u j † ) 2 v j 2 ]
+ ( v j † v j ) 2 + u j † u j ( u j v j † + u j † v j ) + ( u j v j † + u j † v j ) u j † u j
+ 2 u j † u j v j † v j + ( u j v j † + u j † v j ) v j † v j + v j † v j ( u j v j † + u j † v j ) .
〈 q j 〉 = 〈 u j e − i ϕ + u j † e i ϕ 〉 ⁄ 2 ,
〈 δ q j 2 〉 = 〈 δ q 2 ( u j ) 〉 + 〈 δ q 2 ( v j ) 〉 ,
〈 δ q 2 ( v j ) 〉 = ∑ k > 1 ∣ λ j k ∣ 2 ⁄ 4 ,
〈 n j 〉 = 〈 u j † u j 〉 + 〈 v j † v j 〉 ,
〈 δ n j 2 〉 = 〈 δ n 2 ( u j ) 〉 + 〈 u j 2 ( v j † ) 2 + u j u j † v j † v j
+ u j † u j v j v j † + ( u j † ) 2 v j 2 〉 + 〈 δ n 2 ( v j ) 〉 ,
〈 v j 2 〉 = ∑ k > 1 μ j k ν j k ,
〈 v j v j † 〉 = ∑ k > 1 ∣ μ j k ∣ 2 ,
〈 v j v j † 〉 = ∑ k > 1 ∣ ν j k ∣ 2 ,
〈 ( v j † ) 2 〉 = ∑ k > 1 μ j k * ν j k *
〈 δ n 2 ( v j ) 〉 = 2 ∑ k > 1 ∣ μ j k ν j k ∣ 2 + ∑ k > 1 ∑ l > k ∣ μ j k * ν j l + μ j l * ν j k ∣ 2 .
〈 q ( u j ) 〉 = ( α j e − i ϕ + α j * e i ϕ ) ⁄ 2 ,
〈 δ q 2 ( u j ) 〉 = ∣ λ j 1 ∣ 2 ⁄ 4 ,
〈 n ( u j ) 〉 = ∣ α j ∣ 2 + ∣ ν j 1 ∣ 2 ,
〈 δ n 2 ( u j ) 〉 = ∣ α j ∣ 2 ∣ λ j 1 ' ∣ 2 + 2 ∣ μ j 1 ν j 1 ∣ 2 ,
〈 δ n j 2 〉 = ∣ α j ∣ 2 ∣ λ j 1 ' ∣ 2 + 2 ∣ μ j 1 ν j 1 ∣ 2
+ ( α j 2 + μ j 1 ν j 1 ) ∑ k > 1 ( μ j k ν j k ) * + ( ∣ α j ∣ 2 + ∣ μ j 1 ∣ 2 ) ∑ k > 1 ∣ ν j k ∣ 2
+ ( ∣ α j ∣ 2 + ∣ ν j 1 ∣ 2 ) ∑ k > 1 ∣ μ j k ∣ 2 + ( α j 2 + μ j 1 ν j 1 ) * ∑ k > 1 μ j k ν j k
+ 2 ∑ k > 1 ∣ μ j k ν j k ∣ 2 + ∑ k > 1 ∑ l > k ∣ μ j k * ν j l + μ j l * ν j k ∣ 2 .
∣ α j ∣ 2 ∣ λ j k ' ∣ 2 = α j 2 μ j k * ν j k * + ∣ α j ∣ 2 ∣ μ j k ∣ 2 + ∣ α j ∣ 2 ∣ ν j k ∣ 2 + ( α j * ) 2 μ j k ν j k ,
∣ μ j 1 * ν j l + μ j l * ν j 1 ∣ 2 = μ j 1 ν j 1 μ j l * ν j l * + ∣ μ j 1 ∣ 2 ∣ ν j l ∣ 2 + ∣ μ j l ∣ 2 ∣ ν j 1 ∣ 2 + μ j 1 * ν j 1 * μ j l ν j l ,
u j ( z ) = ∑ j [ μ j l ( z ) a i ( 0 ) + ν j i ( z ) a i † ( 0 ) ] ,
v j ( z ) = ∑ k ≠ i [ μ j k ( z ) a k ( 0 ) + μ j k ( z ) a k † ( 0 ) ] .
a 1 ( z 1 " ) = μ μ ̄ a 1 ( z 0 ) + ν a 2 † ( z ' 1 ) + μ ν ̄ a 3 ( z 0 ) .
a 1 ( z 2 ″ ) = ( μ μ ̄ ) 2 a 1 ( z 0 ) + ( μ μ ̄ ) ν a 2 † ( z 1 ′ ) + ( μ μ ̄ ) μ ν ̄ a 3 ( z 0 ) + ν a 4 † ( z 2 ′ ) + μ ν ̄ a 5 ( z 1 ″ ) .
a 1 ( z s " ) = ( μ μ ̄ ) s a 1 ( z 0 ) + ∑ r = 1 s ( μ μ ̄ ) s − r [ ν a 2 r ′ ( z ' r ) + μ ν ̄ a 2 r + 1 ( z " r − 1 ) ] .
a 1 ( z 1 " ) = μ ̄ μ a 1 ( z 0 " ) + μ ̄ * ν a 1 † ( z 0 " ) + ν ̄ μ a 2 ( z 0 " ) + ν ̄ * ν a 2 † ( z 0 " ) .
a 1 ( z 2 ″ ) = μ ̄ 2 ( μ 2 + ν 2 ) a 1 ( z 0 ″ ) + μ ̄ 2 ( 2 μ ν ) a 1 † ( z 0 ″ ) + μ ̄ ν ̄ ( μ 2 + ν 2 ) a 2 ( z 0 ″ )
+ μ ̄ ν ̄ ( 2 μ ν ) a 2 † ( z 0 ″ ) + ν ̄ μ a 3 ( z 1 " ) + ν ̄ ν a 3 † ( z 1 " ) .
a 1 ( z 2 " ) = μ ̄ s [ p s ( μ , ν ) a 1 ( z 0 " ) + q s ( μ , ν ) a 1 † ( z 0 " ) ] + ∑ r = 1 s μ ̄ s − r ν ̄
× [ p s + 1 − r ( μ , ν ) a r + 1 ( z ″ r − 1 ) + q s + 1 − r ( μ , ν ) a r + 1 † ( z ″ r − 1 ) ] ,
∑ r = 0 s ∣ ν 1 r + 1 ∣ 2 = [ s ( L − 1 ) − ( 1 − T 2 s ) ⁄ ( 1 + T ) ] ⁄ 4 .
∑ r = 0 s ∣ μ 1 r + 1 ν 1 r + 1 ∣ 2 = { ( 1 − T 2 s ) 2 + ( L − 1 ) 2 [ s − 2 T 2 ( 1 − T 2 s ) ⁄ ( 1 − T 2 )
+ T 4 ( 1 − T 4 s ) ⁄ ( 1 − T 4 ) ] } ⁄ 16 .
∑ r = 1 s ∣ μ 11 ν 1 r + 1 + μ 1 r + 1 ν 11 ∣ 2 = ( L − 1 ) [ s − 2 T s + 1 ( 1 − T s ) ⁄ ( 1 − T )
+ T 2 s + 2 ( 1 − T 2 s ) ⁄ ( 1 − T 2 ) ] ⁄ 4 .
∑ r = q + 1 s ∣ μ 1 q + 1 ν 1 r + 1 + μ 1 r + 1 ν 1 q + 1 ∣ 2 = ( L − 1 ) 2 [ ( s − q ) − 2 T s + 2 − q ( 1 − T s − q ) ⁄ ( 1 − T )
+ T 2 s + 4 − 2 q ( 1 − T 2 s − 2 q ) ⁄ ( 1 − T 2 ) ] ⁄ 4 ,
2 ∑ k ∣ μ 1 k ν 1 k ∣ 2 + ∑ k ∑ l > k ∣ μ 1 k ν 1 l + μ 1 l ν 1 k ∣ 2 ≈ { [ 1 + s ( L − 1 ) ] 2 − 2 } ⁄ 8 ,
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