Abstract

The formalism for computing the signal-to-noise ratio (SNR) for laser radar is reviewed and applied to the tasks of target detection, direction finding, and phase-change estimation with squeezed light. The SNR for heterodyne detection of coherent light using a squeezed local oscillator is lower than that obtained using a coherent local oscillator. This is true for target detection, for phase estimation, and for direction finding with a split detector. Squeezing the local oscillator also lowers SNR in balanced homodyne and heterodyne detection of coherent light. Loss places an upper bound on the improvement that squeezing can bring to direct-detection SNR.

© 2009 Optical Society of America

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References

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  1. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
    [CrossRef]
  2. R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, 1978).
  3. Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
    [CrossRef]
  4. Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
    [CrossRef]
  5. M. A. Rubin and S. Kaushik, “Squeezing the local oscillator does not improve signal-to-noise ratio in heterodyne laser radar,” Opt. Lett. 32, 1369-1371 (2007).
    [CrossRef] [PubMed]
  6. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  7. P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1970).
  8. C. G. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press, 2005).
  9. H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).
  10. C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, 2004), p. 2004.
  11. C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurements of very small displacements in optical images,” Opt. Lett. 25, 76-78 (2000).
    [CrossRef]
  12. N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
    [CrossRef] [PubMed]
  13. F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988).
  14. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177-179 (1983).
    [CrossRef] [PubMed]
  15. V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
    [CrossRef]

2007 (1)

2002 (1)

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
[CrossRef]

1997 (1)

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

1992 (1)

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
[CrossRef]

1983 (1)

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Andersen, U.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Annovazzi-Lodi, V.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
[CrossRef]

Bachor, H.-A.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Buchler, B.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Chan, V. W. S.

Donati, S.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
[CrossRef]

Edwards, P. J.

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

Fabre, C.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurements of very small displacements in optical images,” Opt. Lett. 25, 76-78 (2000).
[CrossRef]

Fouet, J. B.

Gardiner, C. W.

C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, 2004), p. 2004.

Gerry, C. G.

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

Guzun, D.

Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
[CrossRef]

Haus, H. A.

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Kaushik, S.

Kingston, R. H.

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, 1978).

Knight, P. L.

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

Lam, P. K.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Li, Y.-q.

Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
[CrossRef]

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

Lynam, P.

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

Maître, A.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

C. Fabre, J. B. Fouet, and A. Maître, “Quantum limits in the measurements of very small displacements in optical images,” Opt. Lett. 25, 76-78 (2000).
[CrossRef]

Merlo, S.

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
[CrossRef]

Meyer, P. L.

P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1970).

Rubin, M. A.

Smith, F. G.

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988).

Thomson, J. H.

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988).

Treps, N.

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Xiao, M.

Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
[CrossRef]

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

Yuen, H. P.

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, 2004), p. 2004.

Opt. Lett. (3)

Opt. Quantum Electron. (1)

V. Annovazzi-Lodi, S. Donati, and S. Merlo, “Squeezed states in direct and coherent detection,” Opt. Quantum Electron. 24, 285-301 (1992).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Phys. Rev. Lett. (3)

N. Treps, U. Andersen, B. Buchler, P. K. Lam, A. Maître, H.-A. Bachor, and C. Fabre, “Surpassing the standard quantum limit for optical imaging using nonclassical multimode light,” Phys. Rev. Lett. 88, 203601 (2002).
[CrossRef] [PubMed]

Y.-q. Li, P. Lynam, M. Xiao, and P. J. Edwards, “Sub-shot-noise laser Doppler anemometry with amplitude-squeezed light,” Phys. Rev. Lett. 78, 3105-3108 (1997).
[CrossRef]

Y.-q. Li, D. Guzun, and M. Xiao, “Sub-shot-noise-limited optical heterodyne detection using an amplitude-squeezed local oscillator,” Phys. Rev. Lett. 82, 5225-5228 (1999).
[CrossRef]

Other (7)

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1970).

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

H. A. Haus, Electromagnetic Noise and Quantum Optical Measurements (Springer, 2000).

C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer, 2004), p. 2004.

F. G. Smith and J. H. Thomson, Optics, 2nd ed. (Wiley, 1988).

R. H. Kingston, Detection of Optical and Infrared Radiation (Springer, 1978).

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Equations (180)

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G = k = 1 M g ( x k )
Q 0 erfc ( ξ ) , Q d erfc ( ξ D g M 1 / 2 ) ,
D g 2 = [ E ( g | H 1 ) E ( g | H 0 ) ] 2 Var 0 g
Var 0 g = E ( g 2 | H 0 ) [ E ( g | H 0 ) ] 2
E ( S | H i ) = ψ i | S ^ | ψ i ,
Var i S = ψ i | S ^ 2 | ψ i ψ i | S ^ | ψ i 2 .
D S 2 = ( ψ 1 | S ^ | ψ 1 ψ 0 | S ^ | ψ 0 ) 2 ψ 0 | S ^ 2 | ψ 0 ψ 0 | S ^ | ψ 0 2 .
E ^ μ ( + ) ( t ) = k , ζ i ( ω k 2 ε 0 V ) 1 / 2 a ^ k , ζ e i ω k t ε k , ζ , μ ,
I ^ ( t ) = ν , μ s ν μ E ^ ν ( ) ( t ) E ^ μ ( + ) ( t ) ,
E ^ μ ( ) ( t ) = ( E ^ μ ( + ) ( t ) ) ,
I ^ ( t ) = k , ζ , l , ρ , μ , ν 2 ε 0 V ( ω k ω k ) 1 / 2 a ^ l , ρ a ^ k , ζ e i ( ω l ω k ) t ε l , ρ , ν ε k , ζ , μ s ν μ .
S ^ = τ 1 0 τ d t cos ( ω H t + θ H ) I ^ ( t ) .
S ^ = 4 ε 0 V k , ζ , l , ρ , μ , ν s . t . | ω l ω k | = ω H ( ω l ω k ) 1 / 2 a ^ l , ρ a ^ k , ζ e i ε ( ω l ω k ) θ H ε l , ρ ν ε k , ζ μ s ν μ ,
ε ( x ) = sign of   x .
S ^ 2 = ( 4 ε 0 V ) 2 k , ζ , l , ρ , μ , ν s . t . | ω l ω k | = ω H k , ζ , l , ρ , μ , ν s . t . | ω l ω k | = ω H ( ω l ω k ω l ω k ) 1 / 2 a ^ l , ρ a ^ k , ζ a ^ l , ρ a ^ k , ζ e i [ ε ( ω l ω k ) + ε ( ω l ω k ) ] θ H ε l , ρ , ν ε k , ζ μ s ν μ ε l , ρ , ν ε k , ζ , μ s v μ .
S = τ 1 0 τ d t cos ( ω H t + θ H ) I ( t ) .
S = τ 1 0 τ d t cos ( ω H t + θ H ) I ( t ) .
ω H = ω H ,
θ H = θ H .
| α , ξ k , ζ = D ^ ( α ) k , ζ Q ^ ( ξ ) k , ζ | 0 k , ζ ,
D ^ ( α ) k , ζ = exp ( α a ^ k , ζ α * a ^ k , ζ ) ,
Q ^ ( ξ ) k , ζ = exp [ 1 2 ( ξ * a ^ k , ζ 2 ξ ( a ^ k , ζ ) 2 ) ] .
| α k , ζ = | α , 0 k , ζ = D ^ ( α ) k , ζ | 0 k , ζ .
| ψ 0 = | α , ξ L O k , ζ L O | 0 k , ζ .
| ψ 1 = | β T | α , ξ L O k , ζ T , L O | 0 k , ζ .
ψ 0 | S ^ | ψ 0 = 0 ,
ψ 1 | S ^ | ψ 1 = κ 2 ε 0 V ( ω T ω L O ) 1 / 2 | α | | β | cos ( θ T θ L O + θ H ) .
ω H ω T , ω H ω L O ,
ω T ω L O ω ,
ψ 1 | S ^ | ψ 1 = κ ω 2 ε 0 V | α | | β | cos ( θ T θ L O + θ H ) .
θ T = arg β , θ L O = arg α ,
κ = ν , μ ε L O , ν ε T , μ s ν μ .
κ = ν , μ ε L O , ν ε L O , μ s ν μ .
ψ 0 | S 2 ^ | ψ 0 = ( 4 ε 0 V ) 2 k , ζ , μ , ν s . t . | ω L O ω k | = ω H     l , ρ , μ , ν s . t . | ω l ω L O | = ω H     ω L O ( ω k ω l ) 1 / 2 ψ 0 | a ^ L O a ^ k , ζ a ^ l , ρ a ^ L O | ψ 0 e i [ ε ( ω L O ω k ) + ε ( ω l ω L O ) ] θ H ε L O , ν ε k , ζ μ s ν μ ε l , ρ , ν ε L O , μ s v μ .
ψ 0 | S 2 ^ | ψ 0 = ( 4 ε 0 V ) 2 k , ζ , s . t . | ω L O ω k | = ω H ω L O ω k n ¯ L O ( ν , μ ε L O , ν ε k , ζ , μ s μ ν ) 2 ,
ψ 0 | S 2 ^ | ψ 0 = 2 ( ω L O 4 ε 0 V ) 2 n ¯ L O k , ζ , s . t . ω k = ω L O ( ν , μ ε L O , ν ε k , ζ , μ s μ ν ) 2 ,
n ¯ L O = LO α , ξ | n ^ L O | α , ξ L O ,
n ^ L O = a ^ L O a ^ L O .
( κ ) 2 = k , ζ , s . t . ω k = ω L O ( ν , μ ε L O , ν ε k , ζ , μ s μ ν ) 2 .
( κ ) 2 ( κ ) 2 ,
( κ ) 2 = κ 2 .
( κ ) 2 κ 2 N ,
N = k s . t . ω k = ω L O max ( | k | 2 Ω / ( ( 2 π ) / L t r ) 2 , 1 ) = max ( ( L t r / λ ) 2 Ω , 1 ) ,
λ = 2 π c / ω
[ x ] = integral part of   x .
N = 1 ,
ψ 0 | S 2 ^ | ψ 0 = 2 ( κ ω L O 4 ε 0 V ) 2 n ¯ L O .
D S 2 = 2 | α | 2 | β | 2 cos 2 ( θ T θ L O + θ H ) n ¯ L O = 2 ( 1 sinh 2 ( r ) n ¯ L O ) n ¯ T cos 2 ( θ T θ L O + θ H ) ,
n ¯ T = T β | n ^ T | β T ,
n ^ T = a ^ T a ^ T ,
r = | ξ | .
n ¯ L O = | α | 2 + sinh 2 ( r ) ,
n ¯ T = | β | 2 ,
sinh 2 ( r ) n ¯ L O .
ψ 1 | S ^ | ψ 1 2 = 1 c 2 A 2 ( κ 2 ε 0 ) 2 P L O P T cos 2 ( θ T θ L O + θ H ) ( 1 sinh 2 ( r ) n ¯ L O ) .
L = V / A ,
T = L / c = V / c A .
B = 1 / 2 T ,
ψ 0 | S 2 ^ | ψ 0 = ( κ 4 ε 0 ) 2 4 B ω c 2 A 2 P L O ,
D S 2 = P T ω B cos 2 ( θ T θ L O + θ H ) ( 1 sinh 2 ( r ) n ¯ L O ) .
a ^ k , ζ = t k , ζ a ^ ( i n ) k , ζ + r k , ζ a ^ ( v a c ) k , ζ ,
S ^ B = 4 ε 0 V k , ζ , l , ρ , μ , ν s . t . | ω l ω k | = ω H ( ω l ω k ) 1 / 2 ( t l , ρ * a ^ ( i n ) l , ρ + r l , ρ * a ^ ( v a c ) l , ρ ) ( t k , ζ a ^ ( i n ) k , ζ + r k , ζ a ^ ( v a c ) k , ζ ) e i ε ( ω l ω l ) θ H ε l , ρ , ν ε k , ζ , μ s ν μ .
| ψ B , 0 = | ψ ( i n ) , 0 | ψ ( v a c ) , 0 ,
| ψ B , 1 = | ψ ( i n ) , 1 | ψ ( v a c ) , 0 ,
| ψ ( v a c ) , 0 = k , ζ | 0 ( v a c ) ,
| ψ ( i n ) , 0 = k , ζ L O | 0 ( i n ) , k , ζ | α , ξ ( i n ) , L O ,
| ψ ( i n ) , 1 = k , ζ L O , T | 0 ( i n ) , k , ζ | α , ξ ( i n ) , L O | β ( i n ) , T .
θ B = arg ( t L O * t T ) ,
ψ B , 0 | S ^ B | ψ B , 0 = 0 ,
ψ B , 1 | S ^ B | ψ B , 1 = | t L O t T | ( κ ω 2 ε 0 V ) ( n ¯ sinh 2 ( r ) ) 1 / 2 n ¯ T 1 / 2 cos ( θ T θ L O + θ H + θ B ) ,
ψ B , 0 | S ^ B 2 | ψ B , 0 = 2 | t L O | 2 ( κ ω 4 ε 0 V ) n ¯ L O .
Var 0 S B = ψ B , 0 | S ^ B 2 | ψ B , 0 ψ B , 0 | S ^ B | ψ B , 0 2 = 2 | t L O | 2 ( κ ω 4 ε 0 V ) n ¯ L O .
D S B 2 = ( ψ B , 1 | S ^ B | ψ B , 1 ψ B , 0 | S ^ B | ψ B , 0 2 / Var 0 S B = | t T | 2 2 ( 1 sinh 2 ( r ) n ¯ L O ) n ¯ T cos 2 ( θ T θ LO + θ H + θ B ) .
η = | t T | 2 .
E ^ μ ( + ) ( t , x ) = i k , ζ , m ( ω k 2 ε 0 V ) 1 / 2 a ^ k , ζ , m ( x ) e i ω k t u k , ζ , m ( x ) ε k , ζ , μ ,
I ^ ( t , x ) = ν μ s ˜ ν , μ E ^ ν ( ) ( t , x ) E ^ μ ( + ) ( t , x ) ,
s ˜ ν , μ = s ν , μ / W ,
E ^ μ ( ) ( t , x ) = ( E ^ μ ( + ) ( t , x ) ) .
I ^ ( t , x ) = l ^ , ρ , n , k ^ , ζ , m , ν , μ 2 ε 0 V ( ω l ω k ) 1 / 2 a ^ l , ρ , n a ^ k , ζ , m e i ( ω l ω k ) t u l , ρ , n * ( x ) u k , ζ , m ( x ) ε l , ρ , ν ε k , ζ , μ s ν μ .
u 0 ( x ) = 1 , W / 2 x W / 2 = 0 otherwise .
u 1 ( x ) = 1 , W / 2 x 0 = 1 , 0 < x W / 2 = 0 otherwise .
S ^ s p = d x ε ( x ) 1 τ 0 τ d t I ^ ( t , x ) .
S ^ s p = l , ρ , n , k , ζ , m , ν , μ s . t . ω l = ω k ω l 2 ε 0 V a ^ l , ρ , n a ^ k , ζ , m d x ε ( x ) u l , ρ , n * ( x ) u k , ζ , m ( x ) ε l , ρ , n ε k , ζ , μ s ˜ ν μ
| ψ s p , 1 = l , ρ , n T | 0 l , ρ , n | α , ξ T .
u T 1 ( x ) = u 0 ( x δ ) .
| ψ s p , 0 = | ψ s p , 1 δ = 0 .
d x ε ( x ) | u T 1 ( x ) | 2 = 2 δ .
ψ s p , 0 | S ^ s p | ψ s p , 0 = 0 ,
ψ s p , 1 | S ^ s p | ψ s p , 1 = 2 δ κ ˜ ω T 2 ε 0 V n ¯ T ,
κ ˜ = κ / W .
ψ s p , 0 | S ^ 2 s p | ψ s p , 0 = k , ζ , m , ν , μ s . t . ω k = ω T     l , ρ , n , ν , μ s . t . ω l = ω T l , ρ , n T l , ρ , n 0 | T α , ξ | a ^ T a ^ k ζ , m a ^ l ρ , n a ^ T k , ζ , m T | 0 k , ζ , m | α , ξ T ( d x ε ( x ) u T * ( x ) u k , ζ , m ( x ) ) ( d x ε ( x ) u l , ρ , n * ( x ) u T ( x ) ) ε T , ν ε k , ζ , μ s ˜ ν μ ε l , ρ , ν ε T , μ s ˜ ν μ .
ψ s p , 0 | S ^ 2 s p | ψ s p , 0 = W 2 ( κ ˜ ω T 2 ε 0 V ) n ¯ T .
Var 0 S s p = ψ s p , 0 | S ^ 2 s p | ψ s p , 0 ψ s p , 0 | S ^ s p | ψ s p , 0 2 = W 2 ( κ ˜ ω T 2 ε 0 V ) 2 n ¯ T .
D S s p 2 = ( 2 δ W ) 2 n ¯ T .
δ = f Δ θ ,
W = f λ / d ,
D S s p 2 = ( 2 d Δ θ λ ) 2 n ¯ T .
Δ θ min = 1 2 n ¯ T 1 / 2 ( λ d ) .
| ψ s p , 1 = l , ρ , n L O , T | 0 l , ρ , n | α , ξ L O | β T .
u L O ( x ) = u 0 ( x ) ,
u T ( x ) = u 1 ( x δ ) .
| ψ s p , 0 = | ψ s p , 1 δ = 0 .
d x ε ( x ) u L O * ( x ) u L O ( x ) = 0 ,
d x ε ( x ) u L O * ( x ) u T ( x ) = d x ε ( x ) u T * ( x ) u L O ( x ) = W 3 δ ,
d x ε ( x ) u T * ( x ) u T ( x ) = 2 δ .
ψ s p , 0 | S ^ s p | ψ s p , 0 = 2 W ( κ ˜ ℏω 2 ε 0 V ) ( n ¯ L O sinh 2 ( r ) ) 1 / 2 n ¯ 1 / 2 cos ( θ T θ L O ) ,
ψ s p , 1 | S ^ s p | ψ s p , 1 = ( κ ˜ ℏω 2 ε 0 V ) ( 2 δ n ¯ T + 2 ( W 3 δ ) ( n ¯ L O sinh 2 ( r ) ) 1 / 2 n ¯ 1 / 2 cos ( θ T θ L O ) ) ,
Var 0 S s p = ψ s p , 1 | S ^ 2 s p | ψ s p , 1 ψ s p , 1 | S ^ s p | ψ s p , 1 2 = ( κ ˜ ℏω 2 ε 0 V ) W 2 ( n ¯ L O + n ¯ T { 1 + 2 sinh ( r ) [ sinh ( r ) cosh ( r ) cos ( 2 θ T θ s q ) ] } ) .
cos ( 2 θ T θ s q ) = 1 ,
Var 0 S s p = ( κ ˜ ω 2 ε 0 V ) W 2 ( n ¯ L O + n ¯ T e 2 r ) .
D S s p 2 = ( 2 δ W ) 2 ( n ¯ T 3 ( n ¯ L O sinh 2 ( r ) ) 1 / 2 n ¯ T 1 / 2 cos ( θ T θ L O ) ) 2 ( n ¯ L O + n ¯ T e 2 r ) .
n ¯ L O = sinh 2 ( r ) .
n ¯ L O 1 ,
e 2 r 1 4 n ¯ L O .
D S s p 2 ( 2 δ W ) 2 n ¯ T ( n ¯ T n ¯ L O ) .
u L O ( x ) = u 1 ( x ) ,
u T ( x ) = u 0 ( x δ ) .
S ^ s p = d x ε ( x ) 1 τ 0 τ d t cos ( ω H t + θ H ) I ^ ( t ) .
S ^ s p = l , ρ , n , k , ζ , m , ν , μ s . t . | ω l ω k | = ω H ( 4 ε 0 V ) ( ω l ω k ) 1 / 2 a ^ l , ρ , n a ^ k , ζ , m e i ε ( ω l ω k ) θ H ( d x ε ( x ) u l , ρ , n * ( x ) u k , ζ , m ( x ) ) ε l , ρ , ν ε k , ζ , μ s ν μ .
ψ s p , 0 | S ^ s p | ψ s p , 0 = W κ ˜ ℏω 2 ε 0 V ( n ¯ L O sinh 2 ( r ) ) 1 / 2 n ¯ t 1 / 2 cos ( θ t θ L O + θ H ) ,
ψ s p , 1 | S ^ s p | ψ s p , 1 = ( W δ ) κ ˜ ℏω 2 ε 0 V ( n ¯ L O sinh 2 ( r ) ) 1 / 2 n ¯ t 1 / 2 cos ( θ t θ L O + θ H ) ,
Var 0 S s p = W 2 ( κ ˜ ℏω 4 ε 0 V ) 2 { 2 n ¯ L O + 2 n ¯ T [ 1 sinh ( r ) ( sinh ( r ) + cosh ( r ) cos ( 2 θ θ s q + 2 θ H ) ) ] } .
2 θ θ s q + 2 θ H = 2 π n , n = 0 , ± 1 , ± 2 , ,
Var 0 S s p = 2 W 2 ( κ ˜ ℏω 4 ε 0 V ) 2 [ n ¯ L O + n ¯ T 2 ( 1 + e 2 r ) ] .
D S s p 2 = ( ψ s p , 1 | S ^ s p | ψ s p , 1 ψ s p , 0 | S ^ s p | ψ s p , 0 ) 2 / Var 0 S s p = 2 ( d Δ θ λ ) 2 ( 1 sinh 2 ( r ) n ¯ L O 1 + n ¯ T 2 n ¯ L O ( 1 + e 2 r ) ) n ¯ T cos 2 ( θ T θ L O + θ H ) ,
n ¯ T / n ¯ L O 0 ,
sinh 2 ( r ) / n ¯ L O 0.
D S s p 2 e 2 = 2 ( d Δ θ λ ) 2 ( 1 sinh 2 ( r ) n ¯ L O 1 + n ¯ T n ¯ L O ( 1 + e 2 r ) ) n ¯ T cos 2 ( θ T θ L O + θ H ) .
| ψ 0 , ph = | β T | α , ξ L O k , ζ T , L O | 0 k , ζ .
| ψ 1 , ph = | β e i δ θ T T | α , ξ L O k , ζ T , L O | 0 k , ζ ;
D p h 2 = ( ψ 1 , ph | S ^ | ψ 1 , p h ψ 0 , p h | S ^ | ψ 0 , p h ) 2 / Var 0 , p h S ,
Var 0 , p h S = ψ 1 , p h | S ^ 2 | ψ 1 , p h ψ 1 , p h | S ^ | ψ 1 , p h 2 .
ψ 1 , p h | S ^ | ψ 1 , p h ψ 0 , p h | S ^ | ψ 0 , p h = κ ω 2 ε 0 V | α | | β | ( cos ( θ T + δ θ T θ L O + θ H ) cos ( θ T θ L O + θ H ) ) δ θ T κ ω 2 ε 0 V | α | | β | sin ( θ T θ L O + θ H ) .
Var 0 , p h S = 2 ( κ ω 4 ε 0 V ) 2 ( n ¯ L O + n ¯ T { 1 + sinh ( r ) [ sinh ( r ) cosh ( r ) cos ( 2 θ T θ sq + 2 θ H ) ] } ) .
2 θ T θ s q + θ H = 2 π n , n = 0 , ± 1 , ± 2 , ,
Var 0 , p h S = 2 ( κ ω 4 ε 0 V ) 2 [ n ¯ L O + n ¯ T 2 ( 1 + e 2 r ) ] .
D p h 2 = 2 ( δ θ T ) 2 ( 1 sinh 2 ( r ) n ¯ L O 1 + n ¯ T 2 n ¯ L O ( 1 + e 2 r ) ) n ¯ T sin 2 ( θ T θ L O + θ H ) .
a ^ ( r e f ) , k , ζ = r a ^ ( T ) , k , ζ + t a ^ ( L O ) , k , ζ ,
a ^ ( t r ) , k , ζ = t a ^ ( T ) , k , ζ + r a ^ ( L O ) , k , ζ .
I ^ ( t r ) ( t ) = l ρ , k ζ , ν μ 2 ε 0 V ( ω l ω l ) 1 / 2 e i ( ω l ω k ) t ε l , ρ , ν ε k , ζ , μ s ν μ ( | t | 2 a ^ ( T ) l , ρ a ^ ( T ) k , ζ + r * t a ^ ( L O ) l ρ a ^ ( T ) k ζ + t * r a ^ ( T ) l ρ a ^ ( L O ) k ζ + | r | 2 a ^ ( L O ) l , ρ a ^ ( L O ) k , ζ ) ,
I ^ ( r e f ) ( t ) = l ρ , k ζ , ν μ 2 ε 0 V ( ω l ω l ) 1 / 2 e i ( ω l ω k ) t ε l , ρ , ν ε k , ζ , μ s ν μ ( | r | 2 a ^ ( T ) l , ρ a ^ ( T ) k , ζ + t * r a ^ ( L O ) l ρ a ^ ( T ) k ζ + r * t a ^ ( T ) l ρ a ^ ( L O ) k ζ + | t | 2 a ^ ( LO ) l , ρ a ^ ( L O ) k , ζ ) .
I ^ ( d i f f ) ( t ) = I ^ ( t r ) ( t ) I ^ ( r e f ) ( t ) .
I ^ ( d i f f ) ( t ) = l ρ , k ζ , ν μ 2 ε 0 V ( ω l ω l ) 1 / 2 e i ( ω l ω k ) t ε l , ρ , ν ε k , ζ , μ s ν μ ( ( | t | 2 | r | 2 ) a ^ ( T ) l , ρ a ^ ( T ) k , ζ + ( r * t t * r ) a ^ ( L O ) l ρ a ^ ( T ) k ζ ( t * r r * t ) a ^ ( T ) l ρ a ^ ( L O ) k ζ + ( | r | 2 | t | 2 ) a ^ ( L O ) l , ρ a ^ ( L O ) k , ζ ) ,
I ^ ( b a l ) ( t ) = i l ρ , k ζ , ν μ 2 ε 0 V ( ω l ω k ) 1 / 2 e i ( ω l ω k ) t ε l , ρ , ν ε k , ζ , μ s ν μ ( a ^ ( L O ) l ρ a ^ ( T ) k ζ a ^ ( T ) l ρ a ^ ( L O ) k ζ )
t = t = 1 / 2 ,
r = r = i / 2 .
S ^ b a l h e t = 1 τ 0 τ d t cos ( ω H + θ H ) I ^ ( b a l ) ( t ) = i l ρ , k ζ , ν μ s . t . | ω l ω k | = ω H 4 ε 0 V ( ω l ω k ) 1 / 2 e i ε ( ω l ω k ) θ H ε l , ρ , ν ε k , ζ , μ s ν μ ( a ^ ( L O ) l , ρ a ^ ( T ) k , ζ a ^ ( T ) l , ρ a ^ ( L O ) k , ζ ) .
| ψ b a l , 0 = ( l , ρ | 0 ( T ) l , ρ ) ( k , ζ L O | 0 ( L O ) k , ζ ) | α , β ( L O ) L O ,
| ψ b a l , 1 = ( l , ρ T | 0 ( T ) l , ρ ) ( k , ζ L O | 0 ( L O ) k , ζ ) | α , β ( L O ) L O | β ( T ) T .
D b a l h e t 2 = 2 ( 1 sinh 2 ( r ) n ¯ L O ) n ¯ T sin 2 ( θ T θ L O + θ H ) .
S ^ b a l h o m = 1 τ 0 τ d t I ^ ( b a l ) ( t ) = i l ρ , k ζ , ν μ s . t . ω l = ω k ω l 2 ε 0 V ε l , ρ , ν ε k , ζ , μ s ν μ ( a ^ ( L O ) l , ρ a ^ ( T ) k , ζ a ^ ( T ) l , ρ a ^ ( L O ) k , ζ ) .
D b a l h o m 2 = 4 ( 1 sinh 2 ( r ) n ¯ L O ) n ¯ T sin 2 ( θ T θ L O ) .
| ψ 0 = k , ζ | 0 k , ζ ,
| ψ 1 = | α , ξ T k , ζ T | 0 k , ζ .
S ^ = τ 1 0 τ d t I ^ ( t ) = l , ρ , k , ζ , μ , ν s . t . ω l = ω k ω k 2 ε 0 V a ^ l , ρ a ^ k , ζ ε l , ρ ν ε k , ζ μ s ν μ .
D g 2 = [ E ( g | H 1 ) E ( g | H 0 ) ] 2 Var 1 g ,
Var 1 g = E ( g 2 | H 1 ) [ E ( g | H 1 ) ] 2 .
D S 2 = n ¯ T 2 var s q ( n T ) ,
var s q ( n T ) = T α , ξ | n ^ T 2 | α , ξ T ( T α , ξ | n ^ T | α , ξ T ) 2 .
var s q ( n T ) = ( n ¯ T sinh 2 ( r ) ) | cosh ( r ) e i ( θ s q 2 θ T ) sinh ( r ) | 2 + 2 cosh 2 ( r ) sinh 2 ( r )
ξ = r e i θ s q .
θ s q 2 θ T = 2 π , n = 0 , ± 1 , ± 2 , ,
var s q ( n T ) = ( n ¯ T sinh 2 ( r ) ) e 2 r + 2 cosh 2 ( r ) sinh 2 ( r ) ,
D S 2 = n ¯ T 2 ( n ¯ T sinh 2 ( r ) ) e 2 r + 2 cosh 2 ( r ) sinh 2 ( r ) .
sinh 2 ( r ) n ¯ T .
D S 2 = n ¯ T e 2 r .
sinh 2 ( r ) cosh 2 ( r ) 1 4 e 2 r ,
e 2 r 4 n ¯ T .
D S 2 4 n ¯ T 2 .
S ^ B = l , ρ , k , ζ , μ , ν s . t . ω l = ω k ( ω k 2 ε 0 V ) ( t l , ρ * a ^ ( i n ) l , ρ + r l , ρ * a ^ ( v a c ) l , ρ ) ( t k , ζ a ^ ( i n ) k , ζ + r k , ζ a ^ ( v a c ) k , ζ ) ε l , ρ , ν ε k , ζ , μ s ν μ .
| ψ B , 0 = | ψ ( i n ) , 0 | ψ ( v a c ) , 0 ,
| ψ B , 1 = | ψ ( i n ) , 1 | ψ ( v a c ) , 0 ,
| ψ ( i n ) , 0 = k , ζ | 0 ( i n ) , k , ζ ,
| ψ ( i n ) , 1 = k , ζ T | 0 ( i n ) , k , ζ | α , ξ ( i n ) , T ,
D S B 2 = | t T | 2 n ¯ ( i n ) , T 2 | t T | 2 var s q n ( i n ) , T + ( 1 | t T | 2 ) n ¯ ( i n ) , T ,
var s q n ( i n ) , T var coh n ( i n ) , T = n ¯ ( i n ) , T ,
D S B 2 D S B c o h 2 = | t T | 2 n ¯ ( i n ) , T .
D S B 2 D S B c o h 2 = n ¯ ( i n ) , T | t T | 2 var s q n ( i n ) , T + ( 1 | t T | 2 ) n ¯ ( i n ) , T 1 1 | t T | 2 ,
D S B 2 D S B c o h 2 1 L ,
L = 1 | t T | 2 = 1 η

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