Abstract

The time-dependent, linearly driven Fokker–Planck equation for the degenerate parametric amplifier is derived when the pump depletion is present both above and below threshold. A model that the driving force εα1 is a product of the classical quantity ε and quantum operator α1 is proposed. It is found that below or near threshold, our results conform to the linear theory or perturbation series expansion, and above threshold, the short-time behavior of our solution is close to the linearization approximation; with the increase of interaction time τ, the long-time behavior of our solution shows that the squeezing is always quite different from the linear theory. It is shown that the maximal squeezing can be obtained at large pump parameter μ=εk and interaction time Δτ.

© 2006 Optical Society of America

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  1. B. Yurke, "Squeezed-coherent-state generation via four-wave mixers and detection via homodyne detectors," Phys. Rev. A 32, 300-310 (1985).
    [CrossRef] [PubMed]
  2. C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
    [CrossRef] [PubMed]
  3. M. J. Collett and D. F. Walls, "Squeezing spectra for nonlinear optical systems," Phys. Rev. A 32, 2887-2892 (1985).
    [CrossRef] [PubMed]
  4. L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
    [CrossRef] [PubMed]
  5. P. Kinsler and P. D. Drummond, "Quantum dynamics of the parametric oscillator," Phys. Rev. A 43, 6194-6208 (1991).
    [CrossRef] [PubMed]
  6. P. Kinsler and P. D. Drummond, "Critical fluctuations in the quantum parametric oscillator," Phys. Rev. A 52, 783-790 (1995).
    [CrossRef] [PubMed]
  7. L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
    [CrossRef] [PubMed]
  8. P. D. Drummond and C. W. Gardiner, "Generalised P-representations in quantum optics," J. Phys. A 13, 2353-2368 (1980).
    [CrossRef]
  9. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994), p. 178.
  10. R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
    [CrossRef]
  11. P. Kumar and J. H. Shapiro, "Squeezed-state generation via forward degenerate four-wave mixing," Phys. Rev. A 30, 1568-1571 (1984).
    [CrossRef]
  12. M. Wolinsky and H. J. Carmichael, "Squeezing in the degenerate parametric oscillator," Opt. Commun. 55, 138-142 (1985).
    [CrossRef]
  13. W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
    [CrossRef]
  14. C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
    [CrossRef] [PubMed]
  15. C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
    [CrossRef] [PubMed]
  16. O. Veits and M. Fleischhauer, "Quantum fluctuations in the optical parametric oscillator in the limit of a fast decaying subharmonic mode," Phys. Rev. A 52, R4344-R4347 (1995).
    [CrossRef] [PubMed]
  17. L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
    [CrossRef] [PubMed]
  18. C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).
  19. S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
    [CrossRef]
  20. P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
    [CrossRef]
  21. B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
    [CrossRef]
  22. G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Phys. Rev. 36, 823-841 (1930).
    [CrossRef]
  23. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 1983).

2005

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

2003

C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).

2002

S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
[CrossRef]

P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
[CrossRef]

1995

O. Veits and M. Fleischhauer, "Quantum fluctuations in the optical parametric oscillator in the limit of a fast decaying subharmonic mode," Phys. Rev. A 52, R4344-R4347 (1995).
[CrossRef] [PubMed]

P. Kinsler and P. D. Drummond, "Critical fluctuations in the quantum parametric oscillator," Phys. Rev. A 52, 783-790 (1995).
[CrossRef] [PubMed]

1994

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

1993

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
[CrossRef] [PubMed]

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
[CrossRef] [PubMed]

1991

P. Kinsler and P. D. Drummond, "Quantum dynamics of the parametric oscillator," Phys. Rev. A 43, 6194-6208 (1991).
[CrossRef] [PubMed]

1987

W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
[CrossRef]

1986

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

1985

B. Yurke, "Squeezed-coherent-state generation via four-wave mixers and detection via homodyne detectors," Phys. Rev. A 32, 300-310 (1985).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

M. J. Collett and D. F. Walls, "Squeezing spectra for nonlinear optical systems," Phys. Rev. A 32, 2887-2892 (1985).
[CrossRef] [PubMed]

M. Wolinsky and H. J. Carmichael, "Squeezing in the degenerate parametric oscillator," Opt. Commun. 55, 138-142 (1985).
[CrossRef]

1984

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

P. Kumar and J. H. Shapiro, "Squeezed-state generation via forward degenerate four-wave mixing," Phys. Rev. A 30, 1568-1571 (1984).
[CrossRef]

1980

P. D. Drummond and C. W. Gardiner, "Generalised P-representations in quantum optics," J. Phys. A 13, 2353-2368 (1980).
[CrossRef]

1930

G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Phys. Rev. 36, 823-841 (1930).
[CrossRef]

Bondurant, R. S.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

Carmichael, H. J.

M. Wolinsky and H. J. Carmichael, "Squeezing in the degenerate parametric oscillator," Opt. Commun. 55, 138-142 (1985).
[CrossRef]

Chaturvedi, S.

S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
[CrossRef]

P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
[CrossRef]

Collett, M. J.

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

M. J. Collett and D. F. Walls, "Squeezing spectra for nonlinear optical systems," Phys. Rev. A 32, 2887-2892 (1985).
[CrossRef] [PubMed]

da Silva, L. F.

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

Dechoum, K.

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
[CrossRef]

P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
[CrossRef]

dos Santos, B. C.

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

Drummond, P. D.

S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
[CrossRef]

P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
[CrossRef]

P. Kinsler and P. D. Drummond, "Critical fluctuations in the quantum parametric oscillator," Phys. Rev. A 52, 783-790 (1995).
[CrossRef] [PubMed]

P. Kinsler and P. D. Drummond, "Quantum dynamics of the parametric oscillator," Phys. Rev. A 43, 6194-6208 (1991).
[CrossRef] [PubMed]

P. D. Drummond and C. W. Gardiner, "Generalised P-representations in quantum optics," J. Phys. A 13, 2353-2368 (1980).
[CrossRef]

Fleischhauer, M.

O. Veits and M. Fleischhauer, "Quantum fluctuations in the optical parametric oscillator in the limit of a fast decaying subharmonic mode," Phys. Rev. A 52, R4344-R4347 (1995).
[CrossRef] [PubMed]

Gardiner, C. W.

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

P. D. Drummond and C. W. Gardiner, "Generalised P-representations in quantum optics," J. Phys. A 13, 2353-2368 (1980).
[CrossRef]

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 1983).

Guo, Q. Z.

C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).

Hall, J. L.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

Kennedy, T. A. B.

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
[CrossRef] [PubMed]

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
[CrossRef] [PubMed]

Khoury, A. Z.

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

Kinsler, P.

P. Kinsler and P. D. Drummond, "Critical fluctuations in the quantum parametric oscillator," Phys. Rev. A 52, 783-790 (1995).
[CrossRef] [PubMed]

P. Kinsler and P. D. Drummond, "Quantum dynamics of the parametric oscillator," Phys. Rev. A 43, 6194-6208 (1991).
[CrossRef] [PubMed]

Kumar, P.

P. Kumar and J. H. Shapiro, "Squeezed-state generation via forward degenerate four-wave mixing," Phys. Rev. A 30, 1568-1571 (1984).
[CrossRef]

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

Li, Y. F.

W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
[CrossRef]

Maeda, M.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

Mertens, C. J.

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
[CrossRef] [PubMed]

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
[CrossRef] [PubMed]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994), p. 178.

Olsen, M. K.

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

Ornstein, L. S.

G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Phys. Rev. 36, 823-841 (1930).
[CrossRef]

Plimak, L. I.

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

Shapiro, J. H.

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

P. Kumar and J. H. Shapiro, "Squeezed-state generation via forward degenerate four-wave mixing," Phys. Rev. A 30, 1568-1571 (1984).
[CrossRef]

Swain, S.

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
[CrossRef] [PubMed]

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
[CrossRef] [PubMed]

Tan, W. H.

C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).

W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
[CrossRef]

Uhlenbeck, G. E.

G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Phys. Rev. 36, 823-841 (1930).
[CrossRef]

Veits, O.

O. Veits and M. Fleischhauer, "Quantum fluctuations in the optical parametric oscillator in the limit of a fast decaying subharmonic mode," Phys. Rev. A 52, R4344-R4347 (1995).
[CrossRef] [PubMed]

Walls, D. F.

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

M. J. Collett and D. F. Walls, "Squeezing spectra for nonlinear optical systems," Phys. Rev. A 32, 2887-2892 (1985).
[CrossRef] [PubMed]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994), p. 178.

Wolinsky, M.

M. Wolinsky and H. J. Carmichael, "Squeezing in the degenerate parametric oscillator," Opt. Commun. 55, 138-142 (1985).
[CrossRef]

Wu, H.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

Wu, L. A.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

Yurke, B.

B. Yurke, "Squeezed-coherent-state generation via four-wave mixers and detection via homodyne detectors," Phys. Rev. A 32, 300-310 (1985).
[CrossRef] [PubMed]

Zhang, W. P.

W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
[CrossRef]

Zhao, C. Y.

C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).

Acta Phys. Sin.

C. Y. Zhao, W. H. Tan, and Q. Z. Guo, "The solution of the Fokker-Planck equation of non-degenerate parametric amplification system for generation of squeezed light," Acta Phys. Sin. 52, 2694-2699 (2003) (in Chinese).

J. Phys. A

P. D. Drummond and C. W. Gardiner, "Generalised P-representations in quantum optics," J. Phys. A 13, 2353-2368 (1980).
[CrossRef]

Opt. Commun.

M. Wolinsky and H. J. Carmichael, "Squeezing in the degenerate parametric oscillator," Opt. Commun. 55, 138-142 (1985).
[CrossRef]

W. H. Tan, Y. F. Li, and W. P. Zhang, "The solution of the Fokker-Planck equation with zero or negative diffusion coefficients in quantum optics," Opt. Commun. 64, 195-199 (1987).
[CrossRef]

Phys. Rev.

G. E. Uhlenbeck and L. S. Ornstein, "On the theory of the Brownian motion," Phys. Rev. 36, 823-841 (1930).
[CrossRef]

Phys. Rev. A

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body quantum theory of the optical parametric oscillator," Phys. Rev. A 48, 2374-2385 (1993).
[CrossRef] [PubMed]

O. Veits and M. Fleischhauer, "Quantum fluctuations in the optical parametric oscillator in the limit of a fast decaying subharmonic mode," Phys. Rev. A 52, R4344-R4347 (1995).
[CrossRef] [PubMed]

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, "Degenerate four-wave mixing as a possible source of squeezed-state light," Phys. Rev. A 30, 343-353 (1984).
[CrossRef]

P. Kumar and J. H. Shapiro, "Squeezed-state generation via forward degenerate four-wave mixing," Phys. Rev. A 30, 1568-1571 (1984).
[CrossRef]

S. Chaturvedi, K. Dechoum, and P. D. Drummond, "Limits to squeezing in the degenerate optical parametric oscillator," Phys. Rev. A 65, 033805 (2002).
[CrossRef]

P. D. Drummond, K. Dechoum, and S. Chaturvedi, "Critical quantum fluctuations in the degenerate parametric oscillator," Phys. Rev. A 65, 033806 (2002).
[CrossRef]

B. C. dos Santos, K. Dechoum, A. Z. Khoury, L.F. da Silva, and M. K. Olsen, "Quantum analysis of the non-degenerate optical parametric oscillator with injected signal," Phys. Rev. A 72, 033820 (2005).
[CrossRef]

P. Kinsler and P. D. Drummond, "Quantum dynamics of the parametric oscillator," Phys. Rev. A 43, 6194-6208 (1991).
[CrossRef] [PubMed]

P. Kinsler and P. D. Drummond, "Critical fluctuations in the quantum parametric oscillator," Phys. Rev. A 52, 783-790 (1995).
[CrossRef] [PubMed]

L. I. Plimak and D. F. Walls, "Dynamical restrictions to squeezing in a degenerate optical parametric oscillator," Phys. Rev. A 50, 2627-2641 (1994).
[CrossRef] [PubMed]

B. Yurke, "Squeezed-coherent-state generation via four-wave mixers and detection via homodyne detectors," Phys. Rev. A 32, 300-310 (1985).
[CrossRef] [PubMed]

C. W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985).
[CrossRef] [PubMed]

M. J. Collett and D. F. Walls, "Squeezing spectra for nonlinear optical systems," Phys. Rev. A 32, 2887-2892 (1985).
[CrossRef] [PubMed]

Phys. Rev. Lett.

L. A. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520-2523 (1986).
[CrossRef] [PubMed]

C. J. Mertens, T. A. B. Kennedy, and S. Swain, "Many-body theory of quantum noise," Phys. Rev. Lett. 71, 2014-2017 (1993).
[CrossRef] [PubMed]

Other

D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994), p. 178.

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, 1983).

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Figures (5)

Fig. 1
Fig. 1

(a). The curve of α 1 versus τ. (b). The curve of α 2 versus τ.

Fig. 2
Fig. 2

Deviation Δ versus interaction time τ with γ r = 0.5 (dashed curve) and 5 (solid curve).

Fig. 3
Fig. 3

Relation curve between β and i β ̃ .

Fig. 4
Fig. 4

(a). Quantum fluctuation ( Δ y ) 2 versus interaction time τ with (a) μ = 0.8 , α 10 = 1 , (b) μ = 1 , α 10 = 1 , and (c) μ = 2 , α 10 = 1 .

Fig. 5
Fig. 5

Quantum fluctuation ( Δ y ) 2 versus τ with μ = 2 , α 10 = 0.001 .

Equations (32)

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t P ( t , α 1 , α 1 * , α 2 , α 2 * ) = { α 1 ( γ 1 α 1 χ α 1 * α 2 ) + α 1 * ( γ 1 α 1 * χ α 1 α 2 * ) + α 2 ( γ 2 α 2 ε ¯ + χ 2 α 1 2 ) + α 2 * ( γ 2 α 2 * ε ¯ * + χ 2 α 1 * 2 ) + 1 2 [ 2 α 1 2 ( χ α 2 ) + 2 α 1 * 2 ( χ α 2 * ) ] } P ( t , α 1 , α 1 * , α 2 , α 2 * ) ,
[ α 2 ( γ 2 α 2 ε ¯ + χ 2 α 1 2 ) + α 2 * ( γ 2 α 2 * ε ¯ * + χ 2 α 1 * 2 ) ] P d 2 α 2 = 0 ,
p t = [ γ 1 ( α 1 α 1 + α 1 * α 1 * ) ε ( α 1 * α 1 + α 1 α 1 * ) + ε 2 ( 2 α 1 2 + 2 α 1 * 2 ) ] p ,
d α 1 d t = γ 1 α 1 + χ α 1 * α 2 + χ α 2 η ( t ) ,
d α 2 d t = γ 2 α 2 + ε ¯ χ 2 α 1 2 .
η ( t ) = η * ( t ) = 0 ,
η ( t ) η * ( t ) = δ ( t t ) ,
d α 1 d τ = α 1 + χ γ 1 α 2 α 1 * ,
d α 2 d τ = γ r α 2 + ε ¯ γ 1 χ 2 γ 1 α 1 2 .
α 1 = χ γ 1 α 2 α 1 * , α 2 = 1 γ 2 ( ε ¯ χ 2 α 1 2 ) .
α ¯ 1 = ± 2 χ ( ε ¯ ε ¯ c ) = 2 χ γ 1 γ 2 χ ( μ 1 ) = μ 1 η ,
α ¯ 2 = 1 γ r [ ε ¯ γ 1 1 2 χ γ 1 μ 1 η ] = 1 2 γ r η = γ 1 χ .
p t = x ( k x p ) + 1 2 D 2 x 2 p , p = p ( x , t x 0 , 0 ) .
ϕ ( s , t ) = exp ( i s x ) p ( x , t x 0 , 0 ) d x ,
ϕ = exp { D s 2 4 k [ 1 exp ( 2 k t ) ] + i s x 0 exp ( k t ) } .
ln ϕ t + k s s ln ϕ = D 2 s 2 .
A t + 2 k A = D 2 , A 0 = D 4 k t = 0 ,
A = A 0 exp ( 2 0 t k d t ) + exp ( 2 0 t k d t ) 0 t exp ( 2 0 t k d t ) D 2 d t .
ϕ = exp ( A s 2 ) g ( s exp [ 0 t k d t ] ) .
ϕ = exp { s 2 [ A A 0 exp ( 2 0 t k d t ) ] + i s x 0 exp ( 0 t k d t ) } .
p ( x , t x 0 , t ) = 1 4 π ( A A 0 exp [ 2 0 t k d t ] ) exp [ ( x x 0 exp [ 0 t k d t ] ) 2 4 ( A A 0 exp [ 2 0 t k d t ] ) ] .
p t = [ k ( α α + α * α * ) ε ( α * α + α α * ) + ε 2 ( 2 α 2 + 2 α * 2 ) ] p .
p ( β ) t = [ ( k ε ) β β + ε 2 2 β 2 ] p ( β ) ,
p ̃ ( i β ̃ ) t = [ ( k + ε ) ( i β ̃ ) ( i β ̃ ) + ε 2 2 ( i β ̃ ) 2 ] p ̃ ( i β ̃ ) .
p ( β ) = exp [ 1 4 { β β 0 exp [ 0 t ( k ε ) d t ] } 2 A A 0 exp [ 2 0 t ( k ε ) d t ] ] 4 π { A A 0 exp [ 2 0 t ( k ε ) d t ] } ,
p ̃ ( i β ̃ ) = exp ( 1 4 { i β ̃ i β ̃ 0 exp [ 0 t ( k + ε ) d t ] } 2 A ̃ A ̃ 0 exp [ 2 0 t ( k + ε ) d t ] ) 4 π { A ̃ A ̃ 0 exp [ 2 0 t ( k + ε ) d t ] } .
A = A 0 exp [ 2 0 t ( k ε ) d t ] + exp [ 2 0 t ( k ε ) d t ] 0 t exp [ 2 0 t ( k ε ) d t ] ε 2 d t = α 2 [ A 0 + 0 t α 2 2 ( d ln α d t + k ) d t ] = α 2 ( A 0 + 1 4 ) 1 4 + α 2 k 2 0 t α 2 d t .
A ̃ = α ̃ ( t ) 2 ( A ̃ 0 + 1 4 ) 1 4 + α ̃ ( t ) 2 k 2 0 t α ̃ ( t ) 2 d t .
( Δ β ) 2 = ( β β 0 exp [ 0 t ( k ε ) d t ] ) 2 = 2 { A A 0 exp [ 2 0 t ( k ε ) d t ] } = 2 [ A A 0 α 2 ( t ) ] ,
( Δ i β ̃ ) 2 = ( i β ̃ i β ̃ 0 exp [ 0 t ( k + ε ) d t ] ) 2 = 2 { A ̃ A ̃ 0 exp [ 2 0 t ( k + ε ) d t ] } = 2 [ A ̃ A ̃ 0 α ̃ 2 ( t ) ] .
( Δ x ) 2 = 1 4 + : ( Δ x ) 2 : = 1 4 + ( Δ β ) 2 2 = 1 4 + [ A A 0 α 2 ( t ) ] ,
( Δ y ) 2 = 1 4 + : ( Δ y ) 2 : = 1 4 + ( Δ i β ̃ ) 2 2 = 1 4 + [ A ̃ A ̃ 0 α ̃ 2 ( t ) ] .

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