Abstract

Nonlinear optical cavities are crucial both in classical and quantum optics; in particular, nowadays optical parametric oscillators are one of the most versatile and tunable sources of coherent light, as well as the sources of the highest quality quantum-correlated light in the continuous variable regime. Being nonlinear systems, they can be driven through critical points in which a solution ceases to exist in favour of a new one, and it is close to these points where quantum correlations are the strongest. The simplest description of such systems consists in writing the quantum fields as the classical part plus some quantum fluctuations, linearizing then the dynamical equations with respect to the latter; however, such an approach breaks down close to critical points, where it provides unphysical predictions such as infinite photon numbers. On the other hand, techniques going beyond the simple linear description become too complicated especially regarding the evaluation of two-time correlators, which are of major importance to compute observables outside the cavity. In this article we provide a regularized linear description of nonlinear cavities, that is, a linearization procedure yielding physical results, taking the degenerate optical parametric oscillator as the guiding example. The method, which we call self-consistent linearization, is shown to be equivalent to a general Gaussian ansatz for the state of the system, and we compare its predictions with those obtained with available exact (or quasi-exact) methods. Apart from its operational value, we believe that our work is valuable also from a fundamental point of view, especially in connection to the question of how far linearized or Gaussian theories can be pushed to describe nonlinear dissipative systems which have access to non-Gaussian states.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
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  7. 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).
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  18. P. Kinsler, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
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  19. P. Kinsler and P. D. Drummond, “Critical fluctuations in the quantum parametric oscillator,” Phys. Rev. A 52, 783–790 (1995).
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    [Crossref]
  21. S. Chaturvedi and P. D. Drummond, “Stochastic diagrams for critical point spectra,” Eur. Phys. J. B. 8, 251–267 (1999).
    [Crossref]
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    [Crossref]
  23. M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836–1839 (1988).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  29. In particular, the singular character of the linear stability matrix occurs because one of its eigenvalues becomes zero at σ= 1 (it changes from negative to positive, corresponding to the destabilization of the β̄s = 0 solution when crossing the threshold); on the other hand, it is clear that within this linear description, the correlators of quantum fluctuations are inversely proportional to combinations of these eigenvalues, and hence some might diverge at threshold, what we will show later explicitly.
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    [Crossref] [PubMed]
  31. P. Kinsler and P. D. Drummond, “Quantum dynamics of the parametric oscillator,” Phys. Rev. A 43, 6194–6208 (1991).
    [Crossref] [PubMed]
  32. M. D. Reid and P. D. Drummond, “quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
    [Crossref] [PubMed]
  33. P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
    [Crossref] [PubMed]
  34. C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, “noncritically squeezed light via spontaneous rotational symmetry breaking,” Phys. Rev. Lett. 100, 203601 (2008).
    [Crossref] [PubMed]
  35. C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
    [Crossref]
  36. P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A: Math. Gen. 13, 725–741 (1980).
    [Crossref]
  37. K. Vogel and H. Risken, “Quantum-tunneling rates and stationary solutions in dispersive optical bistability,” Phys. Rev. A 38, 2409–2422 (1988).
    [Crossref] [PubMed]
  38. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
    [Crossref] [PubMed]
  39. P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).
  40. L. A. Lugiato, G. Strini, and F. De Martini, “Squeezed states in second-harmonic generation,” Opt. Lett. 8, 256–258 (1983).
    [Crossref] [PubMed]

2012 (1)

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

2010 (2)

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

2009 (1)

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
[Crossref]

2008 (3)

C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, “noncritically squeezed light via spontaneous rotational symmetry breaking,” Phys. Rev. Lett. 100, 203601 (2008).
[Crossref] [PubMed]

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (2)

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

2003 (1)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

2002 (2)

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]

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

2000 (1)

D. T. Pope, P. D. Drummond, and S. Chaturvedi, “Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator,” Phys. Rev. A 62, 042108 (2000).
[Crossref]

1999 (1)

S. Chaturvedi and P. D. Drummond, “Stochastic diagrams for critical point spectra,” Eur. Phys. J. B. 8, 251–267 (1999).
[Crossref]

1997 (1)

O. Veits and M. Fleischhauer, “Effects of finite-system size in nonlinear optical systems: A quantum many-body approach to parametric oscillation,” Phys. Rev. A 55, 3059–3072 (1997).
[Crossref]

1995 (1)

P. Kinsler and P. D. Drummond, “Critical fluctuations in the quantum parametric oscillator,” Phys. Rev. A 52, 783–790 (1995).
[Crossref] [PubMed]

1994 (1)

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

1993 (3)

P. Kinsler, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
[Crossref] [PubMed]

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

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

1991 (1)

P. Kinsler and P. D. Drummond, “Quantum dynamics of the parametric oscillator,” Phys. Rev. A 43, 6194–6208 (1991).
[Crossref] [PubMed]

1990 (1)

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[Crossref] [PubMed]

1989 (1)

P. D. Drummond and P. Kinsler, “Quantum tunneling and thermal activation in the parametric oscillator,” Phys. Rev. A 40, 4813–4816(R) (1989).
[Crossref] [PubMed]

1988 (3)

M. D. Reid and P. D. Drummond, “quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[Crossref] [PubMed]

M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836–1839 (1988).
[Crossref] [PubMed]

K. Vogel and H. Risken, “Quantum-tunneling rates and stationary solutions in dispersive optical bistability,” Phys. Rev. A 38, 2409–2422 (1988).
[Crossref] [PubMed]

1984 (1)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386 (1984).
[Crossref]

1983 (1)

1981 (2)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub-second harmonic generation, II: Quantum theory,” J. Mod. Opt. 28, 211–225 (1981).

L. A. Lugiato and G. Strini, “On the squeezing obtainable in parametric oscillators and bistable absorption,” Opt. Commun. 41, 67–70 (1981).
[Crossref]

1980 (3)

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A: Math. Gen. 13, 2353–2368 (1980).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A: Math. Gen. 13, 725–741 (1980).
[Crossref]

Adhikari, R.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[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]

Bachor, H.-A.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

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]

Bauchrowitz, J.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

Bourzeix, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Bowen, W. P.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

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]

Carmichael, H. J.

M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836–1839 (1988).
[Crossref] [PubMed]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Chaturvedi, S.

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

D. T. Pope, P. D. Drummond, and S. Chaturvedi, “Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator,” Phys. Rev. A 62, 042108 (2000).
[Crossref]

S. Chaturvedi and P. D. Drummond, “Stochastic diagrams for critical point spectra,” Eur. Phys. J. B. 8, 251–267 (1999).
[Crossref]

Chelkowski, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

Collett, M. J.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386 (1984).
[Crossref]

Danzmann, K.

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

De Martini, F.

de Valcárcel, G. J.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
[Crossref]

C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, “noncritically squeezed light via spontaneous rotational symmetry breaking,” Phys. Rev. Lett. 100, 203601 (2008).
[Crossref] [PubMed]

Dechoum, K.

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

Drummond, P. D.

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

D. T. Pope, P. D. Drummond, and S. Chaturvedi, “Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator,” Phys. Rev. A 62, 042108 (2000).
[Crossref]

S. Chaturvedi and P. D. Drummond, “Stochastic diagrams for critical point spectra,” Eur. Phys. J. B. 8, 251–267 (1999).
[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, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
[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 M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[Crossref] [PubMed]

P. D. Drummond and P. Kinsler, “Quantum tunneling and thermal activation in the parametric oscillator,” Phys. Rev. A 40, 4813–4816(R) (1989).
[Crossref] [PubMed]

M. D. Reid and P. D. Drummond, “quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[Crossref] [PubMed]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub-second harmonic generation, II: Quantum theory,” J. Mod. Opt. 28, 211–225 (1981).

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A: Math. Gen. 13, 2353–2368 (1980).
[Crossref]

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A: Math. Gen. 13, 725–741 (1980).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).

Eberle, T.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

Fabre, C.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

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, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Fernée, M.

P. Kinsler, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
[Crossref] [PubMed]

Fleischhauer, M.

O. Veits and M. Fleischhauer, “Effects of finite-system size in nonlinear optical systems: A quantum many-body approach to parametric oscillation,” Phys. Rev. A 55, 3059–3072 (1997).
[Crossref]

Franzen, A.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

Furusawa, A.

García-Patron, R.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Gardiner, C. W.

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386 (1984).
[Crossref]

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A: Math. Gen. 13, 2353–2368 (1980).
[Crossref]

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 1991).
[Crossref]

Giacobino, E.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Goda, K.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Gossler, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

Hage, B.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

Handchen, V.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

Heidmann, A.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Kennedy, T. B. A.

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

C. J. Mertens, T. B. A. Kennedy, and S. Swain, “Many-body quantum theory of the optical parametric oscillator,” Phys. Rev. A 48, 2374–2385 (1993).
[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, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
[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 P. Kinsler, “Quantum tunneling and thermal activation in the parametric oscillator,” Phys. Rev. A 40, 4813–4816(R) (1989).
[Crossref] [PubMed]

Lam, P. K.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

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]

Lastzka, N.

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Lugiato, L. A.

L. A. Lugiato, G. Strini, and F. De Martini, “Squeezed states in second-harmonic generation,” Opt. Lett. 8, 256–258 (1983).
[Crossref] [PubMed]

L. A. Lugiato and G. Strini, “On the squeezing obtainable in parametric oscillators and bistable absorption,” Opt. Commun. 41, 67–70 (1981).
[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]

Mavalvala, N.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

McKenzie, K.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

McNeil, K. J.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub-second harmonic generation, II: Quantum theory,” J. Mod. Opt. 28, 211–225 (1981).

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).

Mehmet, M.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

Mertens, C. J.

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

Mikhailov, E. E.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Milburn, G. J.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
[Crossref]

Miyakawa, O.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Muller-Ebhardt, H.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

Navarrete-Benlloch, C.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
[Crossref]

C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, “noncritically squeezed light via spontaneous rotational symmetry breaking,” Phys. Rev. Lett. 100, 203601 (2008).
[Crossref] [PubMed]

Pinard, M.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Pope, D. T.

D. T. Pope, P. D. Drummond, and S. Chaturvedi, “Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator,” Phys. Rev. A 62, 042108 (2000).
[Crossref]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Reid, M. D.

P. D. Drummond and M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[Crossref] [PubMed]

M. D. Reid and P. D. Drummond, “quantum correlations of phase in nondegenerate parametric oscillation,” Phys. Rev. Lett. 60, 2731–2733 (1988).
[Crossref] [PubMed]

Reynaud, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Risken, H.

K. Vogel and H. Risken, “Quantum-tunneling rates and stationary solutions in dispersive optical bistability,” Phys. Rev. A 38, 2409–2422 (1988).
[Crossref] [PubMed]

Roldán, E.

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
[Crossref]

C. Navarrete-Benlloch, E. Roldán, and G. J. de Valcárcel, “noncritically squeezed light via spontaneous rotational symmetry breaking,” Phys. Rev. Lett. 100, 203601 (2008).
[Crossref] [PubMed]

Sánchez-Morcillo, V. J.

K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2002).

Saraf, S.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Schnabel, R.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Staliunas, K.

K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2002).

Steinlechner, S.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

Strini, G.

L. A. Lugiato, G. Strini, and F. De Martini, “Squeezed states in second-harmonic generation,” Opt. Lett. 8, 256–258 (1983).
[Crossref] [PubMed]

L. A. Lugiato and G. Strini, “On the squeezing obtainable in parametric oscillators and bistable absorption,” Opt. Commun. 41, 67–70 (1981).
[Crossref]

Swain, S.

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

Takeno, Y.

Treps, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A quantum laser pointer,” Science 301, 940–943 (2003).
[Crossref] [PubMed]

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]

Vahlbruch, H.

T. Eberle, S. Steinlechner, J. Bauchrowitz, V. Handchen, H. Vahlbruch, M. Mehmet, H. Muller-Ebhardt, and R. Schnabel, “quantum enhancement of the zero-area sagnac interferometer topology for gravitational wave detection,” Phys. Rev. Lett. 104, 251102 (2010).
[Crossref] [PubMed]

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Gossler, K. Danzmann, and R. Schnabel, “observation of squeezed light with 10-db quantum-noise reduction,” Phys. Rev. Lett. 100, 033602 (2008).
[Crossref] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[Crossref] [PubMed]

van Loock, P.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

Vass, S.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Veits, O.

O. Veits and M. Fleischhauer, “Effects of finite-system size in nonlinear optical systems: A quantum many-body approach to parametric oscillation,” Phys. Rev. A 55, 3059–3072 (1997).
[Crossref]

Vogel, K.

K. Vogel and H. Risken, “Quantum-tunneling rates and stationary solutions in dispersive optical bistability,” Phys. Rev. A 38, 2409–2422 (1988).
[Crossref] [PubMed]

Walls, D. F.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub-second harmonic generation, II: Quantum theory,” J. Mod. Opt. 28, 211–225 (1981).

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A: Math. Gen. 13, 725–741 (1980).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 1994).
[Crossref]

Ward, R.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Weedbrook, C.

C. Weedbrook, S. Pirandola, R. García-Patron, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84, 621–669 (2012).
[Crossref]

Weinstein, A. J.

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Wolinsky, M.

M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836–1839 (1988).
[Crossref] [PubMed]

Yonezawa, H.

Yukawa, M.

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 1991).
[Crossref]

Eur. Phys. J. B. (1)

S. Chaturvedi and P. D. Drummond, “Stochastic diagrams for critical point spectra,” Eur. Phys. J. B. 8, 251–267 (1999).
[Crossref]

J. Mod. Opt. (2)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub-second harmonic generation, II: Quantum theory,” J. Mod. Opt. 28, 211–225 (1981).

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” J. Mod. Opt. 27, 321–335 (1980).

J. Phys. A: Math. Gen. (2)

P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A: Math. Gen. 13, 2353–2368 (1980).
[Crossref]

P. D. Drummond and D. F. Walls, “Quantum theory of optical bistability. I. Nonlinear polarisability model,” J. Phys. A: Math. Gen. 13, 725–741 (1980).
[Crossref]

Nat. Phys. (1)

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, “A quantum-enhanced prototype gravitational-wave detector,” Nat. Phys. 4, 472–476 (2008).
[Crossref]

Opt. Commun. (1)

L. A. Lugiato and G. Strini, “On the squeezing obtainable in parametric oscillators and bistable absorption,” Opt. Commun. 41, 67–70 (1981).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (14)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386 (1984).
[Crossref]

P. Kinsler, M. Fernée, and P. D. Drummond, “Limits to squeezing and phase information in the parametric amplifier,” Phys. Rev. A 48, 3310–3320 (1993).
[Crossref] [PubMed]

P. Kinsler and P. D. Drummond, “Critical fluctuations in the quantum parametric oscillator,” Phys. Rev. A 52, 783–790 (1995).
[Crossref] [PubMed]

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

M. Mehmet, H. Vahlbruch, N. Lastzka, K. Danzmann, and R. Schnabel, “Observation of squeezed states with strong photon-number oscillations,” Phys. Rev. A 81, 013814 (2010).
[Crossref]

K. Vogel and H. Risken, “Quantum-tunneling rates and stationary solutions in dispersive optical bistability,” Phys. Rev. A 38, 2409–2422 (1988).
[Crossref] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

C. Navarrete-Benlloch, G. J. de Valcárcel, and E. Roldán, “Generating highly squeezed hybrid Laguerre-Gauss modes in large-Fresnel-number degenerate optical parametric oscillators,” Phys. Rev. A 79, 043820 (2009).
[Crossref]

P. D. Drummond and P. Kinsler, “Quantum tunneling and thermal activation in the parametric oscillator,” Phys. Rev. A 40, 4813–4816(R) (1989).
[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 M. D. Reid, “Correlations in nondegenerate parametric oscillation. II. Below threshold results,” Phys. Rev. A 41, 3930–3949 (1990).
[Crossref] [PubMed]

D. T. Pope, P. D. Drummond, and S. Chaturvedi, “Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator,” Phys. Rev. A 62, 042108 (2000).
[Crossref]

C. J. Mertens, T. B. A. 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, “Effects of finite-system size in nonlinear optical systems: A quantum many-body approach to parametric oscillation,” Phys. Rev. A 55, 3059–3072 (1997).
[Crossref]

Phys. Rev. Lett. (8)

M. Wolinsky and H. J. Carmichael, “Quantum noise in the parametric oscillator: From squeezed states to coherent-state superpositions,” Phys. Rev. Lett. 60, 1836–1839 (1988).
[Crossref] [PubMed]

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In particular, the singular character of the linear stability matrix occurs because one of its eigenvalues becomes zero at σ= 1 (it changes from negative to positive, corresponding to the destabilization of the β̄s = 0 solution when crossing the threshold); on the other hand, it is clear that within this linear description, the correlators of quantum fluctuations are inversely proportional to combinations of these eigenvalues, and hence some might diverge at threshold, what we will show later explicitly.

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Figures (3)

Fig. 1
Fig. 1 Mean-field steady-state intensities of the pump (a) and signal (b) modes as a function of the injection parameter σ. We have chosen κ = 1 and g = 0.01, but similar figures are found for any other choice of these parameters. The thin-solid light-grey curve corresponds to the usual linearized description, while the solid blue line and dashed-dotted red line correspond to our self-consistent method below and above threshold, respectively. In (a) the dashed yellow line corresponds to the predictions of Drummond and collaborators’ perturbative analysis [19, 20] (see Section 5), while in (b) it corresponds to the number of (normalized) signal photons obtained from our self-consistent below-threshold solution, showing how it is not divergent at threshold, in contrast with the predictions found with the usual linearization approach.
Fig. 2
Fig. 2 Variances of the squeezed (a) and anti-squeezed (b) quadratures as a function of the injection parameter σ, for κ = 1 and g = 0.01 (similar figures are found for any other choice). As in Fig. 1, the thin-solid light-grey curve corresponds to the usual linearized description; the solid blue line and dashed-dotted red line correspond to our self-consistent method below and above threshold, respectively; and the dashed yellow line corresponds to the predictions of Drummond and collaborators’ perturbative analysis [19, 20].
Fig. 3
Fig. 3 Marginal p(x+) corresponding to the positive P distribution of the signal field in the κ → ∞ limit as obtained from the exact solution (thin-solid light-grey), and our Gaussian ansatzes below (solid blue) and above (dashed-dotted red) threshold. We have picked the value g = 0.01 and show five values of σ : 1 − g (a), 1 − g2/4 (b), 1 + g2/4 (c), 1 +g (d), and 1 + 2g (e). In (f) we show the marginal q(x) for σ = 1 + g; in the case of this marginal, similar figures are found for any other value of σ around the classical threshold.

Equations (32)

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H ^ inj = i h ¯ p ( a ^ p a ^ p ) , H ^ PDC = i h ¯ χ 2 ( a ^ p a ^ s 2 a ^ p a ^ s 2 ) ,
d ρ ^ d t = [ H ^ DOPO i h ¯ , ρ ^ ] + j = p , s γ j ( 2 a ^ j ρ ^ a ^ j a ^ j a ^ j ρ ^ ρ ^ a ^ j a ^ j ) ,
d a ^ p d t = p γ p a ^ p χ 2 a ^ s 2 + 2 γ p a ^ p , in ( t ) , d a ^ s d t = γ s a ^ s + χ a ^ p a ^ s + 2 γ s a ^ s , in ( t ) ,
a ^ j , in ( t ) = a ^ j , in ( t ) a ^ l , in ( t ) = 0 , a ^ j , in ( t ) a ^ l , in ( t ) = δ j l δ ( t t ) ,
σ = p χ / γ p γ s , κ = γ p / γ s , g = χ / γ p γ s ,
τ = γ s t , b ^ s = g a ^ s , b ^ p = κ g a ^ p , b ^ j , in ( τ ) = γ s 1 / 2 a ^ j , in ( γ s 1 τ ) ,
1 κ d b ^ p d τ = σ b ^ p 1 2 b ^ s 2 + 2 g b ^ p , in ( τ ) , d b ^ s d τ = b ^ s + b ^ p b ^ s + 2 g b ^ s , in ( τ ) ;
1 κ d β p d τ = σ β p 1 2 β s 2 1 2 δ b ^ s 2 , d β s d τ = β s + β p β s * + δ b ^ p δ b ^ s .
1 κ d δ b ^ p d τ = δ b ^ p 1 2 β s δ b ^ s 1 2 ( δ b ^ s 2 δ b ^ s 2 ) + 2 g b ^ p , in ( τ ) , d δ b ^ s d τ = δ b ^ s + β p δ b ^ s + β s * δ b ^ p + ( δ b ^ p δ b ^ s δ b ^ p δ b ^ s ) + 2 g b ^ s , in ( τ ) .
d d τ δ b ^ = ( β s , β p ) δ b ^ + 2 g b ^ in ( τ ) ,
= ( κ 0 κ β s 0 0 κ 0 κ β s * β s * 0 1 β p 0 β s β p * 1 ) ;
lim τ δ b ^ s 2 = g 2 β p [ I p ( 1 + κ ) ( 1 + I s ) ( 1 + κ + I s ) ] 2 [ I p ( 1 + κ ) 2 ] [ I p ( 1 + I s ) 2 ] , lim τ δ b ^ p δ b ^ s = g 2 κ β s I p ( 2 + κ + I s ) 2 [ I p ( 1 + κ ) 2 ] [ I p ( 1 + I s ) 2 ] ,
σ = β ¯ p + 1 2 β ¯ s 2 + g 2 β ¯ p [ I ¯ p ( 1 + κ ) ( 1 + I ¯ s ) ( 1 + κ + I ¯ s ) ] 4 [ I ¯ p ( 1 + κ ) 2 ] [ I ¯ p ( 1 + I ¯ s ) 2 ] ,
β ¯ p β ¯ s * = ( 1 + g 2 κ I ¯ p ( 2 + κ + I ¯ s ) 2 [ I ¯ p ( 1 + κ ) 2 ] [ I ¯ p ( 1 + I ¯ s ) 2 ] ) β ¯ s ,
I ¯ s = I ¯ p R ( I ¯ p ) + κ g 2 I ¯ p 4 ( I ¯ p 1 ) [ ( 1 + κ ) 2 I ¯ p ] 1 ,
R ( I ¯ p ) = { 4 ( I ¯ p 1 ) [ ( 1 + κ ) 2 I ¯ p ] κ ( 1 + κ ) g 4 } 2 κ 2 g 4 [ ( 1 + κ ) 2 I ¯ p ] ;
σ = 1 2 I ¯ s + I ¯ p { 1 + g 2 4 I ¯ p ( 1 + κ ) ( 1 + I ¯ s ) ( 1 + κ + I ¯ s ) [ ( 1 + κ ) 2 I ¯ p ] [ I ¯ p ( 1 + I ¯ s ) 2 ] } ;
g 2 d ρ ^ d τ = [ A ^ , ρ ^ ] + j = p , s ( 2 b ^ j ρ ^ b ^ j b ^ j b ^ j ρ ^ ρ ^ b ^ j b ^ j ) ,
g 2 d d τ B ^ = g 2 tr { B ^ d ρ ^ d τ } = [ B ^ , A ^ ] + j = p , s ( [ b ^ j , B ^ ] b ^ j + b ^ j [ B ^ , b ^ j ] ) .
m = col ( δ b ^ p 2 , δ b ^ p 2 * , δ b ^ p δ b ^ p , δ b ^ p δ b ^ s , δ b ^ p δ b ^ s * , δ b ^ p δ b ^ s , δ b ^ p δ b ^ s * , δ b ^ s 2 , δ b ^ s 2 * , δ b ^ s δ b ^ s ) ,
d m d τ = ( β p , β s ) m + n ( β p ) ,
= ( 2 κ 0 0 2 κ β s 0 0 0 0 0 0 0 2 κ 0 0 2 κ β s * 0 0 0 0 0 0 0 2 κ 0 0 κ β s * κ β s 0 0 0 β s * 0 0 κ ps 0 β p 0 κ β s 0 0 0 β s 0 0 κ ps 0 β p * 0 κ β s * 0 0 0 β s β p 0 κ ps 0 0 0 κ β s 0 0 β s * 0 β p 0 κ ps 0 0 κ β s * 0 0 0 2 β s * 0 0 0 2 0 2 β p 0 0 0 0 2 β s * 0 0 0 2 2 β p * 0 0 0 0 0 β s * β s β p * β p 2 ) ,
n = g 2 col ( 0 , 0 , 0 , 0 , 0 , 0 , 0 , β p , β p * , 0 ) ;
lim t δ x ^ s 2 = ( 1 + κ ) ( 1 + I ¯ s ) I p ( 1 + I ¯ s I p ) ( 1 + κ I p ) , lim t δ y ^ s 2 = ( 1 + κ ) ( 1 + I ¯ s ) + I p ( 1 + I ¯ s + I p ) ( 1 + κ + I p ) .
lim t δ x ^ s 2 2 2 g , and lim t δ y ^ s 2 0.5 + g 8 2 ,
lim t b ^ p σ g 4 2 x 2 ,
lim t δ x ^ s 2 2 g x 2 , lim t δ y ^ s 2 3 σ 4 + g 16 2 ( 2 + 3 κ 2 + κ ) x 2 ,
D ( x ) = d exp [ σ 1 2 g x 2 x 4 16 ] ,
lim t δ x ^ s 2 | Drummond et : al lim t δ x ^ s 2 | self consistent 2 3 ,
lim t δ y ^ s 2 | Drummond et : al 0.5 lim t δ y ^ s 2 | self consistent 0.5 2 3 ( 1 + 2 κ 2 + κ ) .
P ( α s , α s + ) = { K [ ( α s 2 2 σ g 2 ) ( α s + 2 2 σ g 2 ) ] 1 + 2 / g 2 e 2 α s α s + for | α s | , | α s + | 2 σ / g 0 for | α s | , | α s + | > 2 σ / g ,
lim t a ^ s m a ^ s n = 2 d α s d α s + P ( α s , α s + ) α s + m α s n .

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