Abstract

We study the nondegenerate optical parametric oscillator in a planar interferometer near threshold, where critical phenomena are expected. These phenomena are associated with nonequilibrium quantum dynamics that are known to lead to quadrature entanglement and squeezing in the oscillator field modes. We obtain a universal form for the equation describing this system, which allows a comparison with other phase transitions. We find that the unsqueezed quadratures of this system correspond to a two-dimensional XY-type model with a tricritical Lifshitz point. This leaves open the possibility of a controlled experimental investigation into this unusual class of statistical models. We evaluate the correlations of the unsqueezed quadrature using both an exact numerical simulation and a Gaussian approximation, and obtain an accurate numerical calculation of the non-Gaussian correlations.

© 2016 Optical Society of America

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References

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2015 (2)

A. Bonanno and D. Zappalà, “Isotropic Lifshitz critical behavior from the functional renormalization group,” Nucl. Phys. B 893, 501–511 (2015).
[Crossref]

P. C. Hohenberg and A. P. Krekhov, “An introduction to the Ginzburg–Landau theory of phase transitions and nonequilibrium patterns,” Phys. Rep. 572, 1–42 (2015).
[Crossref]

2014 (1)

P. D. Drummond, “Fundamentals of higher order stochastic equations,” J. Phys. A 47, 335001 (2014).
[Crossref]

2013 (1)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

2010 (3)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 28, 211–225 (2010).
[Crossref]

V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
[Crossref]

2009 (2)

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

J. Yin and D. P. Landau, “Phase diagram and critical behavior of the square-lattice Ising model with competing nearest-neighbor and next-nearest-neighbor interactions,” Phys. Rev. E 80, 051117 (2009).
[Crossref]

2008 (1)

2007 (2)

A. S. Villar, K. N. Cassemiro, K. Dechoum, A. Z. Khoury, M. Martinelli, and P. Nussenzveig, “Entanglement in the above-threshold optical parametric oscillator,” J. Opt. Soc. Am. B 24, 249–256 (2007).
[Crossref]

G. Kozyreff and M. Tlidi, “Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems,” Chaos 17, 037103 (2007).
[Crossref]

2005 (5)

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[Crossref]

J. Laurat, L. Longchambon, C. Fabre, and T. Coudreau, “Experimental investigation of amplitude and phase quantum correlations in a type II optical parametric oscillator above threshold: from nondegenerate to degenerate operation,” Opt. Lett. 30, 1177–1179 (2005).
[Crossref]

V. D’Auria, A. Chiummo, M. De Laurentis, A. Porzio, S. Solimeno, and M. G. A. Paris, “Tomographic characterization of OPO sources close to threshold,” Opt. Express 13, 948–956 (2005).
[Crossref]

P. D. Drummond and K. Dechoum, “Universality of quantum critical dynamics in a planar optical parametric oscillator,” Phys. Rev. Lett. 95, 083601 (2005).
[Crossref]

K. Staliunas and M. Tlidi, “Hyperbolic transverse patterns in nonlinear optical resonators,” Phys. Rev. Lett. 94, 133902 (2005).
[Crossref]

2004 (1)

K. Dechoum, P. D. Drummond, S. Chaturvedi, and M. D. Reid, “Critical fluctuations and entanglement in the nondegenerate parametric oscillator,” Phys. Rev. A 70, 053807 (2004).
[Crossref]

2003 (5)

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]

R. Zambrini, S. M. Barnett, P. Colet, and M. San Miguel, “Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q-representation,” Eur. Phys. J. D 22, 461–471 (2003).
[Crossref]

R. Zambrini, A. Gatti, L. Lugiato, and M. San Miguel, “Polarization quantum properties in a type-II optical parametric oscillator below threshold,” Phys. Rev A 68, 063809 (2003).
[Crossref]

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262–267 (2003).

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

2002 (5)

G. Izús, M. San Miguel, and D. Walgraef, “Polarization coupling and pattern selection in a type-II optical parametric oscillator,” Phys. Rev. E 66, 36228 (2002).
[Crossref]

C. Lamprecht, M. K. Olsen, P. D. Drummond, and H. Ritsch, “Positive-P and Wigner representations for quantum-optical systems with nonorthogonal modes,” Phys. Rev. A 65, 053813 (2002).
[Crossref]

M. Santagiustina, E. Hernandez-Garcia, M. San-Miguel, A. J. Scroggie, and G.-L. Oppo, “Polarization patterns and vectorial defects in type-II optical parametric oscillators,” Phys. Rev. E 65, 036610 (2002).
[Crossref]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176–S183 (2002).
[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]

2001 (4)

G. R. Collecutt and P. D. Drummond, “Xmds: eXtensible multi-dimensional simulator,” Comput. Phys. Commun. 142, 219–223 (2001).
[Crossref]

S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[Crossref]

K. Staliunas, “Spatial and temporal noise spectra of spatially extended systems with order disorder phase transitions,” Int. J. Bifurcation Chaos 11, 2845–2852 (2001).
[Crossref]

K. Staliunas, “Spatial and temporal spectra of noise driven stripe patterns,” Phys. Rev. E 64, 066129 (2001).
[Crossref]

2000 (1)

G. J. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[Crossref]

1999 (2)

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

J. P. Gollub and J. S. Langer, “Pattern formation in nonequilibrium physics,” Rev. Mod. Phys. 71, S396 (1999).
[Crossref]

1998 (3)

C. Bolman and A. C. Newell, “Natural patterns and wavelets,” Rev. Mod. Phys. 70, 289–301 (1998).
[Crossref]

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[Crossref]

M. A. M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, “Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects,” Opt. Express 3, 71–80 (1998).
[Crossref]

1997 (5)

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[Crossref]

A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express 1, 21–30 (1997).
[Crossref]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comp. Phys. 132, 312–326 (1997).
[Crossref]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[Crossref]

K. Staliunas, G. Slekys, and C. O. Weiss, “Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns,” Phys. Rev. Lett. 79, 2658–2661 (1997).
[Crossref]

1996 (2)

S. Longhi and A. Geraci, “Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[Crossref]

S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 43, 1569–1575 (1996).

1995 (2)

K. Staliunas, “Transverse pattern formation in optical parametric oscillators,” J. Mod. Opt. 42, 1261–1269 (1995).
[Crossref]

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675–1690 (1995).
[Crossref]

1994 (4)

G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[Crossref]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[Crossref]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[Crossref]

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref]

1993 (3)

K. Staliunas, “Laser Ginzburg-Landau equation and laser hydrodynamics,” Phys. Rev. A 48, 1573–1581 (1993).
[Crossref]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

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

1992 (1)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref]

1991 (1)

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comp. Phys. 93, 144–170 (1991).
[Crossref]

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]

1989 (2)

M. D. Reid and P. D. Drummond, “Correlations in nondegenerate parametric oscillation: Squeezing in the presence of phase diffusion,” Phys. Rev. A 40, 4493–4506 (1989).
[Crossref]

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
[Crossref]

1988 (2)

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

P. D. Drummond and M. D. Reid, “Laser bandwidth effects on squeezing in intracavity parametric oscillation,” Phys. Rev. A 37, 1806–1808 (1988).
[Crossref]

1986 (1)

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]

1985 (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]

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

1984 (2)

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
[Crossref]

1980 (2)

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

R. M. Hornreich, “The Lifshitz point: Phase diagrams and critical behavior,” J. Magn. Mater. 15, 387–392 (1980).
[Crossref]

1979 (1)

R. M. Hornreich, R. Liebmann, H. G. Schuster, and W. Selke, “Lifshitz points in ising systems,” Z. Phys. B 35, 91–97 (1979).
[Crossref]

1977 (3)

A. Michelson, “Phase diagrams near the Lifshitz point. I. Uniaxial magnetization,” Phys. Rev. B 16, 577–584 (1977).
[Crossref]

P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod. Phys. 49, 435–479 (1977).
[Crossref]

J. Swift and P. C. Hohenberg, “Hydrodynamic fluctuations at the convective instability,” Phys. Rev. A 15, 319–328 (1977).
[Crossref]

1975 (1)

R. M. Hornreich, M. Luban, and S. Shtrikman, “Critical behavior at the onset of k⃗-space instability on the λ line,” Phys. Rev. Lett. 35, 1678–1681 (1975).
[Crossref]

1973 (1)

J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203 (1973).
[Crossref]

1972 (2)

V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. quantum systems,” Sov. Phys. J. Exp. Theor. Phys. 34, 610–616 (1972).

E. K. Riedel and F. J. Wegner, “Tricritical exponents and scaling fields,” Phys. Rev. Lett. 29, 349–352 (1972).
[Crossref]

1970 (2)

R. Graham and H. Haken, “Laserlight -First example of a second order phase transition far from thermal equilibrium,” Z. Phys. 237, 31–46 (1970).
[Crossref]

V. deGiorgio and M. O. Scully, “Analogy between the laser threshold region and a second-order phase transition,” Phys. Rev. A 2, 1170–1177 (1970).
[Crossref]

1966 (1)

N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).
[Crossref]

1963 (2)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

1940 (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Andersen, U. L.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Anwar, J.

V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
[Crossref]

Bachor, H. A.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

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]

Barnett, S. M.

R. Zambrini, S. M. Barnett, P. Colet, and M. San Miguel, “Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q-representation,” Eur. Phys. J. D 22, 461–471 (2003).
[Crossref]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[Crossref]

Bartosch, L.

P. Kopietz, L. Bartosch, and F. Schtz, Introduction to the Functional Renormalization Group (Springer, 2010).

Berezinskii, V. L.

V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. quantum systems,” Sov. Phys. J. Exp. Theor. Phys. 34, 610–616 (1972).

Berzanskis, A.

Bolman, C.

C. Bolman and A. C. Newell, “Natural patterns and wavelets,” Rev. Mod. Phys. 70, 289–301 (1998).
[Crossref]

Bonanno, A.

A. Bonanno and D. Zappalà, “Isotropic Lifshitz critical behavior from the functional renormalization group,” Nucl. Phys. B 893, 501–511 (2015).
[Crossref]

Bowen, W. P.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

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]

Brambilla, E.

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176–S183 (2002).
[Crossref]

Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[Crossref]

G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[Crossref]

Camesasca, D.

G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[Crossref]

Carmichael, H. J.

H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, 2002).

Carusotto, I.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

Cassemiro, K. N.

Cavalcanti, E. G.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Chaikin, P. M.

P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, 1995).

Chaturvedi, S.

K. Dechoum, P. D. Drummond, S. Chaturvedi, and M. D. Reid, “Critical fluctuations and entanglement in the nondegenerate parametric oscillator,” Phys. Rev. A 70, 053807 (2004).
[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]

Chiummo, A.

Ciuti, C.

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

Colet, P.

R. Zambrini, S. M. Barnett, P. Colet, and M. San Miguel, “Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q-representation,” Eur. Phys. J. D 22, 461–471 (2003).
[Crossref]

Collecutt, G. R.

G. R. Collecutt and P. D. Drummond, “Xmds: eXtensible multi-dimensional simulator,” Comput. Phys. Commun. 142, 219–223 (2001).
[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]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

Coudreau, T.

G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
[Crossref]

J. Laurat, L. Longchambon, C. Fabre, and T. Coudreau, “Experimental investigation of amplitude and phase quantum correlations in a type II optical parametric oscillator above threshold: from nondegenerate to degenerate operation,” Opt. Lett. 30, 1177–1179 (2005).
[Crossref]

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[Crossref]

J. Laurat, T. Coudreau, and C. Fabre, Quantum Information with Continuous Variables of Atoms and Light, N. J. Cerf, G. Leuchs, and E. S. Polzik, eds. (World Scientific Publishing, 2007).

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

D’Auria, V.

De Laurentis, M.

de Lisio, C.

V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
[Crossref]

de Valcárcel, G. J.

G. J. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[Crossref]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[Crossref]

Dechoum, K.

A. S. Villar, K. N. Cassemiro, K. Dechoum, A. Z. Khoury, M. Martinelli, and P. Nussenzveig, “Entanglement in the above-threshold optical parametric oscillator,” J. Opt. Soc. Am. B 24, 249–256 (2007).
[Crossref]

P. D. Drummond and K. Dechoum, “Universality of quantum critical dynamics in a planar optical parametric oscillator,” Phys. Rev. Lett. 95, 083601 (2005).
[Crossref]

K. Dechoum, P. D. Drummond, S. Chaturvedi, and M. D. Reid, “Critical fluctuations and entanglement in the nondegenerate parametric oscillator,” Phys. Rev. A 70, 053807 (2004).
[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]

deGiorgio, V.

V. deGiorgio and M. O. Scully, “Analogy between the laser threshold region and a second-order phase transition,” Phys. Rev. A 2, 1170–1177 (1970).
[Crossref]

Di Trapani, P.

Drummond, P. D.

P. D. Drummond, “Fundamentals of higher order stochastic equations,” J. Phys. A 47, 335001 (2014).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 28, 211–225 (2010).
[Crossref]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

P. D. Drummond and K. Dechoum, “Universality of quantum critical dynamics in a planar optical parametric oscillator,” Phys. Rev. Lett. 95, 083601 (2005).
[Crossref]

K. Dechoum, P. D. Drummond, S. Chaturvedi, and M. D. Reid, “Critical fluctuations and entanglement in the nondegenerate parametric oscillator,” Phys. Rev. A 70, 053807 (2004).
[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]

C. Lamprecht, M. K. Olsen, P. D. Drummond, and H. Ritsch, “Positive-P and Wigner representations for quantum-optical systems with nonorthogonal modes,” Phys. Rev. A 65, 053813 (2002).
[Crossref]

G. R. Collecutt and P. D. Drummond, “Xmds: eXtensible multi-dimensional simulator,” Comput. Phys. Commun. 142, 219–223 (2001).
[Crossref]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comp. Phys. 132, 312–326 (1997).
[Crossref]

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comp. Phys. 93, 144–170 (1991).
[Crossref]

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

M. D. Reid and P. D. Drummond, “Correlations in nondegenerate parametric oscillation: Squeezing in the presence of phase diffusion,” Phys. Rev. A 40, 4493–4506 (1989).
[Crossref]

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

P. D. Drummond and M. D. Reid, “Laser bandwidth effects on squeezing in intracavity parametric oscillation,” Phys. Rev. A 37, 1806–1808 (1988).
[Crossref]

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

S. Kiesewetter, R. Polkinghorne, B. Opanchuk, and P. D. Drummond, “xSPDE: extensible software for stochastic equations,” SoftwareX, to be published.

P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics (Cambridge University, 2014).

Ducci, S.

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[Crossref]

Fabre, C.

G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
[Crossref]

J. Laurat, L. Longchambon, C. Fabre, and T. Coudreau, “Experimental investigation of amplitude and phase quantum correlations in a type II optical parametric oscillator above threshold: from nondegenerate to degenerate operation,” Opt. Lett. 30, 1177–1179 (2005).
[Crossref]

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[Crossref]

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]

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[Crossref]

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

M. A. M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, “Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects,” Opt. Express 3, 71–80 (1998).
[Crossref]

J. Laurat, T. Coudreau, and C. Fabre, Quantum Information with Continuous Variables of Atoms and Light, N. J. Cerf, G. Leuchs, and E. S. Polzik, eds. (World Scientific Publishing, 2007).

Feng, S.

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262–267 (2003).

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]

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

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

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Gatti, A.

R. Zambrini, A. Gatti, L. Lugiato, and M. San Miguel, “Polarization quantum properties in a type-II optical parametric oscillator below threshold,” Phys. Rev A 68, 063809 (2003).
[Crossref]

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176–S183 (2002).
[Crossref]

M. A. M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, “Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects,” Opt. Express 3, 71–80 (1998).
[Crossref]

A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express 1, 21–30 (1997).
[Crossref]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[Crossref]

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675–1690 (1995).
[Crossref]

G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[Crossref]

Geraci, A.

S. Longhi and A. Geraci, “Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[Crossref]

Gigan, S.

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

Glauber, R. J.

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Gollub, J. P.

J. P. Gollub and J. S. Langer, “Pattern formation in nonequilibrium physics,” Rev. Mod. Phys. 71, S396 (1999).
[Crossref]

Gradshteyn, I. S.

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Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
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M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
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C. Lamprecht, M. K. Olsen, P. D. Drummond, and H. Ritsch, “Positive-P and Wigner representations for quantum-optical systems with nonorthogonal modes,” Phys. Rev. A 65, 053813 (2002).
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J. Yin and D. P. Landau, “Phase diagram and critical behavior of the square-lattice Ising model with competing nearest-neighbor and next-nearest-neighbor interactions,” Phys. Rev. E 80, 051117 (2009).
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M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
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S. Kiesewetter, R. Polkinghorne, B. Opanchuk, and P. D. Drummond, “xSPDE: extensible software for stochastic equations,” SoftwareX, to be published.

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M. Santagiustina, E. Hernandez-Garcia, M. San-Miguel, A. J. Scroggie, and G.-L. Oppo, “Polarization patterns and vectorial defects in type-II optical parametric oscillators,” Phys. Rev. E 65, 036610 (2002).
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A. Gatti, L. A. Lugiato, G.-L. Oppo, R. Martin, P. Di Trapani, and A. Berzanskis, “From quantum to classical images,” Opt. Express 1, 21–30 (1997).
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V. D’Auria, A. Chiummo, M. De Laurentis, A. Porzio, S. Solimeno, and M. G. A. Paris, “Tomographic characterization of OPO sources close to threshold,” Opt. Express 13, 948–956 (2005).
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Polkinghorne, R.

S. Kiesewetter, R. Polkinghorne, B. Opanchuk, and P. D. Drummond, “xSPDE: extensible software for stochastic equations,” SoftwareX, to be published.

Porzio, A.

V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
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V. D’Auria, A. Chiummo, M. De Laurentis, A. Porzio, S. Solimeno, and M. G. A. Paris, “Tomographic characterization of OPO sources close to threshold,” Opt. Express 13, 948–956 (2005).
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M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
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M. A. M. Marte, H. Ritsch, K. Petsas, A. Gatti, L. Lugiato, C. Fabre, and D. Leduc, “Spatial patterns in optical parametric oscillators with spherical mirrors: classical and quantum effects,” Opt. Express 3, 71–80 (1998).
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V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
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K. Staliunas, “Spatial and temporal noise spectra of spatially extended systems with order disorder phase transitions,” Int. J. Bifurcation Chaos 11, 2845–2852 (2001).
[Crossref]

K. Staliunas, “Spatial and temporal spectra of noise driven stripe patterns,” Phys. Rev. E 64, 066129 (2001).
[Crossref]

G. J. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[Crossref]

K. Staliunas, G. Slekys, and C. O. Weiss, “Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns,” Phys. Rev. Lett. 79, 2658–2661 (1997).
[Crossref]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[Crossref]

K. Staliunas, “Transverse pattern formation in optical parametric oscillators,” J. Mod. Opt. 42, 1261–1269 (1995).
[Crossref]

K. Staliunas, “Laser Ginzburg-Landau equation and laser hydrodynamics,” Phys. Rev. A 48, 1573–1581 (1993).
[Crossref]

K. Staliunas and V. J. Sanchez-Morcillo, “Transverse patterns in nonlinear optical resonators,” in Springer Tracts Mod. Phys. (Springer, 2003).

Sudarshan, E. C. G.

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

Swain, S.

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

Swift, J.

J. Swift and P. C. Hohenberg, “Hydrodynamic fluctuations at the convective instability,” Phys. Rev. A 15, 319–328 (1977).
[Crossref]

Thouless, D. J.

J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203 (1973).
[Crossref]

Tlidi, M.

G. Kozyreff and M. Tlidi, “Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems,” Chaos 17, 037103 (2007).
[Crossref]

K. Staliunas and M. Tlidi, “Hyperbolic transverse patterns in nonlinear optical resonators,” Phys. Rev. Lett. 94, 133902 (2005).
[Crossref]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[Crossref]

Treps, N.

G. Keller, V. D’Auria, N. Treps, T. Coudreau, J. Laurat, and C. Fabre, “Experimental demonstration of frequency-degenerate bright EPR beams with a self-phase-locked OPO,” Opt. Express 16, 9351–9356 (2008).
[Crossref]

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[Crossref]

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]

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[Crossref]

Vaupel, M.

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

Villar, A. S.

Wagner, H.

N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).
[Crossref]

Walgraef, D.

G. Izús, M. San Miguel, and D. Walgraef, “Polarization coupling and pattern selection in a type-II optical parametric oscillator,” Phys. Rev. E 66, 36228 (2002).
[Crossref]

Walls, D. F.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 28, 211–225 (2010).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
[Crossref]

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2008).

Wegner, F. J.

E. K. Riedel and F. J. Wegner, “Tricritical exponents and scaling fields,” Phys. Rev. Lett. 29, 349–352 (1972).
[Crossref]

Weiss, C. O.

K. Staliunas, G. Slekys, and C. O. Weiss, “Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns,” Phys. Rev. Lett. 79, 2658–2661 (1997).
[Crossref]

Werner, M. J.

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comp. Phys. 132, 312–326 (1997).
[Crossref]

Wiedemann, H.

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[Crossref]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[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]

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]

Yin, J.

J. Yin and D. P. Landau, “Phase diagram and critical behavior of the square-lattice Ising model with competing nearest-neighbor and next-nearest-neighbor interactions,” Phys. Rev. E 80, 051117 (2009).
[Crossref]

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]

B. Yurke, Quantum Squeezing, P. D. Drummond and Z. Ficek, eds. (Springer, 2004).

Zambrini, R.

R. Zambrini, A. Gatti, L. Lugiato, and M. San Miguel, “Polarization quantum properties in a type-II optical parametric oscillator below threshold,” Phys. Rev A 68, 063809 (2003).
[Crossref]

R. Zambrini, S. M. Barnett, P. Colet, and M. San Miguel, “Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q-representation,” Eur. Phys. J. D 22, 461–471 (2003).
[Crossref]

Zappalà, D.

A. Bonanno and D. Zappalà, “Isotropic Lifshitz critical behavior from the functional renormalization group,” Nucl. Phys. B 893, 501–511 (2015).
[Crossref]

Zoller, P.

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

Chaos (1)

G. Kozyreff and M. Tlidi, “Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems,” Chaos 17, 037103 (2007).
[Crossref]

Comput. Phys. Commun. (1)

G. R. Collecutt and P. D. Drummond, “Xmds: eXtensible multi-dimensional simulator,” Comput. Phys. Commun. 142, 219–223 (2001).
[Crossref]

Eur. Phys. J. D (1)

R. Zambrini, S. M. Barnett, P. Colet, and M. San Miguel, “Non-classical behavior in multimode and disordered transverse structures in OPO. Use of the Q-representation,” Eur. Phys. J. D 22, 461–471 (2003).
[Crossref]

Int. J. Bifurcation Chaos (1)

K. Staliunas, “Spatial and temporal noise spectra of spatially extended systems with order disorder phase transitions,” Int. J. Bifurcation Chaos 11, 2845–2852 (2001).
[Crossref]

J. Comp. Phys. (2)

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comp. Phys. 93, 144–170 (1991).
[Crossref]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comp. Phys. 132, 312–326 (1997).
[Crossref]

J. Magn. Mater. (1)

R. M. Hornreich, “The Lifshitz point: Phase diagrams and critical behavior,” J. Magn. Mater. 15, 387–392 (1980).
[Crossref]

J. Mod. Opt. (3)

S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 43, 1569–1575 (1996).

K. Staliunas, “Transverse pattern formation in optical parametric oscillators,” J. Mod. Opt. 42, 1261–1269 (1995).
[Crossref]

G.-L. Oppo, M. Brambilla, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (1994).
[Crossref]

J. Opt. B (2)

S. Feng and O. Pfister, “Stable nondegenerate optical parametric oscillation at degenerate frequencies in Na:KTP,” J. Opt. B 5, 262–267 (2003).

L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176–S183 (2002).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. A (2)

P. D. Drummond, “Fundamentals of higher order stochastic equations,” J. Phys. A 47, 335001 (2014).
[Crossref]

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

J. Phys. C (1)

J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203 (1973).
[Crossref]

Nucl. Phys. B (1)

A. Bonanno and D. Zappalà, “Isotropic Lifshitz critical behavior from the functional renormalization group,” Nucl. Phys. B 893, 501–511 (2015).
[Crossref]

Opt. Acta (2)

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
[Crossref]

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 28, 211–225 (2010).
[Crossref]

Opt. Commun. (2)

S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
[Crossref]

G. J. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rep. (1)

P. C. Hohenberg and A. P. Krekhov, “An introduction to the Ginzburg–Landau theory of phase transitions and nonequilibrium patterns,” Phys. Rep. 572, 1–42 (2015).
[Crossref]

Phys. Rev A (1)

R. Zambrini, A. Gatti, L. Lugiato, and M. San Miguel, “Polarization quantum properties in a type-II optical parametric oscillator below threshold,” Phys. Rev A 68, 063809 (2003).
[Crossref]

Phys. Rev. (2)

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[Crossref]

Phys. Rev. A (23)

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
[Crossref]

C. Lamprecht, M. K. Olsen, P. D. Drummond, and H. Ritsch, “Positive-P and Wigner representations for quantum-optical systems with nonorthogonal modes,” Phys. Rev. A 65, 053813 (2002).
[Crossref]

A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675–1690 (1995).
[Crossref]

A. Gatti, H. Wiedemann, L. A. Lugiato, I. Marzoli, G.-L. Oppo, and S. M. Barnett, “Langevin treatment of quantum fluctuations and optical patterns in optical parametric oscillators below threshold,” Phys. Rev. A 56, 877–897 (1997).
[Crossref]

K. Staliunas, “Laser Ginzburg-Landau equation and laser hydrodynamics,” Phys. Rev. A 48, 1573–1581 (1993).
[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]

K. Dechoum, P. D. Drummond, S. Chaturvedi, and M. D. Reid, “Critical fluctuations and entanglement in the nondegenerate parametric oscillator,” Phys. Rev. A 70, 053807 (2004).
[Crossref]

V. D’Auria, C. de Lisio, A. Porzio, S. Solimeno, J. Anwar, and M. G. A. Paris, “Non-Gaussian states produced by close-to-threshold optical parametric oscillators: Role of classical and quantum fluctuations,” Phys. Rev. A 81, 033846 (2010).
[Crossref]

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[Crossref]

J. Laurat, T. Coudreau, G. Keller, N. Treps, and C. Fabre, “Effects of mode coupling on the generation of quadrature Einstein-Podolsky-Rosen entanglement in a type-II optical parametric oscillator below threshold,” Phys. Rev. A 71, 022313 (2005).
[Crossref]

V. J. Sánchez-Morcillo, E. Roldán, G. J. de Valcárcel, and K. Staliunas, “Generalized complex Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 56, 3237–3244 (1997).
[Crossref]

M. Martinelli, N. Treps, S. Ducci, S. Gigan, A. Maitre, and C. Fabre, “Experimental study of the spatial distribution of quantum correlations in a confocal optical parametric oscillator,” Phys. Rev. A 67, 023808 (2003).
[Crossref]

S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
[Crossref]

J. Swift and P. C. Hohenberg, “Hydrodynamic fluctuations at the convective instability,” Phys. Rev. A 15, 319–328 (1977).
[Crossref]

M. D. Reid and P. D. Drummond, “Correlations in nondegenerate parametric oscillation: Squeezing in the presence of phase diffusion,” Phys. Rev. A 40, 4493–4506 (1989).
[Crossref]

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

M. J. Collett and C. W. Gardiner, “Squeezing of intracavity and traveling-wave light fields produced in parametric amplification,” Phys. Rev. A 30, 1386–1391 (1984).
[Crossref]

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]

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

V. deGiorgio and M. O. Scully, “Analogy between the laser threshold region and a second-order phase transition,” Phys. Rev. A 2, 1170–1177 (1970).
[Crossref]

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (1989).
[Crossref]

S. Longhi and A. Geraci, “Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
[Crossref]

P. D. Drummond and M. D. Reid, “Laser bandwidth effects on squeezing in intracavity parametric oscillation,” Phys. Rev. A 37, 1806–1808 (1988).
[Crossref]

Phys. Rev. B (1)

A. Michelson, “Phase diagrams near the Lifshitz point. I. Uniaxial magnetization,” Phys. Rev. B 16, 577–584 (1977).
[Crossref]

Phys. Rev. E (4)

G. Izús, M. San Miguel, and D. Walgraef, “Polarization coupling and pattern selection in a type-II optical parametric oscillator,” Phys. Rev. E 66, 36228 (2002).
[Crossref]

M. Santagiustina, E. Hernandez-Garcia, M. San-Miguel, A. J. Scroggie, and G.-L. Oppo, “Polarization patterns and vectorial defects in type-II optical parametric oscillators,” Phys. Rev. E 65, 036610 (2002).
[Crossref]

J. Yin and D. P. Landau, “Phase diagram and critical behavior of the square-lattice Ising model with competing nearest-neighbor and next-nearest-neighbor interactions,” Phys. Rev. E 80, 051117 (2009).
[Crossref]

K. Staliunas, “Spatial and temporal spectra of noise driven stripe patterns,” Phys. Rev. E 64, 066129 (2001).
[Crossref]

Phys. Rev. Lett. (14)

K. Staliunas and M. Tlidi, “Hyperbolic transverse patterns in nonlinear optical resonators,” Phys. Rev. Lett. 94, 133902 (2005).
[Crossref]

K. Staliunas, G. Slekys, and C. O. Weiss, “Nonlinear pattern formation in active optical systems: shocks, domains of tilted waves, and cross-roll patterns,” Phys. Rev. Lett. 79, 2658–2661 (1997).
[Crossref]

M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
[Crossref]

E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
[Crossref]

N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models,” Phys. Rev. Lett. 17, 1133–1136 (1966).
[Crossref]

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

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref]

J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref]

R. M. Hornreich, M. Luban, and S. Shtrikman, “Critical behavior at the onset of k⃗-space instability on the λ line,” Phys. Rev. Lett. 35, 1678–1681 (1975).
[Crossref]

E. K. Riedel and F. J. Wegner, “Tricritical exponents and scaling fields,” Phys. Rev. Lett. 29, 349–352 (1972).
[Crossref]

M. Vaupel, A. Maitre, and C. Fabre, “Observation of pattern formation in optical parametric oscillators,” Phys. Rev. Lett. 83, 5278–5281 (1999).
[Crossref]

P. D. Drummond and K. Dechoum, “Universality of quantum critical dynamics in a planar optical parametric oscillator,” Phys. Rev. Lett. 95, 083601 (2005).
[Crossref]

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

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]

Proc. Phys. Math. Soc. Jpn. (1)

K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).

Rev. Mod. Phys. (6)

I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
[Crossref]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

C. Bolman and A. C. Newell, “Natural patterns and wavelets,” Rev. Mod. Phys. 70, 289–301 (1998).
[Crossref]

J. P. Gollub and J. S. Langer, “Pattern formation in nonequilibrium physics,” Rev. Mod. Phys. 71, S396 (1999).
[Crossref]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851–1112 (1993).
[Crossref]

P. C. Hohenberg and B. I. Halperin, “Theory of dynamic critical phenomena,” Rev. Mod. Phys. 49, 435–479 (1977).
[Crossref]

Science (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]

Sov. Phys. J. Exp. Theor. Phys. (1)

V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. quantum systems,” Sov. Phys. J. Exp. Theor. Phys. 34, 610–616 (1972).

Z. Phys. (1)

R. Graham and H. Haken, “Laserlight -First example of a second order phase transition far from thermal equilibrium,” Z. Phys. 237, 31–46 (1970).
[Crossref]

Z. Phys. B (1)

R. M. Hornreich, R. Liebmann, H. G. Schuster, and W. Selke, “Lifshitz points in ising systems,” Z. Phys. B 35, 91–97 (1979).
[Crossref]

Other (15)

E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics, Vol. 10 of Course of Theoretical Physics (Pergamon, 1981).

P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, 1995).

J. Laurat, T. Coudreau, and C. Fabre, Quantum Information with Continuous Variables of Atoms and Light, N. J. Cerf, G. Leuchs, and E. S. Polzik, eds. (World Scientific Publishing, 2007).

C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

B. Yurke, Quantum Squeezing, P. D. Drummond and Z. Ficek, eds. (Springer, 2004).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2008).

P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics (Cambridge University, 2014).

H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, 2002).

H. Haken, Laser Theory (Springer, 1984).

S. Kiesewetter, R. Polkinghorne, B. Opanchuk, and P. D. Drummond, “xSPDE: extensible software for stochastic equations,” SoftwareX, to be published.

K. Staliunas and V. J. Sanchez-Morcillo, “Transverse patterns in nonlinear optical resonators,” in Springer Tracts Mod. Phys. (Springer, 2003).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

E. M. Lifshitz and L. P. Pitaevskii, “Statistical physics, part 2,” in The Course of Theoretical Physics (Pergamon, 1980), Vol. 9.

S.-K. Ma, Modern Theory of Critical Phenomena (W. A. Benjamin, 1976).

P. Kopietz, L. Bartosch, and F. Schtz, Introduction to the Functional Renormalization Group (Springer, 2010).

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Figures (6)

Fig. 1.
Fig. 1. Dimensionless intensity versus dimensionless pump μ in the vicinity of the critical point. The point μ = 1 corresponds to η 1 = η 2 = 0 and η 3 = 1 2 . Here we have used the parameter g = 0.01 . Results were obtained from a simulation with 300 samples on a 50 × 20 × 20 numerical grid of 30000 × 96 × 96 points.
Fig. 2.
Fig. 2. Dimensionless intensity versus detuning Δ . The Lifshitz point corresponds to zero detuning. Here we have used μ = 1 and g = 0.01 . Results were obtained from a simulation with 60 samples on a 200 × 50 × 50 numerical grid of 15000 × 96 × 96 points.
Fig. 3.
Fig. 3. Dimensionless intensity versus detuning Δ and transverse momentum k x . For negative detuning there are fluctuations for lower momentum values. Results were obtained from a simulation with 800 samples on a 100 × 50 × 50 numerical grid of 15000 × 96 × 96 points. Here we have used μ = 1 and g = 0.01 .
Fig. 4.
Fig. 4. Left: Dimensionless intensity versus transverse momentum k for a fixed value of the detuning Δ = 0.5 . Here we have used μ = 1 and g = 0.01 . A ring is formed for lower values of the momentum. The fluctuations (peaks) are not symmetric due to noise. Right: Dimensionless intensity versus transverse momentum k x for a fixed value of the detuning Δ = 0.45 , and k y = 0 . Other parameters are as in Fig. 3.
Fig. 5.
Fig. 5. Growth of non-Gaussian correlations, | X | 2 | X ˜ | 2 versus time τ , starting from X = 0 . Error bars were obtained from comparing a coarse (5000 step) and fine (10000 step) simulation. The sampling error is indicated by the upper and lower solid lines, with a standard deviation of ± 0.00025 . This gives the steady-state correlation result X · X = | X | 2 = 0.2574 ± 0.0003 . Results were obtained from a simulation with 3200 samples on a 10 × 20 × 20 fine numerical grid of 10000 × 48 × 48 points.
Fig. 6.
Fig. 6. Left: Steady-state momentum correlations, | X ( k ) | 2 versus transverse momentum k , starting from X = 0 . Right: Steady-state non-Gaussian correlations in momentum space, Δ ln | X ( k ) | 2 = ln | X ( k ) | 2 ln | X ˜ ( k ) | 2 versus momentum k . Here we consider η 1 = η 2 = 0 and η 3 = 1 2 . Other parameters are as in Fig. 5.

Equations (73)

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H ^ = H ^ free + H ^ int + H ^ pump + H ^ res ,
[ A ^ i ( x , t ) , A ^ j ( x , t ) ] = δ i j δ 2 ( x x ) .
H ^ free = i = 0 2 d 2 x A ^ i [ ω i v i 2 2 ω i 2 ] A ^ i .
H ^ int = i d 2 x [ χ A ^ 0 A ^ 1 A ^ 2 χ * A ^ 0 A ^ 1 A ^ 2 ] .
H ^ pump = i d 2 x [ E * ( x ) e 2 i ω L t A ^ 0 E ( x ) e 2 i ω L t A ^ 0 ] .
H ^ res = i = 0 2 d 2 x [ Γ ^ j A ^ j + Γ ^ j A ^ j ] + H ^ res 0 .
H ^ 0 = j d 2 x ω j 0 A ^ j A ^ j ,
ω 1 0 = ω L + ϵ ω 1 , ω 2 0 = ω L ϵ ω 2 , ω 0 0 = 2 ω L ω 0 .
ρ ^ t = 1 i [ H ^ , ρ ^ ] + i = 0 2 γ i L i [ ρ ^ ] ,
L i [ ρ ^ ] = d 2 x [ 2 A ^ i ρ ^ A ^ i ρ ^ A ^ i A ^ i A ^ i A ^ i ρ ^ ] ,
A 0 t = γ ˜ 0 A 0 + E ( x ) χ * A 1 A 2 + i v 0 2 2 ω 0 2 A 0 , A 1 t = γ ˜ 1 A 1 + χ A 0 A 2 + + i v 1 2 2 ω 1 2 A 1 + χ A 0 ξ 1 , A 2 t = γ ˜ 2 A 2 + χ A 0 A 1 + + i v 2 2 2 ω 2 2 A 2 + χ A 0 ξ 2 .
ξ 1 ( x , t ) ξ 2 ( x , t ) = δ 2 ( x x ) δ ( t t ) , ξ 1 + ( x , t ) ξ 2 + ( x , t ) = δ 2 ( x x ) δ ( t t ) .
A 0 t = γ ˜ 0 A 0 + E ( x ) χ * A 1 A 2 + i v 0 2 2 ω 0 2 A 0 , A i t = γ ˜ i A i + χ A 0 A 3 i * + i v i 2 2 ω i 2 A i .
A 0 = ( E χ * A 1 A 2 ) / γ ˜ 0 , A i = χ A 0 A 3 i * / γ ˜ i .
A 1 A 2 * = A 1 A 2 * | χ A 0 | 2 γ ˜ 1 γ ˜ 2 * .
| E c | 2 = γ ¯ 2 | γ ˜ 0 / χ | 2 .
A 0 = ( E | χ 2 | I 1 A 0 / γ ˜ 2 ) / γ ˜ 0 .
| χ A 0 | 2 = γ ¯ 2 = | γ ˜ 2 χ E | 2 | γ ˜ 0 γ ˜ 2 + | χ 2 | I 1 | 2 .
I 1 = 1 | χ 2 | [ z ± | χ E | 2 + ( z ) 2 | z | 2 ] ,
A 0 = A ¯ 0 E χ * A 1 A 2 γ ˜ 0 .
A i t ( x , t ) = γ ˜ i A i + χ γ ˜ 0 ( E χ * A 1 A 2 ) A 3 i + + i v i 2 2 ω i 2 A i + χ A ¯ 0 ξ i ( x , t ) .
1 t 0 α i τ = γ ˜ α i + χ A ¯ 0 α 3 i + + i v 2 2 ω x 0 2 r 2 α i + x 0 χ A ¯ 0 ξ i .
μ ˜ = χ E γ 0 γ = μ e i ϕ ,
g 2 = | χ | 2 4 γ 0 γ x 0 2 .
x 0 2 = v 2 2 γ g ω .
g = ( | χ | 2 ω 2 γ 0 v 2 ) 2 / 3 .
ζ 1 ( r , τ ) ζ 2 ( r , τ ) = δ 2 ( r r ) δ ( τ τ ) , ζ 1 + ( r , τ ) ζ 2 + ( r , τ ) = δ 2 ( r r ) δ ( τ τ ) .
μ ( α⃗ ) = μ 4 g 2 α 1 α 2 , μ + ( α⃗ ) = μ 4 g 2 α 1 + α 2 + ,
g α i τ = μ ( α⃗ ) α 3 i + ( 1 + i Δ ) α i + g [ i r 2 α i + μ ( α⃗ ) ζ i ( r , τ ) ] ,
g τ ( α 1 α 2 + ) = ( ( 1 + i Δ ) μ μ ( 1 i Δ ) ) ( α 1 α 2 + ) ,
u⃗ ± = ( μ i Δ ± Δ 2 + μ 2 ) .
X = g ( α 1 + α 2 + ) , X + = g ( α 2 + α 1 + ) , Y = 1 i ( α 1 α 2 + ) , Y + = 1 i ( α 2 α 1 + ) .
X τ = μ g X + D + Y ( X 2 + g Y 2 ) X + + ζ + , X + τ = μ g X + + D + Y + ( X + 2 + g Y + 2 ) X + ζ + * , g Y τ = μ + Y + D X g ( X 2 + g Y 2 ) Y + i g ζ , g Y + τ = μ + Y + + D X + g ( X + 2 + g Y + 2 ) Y i g ζ * .
D ± = ± Δ g r 2 μ ± = μ ± 1 ,
ζ ± = ζ 1 ± ζ 2 + = ( ζ 2 ± ζ 1 + ) * ,
X τ = ( 1 μ g ) X + ( Δ g r 2 ) Y X 2 X + + ζ + , X + τ = ( 1 μ g ) X + + ( Δ g r 2 ) Y + X + 2 X + ζ + * , 0 = ( 1 + μ ) Y + r 2 X , 0 = ( 1 + μ ) Y + + r 2 X + .
Y ( + ) = 2 X ( + ) 1 + μ ,
X τ = D ˜ X X 2 X + + ζ + , X + τ = D ˜ X ( X + ) 2 X + ζ + * .
D ˜ D ˜ r = η 1 + η 2 r 2 η 3 r 4 ,
η 1 = ( 1 μ g ) , η 2 = Δ ( 1 + μ ) g , η 3 = 1 1 + μ .
X τ = D ˜ X X | X | 2 + ζ + .
X 1 = X + X * 2 , X 2 = X X * 2 i .
X = X 1 + i X 2 , X * = X 1 i X 2 .
X = ( X 1 X 2 ) , ζ ˜ = 1 2 ( ζ + + ζ + * i ( ζ + ζ + * ) ) .
X τ = D ˜ X | X | 2 X + ζ ˜ ,
ζ ˜ i ( r , τ ) ζ ˜ j ( r , τ ) = δ i j δ ( r r ) δ ( τ τ ) .
P τ ( X , τ ) = i δ δ X i [ ( | X | 2 D ˜ ) X i + 1 2 δ δ X i ] P ( X , τ ) ,
P ( X ) = N exp [ d 2 x ( η 1 X · X + 1 2 ( X · X ) 2 + η 2 X · X + η 3 2 X · 2 X ) ] .
X i ( r , τ ) X j ( r , τ ) τ = D ˜ r X i ( r , τ ) X j ( r , τ ) X i ( r , τ ) | X ( r , τ ) | 2 X j ( r , τ ) .
G 1 j τ = D ˜ r G 1 j G 1 j ( 3 X 1 2 + X 2 2 ) .
G 1 j τ = [ D ˜ r 2 ( X 1 2 + X 2 2 ) ] G 1 j .
X ˜ ( r , τ ) τ = D ˜ X ˜ ( r , τ ) 2 X ˜ · X ˜ X ˜ ( r , τ ) + ζ ˜ ( r , τ ) .
X ˜ ( r , τ ) τ = ( η 1 + η 2 r 2 η 3 r 4 ) X ˜ ( r , τ ) + ζ ˜ ( r , τ ) .
X ˜ ( r , t ) = 1 2 π e i k⃗ · r X ˜ ( k⃗ , t ) d 2 k⃗ ,
X ˜ ( k , τ ) τ = ( η 1 + η 2 k 2 + η 3 k 4 ) X ˜ ( k , τ ) + ζ ˜ ( k , τ ) .
X ˜ ( k , τ ) = τ ζ ˜ ( k , τ ) e ( η 1 + k 4 2 ) ( τ τ ) d τ .
X ˜ · X ˜ = 1 4 π 2 0 2 π k d k η 1 + k 4 2 .
X ˜ · X ˜ = 1 4 2 ( η 1 + 2 X ˜ · X ˜ ) .
X ˜ · X ˜ = | X ˜ | 2 = 0.25 .
X ˜ ( k ) [ X ˜ ( k ) ] * = 1 2 δ ( k + k ) η 1 + η 2 k 2 + η 3 k 4 .
X ˜ ( r ) X ˜ ( r ) = 1 8 π 2 d 2 k e i k · ( r r ) η 1 + η 2 k 2 + η 3 k 4 .
X ˜ ( r ) X ˜ ( r ) = 1 4 π 0 d k k J 0 ( k | r r | ) η 1 + η 2 k 2 + η 3 k 4 ,
X ˜ ( r ) X ˜ ( r ) = η 3 η 1 1 4 π η 3 k e i ( ( η 1 η 3 ) 1 / 4 | r r | ) ,
X ˜ ( r ) X ˜ ( r ) | r r | 0.5 .
X ˜ τ = D ˜ X ˜ 2 X ˜ | X ˜ | 2 2 X ˜ * X ˜ 2 + ζ + ,
Δ X τ = D ˜ X X | X | 2 + ζ + X ˜ τ .
A ^ 0 t = γ ˜ 0 A ^ 0 + E ( x ) χ * A ^ 1 A ^ 2 + i v 0 2 2 ω 0 2 A ^ 0 + 2 γ 0 A ^ 0 in , A ^ 1 t = γ ˜ 1 A ^ 1 + χ A ^ 0 A ^ 2 + i v 1 2 2 ω 1 2 A ^ 1 + 2 γ 1 A ^ 1 in , A ^ 2 t = γ ˜ 2 A ^ 2 + χ A ^ 0 A ^ 1 + i v 2 2 2 ω 2 2 A ^ 2 + 2 γ 2 A ^ 2 in .
A ^ i in ( x , t ) A ^ j in ( x , t ) = ( n ¯ i th + 1 ) δ i j δ ( x x ) , A ^ i in ( x , t ) A ^ j in ( x , t ) = n ¯ i th δ i j δ ( x x ) .
a ^ i ( k ) | α ˜ = α ˜ i ( k ) | α ˜ .
ρ ^ = d 6 M α ˜ d 6 M α ˜ + | α ˜ α ˜ + * | α ˜ + * | α ˜ P ( α ˜ , α ˜ + ) .
A i ( x , t ) = 1 L k e i k · x α ˜ i ( k , t ) ,
ξ 1 , 2 ( x , t ) = [ ξ x ( x , t ) ± i ξ y ( x , t ) ] / 2 , ξ 1 , 2 + ( x , t ) = [ ξ x + ( x , t ) ± i ξ y + ( x , t ) ] / 2 .
ξ 1 ( x , t ) = ξ 2 * ( x , t ) , and ξ 1 + ( x , t ) = ( ξ 2 + ( x , t ) ) * .

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