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

1. INTRODUCTION

Nonequilibrium pattern formation occurs in many physical systems, giving rise to the emergence of order on macroscopic scales [1,2]. The theory of hydrodynamics is a paradigm for understanding these phenomena, and is applicable to many branches of physics, chemistry, biology, astrophysics, and other sciences [3,4]. This is often applied to many body systems subject to nonlinear coupling in a dissipative environment with external fluxes. In physics, one of the most studied hydrodynamic effects is the theory of fluid flows near the Rayleigh–Bénard instability [1,2]. The simplest nontrivial model for fluid convection that displays pattern formation due to convective instabilities was derived by Swift and Hohenberg [5], where nonlinear coupling of fluctuations was included to demonstrate the failure of mean field theory for critical exponents.

Here, we treat pattern formation and universality in a paradigmatic nonequilibrium quantum system: the nondegenerate parametric oscillator (OPO). Theories of a similar nature have been applied to nonequilibrium spatially extended structures in lasers and other related systems, with an emphasis on universal behavior of phase transitions, pattern formation, and self-organization [6,7]. Yet downconversion in a nondegenerate parametric system can display new possibilities not found in these simpler cases. In particular, one can have entanglement and Einstein–Podolsky–Rosen (EPR) paradoxes [811].

In the present paper, we investigate a new type of critical point phase transition in this nonequilibrium quantum system in order to understand the universality class. In a type II OPO, there are two downconverted fields with orthogonal polarization. Hence, the order parameter is a complex or vector field in two dimensions. The two components of this vector field are associated with the polarization degrees of freedom of the downconverted radiation field. This is a quantum system driven to a phase transition far from thermal equilibrium. It is also known to display strong quantum entanglement and EPR correlations [810] in the case where there are two correlated output modes. We wish to understand the behavior of this phase transition using a first-principles analysis [12,13] of the relevant master equation.

Experimentally, the OPO is now a mature technology with both commercial and fundamental applications. Following initial theoretical predictions [1416], the quantum limited type I OPO was investigated experimentally by Wu et al. [17], demonstrating quantum squeezing. Later the type II case, in a triply resonant cavity, was used to experimentally demonstrate continuous variable EPR correlations [11], also originally predicted theoretically [12,13]. These initial experimental investigations were in few-mode devices. Both below- and above-threshold experiments have been carried out, as close as ±1% of the critical point [1823], confirming predictions of thresholds and conversion efficiency. Operation at the critical point results in low-frequency critical fluctuations and non-Gaussian behavior [24,25].

Spatially extended pattern formation in a degenerate or type I OPO has also been analyzed previously [26]. It is related to the Lifshitz phase transition [27]. This is a model used to describe the phase transition to a modulated magnetic phase [2830]. In this simplest case one has a two-dimensional, planar system with a scalar-order parameter [31], which has known universality properties. A Swift–Hohenberg equation was derived for spatially extended nondegenerate type I OPOs with flat end mirrors [32], but ignoring fluctuations. OPO experiments have been recently extended to these types of multimode devices [3335], with type II as well as type I parametric downconversion. The experimental situation is that while multimode experiments have mostly used confocal mirrors, the first type II multimode planar mirror experiments have been carried out [35].

Studies of the Swift–Hohenberg equation in the neighborhood of the critical point, the Lifshitz point, as well as pattern formation for lasers have been discussed [36]. The Lifshitz point is similar to the tricritical point [37]. Tricritical points occur in different physical systems. They correspond to a point in a phase diagram where two lines of ordinary critical points meet and terminate [3840]. For critical points and tricritical points there are two critical dimensions. The upper critical dimension refers to the one above which the critical exponents have classical values [37]. The lower critical dimension refers to the smallest dimension for which there is a true phase transition, due to increasing fluctuations as the dimension is decreased.

Systems like these can entangle large numbers of modes, with quantum entanglement increasing as threshold is approached. Currently, experiments near the critical point are sensitive to classical fluctuations [24,25] and heating effects [35]. With improved stabilization methods, we expect that these technical problems can be overcome. However, noise due to quantum fluctuation effects will remain.

Theoretically, the usual approach for two-dimensional nonequilibrium problems is the Landau–Ginzburg (LG) equation [41,42], which was first used for understanding superconductivity near threshold. Normally, the LG equation is derived using approximations like adiabatic elimination, and is taken as an effective equation for the physical system. There is now an increased interest in extended spatial and multimode structures [33,34,4346] in the quantum optical parametric oscillator (OPO) [17,47,48], due to availability of multiple transverse mode cavities [49] and quantum imaging control [50]. It is important in these cases to understand how quantum noise enters the dynamical equations [51].

We show that planar type II downconversion creates a nonequilibrium system with a similar general type of symmetry to the Berezinskii [52] and Kosterlitz–Thouless (BKT) models [53], but with an isotropic Lifshitz point, first studied in magnetic systems by Hornreich et al. [37]. Physically, this is not a classical fluid or magnetic system, but rather is a quantum system, driven into a nonequilibrium critical point [54,55] far from thermal equilibrium. The nondegenerate planar OPO is fundamentally different to the usual BKT model, having a quartic rather than quadratic momentum dependence in the linear response function. This places it in a similar category to the Swift–Hohenberg model and next-nearest neighbor lattice models. Our model can also display strong EPR entanglement and other nonclassical properties in addition to the Lifshitz point behavior. The parametric model therefore provides a novel path to the investigation of unusual classical and quantum noise effects.

Phase transitions with this general type of symmetry are continuous yet break no symmetries. There is also no ferromagnetism in this system, as this is prohibited by the Mermin–Wagner theorem [56]. Berezinskii predicted a new type of phase with correlations that decay slowly with distance with a power law. The phase transition for a ferromagnetic phase is prevented by the appearance of vortex and antivortex pairs. Many quantum phase transitions in two dimensions belong to this class. The continuous version of the XY or Ising models are often used to model systems that possess order parameters with a symmetry of this type, e.g., superfluid helium, liquid crystals, two-dimensional Bose–Einstein condensate (BEC), and others. While having the same order parameter and space dimension as this case, we show that the type II OPO has a different universality class.

Our approach is to start from the usual master equation model that describes a type II OPO with transverse modes. Here we consider that the OPO is nondegenerate in polarization. We then map the coupled Heisenberg equations into the positive P-representation [57] of the density matrix. This choice is made because the positive P-representation allows us to exactly map the density matrix evolution into an equivalent stochastic equation.

We organize this paper in the following way. First we describe the Hamiltonian model and derive the equation of motion for this open system. Next, we use the positive P-representation for mapping these equations into a Langevin type, which can either be treated numerically or via analytic approximations. We will mostly describe the unsqueezed quadratures of the field, which have a large similarity with the corresponding magnetic system. At this point we can recognize the universality class. We give both a Gaussian approximation to the correlation functions of the unsqueezed quadratures, and high-precision numerical simulations of the non-Gaussian corrections. We show that, even though the system is far below its upper critical dimension, the non-Gaussian character of the fluctuations is relatively small, and the intensity fluctuations are nearly factorizable.

In a following paper we will focus our attention on the squeezed field quadratures, to give a spatial map of quantum squeezing and EPR entanglement.

2. MODEL

The system of interest comprises an optically driven planar Fabry–Perot cavity or interferometer with a nonlinear medium that possess a parametric nonlinearity. The nonlinear crystal is cut to give a type II phase matching that couples a pump field to two downconverted fields having an orthogonal polarization. The cavity is pumped with a spatially extended coherent light with frequency ω0, with a transverse spatial profile. In the simplest case we consider this pump to be a plane wave.

A. Quantum Hamiltonian

The outgoing downconverted light, amplified inside the cavity, develops structures and patterns due to diffraction, nonlinear coupling, and detuning between the wavelength of the downconverted field and the cavity size. This depends on the modal decomposition of this cavity. The mirrors have parameters that can be controlled by experimentalists. The tunable parameters include the reflection coefficient for each mode, and the cavity detunings for each mode.

Our model for this system is similar to many earlier treatments of driven nonlinear optical cavities [12,14,15,4346,5863]. It includes a linear coupling between the external electromagnetic field modes and the internal cavity modes, owing to a partially transmitting mirror. The cavity and mirror parameters determine both the coupling to the driving field and the decay rate of the cavity or interferometer. We note that for a low-Q device, it is important to use nonorthogonal quasi-modes [59]. Here, we assume the opposite case of a thin, high-Q extended planar cavity, so that the external modes are simply plane-wave modes.

The quantum Hamiltonian in the interaction picture has four main terms that can be summarized by the following expression,

H^=H^free+H^int+H^pump+H^res,
where there are three fundamental Bose fields, A^i(x,t) for i=0,2. The boson fields are the transverse internal modes of the planar cavity, which is assumed to have a single longitudinal mode in the direction normal to the mirrors. We note that this planar polariton field model is also used in the theory of a polaritonic BEC [64]. The boson fields have two orthogonal polarizations, i=1,2, and obey the usual equal time commutation relation:
[A^i(x,t),A^j(x,t)]=δijδ2(xx).
Here δ2(xx) is a Dirac delta function in the two-dimensional transverse plane of x=(x,y). The cavity fields are defined in terms of polaritonic annihilation and creation operators as A^i(x,t)=keik·xa^i(k,t)/L, where a^i(k,t) represents the annihilation operator for a free polariton mode with transverse momentum k, and the summation is over a set of M discrete modes with periodic boundary conditions, on a large area L2.

The free-evolution Hamiltonian that accounts for diffraction inside the planar cavity is

H^free=i=02d2xA^i[ωivi22ωi2]A^i.
This Hamiltonian describes a planar cavity with intracavity resonant frequencies ωi and group velocities vi for the three field envelopes. The resonant frequencies have the relation ω0ω1+ω2, where ω0 is the fundamental mode, and ω1 and ω2 stand for the downconverted light. The case of spherical mirrors can also be analyzed in a similar way [49], but is not treated here. We suppress the field space–time arguments to obtain more compact expressions in the integrals. The two-dimensional Laplacian is, as usual, 2=2/x2+2/y2. A one-dimensional system can be treated by simply dropping one of the dimensions.

The interaction Hamiltonian representing the coupled modes inside a crystal with χ(2) nonlinearity is given by [65]

H^int=id2x[χA^0A^1A^2χ*A^0A^1A^2].
This term may represent a photon of frequency ω0 converted in two photons of distinct frequencies ω1 and ω2, or photons with orthogonal polarizations, or both.

From now on, for definiteness, labels 1 and 2 stand for polarizations, and we shall focus on the degenerate frequency case with nondegenerate polarization. For dimensional reasons, χχ(2)/, where χ(2) is the Bloembergen nonlinear polarizability coefficient and is the intracavity longitudinal mirror spacing.

Following standard input–output theory derivations [1416,6063], the Hamiltonian term associated with the input laser pumping in a rotating frame at frequency ωL is

H^pump=id2x[E*(x)e2iωLtA^0E(x)e2iωLtA^0].
While it is possible to choose any shape carrying spatial structure for the input pump, here for simplicity we will assume a plane-wave input. The reservoir Hamiltonian is assumed to have the structure
H^res=i=02d2x[Γ^jA^j+Γ^jA^j]+H^res0.
Hence, there are local coupling terms to independent external free-field reservoirs for each polarization and each spatial mode. The external fields are described by a free Hamiltonian H^res0.

B. Master Equation

The nonunitary evolution of the system comes from the coupling between the cavity modes and the output modes. This can be treated as a quantum Markovian process that simulates a bath interaction. We carry out this calculation in a type of interaction picture so that the interaction picture operators evolve according to a reference Hamiltonian H^0, given by

H^0=jd2xωj0A^jA^j,
where the reference frequencies ω10 are chosen so that
ω10=ωL+ϵω1,ω20=ωLϵω2,ω00=2ωLω0.
We note that while the choice of ωL is determined by the pump frequency, the choice of ϵ is arbitrary, as long as the Markovian approximation is still valid. Our choice of reference Hamiltonian is not determined by the intracavity frequencies ωi, which leads to detuning terms occurring in the resulting equations of motion. This allows us to have some freedom of choice in defining the detuning parameters. The choice will be made definite in the following sections. Following standard techniques for deriving the master equations [58,60,66], we can write the master equation for the density operator in the generalized Lindblad form,
ρ^t=1i[H^,ρ^]+i=02γiLi[ρ^],
where the dissipative Liouville superoperator,
Li[ρ^]=d2x[2A^iρ^A^iρ^A^iA^iA^iA^iρ^],
describes the output coupling of the i-th intracavity mode with the external bath.

We note that, although we focus on the master equation approach here, it is sometimes useful to write the time evolution in a complementary quantum Langevin formulation. This formulation is given in Appendix A.

3. STOCHASTIC EQUATIONS IN THE POSITIVE P-REPRESENTATION

As nonlinear operator equations are not generally soluble, it is more manageable to map the operator equations into c-number form. In this approach, a master equation is transformed into a positive-definite Fokker–Planck equation using operator identities that map the operator terms in the master equation into differential operators [60,63,66]. To do this we have to use a phase-space representation of the master equation.

Phase-space representations in a classical phase space do not give a positive-definite equation. An example is the Wigner representation [67]. While this is exact, it is not able to be mapped into Langevin equations, unless one either truncates or uses higher-order noise [68]. One can also linearize the Hamiltonian and obtain an approximate Wigner diffusion [54,55]. Another approach is the Husimi Q-function [69], where, in order to obtain a positive Fokker–Planck equation, an unphysical constraint on the phase-space trajectories has to be used [70].

Here we wish to have the ability to treat nonequilibrium structures without restrictions. For this purpose, the most useful representation is the positive P-representation [57], which is an extension of the Glauber–Sudarshan P-representation [71,72] into a phase space of double the classical dimensions. Unlike the P-representation, which is singular for nonclassical states, the positive P-representation is well defined, positive, and nonsingular for any quantum state. This approach allows us to map the density matrix equation into a Fokker–Planck equation on a nonclassical phase space. In the positive P-representation stochastic averages give normally ordered quantum expectation values. A brief description is given in Appendix B.

The stochastic field partial differential equations are given by an extension of our earlier work [31]:

A0t=γ˜0A0+E(x)χ*A1A2+iv022ω02A0,A1t=γ˜1A1+χA0A2++iv122ω12A1+χA0ξ1,A2t=γ˜2A2+χA0A1++iv222ω22A2+χA0ξ2.
The three equations that correspond to the Hermitian conjugate fields, Ai+, are obtained by conjugating the constant terms, and replacing stochastic and noise fields so that AiAi+ and ξiξi+, where ξi and ξi+ are independent Gaussian complex noises. These are equivalent in the mean to the conjugated Heisenberg equations, but are independent c-number equations. They are not conjugate in every realization. We also note that Ai and Ai+ are six independent, complex c-number fields.

The stochastic fields ξk that describe the quantum noise are complex and Gaussian, whose nonvanishing correlations are

ξ1(x,t)ξ2(x,t)=δ2(xx)δ(tt),ξ1+(x,t)ξ2+(x,t)=δ2(xx)δ(tt).
This means that ξk(x,t), ξk+(x,t) represent four independent, delta-correlated, complex c-number Gaussian stochastic fields with zero mean. They are completely characterized by the specified correlations. From the stochastic Eqs. (10), it is clear that the amplitude of the stochastic fluctuations that act on the converted modes depend on the pump field dynamics. A brief discussion of the noises is given in Appendix B.

A. Critical Driving Field

As a first investigation, we treat the classical approximation, which has also been analyzed in some earlier work [32,4346,7381]. Here one assumes that all noise is negligible, so that Ai+=Ai*, which gives equations in the form

A0t=γ˜0A0+E(x)χ*A1A2+iv022ω02A0,Ait=γ˜iAi+χA0A3i*+ivi22ωi2Ai.
The phases of E and χ are essentially arbitrary, as they depend on the phase definition for the field amplitudes Ai, which in turn depend on arbitrary mode phases. We will use this freedom later on to simplify the equations. If in addition, we assume that the input is a plane wave, and we ignore possible spatial instabilities, diffraction can be neglected as well. A steady-state result involves setting the time derivatives to zero, so
A0=(Eχ*A1A2)/γ˜0,Ai=χA0A3i*/γ˜i.
Hence, the defining equation for a steady state is
A1A2*=A1A2*|χA0|2γ˜1γ˜2*.
We can always choose the interaction picture detuning ϵ so that γ˜1γ˜2*=γ¯2 is real, otherwise the above threshold solutions will be oscillatory rather than stable. There are two types of steady-state solution. Either A1A2*=0, or else |χA0|=γ¯. The first is called a below-threshold solution, the second an above-threshold solution. There is a driving field where both the solutions coincide, at a critical pump intensity of
|Ec|2=γ¯2|γ˜0/χ|2.
More generally, suppose we are in the above-threshold regime. Using Eq. (13), and defining A1*A1=I1, one obtains
A0=(E|χ2|I1A0/γ˜2)/γ˜0.
On rearranging the equation, and using the above-threshold solution |χA0|=γ¯, this result becomes
|χA0|2=γ¯2=|γ˜2χE|2|γ˜0γ˜2+|χ2|I1|2.
The roots of the resulting quadratic are
I1=1|χ2|[z±|χE|2+(z)2|z|2],
where z=z+iz=γ˜0γ˜2.

Solutions with negative intensities are unphysical. There is a positive, above-threshold solution if |χE|>|z|, which gives an identical critical field to Eq. (15). For |E|>Ec there is a transfer of energy from the pump to the signal and idler modes, which develop a finite mean intensity. It is the vicinity and just above this critical point that is the main regime of interest in this paper. We note that above threshold, further instabilities exist, including limit cycles and spatial pattern formation [75].

4. ADIABATIC ELIMINATION OF THE PUMP MODE

We now return to the full quantum behavior given by the stochastic equations obtained above. One limit that has an especially simple behavior is found in the case of a rapidly decaying pump mode. We can treat this by means of an adiabatic elimination procedure. Assuming that γ˜0γ˜1γ˜2, and that E is spatially uniform (that is, we are neglecting pump diffraction), we can perform an adiabatic elimination by using the stationary solution for the pump mode, so that

A0=A¯0Eχ*A1A2γ˜0.

A. Signal and Idler Equations

The resulting equations for the downconverted modes—often called the signal and idler equations—are, for i=1,2,

Ait(x,t)=γ˜iAi+χγ˜0(Eχ*A1A2)A3i++ivi22ωi2Ai+χA¯0ξi(x,t).
We see that the main effect of detuning the pump is to reduce the effective intracavity pump intensity. So far, we have treated the case of general frequencies and group velocities. An important special case is obtained when the two downconverted frequencies are equal. In this case, the modes are still nondegenerate, as they can have different polarizations. From now on, we consider this special case for simplicity, so we take γ˜1=γ˜2=γ˜, v1=v2=v, and ω1=ω2=ω. This implies that Δ1=Δ2=Δ. We will also assume that Δ0=0, i.e., that the pump is on-resonance with the cavity, even when the downconverted fields may be off-resonant from their cavity resonance frequencies.

Although these assumptions simplify the algebra, they are not essential for our main conclusions, which mostly rely simply on the fact that we now have a two-dimensional order parameter rather than a one-dimensional order parameter as found in the degenerate case. We should note that a nonzero pump detuning can excite another mode different from ω0. This could give rise to nonlinear phenomena like bistabilities, nonlinear resonances, and subcritical bifurcation, especially for the case of large detuning [32]. In the case where the diffraction terms are different, there will be a new term which is proportional to the difference of the two diffraction terms, as has been studied elsewhere [32]. This term will be present in the equations even for the case of zero detuning. Since we are interested in the universal behavior of the system we do not include these cases, but we point out that they can give rise to nonlinear behavior.

B. Dimensionless Form

Equation (20) will now be transformed into a dimensionless form, which allows comparisons with other types of phase transitions. First, we define the dimensionless variables τ=t/t0, r=x/x0, with a scaled Laplacian r2=2/r12+2/r22. It is useful to also define a dimensionless field αi=x0Ai as well. This has an intuitive interpretation as the coherent amplitude in real space, defined relative to a physical area of x02. After this transformation, one obtains

1t0αiτ=γ˜αi+χA¯0α3i++iv22ωx02r2αi+x0χA¯0ξi.
We define γ˜=γ(1+iΔ) and introduce the dimensionless pump amplitude,
μ˜=χEγ0γ=μeiϕ,
where μ is real and positive. Since the phase of the driving field E is arbitrary, and the equations are invariant under phase changes of E, we will choose ϕ=0 with no loss of generality. We also define the time scale t0=1/gγ as the scaling time for critical slowing down, where
g2=|χ|24γ0γx02.
The length scale x0 is now chosen so that
x02=v22γgω.
Combining Eqs. (23) and (24), one can write the dimensionless coupling g in the form
g=(|χ|2ω2γ0v2)2/3.
Similarly, we introduce dimensionless complex noises ζi=ξix0t0, so that
ζ1(r,τ)ζ2(r,τ)=δ2(rr)δ(ττ),ζ1+(r,τ)ζ2+(r,τ)=δ2(rr)δ(ττ).
Finally, defining the driving field saturation factor as
μ(α⃗)=μ4g2α1α2,μ+(α⃗)=μ4g2α1+α2+,
we obtain the following dimensionless form of the scaled equations for i=1,2,
gαiτ=μ(α⃗)α3i+(1+iΔ)αi+g[ir2αi+μ(α⃗)ζi(r,τ)],
together with a conjugate equation for αi+. Although this is true in general, we are mostly interested here in the regime of g1, which allows us to make an expansion in powers of the coupling. For small g, the classical approximation gives the leading order term, while the diffraction and noise terms provide the next order in an expansion in g.

5. STABILITY PROPERTIES AND QUADRATURE EQUATIONS

We now wish to transform these equations into quadrature equations that are simpler to investigate. There are very different stability properties for the orthogonal quadratures near the critical point. To investigate this, as a first approximation, we will ignore noise and nonlinear terms of order g and smaller. The stability of the equations near αi=0, to leading order in g, is

gτ(α1α2+)=((1+iΔ)μμ(1iΔ))(α1α2+),
which has eigenvalues λ±=1±Δ2+μ2, and with a similar equation coupling α2 and α1+. This gives an unstable eigenvalue, leading classically to growth of the signal and idler terms if μ2Δ2>1, as expected from the analysis in the previous section. The resulting eigenvectors, u⃗±, are
u⃗±=(μiΔ±Δ2+μ2).
There is clearly a line of critical points where the stability has a continuous change at μ2Δ2=1. The tricritical point then occurs when μ=1, Δ=0, as we will show in later sections.

A. Quadrature Field Variables

To understand the behavior of Eq. (28) in the neighborhood of the critical point, we define complex, dimensionless scaled quadrature fields that are proportional to the critical eigenvectors, as follows [31,54,55]:

X=g(α1+α2+),X+=g(α2+α1+),Y=1i(α1α2+),Y+=1i(α2α1+).

Next, we consider the case μ1g2|α1α2|, to simplify the noise term. As this is both relatively small and nearly constant in the neighborhood of the critical point, the resulting terms are of higher order in g than the leading terms we wish to include. The resulting equations for these quadratures in the positive P-representation near the critical point are

Xτ=μgX+D+Y(X2+gY2)X++ζ+,X+τ=μgX++D+Y+(X+2+gY+2)X+ζ+*,gYτ=μ+Y+DXg(X2+gY2)Y+igζ,gY+τ=μ+Y++DX+g(X+2+gY+2)Yigζ*.
Here we have defined a modified Laplacian and driving term as
D±=±Δgr2μ±=μ±1,
and new Gaussian noise terms according to
ζ±=ζ1±ζ2+=(ζ2±ζ1+)*,
where we have used the result of Eq. (B5). This shows that the noise terms driving the X, X+ fields are conjugate, while those driving the Y, Y+ fields change sign on conjugation, so that Y, Y+ will not remain conjugate during time evolution.

At this stage, we can make the following remarks. The stochastic quadrature fields X,X+ are both complex fields, so they have four degrees of freedom between them, and similarly for Y,Y+. They have a correspondence with non-Hermitian operator fields X^,X^ and Y^,Y^. In general, Y,Y+ are not complex conjugate except in the mean, and neither are X,X+, since they are driven by the Y,Y+ fields. However, as we will show, this picture simplifies when one considers an expansion near the critical point.

B. Critical Point Adiabatic Elimination

We can now perform a second type of adiabatic elimination, which is valid in the neighborhood of the critical point. This takes into account the fact that the fluctuations in the X quadrature become very slow near threshold, while the Y quadrature still responds on the fast relative time scale 1/γ. Formally, we can drop terms of O(g) where g1, and approximate Eq. (32) as follows [31],

Xτ=(1μg)X+(Δgr2)YX2X++ζ+,X+τ=(1μg)X++(Δgr2)Y+X+2X+ζ+*,0=(1+μ)Y+r2X,0=(1+μ)Y++r2X+.
Here we have considered that the noise term of the quadrature variables Y, Y+ has been neglected since it scales as g and we are considering the limit g1. If this limit is not taken, the noise term would appear in the above equations. To lowest order in g, we can eliminate the fast or noncritical quadrature Y,Y+ variables by setting
Y(+)=2X(+)1+μ,
which gives the result that
Xτ=D˜XX2X++ζ+,X+τ=D˜X(X+)2X+ζ+*.
Here we have introduced a linear differential operator that describes the linear gain, loss, and diffraction terms,
D˜D˜r=η1+η2r2η3r4,
where we have defined the following parameters:
η1=(1μg),η2=Δ(1+μ)g,η3=11+μ.
We note that since the nonconjugate variables Y,Y+ do not appear in these equations, it follows that variables X,X+ will remain conjugate if they are conjugate initially, as, for example, in an initial thermal or coherent state. Hence, to leading order, we can set X+=X* near the critical point, and write one complex equation for the critical quadratures, which is valid near threshold:
Xτ=D˜XX|X|2+ζ+.
While this equation is similar to a Landau–Ginzburg equation for a complex order parameter [82], it is not identical, owing to the presence of the fourth-order Laplacian term in D˜. In the next section we will show that the equations are similar to a vector Swift–Hohenberg equation [3,4].

C. Vector Swift–Hohenberg Equations

We will now show that these near-threshold equations are actually coupled or vector Swift–Hohenberg equations [3,4] that represent the leading-order dynamics near threshold of the downconverted modes with the same frequency but orthogonal polarization. To demonstrate this, it is convenient to make the following change of variables:

X1=X+X*2,X2=XX*2i.
After this change of variables it is possible to write Eq. (40) as a single vector equation, as we now show. On inverting this equation we get
X=X1+iX2,X*=X1iX2.
Next we define two real vectors X and ζ˜ as
X=(X1X2),ζ˜=12(ζ++ζ+*i(ζ+ζ+*)).
Using the above expressions we can write Eq. (40) as a single vector equation of the form
Xτ=D˜X|X|2X+ζ˜,
where X is a two real component vector whose elements are X1 and X2, and ζ˜ is also a two real component Gaussian noise vector, with correlations given by
ζ˜i(r,τ)ζ˜j(r,τ)=δijδ(rr)δ(ττ).
In Fig. 1 we plot the modulus squared of the two-dimensional order parameter X versus the dimensionless pump parameter μ, averaged over a transverse area of dimensionless size 20×20.

 figure: Fig. 1.

Fig. 1. Dimensionless intensity versus dimensionless pump μ in the vicinity of the critical point. The point μ=1 corresponds to η1=η2=0 and η3=12. 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.

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Direct simulations were employed, using both a central partial difference algorithm in the interaction picture [83,84] and a fourth-order interaction picture Runge–Kutta method. Two public domain software packages were used and checked against each other [85,86]. This figure is obtained by solving the stochastic equations given in Eq. (43) at μ=0.9, equilibrating for t=50 units then scanning the driving field μ adiabatically (using μ=0.9+0.004t) through the critical point until μ=1.1. There is a rate-independent critical region at the threshold value of μ=1, where the transition is smooth rather than discontinuous, as it would be classically.

In the case of a one-dimensional, real order parameter, this equation was first derived by Swift and Hohenberg [35,87,88]. Here it appears as a two-dimensional vector equation, since the order parameter is two-dimensional. This was used to explain the convective roll patterns generated by the Rayleigh–Bénard instability, where the order parameter is the vertical fluid velocity. The real order parameter case is also similar to the Ising model for magnets in two dimensions with next-nearest neighbor interactions [89], where competition between the nearest and next-nearest interactions generates a magnetic modulated phase called the Lifshitz phase [29,30].

Higher-dimensional or complex order parameters, as in the present analysis, are described using a generalized Landau-Ginzburg-Wilson Hamiltonian [37], where the authors also introduce the Lifshitz point. For the present case, the upper critical dimension for classical behavior is at d=5, and the classical location of the Lifshitz point is at η1=η2=0. Low-dimensional cases should have enhanced fluctuations, without spontaneous magnetization or symmetry breaking, as expected from the Mermin–Wagner theorem [56]. This applies to Heisenberg-type models with higher-dimensional order parameters in two dimensions. As it is valid for general, finite-range interactions, it also holds in our case.

6. CORRELATIONS

All physical quantities that we wish to understand in the double adiabatic limit treated in the previous section come from the solution of Eq. (43). This allows us to calculate the expectation value of any other observable in the vicinity of the critical point. Another way to obtain expectation values is to write the functional probability as a solution of the master equation. Below we develop both methods.

A. Stationary Solution of the Fokker–Planck Equation

We start with Eq. (43) and note that it is possible to write a functional Fokker–Planck equation for the probability density P(X,τ),

Pτ(X,τ)=iδδXi[(|X|2D˜)Xi+12δδXi]P(X,τ),
and look for the equilibrium distribution in the form P(X)=Nexp[H(X)], where H is a potential function. The solution for the distribution of X is given by
P(X)=Nexp[d2x(η1X·X+12(X·X)2+η2X·X+η32X·2X)].
This expression is similar to the Landau–Ginzburg free energy of a next-nearest neighbor interaction in a planar magnetic interaction, with X playing the role of a two-component vector order parameter. Owing to this parallel, the planar type II parametric system can provide a superb model platform for investigating fluctuations and universal behavior in this paradigmatic system.

B. Stochastic Moments in the Gaussian Approximation

In order to evaluate the moments and spatial correlations we will approximate the nonlinear terms of Eq. (43) using a Gaussian approximation together with a Green’s function approach. Hence we can write Eq. (43) as follows:

Xi(r,τ)Xj(r,τ)τ=D˜rXi(r,τ)Xj(r,τ)Xi(r,τ)|X(r,τ)|2Xj(r,τ).
Here, denotes the stochastic average over the Gaussian fluctuations and we have used the notation D˜r in order to avoid ambiguities. We explicitly denote that the operator D˜ acts on the spatial coordinate r. In the Gaussian approximation, fluctuations around the most probable configuration are approximately treated as independent modes with Gaussian distributions [90,91]. This allows us to replace ensemble averages by a Gaussian ansatz, in which higher-order moments are approximated by the expressions for a Gaussian distribution, e.g., X43X22. On defining GijGij(r,r)=Xi(r,τ)Xj(r,τ) we can write the above equation, for i=1, as
G1jτ=D˜rG1jG1j(3X12+X22).
Here we have assumed rotational symmetry of the problem in the (X1,X2) plane, so that X1X2=0. Using again the assumption of rotational symmetry in the X1X2 plane, we can also write the Green’s function results as
G1jτ=[D˜r2(X12+X22)]G1j.
This corresponds to an equivalent stochastic equation, valid in the steady state for a rotationally symmetric system:
X˜(r,τ)τ=D˜X˜(r,τ)2X˜·X˜X˜(r,τ)+ζ˜(r,τ).
We use the variable X˜ to denote that we are using the Gaussian approximation. In order to evaluate the Gaussian correlation functions in the near and far field, we will define a parameter η1=η1+2X˜·X˜. Using the expression for D˜ of Eq. (38) we can write the above equation as
X˜(r,τ)τ=(η1+η2r2η3r4)X˜(r,τ)+ζ˜(r,τ).
Next, we note that because of translational symmetry the term X˜·X˜ is independent of the spatial coordinate so that X˜·X˜=|X˜|2 can be calculated analytically. In order to evaluate it, we use the Fourier transform
X˜(r,t)=12πeik⃗·rX˜(k⃗,t)d2k⃗,
so that in momentum space we write Eq. (50) as
X˜(k,τ)τ=(η1+η2k2+η3k4)X˜(k,τ)+ζ˜(k,τ).

C. Lifshitz Point

We now consider the line of points where η2=0 and η3=12. These are the points we have defined as corresponding physically to zero detuning, with a pump in the vicinity of the μ=1. In this case, the solution for X˜(k⃗,t) is given by

X˜(k,τ)=τζ˜(k,τ)e(η1+k42)(ττ)dτ.
In this way, we obtain
X˜·X˜=14π202πkdkη1+k42.
On performing the integration there is a resulting self-consistency condition:
X˜·X˜=142(η1+2X˜·X˜).
At the Lifshitz point, where η1=0, we find that
X˜·X˜=|X˜|2=0.25.
As explained below, this is remarkably close to accurate numerical simulations of the correlations, including non-Gaussian fluctuations. We also consider the case where μ=1, η20, and η3=12. This corresponds to the case of nonzero detuning Δ. In Fig. 2 we plot the modulus squared of the two-dimensional order parameter X versus the detuning Δ.

 figure: 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.

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In order to obtain this figure, we solve the stochastic equations given in Eq. (43), and scan the detuning which is proportional to the parameter η2 defined in Eq. (39), so that Δ=0.5+0.005t. We notice that the fluctuations depend on the detuning. For a positive detuning there is a decrease of the fluctuations while for a negative detuning the fluctuations increase. A classical Swift–Hohenberg equation with a complex order parameter and nonzero detuning has been treated [92,93], as has the hyperbolic complex Swift–Hohenberg equation [94] and other related studies [95,96]. For negative detuning there is a ring with strong fluctuations, shown in Figs. 3 and 4.

 figure: Fig. 3.

Fig. 3. Dimensionless intensity versus detuning Δ and transverse momentum kx. 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.

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 figure: 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 kx for a fixed value of the detuning Δ=0.45, and ky=0. Other parameters are as in Fig. 3.

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D. Spatial Correlations in the Gaussian Approximation

From Eq. (53), the Gaussian correlation function in the momentum space (far field), in the stationary regime is therefore given by

X˜(k)[X˜(k)]*=12δ(k+k)η1+η2k2+η3k4.
This should be readily observable experimentally, since the far-field region of the output field from the planar cavity is simply the Fourier transform of the internal cavity field. The correlation function for the enhanced quadrature in configuration space, or near field, is
X˜(r)X˜(r)=18π2d2keik·(rr)η1+η2k2+η3k4.
On integrating over the angle variable we obtain
X˜(r)X˜(r)=14π0dkkJ0(k|rr|)η1+η2k2+η3k4,
where J0 is the Bessel function of zero order. The result of the above integration is, in the limit η2=0,
X˜(r)X˜(r)=η3η114πη3kei((η1η3)1/4|rr|),
where kei(x) is Thomson’s function [97]. An interesting remark is that as η10, the spatial correlation decays with a power law:
X˜(r)X˜(r)|rr|0.5.
We note, however, that this system is predicted to have an upper critical dimension [37] with mean-field critical exponents at spatial dimension d>5. Since the system is well below this critical dimension, one may expect non-Gaussian behavior that is not predicted by the approximations used in this section.

E. Non-Gaussian Behavior and Universality

In order to verify these analytic results and investigate non-Gaussian correlations, numerical simulations were carried out of the original stochastic partial differential equations of Eq. (43). Small time steps are needed to treat the quartic growth of the squared Laplacian term in momentum, together with large sample numbers to obtain a low sampling error. In initial investigations, we used a 10×10×10 numerical grid of 8000×64×64 points in t, x, y, respectively. Employing a fine numerical grid of 16000×64×64 points to check convergence in time step, the final steady-state correlation result converged to X·X=0.264±0.005. This was close to the Gaussian value, but with a relatively large sampling error.

In order to understand the quantitative difference between the exact and Gaussian results, a more precise differencing technique was used. This variance reduction or differencing technique simulates the difference between the full sample path and the Gaussian approximation [54,55]. The results were in agreement with a direct simulation, but gave much more rapid convergence. This was carried out as follows. First a mean field variable, X˜ was simulated, in the Gaussian approximation, using

X˜τ=D˜X˜2X˜|X˜|22X˜*X˜2+ζ+,
where the averages were carried out both over a spatial numerical grid and a set of parallel trajectories. From the analytic calculations above, this should converge precisely to the analytic result of |X˜|2=0.25, which was confirmed to a numerical accuracy of ±5×103. Here we have considered that η1=η2=0 and η3=12, corresponding to the classical Lifshitz point. Next, the full variable X was simulated. This was achieved by introducing a difference variable defined as ΔX=XX˜, which has the stochastic equation
ΔXτ=D˜XX|X|2+ζ+X˜τ.
By using an identical noise source to those in the equations for the mean-field X˜, the difference simulation permits a more precise calculation with reduced variance. The result, shown in Figs. 5 and 6, was that at the critical point, X·X=0.2574±0.0003. This result, of much greater accuracy, only required 3200 samples with a 10×20×20 numerical grid of 10000×48×48 points. The discretization error was estimated from using several grids with different transverse sizes and time steps.

 figure: 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.

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In summary, the full statistical calculation gives increased critical fluctuations due to non-Gaussian effects, but this increase is relatively small. At large transverse momentum there is no measurable difference between the Gaussian and exact results, shown in Fig. 6. The deviation from the Gaussian approximation vanishes rapidly as higher-order transverse momenta are investigated. Small values of this difference are observed only at zero transverse direction.

 figure: 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, Δln|X(k)|2=ln|X(k)|2ln|X˜(k)|2 versus momentum k. Here we consider η1=η2=0 and η3=12. Other parameters are as in Fig. 5.

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7. CONCLUSION

We have shown that parametric downconversion in a type II parametric planar cavity leads to a Swift and Hohenberg type of stochastic equation for the leading terms in the critical fluctuations, but with a vector order parameter. This combines the rotationally invariant symmetry properties of the X–Y model with the higher-order Laplacian of a Lifshitz magnetic phase transition. Surprisingly, these fluctuations are not thermal in origin, but come instead from the quantum fluctuations associated with parametric amplification.

This model can be approximately treated for the critical fluctuations with a Gaussian factorization. However, a careful numerical treatment shows that non-Gaussian critical fluctuations occur. These are responsible for enhanced intensity correlations, but are reduced at large transverse momenta due to the momentum dependence of the linear propagator. As techniques improve, we expect that this novel, nonequilibrium critical point will become accessible to experimental studies.

APPENDIX A: QUANTUM LANGEVIN FORM

In the quantum Langevin form the corresponding operator equations of the system would be

A^0t=γ˜0A^0+E(x)χ*A^1A^2+iv022ω02A^0+2γ0A^0in,A^1t=γ˜1A^1+χA^0A^2+iv122ω12A^1+2γ1A^1in,A^2t=γ˜2A^2+χA^0A^1+iv222ω22A^2+2γ2A^2in.
Here we use a rotating frame such that the three field operators are treated as in a frame rotating with frequency ωi0. The relative detuning between the pump laser at 2ωL and the intracavity pumped mode ω0 is Δ0=(ω02ωL)/γ0, and the downconverted modes have relative detunings Δi=(ωiωi0)/γi. The terms γ˜i=γi(1+iΔi) represent the complex cavity decay for each mode, including detunings. However, the quantum Langevin approach has the drawback that it deals with operator equations that are intractable analytically. It was shown in Section 3 that the phase-space representation method generates similar equations, but with a more useful c-number form.

The input–output relations that describe the external modes outside the cavity are [1416,6063] A^iout=2γiA^iA^iin, where A^iin and A^iout are the corresponding input and output fields, with input correlations

A^iin(x,t)A^jin(x,t)=(n¯ith+1)δijδ(xx),A^iin(x,t)A^jin(x,t)=n¯ithδijδ(xx).
In our calculations we assume that the reservoirs are in the vacuum state. However, nonzero reservoir temperatures can be readily included. While we do not use these input–output equations here, we note that they are important when dealing with external measurements.

APPENDIX B: POSITIVE P-REPRESENTATION

The positive P-representation generates a genuine (second-order) Fokker–Planck equation with positive-definite diffusion, provided the distribution vanishes sufficiently rapidly at the phase-space boundaries. This can then be mapped into a set of c-number Langevin equations similar to the quantum Heisenberg equations, except for additional stochastic terms.

This approach uses a multimode coherent state |α˜,α˜1,α˜2|α˜, defined as an eigenstate of the annihilation operators a^i(k), where α˜iα˜i(k) so that the following eigenvalue equation is obtained:

a^i(k)|α˜=α˜i(k)|α˜.
The positive P-representation then expands the density matrix in terms of coherent-state projection operators [57], which is always possible as a positive distribution:
ρ^=d6Mα˜d6Mα˜+|α˜α˜+*|α˜+*|α˜P(α˜,α˜+).
In the positive P-representation the coherent amplitudes α˜i(k,t) satisfy time-dependent stochastic differential equations. It is simplest to write these equations in a form analogous to classical equations by introducing stochastic fields Ai, Ai+ defined as
Ai(x,t)=1Lkeik·xα˜i(k,t),
together with a stochastic conjugate Ai+, which is a c-number, rather than an operator field. It is only conjugate to Ai in the mean: it is stochastically equivalent to the conjugate operator.

A. Noises of the Stochastic Equations

We note that while our derivation of the set of stochastic equations given in Eq. (10) is formally based on the Itô stochastic calculus, in this case either Itô or Stratonovich stochastic calculus gives identical results. These complex noise terms can be constructed from four delta-correlated real Gaussian noise fields (ξx,ξy,ξx+,ξy+), with the mapping

ξ1,2(x,t)=[ξx(x,t)±iξy(x,t)]/2,ξ1,2+(x,t)=[ξx+(x,t)±iξy+(x,t)]/2.
It follows that the stochastic fields in the positive P-representation for the ξ1 and ξ2 fields are complex conjugate, i.e.,
ξ1(x,t)=ξ2*(x,t),andξ1+(x,t)=(ξ2+(x,t))*.
The physics of the noise is that it describes a departure from coherent behavior. The deterministic terms correspond to the evolution of coherent states. However, the true quantum state does not remain coherent. Instead it develops, via the noise terms, into squeezed, entangled, and even more complex states. Despite this complexity, there are simple, universal properties caused by these noise terms at the critical point.

We note that there are no “normal” noise correlations, that is, ξi(x,t)ξk+(x,t)=0. This is due to our assumption that the optical reservoirs are at zero temperature, which is an excellent approximation at optical frequencies. If there are thermal reservoirs, as can occur in microwave devices, then additional reservoir correlations must be included, which are proportional to the thermal occupation number. More generally, our model includes only the minimal noise due to fundamental quantum effects, but there can be a range of additional technical noise sources in practical devices, caused by temperature fluctuations, laser intensity fluctuations, and laser phase noise [12,98].

Funding

Australian Research Council (ARC); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

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52. 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).

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

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

55. 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]  

56. 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]  

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

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

59. 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]  

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

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

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

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

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

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

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

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

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

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

70. 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]  

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

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

73. 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]  

74. 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]  

75. 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]  

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

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

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

79. 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]  

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

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

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

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

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

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

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

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

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

89. 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]  

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

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

92. 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]  

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

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

95. 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]  

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

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

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

References

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  32. 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).
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  35. S. Ducci, N. Treps, A. Maître, and C. Fabre, “Pattern formation in optical parametric oscillators,” Phys. Rev. A 64, 023803 (2001).
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  37. 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).
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  39. P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, 1995).
  40. A. Bonanno and D. Zappalà, “Isotropic Lifshitz critical behavior from the functional renormalization group,” Nucl. Phys. B 893, 501–511 (2015).
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  41. K. Staliunas, “Laser Ginzburg-Landau equation and laser hydrodynamics,” Phys. Rev. A 48, 1573–1581 (1993).
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  42. 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).
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  43. A. Gatti and L. Lugiato, “Quantum images and critical fluctuations in the optical parametric oscillator below threshold,” Phys. Rev. A 52, 1675–1690 (1995).
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  44. 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).
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  45. 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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  46. L. A. Lugiato, A. Gatti, and E. Brambilla, “Quantum imaging,” J. Opt. B 4, S176–S183 (2002).
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  47. P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
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  48. 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]
  49. 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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  50. 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).
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  51. C. J. Mertens, T. A. B. Kennedy, and S. Swain, “Many body theory of quantum noise,” Phys. Rev. Lett. 71, 2014–2017 (1993).
    [Crossref]
  52. 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).
  53. J. M. Kosterlitz and D. J. Thouless, “Ordering, metastability and phase transitions in two-dimensional systems,” J. Phys. C 6, 1181–1203 (1973).
    [Crossref]
  54. S. Chaturvedi, K. Dechoum, and P. D. Drummond, “Limits to squeezing in the degenerate optical parametric oscillator,” Phys. Rev. A 65, 033805 (2002).
    [Crossref]
  55. 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]
  56. 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]
  57. P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
    [Crossref]
  58. H. Haken, Laser Theory (Springer, 1984).
  59. 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]
  60. C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).
  61. B. Yurke, Quantum Squeezing, P. D. Drummond and Z. Ficek, eds. (Springer, 2004).
  62. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, 2008).
  63. P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics (Cambridge University, 2014).
  64. I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys. 85, 299–366 (2013).
    [Crossref]
  65. M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
    [Crossref]
  66. H. J. Carmichael, Statistical Methods in Quantum Optics 1 (Springer, 2002).
  67. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  68. P. D. Drummond, “Fundamentals of higher order stochastic equations,” J. Phys. A 47, 335001 (2014).
    [Crossref]
  69. K. Husimi, “Some formal properties of the density matrix,” Proc. Phys. Math. Soc. Jpn. 22, 264–314 (1940).
  70. 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]
  71. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [Crossref]
  72. E. C. G. Sudarshan, “Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams,” Phys. Rev. Lett. 10, 277–279 (1963).
    [Crossref]
  73. 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]
  74. 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]
  75. 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]
  76. S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 43, 1569–1575 (1996).
  77. G. J. de Valcárcel, E. Roldán, and K. Staliunas, “Cavity solitons in nondegenerate optical parametric oscillation,” Opt. Commun. 181, 207–213 (2000).
    [Crossref]
  78. K. Staliunas, “Transverse pattern formation in optical parametric oscillators,” J. Mod. Opt. 42, 1261–1269 (1995).
    [Crossref]
  79. 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]
  80. S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
    [Crossref]
  81. S. Longhi and A. Geraci, “Swift-Hohenberg equation for optical parametric oscillators,” Phys. Rev. A 54, 4581–4584 (1996).
    [Crossref]
  82. E. M. Lifshitz and L. P. Pitaevskii, “Statistical physics, part 2,” in The Course of Theoretical Physics (Pergamon, 1980), Vol. 9.
  83. P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comp. Phys. 93, 144–170 (1991).
    [Crossref]
  84. M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comp. Phys. 132, 312–326 (1997).
    [Crossref]
  85. G. R. Collecutt and P. D. Drummond, “Xmds: eXtensible multi-dimensional simulator,” Comput. Phys. Commun. 142, 219–223 (2001).
    [Crossref]
  86. S. Kiesewetter, R. Polkinghorne, B. Opanchuk, and P. D. Drummond, “xSPDE: extensible software for stochastic equations,” SoftwareX, to be published.
  87. G. Kozyreff and M. Tlidi, “Nonvariational real Swift-Hohenberg equation for biological, chemical, and optical systems,” Chaos 17, 037103 (2007).
    [Crossref]
  88. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. 73, 640–643 (1994).
    [Crossref]
  89. 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]
  90. S.-K. Ma, Modern Theory of Critical Phenomena (W. A. Benjamin, 1976).
  91. P. Kopietz, L. Bartosch, and F. Schtz, Introduction to the Functional Renormalization Group (Springer, 2010).
  92. 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]
  93. K. Staliunas, “Spatial and temporal spectra of noise driven stripe patterns,” Phys. Rev. E 64, 066129 (2001).
    [Crossref]
  94. K. Staliunas and M. Tlidi, “Hyperbolic transverse patterns in nonlinear optical resonators,” Phys. Rev. Lett. 94, 133902 (2005).
    [Crossref]
  95. 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]
  96. K. Staliunas and V. J. Sanchez-Morcillo, “Transverse patterns in nonlinear optical resonators,” in Springer Tracts Mod. Phys. (Springer, 2003).
  97. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  98. P. D. Drummond and M. D. Reid, “Laser bandwidth effects on squeezing in intracavity parametric oscillation,” Phys. Rev. A 37, 1806–1808 (1988).
    [Crossref]

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, D. Camesasca, A. Gatti, and L. A. Lugiato, “Spatiotemporal dynamics of optical parametric oscillators,” J. Mod. Opt. 41, 1151–1162 (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]

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, 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]

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).
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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]

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

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

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

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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]

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]

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]

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).

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

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P. D. Drummond and C. W. Gardiner, “Generalised P-representations in quantum optics,” J. Phys. A 13, 2353–2368 (1980).
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C. W. Gardiner and P. Zoller, Quantum Noise (Springer, 2000).

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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]

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

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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).
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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).
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M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
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P. D. Drummond and M. Hillery, The Quantum Theory of Nonlinear Optics (Cambridge University, 2014).

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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).
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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).
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Khoury, A. Z.

Kiesewetter, S.

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

Kimble, H. J.

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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Lam, P. K.

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]

Lamprecht, C.

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

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

Leuchs, 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]

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

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

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

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S. Longhi, “Spatial solitary waves in nondegenerate optical parametric oscillators near an inverted bifurcation,” Opt. Commun. 149, 335–340 (1998).
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S. Longhi, “Alternating rolls in non-degenerate optical parametric oscillators,” J. Mod. Opt. 43, 1569–1575 (1996).

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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).
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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]

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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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]

Lugiato, L. A.

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

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]

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

Maître, A.

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

Mandel, P.

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

Martin, R.

Martinelli, M.

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[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]

Marzoli, I.

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]

McNeil, K. J.

P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 28, 211–225 (2010).
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P. D. Drummond, K. J. McNeil, and D. F. Walls, “Non-equilibrium transitions in sub/second harmonic generation,” Opt. Acta 27, 321–335 (2010).
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M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
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J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref]

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P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comp. Phys. 93, 144–170 (1991).
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C. Bolman and A. C. Newell, “Natural patterns and wavelets,” Rev. Mod. Phys. 70, 289–301 (1998).
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J. Lega, J. V. Moloney, and A. C. Newell, “Swift-Hohenberg equation for lasers,” Phys. Rev. Lett. 73, 2978–2981 (1994).
[Crossref]

Nussenzveig, P.

Olsen, M. K.

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]

Opanchuk, B.

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

Oppo, G.-L.

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]

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]

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]

Ou, Z. Y.

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]

Paris, M. G. 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).
[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]

Peng, K. C.

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]

Pereira, S. F.

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]

Petsas, K.

Pfister, O.

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

Pitaevskii, L. P.

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

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

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).
[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]

Reid, M. D.

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]

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]

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, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40, 913–923 (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]

Riedel, E. K.

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

Ritsch, H.

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. 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]

Roldán, E.

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]

Ryzhik, I. M.

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

San Miguel, M.

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]

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]

Sanchez-Morcillo, V. J.

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

Sánchez-Morcillo, V. J.

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]

San-Miguel, M.

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]

Santagiustina, M.

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]

Schtz, F.

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

Schuster, H. G.

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

Scroggie, A. J.

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]

Scully, M. O.

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]

Selke, W.

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

Shtrikman, S.

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]

Slekys, G.

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]

Solimeno, S.

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]

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]

Staliunas, K.

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

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]

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]

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]

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]

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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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