The Ising–Bloch transition in degenerate optical parametric oscillators

Domain walls in type I degenerate optical parametric oscillators are numerically investigated. Both steady Ising and moving Bloch walls are found, bifurcating one into another through a nonequilibrium Ising--Bloch transition. Bloch walls are found that connect either homogeneous or roll planforms. Secondary bifurcations affecting Bloch wall movement are characterized that lead to a transition from a steady drift state to a temporal chaotic movement as the system is moved far from the primary, Ising--Bloch bifurcation. Two kinds of routes to chaos are found, both involving tori: a usual Ruelle-Takens and an intermittent scenarios.

Subdiffractive solitons in bose-einstein condensates

We predict the disappearance of diffraction (the increase of the mass) of Bose-Einstein condensates in counter-moving periodic potentials. We demonstrate subdiffractive solitons (stable droplets of the condensate) in the vicinity of this zero diffraction point.

Quantum noise properties of cavity solitons

General method for studying quantum fluctuations of dissipative structures formed in nonlinear optical cavities is presented. Application to cavity soliton supported by degenerate optical parametric oscillator is presented. Squeezing and intensity fluctuations spectra are discussed.

Experimental demonstration of hyperbolic patterns.

We give experimental evidence of hyperbolic patterns in a nonlinear optical resonator. Such transverse patterns are a new kind of 2D dissipative structures, characterized by a distribution of the active modes along hyperbolas in the transverse wave-vector domain, in contrast with the usual (elliptic) patterns where the active modes distribute along rings. The hyperbolic character is realized by manipulating diffraction inside the optical resonator with cylindrical lenses. We also investigate theoretically hyperbolic patterns in corresponding Swift-Hohenberg models.

Generating highly squeezed Hybrid Laguerre-Gauss modes in large-Fresnel-number Degenerate Optical Parametric Oscillators

We theoretically describe the quantum properties of a large Fresnel number degenerate optical parametric oscillator with spherical mirrors that is pumped by a Gaussian beam. The resonator is tuned so that the resonance frequency of a given transverse mode family coincides with the down-converted frequency. After demonstrating that only the lower orbital angular momentum (OAM) Laguerre-Gauss modes are amplified above threshold, we focus on the quantum properties of the rest of (classically empty) modes. We find that combinations of opposite OAM (Hybrid Laguerre-Gauss modes) can exhibit arbitrary large quadrature squeezing for the lower OAM non amplified modes.

Squeezed Light Generation via Spatial Symmetry Breaking

The spontaneous spatial symmetry breaking occurring in the transverse section of the light beam emitted by a degenerate optical parametric oscillator is shown to give rise to perfectly squeezed light. Such phenomenon occurs at any operating conditions, unlike conventional squeezing.

Domain wall dynamics in an optical Kerr cavity

An anisotropic (dichroic) optical cavity containing a self-focusing Kerr medium is shown to display a bifurcation between static --Ising-- and moving --Bloch-- domain walls, the so-called nonequilibrium Ising-Bloch transition (NIB). Bloch walls can show regular or irregular temporal behaviour, in particular, bursting and spiking. These phenomena are interpreted in terms of the spatio-temporal dynamics of the extended patterns connected by the wall, which display complex dynamical behaviour as well. Domain wall interaction, including the formation of bound states is also addressed.

Controlled Observation of a Nonequilibrium Ising-Bloch Transition in a Nonlinear Optical Cavity

We describe the controlled observation of the nonequilibrium Ising-Bloch transition in a broad area nonlinear optical cavity, namely, a quasi-1D single longitudinal-mode photorefractive oscilator in a degenerate four-wave mixing configuration. Our experimental technique allows for the controlled injection of the domain walls. We use cavity detuning as control parameter and find that both Ising and Bloch walls can exist for the same detuning values within a certain interval of detunings, i.e., the Ising-Bloch transition is hysteretic in our case. A complex Ginzburg-Landau model is used for supporting the observations.

Noncritical quadrature squeezing in two-transverse-mode optical parametric oscillators

In this article we explore the quantum properties of a degenerate optical parametric oscillator when it is tuned to the first family of transverse modes at the down-converted frequency. Recently we found [C. Navarrete-Benlloch et al., Phys. Rev. Lett. 100, 203601 (2008)] that above threshold a TEM${}_{10}$ mode following a random rotation in the transverse plane emerges in this system (we denote it as the bright mode), breaking thus its rotational invariance. Then, owing to the mode orientation being undetermined, we showed that the phase quadrature of the transverse mode orthogonal to this one (denoted as the dark mode) is perfectly squeezed at any pump level and without an increase in the…

Active locking and entanglement in type II optical parametric oscillators

Type II optical parametric oscillators are amongst the highest-quality sources of quantum-correlated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonally-polarized beams with strong continuous-variable entanglement. However, these sources are of limited practical use, because the entangled beams emerge with different frequencies and a diffusing phase-difference. It has been proven that the use of an internal wave-plate coupling the modes with orthogonal polarization is capable of locking the frequencies of the emerging beams to half the pump frequency, as well as reducing the phase-difference diffusion, at the expense of reducing the entangl…

Cavity solitons in nondegenerate optical parametric oscillation

Abstract We find analytically cavity solitons in nondegenerate optical parametric oscillators. These solitons are exact localised solutions of a pair of coupled parametrically driven Ginzburg–Landau equations describing the system for large pump detuning. We predict the existence of a Hopf bifurcation of the soliton resulting in a periodically pulsing localised structure. We give numerical evidence of the analytical results and address the problem of cavity soliton interaction.

Impact of anisotropy on the noncritical squeezing properties of two-transverse-mode optical parametric oscillators

In a series of articles we studied the quantum properties of a degenerate optical parametric oscillator tuned to the first family of transverse modes at the subharmonic. We found that, for a cavity having rotational symmetry respect to the optical axis, a TEM$_{10}$ mode with an arbitrary orientation in the transverse plane is emitted above threshold. We proved then that quantum noise induces a random rotation of this bright TEM$_{10}$ mode in the transverse plane, while the mode orthogonal to it, the so-called dark mode, has perfect quadrature squeezing irrespective of the distance to threshold (noncritical squeezing). This result was linked to the spontaneous rotational symmetry breaking …

Stabilizing and controlling domain walls and dark-ring cavity solitons.

We demonstrate two alternative techniques for controlling and stabilizing domain walls (DW) in phase-sensitive, nonlinear optical resonators. The first of them uses input pumps with spatially modulated phase and can be applied also to dark-ring cavity solitons. An optical memory based on the latter is demonstrated. Here the physical mechanism of control relies on the advection caused to any feature by the phase gradients. The second technique uses a plane wave input pump with holes of null intensity across its transverse plane, which are able to capture DWs. Here the physical mechanism of control is of topological nature. When distributed as a regular array, these holes delimit spatial opti…

Faraday patterns in bose-Einstein condensates.

Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns of atom density through a parametric resonance. The dominating wavelength of the spatial structures is shown to be primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent macroscopic quantum analogues of the Faraday waves excited in vertically shaken liquids.

Theory of quantum fluctuations of optical dissipative structures - Application to the study of squeezing and intensity fluctuations of DOPO cavity solitons

We present a general theory of quantum fluctuations of dissipative structures in nonlinear optical cavities with transverse translation invariance. Perfect squeezing of the transverse momentum, detectable under homodyning, occurs irrespectively of the system parameters.

Noncritically squeezed light via spontaneous rotational symmetry breaking.

We theoretically address squeezed light generation through the spontaneous breaking of the rotational invariance occuring in a type I degenerate optical parametric oscillator (DOPO) pumped above threshold. We show that a DOPO with spherical mirrors, in which the signal and idler fields correspond to first order Laguerre-Gauss modes, produces a perfectly squeezed vacuum with the shape of a Hermite-Gauss mode, within the linearized theory. This occurs at any pumping level above threshold, hence the phenomenon is non-critical. Imperfections of the rotational symmetry, due e.g. to cavity anisotropy, are shown to have a small impact, hence the result is not singular.

Theory of quantum fluctuations of optical dissipative structures and its application to the squeezing properties of bright cavity solitons

We present a method for the study of quantum fluctuations of dissipative structures forming in nonlinear optical cavities, which we illustrate in the case of a degenerate, type I optical parametric oscillator. The method consists in (i) taking into account explicitly, through a collective variable description, the drift of the dissipative structure caused by the quantum noise, and (ii) expanding the remaining -internal- fluctuations in the biorthonormal basis associated to the linear operator governing the evolution of fluctuations in the linearized Langevin equations. We obtain general expressions for the squeezing and intensity fluctuations spectra. Then we theoretically study the squeezi…

Coherent master equation for laser modelocking

Modelocked lasers constitute the fundamental source of optically-coherent ultrashort-pulsed radiation, with huge impact in science and technology. Their modeling largely rests on the master equation (ME) approach introduced in 1975 by Hermann A. Haus. However, that description fails when the medium dynamics is fast and, ultimately, when light-matter quantum coherence is relevant. Here we set a rigorous and general ME framework, the coherent ME (CME), that overcomes both limitations. The CME predicts strong deviations from Haus ME, which we substantiate through an amplitude-modulated semiconductor laser experiment. Accounting for coherent effects, like the Risken-Nummedal-Graham-Haken multim…

Four-phase patterns in a forced nonlinear optical oscillator

We present preliminary theoretical and experimental results indicating that a high Fresnel number nonlinear optical oscillator with planar mirrors can display four-phase multistability, eventually leading to four-phase patterns. Such situation is similar to that emerging in extended oscillatory systems forced within a 4:1 resonance and, to the best of our knowledge, has not been predicted nor observed previously in an optical system.

Generalized rate equations for multimode lasers

Abstract A generalized rate equations model for class B lasers outside the uniform field limit is presented. This model allows to correctly describe the Risken–Nummedal–Graham–Haken instability and its associated multimode dynamics. When parameters suitable for fiber lasers – where the instability has been observed – are adopted, the computation time is shown to be greatly reduced with respect to the complete model based on the full set of Maxwell–Bloch (MB) equations.

Four-wave mixing and vacuum squeezing in polariton microcavities

In a recent paper [1] it has been shown how a bichromatic fast driving of optomechanical (optical domain) and superconducting circuit systems (microwave domain), operating in a limit where they present a non-linear Kerr-type interaction, can give rise to very strong vacuum squeezing. The driving with two close frequencies of a Kerr cavity changes the usual bistability bifurcation behaviour that takes place under monochromatic driving, into a degenerate four-wave mixing bifurcation, where a phase-bistable component starts oscillating spontaneously at a frequency that lies halfway between the two driving frequencies [2]. This resembles the physics of the optical parametric oscillator threshol…

Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance.

We show that a self-oscillatory system, driven at two frequencies close to that of the unforced system (resonance 1:1), becomes phase locked and exhibits two equivalent stable states of opposite phases. For spatially extended systems this phase bistability results in patterns characteristic for real order parameter systems, such as phase domains, labyrinths, and phase spatial solitons. In variational cases, the phase-locking mechanism is interpreted as a result of the periodic "rocking" of the system potential. Rocking could be tested experimentally in lasers and in oscillatory chemical reactions.

Domain walls and ising-BLOCH transitions in parametrically driven systems.

Parametrically driven systems sustaining sech solitons are shown to support a new kind of localized state. These structures are walls connecting two regions oscillating in antiphase that form in the parameter domain where the sech soliton is unstable. Depending on the parameter set the oppositely phased domains can be either spatially uniform or patterned. Both chiral (Bloch) and nonchiral (Ising) walls are found, which bifurcate one into the other via an Ising-Bloch transition. While Ising walls are at rest Bloch walls move and may display secondary bifurcations leading to chaotic wall motion.

Faraday patterns in low-dimensional Bose-Einstein condensates

We show that Faraday patterns can be excited in the weak confinement space of low-dimensional Bose-Einstein condensates by temporal modulation of the trap width, or equivalently of the trap frequency Omega_tight, in the tight confinement space. For slow modulation, as compared with Omega_tight, the low-dimensional dynamics of the condensate in the weak confinement space is described by a Gross-Pitaevskii equation with time modulated nonlinearity coefficient. For increasing modulation frequencies a noticeable reduction of the pattern formation threshold is observed close to 2*Omega_tight, which is related to the parametric excitation of the internal breathing mode in the tight confinement sp…

Coherent effects in the multimode dynamics of inhomogeneously broadened ring lasers

We investigate under which conditions coherent effects manifest in the multimode dynamics of inhomogeneously broadened ring lasers. In particular, we demonstrate that for long enough cavities standard rate equations for class-B lasers fail in describing the multimode dynamics.

Risken–Nummedal–Graham–Haken instability in class-B lasers

We determine analytical expressions for the Risken-Nummedal-Graham-Haken multimode laser instability outside the uniform field limit in the case of very fast polarization decay (class-B laser). A new condition for the observability of that instability, concerning the value of the cavity mirrors reflectivity, is predicted.

Bistable phase locking in a laser with injected signal

Unveiling two-dimensional discrete quantum walks dynamics via dispersion relations

The discrete, or coined, quantum walk (QW) [1] is a process originally introduced as the quantum counterpart of the classical random walk (RW). In both cases there is a walker and a coin: at every time step the coin is tossed and the walker moves depending on the toss output. Unlike the RW, in the QW the walker and coin are quantum in nature what allows the coherent superpositions right/left and head/tail happen. This feature endows the QW with outstanding properties, such as making the standard deviation of the position of an initially localized walker grow linearly with time t, unlike the RW in which this growth goes as t1/2. This has strong consequences in algorithmics and is one of the …

Universal description of pattern formation in optical oscillators under bichromatic injection

We study pattern formation in a complex Swift–Hohenberg equation with phase-sensitive (parametric) gain. Such an equation serves as a universal order parameter equation describing the onset of spontaneous oscillations in extended systems submitted to a bichromatic injection when the instability is toward long (transverse) wavelengths. Applications include two-level lasers and photorefractive oscillators. Under such an injection, the original continuous phase symmetry of the system is replaced by a discrete one and phase bistability emerges. This leads to the spontaneous formation of phase-locked spatial structures, such as phase domains and dark-ring (phase) cavity solitons. The stability o…

General Linearized Theory of Quantum Fluctuations around Arbitrary Limit Cycles

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, which is the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical long-time solution is time dependent, a most prominent example being spontaneous limit-cycle formation. H…

Multimode instability in inhomogeneously broadened class-Bring lasers: Beyond the uniform-field limit

The multimode emission threshold of class-$B$ ring lasers is analytically investigated taking into account the localized nature of the cavity losses and the inhomogeneous broadening of the amplifying medium. This analysis finds a relevant application to erbium-doped fiber lasers (EDFL's). The main conclusion is that the predictions of the simplest models are deeply modified both quantitatively and qualitatively. Thus, any attempt to interpret multimode emission in EDFL's must incorporate the two considered factors. Two main results are: (i) While in homogeneously broadened lasers instabilities are inhibited for values of the mirrors' reflectivity $\mathcal{R}l0.54ca,$ this limitation disapp…

Control and steering of phase domain walls

We show experimentally the feasibility of optically controlled location, individual addressing/erasure and steering of phase domain walls by injection of coherent addressing pulses into a phase-locked four-wave-mixing photorefractive oscillator.

Dissipative structures in optomechanical cavities

Motivated by the increasing interest in the properties of multimode optomechanical devices, here we study a system in which a driven mode of a large-area optical cavity is despersively coupled to a deformable mechanical element. Two different models naturally appear in such scenario, for which we predict the formation of periodic patterns, localized structures (cavity solitons), and domain walls, among other complex nonlinear phenomena. Further, we propose a realistic design based on intracavity membranes where our models can be studied experimentally. Apart from its relevance to the field of nonlinear optics, the results put forward here are a necessary step towards understanding the quant…

Polarization instability in anisotropic-cavity degenerate four-wave mixing

Abstract The emission and stability properties of a plane-wave model of intracavity degenerate four-wave mixing including self- and cross-phase modulation are studied. A Kerr medium inside an anisotropic cavity in which a linearly polarized field is injected is considered. Cavity anisotropy leads to qualitative new phenomena such as a subcritical polarization instability.

Strong vacuum squeezing from bichromatically driven Kerrlike cavities: from optomechanics to superconducting circuits

AbstractSqueezed light, displaying less fluctuation than vacuum in some observable, is key in the flourishing field of quantum technologies. Optical or microwave cavities containing a Kerr nonlinearity are known to potentially yield large levels of squeezing, which have been recently observed in optomechanics and nonlinear superconducting circuit platforms. Such Kerr-cavity squeezing however suffers from two fundamental drawbacks. First, optimal squeezing requires working close to turning points of a bistable cycle, which are highly unstable against noise thus rendering optimal squeezing inaccessible. Second, the light field has a macroscopic coherent component corresponding to the pump, ma…

Diffraction management and sub-diffractive solitons in periodically driven Bose–Einstein condensates

Abstract We theoretically investigate the diffraction management in Bose–Einstein condensates (BECs) in one- (1D), two- (2D) and three-dimensional (3D) geometries. The management technique is based on the superposition of harmonic lattices’ potentials moving at a common speed but in different directions, leading to a harmonic spatio-temporal modulation of the potential. In this way a reduction in, and eventually the disappearance of usual diffraction and emergence of fourth-order diffraction are achieved. We show sub-diffractive solitons in such a diffraction managed system and demonstrate their stability in 1D, 2D and 3D. In 2D and 3D cases we investigate diffraction management by lattices…

Quadrature and polarization squeezing in a dispersive optical bistability model

We theoretically study quadrature and polarization squeezing in dispersive optical bistability through a vectorial Kerr cavity model describing a nonlinear cavity filled with an isotropic chi(3) medium in which self-phase and cross-phase modulation, as well as four--wave mixing, occur. We derive expressions for the quantum fluctuations of the output field quadratures as a function of which we express the spectrum of fluctuations of the output field Stokes parameters. We pay particular attention to study how the bifurcations affecting the non-null linearly polarized output mode squeezes the orthogonally polarized vacuum mode, and show how this produces polarization squeezing.

Noncritical quadrature squeezing through spontaneous polarization symmetry breaking

We discuss the possibility of generating noncritical quadrature squeezing by spontaneous polarization symmetry breaking. We first consider Type II frequency-degenerate optical parametric oscillators but discard them for a number of reasons. Then we propose a four-wave-mixing cavity, in which the polarization of the output mode is always linear but has an arbitrary orientation. We show that in such a cavity, complete noise suppression in a quadrature of the output field occurs, irrespective of the parameter values.

Lorenz–Haken instability in a laser with arbitrary mirrors reflectivity

Abstract We study the Lorenz–Haken instability in a laser with arbitrary mirrors reflectivity R . In the limit of slow population difference we demonstrate that in order to observe the instability the reflectivity must exceed the critical value R c ≃0.5379, which coincides with that found recently for the multimode (Risken–Nummedal–Graham–Haken) instability. Several other differences with respect to the uniform field limit R →1 are presented.

Sub-Diffractive Band-Edge Solitons in Bose-Einstein Condensates in Periodic Potentials

A new type of matter wave diffraction management is presented that leads to sub-diffractive soliton-like structures. The proposed management technique uses two counter-moving, identical periodic potentials (e.g. optical lattices). For suitable lattice parameters a novel type of atomic band-gap structure appears in which the effective atomic mass becomes infinite at the lowest edge of an energy band. This way normal matter-wave diffraction (proportional to the square of the atomic momentum) is replaced by fourth-order diffraction, and hence the evolution of the system becomes sub-diffractive.

Phase-bistable Kerr cavity solitons and patterns

We study pattern formation in a passive nonlinear optical cavity on the basis of the classic Lugiato-Lefever model with a periodically modulated injection. When the injection amplitude sign alternates, e.g., following a sinusoidal modulation in time or in space, a phase-bistable response emerges, which is at the root of the spatial pattern formation in the system. An asymptotic description is given in terms of a damped nonlinear Schr\"odinger equation with parametric amplification, which allows gaining insight into the basic spatiotemporal dynamics of the system. One- and two-dimensional phase-bistable spatial patterns, such as bright and dark-ring cavity solitons and labyrinths, are demons…

Cavity solitons in lasers with spatially modulated injected signal

The injection of a monochromatic signal into a laser is a well-known technique for locking the laser phase to that of the injection. Some years ago another type of injection, called rocking [1,2], was introduced to render the laser phase-locking bistable. Rocking consists of the modulation of the amplitude injection so that its sign changes periodically, or even randomly [3], in time. Here we present an alternative to rocking that leads to the same type of behaviour, namely the appearance of bistable phase locking and, in the case of large Fresnel number lasers, to stable (phase bistable) cavity solitons and extended patterns. The new type of injection we present here is monochromatic, unli…

Bistable phase locking of a laser via monochromatic signal injection

In free running lasers the field phase is not fixed and any value possible value is equally likely (invariant), but can be locked to an external reference by injecting a monochromatic signal field into the cavity. In this way the phase of the slave laser locks to a single value resulting in a monostable phase locking. It could be however of practical interest that the laser field be locked not to a single value but to two different possible values, hence the name bistable phase locking.

Pattern formation through phase bistability in oscillatory systems with space-modulated forcing.

We propose a novel forcing technique of spatially extended self-oscillatory systems able to excite phase bistability and the dissipative structures associated with it. The forcing is time periodic at a frequency close to the oscillators' frequency and is spatially modulated. The effects of this type of forcing are demonstrated analytically and numerically in a directly driven complex Ginzburg-Landau equation. Both spatially periodic and spatially random drives prove to be effective.

Quantum fluctuations in cavity solitons

Quantum fluctuations of degenerate optical parametric oscillators' cavity solitons (CS) are studied. We show that CSs are sources of perfectly squeezed light that exhibit photon fluctuations below the shot-noise level as well.

Phase-bistable pattern formation in oscillatory systems via rocking: application to nonlinear optical systems

We present a review, together with new results, of a universal forcing of oscillatory systems, termed ‘rocking’, which leads to the emergence of a phase bistability and to the kind of pattern formation associated with it, characterized by the presence of phase domains, phase spatial solitons and phase-bistable extended patterns. The effects of rocking are thus similar to those observed in the classic 2 : 1 resonance (the parametric resonance) of spatially extended systems of oscillators, which occurs under a spatially uniform, time-periodic forcing at twice the oscillations' frequency. The rocking, however, has a frequency close to that of the oscillations (it is a 1 : 1 resonant forcing) …

N-dimensional alternate coined quantum walks from a dispersion-relation perspective

We propose an alternative definition of an N-dimensional coined quantum walk by generalizing a recent proposal [Di Franco et al., Phys. Rev. Lett. 106, 080502 (2011)]. This N-dimensional alternate quantum walk, AQW_N, in contrast with the standard definition of the N-dimensional quantum walk, QW_N, requires only a coin-qubit. We discuss the quantum diffusion properties of AQW_2 and AQW_3 by analyzing their dispersion relations that reveal, in particular, the existence of diabolical points. This allows us to highlight interesting similarities with other well known physical phenomena. We also demonstrate that AQW_3 generates genuine multipartite entanglement. Finally we discuss the implementa…

Bistable phase locking of a nonlinear optical cavity via rocking: Transmuting vortices into phase patterns.

We report experimental observation of the conversion of a phase-invariant nonlinear system into a phase-locked one via the mechanism of rocking [G. J. de Valcarcel and K. Staliunas, Phys. Rev. E 67, 026604 (2003)]. This conversion results in that vortices of the phase-invariant system are being replaced by phase patterns such as domain walls. The experiment is carried out on a photorefractive oscillator in two-wave mixing configuration.A model for the experimental device is given that reproduces the observed behavior.