Search

Quantum fluids of light are an emerging tool employed in quantum many-body physics. Their amazing properties and versatility allow using them in a wide variety of fields including gravitation, quantum information and simulation. However the implications of the quantum nature of light in the nonlinear optical propagation are still missing many features. We theoretically predict classical spontaneous squeezing of a photon fluid in a nonlocal nonlinear medium. By using the so called Gamow vectors, we show that the quadratures of a coherent state get squeezed and that a maximal squeezing power exists. Our analysis holds true for temporal and spatial optical propagation in highly nonlocal regime. These results open a new scenario in quantum photon fluids and may lead to novel applications in fields like metrology and analogues of quantum gravity., Comment: 6 pages, 2 figures

We report on the characterization of silica aerogel thermal optical nonlinearity, obtained by z-scan technique. The results show that typical silica aerogels have nonlinear optical coefficient similar to that of glass $(\simeq 10^{-12} $m$^2/$W), with negligible optical nonlinear absorption. The non\-li\-near coefficient can be increased to values in the range of $10^{-10} $m$^2/$W by embedding an absorbing dye in the aerogel. This value is one order of magnitude higher than that observed in the pure dye and in typical highly nonlinear materials like liquid crystals., Comment: 5 pages, 5 figures, Published by AIP Publishing

We unveil the relation between the linear Anderson localisation process and nonlinear modulation instability. Anderson localised modes are formed in certain temporal intervals due to the random background noise. Such localised modes seed the formation of solitary waves that will appear during the modulation instability process at those preferred intervals. Afterwards, optical-event horizon effects between dispersive waves and solitons produce an artificial collective acceleration that favours the collision of solitons, which could eventually lead to a rogue-soliton generation.

We have very little experience of the quantum dynamics of the ubiquitous nonlinear waves. Observed phenomena in high energy physics are perturbations to linear waves, and classical nonlinear waves, like solitons, are barely affected by quantum effects. We know that solitons, immutable in classical physics, exhibit collapse and revivals according to quantum mechanics. However this effect is very weak and has never been observed experimentally. By predicting black hole evaporation Hawking first introduced a distinctly quantum effect in nonlinear gravitational physics.Here we show the existence of a general and universal quantum process whereby a soliton emits quantum radiation with a specific frequency content, and a temperature given by the number of quanta, the soliton Schwarzschild radius, and the amount of nonlinearity, in a precise and surprisingly simple way. This result may ultimately lead to the first experimental evidence of genuine quantum black hole evaporation. In addition, our results show that black hole radiation occurs in a fully quantised theory, at variance with the common approach based on quantum field theory in a curved background; this may provide insights into quantum gravity theories. Our findings also have relevance to the entire field of nonlinear waves, including cold atomic gases and extreme phenomena such as shocks and rogue-waves. Finally, the predicted effect may potentially be exploited for novel tunable quantum light sources., Comment: 5 pages, 3 figures, minor changes to text and references

It is well known that a state with complex energy cannot be the eigenstate of a self-adjoint operator, like the Hamiltonian. Resonances, i.e. states with exponentially decaying observables, are not vectors belonging to the conventional Hilbert space. One can describe these resonances in an unusual mathematical formalism, based on the so-called Rigged Hilbert Space (RHS). In the RHS, the states with complex energy are denoted as Gamow Vectors (GV), and they model decay processes. We study GV of the Reversed Harmonic Oscillator (RHO), and we analytically and numerically investigate the unstable evolution of wave packets. We introduce the background function to study initial data not composed only by a summation of GV and we analyse different wave packets belonging to specific function spaces. Our work furnishes support to the idea that irreversible wave propagations can be investigated by means of Rigged Hilbert Space Quantum Mechanics and provides insights for the experimental investigation of irreversible dynamics., Comment: 16 pages, 19 figures, revised version

Multi-level quantum protocols may potentially supersede standard quantum optical polarization-encoded protocols in terms of amount of information transmission and security. However, for free space telecomunications, we do not have tools for limiting loss due to diffraction and perturbations, as for example turbulence in air. Here we study propagation invariant quantum X-waves with angular momentum; this representation expresses the electromagnetic field as a quantum gas of weakly interacting bosons. The resulting spatio-temporal quantized light pulses are not subject to diffraction and dispersion, and are intrinsically resilient to disturbances in propagation. We show that spontaneous down-conversion generates squeezed X-waves useful for quantum protocols. Surprisingly the orbital angural momentum affects the squeezing angle, and we predict the existence of a characteristic axicon aperture for maximal squeezing. There results may boost the applications in free space of quantum optical transmission and multi-level quantum protocols, and may also be relevant for novel kinds of interferometers, as satellite-based gravitational wave detectors.

The design of optical systems capable of processing and manipulating ultra-short pulses and ultra-focused beams is highly challenging with far reaching fundamental technological applications. One key obstacle routinely encountered while implementing sub-wavelength optical schemes is how to overcome the limitations set by standard Fourier optics. A strategy to overcome these difficulties is to utilize the concept of generalized uncertainty principle (G-UP) that has been originally developed to study quantum gravity. In this paper we propose to use the concept of G-UP within the framework of optics to show that the generalized Schrodinger equation describing short pulses and ultra-focused beams predicts the existence of a minimal spatial or temporal scale which in turn implies the existence of maximally localized states. Using a Gaussian wavepacket with complex phase, we derive the corresponding generalized uncertainty relation and its maximally localized states. We numerically show that the presence of nonlinearity helps the system to reach its maximal localization. Our results may trigger further theoretical and experimental tests for practical applications and analogues of fundamental physical theories., Comment: 10 pages, 5 figures

The use of virtually lossless topologically isolated edge states may lead to a novel class of thresholdless lasers operating without inversion. One needs however to understand if topological states may be coupled to external radiation and act as active cavities. We study a two-level topological insulator and show that self-induced transparency pulses can directly excite edge states. We simulate laser emission by a suitable designed topological cavity, and show that it can emit tunable radiation. For a configuration of sites following the off-diagonal Aubry-Andre-Harper model we solve the Maxwell-Bloch equations in the time domain and provide a first principle confirmation of topological lasers. Our results open the road to a new class of light emitters with topological protection for applications ranging from low-cost energetically-effective integrated lasers sources, also including silicon photonics, to strong coupling devices for studying ultrafast quantum processes with engineered vacuum.

We extend the concept of radially and azimuthally polarized optical beams to the polychromatic domain by introducing cylindrically polarized nondiffracting optical pulses. In particular, we discuss in detail the case of cylindrically polarized X-waves, both in the paraxial and nonparaxial regime. The explicit expressions for the electric and magnetic fields of cylindrically polarized X-waves is also reported.

The description of shock waves beyond the shock point is a challenge in nonlinear physics. Finding solutions to the global dynamics of dispersive shock waves is not always possible due to the lack of integrability. Here we propose a new method based on the eigenstates (Gamow vectors) of a reversed harmonic oscillator in a rigged Hilbert space. These vectors allow analytical formulation for the development of undular bores of shock waves in a nonlinear nonlocal medium. Experiments by a photothermal induced nonlinearity confirm theoretical predictions: as the undulation period as a function of power and the characteristic quantized decays of Gamow vectors. Our results demonstrate that Gamow vector are a novel and effective paradigm for describing extreme nonlinear phenomena., Comment: 5 pages, 3 figures

Dispersive shock waves dominate wave-breaking phenomena in Hamiltonian systems. In the absence of loss, these highly irregular and disordered waves are potentially reversible. However, no experimental evidence has been given about the possibility of inverting the dynamics of a dispersive shock wave and turn it into a regular wave-front. Nevertheless, the opposite scenario, i.e., a smooth wave generating turbulent dynamics is well studied and observed in experiments. Here we introduce a new theoretical formulation for the dynamics in a highly nonlocal and defocusing medium described by the nonlinear Schroedinger equation. Our theory unveils a mechanism that enhances the degree of irreversibility. This mechanism explains why a dispersive shock cannot be reversed in evolution even for an arbitrarirly small amount of loss. Our theory is based on the concept of nonlinear Gamow vectors, i.e., power dependent generalizations of the counter-intuitive and hereto elusive exponentially decaying states in Hamiltonian systems. We theoretically show that nonlinear Gamow vectors play a fundamental role in nonlinear Schroedinger models: they may be used as a generalized basis for describing the dynamics of the shock waves, and affect the degree of irreversibility of wave-breaking phenomena. Gamow vectors allow to analytically calculate the amount of breaking of time-reversal with a quantitative agreement with numerical solutions. We also show that a nonlocal nonlinear optical medium may act as a simulator for the experimental investigation of quantum irreversible models, as the reversed harmonic oscillator., Comment: 6 pages, 4 figure, Editors'Suggestion in Physical Review A

We introduce a new class of nondiffracting optical pulses possessing orbital angular momentum. By generalizing the X-waves solution of the Maxwell equation, we discover the coupling between angular momentum and the temporal degrees of freedom of ultra-short pulses. The spatial twist of propagation invariant light pulse turns out to be directly related to the number of optical cycles. Our results may trigger the development on novel multi-level classical and quantum transmission channels free of dispersion and diffraction, may also find application in the manipulation of nano-structured objects by ultra-short pulses, and for novel approaches to the spatio-temporal measurements in ultrafast photonics.

A novel comparison between the data and the theory is proposed for the nonmesonic (NM) weak decay of hypernuclei. Instead of confronting the primary decay rates, as is usually done, we focus attention on the effective decay rates that are straightforwardly related with the number of emitted particles. Proton kinetic energy spectra of $^5_\Lambda$He, $^7_\Lambda$Li, $^9_\Lambda$Be, $^{11}_\Lambda$B, $^{12}_{\Lambda}$C, $^{13}_\Lambda$C, $^{15}_{\Lambda}$N and $^{16}_{\Lambda}$O, measured by FINUDA, are evaluated theoretically. The Independent Particle Shell Model (IPSM) is used as the nuclear structure framework, while the dynamics is described by the One-Meson-Exchange (OME) potential. Only for the $^{5}_{\Lambda}$He, $^{7}_{\Lambda}$Li, and $^{12}_{\Lambda}$C hypernuclei is it possible to make a comparison with the data, since for the rest there is no published experimental information on number of produced hypernuclei. Considering solely the one-nucleon-induced ($1N$-NM) decay channel, the theory reproduces correctly the shapes of all three spectra at medium and high energies ($E_p \geq 40 $ MeV). Yet, it greatly overestimates their magnitudes, as well as the corresponding transition rates when the full OME ($\pi+K+ \eta+\rho+\omega+K^*$) model is used. The agreement is much improved when only the $\pi+K$ mesons with soft dipole cutoff parameters participate in the decay process. We find that the IPSM is a fair first order approximation to disentangle the dynamics of the $1N$-NM decay, the knowledge of which is indispensable to inquire about the baryon-baryon strangeness-flipping interaction. It is shown that the IPSM provides very useful insights regarding the determination the $2N$-NM decay rate. In a new analysis of the FINUDA data, we derive two results for this quantity with one of them close to that obtained previously.

We investigate the effect of non-paraxiality in the dynamics of dispersive shock waves in the defocusing nonlinear Schroedinger equation. We show that the problem can be described in terms of a relativistic particle moving in a potential. Lowest order corrections enhance the wave-breaking and impose a limit to the highest achievable spectrum in an amount experimentally testable., Comment: 6 pages, 4 figures

We present a theoretical analysis for the \GH and \IF shifts experienced by an X-wave upon reflection from a dielectric interface. We show that the temporal chirp, as well as the bandwidth of the X-wave directly affect the spatial shifts in a way that can be experimentally observed, while the angular shifts do not depend on the spectral features of the X-Wave. A dependence of the spatial shifts on the spatial structure of the X-wave is also discussed.

Using light to control the movement of nano-structured objects is a great challenge. This challenge involves fields like optical tweezing, Casimir forces, integrated optics, bio-physics, and many others. Photonic "robots" could have uncountable applications. However, if the complexity of light-activated devices increases, structural disorder unavoidably occurs and, correspondingly, light scattering, diffusion and localization. Are optically-driven mechanical forces affected by disorder-induced effects? A possible hypothesis is that light scattering reduces the optomechanical interaction. Conversely, we show that disorder is a mechanism that radically enhances the mechanical effect of light. We determine the link between optical pressure and the light diffusion coefficient, and unveil that when the Thouless conductivity becomes smaller than the unity, at the so-called Anderson transition, optical forces and their statistical fluctuations reach a maximum. Recent advances in photonics demonstrate the possibility of harnessing disorder for fundamental physics and applications. Designing randomness allows new materials with innovative optical properties. Here we show that disorder and related phenomena may be exploited for optomechanical devices., Comment: 7 pages, 4 figures

Anderson localization is a regime in which diffusion is inhibited and waves (also electromagnetic waves) get localized. Here we exploit adaptive optics to achieve focusing in disordered optical fibers in the Anderson regime. By wavefront shaping and optimization, we observe the generation of a propagation invariant beam, where light is trapped transversally by disorder, and show that Anderson localizations can be also excited by extended speckled beams. We demonstrate that disordered fibers allow a more efficient focusing action with respect to standard fibers in a way independent of their length, because of the propagation invariant features and cooperative action of transverse localizations.

We report the experimental observation of the interaction and attraction of many localized modes in a two dimensional (2D) system realized by a disordered optical fiber supporting transverse Anderson localization. We show that a nonlocal optically nonlinear response of thermal origin alters the localization length by an amount determined by the optical power and also induces an action at a distance between the localized modes and their spatial migration. Evidence of a collective and strongly interacting regime is given.

We present a unified approach to the theory of multimodal laser cavities including a variable amount of structural disorder. A general mean-field theory is studied for waves in media with variable non-linearity and randomness. Phase diagrams are reported in terms of optical power, degree of disorder and degree of non-linearity, tuning between closed and open cavity scenario's. In the thermodynamic limit of infinitely many modes the theory predicts four distinct regimes: a continuous wave behavior for low power, a standard mode-locking laser regime for high power and weak disorder, a random laser for high pumped power and large disorder and an intermediate regime of phase locking occurring in presence of disorder below the lasing threshold., Comment: 9 pages, 3 figures

We show that an analog of the physics at the Planck scale can be found in the propagation of tightly focused laser beams. Various equations that occur in generalized quantum mechanics are formally identical to those describing the nonlinear nonlocal propagation of nonparaxial laser beams. The analysis includes a generalized uncertainty principle and shows that the nonlinear focusing of a light beam with dimensions comparable to the wavelength corresponds to the spontaneous excitation of the so-called maximally localized states. The approach, driven by the ideas of the quantum gravity physics, allows one to predict the existence of self-trapped subwavelength solitary waves for both focusing and defocusing nonlinearities, and opens the way to laboratory simulations of phenomena that have been considered to be inaccessible.

We consider a microcavity made by a graded-index (GRIN) glass, doped by dye molecules, placed within two planar mirrors and study Bose-Einstein condensation (BEC) of photons. The presence of the mirrors leads to an effective photon mass, and the index grading provides an effective trapping frequency; the photon gas becomes formally equivalent to a two dimensional Bose gas trapped in an isotropic harmonic potential. The inclusion of nonlinear effects provides an effective interaction between photons. We discuss, in particular, thermal lensing effects and nonlocal nonlinearity, and quantitatively compare our results with the reported experimental data., Comment: 13 pages, 6 figures

Non-deterministic giant waves, denoted as rogue, killer, monster or freak waves, have been reported in many different branches of physics. Their origin is however still unknown: despite the massive numerical and experimental evidence, the ultimate reason for their spontaneous formation has not been identified yet. Here we show that rogue waves in optical fibres actually result from a complex dynamic process very similar to well known mechanisms such as glass transitions and protein folding. We describe how the interaction among optical solitons produces an energy landscape in a highly-dimensional parameter space with multiple quasi-equilibrium points. These configurations have the same statistical distribution of the observed rogue events and are explored during the light dynamics due to soliton collisions, with inelastic mechanisms enhancing the process. Slightly different initial conditions lead to very different dynamics in this complex geometry; a rogue soliton turns out to stem from one particular deep quasi-equilibrium point of the energy landscape in which the system may be transiently trapped during evolution. This explanation will prove fruitful to the wide community interested in freak waves.

A transparent material exhibits ultra-fast optical nonlinearity and is subject to optical pressure if irradiated by a laser beam. However, the effect of nonlinearity on optical pressure is often overlooked, even if a nonlinear optical pressure may be potentially employed in many applications, as optical manipulation, biophysics, cavity optomechanics, quantum optics, optical tractors, and is relevant in fundamental problems as the Abraham-Minkoswky dilemma, or the Casimir effect. Here we show that an ultra-fast nonlinear polarization gives indeed a contribution to the optical pressure that also is negative in certain spectral ranges; the theoretical analysis is confirmed by first-principles simulations. An order of magnitude estimate shows that the effect can be observable by measuring the deflection of a membrane made by graphene., Comment: 10 pages, 6 figures, minor corrections to text and references, Physical Review A, to be published

We investigate beam diffraction and spatial modulation instability of coherent light beams propagating in the non-paraxial regime in a nonlinear Kerr medium. We study the instability of plane wave solutions in terms of the degree of non-paraxiality, beyond the regime of validity of the nonlinear Schroedinger equation. We also numerically analyze the way non-paraxial terms breaks the integrability and affect the periodical evolution of higher order soliton solutions. We discuss non-locality and effective dissipation introduced by the non-linear coupling with evanescent waves., Comment: 8 pages, 6 figures

We investigate spatial localization in a quadratic nonlinear medium in the presence of randomness. By means of numerical simulations and theoretical analyses we show that, in the down conversion regime, the transverse random modulation of the nonlinear susceptibility generates localizations of the fundamental wave that grow exponentially in propagation. The localization length is optically controlled by the pump intensity which determines the amplification rate. The results also apply to cubic nonlinearities., Comment: 5 pages, 5 figures

Because of the huge commercial importance of granular systems, the second-most used material in industry after water, intersecting the industry in multiple trades, like pharmacy and agriculture, fundamental research on grain-like materials has received an increasing amount of attention in the last decades. In photonics, the applications of granular materials have been only marginally investigated. We report the first phase-diagram of a granular as obtained by laser emission. The dynamics of vertically-oscillated granular in a liquid solution in a three-dimensional container is investigated by employing its random laser emission. The granular motion is function of the frequency and amplitude of the mechanical solicitation, we show how the laser emission allows to distinguish two phases in the granular and analyze its spectral distribution. This constitutes a fundamental step in the field of granulars and gives a clear evidence of the possible control on light-matter interaction achievable in grain-like system., Comment: 16 pages, 7 figures

We investigate disorder induced localization in the presence of nonlinearity and curvature. We numerically analyze the time-resolved three-dimensional expansion of a wave-packet in a bended cigar shaped potential with a focusing Kerr-like interaction term and Gaussian disorder. We report on a self-consistent analytical theory in which randomness, nonlinearity and geometry are determined by a single scaling parameter, and show that curvature enhances localization.

Controlling the flow of energy in a random medium is a research frontier with a wide range of applications. As recently demonstrated, the effect of disorder on the transmission of optical beams, may be partially compensated by wavefront shaping, but losing control over individual light paths. Here we report on a novel physical effect whereby energy is spatially and spectrally transferred inside a disordered active medium by the coupling between individual lasing modes. We show that is possible to transmit an optical resonance to a remote point by employing specific control over optical excitations. The observed nonlinear transport bears some analogies to a field-effect transistor for light, which acts as a switch and as an amplifier.

The discovery of the spontaneous mode-locking of lasers, i.e., the synchronous oscillation of electromagnetic modes in a cavity, has been a milestone of photonics allowing the realization of oscillators delivering ultra-short pulses. This process is so far known to occur only in standard ordered lasers with meter size length and only in the presence of a specific device (the saturable absorber). Here we demonstrate that mode-locking can spontaneously arise also in random lasers composed by micronsized laser resonances dwelling in intrinsically disordered, self-assembled clusters of nanometer-sized particles. Moreover by engineering a novel mode-selective pumping mechanism we show that it is possible to continuously drive the system from a configuration in which the various excited electromagnetic modes oscillate in the form of several, weakly interacting, resonances to a collective strongly interacting regime. By realizing the smallest mode-locking device ever fabricated, we open the way to novel generation of miniaturized and all-optically controlled light sources.

We investigate the dynamics of a Bose-Einstein condensate in a progressively bended three dimensional cigar shaped potential. The interplay between geometry and nonlinearity is considered. At high curvature, topological localization occurs and becomes frustrated by the generation of curved dispersive shock-waves when the strength of nonlinearity is increased. The analysis is supported by four-dimensional parallel simulations., Comment: 5 pages, 4 figures, minor corrections

We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedinger equation and that of the disorder induced localization in the presence of a Gaussian random potential. With emphasis on the focusing nonlinearity, we consider the bound states of the nonlinear Schroedinger equation with a random potential; for the state exhibiting the highest degree of localization, we derive explicit expressions for the nonlinear eigenvalue and for the localization length by using perturbation theory and a variational approach following the methods of statistical mechanics of disordered systems. We numerically investigate the linear stability and "superlocalizations". The profile of the disorder averaged Anderson localization is found to obey a nonlocal nonlinear Schroedinger equation, Comment: 5 pages, 3 figures

Localized surface plasmon resonances have recently attracted considerable attention due to their ability to dramatically enhance near-field optical intensities and boost nanoscale light-matter interactions. Here we demonstrate unambiguously that polyhedral silver nanoparticles can be tailored to promote enhanced coherent emission from organic dye dispersions in a cavity. In particular, we observe that nanoparticles supporting resonances close to the absorption peak of the dye yield to more efficient lasing, up to one order of magnitude, which is also evidenced by the narrowing of the emission spectral line down to 3 nm. Moreover we observe emission saturation and energy dependent spectral shifts associated with Kerr effects, which demonstrate the general relevance of non-resonant nonlinearities in plasmonic enhanced laser emissions. Works in this area will contribute to the design of novel kind of laser sources providing ultra-short pulsed operation and structural tunability.

We show that the degree of localization for the modes of a random laser (RL) is affected by the inter mode interaction that is controlled by shaping the spot of the pump laser. By experimentally investigating the spatial properties of the lasing emission we infer that strongly localized modes are activated in the low interacting regime while in the strongly interacting one extended modes are found lasing. Thus we demonstrate that the degree o localization may be finely tuned at the micrometer level.

Using the history-dependence of a dipolar glass hosted in a compositionally-disordered lithium-enriched potassium-tantalate-niobate (KTN:Li) crystal, we demonstrate scale-free optical propagation at tunable temperatures. The operating equilibration temperature is determined by previous crystal spiralling in the temperature/cooling-rate phase-space.

Granular materials have been studied for decades, also driven by industrial and technological applications. These very simple systems, composed by agglomerations of mesoscopic particles, are characterized, in specific regimes, by a large number of metastable states and an extreme sensitivity (e.g., in sound transmission) on the arrangement of grains; they are not substantially affected by thermal phenomena, but can be controlled by mechanical solicitations. Laser emission from shaken granular matter is so far unexplored; here we provide experimental evidence that it can be affected and controlled by the status of motion of the granular, we also find that competitive random lasers can be observed. We hence demonstrate the potentialities of gravity affected moving disordered materials for optical applications, and open the road to a variety of novel interdisciplinary investigations, involving modern statistical mechanics and disordered photonics., Comment: 4 pages, 3 figures. To be published in Physical Review Letters

We theoretically investigate the stability and dynamics of self-channelled beams that form via nonlocal optomechanical interactions in dual-nanoweb microstructured fibers. These "optomechanicons" represent a novel class of spatial soliton.

We have evaluated the nonmesonic weak decay spectra within the independent-particle shell-model, and compared them with the recent measurements of: i) the single and double coincidence nucleon spectra in ${\mathrm{^{12}_{\Lambda}C}}$ performed at KEK, and ii) proton kinetic energy spectra in ${\mathrm{^{5}_{\Lambda}He}}$, ${\mathrm{^{7}_{\Lambda}Li}}$, ${\mathrm{^{9}_{\Lambda}Be}}$, ${\mathrm{^{11}_{\Lambda}B}}$, ${\mathrm{^{12}_{\Lambda}C}}$, ${\mathrm{^{13}_{\Lambda}C}}$, ${\mathrm{^{15}_{\Lambda}N}}$ and ${\mathrm{^{16}_{\Lambda}O}}$ done by FINUDA. Based on this comparison we argue that the extraction from the data of the three-body $\Lambda NN \rightarrow nNN$ induced decay rate, as done in these works, could be questionable., Comment: 15 pages, 8 figures

We perform a self-consistent relativistic QRPA (RQRPA) calculation of $2\nu\b\b$-decay based on relativistic BCS (RBCS) mean field theory results for odd-odd intermediate nuclei $^{48}$Sc, $^{76}$As, $^{82}$Br, $^{100}$Tc, $^{128}$I, and $^{130}$I. The RBCS equations that resemble the non-relativistic ones are constructed from a Dirac-Gorkov variational functional. We use the parameter set $NL1$ for the $\sigma$, $\omega$ and $\rho$ mesons. The RQRPA equations are solved for the residual $\pi+\rho$ interaction by employing the same parameters used in the RBCS for the latter meson, and experimental values for the pion and nucleon. The RQRPA results for the ${2\nu}\b\b$ matrix elements are similar to those obtained within the QRPA and the shell model., Comment: 8 pages, XXXIV edition of the Brazilian Workshop on Nuclear Physics, 5-10 June 2011, Foz de Igua\c{c}u, Paran\'a state, Brasil