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We study modulational instability (MI) in optical fibers with random group-velocity dispersion (GVD). We consider Gaussian and dichotomous colored stochastic processes. We resort to different analytical methods (namely, the cumulant expansion and the functional approach) and assess their reliability in estimating the MI gain of stochastic origin. If the power spectral density (PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-frequency MI sidelobes which converge to those given by a white noise perturbation when the correlation length tends to 0. If instead the stochastic processes are modulated in space, one or more MI sidelobe pairs corresponding to the well-known parametric resonance (PR) condition can be found. A transition from small and broad sidelobes to peaks nearly indistinguishable from PR-MI is predicted, in the limit of large perturbation amplitudes and correlation lengths of the random process. We find that the cumulant expansion provides good analytical estimates for small PSD values and small correlation lengths, when the MI gain is very small. The functional approach is rigorous only for the dichotomous processes, but allows us to model a wider range of parameters and to predict the existence of MI sidelobes comparable to those observed in homogeneous fibers of anomalous GVD, 12 pages, 6 figures submitted

International audience; Given two orthonormal bases in a d-dimensional Hilbert space, one associates to each state its Kirkwood-Dirac (KD) quasi-probability distribution. KD-nonclassical states - for which the KD-distribution takes on negative and/or nonreal values - have been shown to provide a quantum advantage in quantum metrology and information, raising the question of their identification. Under suitable conditions of incompatibility between the two bases, we provide sharp lower bounds on the support uncertainty of states that guarantee their KD-nonclassicality. In particular, when the bases are completely incompatible, a notion we introduce, states whose support uncertainty is not equal to its minimal value d+1 are necessarily KD-nonclassical. The implications of these general results for various commonly used bases, including the mutually unbiased ones, and their perturbations, are detailed.

The quantum nature of the state of a bosonic quantum field manifests itself in its entanglement, coherence, or optical nonclassicality which are each known to be resources for quantum computing or metrology. We provide quantitative and computable bounds relating entanglement measures with optical nonclassicality measures. These bounds imply that strongly entangled states must necessarily be strongly optically nonclassical. As an application, we infer strong bounds on the entanglement that can be produced with an optically nonclassical state impinging on a beam splitter. For Gaussian states, we analyze the link between the logarithmic negativity and a specific nonclassicality witness called "quadrature coherence scale"., Comment: 13 pages, 2 figures, change of notation in v2

We introduce, for each state of a bosonic quantum field, its quadrature coherence scale (QCS), a measure of the range of its quadrature coherences. Under coupling to a thermal bath, the purity and QCS are shown to decrease on a time scale inversely proportional to the QCS squared. The states most fragile to decoherence are therefore those with quadrature coherences far from the diagonal. We further show a large QCS is difficult to measure since it induces small scale variations in the state's Wigner function. These two observations imply a large QCS constitutes a mark of "macroscopic coherence". Finally, we link the QCS to optical classicality: optical classical states have a small QCS and a large QCS implies strong optical nonclassicality., 12 pages, 5 figures. New version to match the published version. Minor errors were corrected

Entanglement, coherence and optical nonclassicality, all reveal the quantum nature of a state and can be viewed as a resource for quantum computing. We provide quantitative and computable bounds relating those quantities.

In the last decades much attention has been attracted to the photon heralding technique [1], which consists in generating a photon pair in two modes and detecting a photon of one (idler) mode, preparing thus the other (signal) mode in a single-photon state. In this work we consider spontaneous parametric downconversion (SPDC) as a source of photon pairs and find the density matrix of the signal mode conditioned by a photon detection in the idler one. We take into account three physical parameters of the scheme: the degree of two-mode squeezing r produced by SPDC, the transmittance of the idler channel η, which includes the quantum efficiency of the photodetector, and the transmittance of the signal channel μ, see Fig. 1(a). We show that the conditional state of the signal mode is a weighted difference of two thermal states $\rho^{(-)} (q,p,\gamma) = \mathscr{N} \left({\rho_{\mathrm{th}} (q) \over 1-q} - \gamma {\rho_{\mathrm{th}}(q p) \over 1-qp} \right) = \mathscr{N} \sum^{\infty}_{n=0} (d^{n}- \gamma (qp)^{n}) \vert n \rangle \langle n \vert, \quad (1)$

International audience; We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasi-probabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyse the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

International audience; We review a recently proposed measure of the nonclassicality of a bosonic field, based on the sensitivity of its quasi-probability distributions to ordering of the creation and annihilation operators. We illustrate the new measure by several concrete examples and show its advantages compared to other measures of nonclassicality such as the Wigner function negativity and the entanglement potential.

An otherwise free classical particle moving through an extended spatially homogeneous medium with which it may exchange energy and momentum will undergo a frictional drag force in the direction opposite to its velocity with a magnitude which is typically proportional to a power of its speed. We study here the quantum equivalent of a classical Hamiltonian model for this friction phenomenon that was proposed in [11]. More precisely, we study the spectral properties of the quantum Hamiltonian and compare the quantum and classical situations. Under suitable conditions on the infrared behaviour of the model, we prove that the Hamiltonian at fixed total momentum has no ground state except when the total momentum vanishes, and that its spectrum is otherwise absolutely continuous., Comment: 40 pages

International audience; We investigate the nonlinear stage of modulational instability in dispersion oscillating fibers in normal dispersion regime. We unveil a heteroclinic structure leading to the excitability of superbreathers and parametric amplification outside the linear gain bandwidth.

Demonstrating how microscopic dynamics cause large systems to approach thermal equilibrium remains an elusive, longstanding, and actively-pursued goal of statistical mechanics. We identify here a dynamical mechanism for thermalization in a general class of two-component dynamical Lorentz gases, and prove that each component, even when maintained in a non-equilibrium state itself, can drive the other to a thermal state with a well-defined effective temperature., 5 pages, 5 figures

International audience; We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.

We introduce a three-parameter family of single-mode optical states whose density operator is a weighted difference of two thermal states, the thermal-difference states. We identify the parameter values for which these states have a negative non-singular P-function, implying they are nonclassical. We show that the states of the “heralded photons” generated via spontaneous parametric downconversion belong to this family, with the three parameters corresponding to the nonlinear gain and the losses in the signal and the idler channels. The thermal-difference states yield new benchmark states for the analysis of nonclassicality and quantum macroscopicity criteria.

International audience; We are interested in a kinetic equation intended to describe the interaction of particles with their environment. The environment is modeled by a collection of local vibrational degrees of freedom. We establish the existence of weak solutions for a wide class of initial data and external forces. We also identify a relevant regime which allows us to derive, quite surprisingly, the attractive Vlasov–Poisson system from the coupled Vlasov-Wave equations.

International audience; We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

International audience; It is shown that sufficiently large periodic modulations in the coefficients of a nonlinear Schrödinger equation can drastically impact the spatial shape of the Peregrine soliton solutions: they can develop multiple compression points of the same amplitude, rather than only a single one, as in the spatially homogeneous focusing nonlinear Schrödinger equation. The additional compression points are generated in pairs forming a comblike structure. The number of additional pairs depends on the amplitude of the modulation but not on its wavelength, which controls their separation distance. The dynamics and characteristics of these generalized Peregrine solitons are analytically described in the case of a completely integrable modulation. A numerical investigation shows that their main properties persist in nonintegrable situations, where no exact analytical expression of the generalized Peregrine soliton is available. Our predictions are in good agreement with numerical findings for an interesting specific case of an experimentally realizable periodically dispersion modulated photonic crystal fiber. Our results therefore pave the way for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in the wide class of physical systems modeled by the nonlinear Schrödinger equation.

We introduce and study rigorously a Hamiltonian model of a classical particle moving through a homogeneous dissipative medium at zero temperature in such a way that it experiences an effective linear friction force proportional to its velocity (at small speeds). The medium consists at each point in a space of a vibration field modelling an obstacle with which the particle exchanges energy and momentum in such a way that total energy and momentum are conserved. We show that in the presence of a constant (not too large) external force, the particle reaches an asymptotic velocity proportional to this force. In a potential well, on the other hand, the particle comes exponentially fast to rest in the bottom of the well. The exponential rate is in both cases an explicit function of the model parameters and independent of the potential.

We study the behaviour, in the simultaneous limits \hbar going to 0, t going to \infty, of the Husimi and Wigner distributions of initial coherent states and position eigenstates, evolved under the quantized hyperbolic toral automorphisms and the quantized baker map. We show how the exponential mixing of the underlying dynamics manifests itself in those quantities on time scales logarithmic in \hbar. The phase space distributions of the coherent states, evolved under either of those dynamics, are shown to equidistribute on the torus in the limit \hbar going to 0, for times t between |\log\hbar|/(2\gamma) and |\log|\hbar|/\gamma, where \gamma is the Lyapounov exponent of the classical system. For times shorter than |\log\hbar|/(2\gamma), they remain concentrated on the classical trajectory of the center of the coherent state. The behaviour of the phase space distributions of evolved position eigenstates, on the other hand, is not the same for the quantized automorphisms as for the baker map. In the first case, they equidistribute provided t goes to \infty as \hbar goes to 0, and as long as t is shorter than |\log\hbar|/\gamma. In the second case, they remain localized on the evolved initial position at all such times.

Resume Nous prouvons que les valeurs propres (quasi-energies) des automorphismes hyperboliques du tore quantifies, ainsi que celles de leurs perturbations hamiltoniennes lisses sont, dans la limite semi-classique, equidistribuees sur le cercle unite. Nous en deduisons l'equipartition des fonctions propres dont les quasi-energies sont restreintes a un arc de ce cercle.

Quantum field theory is the study of quantum systems with an infinite number of degrees of freedom. The simplest quantum field theories are the free bose fields, which are essentially assemblies of an infinite number of coupled oscillators. Examples include the quantized electromagnetic field, lattice vibrations in solid state physics, and the Klein-Gordon field, some of which are relativistic while others are not. Particles show up in quantum field theory as “field quanta.” As we will see, these do not quite have all the properties of the usual particles of Newtonian physics or of Schrodinger quantum mechanics. I will in particular show (Section 3) that the quanta of free bose fields, relativistic or not, can not be perfectly localized in a bounded subset of space. This, in my opinion, shows conclusively that the difficulties encountered when attempting to define a position observable for the field quanta of relativistic fields, that continue to be the source of regular debate in the literature [20] [21] [22] [3] [10] [15] [18] [19] [48] [47], do not find their origin in any form of causality violation, as seems to be generally thought. Instead, they result from an understandable but ill-fated attempt to force too stringent a particle interpretation on the states of the quantum field containing a finite number of quanta.

We study the closed Hamiltonian dynamics of a free particle moving on a ring, over one section of which it interacts linearly with a single harmonic oscillator. On the basis of numerical and analytical evidence, we conjecture that at small positive energies the phase space of our model is completely chaotic except for a single region of complete integrability with a smooth sharp boundary showing no KAM-type structures of any kind. This results in the cleanest mixed phase space structure possible, in which motions in the integrable region and in the chaotic region are clearly separated and independent of one another. For certain system parameters, this mixed phase space structure can be tuned to make either of the two components disappear, leaving a completely integrable or completely chaotic phase space. For other values of the system parameters, additional structures appear, such as KAM-like elliptic islands, and one parameter families of parabolic periodic orbits embedded in the chaotic sea. The latter are analogous to bouncing ball orbits seen in the stadium billiard. The analytical part of our study proceeds from a geometric description of the dynamics, and shows it to be equivalent to a linked twist map on the union of two intersecting disks., Comment: 17 pages, 11 figures Typos corrected to display section labels

We study the Hamiltonian dynamics of a free particle injected onto a chain containing a periodic array of harmonic oscillators in thermal equilibrium. The particle interacts locally with each oscillator, with an interaction that is linear in the oscillator coordinate and independent of the particle's position when it is within a finite interaction range. At long times the particle exhibits diffusive motion, with an ensemble averaged mean-squared displacement that is linear in time. The diffusion constant at high temperatures follows a power law D ~ T^{5/2} for all parameter values studied. At low temperatures particle motion changes to a hopping process in which the particle is bound for considerable periods of time to a single oscillator before it is able to escape and explore the rest of the chain. A different power law, D ~ T^{3/4}, emerges in this limit. A thermal distribution of particles exhibits thermally activated diffusion at low temperatures as a result of classically self-trapped polaronic states., Comment: 15 pages, 4 figures Submitted to Physical Review E

We show that in the semi-classical limit the eigenfunctions of quantized ergodic symplectic toral automorphisms can not concentrate in measure on a finite number of closed orbits of the dynamics. More generally, we show that, if the pure point component of the limit measure has support on a finite number of such orbits, then the mass of this component must be smaller than two thirds of the total mass. The proofs use only the algebraic (i.e. not the number theoretic) properties of the toral automorphisms together with the exponential instability of the dynamics and therefore work in all dimensions., Comment: Latex file, 19 pages

International audience; In the single band tight-binding approximation, we consider the transport properties of an electron in a homogeneous static electric field. We show that repeated interactions of the electron with two-level systems in thermal equilibrium suppress the Bloch oscillations and induce a steady current, the statistical properties of which we study.

This thesis is devoted to the study of traveling waves of the nonlocal Gross-Pitaevskii equation with nonzero conditions at infinity. The Gross-Pitaevskii equation is a Hamiltonian equation and arises in several areas of quantum physics such as nonlinear optics, superfluidity and Bose-Einstein condensation. There have been extensive studies concerning the traveling waves, particularly in the local case, since the Jones-Roberts programme in 1982. In order to describe more realistic physical interactions, we consider the nonlocal Gross-Pitaevskii equation. The first chapter is devoted to the numerical and theoretical aspects of the nonzero conditions at infinity, in the case of the linear Schrödinger equation. We show that the solution of the linear equation shows a quasi-universal behaviour and we illustrate it with numerical simulations. Then, we provide conditions on the nonlocal interaction such that there exists a branch of nontrivial traveling waves. We also show that this branch is orbitally stable. Our results generalize the local case and rely on a minimisation under constraints approach, the study of the minimizing curve and a concentration-compactness argument. Moreover, we generalize the properties of the minimizing curve in dimension N. Finally, we propose and implement a gradient method in dimension 1 and a penalty method in dimension 2 to numerically compute the traveling waves and the energy curve for nonlocal potentials. In each method, the nonlocal term is treated by the Fast Fourier Transform.; Cette thèse est consacrée à l’étude des ondes progressives de l’équation Gross–Pitaevskii non localeavec des conditions non nulles à l’infini. L’équation de Gross–Pitaevskii est une équation hamiltonienneapparaissant dans divers domaines de la physique tels que l’optique non linéaire, la superfluidité ou lacondensation de Bose-Einstein. L’étude des ondes progressives pour l’équation de Gross–Pitaevskii faitl’objet de nombreux travaux depuis les résultats de Jones et Roberts en 1982, principalement dans lecas local. Afin de modéliser des interactions plus réalistes, il est intéressant de considérer l’équation deGross–Pitaevskii non locale. Avant de traiter la question des ondes progressives, on consacre le premierchapitre à l’étude des conditions non nulles à l’infini d’un point de vue numérique et théorique, dans lecas de l’équation de Schrödinger linéaire. Nous montrons que la solution de l’équation linéaire présenteun comportement asymptotique quasi-universel dans ce cas, ce que l’on illustre numériquement. Ensuite,nous montrons que, pour une famille d’interaction non locales, il existe une branche d’ondes progressivesnon triviales, orbitalement stable, en dimension 1. Notre résultat généralise le cas local et la preuveest basée sur un argument de minimisation sous contraintes, l’étude de la courbe minimisante et leprincipe de concentration compacité. En outre, on généralise les propriétés de la courbe minimisanteen dimension N, dans le cas non local. Enfin, dans le dernier chapitre, nous proposons une méthodede gradient avec projection en dimension 1 et une méthode de pénalisation en dimension 2 afin decalculer numériquement les ondes progressives et la courbe d’énergie pour certains noyaux. Dans cesdeux méthodes, l’utilisation de la transformée de Fourier rapide est cruciale afin de traiter l’interactionnon locale.

In this thesis, we study the dynamics of a particle in an inelastic environment composed of scatterer which is commonly known as inelastic Lorentz gas. In the inert case, the environment is not affected by the particle. The kinetic energy of the latter grows with the time and this phenomenon is called « stochastic acceleration ». We approximate the particle's motion by a Markov chain where each step corresponds to a unique collision of the particle with a scatterer. We show that the particle's averaged kinetic energy grows with the time with the exponent 2/5. The result is proved by using probabilistic arguments, bringing into weak convergence theorems of Markov chain as well as the weak convergence of the chain, correctly rescaled in time and space, to a Bessel process. We thus obtain a convergence result for the velocity vector. Under a different rescaling that the one used for the kinetic energy, the latter converges weakly to a spherical brownian motion. In the dynamical case, the evolution of the degrees of freedom of the Lorentz gas is affected by the particle and the dynamical system considered is constitued of the particle and the environment. In such a system, the stochastic acceleration phenomenon cannot be observed. However, we show that the velocity distribution admits a stationnary state.; Dans cette thèse, nous étudions la dynamique d'une particule dans un milieu inélastique composé de diffuseur et communément appelé gaz de Lorentz inélastique. Dans le cas inerte, le milieu n'est pas affecté par le passage de la particule. L'énergie cinétique de celle-ci croît avec le temps et ce phénomène est appelé « accélération stochastique ». Nous approximons le mouvement de la particule par une chaîne de Markov dont chaque pas correspond à une unique collision entre la particule et un diffuseur. Nous montrons que l'énergie cinétique moyenne de la particule croît avec le temps avec pour exposant 2/5. Ce résultat est montré en utilisant des arguments probabilistes, utilisant des théorèmes de convergence de chaînes de Markov ainsi que la convergence en distribution de la chaîne, correctement changée d'échelle en temps et en espace, vers un processus de Bessel. Nous obtenons également un résultat de convergence pour le vecteur vitesse. Sous un changement d'échelle différent de celui utilisé pour l'énergie cinétique, celui-ci converge en distribution vers un mouvement brownien sphérique. Dans le cas dynamique, l'évolution des degrés de liberté du gaz de Lorentz est affecté par le passage de la particule et le système dynamique considéré est composé de la particule et du milieu. Dans un tel système, le phénomène d'accélération stochastique ne peut pas être observé. En revanche nous montrons que la distribution des vitesses admet un état stationnaire.