S. De Bièvre

Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation

We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.

Entanglement of quantum circular states of light

We present a general approach to calculating the entanglement of formation for superpositions of two-mode coherent states, placed equidistantly on a circle in the phase space. We show that in the particular case of rotationally-invariant circular states the Schmidt decomposition of two modes, and therefore the value of their entanglement, are given by analytical expressions. We analyse the dependence of the entanglement on the radius of the circle and number of components in the superposition. We also show that the set of rotationally-invariant circular states creates an orthonormal basis in the state space of the harmonic oscillator, and this basis is advantageous for representation of other circular states of light.

Operating modes distinguishability condition in switching systems

In this paper, we consider a switching system with several operating modes which represent normal or faulty behavior. The objective is to find a necessary and sufficient condition to distinguish these modes through the input/output data. If faulty operation modes are considered the distinguishability condition corresponds to a fault detectability condition. As in [1], we define in this paper two notions of distinguishability: the distinguishability through the input/output data of the system and the distinguishability through the parity residuals of each mode. The Necessary and Sufficient Condition (NSC) of distinguishability through parity-space residuals in [1] was obtained in the case where the parity-space orders are identical in all modes. Here a more generalized NSC of distinguishability is provided for systems which could have different parity-space orders. We show in this context two main results: a NSC of distinguishability through parity residuals which generalizes the NSC in [1] and the equivalence between the two notions of distinguishability. We also demonstrate that the distinguishability definition in [1] is linked to the notion of equivalent representations used in the theory of realization.

Entanglement of two-mode Schrödinger cats

Quantum superpositions of coherent states are produced both in microwave and optical domains, and are considered realizations of the famous “Schroedinger cat” state. The recent progress shows an increase in the number of components and the number of modes involved. Our work gives a theoretical treatment of multicomponent two-mode Schroedinger cat states. We consider a class of single-mode states, which are superpositions of N coherent states lying on a circle in the phase space. In this class we consider an orthonormal basis created by rotationally-invariant circular states (RICS). A two-mode extension of this basis is created by splitting a single-mode RICS on a balanced beam-splitter. After performing a symmetric (Loewdin) orthogonalization of the sets of coherent states in both modes we obtain the Schmidt decomposition of the two-mode state, and therefore an analytic expression for its entanglement. We show that the states obtained by splitting a RICS are generalizations of Bell states of two qubits to the case of N -level systems encoded into superpositions of coherent states on the circle, and we propose for them the name of generalized quasi-Bell states. We show that an exact probabilistic teleportation of arbitrary superposition of coherent states on the circle is possible with such a state used as shared resource.

Nonlinear stage of modulation instability in dispersion oscillating fibers

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.

Topographic Optical Fibers: A New Degree of Freedom in Nonlinear Optics

We investigate theoretically and experimentally basic nonlinear effects such as soliton propagation or modulation instability in what we called topographic optical fibers. We show that in these fibers which parameters are longitudinally modulated over a scale of a few meters, new dynamics are observed. As a consequence it adds a new degree of freedom in nonlinear optics and allows to experimentally explore original phenomena.

On the Distinguishability of Positive Linear Time-Invariant Systems with Affine Parametric Uncertainties K. Motchon

Abstract Studies of distinguishability have focused on the case of Linear Time-Invariant systems without uncertainties. In this work, distinguishability is studied for Positive Linear Time-Invariant systems with affine parametric uncertainties in the state space model. We propose a definition of distinguishability adapted to this new context and give a characterization of this notion. The approach used is based on the estimate of the reachable output space of the systems. Under suitable assumptions, a sufficient condition for distinguishability is established.