On the continuous resonant equation for NLS: II. Statistical study

We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schr{\"o}dinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of NLS in confined regimes (e.g. on a compact domain or with a trapping potential). The system was derived and studied from a deterministic viewpoint in several earlier works, which uncovered many of its striking properties. This manuscript is devoted to a probabilistic study of this system. Most notably, we construct global solutions in negative Sobolev spaces, which leave Gibbs and white noise measures invariant. Invariance of white noise measure seems particularly interesting in view of the absence of similar results for NLS.Comment: 23 page

Experimental demonstration of intermodulation effects in a continuous cesium fountain microwave frequency standard

International audienceThe short-term stability of passive atomic frequency standards, particularly ones operated sequentially, is often limited by local oscillator noise via intermodulation effects. This article describes an experimental demonstration of the intermodulation effect on the frequency stability of a continuous atomic fountain standard usually imperceptible under normal operating conditions. To make the effect observable, we increase the phase instability of the microwave field interrogating the clock transition.We measure the frequency stability of the locked, commercial local oscillator, for both square-wave phase modulation and squarewave frequency modulation of the microwave field. The observed degradation of the stability depends on the modulation frequency in a way that agrees with our earlier theoretical predictions. Most significantly, no degradation is observed when the modulation frequency is made equal to the Ramsey linewidth. When no extra phase noise is added, the frequency instability, currently 2.0x10-13 at 1 s, is limited only by atomic shot-noise. This shows the potential to reduce it via the use of a higher atomic flux

On the continuous resonant equation for NLS: I. Deterministic analysis

International audienceWe study the continuous resonant (CR) equation which was derived by Faou-Germain-Hani as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces of the harmonic oscillator, of the rotation operator, and the Bargmann-Fock space. We focus on stationary waves and their stability

Nonlinear quasimodes near elliptic periodic geodesics

International audienceWe consider the nonlinear Schrödinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schrödinger evolution of such a quasimode remains localized near the geodesic, at least for short times

Design Details of FOCS-2, an Improved Continuous Cesium Fountain Frequency Standard

International audienceWe report on the design, construction and current status of FOCS-2, the second continuous fountain microwave cesium frequency standard after FOCS-1. Both incorporate velocity-selective light traps driven by an electrostatic motor. FOCS-2 will take fuller advantage of the continuous fountain approach to gain in shot-noise-limited stability without loss of accuracy via the use of a higher flux. This is obtained via the implementation of a novel slow-atom pre-source and better collimation of the atomic beam. A detailed description of the apparatus is provided and compared with FOCS-1 to highlight improvements. In addition, we present results from related experiments on collimation in a 2D optical lattice. The goals for this new standard are a short-term stability of < 4x10-14 $\tau$-1/2 and a relative frequency uncertainty of < 1 x 10-15

Combined quantum state preparation and laser cooling of a continuous beam of cold atoms

We use two-laser optical pumping on a continuous atomic fountain in order to prepare cold cesium atoms in the same quantum ground state. A first laser excites the F=4 ground state to pump the atoms toward F=3 while a second pi-polarized laser excites the F=3 -> F'=3 transition of the D2 line to produce Zeeman pumping toward m=0. To avoid trap states, we implement the first laser in a 2D optical lattice geometry, thereby creating polarization gradients. This configuration has the advantage of simultaneously producing Sisyphus cooling when the optical lattice laser is tuned between the F=4 -> F'=4 and F=4 -> F'=5 transitions of the D2 line, which is important to remove the heat produced by optical pumping. Detuning the frequency of the second pi-polarized laser reveals the action of a new mechanism improving both laser cooling and state preparation efficiency. A physical interpretation of this mechanism is discussed.Comment: Minor changes according to the recommendations of the referee: - Corrected Fig.1. - Split the graph of Fig.6 for clarity. - Added one reference. - Added two remarks in the conclusion. - Results unchange