E. MATHIEU. — Mémoire sur le mouvement vibratoire des cloches (Journal de l'École Polytechnique, LIe Cahier, 1882)

Mathieu, Emmanuelle

doi:10.1051/jphystap:01883002003200

Cited by 15 publications

(23 citation statements)

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“…The ion full-motion into the quadrupole can be composed of two components: the micro-motion and the macro-motion [10,16]. The transverse ion motions are governed by the so-called Mathieu equations [11,12,17] and their stability is insured by the Mathieu parameter 0 < q < 0.908 which is defined by the following formula [12,16]:…”

Section: Introductionmentioning

confidence: 99%

Simulations of high intensity ion beam RFQ cooler for DESIR/SPIRAL 2: SHIRaC

et al. 2014

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ABSTRACT:The transport simulation of a high intensity ions beam with a radio-frequency cooler within the SPIRAL 2/GANIL High Intensity Radiofrequency Cooler, SHIRAC, installed at the SPIRAL 2/DESIR facility, is presented. Two simulation methods of cooling process in presence of space charge effect will be studied. The beam properties degradation by the coulomb repulsion and the buffer gas diffusion will be discussed. Finally, a comparison in term of transmission between simulated and experimental results for 133 Cs + ions beam of intensity going up to 1µA, will be outlined.

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Section: Introductionmentioning

confidence: 99%

Simulations of high intensity ion beam RFQ cooler for DESIR/SPIRAL 2: SHIRaC

et al. 2014

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“…At later delay times, i.e. t > 0.5 ns, when the parameters of the LLG equation become time-independent, it can be reduced to the oscillator equation where Equation represents the magnetoelastic analogue of the Mathieu equation . The well-known condition q ε xx > Γω SAW to observe the lowest-order parametric resonance at in the case of the experimental results shown in Figure reads Thus, we assign the observed peak at the 2.9 GHz frequency to the parametric resonance .…”

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confidence: 63%

“…Here we demonstrate two interesting examples of the interaction of the acoustic waves with the magnetization: The first example is the resonant excitation of the FMR precession in the Ni/fused silica grating with Λ = 1000 nm by the spatial third harmonic of a SAW with acoustic wavelength Λ/3 = 333 nm. The second example is the observation of the fingerprints of a parametric resonance at the 1/2 SAW frequency, described by the magnetoelastic Mathieu equation , in the experiments performed on the Ni/Si grating with Λ = 800 nm. The quantitative agreement between time-resolved MOKE measurements and the solutions of the magnetoelastically driven Landau–Lifshitz–Gilbert (LLG) equations corroborates their ability to predict ultrafast magnetoelastic responses of magnetic-grating-based metasurfaces.…”

mentioning

confidence: 99%

“…Second, the analysis of the Ni grating on silicon with Λ = 800 nm revealed a weak and narrow ferromagnetic resonance peak at the 1 2 ω normalS normalA normalW frequency, which has been accurately reproduced by the numerical LLG-modeling. Interestingly, the analysis of the underlying magnetoelastic oscillator equation made it possible to interpret this peak as a fingerprint of the parametric resonance of the Mathieu equation …”

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confidence: 99%

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Quantitative Ultrafast Magnetoacoustics at Magnetic Metasurfaces

Alekhin,

Lomonosov,

Leo

et al. 2023

Nano Lett.

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Femtosecond (fs) time-resolved magneto-optics is applied to investigate laser-excited ultrafast dynamics of onedimensional nickel gratings on fused silica and silicon substrates for a wide range of periodicities Λ = 400−1500 nm. Multiple surface acoustic modes with frequencies up to a few tens of GHz are generated. Nanoscale acoustic wavelengths Λ/n have been identified as n th -spatial harmonics of Rayleigh surface acoustic wave (SAW) and surface skimming longitudinal wave (SSLW), with acoustic frequencies and lifetimes being in agreement with theoretical calculations. Resonant magnetoelastic excitation of the ferromagnetic resonance (FMR) by SAW's third spatial harmonic, and, most interestingly fingerprints of the parametric resonance at 1/2 SAW frequency have been observed. Numerical solutions of Landau−Lifshitz−Gilbert (LLG) equation magnetoelastically driven by complex polychromatic acoustic fields quantitatively reproduce all resonances at once. Thus, our results provide a solid experimental and theoretical base for a quantitative understanding of ultrafast fs-laser-driven magnetoacoustics and tailoring the magnetic-gratingbased metasurfaces at the nanoscale.

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“…Much discussion is currently focused on the necessary and sufficient conditions to fulfill the three criteria listed above [22][23][24][25][26][27]. For example, discrete time-translation symmetry breaking (DTTSB) leading to (i) has long been known in the field of nonlinear dynamics and is, for instance, a prominent phenomenon in Kerr Parametric Oscillators (KPOs) [28][29][30][31][32][33]. In this context, the many-body aspect (iii) naturally emerges from the existence of normal modes for both weak and strong coupling between local degrees of freedom [34].…”

Section: Introductionmentioning

confidence: 99%

The role of fluctuations in quantum and classical time crystals

Heugel¹,

Eichler²,

Chitra³

et al. 2022

Preprint

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Discrete time crystals (DTCs) are a many-body state of matter whose dynamics are slower than the forces acting on it. The same is true for classical systems with period-doubling bifurcations. Hence, the question naturally arises what differentiates classical from quantum DTCs. Here, we analyze a variant of the Bose-Hubbard model, which describes a plethora of physical phenomena and has both a classical and a quantum time-crystalline limit. We study the role of fluctuations on the stability of the system and find no distinction between quantum and classical DTCs. This allows us to probe the fluctuations in an experiment using two strongly coupled parametric resonators subject to classical noise.

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E. MATHIEU. — Mémoire sur le mouvement vibratoire des cloches (Journal de l'École Polytechnique, LIe Cahier, 1882)

Cited by 15 publications

References 0 publications

Simulations of high intensity ion beam RFQ cooler for DESIR/SPIRAL 2: SHIRaC

Simulations of high intensity ion beam RFQ cooler for DESIR/SPIRAL 2: SHIRaC

Quantitative Ultrafast Magnetoacoustics at Magnetic Metasurfaces

The role of fluctuations in quantum and classical time crystals

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