New adaptive optics prototype system for the ESO 3.6-m telescope: Come-on-Plus
Following the successful astronomical runs of the Come-On adaptive optics prototype on the ESO 3.... more Following the successful astronomical runs of the Come-On adaptive optics prototype on the ESO 3.6 m telescope in La Silla, Chile an upgraded version called Come-On-Plus is currently being constructed and was set up in December 1992. This paper describes the main improvements of this new system. In particular, the 52 actuator deformable mirror with 30 Hz closed loop bandwidth, the modal control, the high detectivity wavefront sensor channel, and the infrared imaging channel are presented. Finally, the laboratory tests of each subsystem are analyzed. This second prototype is dedicated to routine astronomical observing as well as providing design parameters for the adaptive optics for the ESO Very Large Telescope (VLT).
We study the Floquet Hamiltonian −i∂ t + H + V (ωt), acting in L 2 ([ 0, T ], H, dt), as dependin... more We study the Floquet Hamiltonian −i∂ t + H + V (ωt), acting in L 2 ([ 0, T ], H, dt), as depending on the parameter ω = 2π/T . We assume that the spectrum of H in H is discrete, Spec(H) = {h m } ∞ m=1 , but possibly degenerate, and that t → V (t) ∈ B(H) is a 2π-periodic function with values in the space of Hermitian operators on H. Let J > 0 and set Ω 0 = [ 8 9 J, 9 8 J ]. Suppose that for some σ > 0 it holds true that hm>hn µ mn (h m − h n ) −σ < ∞ where µ mn = (min{M m , M n }) 1/2 M m M n and M m is the multiplicity of h m . We show that in that case there exist a suitable norm to measure the regularity of V , denoted ǫ V , and positive constants, ǫ ⋆ and δ ⋆ , with the property: if ǫ V < ǫ ⋆ then there exists a measurable subset Ω ∞ ⊂ Ω 0 such that its Lebesgue measure fulfills |Ω ∞ | ≥ |Ω 0 | − δ ⋆ ǫ V and the Floquet Hamiltonian has a pure point spectrum for all ω ∈ Ω ∞ .
The main motivation of this article is to derive sufficient conditions for dynamical stability of... more The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e., conditions under which it holds sup t∈R | ψ t , H(t)ψ t | < ∞ where ψ t denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next we show, under certain assumptions, that if the spectrum of the monodromy operator U (T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of energy is uniformly bounded in the course of time. Further we show that if the propagator admits a differentiable Floquet decomposition then H(t)ψ t is bounded in time for any initial condition ψ 0 , and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.
Guiding-center reduction is studied using gyro-gauge-independent coordinates. The Lagrangian 1-fo... more Guiding-center reduction is studied using gyro-gauge-independent coordinates. The Lagrangian 1-form of charged particle dynamics is Lie transformed without introducing a gyro-gauge, but using directly the unit vector of the component of the velocity perpendicular to the magnetic field as the coordinate corresponding to Larmor gyration. The reduction is shown to provide a maximal reduction for the Lagrangian and to work for all orders in the Larmor radius, following exactly the same procedure as when working with the standard gauge-dependent coordinate.
In guiding center theory, the standard gyro-angle coordinate is associated with gyrogauge depende... more In guiding center theory, the standard gyro-angle coordinate is associated with gyrogauge dependence, the global existence problem for unit vectors perpendicular to the magnetic field, and the notion of anholonomy, which is the failure of the gyro-angle to return to its original value after being transported around a loop in configuration space. We analyse these three intriguing topics through the lens of a recently proposed, global, gauge-independent gyro-angle. This coordinate is constrained, and therefore necessitates the use of a covariant derivative. It also highlights the intrinsic meaning and physical content of gyro-gauge freedom and anholonomy. There are, in fact, many possible covariant derivatives compatible with the intrinsic gyro-angle, and each possibility corresponds to a different notion of gyro-angle transport. This observation sheds new light on Littlejohn's notion of gyro-angle transport and suggests a new derivation of the recently-discovered global existence condition for unit vectors perpendicular to the magnetic field. We also discuss the relationship between Cartesian position-momentum coordinates and the intrinsic gyro-angle.
We introduce a gyro-gauge independent formulation of a simplified guiding-center reduction, which... more We introduce a gyro-gauge independent formulation of a simplified guiding-center reduction, which removes the fast time-scale from particle dynamics by Lietransforming the velocity vector field. This is close to Krylov-Bogoliubov method of averaging the equations of motion, although more geometric. At leading order, the Lie-transform consists in the generator of Larmor gyration, which can be explicitly inverted, while working with gauge-independent coordinates and operators, by using the physical gyro-angle as a (constrained) coordinate. This brings both the change of coordinates and the reduced dynamics of the minimal guiding-center reduction order by order in a Larmor radius expansion. The procedure is algorithmic and the reduction is systematically derived up to full second order, in a more straightforward way than when Lie-transforming the phase-space Lagrangian or averaging the equations of motion. The results write up some structures in the guiding-center expansion. Extensions and limitations of the method are considered.
Journal of Physics A: Mathematical and General, 1997
We consider a perturbed Floquet Hamiltonian −i∂t + H + βV (ωt) in the Hilbert space L 2 ([0, T ],... more We consider a perturbed Floquet Hamiltonian −i∂t + H + βV (ωt) in the Hilbert space L 2 ([0, T ], H, dt). Here H is a self-adjoint operator in H with a discrete spectrum obeying a growing gap condition, V (t) is a symmetric bounded operator in H depending on t 2π-periodically, ω = 2π/T is a frequency and β is a coupling constant. The spectrum Spec(−i∂t + H) of the unperturbed part is pure point and dense in R for almost every ω. This fact excludes application of the regular perturbation theory. Nevertheless we show, for almost all ω and provided V (t) is sufficiently smooth, that the perturbation theory still makes sense, however, with two modifications. First, the coupling constant is restricted to a set I which need not be an interval but 0 is still a point of density of I. Second, the Rayleigh-Schrodinger series are asymptotic to the perturbed eigen-value and the perturbed eigen-vector.
Journal of Physics A: Mathematical and General, 2004
We consider a perturbation of an 'integrable' Hamiltonian and give an expression for the canonica... more We consider a perturbation of an 'integrable' Hamiltonian and give an expression for the canonical or unitary transformation which 'simplifies' this perturbed system. The problem is to invert a functional defined on the Liealgebra of observables. We give a bound for the perturbation in order to solve this inversion, and apply this result to a particular case of the control theory, as a first example, and to the 'quantum adiabatic transformation', as another example.
Series on Stability, Vibration and Control of Systems, Series B, 2010
where x = (x, y) represents the spatial coordinates of the transversal section to the confining t... more where x = (x, y) represents the spatial coordinates of the transversal section to the confining toroidal magnetic field, B the norm of the magnetic field B, c the velocity of light and V is the electric turbulent potential, that is E = −∇V . We notice that the resulting dynamics is of Hamiltonian nature with a pair of canonically conjugate variables (x, y) which consists of the position of the guiding center. Since the potential which plays the role of the Hamiltonian is time-dependent, it is expected that the dynamics is chaotic (with one and a half degrees of freedom).
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2014
We study the motion of a charged particle in a tokamak magnetic field and discuss its chaotic nat... more We study the motion of a charged particle in a tokamak magnetic field and discuss its chaotic nature. Contrary to most of recent studies, we do not make any assumption on any constant of the motion and solve numerically the cyclotron gyration using Hamiltonian formalism. We take advantage of a symplectic integrator allowing us to make long-time simulations. First considering an idealized magnetic configuration, we add a non generic perturbation corresponding to a magnetic ripple, breaking one of the invariant of the motion. Chaotic motion is then observed and opens questions about the link between chaos of magnetic field lines and chaos of particle trajectories. Second, we return to an axisymmetric configuration and tune the safety factor (magnetic configuration) in order to recover chaotic motion. In this last setting with two constants of the motion, the presence of chaos implies that no third global constant exists, we highlight this fact by looking at variations of the first order of the magnetic moment in this chaotic setting. We are facing a mixed phase space with both regular and chaotic regions and point out the difficulties in performing a global reduction such as gyrokinetics.
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Papers by M. Vittot