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Photons are the elementary quantum excitations of the electromagnetic field. Quantization is usually constructed on the basis of an expansion in eigenmodes, in the form of plane waves. Since they form a basis, other electromagnetic configurations can be constructed by linear combinations. In this presentation we discuss a formulation constructed in the general formalism of bosonic Fock space, in which the quantum excitation can be constructed directly on localized pulses of arbitrary shape. Although the two formulations are essentially equivalent, the direct formulation in terms of pulses has some conceptual and practical advantages, which we illustrate with some examples. The first one is the passage of a single photon pulse through a beam splitter. The analysis of this formulation in terms of pulses in Fock space shows that there is no need to introduce "vacuum fluctuations entering through the unused port", as is often done in the literature. Another example is the Hong-Ou-Mandel effect. It is described as a time dependent process in the Schr\"odinger representation in Fock space. The analysis shows explicitly how the two essential ingredients of the Hong-Ou-Mandel effect are the same shape of the pulses and the bosonic nature of photons. This formulation shows that all the phenomena involving linear quantum optical devices can be described and calculated on the basis of the time dependent solution of the corresponding classical Maxwell's equations for pulses, from which the quantum dynamics in Fock space can be immediately constructed., Comment: 26 pages, 3 figures

Quantum plasmonics extends cavity quantum electrodynamics (cQED) concepts to the nanoscale, taking benefit from the strongly subwavelength confinement of the plasmon modes supported by metal nanostructures. In this work, we describe in detail collective strong coupling to a plasmonic nanocavity. Similarities and differences to cQED are emphasized. We notably observe that the Rabi splitting can strongly deviate from the standard $\sqrt{N_e}\Delta \Omega_1$ law, where $N_e$ is the number of emitters and $\Delta \Omega_1$ the Rabi splitting for a single emitter. In addition, we discuss the collective Lamb shift and the role of quantum corrections to the emission spectra.

We provide a critical analysis of some of the commonly used theoretical models to describe quantum plasmons. We summarize the standard approach based on a Fano diagonalization and we show explicit discrepancies in the obtained results by taking the limit of vanishing coupling between the electromagnetic field and the material medium. We then discuss the derivation of spontaneous emission in a plasmonic environment, which usually relies on a Green tensor and is based on an incomplete identity. The effect of the missing terms is calculated in a one-dimensional model.

Plasmonic superradiance originates from the plasmon mediated strong correlation that builds up between dipolar emitters coupled to a metal nanoparticle. This leads to a fast burst of emission so that plasmonic superradiance constitutes ultrafast and extremely bright optical nanosources of strong interest for integrated quantum nano-optics platforms. We elucidate the superradiance effect by establishing the dynamics of the system, including all features like the orientation of the dipoles, their distance to the particle and the number of active plasmon modes. We determine an optimal configuration for Purcell enhanced superradiance. We also show superradiance blockade at small distances.

The quantization of plasmons has been analyzed mostly under the assumption of an infinite-sized bulk medium interacting with the electromagnetic field. We reformulate it for finite-size media, such as metallic or dielectric nano-structures, highlighting sharp differences. By diagonalizing the Hamiltonian by means of a Lippmann-Schwinger equation, we show the contribution of two sets of bosonic operators, one stemming from medium fluctuations, and one from the electromagnetic field. The results apply to general models including dissipative and dispersive responses.

We derive effective Hamiltonians for a single dipolar emitter coupled to a metal nanoparticle (MNP) with particular attention devoted to the role of losses. For small particles sizes, absorption dominates and a non hermitian effective Hamiltonian describes the dynamics of the hybrid emitter-MNP nanosource. We discuss the coupled system dynamics in the weak and strong coupling regimes offering a simple understanding of the energy exchange, including radiative and non radiative processes. We define the plasmon Purcell factors for each mode. For large particle sizes, radiative leakages can significantly perturbate the coupling process. We propose an effective Fano Hamiltonian including plasmon leakages and discuss the link with the quasi-normal mode description. We also propose Lindblad equations for each situation and introduce a collective dissipator for describing the Fano behaviour.

We construct mode-selective effective models describing the interaction of N quantum emitters (QEs) with the localised surface plasmon polaritons (LSPs) supported by a spherical metal nanoparticle (MNP) in an arbitrary geometric arrangement of the QEs. We develop a general formulation in which the field response in the presence of the nanosystem can be decomposed into orthogonal modes with the spherical symmetry as an example. We apply the model in the context of quantum information, investigating on the possibility of using the LSPs as mediators of an efficient control of population transfer between two QEs. We show that a Stimulated Raman Adiabatic Passage configuration allows such a transfer via a decoherence-free dark state when the QEs are located on the same side of the MNP and very closed to it, whereas the transfer is blocked when the emitters are positioned at the opposite sides of the MNP. We explain this blockade by the destructive superposition of all the interacting plasmonic modes.

We propose a geometry-specific, mode-selective quantization scheme in coupled field-emitter systems which makes it easy to include material and geometrical properties, intrinsic losses as well as the positions of an arbitrary number of quantum emitters. The method is presented through the example of a spherically symmetric, non-magnetic, arbitrarily layered system. We follow it up by a framework to project the system on simpler, effective cavity QED models. Maintaining a well-defined connection to the original quantization, we derive the emerging effective quantities from the full, mode-selective model in a mathematically consistent way. We discuss the uses and limitations of these effective models.

We develop the method of adiabatic tracking for photo- and magneto-association of Bose-Einstein atomic condensates with models that include Kerr type nonlinearities. We show that the inclusion of these terms can produce qualitatively important modifications in the adiabatic dynamics, like the appearance of bifurcations, in which the trajectory that is being tracked loses its stability. As a consequence the adiabatic theorem does not apply and the adiabatic transfer can be strongly degraded. This degradation can be compensated by using fields that are strong enough compared with the values of the Kerr terms. The main result is that, despite these potentially detrimental features, there is always a choice of the detuning that leads to an efficient adiabatic tracking, even for relatively weak fields.

We propose a method of construction of a normalized vibrational Hamiltonian of a highly excited molecular system with $n$ degrees of freedom in the case of a a $p:q$ resonance. We present also the counting of all the independent operators and the counting of all the parameters included in the Hamiltonian (Counting theorems 1 to 8). The method introduces, on a systematic way, all the operators, in particular the coupling operators, that can be built from the polynomials formed by products of powers of the generators of a Lie algebra: the algebra of the invariant polynomials built in classical mechanics from the the kernel $Ker \,ad_{\mathcal{H}_{0}}$ of the adjoint operator $ad_{\mathcal{H}_{0}}$ (see [6] or [4],[5]). Application to the non-linear triatomic molecule ClOH is then given, taking into account the Fermi resonance between the O-Cl stretching oscillators and the bending motion. The study of this molecular system in highly excited vibrational states (until almost the dissociation limit) has been realized in [2], with a fit of 725 levels of energy. On the 86 coefficients (among which 31 coupling coefficients) that we count, and completely compatible with [2], the smallest rms value leads to keep only 28 non-zero coefficients. In the appendix, we explain the vocabulary and the strategy employed in order to demonstrate the theorems of coupling operators included in the Hamiltonian., Comment: 26 pages

The present paper is the first of two articles aimed at constructing $n$-degree-of-freedom Hamiltonian systems by an algebraic approach. In molecular spectroscopy, the construction of vibrational Hamiltonian for strongly excited molecular systems by using an algebraic formalism requires the introduction by hand the operators describing the change in energy by numerous quanta and it is tedious to predict in advance the total number of operators appearing in the development. The goal of the two articles is not only to propose in the local limit a systematic method for constructing a normalized vibrational Hamiltonian for a strongly excited $n$-degree-of-freedom molecular system from the generators of the Lie algebra, the algebra of polynomial invariants, but also to enumerate the number of independent operators needed for the construction of the Hamiltonian developed in the base of these generators up to the given order $N$. The first article introduces the theoretical tools used in the both papers (section \ref{norm}), and presents the method of construction in case of absence of resonance (section \ref{const}). Finally, an application for a triatomic non-linear ClOH molecule is considered in case close to the dissociation limit. (section \ref{Appli})., Comment: 7 pages

We consider the population transfer process in a Lambda-type atomic medium of unequal oscillator strengths by stimulated Raman adiabatic passage via bright-state (b-STIRAP) taking into account propagation effects. Using both analytic and numerical methods we show that the population transfer efficiency is sensitive to the ratio q_p/q_s of the transition oscillator strengths. We find that the case q_p>q_s is more detrimental for population transfer process as compared to the case where $q_p \leq q_s$. For this case it is possible to increase medium dimensions while permitting efficient population transfer. A criterion determining the interaction adiabaticity in the course of propagation process is found. We also show that the mixing parameter characterizing the population transfer propagates superluminally.

We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between classical and quantum systems. We give a summary of the main tools of Berezin-Toeplitz and geometric quantization, that provide a relation between the classical and the quantum models, based essentially on the selection of a subspace of the classical Hilbert space. Coherent states provide a systematic tool for the inverse process, called dequantization, that associates a classical Hamiltonian system to a given quantum dynamics through the choice of a complete set of coherent states.

We present a mechanism to produce indistinguishable single-photon pulses on demand from an optical cavity. The sequences of two laser pulses generate, at the two Raman transitions of a four-level atom, the same cavity-mode photons without repumping of the atom between photon generations. Photons are emitted from the cavity with near-unit efficiency in well-defined temporal modes of identical shapes controlled by the laser fields. The second order correlation function reveals the single-photon nature of the proposed source. A realistic setup for the experimental implementation is presented., Comment: 5 pages, 3 figures, submitted to Phys. Rev. A

We study the strong coupling limit of a quadratic-nonlinear Landau-Zener problem for coherent photo- and magneto-association of cold atoms taking into account the atom-atom, atom-molecule, and molecule-molecule elastic scattering. Using an exact third-order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct a highly accurate and simple analytic approximation describing the time dynamics of the coupled atom-molecule system. We show that the approximation describing time evolution of the molecular state probability can be written as a sum of two distinct terms; the first one, being a solution to a limit first-order nonlinear equation, effectively describes the process of the molecule formation while the second one, being a scaled solution to the linear Landau-Zener problem (but now with negative effective Landau-Zener parameter as long as the strong coupling regime is considered), corresponds to the remaining oscillations which come up when the process of molecule formation is over., Comment: 19 pages, 7 figures, accepted for publication in Eur. Phys. J. D

We study a nonlinear generalization of the Landau-Zener resonance-crossing problem relevant to coherent photo- and magneto-association of ultracold atoms. Due to the structure of the corresponding classical phase space, the adiabatic theorem breaks down even at very small sweep rates, and the adiabatic approximation diverges because of the crossing of a separatrix. First, by introducing a complex term into the Hamiltonian of the system, we eliminate this divergence and construct a valid zero-order approximation. Further, taking into account that the molecular conversion efficiency and the change of the classical adiabatic invariant at the separatrix crossing are related quantities, we calculate the change of the action for the situation when the system starts from the all-atomic state that corresponds to the case of zero initial action. The absolute error of the presented formula for the change in the action is of the order of or less than 10^-4., Comment: 28 pages, 7 figures; submitted to Physica D

We present a general and versatile technique of population transfer based on {\it parallel adiabatic passage} by femtosecond shaped pulses. Their amplitude and phase are specifically designed to optimize the adiabatic passage corresponding to parallel eigenvalues at all times. We show that this technique allows the robust adiabatic population transfer in a Raman system with the total pulse area as low as 3 $\pi$, corresponding to a fluence of one order of magnitude below the conventional stimulated Raman adiabatic passage process. This process of short duration, typically pico- and subpicosecond, is easily implementable with the modern pulse shaper technology and opens the possibility of ultrafast robust population transfer with interesting applications in quantum information processing., Comment: 5 pages, 4 figures

An investigation of field-free molecular alignment produced by elliptically polarized laser pulses is reported. Experiments are conducted in CO$_2$ at room temperature. A non invasive all-optical technique, based on the cross defocusing of a probe pulse, is used to measure the alignment along two orthogonal directions that is sufficient to provide a 3 dimensional characterization. The field-free molecular alignment produced by a laser of elliptical polarization is in good agreement in terms of amplitude and shape with theoretical predictions. It turns out to be almost equivalent to the superposition of the effects that one would obtain with two individual cross-polarized pulses. The investigation highlights notably the occurrence of field-free two-direction alignment alternation for a suitably chosen degree of ellipticity. The analogy between this specific ellipticity and the well known "magic angle" used in time resolved spectroscopy to prevent rotational contributions is discussed., Comment: 6 pages, 4 figures

Fractional Hamiltonian Monodromy is a generalization of the notion of Hamiltonian Monodromy, recently introduced by N. N. Nekhoroshev, D. A. Sadovskii and B. I. Zhilinskii for energy-momentum maps whose image has a particular type of non-isolated singularities. In this paper, we analyze the notion of Fractional Hamiltonian Monodromy in terms of the Gauss-Manin Monodromy of a Riemann surface constructed from the energy-momentum map and associated to a loop in complex space which bypasses the line of singularities. We also prove some propositions on Fractional Hamiltonian Monodromy for 1:-n and m:-n resonant systems., Comment: 39 pages, 24 figures. submitted to J. Math. Phys

We propose an analysis of the time-optimal control of a dissipative two-level quantum system whose dynamics is governed by the Lindblad equation. This simple system allows one to use tools of geometric control theory and to construct its optimal synthesis, i.e. to determine the set of all the optimal trajectories starting from a given initial point. We study different processes such as conversion of a pure state into a mixed state and purification of a mixed state. In particular cases, we show that dissipation is not undesirable and can help accelerating the control., Comment: 25 pages, 10 figures

We propose a robust scheme involving atoms fixed in an optical cavity to directly implement the universal controlled-unitary gate. The present technique based on adiabatic passage uses novel dark states well suited for the controlled-rotation operation. We show that these dark states allow the robust implementation of a gate that is a generalisation of the controlled-unitary gate to the case where the control qubit can be selected to be an arbitrary state. This gate has potential applications to the rapid implementation of quantum algorithms such as of the projective measurement algorithm. This process is decoherence-free since excited atomic states and cavity modes are not populated during the dynamics., Comment: 6 pages, 6 figure, submitted to Phys. Rev. A

We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties., Comment: 17 pages, no figure

A density matrix approach is developped for the control of a mixed-state quantum system using a time-dependent external field such as a train of pulses. This leads to the definition of a target density matrix constructed in a reduced Hilbert space as a specific combination of the eigenvectors of a given observable through weighting factors related with the initial statistics of the system. A train of pulses is considered as a possible strategy to reach this target. An illustration is given by considering the laser control of molecular alignment / orientation in thermal equilibrium., Comment: 10 pages, 8 figures

Starting from an initial pure quantum state, we present a strategy for reaching a target state corresponding to the extremum (maximum or minimum) of a given observable. We show that a sequence of pulses of moderate intensity, applied at times when the average of the observable reaches its local or global extremum, constitutes a strategy transferable to different control issues. Among them, post-pulse molecular alignment and orientation are presented as examples. The robustness of such strategies with respect to experimentally relevant parameters is also examined., Comment: 16 pages, 9 figures

We propose a scheme for the construction of a CNOT gate by adiabatic passage in an optical cavity. In opposition to a previously proposed method, the technique is not based on fractional adiabatic passage, which requires the control of the ratio of two pulse amplitudes. Moreover, the technique constitutes a decoherence-free method in the sense that spontaneous emission and cavity damping are avoided since the dynamics follows dark states., Comment: 6 pages, 4 figures, submitted to EPJD

We show that a linear molecule subjected to a short specific elliptically polarized laser field yields postpulse revivals exhibiting alignment alternatively located along the orthogonal axis and the major axis of the ellipse. The effect is experimentally demonstrated by measuring the optical Kerr effect along two different axes. The conditions ensuring an optimal field-free alternation of high alignments along both directions are derived., Comment: 5 pages, 4 color figures

We show that a combination of a half-cycle pulse and a short nonresonant laser pulse produces a strongly enhanced postpulse orientation. Robust transients that display both efficient and long-lived orientation are obtained. The mechanism is analyzed in terms of optimal oriented target states in finite Hilbert subspaces and shows that hybrid pulses can prove useful for other control issues., Comment: 4 pages, 4 color figures

We present a process for the construction of a SWAP gate which does not require a composition of elementary gates from a universal set. We propose to employ direct techniques adapted to the preparation of this specific gate. The mechanism, based on adiabatic passage, constitutes a decoherence-free method in the sense that spontaneous emission and cavity damping are avoided., Comment: 5 pages, 4 figures, submitted to Phys. Rev

We propose a relatively robust scheme to generate maximally entangled states of (i) an atom and a cavity photon, (ii) two atoms in their ground states, and (iii) two photons in two spatially separate high-Q cavities. It is based on the interaction via fractional adiabatic passage of a three-level atom traveling through a cavity mode and a laser beam. The presence of optical phases is emphasized., Comment: 6 pages, 7 figures. We have changed the title, the abstract and the text. The references have been updated. (Accepted by Phys. Rev. A)

We propose a robust scheme to generate multi-photon Fock states in an atom-maser-cavity system using adiabatic passage techniques and topological properties of the dressed eigenenergy surfaces. The mechanism is an exchange of photons from the maser field into the initially empty cavity by bichromatic adiabatic passage. The number of exchanged photons depends on the design of the adiabatic dynamics through and around the conical intersections of dressed eigenenergy surfaces., Comment: 6 pages, 4 figures

We review various unitary time-dependent perturbation theories and compare them formally and numerically. We show that the Kolmogorov-Arnold-Moser technique performs better owing to both the superexponential character of correction terms and the possibility to optimize the accuracy of a given level of approximation which is explored in details here. As an illustration, we consider a two-level system driven by short pulses beyond the sudden limit., Comment: 15 pages, 5 color figures

We apply a time-dependent perturbation theory based on unitary transformations combined with averaging techniques, on molecular orientation dynamics by ultrashort pulses. We test the validity and the accuracy of this approach on LiCl described within a rigid-rotor model and find that it is more accurate than other approximations. Furthermore, it is shown that a noticeable orientation can be achieved for experimentally standard short laser pulses of zero time average. In this case, we determine the dynamically relevant parameters by using the perturbative propagator, that is derived from this scheme, and we investigate the temperature effects on the molecular orientation dynamics., Comment: 16 pages, 6 figures

We study the quantum dynamics of a two-level system driven by a pulse that starts near-resonant for small amplitudes, yielding nonadiabatic evolution, and induces an adiabatic evolution for larger amplitudes. This problem is analyzed in terms of lifting of degeneracy for rising amplitudes. It is solved exactly for the case of linear and exponential rising. Approximate solutions are given in the case of power law rising. This allows us to determine approximative formulas for the lineshape of resonant excitation by various forms of pulses such as truncated trig-pulses. We also analyze and explain the various superpositions of states that can be obtained by the Half Stark Chirped Rapid Adiabatic Passage (Half-SCRAP) process., Comment: 21 pages, 12 figures

We construct a non-perturbative approach based on quantum averaging combined with resonant transformations to detect the resonances of a given Hamiltonian and to treat them. This approach, that generalizes the rotating-wave approximation, takes into account the resonances at low field and also at high field (non-linear resonances). This allows to derive effective Hamiltonians that contain the qualitative features of the spectrum, i.e. crossings and avoided crossings, as a function of the coupling constant. At a second stage the precision of the spectrum can be improved quantitatively by standard perturbative methods like contact transformations. We illustrate this method to determine the spectrum of a two-level atom interacting with a single mode of a quantized field., Comment: 11 pages, 4 figures

We present a time-dependent perturbative approach adapted to the treatment of intense pulsed interactions. We show there is a freedom in choosing secular terms and use it to optimize the accuracy of the approximation. We apply this formulation to a unitary superconvergent technique and improve the accuracy by several orders of magnitude with respect to the Magnus expansion., Comment: 4 pages, 2 figures

We present a strategy for post-pulse orientation aiming both at efficiency and maximal duration within a rotational period. We first identify the optimally oriented states which fulfill both requirements. We show that a sequence of half-cycle pulses of moderate intensity can be devised for reaching these target states., Comment: 4 pages, 3 figures

We present a superconvergent Kolmogorov-Arnold-Moser type of perturbation theory for time-dependent Hamiltonians. It is strictly unitary upon truncation at an arbitrary order and not restricted to periodic or quasiperiodic Hamiltonians. Moreover, for pulse-driven systems we construct explicitly the KAM transformations involved in the iterative procedure. The technique is illustrated on a two-level model perturbed by a pulsed interaction for which we obtain convergence all the way from the sudden regime to the opposite adiabatic regime., Comment: 10 pages, 3 figures

Preparation of entangled pairs of coupled two-state systems driven by a bichromatic external field is studied. We use a system of two coupled spin-1/2 that can be translated into a three-state ladder model whose intermediate state represents the entangled state. We show that this entangled state can be prepared in a robust way with appropriate fields. Their frequencies and envelopes are derived from the topological properties of the model., Comment: 10 pages, 9 figures

We examine the topology of eigenenergy surfaces characterizing the population transfer processes based on adiabatic passage. We show that this topology is the essential feature for the analysis of the population transfers and the prediction of its final result. We reinterpret diverse known processes, such as stimulated Raman adiabatic passage (STIRAP), frequency-chirped adiabatic passage and Stark-chirped rapid adiabatic passage (SCRAP). Moreover, using this picture, we display new related possibilities of transfer. In particular, we show that we can selectively control the level which will be populated in STIRAP process in Lambda or V systems by the choice of the peak amplitudes or the pulse sequence.

We consider a class of Hamiltonians with three degrees of freedom that can be mapped into quasi-periodically driven pendulums. The purpose of this paper is to determine the threshold of the break-up of invariant tori with a specific frequency vector. We apply two techniques: the frequency map analysis and renormalization-group methods. The renormalization transformation acting on a Hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the Hamiltonian and a rescaling of phase space around the considered torus. We give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break-up of invariant tori. Furthermore, we obtain by these methods numerical values of the threshold of the break-up of the last invariant torus., Comment: 18 pages, 4 figures

We analyze the stability of invariant tori for Hamiltonian systems with two degrees of freedom by constructing a transformation that combines Kolmogorov-Arnold-Moser theory and renormalization-group techniques. This transformation is based on the continued fraction expansion of the frequency of the torus. We apply this transformation numerically for arbitrary frequencies that contain bounded entries in the continued fraction expansion. We give a global picture of renormalization flow for the stability of invariant tori, and we show that the properties of critical (and near critical) tori can be obtained by analyzing renormalization dynamics around a single hyperbolic strange attractor. We compute the fractal diagram, i.e., the critical coupling as a function of the frequencies, associated with a given one-parameter family., Comment: 10 pages, 5 figures

We construct an approximate renormalization transformation that combines Kolmogorov-Arnold-Moser (KAM)and renormalization-group techniques, to analyze instabilities in Hamiltonian systems with three degrees of freedom. This scheme is implemented both for isoenergetically nondegenerate and for degenerate Hamiltonians. For the spiral mean frequency vector, we find numerically that the iterations of the transformation on nondegenerate Hamiltonians tend to degenerate ones on the critical surface. As a consequence, isoenergetically degenerate and nondegenerate Hamiltonians belong to the same universality class, and thus the corresponding critical invariant tori have the same type of scaling properties. We numerically investigate the structure of the attracting set on the critical surface and find that it is a strange nonchaotic attractor. We compute exponents that characterize its universality class., Comment: 10 pages typeset using REVTeX, 7 PS figures

We formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered spectral projector, and some geometric hypothesis on the local behaviour of the eigenvalues at the crossings.

We review a formulation of a renormalization-group scheme for Hamiltonian systems with two degrees of freedom. We discuss the renormalization flow on the basis of the continued fraction expansion of the frequency. The goal of this approach is to understand universal scaling behavior of critical invariant tori., Comment: LaTeX2e, 7 pages

We study an approximate renormalization-group transformation to analyze the breakup of invariant tori for three degrees of freedom Hamiltonian systems. The scheme is implemented for the spiral mean torus. We find numerically that the critical surface is the stable manifold of a critical nonperiodic attractor. We compute scaling exponents associated with this fixed set, and find that they can be expected to be universal., Comment: 4 pages, 3 figures, REVTeX

We construct an approximate renormalization scheme for Hamiltonian systems with two degrees of freedom. This scheme is a combination of Kolmogorov-Arnold-Moser (KAM) theory and renormalization-group techniques. It makes the connection between the approximate renormalization procedure derived by Escande and Doveil, and a systematic expansion of the transformation. In particular, we show that the two main approximations, consisting in keeping only the quadratic terms in the actions and the two main resonances, keep the essential information on the threshold of the breakup of invariant tori., Comment: 6 pages, RevTeX

We give a proof of the KAM theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring's approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring's proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring's proof., Comment: 6 pages, RevTex

We analyze the breakup of invariant tori in Hamiltonian systems with two degrees of freedom using a combination of KAM theory and renormalization-group techniques. We consider a class of Hamiltonians quadratic in the action variables that is invariant under the chosen KAM transformations, following the approach of Thirring. The numerical implementation of the transformation shows that the KAM iteration converges up to the critical coupling at which the torus breaks up. By combining this iteration with a renormalization, consisting of a shift of resonances and rescalings of momentum and energy, we obtain a much more efficient method that allows to determine the critical coupling with high accuracy. This transformation is based on the physical mechanism of the breakup of invariant tori. We show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point, and we discuss its universality properties., Comment: 9 pages, 5 figures, RevTex

In this article, we describe a new renormalization-group scheme for analyzing the breakup of invariant tori for Hamiltonian systems with two degrees of freedom. The transformation, which acts on Hamiltonians that are quadratic in the action variables, combines a rescaling of phase space and a partial elimination of irrelevant (non-resonant) frequencies. It is implemented numerically for the case applying to golden invariant tori. We find a nontrivial fixed point and compute the corresponding scaling and critical indices. If one compares flows to maps in the canonical way, our results are consistent with existing data on the breakup of golden invariant circles for area-preserving maps., Comment: 7 pages, 6 figures, RevTex