# Research

Related to my research activity and beyond

## Publication: Capturing Photoelectron Motion with Guiding Centers

*posted Sept 11 2018*

Subjecting atoms or molecules to intense laser fields gives rise to a variety of non-perturbative and highly nonlinear phenomena, such as high-harmonic generation, non-sequential multiple ionization, and high-order above-threshold ionization. All these phenomena are based on the key mechanism of attosecond physics, namely the recollision. A recollision is obtained when an electron tunnel-ionizes, freely travels in the laser field, and then upon return to the ionic core, either recombines into an atomic or molecular bound state, or undergoes inelastic or elastic scattering. By experiencing a strong ion-electron interaction, rescattered electrons probe the atomic or molecular structure, providing essential information to imaging atoms and molecules.

By mapping this problem onto the motion of a guiding center, we derive a reduced model which naturally embeds important Coulomb effects such as focusing and asymmetry, and clearly distinguishes direct versus rescattered electron ionization processes. We demonstrate the power of this tool by unraveling the bifurcation in photoelectron momentum distributions seen in experiments.

Reference: J. Dubois, S. A. Berman, C. Chandre, T. Uzer, *Capturing Photoelectron Motion with Guiding Centers*, Physical Review Letters 121, 113202 (2018)

## Publication: High-order Harmonic Generation (HHG) - What does classical mechanics bring to this fundamentally quantum problem?

*posted Jun 25 2018*

High-order harmonic generation (HHG) is the production of coherent high-frequency radiation observed during the ionization of gases by intense laser pulses, often harnessed to generate attosecond pulses. While the laser pulse propagate through the gas, it undergoes tremendous reshaping due to the radiation emitted by the ionizing atoms, leading, for example, to a blueshift and intensity reduction. The self-consistent interaction between the ionizing atoms and the laser field plays a decisive role in shaping the HHG radiation.

In a recent work, published in Physical Review A, we present a classical model for HHG during the propagation of an intense laser pulse through an atomic gas. Numerical simulations show excellent quantitative agreement with the corresponding quantum model for the blueshift and intensity reduction of the propagating laser pulse over experimentally realistic propagation distances. We observe a significant extension of the HHG cutoff due to propagation effects, the origin of which is uncovered by a phase-space analysis of our classical model.

Reference: S. A. Berman, J. Dubois, C. Chandre, M. Perin, T. Uzer, *Coherent buildup of high-order harmonic radiation: The classical perspective*, Physical Review A 97, 061402(R) (2018)

## TraX International Conference 2018

*posted Apr 20 2018*

The TraX International Conference 2018 is organized within the H2020 European project TraX (Stability and Transitions in Physical Processes).

TraX 2018 will be held on May 9th-10th 2018 at the Instituto de Ciencias Matemáticas (ICMAT), which is located at the Campus de Cantoblanco of the Universidad Autónoma de Madrid (Spain).

The main goal of this workshop is to bring together physicists, chemists and mathematicians working on transition state theory and molecular ionization in order to discuss the last advances on both topics, and establish new lines of research and collaborations.

For more information: http://www.gsc.upm.es/trax2018/

## MSRI semester on Hamiltonian systems, from topology to applications through analysis

August 13, 2018 to December 14, 2018

More on this program at https://www.msri.org/programs/305

Programmatic Workshops

August 16, 2018 - August 17, 2018 Connections for Women: Hamiltonian Systems, from topology to applications through analysis

August 20, 2018 - August 24, 2018 Introductory Workshop: Hamiltonian systems, from topology to applications through analysis

October 08, 2018 - October 12, 2018 Hamiltonian systems, from topology to applications through analysis I

November 26, 2018 - November 30, 2018 Hamiltonian systems, from topology to applications through analysis II

## TraX kickoff meeting

*posted Apr 13, 2017*

The TraX kickoff meeting will take place on May 4th, 2017, at the Georgia Institute of Technology in Atlanta, Georgia, USA.

TraX (Stability and Transitions in Physical Processes) is a European Union H2020 Marie Sklodowska-Curie RISE action (2017-2021).

## Publication: Bowling with electrons and atoms

*posted Feb 28, 2017*

Throwing electrons on target atoms at a speed of about one million meters per second causes the atoms to multiple ionize, i.e., during the collision, one or several outer shell electrons leave the atom. As in bowling and billiards, the mechanisms involve some kind of instability, which makes the game interesting and the outcome not necessarily predictable: Depending on the initial conditions, the impact electron can "knock down" zero, one, two or more electrons from the target by communicating part of its kinetic energy to the outer shell electrons strongly bound by the Coulomb interaction.

We have considered a Hamiltonian dynamical system to model the electron impact of magnesium with two effective outer shell electrons. We have compared the experimentally measures cross-sections with the ones computed from this model, and identified the mechanisms responsible for the single and double ionization of magnesium. In essence, these mechanisms are "chaotic", involving some kind of strong dependence to initial conditions. For double ionization mechanisms, we provide evidence that the Two-Step 2 in which the impact electron "knocks down" the two outer shell electrons sequentially, is dominant over the Two-Step 1 in which the target electron "knocks down" one outer shell electron, and then this outer shell electron "knocks down" the second outer shell electron.

Reference: J. Dubois, S.A. Berman, C. Chandre, T. Uzer, *Single and double ionization of magnesium by electron impact: A classical study*, Physical Review A 95, 022713 (2017); featured in the kaleidoscope images of Physical Review A

## Video: Multi-electron dynamics in HHG

made by Olga Smirnova and Misha Ivanov (MBI)

*posted Jan 10, 2017*

## Hamiltonian reductions in plasma physics

*posted Dec 1, 2016*

The dynamics of a non-collisional plasma is described by a kinetic evolution equation for the particle distribution function in phase space (Vlasov equation) and evolution equations for electromagnetic fields (Poisson, Ampère or Maxwell equation).

My interest lies in the modeling (formulation of the basic equations from first principles) in relation to the numerical codes under development and for which there is an unprecedented effort worldwide, with the long-term goal of an integrated code for magnetic fusion plasmas such as those encountered in tokamaks. This modeling is based on dynamical reductions for the fluid and kinetic descriptions of fusion plasmas by magnetic confinement. Ultimately, the aim is to obtain a hierarchy of models corresponding to the different approximations involved. From these models, the application of Hamiltonian tools will allow us to understand the stability properties of equilibria, the origin of conserved quantities (in relation to symmetries), and dynamical properties. A good understanding of the dynamics will make it possible to envisage control strategies aimed at obtaining significant changes in the dynamics with the aim of an improved confinement of the plasma.

*Keywords*: Hamiltonian systems, Poisson algebras, variational methods, dynamical reductions, kinetic equations.

## Multiple ionizations of atoms and molecules driven by intense laser pulses: A nonlinear dynamics perspective

*posted Nov 29, 2016*

How and when are ionized electrons driven back to the ionic core by an ultrastrong, ultrashort laser pulse? The answer to this question holds the key to future breakthroughs like the real-time imaging of biomolecules with bright short-wave light sources. In a strong field, most ionized electrons drift far from the core and cannot contribute to energy transfer processes. However, a few find their way back to the core, and bring with them the energy they absorbed from the laser, thereby becoming the drivers of the all important phenomena of High Harmonic Generation (HHG) and multiple ionization. That is why predicting which electrons return matters greatly. The recollision scenario states that after ionization, the ionized electron is potentially hurled back to the atomic (or molecular) core by the laser field alone. Upon collision with the core, the kinetic energy gained from the laser field is transferred to other electrons of the core (potentially leading to nonsequential multiple ionization) or emitted as high-frequency electromagnetic radiation (potentially leading to HHG) when the pre-ionized electron recombines with the core.

In order to gain insight into these phenonema, we consider the classical mechanical treatment of the underlying processes. The main advantage of classical mechanics is the power-law scaling of its representation with system size, as compared with the exponential increase of complexity of quantum mechanics. Another advantage of the classical treatment compared to its quantum counterpart is the notion of a trajectory, which allows for an in-depth analysis of the electronic dynamics in phase space on its natural spatial and temporal timescales. By addressing the collective behavior of ensembles of trajectories rather than individual trajectories, nonlinear dynamics can help us to discover the mechanisms at play and thereby provide ways to control these processes.

*Keywords*: electron-electron interaction, high harmonic generation, ionization and recollision processes, invariant manifolds, parabolic manifolds, N-body problem, billiards

## Publication: Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments

*posted Apr 21, 2016*

We consider Hamiltonian closures of the Vlasov equation using the phase-space moments of the distribution function. We provide some conditions on the closures imposed by the Jacobi identity. We completely solve some families of examples. As a result, we show that imposing that the resulting reduced system preserves the Hamiltonian character of the parent model shapes its phase space by creating a set of Casimir invariants as a direct consequence of the Jacobi identity. We exhibit three main families of Hamiltonian models with two, three, and four degrees of freedom aiming at modeling the complexity of the bunch of particles in the Vlasov dynamics.

Reference: C. Chandre and M. Perin, Hamiltonian reductions of the one-dimensional Vlasov equation using phase-space moments, Journal of Mathematical Physics 57 (2016)032902

## Publication: Incomplete Dirac reduction of constrained Hamiltonian systems

*posted Jun 15, 2015*

First-class constraints constitute a potential obstacle to the computation of a Poisson bracket in Dirac's theory of constrained Hamiltonian systems. Using the pseudoinverse instead of the inverse of the matrix defined by the Poisson brackets between the constraints, we show that a Dirac-Poisson bracket can be constructed, even if it corresponds to an incomplete reduction of the original Hamiltonian system. The uniqueness of Dirac brackets is discussed. The relevance of this procedure for infinite dimensional Hamiltonian systems is exemplified.

Reference: C. Chandre, *Incomplete Dirac reduction of constrained Hamiltonian systems*, Annals of Physics 361, 1 (2015)

## A recollision?

*posted on Nov 29, 2016*

An ionized electron, after picking up energy from the field, is hurled back at the ion core upon reversal of the laser field and dislodges the second electron.

## Publication: Recollision scenario without tunneling

posted Mar 10, 2014

Modern laser experiments can ionize atoms with field strengths which rival the electric fields inside atoms, and these processes are rapidly becoming the tool of choice for resolving the structure of matter ranging from atoms to biological complexes. During these events, electrons are first detached (presumably by tunneling) and absorb energy while following the laser, only to be hurled back at the ionic core when the laser reverses direction, where they can ionize more electrons or generate very high harmonics of the driving laser--a crucial source of very short-wave radiation for imaging applications. This so-called "three-step" or "recollision" model is the workhorse in the field of intense laser-atom interactions and current theoretical treatments of this model omit the Coulomb field altogether after the tunneling step. Strikingly enough, the maximum energy the electrons can bring back to the core --and thus the shortest-wavelength harmonics they can generate-- can be found by simply ignoring the Coulomb field, even though the field of the ionic core drives all stages of recollision and high harmonic generation. Here we resolve this long-standing paradox by showing that the tunneling step is not needed for the interpretation of high-harmonic spectra, and that the strong Coulomb field can be fully integrated into a purely classical scenario that explains recollisions. Our argument centers on a set of particular electron trajectories (recolliding periodic orbits) which drive the recollision process. Through them, we connect the crucial quantity, the maximum return energy, to the orbit's unstable and stable manifolds.

Beyond stimulating more realistic theoretical treatments of the recollision process, we anticipate that our classical recollision mechanism extends to all polarizations and wave forms, which in turn opens up a promising avenue to extend the harmonic cutoffs beyond their single-color limit by manipulating the electron orbits and their manifolds.

Reference: A. Kamor, C. Chandre, T. Uzer, F. Mauger, *Recollision scenario without tunneling: Role of the ionic core potential*, Physical Review Letters 112, 133003 (2014)

## Time-frequency analysis of chaotic systems

*posted Jan 10, 2010*

We devised a method for analyzing the phase space structures of Hamiltonian systems, based on a time–frequency decomposition of a trajectory using wavelets. The ridges of the time–frequency landscape of a trajectory, also called instantaneous frequencies, enable us to analyze the phase space structures. In particular, this method detects resonance trappings and transitions and allows a characterization of the notion of weak and strong chaos. We applied this method to examples in atomic physics such as the hydrogen atom in crossed magnetic and elliptically polarized microwave fields.

*Main reference*: Chandre, Wiggins, Uzer, Physica D 181, 171 (2003)

*Keywords*: Time-frequency analysis, wavelets, Hamiltonian systems, resonances

## Control of Hamiltonian chaos

*posted Jan 10, 2010*

Controlling chaotic transport is a key challenge in many branches of physics like particle accelerator physics, free electron lasers or magnetically confined fusion plasmas. One way to control transport would be that of reducing or suppressing chaos. Most of the methods for controlling chaotic systems is done by tilting targeted trajectories. However, for many body experiments like the magnetic confinement of a plasma or the control of turbulent flows, such methods are hopeless due to the high number of trajectories to deal with simultaneously. For these systems, it is desirable to control transport properties without significantly altering the original system under investigation nor its overall chaotic structure. Here we focus on a different strategy which aims at modifying the phase space structures by adding a small apt perturbation or by tuning appropriately the parameters.

*Main reference*: Chandre, Ciraolo, Doveil, Lima, Macor, Vittot, Physical Review Letters 94, 074101 (2005)

*Keywords*: Chaos control, KAM theory, electrostatic turbulence in plasmas

## Renormalization for Hamiltonian flows

*posted Jan 10, 2010*

We investigated the stability of classical Hamiltonian systems with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the long term stability of the system. The objective was to determine the break-up mechanism of invariant tori. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. A simple attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. The analysis of the renormalization flow indicates that the break-up of invariant tori is a universal mechanism.

*Main reference*:* *Chandre, Jauslin, Physics Reports 365, 1 (2002)

*Keywords*: Renormalization group, Hamiltonian systems, KAM theory, break-up of invariant tori