## Section: New Results

### Application of Vlasov codes nanophysics

Participants : Nicolas Crouseilles, Paul-Antoine Hervieux, Giovanni Manfredi, Omar Morandi.

#### Quantum hydrodynamics.

The dynamical properties of a finite system of electrons (thin metal film) have been recently investigated using a semi-classical Vlasov-Poisson model In order to include quantum effects, a quantum hydrodynamic model has been implemented in this context. First, the ground state was determined numerically using a relaxation technique. A systematic comparison with a stationary Schrödinger-type model enabled us to validate the accuracy of the hydrodynamic approach. Second, the electron dynamics was excited by perturbing the computed ground state. The quantum hydrodynamic model was shown to reproduce the most salient features of the Vlasov simulations: the thermal energy initially increases (heating) and, after saturation, low-frequency oscillations appear, corresponding to ballistic electrons traveling through the film. These results published in [23] confirm that the hydrodynamic model is well suited to describe the electron dynamics in finite-size nano-objects.

#### Quantum electron dynamics in thin metal films

In the past few years, we carried out a series of studies on the ultra-fast electron dynamics in thin metal films. Self-consistent simulations of the electron dynamics and transport were performed using a semi-classical Vlasov-Poisson model. In particular, we showed that, after the excitation energy has been absorbed by the film, slow nonlinear oscillations appear, with a period proportional to the film thickness. These oscillations were attributed to non-equilibrium electrons bouncing back and forth on the film surfaces. Recently, we have extended this semi-classical study to the quantum regime, by employing the Wigner-Poisson equations. It was found that the Vlasov and Wigner results coincide for large excitations. In contrast, for small excitations, the period of the oscillations diverges from the semi-classical “ballistic" value obtained previously. Closer inspection of the Wigner functions reveals that, in the fully quantum regime, the phase-space structures that are responsible for the ballistic transport cannot form. Simple analytical arguments are provided in support of the above numerical findings.

#### Ultra-fast magnetization dynamics in diluted magnetic semiconductors

In [31] , we have developed a dynamical model that successfully explains the observed time evolution of the magnetization in diluted magnetic semiconductor quantum wells after weak laser excitation. Based on a many-particle expansion of the exact p- d exchange interaction, our approach goes beyond the usual mean-field approximation. It includes both the sub-picosecond demagnetization dynamics and the slower relaxation processes which restore the initial ferromagnetic order on a nanosecond timescale. In agreement with experimental results, our numerical simulations show that, depending on the value of the initial lattice temperature, a subsequent enhancement of the total magnetization may be observed on a timescale of few hundreds of picoseconds.

#### Quasilinear approximation of the Wigner-Poisson equations

The Wigner equation is the quantum analog of the classical Vlasov equation. Although many important results have been obtained in the field of classical phase-space (or Langmuir) turbulence, comparatively few works have investigated the important question of quantum plasma turbulence. A reasonable strategy to attack these problems would consist in extending well-known techniques issued from the theory of classical plasmas in order to include quantum effects. In this context, the simplest approach is given by the weak turbulence kinetic equations first derived by Vedenov and Drummond in the 1960s – the so-called quasilinear theory. In a recent work [27] , we re-examined the quasilinear theory of the Wigner-Poisson system in one spatial dimension. It was found that quantum effects manifest themselves in transient periodic oscillations of the averaged Wigner function in velocity space. The quantum quasilinear theory was checked against numerical simulations of the bump-on-tail and two-stream instabilities. The predicted wavelength of the oscillations in velocity space agrees well with the numerical results.