Resolving light-matter interactions on the atomic-scale, both in terms of temporal and spatial resolution, promises an *ultimate* understanding of ultrafast optical phenomena. A complete microscopic analysis that would include all optical processes, from internal atomic dynamics to long-range light propagation, is impossible. In fact, rigorous modelling of atomic dynamics is a very demanding quantum mechanical problem that is limited to a few electrons. The situation changes at non-perturbative laser intensities where warm/hot plasmas are created by ionization that can be modelled with classical dynamics. In the following sections, the validity ranges of different approaches to classical light-matter interaction are reviewed. Finally, common tools used for computational classical light-matter interaction are presented and put in context with MicPIC.

#### 2.1 Theoretical foundations of classical light-matter interaction

To understand the physical meaning of a strongly coupled plasma we briefly review the three main realms of plasma physics: weakly coupled, strongly coupled, and quantum (see Fig. 1). The relevant quantities for characterizing the nature and coupling strength of a plasma are the degeneracy parameter

- (1)

and the Debye number

- (2)

where is the Fermi energy, the Debye length, ℏ the Planck constant, the electron mass, the Boltzmann constant, ε_{0} the vacuum dielectric constant, and *e* the elementary electric charge. The parameters *T* and *n* are the electron temperature and number density, respectively. Physically, Θ quantifies the influence of the internal plasma pressure associated with the Pauli exclusion principle. In cold dense plasmas, where , all particles tend to occupy the lowest system eigenstates. As the Pauli principle prohibits Fermions from going into an already occupied state, plasma dynamics is substantially changed by blocking effects and quantum mechanical screening, as compared to a classical plasma. In warm classical plasmas, where , quantum effects of plasma electrons diminish and the relevant screening parameter is the Debye length , that represents the scale over which the electrons screen the electric field, with giving the average number of electrons that participate locally to the process. The Debye length and Debye number are closely related to the plasma parameter that appears in the Coulomb logarithm for the evaluation of classical two-body collision integrals [12, 13]. When , the plasma is hot and dilute. In this limit, plasma microfield fluctuations and particle collisions can be assumed as small perturbations and the system can be described by an effective mean-field one-body density. When , the plasma is warm and dense and the system can only be described by rigorous microscopic *N*-body calculations.

In the following we discuss the transition between the three plasma realms by the example of a metal at room temperature that is exposed to an intense laser field. At equilibrium, the lowest eigenstates of the metal are nearly fully occupied and the Pauli exclusion principle keeps electrons from going into already populated states. However, when the metal is excited by a laser pulse, the electrons absorb energy and the occupation of the lowest eigenstates becomes diluted. As a result, the Pauli exclusion principle loses importance and the dynamics can be approximated by classical equations of motions. During heating, the plasma goes first through the state of a strongly coupled plasmas (warm and dense, with ). This regime characterizes solid-density materials driven by relatively low-intensity lasers: during laser ablation, for example. In such a plasma many-particle collisions and the strongly fluctuating plasma micro-fields are important, which means that the classical equations of motion of all plasma particles have to be accounted for. When the plasma is heated further and expands, it goes over into a weakly coupled plasma (hot and dilute, with ), which can be modelled by mean-field transport equations, such as the Vlasov and Boltzman equations (see below).

An exact semi-classical analysis of light-matter interaction requires the self-consistent solution of the Quantum Liouville (*von Neumann*) equation coupled to the classical (not quantized) Maxwell equations. In a concise form, the Quantum Liouville equation reads [14]:

- (3)

It describes, in the Schrödinger representation, the time evolution of the density matrix (or density operator) of a quantum system characterized by the *N*-particle Hamiltonian . Solving this fundamental set of equations is a formidable task. Therefore, different levels of approximation have been used so far to facilitate calculations. Different approaches, and how they arise from the Quantum-Liouville-Maxwell equations, are summarized in Fig. 2. Background information on statistical physics are found in Schwabl [14] and Landau-Lifshitz [12].

The probabilistic treatment of collisions in phase-space usually relies on the so-called small-angle binary-collisions approximations, which is valid for weakly coupled plasmas in which only two particles are involved in a collision and close encounters resulting in large angle deflections are rare. This is generally justified for gases but not for laser-driven solid-density plasmas that are characterized by strongly-coupled dynamics (see Sec. 2.3 for more details).

The analysis of light-plasma interaction is usually supported by numerical modelling and simulations. Figure 2 relates the various numerical methods to the various levels of approximation of plasma physics. CFD and Computational electromagnetism (CEM) are bound to modelling light propagation in materials *via* the use of macroscopic dielectric and magnetic constants. Microscopic features of weakly coupled plasmas are taken into account by the collisional electromagnetic particle-in-cell (PIC) method [16-18]. Electrostatic molecular dynamics (MD) [19] solves the *N*-body dynamics rigorously but neglects light propagation. MicPIC is effectively the only approach that bridges the microscopic and macroscopic realms of laser-matter interactions by solving rigorously and self-consistently the *N*-body problem corresponding to the classical Liouville equation coupled to the microscopic Maxwell equations. The three main approaches: MD, PIC, and MicPIC, and their relation to each other are described in more detail in the following sections.

#### 2.2 Molecular dynamics

For small objects (and when the laser intensity is fairly weak), relativistic effects and the explicit treatment of the electromagnetic wave propagation can be neglected. In such cases, the classical plasma dynamics are described exactly by MD. This approach solves the equation of motion, (i.e., the Newton equation), for all the individual particles, where the force is the sum over all binary Coulomb forces (for example, see [20]). MD has been proven to work extremely well for small nanoplasmas in few-nanometer clusters, where the dipole and electrostatic approximations are justified. It has revealed the importance of microscopic processes, such as collisions and plasma microfields, in dense non-relativistic plasmas [21, 22, 11]. However, as soon as the object size is comparable to the wavelength in the medium, MD can not be used because light propagation has to be taken into account *explicitly* by solving Maxwell's equations. The onset of this “propagation regime” for the example of a resonantly excited cluster, i.e. the size where the electrostatic approximation breaks down, will be elucidated in detail in Sec. 4.3.

#### 2.3 The particle-in-cell method

Electromagnetic PIC codes are the most common tools used to study wave propagation phenomena in the presence of free charges [16-18]. With this approach, particle motion is coupled to a numerical grid that samples the electromagnetic field. The temporal evolution is then calculated by solving Maxwell's equations *via* the finite-difference time-domain method (FDTD) along with the relativistic equations of motion for the particles [23-25]. This tool has literally revolutionized the understanding of relativistic laser-plasma interactions. Its potential is well exemplified through a major contribution to the development of compact laser-driven plasma accelerators [26, 27], to cite one success story in particular. However, PIC is also often used in cases where it is not justified and where only qualitative insight can be provided.

The general idea behind the PIC algorithm is to solve the Vlasov equation by decomposing the single-particle distribution into phase-space subvolumes, where each subvolume effectively corresponds to a local collection of particles that have the same charge-to-mass ratio and move with the same velocity. When each macroparticle contains a fairly large number of physical particles, typically 10^{4} − 10^{6}, they can be represented by continuous charge distributions. In this context, the collision between two PIC particles represents, in fact, a collision between two clouds of charged particles. This statistical continuum, or “coarse-grain”, approach eliminates the microscopic features of the laser-plasma interaction. This is justified in the highly relativistic regime and for dilute systems where collisions can be neglected. To some extent, microscopic interactions can be reintroduced in the form of Monte-Carlo binary collisions between plasma particles [28, 29]. However, those “collisional” PIC codes remain bound to the weakly coupled regime, where microfield fluctuations are negligible and microscopic interactions are limited to small-angle binary collisions [30-33].