Jean Michel D. Sellier

Wigner quasi-particle attributes—An asymptotic perspective

Wigner quantum mechanics is reformulated in a discrete momentum space and analyzed within a Monte Carlo approach for solving integral equations and thus associated with a particle picture. General quantum phenomena may thereby be modeled in terms of quasi-particles involving attributes such as drift, generation, sign, and annihilation on a phase space grid. The model is examined in an ultimate regime, where classical and quantum dynamics become equivalent. The peculiarities of the transport in this asymptotic regime are analyzed within simulations, benchmarking the behavior of the Wigner function.

A benchmark study of the Wigner Monte Carlo method

Abstract. The Wigner equation is a promising full quantum model for the simulation of nanodevices. It is also a challenging numerical problem. Two basic Monte Carlo approaches to this model exist exploiting, in the time-dependent case, the so-called particle affinity and, in the stationary case, integer particle signs. In this paper we extend the second approach for time-dependent simulations and present a validation against a well-known benchmark model, the Schrödinger equation. Excellent quantitative agreement is demonstrated by the compared results despite the very different numerical properties of the utilized stochastic and deterministic approaches.

The many-body Wigner Monte Carlo method for time-dependent ab-initio quantum simulations

The aim of ab-initio approaches is the simulation of many-body quantum systems from the first principles of quantum mechanics. These methods are traditionally based on the many-body Schrodinger equation which represents an incredible mathematical challenge. In this paper, we introduce the many-body Wigner Monte Carlo method in the context of distinguishable particles and in the absence of spin-dependent effects. Despite these restrictions, the method has several advantages. First of all, the Wigner formalism is intuitive, as it is based on the concept of a quasi-distribution function. Secondly, the Monte Carlo numerical approach allows scalability on parallel machines that is practically unachievable by means of other techniques based on finite difference or finite element methods. Finally, this method allows time-dependent ab-initio simulations of strongly correlated quantum systems. In order to validate our many-body Wigner Monte Carlo method, as a case study we simulate a relatively simple system consisting of two particles in several different situations. We first start from two non-interacting free Gaussian wave packets. We, then, proceed with the inclusion of an external potential barrier, and we conclude by simulating two entangled (i.e. correlated) particles. The results show how, in the case of negligible spin-dependent effects, the many-body Wigner Monte Carlo method provides an efficient and reliable tool to study the time-dependent evolution of quantum systems composed of distinguishable particles.

A Wigner Monte Carlo Approach to Density Functional Theory

In order to simulate quantum N-body systems, stationary and time-dependent density functional theories rely on the capacity of calculating the single-electron wave-functions of a system from which one obtains the total electron density (Kohn–Sham systems). In this paper, we introduce the use of the Wigner Monte Carlo method in ab-initio calculations. This approach allows time-dependent simulations of chemical systems in the presence of reflective and absorbing boundary conditions. It also enables an intuitive comprehension of chemical systems in terms of the Wigner formalism based on the concept of phase-space. Finally, being based on a Monte Carlo method, it scales very well on parallel machines paving the way towards the time-dependent simulation of very complex molecules. A validation is performed by studying the electron distribution of three different systems, a Lithium atom, a Boron atom and a hydrogenic molecule. For the sake of simplicity, we start from initial conditions not too far from equilibrium and show that the systems reach a stationary regime, as expected (despite no restriction is imposed in the choice of the initial conditions). We also show a good agreement with the standard density functional theory for the hydrogenic molecule. These results demonstrate that the combination of the Wigner Monte Carlo method and Kohn–Sham systems provides a reliable computational tool which could, eventually, be applied to more sophisticated problems.

On the simulation of indistinguishable fermions in the many-body Wigner formalism

The simulation of quantum systems consisting of interacting, indistinguishable fermions is an incredible mathematical problem which poses formidable numerical challenges. Many sophisticated methods addressing this problem are available which are based on the many-body Schrodinger formalism. Recently a Monte Carlo technique for the resolution of the many-body Wigner equation has been introduced and successfully applied to the simulation of distinguishable, spinless particles. This numerical approach presents several advantages over other methods. Indeed, it is based on an intuitive formalism in which quantum systems are described in terms of a quasi-distribution function, and highly scalable due to its Monte Carlo nature. In this work, we extend the many-body Wigner Monte Carlo method to the simulation of indistinguishable fermions. To this end, we first show how fermions are incorporated into the Wigner formalism. Then we demonstrate that the Pauli exclusion principle is intrinsic to the formalism. As a matter of fact, a numerical simulation of two strongly interacting fermions (electrons) is performed which clearly shows the appearance of a Fermi (or exchange-correlation) hole in the phase-space, a clear signature of the presence of the Pauli principle. To conclude, we simulate 4, 8 and 16 non-interacting fermions, isolated in a closed box, and show that, as the number of fermions increases, we gradually recover the Fermi-Dirac statistics, a clear proof of the reliability of our proposed method for the treatment of indistinguishable particles.

The Wigner–Boltzmann Monte Carlo method applied to electron transport in the presence of a single dopant

Abstract The Wigner–Boltzmann model is a partial integro-differential equation which describes the time dependent dynamics of quantum mechanical phenomena including the effects of lattice vibration as a second-order approximation. Recently a Monte Carlo technique exploiting the concept of signed particles has been developed for its ballistic counterpart, in one and two-dimensional space. In this work, we introduce an extension to the Wigner–Boltzmann model in three-dimensional geometries adapted for the treatment of the scattering term. As an application, we study the dynamics of an electron wave packet in proximity of a Coulombic potential in the presence of absorbing boundary conditions. This mimics the presence of a dopant atom buried in a semiconductor substrate. By using this method, one can observe how the lattice temperature eventually affects the dynamics of the wave packet.

A signed particle formulation of non-relativistic quantum mechanics

A formulation of non-relativistic quantum mechanics in terms of Newtonian particles is presented in the shape of a set of three postulates. In this new theory, quantum systems are described by ensembles of signed particles which behave as field-less classical objects which carry a negative or positive sign and interact with an external potential by means of creation and annihilation events only. This approach is shown to be a generalization of the signed particle Wigner Monte Carlo method which reconstructs the time-dependent Wigner quasi-distribution function of a system and, therefore, the corresponding Schrodinger time-dependent wave-function. Its classical limit is discussed and a physical interpretation, based on experimental evidences coming from quantum tomography, is suggested. Moreover, in order to show the advantages brought by this novel formulation, a straightforward extension to relativistic effects is discussed. To conclude, quantum tunnelling numerical experiments are performed to show the validity of the suggested approach.

A sensitivity study of the Wigner Monte Carlo method

Recently a Wigner Monte Carlo technique exploiting the concept of signed particles has been developed for time dependent, multi-dimensional simulations of quantum mechanical effects in the ballistic regime. This method is based on the introduction of a semi-discrete phase-space which involves a free parameter L C defining the discretization of the space of momenta. A systematic study to understand how the quality and reliability of the solution is influenced by this parameter is necessary. In this work, we analyze the sensitivity of the Wigner Monte Carlo method on L C . To this aim, three quality measures are introduced based on a comparison with the Schrodinger equation (considered as a benchmark model in this work). We show that, essentially, the Wigner Monte Carlo method is not affected by the choice of L C . Indeed, a large range of valid choices is available which demonstrates the robustness and reliability of the method.

The Role of Annihilation in a Wigner Monte Carlo Approach

The Wigner equation provides an interesting mathematical limit, which recovers the constant field, ballistic Boltzmann equation. The peculiarities of a recently proposed Monte Carlo approach for solving the transient Wigner problem, based on generation and annihilation of particles are summarized. The annihilation process can be implemented at consecutive time steps to improve the Monte Carlo resolution. We analyze theoretically and numerically this process applied to the simulation of important quantum phenomena, such as time-dependent tunneling of a wave packet through potential barriers.

Comparison of deterministic and stochastic methods for time-dependent Wigner simulations

Recently a Monte Carlo method based on signed particles for time-dependent simulations of the Wigner equation has been proposed. While it has been thoroughly validated against physical benchmarks, no technical study about its numerical accuracy has been performed. To this end, this paper presents the first step towards the construction of firm mathematical foundations for the signed particle Wigner Monte Carlo method. An initial investigation is performed by means of comparisons with a cell average spectral element method, which is a highly accurate deterministic method and utilized to provide reference solutions. Several different numerical tests involving the time-dependent evolution of a quantum wave-packet are performed and discussed in deep details. In particular, this allows us to depict a set of crucial criteria for the signed particle Wigner Monte Carlo method to achieve a satisfactory accuracy.