7.2 Summary of contributions

Chapters 1 to 3 provide the contextual and theoretical background of this thesis. The existing body of literature has been reviewed and summarized to establish the state of the art from where this research started from. In Chapters 4 to 6 the main contributions and achievements made in the scope of this thesis are reported and are reiterated here:

1. Optimized algorithms for the most fundamental building blocks of the signed-particle method were designed and implemented. This entails the Wigner potential, the particle generation and the particle annihilation algorithms:
   1. An algorithm to speed up the calculation of the two-dimensional Wigner potential by at least a factor of five was presented.
   2. An analysis of the physical and computational implications of choosing a finite coherence length was given.
   3. The origin of non-physical behaviour in certain simulations was discovered to be attributable to the Wigner potential, which governs the statistics of particle generation. The application of a tapering window to the potential has been demonstrated to mitigate this problem and greatly improved the quality of the Wigner Monte Carlo simulation results.
   4. A potential statistical biasing when generating the momentum offsets for generated particles was identified and an approach to avoid this was shown.
   5. It was discovered that the particle annihilation process can introduce a numerical diffusion and a method to counteract this has been devised.
   6. Two alternative realizations of the annihilation algorithm have been shown, which greatly reduce, or completely eradicate, the huge memory demands of the annihilation step.
   7. A method to anticipate the increase in the number of particles in a time step was introduced, which has enabled an automatic activation of the annihilation process without user input, greatly simplifying the use of the simulator for non-expert users.
2. A Wigner Ensemble Monte Carlo simulator was developed, which implements the signed-particle method with all the optimized algorithms presented. This entailed:
   1. Writing over 8000 lines of code in C and various scripts for data post-processing and plotting functionality. The usability of the simulator has been considerably improved by facilitating control through input files for simulation parameters, potential profiles, initial conditions and the automatic selection of (some) simulation parameters.
   2. The developed simulator now forms part of the open source suite of ViennaWD tools, which is freely available online and has been published on a website with examples and a user manual for the simulator¹ . The code serves as a reference implementation for the state-of-the-art of the signed-particle method.
   3. A parallelized version of the code for high performance computing was designed. The data structures and logic needed for the parallelization of the code using MPI were implemented and the performance of the parallel code was characterized.
3. The developed simulator has been applied to the simulation of electrostatic lenses, used for electron state control. The most important results include:
   1. Demonstrating the control of the dynamics of wavepackets, making it possible to focus them or split them to create an entangled state, using special potential profiles.
   2. Applying a converging lens to increase the drive-current through a nano-scaled channel.
   3. Approximating a steady-state solution by the periodic injection of wavepackets from a boundary and calculating the corresponding current.