Wigner Transport Dynamics of Spatial Electron Entanglement

Quantum entanglement refers to quantum states of objects to become interdependent. Entanglement has been one of the key exciting quantum processes since the beginning of quantum mechanics in the first half of the 20th century. Historically, photons were primarily utilized for studying entanglement, albeit alternative objects, such as electron spin, are also heavily researched nowadays. However, recent years saw staggering progress in the coherent generation and control of individual electrons, making it possible to utilize the wave nature of electrons in a similar manner as in the photonic world. This opens new research opportunities to study wave-based, spatial electron-electron entanglement and is thus at the core of this research. In addition, the effect of electromagnetic fields on entangled electron transport is of great interest to study the impact of different electromagnetic control and guiding mechanisms. Available modeling and simulation approaches are computationally prohibitive and provide only limited physical intuition about the involved quantum transport processes. We will, therefore, develop a particle Wigner approach to model the transport dynamics of spatially entangled electrons in 2D systems. Our modeling approach will be able to incorporate external electromagnetic fields and will provide an intuitive wave picture of the transport. We will make the developments available in our simulation tool ViennaWD. Our research will enable new ideas for future nanoelectronic systems (e.g., 2D materials) and electron quantum optics systems (e.g., coupled wave guides, interferometers, electron detection).

In the field of electromagnetism, we deal with forces. Opposite poles of a magnet attract each other. Electrostatically charged hairs repel each other and tend to move as far apart as possible, so that they stand perpendicular to the head. The coupling of a mechanical system with an electromagnetic field therefore occurs via forces, or more precisely, via force fields. This fact is reflected in the equations of motion for material bodies, which arise from classical mechanics. This situation changes when considering very small systems to which quantum mechanics applies. However, the Schrödinger equation-the fundamental equation for describing quantum mechanical systems-includes the electrodynamic potentials rather than the electromagnetic force fields. While forces are measurable quantities, potentials are merely mathematical tools. Different potential fields can be constructed for a given electromagnetic field. This indeterminacy is referred to as gauge freedom. In this project, an alternative formulation of quantum mechanics was used, which employs the Wigner equation instead of the Schrödinger equation. The state of a system is described by a function in phase space, which is spanned by the particle positions and momenta. This description exhibits certain formal similarities to the classical phase space description. Since the Wigner equation is derived from the Schrödinger equation, the former also includes the electrodynamic potentials. In this project, we have succeeded in transforming the Wigner equation such that it contains only force fields, which means that the potentials get completely eliminated. Attempts to obtain a gauge-independent formulation of quantum mechanics in this way have been made in the past. However, the results were often very complex and hardly suitable for practical application. Furthermore, in this project, the properties of the newly derived gauge-independent Wigner equation were investigated in greater detail. One can specify a so-called strong formulation of the equation, in which pseudodifferential operators are used, and a weak formulation, in which integral operators are used. Both formulations entail different continuity and asymptotic properties of the solution function. The ability to formulate quantum mechanics in a gauge-independent manner is not merely of fundamental theoretical interest. In practical terms, this means that processes based on interaction with an electromagnetic field-including light-can be interpreted more intuitively. The new formalism is also well-suited for deriving semi-classical models. These are models whose complexity lies between a fully quantum-mechanical description and a simpler, classical one.