Probability Current and a Simulation of Particle Separation

realization of quantum structures since they are easier to handle. Optical and laser cavities have recently been used for studying the concept of space-time reflection symmetry (Brandstetter et al., 2013; Liertzer et al., 2012) developed by Bender, Boettcher and Meisinger (Bender & Boettcher, 1998; Bender, Boettcher, & Meisinger, 1999) as an extension of Hermitian quantum mechanics. Acoustic and microwave cavities have been used to study exceptional points (Dembowski, et al., 2001; Ding, Ma, Xiao, Zhang, & Chan, 2016) which are points where the eigenvalues of two states are equal and their corresponding eigenvectors just differ by a phase (Rotter, 2009). Microwave cavities have also been important in the development of quantum chaos (Sadreev & Berggren, 2005; Stöckmann, 1999); the study of quantum systems that in the classical limit is chaotic i.e. a change in the initial conditions leads to exponential divergence of the trajectories in phase space. Earlier studies, especially of quantum structures, have focused on two-dimensional fields because they are easier to study experimentally. However, a more realistic theoretical model would need to take all three dimensions into account (Ferry, Goodnick, & Bird, 2009). In this paper, we model three-dimensional quantum billiards with different boundary conditions and study the structure of the probability current for states of different excitations. Because different boundary conditions in the modeling were employed, the results are also applicable to the other wave analogues. The probability current is a three-dimensional vector field, which is difficult to visualize. Therefore, we also study the nodal surfaces and nodal lines. Nodal surfaces are surfaces where either the real or the imaginary part of the wave function is zero and the nodal lines are the intersection between the nodal surfaces (i.e., where both the real and imaginary parts are zero). The current will create vortices around these lines (Dirac, 1931; Wyatt, 2005). If the distribution of vortices is known, the overall structure of the current is known. The appearance and location of vortices have been directly connected to minima in the conductance i.e. the transmission through INTRODUCTION Mapping and understanding the structure of wave fields and currents is relevant to a variety of structures such as quantum structures, optical and acoustic cavities, microwave billiards and water waves in a tank (Berggren & Ljung, 2009; Berggren, Yakimenko, & Hakanen, 2010; Blümel, Davidson, Reinhardt, Lin, & Sharnoff, 1992; Stöckmann , 1999; Chen, Liu, Su, Lu, Chen, & Huang, 2007; Ohlin & Berggren, 2016; Panda & Hazra, 2014). Here, cavities and billiards stand for systems with hard walls such that a particle or a wave only scatter from the walls and there is no scattering within the system. Acoustic cavities are metal enclosed cavities where a microphone is used to emit the waves. The underlying physics of all these systems is seemingly different. For example, in acoustic cavities, the waves are pressure differences in the air. In quantum cavities, the waves are the particles themselves. Nevertheless, the wave nature of these phenomena is often equivalent because it is governed by the Helmholtz equation (Stöckmann, 1999). Although there has been much progress within the fabrication of heterostructures, where quantum mechanical phenomena can be directly observed and measured, they are still difficult to manipulate. The different wave analogues are therefore important supplements for the experimental Probability Current and a Simulation of Particle Separation