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Journal of Siberian Federal University Mathematics & Physics

Basic Modeling of Openness, Quantum States and Transport in Two- and Three-Dimensional Ballistic Cavities

Karl‐Fredrik Berggren, Iryna I. Yakimenko

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

A basic model for particle states and current flow in open quantum dots/billiards are investigated. The model is unconventional and extends the use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Attached leads/source drain are represented by complex potentials. Probability densities and currents flows for open 2D quantum dots/billiards are calculated and the results are compared with microwave measurements used to emulate the dot. We also apply the model to a rectangular enclosure and report on helical flows guided by nodal lines and disc-like accumulations of flow lines. The model is of conceptual as well as practical and educational interest

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A basic model for particle states and current flow in open quantum dots/billiards are investigated. The model is unconventional and extends the use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Attached leads/source drain are represented by complex potentials. Probability densities and currents flows for open 2D quantum dots/billiards are calculated and the results are compared with microwave measurements used to emulate the dot. We also apply the model to a rectangular enclosure and report on helical flows guided by nodal lines and disc-like accumulations of flow lines. The model is of conceptual as well as practical and educational interest
A heuristic model for particle states and current flow in open ballistic two-dimensional (2D) quantum dots/wave billiards is proposed. The model makes use of complex potentials first introduced in phenomenological nuclear inelastic scattering theory (the optical model). Here we assume that external input and output leads connecting the system to the source and the drain regions may be represented by complex potentials. In this way, a current may be set up between the two 'pseudo-leads'. Probability densities and current flows for an open quantum dot are analyzed here numerically and the results are compared with the microwave measurements used to emulate the system. The model is of conceptual as well as practical interest. In addition to quantum billiards, it may be used as a tool per se to analyze transport in classical wave analogues, such as microwave resonators, acoustic resonators, effects of leakage on such systems, etc.
A basic quantum-mechanical model for wave functions and current flow in open quantum dots or billiards is investigated. The model involves non-Hertmitian quantum mechanics, parity-time (PT) symmetry, and PT-symmetry breaking. Attached leads are represented by positive and negative imaginary potentials. Thus probability densities, currents flows, etc., for open quantum dots or billiards may be simulated in this way by solving the Schrödinger equation with a complex potential. Here we consider a nominally open ballistic quantum dot emulated by a planar microwave billiard. Results for probability distributions for densities, currents (Poynting vector), and stress tensor components are presented and compared with predictions based on Gaussian random wave theory. The results are also discussed in view of the corresponding measurements for the analogous microwave cavity. The model is of conceptual as well as of practical and educational interest.
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We present an efficient method to solve scattering problems in two-dimensional open billiards with two leads and a complicated scattering region. The basic idea is to transform the scattering region to a rectangle, which will lead to complicated dynamics in the interior, but simple boundary conditions. The method can be specialized to closed billiards, and it allows the treatment of interacting particles in the billiard. We apply this method to quantum echoes measured recently in a microwave cavity, and indicate how it can be used for interacting particles.
We have previously shown that the flow in open quantum dots may be described in terms of meandering open orbits which separate regions of trapped flows around vortices. Furthermore the topology of the flow follows from a simple geometrical construction in which the classical ballistic trajectories are first computed and then all crossings are replaced by hyperbolic flow separation and for which all trapped flows form quantised closed orbits. In the present paper we extend this picture to a computational methodology for developing quantum hydrodynamic descriptions of flows in the presence of quantised vortices by extension from classical hydrodynamics. For coherent transport we again encounter the Bohm quantum potential but the singularity structure is predetermined by applying a fictitious repulsion which prevents trajectories and hence streamlines crossing. This leads to vortex producing terms in the hydrodynamic equations. The discussion is completed by including the effects of non-hermitian potentials into the Hamiltonian in order to simulate dissipation and phase breaking flows.