Quantum stress in chaotic billiards

Abstract

This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T ␣␤ ͑x , y͒ for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as = u + iv. With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T ␣␤ . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T ␣␤ ͑x , y͒ is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.

Basic modelling of transport in 2D wave-mechanical nanodots and billiards with balanced gain and loss mediated by complex potentials

“…Without going into detail here, we conclude that numerical outcomes of the present model for our quantum dot are in surprisingly good agreement with its microwave analogue [5,[10][11][12][13][14]. The model is validated in the same way in a number of additional studies dealing with wave function and current statistics, chaos, quantum stress tensors and other related features [26,27]. In summary, the present non-Hermitian model with PT symmetry appears robust and reliable.…”