Wave transport and statistical properties of an open non-Hermitian quantum dot with parity-time symmetry

Abstract

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.

Spectra, current flow, and wave-function morphology in a modelPT-symmetric quantum dot with external interactions

“…This Hamiltonian is no longer Hermitian, but by enforcing an extra condition it can be made PT symmetric. This extra condition restricts the potential to V (−x) = V * (x), where the asterisk ( * ) denotes the complex conjugation and (−x) denotes inversion through the symmetry axis of the cavity [38]. This means that gain and loss in the system are equal as long as the PT symmetry is unbroken.…”

“…Here we focus on a semiconductor quantum dot and the Schrödinger equation. The description is, however, generic and may be applied to, for example, microwave billiards [37,38]. To motivate the use of an imaginary potential, one may consider the two-dimensional Schrödinger equation for the wave function (x,y,t):…”

Embedding quantum systems with a non-conserved probability in classical environments

“…Nowadays, this field of research is constantly growing. A first general book on the topic has appeared [6]; applications of nonHermitian quantum mechanics involve the study of scattering by complex potentials and quantum transport [7][8][9][10][11][12][13][14][15][16][17], description of metastable states [18][19][20][21][22][23], optical waveguides [24][25][26], multi-photon ionization [27][28][29], and nano-photonic and plasmonic waveguides [30]. The theoretical investigations are also undergoing rapid developments: non-Hermitian quantum mechanics has been investigated within a relativistic framework [31] and it has been adopted by various researchers as a means to describe open quantum systems [32][33][34][35][36][37][38][39][40][41][42].…”

Detecting topological exceptional points in a parity-time symmetric system with cold atoms

“…Very recently, a full dynamical encircling of EP has been realized [29] and a nonreciprocal topological energy transfer due to dynamical encircling of such point has been measurement [30]. In contrast, other theoretical and numerical results suggest in this case the eigenstates change to the other one but both of them obtain a π/2 Berry phase due to the linear dependence of eigenstates [31,32], which has been verified in a recent experiment [33]. * Electronic address: xujian˙328@163.com † Electronic address: zdanwei@126.com…”