Semiclassical Gaussian matrix elements for chaotic quantumwells

Abstract

We derive semiclassical expressions for spectra, weightedby matrix elements of a Gaussian observable, relevantto a range of molecular and mesoscopic systems. We apply theformalism to the particular example of the resonant tunnelling diode(RTD) in tilted fields. The RTD is an experimental realization of amesoscopic system exhibiting a transition to chaos. It has generated muchinterest and several different semiclassical theories for the RTD havebeen proposed recently.Our formalism clarifies the relationship between the differentapproaches and to previous work on semiclassical theories of matrixelements. We introduce three possible levels of approximation in theapplication of the stationary phase approximation, depending on typicallength scales of oscillations of the semiclassical Green function,relative to the degree of localization of the observable. Different typesof trajectories (periodic, normal, closed and saddle orbits) are shownto arise from such considerations. We propose here for the first timea new type of trajectory(`minimal orbits') and show they provide the best real approximation tothe complex saddle points of the stationary phase approximation.We test the semiclassical formulae on quantum calculations andexperimental data. We successfully treat phenomena beyondstandard periodic orbit (PO) theory: `ghost regions' where noreal PO can be found and regions with contributions from non-isolatedPOs. We show that the new types of trajectories (saddle and minimalorbits) provide accurate results. We discuss a divergence of thecontribution of saddle orbits, which suggests the existence ofbifurcation-type phenomena affecting the complex and non-periodic saddleorbits.