Mesoscopic transport and quantum chaos oogle
Rodolfo A. Jalabert, Institut de Physique et Chimie des Matériaux de Strasbourg, CNRS, Université de Strasbourg, France
The field of Quantum Chaos, addressing the quantum manifestations of an underlying classically chaotic dynamics, was developed in the early eighties, mainly from a theoretical perspective. Few experimental systems were initially recognized to exhibit the versatility of being sensitive, at the same time, to their classical and quantum dynamics. Rydberg atoms (Shepelyansky, 2012-i) provided the main testing ground of Quantum Chaos concepts until the early nineties, marked by the development of microwave billiards (Stöckmann, 2010-i), ultra- cold atoms in optical lattices (Raizen, 2011-i), and low-temperature transport in mesoscopic semiconductor structures. The mesoscopic regime is attained in small condensed matter systems at sufficiently low temperatures for the electrons to propagate coherently across the sample. The quantum coherence of electrons, together with the ballistic motion characteristic of ultra-clean microstructures, motivated the proposal (Jalabert, 1990-a) of mesocopic systems as a very special laboratory for performing measurements and testing the theoretical ideas of Quantum Chaos. Experimental realizations (Marcus, 1992-a) and many important developments, reviewed in this article, followed from such a connection.
Contents 1 Introduction 1.1 Basic concepts of Mesoscopic Physics 1.1.1 Quantum coherence 1.1.2 Quantum transport 1.1.3 Disordered systems 1.1.4 Ballistic systems 1.1.5 Mesoscopic samples as "doubly open" quantum systems 1.1.6 Quantum Chaos and Mesoscopic Physics 1.1.7 Summary of characteristic lengths 1.2 Geometry-dependent transport in ballistic microstructures 1.2.1 Quenching of the Hall effect 1.2.2 Conductance fluctuations in the ballistic regime 1.3 The scattering approach to the conductance 1.3.1 The basic components: sample, leads and reservoirs 1.3.2 Reservoir and lead states 1.3.3 Scattering states and scattering matrix 1.3.4 Current-density and electrical current 1.3.5 Conductance is transmission 1.3.6 Shot noise 1.3.7 Polar decomposition of the scattering matrix 1.3.8 Transmission eigenmodes and scattering eigenstates 1.3.9 Scattering amplitudes in terms of the Green function 1.3.10 R -matrix formalism for a quantum dot connected to leads 1.3.11 Scattering phase and Friedel sum-rule 1.4 Chaotic scattering 1.4.1 Transient chaos 1.4.2 Time-delay function 1.4.3 Escape rate and dwell time 1.4.4 Length-distribution in billiards 1.4.5 Effective area distribution 2 Semiclassical description of ballistic transport 2.1 Quantum interference in clean cavities 2.2 Semiclassical scattering amplitudes 2.2.1 Semiclassical Green function 2.2.2 Semiclassical transmission amplitudes