Recent Developments in the Quantum Nonlinear Dimer: Hopf Bifurcations, Time Evolution, and Thermal Stability

Abstract

A simple but rich nonlinear structure that has received a great deal of attention in the last few years is the quantum nonlinear dimer [1] in which a quasiparticle, such as an electron or an excitation, shuttles back and forth between two sites while interacting so strongly with a boson field, e. g. the vibrations of the system, that the quasiparticle evolution becomes nonlinear. The discrete nonlinear Schroedinger equation in various forms has served as the evolution equation for this structure [2]. The advantage of studying this small system lies in the fact that insights into many issues of interest on extended systems can be gained without having to grapple with the considerable additional complexities which extended systems introduce into the problem. Much work has been done on the nonlinear dimer [3–6]. Here we report on a few developments that have occurred recently in our investigation of the system. They are (i) exact solutions for the quantum nonlinear dimer in the absence of dissipation, (ii) a new approach we have constructed to analyze the thermal stability of nonlinear structures, and (iii) bifurcation behaviour we have encountered in the interaction of the dimer with a heat reservoir. The first two will be mentioned in the rest of this section and the third in section 2.