THEORY OF THE PERIODIC ORBITS OF A CHAOTIC QUANTUM WELL

A theory is developed for the periodic orbits of an electron trapped in a rectangular potential well under the influence of an electric field normal to the barriers and a magnetic field. When the magnetic field is parallel to the electric field the dynamics of an electron in the well is integrable; however, when it is tilted by an angle $\ensuremath{\theta}$ the system undergoes a transition to chaos. Motivated by recent experimental and theoretical studies of magnetotunneling in quantum wells that emphasize the role of periodic orbits, we present here a unified theory of all the periodic orbits within the well that are of relevance to experiments. We define the appropriate scaled variables for the problem, which we divide into two qualitatively different cases, the single-barrier model (depending on two parameters) and the double-barrier model (depending on three parameters). We show that in both cases all relevant orbits are related to bifurcations of period-one traversing orbits. A full analytic theory is derived for the period and stability of these traversing orbits; and analytic and numerical results are obtained for the important period-two and period-three orbits. An unusual feature of the classical mechanics of the double-barrier is a discontinuity in the classical Poincar\'e map, which leads to a new type of bifurcation that we term a cusp bifurcation. We show that all the periodic orbits that traverse the well exist only in finite intervals of voltage and magnetic field, appearing and disappearing in bifurcations. These intervals are shown to correspond to the appearance of new resonance peaks in the experimental data, laying the foundation for a quantitative semiclassical treatment of the system.