Chaotic Landau level mixing in classical and quantun wells

Several recent experiments in semiconductor heterostructures have studied the tunneling current of electrons through planar potential barriers in a magnetic field which is tilted at an angle $\theta$ with respect to the normal to the barriers (L.Eaves et al. at Nottingham and G.Boebinger et al. at Bell Labs). When a variable voltage is applied across the barriers these systems exhibit oscillations in the I-V characteristic which were found to have a strong sensitivity to the tilt angle, $\theta$, magnetic field, $B$ and driving voltage, $\Delta V$. Theoretical analysis has shown that tilting the field induces a classical transition from integrability (for $\theta=0$) to chaos; Fromhold et al. showed that Gutzwiller periodic orbit theory of the density of states oscillations could account for the dominant features of resonances in the I-V curve in the strongly chaotic regime. This work has drawn attention to a new dynamical system for the study of classical and quantum chaos which should have other experimental signatures in semiconductor quantum wells and experimental realizations outside of solid-state physics. It therefore seemed worthwhile to analyze the dynamics of this system from the point of view of the global phase space structure and obtain the relevant parametric criteria for the onset of chaos in different regimes [73]. Since the real-space motion is three-dimensional and depends on a large number of parameters ( $\theta,B,\Delta V,E_i$, the initial kinetic energy, and $d$, the distance between the barriers) it is not obvious that this system can be reduced to familiar models which have been previously analyzed. The results obtained in [73] show that in fact the transition to chaos in this system can be described by a two-dimensional map with strong similarities to the Fermi acceleration model in the limit $\Delta V \to 0$, and the Haake ``kicked top'' (see [32]) and the Chirikov standard map for $\Delta V \gg E_i$. In both limits the parametric conditions for the onset of chaos are obtained being in agreement with numerical simulations and experimental results of the above experimental groups. The onset of chaos allows effective energy exchange between Landau levels and longitudinal motion. However, quantum effects suppress energy exchange up to a critical angle determined by the localization transition. The theoretical analysis pointed out many similarities of the above problem with the quantum ping-pong model discussed in section 3.3.4. The problem of dynamics in a tilted magnetic field is still actively studied by different theoretical and experimental groups.