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 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,
, magnetic field,
and driving
voltage,
. Theoretical analysis has shown that tilting the field
induces a classical transition from integrability (for
)
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 (
, the initial kinetic energy, and
, 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
, and the Haake ``kicked top'' (see [32]) and the
Chirikov standard map for
.
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.