Title:

Stochastic webinduced chaotic dynamics in periodic potentials

This thesis describes the dynamics of particles in periodic potentials with a
constant acceleration force and a tilted harmonic trap. The particles exhibit
nonKAM chaos, whose semi classical trajectories are a combination of Bloch and
harmonic oscillations. When the frequencies of these two oscillations are commensurate,
the phase space is threaded by a stochastic web, along which the
particles can diffuse rapidly. vVe find that the stochastic web rings, whose radii
correspond to the roots of Bessel functions , found in the semiclassical analysis
confine the electron to a finite region in real space, both in a semiclassical and a
full quantum picture.
Firstly, we study electron transport in a semiconductor supedattice when an
electric field and a tilted magnetic field are applied. The tiltcd magnetic field
spatially offsets the Landaulike eigenstates, induced by the magnetic field , and
tunnel coupling is allowed. The overlap integral of quantized electron states in
ad.iacent quantum wells, which determines the tunnel coupling and the extent of
the quantum wavefunctions, vanishes at the zeros of the Bessel functions , precisely
the same position where, semi classically, the electron is trapped by the rings of
the stochastic web, suggesting remarkable synergy between the semi classical and
quantum regimes.
Secondly, we consider neutral 23Na atoms falling under the influence of gravity
through a periodic lattice potential, created by the interference pattern of two
counterpropagating laser beams. A harmonic trap is applied tilted to the lattice
axis. The atom trajectories undergo dramatic abrupt extension in the vertical
direction and map out intricate web patterns in momentum phase space when the
Bloch frequency and a frequency corresponding to the harmonic trap arc commensurate.
We find three indicators of a resonant trajectory, which are the maximal
vertical displacement, the longest return time to the origin, and the sign change of
a momentum component. Since the Bloch frequency is determined by the gravitational
acceleration, g, and the appearance of the chaotic resonance point is very
sensitive to the Bloch oscillation frequency, the onset of chaos can measure g with
a small relative uncertainty 6.g/ 9 = 4 X 10  7
. We also investigate the dynamics of
a quantum wavepacket ill the system, using the CrankNicolson method to solve
the timedependent Schrodinger equation with our fast GPU tridiagonal matrix
solver. Estimating the quantum mean trajectory and the wavepacket evolution,
the
