| Summary: | This thesis describes the dynamics of particles in periodic potentials with a constant acceleration force and a tilted harmonic trap. The particles exhibit non-KAM 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 Landau-like eigenstates, induced by the magnetic field , and tunnel coupling is allowed. The overlap integral of quantized electron states in ad.i-acent 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 counter-propagating 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 Crank-Nicolson method to solve the time-dependent Schrodinger equation with our fast GPU tridiagonal matrix solver. Estimating the quantum mean trajectory and the wavepacket evolution, the
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