Directed chaotic transport in Hamiltonian ratchets

Area of Science:

• Physics
• Statistical Mechanics
• Quantum Mechanics

Background:

• Hamiltonian systems with spatial and temporal periodicity can exhibit directed transport without an average force, acting as Hamiltonian ratchets.
• Understanding the conditions and mechanisms for directed transport is crucial for controlling particle motion in such systems.

Purpose of the Study:

• To provide a comprehensive account of directed transport in one-dimensional Hamiltonian ratchets.
• To identify general conditions for directed transport and analyze its behavior in both classical and quantum regimes.

Main Methods:

• Analysis of classical phase space, including mixed phase space conditions.
• Derivation of a sum rule connecting phase-space components to transport.
• Study of quantized Hamiltonian ratchets using wave packet evolution and semiclassical expressions.
• Investigation of dynamical tunneling and transport breakdown in quantum systems.

Main Results:

• Directed transport can occur in the absence of an average force.
• Regular ratchet transport can proceed against an external potential gradient.
• Chaotic ballistic transport is confined to unbiased systems.
• A semiclassical expression for level velocities was derived for quantized ratchets.
• The role of dynamical tunneling and the impact of broken spatial periodicity on transport were discussed.

Conclusions:

• The study elucidates the distinct behaviors of regular and chaotic transport in Hamiltonian ratchets.
• It highlights the importance of phase space structure and quantum effects like dynamical tunneling in determining transport properties.
• Findings provide insights into controlling directed motion in periodically driven quantum systems.