Theory of chaotic Hamiltonian ratchets
Fully chaotic Hamiltonian ratchets were first demonstrated using an asymmetric double-well and a chirped sequence of kicks to provide temporal asymmetry - a necessary condition for directed transport to occur. The hallmark of this ratchet system is the saturation of the classical current to a finite value after a characteristic time, termed the ratchet time. The existence of a finite ratchet current is attributed to differential short time diffusion rates for particles with positive and negative momenta. The first half of this thesis investigates this system in detail. In particular, the diffusion rate and average current are investigated analytically and formulae obtained which give good agreement with numerical simulations. The origin of the differential short time diffusion rates is shown to be due to previously neglected momentum dependent corrections to the standard quasi-linear diffusion. These corrections are found by considering correlations between successive kicks in the sequence. The analytical form for the classical ratchet current is also obtained from these correlations. The second system to be covered in this thesis uses a rocking linear term to create the necessary spatial asymmetry, with temporal asymmetry once again being introduced by a chirped kicking sequence. The system is shown to demonstrate a ratchet effect in a similar fashion to the double well system, illustrating the generic nature of the model. The ability to use this rocking ratchet to preferentially select atoms of a given initial momentum, thus creating a chaotic filter, is also introduced. The diffusion coefficient and average current are once again investigated analytically, and the resulting formulae shown to give excellent agreement with numerical results. Finally, the possibility of performing chaotic Hamiltonian ratchet experiments in pulsed standing waves of light (optical lattices) is discussed, and recent results obtained by the Laser Cooling Group at UCL for the rocking ratchet are shown.