Chaotic quantum ratchets and filters with cold atoms in optical lattices: Properties of Floquet states.

The -kicked rotor is a paradigm of quantum chaos. Its realisation with clouds of cold atoms in pulsed optical lattices demonstrated the well-known quantum chaos phenomenon of 'dynamical localisation'. In those experi ments by several groups world-wide, the £-kicks were applied at equal time intervals. However, recent theoretical and experimental work by the cold atom group at UCL Monteiro et al 2002, Jonckheere et al 2003, Jones et al 2004 showed that novel quantum and classical dynamics arises if the atomic cloud is pulsed with repeating sequences of unequally spaced kicks. In Mon teiro et al 2002 it was found that the energy absorption rates depend on the momentum of the atoms relative to the optical lattice hence a type of chaotic ratchet was proposed. In Jonckheere et al and Jones et al, a possible mechanism for selecting atoms according to their momenta (velocity filter) was investigated. The aim of this thesis was to study the properties of the underlying eigen values and eigenstates. Despite the unequally-spaced kicks, these systems are still time-periodic, so we in fact investigated the Floquet states, which are eigenstates of U(T), the one-period time evolution operator. The Floquet states and corresponding eigenvalues were obtained by diagonalising a ma trix representation of the operator U(T). It was found that the form of the eigenstates enables us to analyse qual itatively the atomic momentum probability distributions, N(p) measured experimentally. In particular, the momentum width of the individual eigen states varies strongly with as expected from the theoretical and ex- perimental results obtained previously. In addition, at specific close to values which in the experiment yield directed motion (ratchet transport), the probability distribution of the individual Floquet states is asymmetric, mirroring the asymmetric N(p) measured in clouds of cesium atoms. In the penultimate chapter, the spectral fluctuations (eigenvalue statis tics) are investigated for one particular system, the double-delta kicked rotor. We computed Nearest Neighbour Spacing (NNS) distributions as well as the number variances (E2 statistics). We find that even in regimes where the corresponding classical dynamics are fully chaotic, the statistics are, unex pectedly, intermediate between fully chaotic (GOE) and fully regular (Pois- son). It is argued that they are analogous to the critical statistics seen in the Anderson metal-insulator transition