DOUBLE-KICKED ATOMS: Experiment and Theory
(1)Atoms in double-kicked periodic potentials: chaos with long-range correlations. PH Jones, M M Stocklin, G.Hur, T S Monteiro Phys.Rev Lett 93 223002 (2004)
(2) Classical Diffusion in double-delta Kicked Particles M M Stocklin and T S Monteiro Phys.Rev.E 74 026210 (2006)
(3)Localization-Delocalization Transition in a System of Quantum Kicked Rotors C.Creffield, G.Hur, T S Monteiro Phys.Rev Lett 96 024103 (2006)
(3) Fractional $\hbar$-scaling for quantum kicked rotors without cantori J. Wang, T S Monteiro, S.Fishman, J.Keating, R.Schubert Phys.Rev Lett 99 234101 (2007).

Atoms subjected to short pulses/'kicks' from optical lattices represent the quantum version of the standard paradigm of Hamiltonian classical chaos, the "Standard Map". There was a lot of excitement when in 1995, a Texas group demonstrated "Dynamical Localization" a phenomenon sometimes called the "quantum suppression of chaotic diffusion". The Figure alongside shows their results: the energy of cloud of atoms subjected to the kicks, initially grows linearly with time, as expected from the chaotic classical dynamics. However, for the quantum case this process stops at the "Break-time". The quantum momentum distribution (which should grow broader as the energy increases) freezes and assumes an exponential shape (the characteristic "triangular shape" using a log plot, as shown in the inset. The width of the distribution, its "localization length" L is proportional to 1/T where T is the kick period.

Experiments by the cold atoms group at UCL (ref[1]) tried instead to apply pairs of closely spaced kicks. The results were surprisingly different from the standard single-kick case. Even in the chaotic regime, the classical phase-space space has a `cellular' structure, with momentum regions partly separated by porous regions where diffusion is slow. The quantum momentum probability distributions have a novel 'staircase' structure superposed on the exponential of the Dynamical localization. The diffusion had a completely different character to the usual Standard Map, because of the "trapping regions" which divide the cells. At first we conjectured that they might be cantori: these represent a sort of fractal dust which arises as the regular trajectories (tori) break up when a Hamiltonian system becomes chaotic. A very nice result of chaos theory shows that the tori which are the most resistant to chaos (the last to break up) are the "golden cantori". These correspond to momenta P/(2*pi)= golden ratio~0.618). As it happens, these would occur in the vicinity of the trapping regions, so at the outset, this seemed a reasonable explanation. The top figure on the right shows that the trapping regions of the DKR2 (middle two figures) coincide with the golden cantori (which surround the period 3 island chain on the Surface of Section for the Standard Map on the far right).

The quantum momentum probability distributions have a novel 'staircase' structure superposed on the exponential of the Dynamical localization. Examples of the staircase distribution are shown in the bottom figure on the right . The width ('Localization Length') of the staircase scales with a fractional power of (-2/3)rds of the period, unlike the usual kicked system, which scales with an integer power of (-1). This too pointed to golden cantori as they have been associated with ratios of (2/3) and (3/4). However, repeating the calculations in regimes with no cantori whatsoever, made no difference to the fractional scaling ratio of (2/3). A closer analysis however found a completely different explanation for this ratio. Its origin resides in a function called an Airy function, which is involved in the mathematical description of the dynamics in the trapping regions.

Colleagues in Singapore (J Wang and J. Gong) later showed that the Double Kicked Rotor has a Hofstadter Butterly spectrum if the total time period T= 4*pi.

Mischa Stocklin won a UCL GSRS Scholarship and graduated in 2008.