Classical diffusion in double-delta-kicked particles

Abstract

We investigate the classical chaotic diffusion of atoms subjected to {\em pairs} of closely spaced pulses (`kicks) from standing waves of light (the $2\delta$-KP). Recent experimental studies with cold atoms implied an underlying classical diffusion of type very different from the well-known paradigm of Hamiltonian chaos, the Standard Map. The kicks in each pair are separated by a small time interval $\epsilon \ll 1$, which together with the kick strength $K$, characterizes the transport. Phase space for the $2\delta$-KP is partitioned into momentum `cells' partially separated by momentum-trapping regions where diffusion is slow. We present here an analytical derivation of the classical diffusion for a $2\delta$-KP including all important correlations which were used to analyze the experimental data. We find a new asymptotic ($t \to \infty$) regime of `hindered' diffusion: while for the Standard Map the diffusion rate, for $K \gg 1$, $D \sim K^2/2[1- J_2(K)..]$ oscillates about the uncorrelated, rate $D_0 =K^2/2$, we find analytically, that the $2\delta$-KP can equal, but never diffuses faster than, a random walk rate. We argue this is due to the destruction of the important classical `accelerator modes' of the Standard Map. We analyze the experimental regime $0.1\lesssim K\epsilon \lesssim 1$, where quantum localisation lengths $L \sim \hbar^{-0.75}$ are affected by fractal cell boundaries. We find an approximate asymptotic diffusion rate $D\propto K^3\epsilon$, in correspondence to a $D\propto K^3$ regime in the Standard Map associated with 'golden-ratio' cantori.