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Chaotic systems, that have a small Lyapunov exponent, do not obey the commonrandom matrix theory predictions within a wide "weak quantum chaos" regime.This leads to a novel prediction for the rate of heating for cold atoms inoptical billiards with vibrating walls. The Hamiltonian matrix of the drivensystem does not look like one from a Gaussian ensemble, but rather it is verysparse. This sparsity can be characterized by parameters $s$ and $g$ thatreflect the percentage of large elements, and their connectivity respectively.For $g$ we use a resistor network calculation that has direct relation to thesemi-linear response characteristics of the system.

Quantum response of weakly chaotic systems