We consider the atom-optical delta-kicked accelerator when the initial momentum distribution is symmetric. We demonstrate the existence of quantum-resonant dynamics, and derive analytic expressions for the system evolution. In particular, we consider the dynamical evolution of the momentum moments and find that all even-ordered momentum moments exhibit a power-law growth. In the ultracold (zero-temperature) limit the exponent is determined by the order of the moment, whereas for a broad, thermal initial momentum distribution the exponent is reduced by 1. To demonstrate the power-law behavior explicitly we consider the evolutions of the second- and fourth-order momentum moments, and cumulants, for an initially Gaussian momentum distribution corresponding to the Maxwell-Boltzmann distribution of an ideal gas at thermal equilibrium
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