The Alcubierre drive is a long shot, but is it possible to create a matter/antimatter reactor at all? I assume that the only thing stopping your grandfather's nuclear reactor from operating the warp drive in Star Trek is the energy demand, a thirst only the warp core can quench.
Remember that in space, nothing is done because it's 'cool'. If the Federation could get away with it, all its exploration would have been done with unmanned probes, which have lower energy demands, no need for expensive artificial gravity and life support, the ability to survive speeds that would kill everybody aboard a starship, and no human losses when the probe is destroyed.
In other words, there has to be some reason why the large, expensive starship, with its fancy matter/antimatter reactor, was chosen over an unmanned probe with lithium-ion batteries.