  We demonstrate that the differential ac-Stark shift of a ground-state
hyperfine transition in an optical trap can be eliminated by using properly
polarized trapping light. We use the vector polarizability of an alkali-metal
atom to produce a polarization-dependent ac-Stark shift that resembles a Zeeman
shift. We study a transition from the |2S1/2,F=2,mF=-2> to the
|2S1/2,F=1,mF=-1> state of 7Li to observe 0.59+-0.02 Hz linewidth with
interrogation time of 2 s and 0.82+-0.06 s coherence time of a superposition
state. Implications of the narrow linewidth and the long coherence time for
precision spectroscopy and quantum information processing using atoms in an
optical lattice are discussed.
