  Decoherence is a major obstacle to any practical implementation of quantum
information processing. One of the leading strategies to reduce decoherence is
dynamical decoupling --- the use of an external field to average out the effect
of the environment. The decoherence rate under any control field can be
calculated if the spectrum of the coupling to the environment is known. We
present a direct measurement of the bath coupling spectrum in an ensemble of
optically trapped ultracold atoms, by applying a spectrally narrow-band control
field. The measured spectrum follows a Lorentzian shape at low frequencies, but
exhibits non-monotonic features at higher frequencies due to the oscillatory
motion of the atoms in the trap. These features agree with our analytical
models and numerical Monte-Carlo simulations of the collisional bath. From the
inferred bath-coupling spectrum, we predict the performance of well-known
dynamical decoupling sequences: CPMG, UDD and CDD. We then apply these
sequences in experiment and compare the results to predictions, finding good
agreement in the weak-coupling limit. Thus, our work establishes experimentally
the validity of the overlap integral formalism, and is an important step
towards the implementation of an optimal dynamical decoupling sequence for a
given measured bath spectrum.
