  In a very interesting recent Letter\cite{machida}, the authors suggested the
possibility of realizing the spatially modulated, or
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state in trapped atomic
fermion systems. The authors this Letter used a 1D mean field solution as
guidance to estimate the parameter range for the existence of the FFLO phase,
and also discussed the possibility of its detection by imaging the atomic
density of the system. In this comment I wish to make two points. (i) In 1D
there exists an exact solution based on bosonization, which fully takes into
account the important quantum fluctuation effects\cite{yang01}; the exact
solution suggests a wider parameter range for the FFLO state than that obtained
from the mean-field solution of the present Letter. (ii) One can detect the
FFLO pairing (in which Cooper pairs carry finite momenta) more directly by
extending the methods used to detect BCS pairing\cite{regal,altman,greiner}.
