  We study the statistical properties of spherical harmonic modes of
temperature maps of the cosmic microwave background. Unlike other studies,
which focus mainly on properties of the amplitudes of these modes, we look
instead at their phases. In particular, we present a simple measure of phase
correlation that can be diagnostic of departures from the standard assumption
that primordial density fluctuations constitute a statistically homogeneous and
isotropic Gaussian random field, which should possess phases that are uniformly
random on the unit circle. The method we discuss checks for the uniformity of
the distribution of phase angles using a non-parametric descriptor based on the
use order statistics, which is known as Kuiper's statistic. The particular
advantage of the method we present is that, when coupled to the judicious use
of Monte Carlo simulations, it can deliver very interesting results from small
data samples. In particular, it is useful for studying the properties of
spherical harmonics at low l for which there are only small number of
independent values of m and which therefore furnish only a small number of
phases for analysis. We apply the method to the COBE-DMR and WMAP sky maps, and
find departures from uniformity in both. In the case of WMAP, our results
probably reflect Galactic contamination or the known variation of
signal-to-noise across the sky rather than primordial non-Gaussianity.
