  The orbital period evolution of X-ray binaries provides fundamental clues to
understanding mechanisms of angular momentum loss from these systems. We
present an X-ray eclipse timing analysis of the transient low mass X-ray binary
AX J1745.6-2901. This system shows full eclipses and thus is one of the few
objects for which accurate orbital evolution studies using this method can be
carried out. We report on XMM-Newton and ASCA observations covering 30 complete
X-ray eclipses spanning an interval of more than 20 years. We improve the
determination of the orbital period to a relative precision of
$2\times10^{-8}$, two orders of magnitudes better than previous estimates. We
determine, for the first time, a highly significant rate of decrease of the
orbital period $\dot{P}_{orb}=-4.03\pm0.32\times10^{-11}$~s/s. This is at least
one order of magnitude larger than expected from conservative mass transfer and
angular momentum losses due to gravitational waves and magnetic breaking, and
might result from non-conservative mass transfer. Imprinted on the long term
evolution of the orbit, we observe highly significant eclipse leads-delays of
~10-20 s, characterised by a clear state dependence in which, on average,
eclipses occur earlier during the hard state.
