  Anderson localization is the ubiquitous phenomenon of inhibition of transport
of classical and quantum waves in a disordered medium. In dimension one, it is
well known that all states are localized, implying that the distribution of an
initially narrow wave-packet released in a disordered potential will, at long
time, decay exponentially on the scale of the localization length. However, the
exact shape of the stationary localized distribution differs from a purely
exponential profile and has been computed almost fifty years ago by Gogolin.
  Using the atomic quantum kicked rotor, a paradigmatic quantum simulator of
Anderson localization physics, we study this asymptotic distribution by two
complementary approaches. First, we discuss the connection of the statistical
properties of the system's localized eigenfunctions and their exponential decay
with the localization length of the Gogolin distribution. Next, we make use of
our experimental platform, realizing an ideal Floquet disordered system, to
measure the long-time probability distribution and highlight the very good
agreement with the analytical prediction compared to the purely exponential one
over 3 orders of magnitude.
