  The parametric ladder climbing (successive Landau-Zener-type transitions) and
the quantum saturation of the threshold for the classical parametric
autoresonance due to the zero point fluctuations at low temperatures are
discussed. The probability for capture into the chirped parametric resonance is
found by solving the Schrodinger equation in the energy basis and the
associated resonant phase space dynamics is illustrated via the Wigner
distribution. The numerical threshold for the efficient capture into the
resonance is compared with the classical and quantum theories in different
parameter regimes.
