  Classical and quantum-mechanical phase locking transition in a nonlinear
oscillator driven by a chirped frequency perturbation is discussed. Different
limits are analyzed in terms of the dimensionless parameters $% P_{1}=\epsilon
/\sqrt{2m\hbar \omega_{0}\alpha}$ and $P_{2}=(3\hbar \beta)/(4m\sqrt{\alpha})$
($\epsilon,$ $\alpha,$ $\beta$ and $\omega_{0}$ being the driving amplitude,
the frequency chirp rate, the nonlinearity parameter and the linear frequency
of the oscillator). It is shown that for $P_{2}\ll P_{1}+1$, the passage
through the linear resonance for $P_{1}$ above a threshold yields classical
autoresonance (AR) in the system, even when starting in a quantum ground state.
In contrast, for $% P_{2}\gg P_{1}+1$, the transition involves
quantum-mechanical energy ladder climbing (LC). The threshold for the
phase-locking transition and its width in $P_{1}$ in both AR and LC limits are
calculated. The theoretical results are tested by solving the Schrodinger
equation in the energy basis and illustrated via the Wigner function in phase
space.
