Quantum versus classical phase-locking transition in a frequency-chirped nonlinear oscillator

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/root 2mh omega(0)alpha and P-2 = (3h beta)/(4m root 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 << 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 >> 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.