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=/√2mω₀α and P₂=(3β)/ (4m√α) (ε, α, β, and ω₀ being the driving amplitude, the frequency chirp rate, the nonlinearity parameter, and the linear frequency of the oscillator). It is shown that, for P₂P ₁+1, the passage through the linear resonance for P₁ above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for P₂P ₁+1, the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in P₁ in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrödinger equation in the energy basis and illustrated via the Wigner function in phase space.