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

Abstract

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 ${P}_{1}=\ensuremath{\varepsilon}/\sqrt{2m\ensuremath{\hbar}{\ensuremath{\omega}}_{0}\ensuremath{\alpha}}$ and ${P}_{2}=(3\ensuremath{\hbar}\ensuremath{\beta})/(4m\sqrt{\ensuremath{\alpha}})$ ($\ensuremath{\varepsilon},$ $\ensuremath{\alpha},$ $\ensuremath{\beta}$, and ${\ensuremath{\omega}}_{0}$ being the driving amplitude, the frequency chirp rate, the nonlinearity parameter, and the linear frequency of the oscillator). It is shown that, for ${P}_{2}\ensuremath{\ll}{P}_{1}+1$, 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 ${P}_{2}\ensuremath{\gg}{P}_{1}+1$, 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 Schr\"odinger equation in the energy basis and illustrated via the Wigner function in phase space.