Crossover from classical to quantum behavior of the Duffing oscillator through a pseudo-Lyapunov-exponent

We discuss the quantum-classical correspondence (QCC) in a specific dissipative chaotic system, the Duffing oscillator. The quantum version of the Duffing oscillator is treated as an open quantum system and analyzed numerically by the use of quantum state diffusion (QSD). We consider a pseudo-Lyapunov exponent and investigate it in detail, varying the Planck constant effectively. We show that there exists a critical stage in which the crossover from classical to quantum behavior occurs. Furthermore, we find that a dissipation effect suppresses the occurrence of chaos in the quantum region, while it, combined with the periodic external force, plays a crucial role in the chaotic behavior of the classical system.

Figure 1

The time evolution of $\Delta \left(\tau \right)$ with $ϵ$ fixed as 0.01. The quantities plotted are dimensionless by definition. The complex Wiener process is used in the QSD. (a) and (b) are obtained with a single realization of the complex Wiener process for each initial condition (20 samples). (c)–(d) are obtained by averaging over 100 realizations of the complex Wiener process for each initial condition (10 samples). (a)–(d) are for $\beta =0.01$, 0.10, 0.40, and 1.00, respectively.

Figure 2

The time evolution of $\Delta \left(\tau \right)$ with $ϵ=\beta$. The quantities plotted are dimensionless by definition. The complex Wiener process is used in the QSD. The asymptotic value of $\Delta \left(\tau \right)$, ${\Delta }_{\mathit{asymp}}$, is indicated by the dotted line. The right-hand side of Eq. (7), $D\left(\tau \right)$, is expressed by the broken dotted line. Figure (a) is obtained with a single realization of the complex Wiener process for each initial condition (20 samples). Figures (b)–(f) are obtained by averaging over 100 realizations of the complex Wiener process for each initial condition (10 samples). Figures (a)–(e), and (f) are for $\beta =0.10$, 0.40, 0.60, 1.00, 1.50, and 2.00, respectively.