Parameter scaling in the decoherent quantum-classical transition for chaotic rf superconducting quantum interference devices

We numerically investigated the quantum-classical transition in rf-superconducting quantum interference device (SQUID) systems coupled to a dissipative environment. It is found that chaos emerges and the degree of chaos, the maximal Lyapunov exponent ${\lambda }_m$, exhibits nonmonotonic behavior as a function of the coupling strength $D$. By measuring the proximity of quantum and classical evolution with the uncertainty of dynamics, we show that the uncertainty is a monotonic function of ${\lambda }_m/D$. In addition, the scaling holds in SQUID systems to a relatively smaller ${\hslash }_{eff}$, suggesting the universality for this scaling.

Figure 1

Poincaré sections for $D=0.25,0.35,0.45$, from top to bottom. From middle panel we can see that points are largely confined in three regions, which indicates a nonmonotonic transition of chaos.

Figure 2

Maximal Lyapunov exponent ${\lambda }_m$ versus $D$.The distinctive dip rightly attests the occurrence of suppression of chaos.

Figure 3

Averaged uncertainty ${\Delta }_a$ as a function of (a) $D$ and (b) a composite parameter ${\lambda }_m/D$. The monotonic increase of ${\Delta }_a$ as a function of ${\lambda }_m/D$ in (b) demonstrated the scaling law.

Figure 4

(Color online) Averaged uncertainty ${\Delta }_a$ as a function of $D$ and ${\hslash }_{eff}$.The parameters for the system with largest effective Planck constant ${\hslash }_{eff}=2.6$ has been shown in text. Other ${\hslash }_{eff}$ and corresponding sets of parameters are listed in Table .

Figure 5

(Color online) ${\Delta }_a$ versus a composite parameter ${\lambda }_m/D$ for different effective Plank constant. It is shown that the scaling law holds for systems with different ${\hslash }_{eff}$. Inset: the curve with ${\hslash }_{eff}=1$ is shown separately.