Parameter Scaling in the Decoherent Quantum-Classical Transition for Chaotic Systems

The quantum to classical transition for a system depends on many parameters, including a scale length for its action, $\hslash$, a measure of its coupling to the environment, $D$, and, for chaotic systems, the classical Lyapunov exponent, $\lambda$. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of $(\hslash ,\lambda ,D)$ and searching for transformations that collapse this hypersurface into a function of a composite parameter $\zeta ={\hslash }^{\alpha }{\lambda }^{\beta }{D}^{\gamma }$. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition.

Figure 1

Top: Maximal Kullback-Liebler distance $K_1^m$ as a function of $\hslash$ and $D$, for the quantum Cat Map. Note that small values reflect strong similarity between classical and quantum evolutions. Bottom: Same data plotted in terms of a composite parameter reflecting scaling behavior.

Figure 2

Top: This measure reflects the generation of fine-scale structure in the dynamics with larger values corresponding to classical dynamics. Bottom: Same data plotted in terms of a composite parameter. Note the same scaling as in Fig. 1 and the coincidence of the transition region.

Figure 3

$K_1^A$, the K-L distance computed between classical and quantal autocorrelation functions for the Duffing oscillator with $\hslash =1$ (solid), $1/\sqrt{2}$ (dashed), $3$ (dotted). The ${\hslash }^2/D$ scaling clearly brings the three disparate curves (inset) together. The Duffing parameters are $a=10$, $b=0.5$, $c=10$, and $\omega =6.07$.