Scaling laws in the quantum-to-classical transition in chaotic systems

We study the quantum-to-classical transition in a chaotic system surrounded by a diffusive environment. First, we analyze the emergence of classicality when it is monitored by the Renyi entropy, a measure of the entanglement of a system with its environment. We show that the Renyi entropy has a transition from quantum to classical behavior that scales with ${\hslash }_{\mathrm{eff}}^2∕D$, where ${\hslash }_{\mathrm{eff}}$ is the effective Planck constant and $D$ is the strength of the noise. However, it was recently shown that a different scaling law controls the quantum-to-classical transition when it is measured comparing the corresponding phase-space distributions. Then, we discuss the meaning of both scalings in the precise definition of a frontier between the classical and quantum behaviors. Finally, we show that there are quantum coherences that the Renyi entropy is unable to detect, which questions its use in studies of decoherence.

Figure 1

Quantum (●) and classical (○) Renyi entropy as a function of the number of the kicks for the KHO with $K=2$ and $\eta =\sqrt{{\hslash }_{\mathrm{eff}}∕2}=0.125$. From bottom to top, the diffusion constants are $D=2.56\times {10}^{-7}$, $6.4\times {10}^{-6}$, $9\times {10}^{-5}$, and $4\times {10}^{-3}$. The dashed line is a linear function with a slope given by the logarithm of the classical expansion coefficient at $(q,p)=(0,0)$.

Figure 2

Slope $\gamma$ of the linear growth of the quantum (solid symbols) and classical (open symbols) Renyi entropies as a function of ${\hslash }_{\mathrm{eff}}^2∕D$ (main plot) and $1∕D$ (inset) ($D$ is the diffusion constant) for $\eta =\sqrt{{\hslash }_{\mathrm{eff}}∕2}=0.5$ (triangles), 0.3125 (diamonds), 0.125 (squares), and 0.0625 (circles). We have also plotted the functions $\gamma \propto D∕{\hslash }_{\mathrm{eff}}^2$ (dotted line) and $\gamma \propto \sqrt{D}∕{\hslash }_{\mathrm{eff}}$ (dashed line). See text for details.

Figure 3

The measure $\sigma \left(n_E\right)$ (see text) of the differences between the quantum and classical Renyi entropies. The initial state is a coherent state centered around the origin of phase space with width $\Delta q=\Delta p=\eta$, with $\eta =\sqrt{2{\hslash }_{\mathrm{eff}}}=0.5$ (triangles), 0.3125 (diamonds), 0.125 (squares), and 0.0625 (circles). For each ${\hslash }_{\mathrm{eff}}$ the Renyi entropies were computed at the logarithmic time $t_E\approx n_E\tau$.

Figure 4

(Color online) Wigner distribution (top panel) and classical distribution [inset (a)] immediately before the kick $n=7$ for $\eta =\sqrt{2{\hslash }_{\mathrm{eff}}}=0.125$ and diffusion constant $D=9\times {10}^{-5}$. Inset (b): Wigner distribution immediately before the kick $n=8$ for $\eta =\sqrt{2{\hslash }_{\mathrm{eff}}}=0.0256$ and diffusion constant $D=9\times {10}^{-5}$. Bottom panel: Wigner distribution (blue dotted line) and classical distribution (red solid line) evaluated on $(q=-2,p)$ (dashed lines in the top panel).