Quantum entropy dynamics for chaotic systems beyond the classical limit

The entropy production rate for an open quantum system with a classically chaotic limit has been previously argued to be independent of $\hslash$ and $D$, the parameter denoting coupling to the environment, and to be equal to the sum of generalized Lyapunov exponents, with these results applying in the near-classical regime. We present results for a specific system going well beyond earlier work, considering how these dynamics are altered for the Duffing problem by changing $\hslash$,$D$ and show that the entropy dynamics have a transition from classical to quantum behavior that scales, at least for a finite time, as a function of ${\hslash }^2∕D$.

Figure 1

(Color online) Entropy production rate ${\dot{S}}_2$ [where $S_2=\ln \left(P\right)$ for states with purity $P$] for the quantum Duffing oscillator with $m=1$, $B=10$, $C=1$, $A=1$, $\omega =5.35$, and $\hslash$,$D$ as indicated, showing a wide variation in behavior. The initial conditions are Gaussians in the chaotic region with ${⟨x⟩}_{t=0}=1.0$, the spread ${\sigma }_x^2=0.05$, ${⟨p⟩}_{t=0}=0.0$, and the spread ${\sigma }_p^2$ is set by the constraint of defining a minimum-uncertainty state and hence by the particular value of $\hslash$.

Figure 2

Evolution of the normalized purity for the same states as Fig. 1 in the quantum Duffing oscillator. Scaling is observed relative to the parameter ${\zeta }_0\equiv {\hslash }^2∕D$. (I) Classical: ${\zeta }_0=2$, (a) $\hslash =0.1$, $D=5\times {10}^{-4}$, (b) $\hslash =0.2$, $D=2\times {10}^{-2}$, and (c) $\hslash =0.5$, $D=0.125$. (II) Semi-classical: ${\zeta }_0=40$, (d) $\hslash ={0.004}^{1∕2}$, $D={10}^{-4}$, (e) $\hslash =0.1$, $D=2.5\times {10}^{-4}$, and (f) $\hslash =0.2$, $D={10}^{-3}$. (III) Quantum: ${\zeta }_0=100$, (g) $\hslash =0.1$, $D={10}^{-4}$, (h) $\hslash =0.5$, $D=2.5\times {10}^{-3}$, and (i) $\hslash =1$, $D={10}^{-2}$.