How complex is quantum motion?

In classical mechanics the complexity of a dynamical system is characterized by the rate of local exponential instability which effaces the memory of initial conditions and leads to practical irreversibility. In striking contrast, quantum mechanics appears to exhibit strong memory of the initial state. Here we introduce a notion of complexity for a quantum system and relate it to its stability and reversibility properties.

Figure 1

(Color online) Root-mean-square radius ${⟨m^2⟩}_t$ of the distribution of harmonics vs time $t$, at ${\omega }_0=1$, $g_0=1.5$, $\delta =0.5$. Squares, diamonds, and triangles correspond to $\hslash =0.01$, 0.1, and 1. In this latter case, $\widehat{\rho }\left(0\right)=\mid 0⟩⟨0\mid$. Empty circles refer to classical dynamics and the dashed line fits these data.

Figure 2

(Color online) Reversibility properties of quantum dynamics. The backward evolution starts at the reversal time $T=50$. We show $\sqrt{{⟨m^2⟩}_t}$ for different values of the perturbation parameter: from bottom to top, $\xi ={\xi }_c\left(T\right)\times \exp (-l∕2)$, $l=8,\dots ,1$, $l=0$ (thick black curve marked by the closed circle), and $l=-1,\dots ,-6$, at ${\omega }_0=1$, $\hslash =1$, $g_0=2$, $\Delta =1$. Circles indicate positions of the minimum on each curve.

Figure 3

(Color online) Dependence of ${⟨m^2⟩}_t$ on the perturbation strength $g_0$ at time $t=3$, for $\hslash =0.01$, ${\omega }_0=1$, $\Delta =0$. Inset: time evolution of $\sqrt{{⟨m^2⟩}_t}$ in the integrable regime, at $g_0=1$, ${\omega }_0=1$, $\delta =0.5$ and, from bottom to top, $\hslash =1,0.1,0.05,0.02,0.01,0.005$.