Quantum-Chaotic Evolution Reproduced from Effective Integrable Trajectories

Classically integrable approximants are here constructed for a family of predominantly chaotic periodic systems by means of the Baker-Hausdorff-Campbell formula. We compare the evolving wave density and autocorrelation function for the corresponding exact quantum systems using semiclassical approximations based alternatively on the chaotic and on the integrable trajectories. It is found that the latter reproduce the quantum oscillations and provide superior approximations even when the initial coherent state is placed in a broad chaotic region. Time regimes are then accessed in which the propagation based on the system’s exact chaotic trajectories breaks down.

Figure 1

Exact (black dots) and effective (red lines) orbits for the coserf map calculated, respectively, via the iterative map (3) and the solutions of Eq. (8) with $\delta ={10}^{-2}$. Panel (a) uses $T=0.1$, for which the map displays solely regular trajectories, but for panel (b) we choose $T=0.6$ and a very large chaotic region is presented. Notice how the effective dynamics is simply an interpolation of the exact map for (a), but for (b) no apparent connection between effective and exact dynamics is seen except for the regular regions near the origin.

Figure 2

Classical propagation of the Wigner function of a coherent state initially centered at ($q=4$, $p=0$). The initial distribution is shown in the inset. The kicking strength is $T=0.6$, as in Fig. 1, and $\delta ={10}^{-2}$ in Tao’s algorithm. Chaotic propagation is shown in black, while its effective approximation is superposed in light red. Panel (a) depicts evolution for $N=38$ in Eq. (3), while for panel (b) $N=87$.

Figure 3

Wave densities for the time evolution of the coherent state (4) obtained via the Herman-Kluk propagation (14) and the quantum map (5) for $T=0.6$. The exact quantum result is displayed as a solid black line, while the semiclassical propagations using effective and chaotic trajectories are shown in solid red and dashed gray lines, respectively. We take $\hslash =1$ and, as in Fig. 2, $\delta ={10}^{-2}$, (a) $N=38$ and (b) $N=87$.

Figure 4

(a) Autocorrelation for the quantum propagation (solid black) and its semiclassical approximations, obtained from the effective (solid red) and the chaotic (dashed gray) trajectories. Although $N$ is discrete, we connect the points for ease of visualization. The parameters are the same as in Fig. 2, and the Ehrenfest time ${\tau }_E$ is near $N=20$. (b) The discrete Fourier transform of the autocorrelation has peaks at the coserf map’s eigenphases, expressed here in the units of the effective system’s eigenenergies.