Short-time Loschmidt gap in dynamical systems with critical chaos

We study the Loschmidt echo $F\left(t\right)$ for a class of dynamical systems showing critical chaos. Using a kicked rotor with singular potential as a prototype model, we found that the classical echo shows a gap (initial drop) $1-F_g$, where $F_g$ scales as $F_g\left(\alpha ,ϵ,\eta \right)=f_{\text{cl}}\left({\chi }_{\text{cl}}\equiv {\eta }^{3-\alpha }/ϵ\right)$; $\alpha$ is the order of singularity of the potential, $\eta$ is the spread of the initial phase-space density, and $ϵ$ is the perturbation strength. Instead, the quantum echo gap is insensitive to $\alpha$, described by a scaling law $F_g=f_q\left({\chi }_q={\eta }^2/ϵ\right)$ which can be captured by a random matrix theory modeling of critical systems. We trace this quantum-classical discrepancy to strong diffraction effects that dominate the dynamics.

Figure 1

(Color online) Quantum (red dashed line) and classical (black solid line) LE $F\left(t\right)$ for the KR with $V\left(q\right)=\text{log}\left|q\right|$ and $K_0=1$ in the nonperturbative regime: (a) torus geometry with $L=1$, $N=2^{17}$, and classical perturbation $ϵ={10}^{-4}$ (corresponding to $\sigma \approx 2.0$). The width of the initial preparation is $\eta \approx 0.04275$. (b) Cylinder geometry, with $L={10}^3$, $N=2^{16}$, and classical perturbation $ϵ=0.4$ (corresponding to $\sigma \approx 4.17$). One observes that after the initial Lyapunov decay (inset; the dashed line indicates an average Lyapunov decay), $F\left(t\right)$ follows a power law decay given by the autocorrelation function [4, 23]. Here we have $\eta \approx 0.15$. In both cases we have used more than ${10}^7$ trajectories for the classical calculation, while an averaging over 800 initial Gaussian wave packets centered at different momenta $p_0$ and $q_0=0$ has been done in the quantum calculation. The filled circles indicate the first (classical or quantum) nontrivial time step of $F_g=F\left(n=2\right)$. The statistical errors are smaller than the symbol size.

Figure 2

(Color online) Classical echo map [Eq. (8)] for the $\alpha =0,K_0=1$ singular potential (cylinder geometry with $L={10}^3$). The initial preparation is a box of size $\eta =\pi /300$ around (0,0). The evolution snapshot is for the shortest nontrivial time scale $n=2$. The thick dashed (red) line is the theoretical prediction of Eq. (9). Upper left inset: The Wigner function representation (torus geometry) of the quantum echo map. The light areas correspond to negative phase-space densities indicating diffraction phenomena. Lower right inset: a typical phase space for $K_0=1$ (20 trajectories involved up to time ${10}^5$ iterations of the map).

Figure 3

(Color online) The $F_g$ for various $\alpha ,\eta ,ϵ$’s of the KR defined by Eqs. (3, 4). In (a) we report the classical $F_g$ against the scaling variable ${\chi }_{\text{cl}}$ [see Eq. (2)], while in (b) we report the corresponding quantum $F_g$ vs the scaling variable ${\chi }_q$ [see Eq. (2)]. At the same subfigure we report the results for the RPKR resulting from the model [Eq. (6)] by a randomization of the phases of the kinetic part of $U_0$. In the insets we report a magnification of the main panels in the regime of $F_g\approx 1$. In all cases an excellent data collapse is observed.