Dynamical transition from localized to uniform scrambling in locally hyperbolic systems

Fast scrambling of quantum correlations, reflected by the exponential growth of out-of-time-order correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works [Phys. Rev. Lett. 123, 160401 (2019)] and [Phys. Rev. Lett. 124, 140602 (2020)], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent ${\lambda }_{\mathrm{q}}$ quantifying the OTOC growth is given, respectively, by ${\lambda }_{\mathrm{q}}=2{\lambda }_{\mathrm{s}}$ or ${\lambda }_{\mathrm{q}}={\lambda }_{\mathrm{s}}$ in terms of the stability exponent ${\lambda }_{\mathrm{s}}$ of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon, and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this crossover is a hallmark of unstable separatrix dynamics in integrable systems, thus opening the possibility to distinguish the latter, on the basis of this particular observable, from genuine chaotic dynamics generally featuring uniform exponential growth of the OTOC.

Figure 1

Expected behavior of an OTOC centered at a FP if ${\tau }_{\mathrm{s}}<{\tau }_{\mathrm{L}}<{\tau }_{\mathrm{E}}$: OTOCs grow polynomial for times shorter ${\tau }_{\mathrm{s}}$, exponential with $2{\lambda }_{\mathrm{s}}$ for times shorter ${\tau }_{\mathrm{L}}$, exponential with ${\lambda }_{\mathrm{s}}$ for times shorter ${\tau }_{\mathrm{E}}$, but greater than ${\tau }_{\mathrm{L}}$. Post-Ehrenfest time scales display oscillatory behavior if the system is integrable [29] and saturation if the system is chaotic [17, 43].

Figure 2

Stability exponent of the off-phase FP over the whole parameter space. A change from stable to unstable behavior occurs at the bifurcation point $\mathrm{\Theta }=arctan2$.

Figure 3

Reduced phase space structure ($z,\phi )$ for ${\mathrm{\Theta }}_{*}$: contour lines of the Hamiltonian correspond to the classical trajectories. There are three elliptic fixed points and one hyperbolic fixed point whose associated separatrix is highlighted in red. Arrows on the separatrix indicate the stable and unstable manifolds.

Figure 4

Top to bottom: OTOC $\mathbf{C}\left(t\right)$ for $N={10}^3,{10}^4,5·{10}^4$, and $\mathrm{\Theta }=1.35$; shaping kink from the $2{\lambda }_{\mathrm{s}}\text{−}{\lambda }_{\mathrm{s}}$ transition at ${\tau }_{\mathrm{L}}={\tau }_{\mathrm{E}}/2$; the classical expressions Eq. (26) and Eq. (27) fit tightly the OTOC in each region showing $exp\left(2{\lambda }_{\mathrm{s}}t\right)$ and $exp\left({\lambda }_{\mathrm{s}}t\right)$ exponential growth rates.

Figure 5

Time evolution of the Husimi distribution for the state $|\vec{\xi }\rangle$ centered at the hyperbolic off-phase FP with $N={10}^3$ particles. We observe scrambling along the unstable manifold on the separatrix until $t\approx {\tau }_{\mathrm{L}}$. The thin lines indicate the classical energy contours to visualize the spreading along the the unstable manifold of the off-phase FP.

Figure 6

Left panel: maximal relative logarithmic deviation, Eq. (30), plotted as a function of the system parameter $\mathrm{\Theta }$ in the instability region. Right panel: relative logarithmic deviation, Eq. (29), plotted as a function of time for $\mathrm{\Theta }=1.35$. The calculations were done for the total particle numbers (from top to bottom) $N=100$, 1000, $10000$, and $50000$.

Figure 7

Left panel: OTOC for a squeezed coherent state for the system parameter $\mathrm{\Theta }=1.35$ and particle number $N={10}^3$. The crossover to ${\lambda }_{\mathrm{s}}$ vanishes. Right panel: Husimi distribution of the squeezed coherent state. The state is distributed along the stable manifold of the off-phase FP and features tight localization, with a width $\sim {\hslash }_{\text{eff}}$, along the unstable manifold.