Saddle-point scrambling without thermalization

Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.

Figure 1

(a) FOTOCs) for driven-dimer chaotic phase space (blue dotted line), a saddle point (thick red line), and a point slightly perturbed from the saddle point (green dashed line) for $N=1000$ particles. The infinite-temperature uniform diagonal ensemble (UDE) prediction is indicated by the horizontal thin orange dashed line. The saddle-point state diagonal ensemble prediction is indicated by the horizontal thin red line. (b) Driven-dimer von Neumann entanglement entropy for a subsystem of $s=N/2$ out of a total of $N=100$ particles. The Page curve prediction is indicated by the thin purple dashed-dot line. (c) Post-Ehrenfest time-averaged von Neumann entanglement entropy for the driven-dimer chaotic (blue error bars) and saddle-perturbed (green error bars) states with two standard deviations indicated by bar height. The Page curve prediction (purple dashed-dot line) coincides with the chaotic state result (blue error bars). Entanglement entropy is averaged over $20\le J_{0}t\le 200$. (d) Driven-dimer semiclassical phase space for the regimes with a saddle point (top: $NU=-2,J=1$) and chaos [bottom: $NU=-1,J\left(t\right)=1+1.5cos\left(0.5t\right)$]. The perturbed state $(z,\varphi /\pi )=(0,0.06)$ is indicated by the green star and the saddle point, and chaotic states $(z,\varphi /\pi )=(0,0)$ are indicated by the red square and blue circle, respectively. The phase-space representation is periodic in $\varphi \in [-\pi ,\pi )$.

Figure 2

(a) Diagonal distributions of coherent states in the driven-dimer chaotic phase space (blue bars), on the saddle point (red bars), and slightly perturbed from the saddle point (green staircase) as in Fig. 1. The distributions are ordered by energy or quasienergy magnitude and indexed by $n$. (b) Closeup of driven-dimer saddle-point diagonal distribution with selected high-overlap energy eigenstate transitions labeled by black markers, corresponding to frequencies identified in Figs. 3 and 3. (c) Diagonal entropies of the initial coherent states used in the driven-dimer simulations, relative to the GOE prediction. (d) Driven-dimer saddle-point eigenstate $Q$ distributions for $N=100$, labeled by $n$. The three $Q$ distributions shown are those with the largest overlap with the saddle-point coherent state. Parameters are as in Fig. 1.

Figure 3

(a) Fourier spectra of FOTOCs for driven-dimer chaotic phase space (blue dotted line), a saddle point (red line), and a point slightly perturbed from the saddle point (green dashed line) for $N=100$ particles. (b) Fourier spectra of driven-dimer von Neumann entanglement entropy for a subsystem of $s=N/2$ particles from a total of $N=100$ particles. In both plots, high-occupancy energy eigenstate transition frequencies for the saddle-point state are indicated by black markers, corresponding to Fig. 2. Parameters are as in Fig. 1. (c) Driven-dimer saddle-point state $Q$ distribution dynamics for $N=100$.

Figure 4

FOTOCs for the saddle point in Fig. 1 for $N={10}^p$ particles, where $p=1–5$ from the bottom. Exact dynamics are indicated by black solid lines, and truncated Wigner approximations are indicated by red dashed lines. Exponential short-time FOTOC growth is evident, and the Ehrenfest (scrambling) time increases with $N$.

Figure 5

(a) FOTOCs for the Dicke-model saddle point for strong (blue dotted line) and weak (red solid line) atom-field couplings, scaled by their respective diagonal ensemble predictions; $C_{\mathrm{DE}}/C_{\mathrm{UDE}}\approx 0.54$ and $C_{\mathrm{DE}}/C_{\mathrm{UDE}}\approx 0.027$ for the strongly and weakly coupled regimes, respectively. The strongly coupled regime corresponds to $\mathrm{\Delta }=0.5(\gamma /{\gamma }_c\approx 2.3)$, and the weakly coupled regime corresponds to $\mathrm{\Delta }=3(\gamma /{\gamma }_c\approx 1.1)$. In both regimes $N=200,\gamma =0.66$, and $\omega =0.5$.