Lyapunov Exponent and Out-of-Time-Ordered Correlator’s Growth Rate in a Chaotic System

It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because, in the semiclassical limit $\hslash \to 0$, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator $C(t)$ for the classical and quantum kicked rotor—a textbook driven chaotic system—and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC’s growth rate and the Lyapunov exponent are, in general, distinct quantities, corresponding to the logarithm of the phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength $K$, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as $K\to 0$, while the OTOC’s growth rate may decrease much slower, showing a higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time $t_E$: transitioning from a time-independent value of $t^{-1}\ln C(t)$ at $t<t_E$ to its monotonic decrease with time at $t>t_E$. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.

Figure 1

The upper panel shows the OTOC $C(t)$ vs $t$ in the semilog scale for various values of the kicking strength ($K=0.5$, 2, 3, 6, 10) and ${\hslash }_{\mathrm{eff}}=2^{-14}$. The lower panel is a plot of the two-point function $B(t)$ vs $t$ at the corresponding parameters (in the linear scale). Averaging is performed over the Gaussian wave packet defined in Eq. (4) with $p_0=0$ and $\sigma =4$.

Figure 2

Red circles: Early-time growth rate of $C(t)$ at ${\hslash }_{\mathrm{eff}}=2^{-14}$ (quantum CGR). The rest of the data are classical. Green solid line: Growth rate of $C_{\mathrm{cl}}(t)$ (classical CGR). Blue triangles: LE calculated numerically. Black dashed line: LE according to the Chirikov analytical formula (5). The main plot and the inset show the same data in the lin-log and linear scales, respectively (and in different ranges). At $K\gtrsim 8$, the difference between the CGR and the LE is constant $\approx \ln \sqrt{2}$. The initial state in $C(t)$ is the Gaussian (4) with $p_0=0$ and $\sigma =4$. Fitting details for extracting the CGR from $C(t)$ and $C_{\mathrm{cl}}(t)$ are given in the main text.

Figure 3

Main plot: $\ln [C(t)]/2t$ vs $t$ in the log-log scale for $K=3$, 4, 7, 10 (from bottom to top line, respectively) and ${\hslash }_{\mathrm{eff}}=2^{-14}$. The flat region at early times quantifies the exponential growth rate of $C(t)$. This flat region persists up to time $t_E$, at which the exponential growth slows down to a power-law growth with a slowly decreasing power. Dotted lines are guides to the eye: Horizontal lines extend the flat regions, and the sloped line is shown for a power comparison. Inset: $\ln [C(t)]/2t$ vs $t$ in the log-log scale for $K=4$ and ${\hslash }_{\mathrm{eff}}=2^{-14},2^{-10},2^{-6},2^{-2}$ (from top to bottom line, respectively). The rate of exponential growth is the same for different values of ${\hslash }_{\mathrm{eff}}$, but $t_E$ shrinks when ${\hslash }_{\mathrm{eff}}$ increases.

Figure 4

Long-time average ${\overline{B}}_{\tau }$ (7) (over various windows $\tau$) of the two-point correlator $B(t)$ as a function of $K$ compared to the regular fraction of the phase space weighted with the initial Wigner distribution $P(x,p)$ (scaled). The trend with increasing $\tau$ shows that, at all $K\ne 0$, the correlations decay in time, but the rate of this decay has a steplike dependence on $K$. At $K>K_{\mathrm{cr}}$, the decay is quite fast, while at $K<K_{\mathrm{cr}}$, it takes ${\overline{B}}_{\tau }$ at least an exponentially large window to vanish. It is not clear from the data whether at small $K\ne 0$ the averaged correlator eventually goes to zero at $\tau \to \infty$ or is bounded from below. The initial state corresponding to $P(x,p)$ is the Gaussian (4) with $p_0=0$ and $\sigma =4$.

Figure 5

Initial Wigner distribution $P(x,p)$ (3D plot) on the top of the classical Lyapunov exponent (shown in color in the horizontal plane; see the color bar for numerical values). The initial state corresponding to $P(x,p)$ is the Gaussian (4) with $p_0=0$ and $\sigma =4$. The Lyapunov exponent is shown for $K=1$.