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

TL;DR

The paper investigates how the growth rate of the out-of-time-ordered correlator (OTOC) in a chaotic quantum system relates to the classical Lyapunov exponent (LE) using the quantum kicked rotor (QKR) as a benchmark. It defines the quantum OTOC C(t) = −⟨[ p̂(t), p̂(0) ]^2⟩ and the two-point correlator B(t), and shows that the early-time exponential growth rate ṫλ (the CGR) generally differs from the classical LE due to the different averaging procedures, with ṫλ > λ in many regimes, especially when chaotic islands are sparse. A cusp at the Ehrenfest time t_E marks the onset of quantum interference, transitioning from exponential to power-law growth, consistent with weak dynamical localization. The two-point correlator B(t) retains signatures of the classical regular-to-chaotic transition even under dynamical localization, illustrating nuanced quantum‑classical correspondence in chaos diagnostics.

Abstract

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 semi-classical limit, $\hbar \to 0$, 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 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 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 monotonous 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.

Figures (7)

• Figure 1: (Color online) The upper panel shows OTOC, $C(t)$, vs $t$ in the semi-log scale for various values of the kicking strength ($K=0.5, 2, 3, 6, 10$) and $\hbar_{\rm eff}=2^{-14}$. 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. (\ref{['eq:Gauss_wf']}) with $p_0 = 0$ and $\sigma=4$.
• Figure 2: (Color online) Red circles: early-time growth rate of $C(t)$ at $\hbar_{\rm eff} = 2^{-14}$ (quantum CGR). The rest of the data is classical. Green solid line: growth rate of $C_{\rm cl}(t)$ (classical CGR). Blue triangles: LE calculated numerically. Black dashed line: LE according to the Chirikov's analytical formula (\ref{['eq:Lyapunov_analytic']}). The main plot and the inset show the same data in lin-log and linear scales, respectively (and in different ranges). At $K \gtrsim 8$, the difference between CGR and LE is constant $\approx\ln\sqrt{2}$. The initial state in $C(t)$ is the Gaussian (\ref{['eq:Gauss_wf']}) with $p_0 = 0$ and $\sigma=4$. Fitting details for extracting CGR from $C(t)$ and $C_{\rm cl}(t)$ are given in the main text.
• Figure 3: (Color online) 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 $\hbar_{\rm eff} = 2^{-14}$. The flat region at early times quantifies the exponential growth rate of $C(t)$. This flat region persists up to the time $t_E$, at which the exponential growth slows down to a power-law growth with a slowly decreasing power. Dotted lines are the eye guides: horizontal lines extend the flat regions, sloped line is shown for power comparison. Inset: $\ln[C(t)]/2t$ vs $t$ in the log-log scale for $K=4$ and $\hbar_{\rm eff} = 2^{-14}, 2^{-10}, 2^{-6}, 2^{-2}$ (from top to bottom line, respectively). The rate of exponential growth is the same for different values of $\hbar_{\rm eff}$, but $t_E$ shrinks when $\hbar_{\rm eff}$ increases.
• Figure 4: (Color online) Long-time average $\overline{B}_\tau$ (\ref{['twopoint']}) (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 $K \neq 0$, the correlations decay in time, but the rate of this decay has a step-like dependence on $K$. At $K > K_{\rm cr}$, the decay is quite fast, while at $K < K_{\rm cr}$, it takes $\overline{B}_\tau$ at least exponentially large window to vanish. It is not clear from the data whether at small $K\neq 0$, averaged correlator eventually goes to zero at $\tau \to \infty$ or is bounded from below. Initial state corresponding to $P(x,p)$ is the Gaussian (\ref{['eq:Gauss_wf']}) with $p_0 = 0$ and $\sigma=4$.
• Figure 5: (Color online) Initial Wigner distribution $P(x,p)$ (3D plot) on the top of the classical Lyapunov exponent (shown in color in the horizontal plane, see colorbar for numerical values). Initial state corresponding to $P(x,p)$ is the Gaussian (\ref{['eq:Gauss_wf']}) with $p_0 = 0$ and $\sigma=4$. Lyapunov exponent is shown for $K = 1$.