Early-Time Exponential Instabilities in Non-Chaotic Quantum Systems

Abstract

The vast majority of dynamical systems in classical physics are chaotic and exhibit the butterfly effect: a minute change in initial conditions can soon have exponentially large effects elsewhere. But this phenomenon is difficult to reconcile with quantum mechanics. One of the main goals in the field of quantum chaos is to establish a correspondence between the dynamics of classical chaotic systems and their quantum counterparts. In isolated systems in the absence of decoherence, there is such a correspondence in dynamics, but it usually persists only over a short time window, after which quantum interference washes out classical chaos. We demonstrate that quantum mechanics can also play the opposite role and generate exponential instabilities in classically non-chaotic systems within this early-time window. Our calculations employ the out-of-time-ordered correlator (OTOC) -- a diagnostic that reduces to the Lyapunov exponent in the classical limit, but is well defined for general quantum systems. Specifically, we show that a variety of classically non-chaotic models, such as polygonal billiards, whose classical Lyapunov exponents are always zero, demonstrate a Lyapunov-like exponential growth of the OTOC at early times with Planck's-constant-dependent rates. This behavior is sharply contrasted with the slow early-time growth of the analog of the OTOC in the systems' classical counterparts. These results suggest that classical-to-quantum correspondence in dynamics is violated in the OTOC even before quantum interference develops.

Figures (7)

• Figure 1: Outer black line: polygonal butterfly-shaped billiard. The area is unit. Inner blue line: effective mathematical billiard hosting a point particle classically equivalent to the outer polygonal billiard hosting a rigid circular particle of radius $r_p = \sigma\sqrt{\hbar_{\rm eff}/2}$ and zero moment of inertia. Note that the inward-pointing corners of the polygonal billiard are rounded into circular arcs or radius $r_p$, making the effective mathematical billiard classically chaotic with positive Lyapunov exponent. Gray shaded region: a close sub-$r_p$ vicinity of the billiard wall: small changes of the billiard geometry within this region do not affect the early-time quantum dynamics. Middle red line: a smoothened billiard used for comparison purposes below.
• Figure 2: Ergodic hierarchy Sinai77ErgTheoryBook provides a nested classification of non-integrable systems. Only K- and B-systems are chaotic and have positive Lyapunov exponents, while merely ergodic and merely mixing systems have no exponential instabilities.
• Figure 3: "Deformed triangular" (quadrilateral) billiard. For a finite-sized particle, the inward-pointing corner gets rounded in the same way as those in the butterfly-shaped billiard in Fig. \ref{['fig:ButtBill']}.
• Figure 4: An example of successive stages of the wave-packet evolution, $\left|\Psi({\bf r},t)\right|^2$, in the butterfly-shaped polygonal billiard. Red arrows indicate the directions of motion of the components. Initial velocity is aimed at an inner corner.
• Figure 5: Main plot -- open blue circles and line: logarithm of the OTOC in the polygonal butterfly-shaped billiard: $\ln\left(C(t)/\hbar_{\rm eff}^2\right) = \ln\left(-\frac{1}{\hbar_{\rm eff}^2}\left<\left[\hat{x}(t), \hat{p}_x(0)\right]^2\right>\right)$. Solid red triangles: the same in the rounded version of this billiard (middle red line in Fig. \ref{['fig:ButtBill']}). A remarkable agreement demonstrates that the growth in both cases is the same, supporting our finite-size-related arguments. In addition, we show the corresponding behavior of an alternative diagnostic, $L(t) = \left<\ln\left(-\frac{1}{\hbar_{\rm eff}^2}\left[\hat{x}(t), \hat{p}_x(0)\right]^2\right)\right>$, that swaps the order of averaging and logarithm to that of the proper definition of the classical Lyapunov exponent. For chaotic systems with uniform phase space, one would expect $L(t) = 2\tilde{\lambda} t + \rm{const}$ at $t<t_E$. Green squares and line: $L(t)$ in the polygonal butterfly-shaped billiard. Pink crosses: $L(t)$ in the rounded billiard. Dashed black lines: linear fits for $\ln(C(t)/\hbar_{\rm eff}^2)$ and $L(t)$ in the polygon. Both show the exponent $2\tilde{\lambda} \approx 3.3$ that is 5 times larger than the inverse time-window, which ensures that the fit is valid. Inset -- the comparison between $C(t)/\hbar_{\rm eff}^2$ and $C_{\rm cl}(t) = \left<\space\left<\,\left\{x(t), p_x(0)\right\}_{\rm Poisson}^2\,\right>\space\right>$ [see Eq. (\ref{['eq:Ccl']})] and between $\exp\left[L(t)\right]$ and $\exp\left[L_{\rm cl}(t)\right] = \exp\left[\,\left<\space\left<\,\ln\left\{x(t), p_x(0)\right\}_{\rm Poisson}^2\,\right>\space\right>\,\right]$ in the polygonal quantum and classical billiards, respectively. $\hbar_{\rm eff} = 2^{-7}$, $\sigma = 1/\sqrt{2}$, $R_s = \frac{\sqrt{2}-1}{16\sqrt{2}}\approx 0.02$.