Early-Time Exponential Instabilities in Nonchaotic Quantum Systems

The majority of classical dynamical systems are chaotic and exhibit the butterfly effect: a minute change in initial conditions has exponentially large effects later on. 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 nonchaotic 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. We show that certain classically nonchaotic models, such as polygonal billiards, 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.

Figure 1

Outer black line: a polygonal butterfly-shaped billiard. Inner blue line: effective mathematical billiard hosting a point particle equivalent to the outer polygonal billiard hosting a solid disk of radius $r_p$. The inward-pointing corners of the 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. Middle red line: a smoothened billiard used for comparison below.

Figure 2

An example of successive stages of the wave-packet evolution, ${|\mathrm{\Psi }(\mathbf{r},t)|}^2$, in the butterfly-shaped billiard. Red arrows indicate the directions of motion of the components. Initial velocity is aimed at the right inner corner.

Figure 3

Main plot. Open blue circles and line: logarithm of the OTOC in the butterfly-shaped billiard: $\ln [C(t)/{\hslash }_{\mathrm{eff}}^2]=\ln [-(1/{\hslash }_{\mathrm{eff}}^2)⟨{[\widehat{x}(t),{\widehat{p}}_x(0)]}^2⟩]$. Solid red triangles: the same in the rounded version of this billiard. A remarkable agreement supports our finite-size smearing arguments. In addition, we show the corresponding behavior of an alternative diagnostic, $L(t)=⟨\ln (-(1/{\hslash }_{\mathrm{eff}}^2){[\widehat{x}(t),{\widehat{p}}_x(0)]}^2)⟩$, that swaps the order of averaging and logarithm to that in the proper definition of the classical Lyapunov exponent. For chaotic systems, one would expect $L(t)=2\tilde{\lambda }t+\text{const}$ 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 $\ln [C(t)/{\hslash }_{\mathrm{eff}}^2]$ and $L(t)$ in the polygon. Both show the exponent $2\tilde{\lambda }\approx 3.3$ that is five times larger than the inverse time window, which ensures the validity of the fit. Inset. The comparison between $C(t)/{\hslash }_{\mathrm{eff}}^2$ and $C_{\mathrm{cl}}(t)$ [see Eq. (4)] and between $\exp [L(t)]$ and $\exp [L_{\mathrm{cl}}(t)]=\exp \{⟪\ln {[\partial x(t)/\partial x(0)]}^2⟫\}$ in the polygonal quantum and classical billiards, respectively. ${\hslash }_{\mathrm{eff}}=2^{-7}$, $\sigma =1/\sqrt{2}$, $R_s=(\sqrt{2}-1/16\sqrt{2})\approx 0.02$.

Figure 4

Main plot. Logarithm of the OTOC in the butterfly-shaped billiard at three different values of ${\hslash }_{\mathrm{eff}}$. The exponential growth hinges on the finite wave-packet size. Inset. Logarithm of the OTOC in the quadrilateral billiard (Fig. 5), averaged over an ensemble of initial conditions as indicated by the bar, $\overline{\dots }$, at the same values of ${\hslash }_{\mathrm{eff}}.\sigma =1/\sqrt{2}$.

Figure 5

Upper blue line and open circles: logarithm of the OTOC in an irrational triangular billiard (shown in the lower right corner). After an initial-condition-dependent delay, the OTOC shows exponential grows, although at a rate lower than that for the nonconvex billiard. Lower red line and asterisks: the $L$-correlator introduced in Fig. 3. Black dashed lines show linear fits with $2\tilde{\lambda }\approx 1$, which is over five times larger than the inverse growth time window, ensuring the validity of the fit.