Classical approach to equilibrium of out-of-time ordered correlators in mixed systems

The out-of-time ordered correlator (OTOC) is a measure of scrambling of quantum information. Scrambling is intuitively considered to be a significant feature of chaotic systems, and thus, the OTOC is widely used as a measure of chaos. For short times exponential growth is related to the classical Lyapunov exponent, sometimes known as the butterfly effect. At long times the OTOC attains an average equilibrium value with possible oscillations. For fully chaotic systems the approach to the asymptotic regime is exponential, with a rate given by the classical Ruelle-Pollicott resonances. In this work, we extend this notion to the more generic case of systems with mixed dynamics, in particular using the standard map, and we are able to show that the relaxation to equilibrium of the OTOC is governed by generalized classical resonances.

Figure 1

Sketch of the typical time dependence of $O_1=\langle W\left(t\right)VW\left(t\right)V\rangle$ for mixed quantum maps exhibiting different behaviors in the short-, intermediate-, and long-time regimes. At short times, namely, for times below the Ehrenfest time $t<t_E$, it has an approximate constant value (blue shading). We name this short-time behavior the Lyapunov regime because of the exponential growth in $C\left(t\right)={\langle \left|[W\left(t\right),V]\right|}^2\rangle =-2(O_1-O_2)$, with $O_2=\langle W{\left(t\right)}^2{V}^2\rangle$ (see the top panel and [24] for more detail). Next, the correlator has an exponential decay related to the Ruelle resonances of the classical map (red shading). The succeeding regime is the power-law regime, which we associate with regular vestiges of the map (green shading). For long times, $O_1$ oscillates around his saturation value (cyan shading).

Figure 2

Sample phase spaces for the standard map (9) and $K=0.4,1,1.8,6.6,17,18.86$ (from left to right, top to bottom). The labels are the same for every panel. The colors in the bottom left and middle panels are meant to correspond to the $K$ values used in Fig. 3. Notice that for very large $K$ (bottom right panel), small regular islands can be seen.

Figure 3

$|O_1|$ versus time for $K=6.6$ (circles) and $K=17$ (squares). These two $K$ values correspond to the bottom left and middle panels in Fig. 2. For $K=6.6$, we observe an exponential decay for $5\lesssim t\lesssim 8$, and after that we find a power-law regime for $9\lesssim t\lesssim 15$. On the other hand, for $K=17$ only the exponential regime is observed in the interval $4\lesssim t\lesssim 7$ before the correlator function saturates for long times. The dashed black lines in the main panel were obtained by fitting $|O_1\left(t\right)|\sim e^{-\gamma t}$. In the inset we show the case of $K=6.6$ in log-log scale, and the black dashed line corresponds to a power-law fit, $|O_1\left(t\right)|\sim t^{\alpha }$.

Figure 4

Decay rate $\gamma$ as a function of kicking strength $K$ obtained from fitting the exponential $|O_1\left(t\right)|\sim e^{-\gamma t}$ in suitably chosen time intervals.

Figure 5

Eigenvalues of the truncated Perron-Frobenius operator computed with the Fourier method. The left panel corresponds to $K=6.6$, and the right panel corresponds to $K=17$. In these cases we can see that there is a gap between the two leading eigenvalues such that $\left|\lambda \right|<1$, which lies on the real axis (big stars). We see that, in general, the larger the chaos parameter $K$ is, the smaller the leading eigenvalues are. The matrix dimension is $D=30$ in both plots.

Figure 6

Absolute value $|{\lambda }_1|$ of the leading eigenvalue of the Perron-Frobenius operator as a function of $K$. The black line corresponds to the momentum-position method, and the red line corresponds to the Fourier method. These are compared to the values obtained from the decay rate of $|O_1|$ as $e^{-\gamma /2}$ (blue circles); the error bars are least squares errors due to the fit. For the Fourier method the dimension of the PF approximation was $D=30$, and the noise parameter $\sigma =0.2$, while for the momentum-position method we used $D=90$ and $s=0.001$.

Figure 7

The black thick line (left $y$ axis) represents an estimation of the area of phase space occupied by regular regions for the standard map as a function of $K$. The light blue line (right $y$ axis) corresponds to the calculation of $|{\lambda }_1|$ using the momentum-position method.

Figure 8

Top: Eigenvalues $|{\lambda }_1|$ of the PF operator computed with the Fourier method versus the chaos parameter for increasing values of $D$ and noise $\sigma =0.2$. Bottom: Eigenvalues $|{\lambda }_1|$ of the PF operator calculated with the Fourier method without noise (blue dashed lines) and $e^{-\gamma /2}$ obtained from decay rates $\gamma$ (black dots and solid line) versus the chaos parameter. The eigenvalues calculated with noise $\sigma =0.2$ are shown by gray lines. The PF discrete matrix has dimension $D=30$.

Figure 9

Ulam eigenvalues versus $K$ for different matrix dimensions $D$. For these values, $D=30,40,50$ (orange, blue, and green lines, respectively), we show the eigenvalues calculated with the addition of noise with normal distribution $\mathcal{N}(0,{\hslash }_{\mathrm{eff}})$ with ${\hslash }_{\mathrm{eff}}=1/2\pi D$. We also show the eigenvalues for $D=30$ without noise (red line).