Border between Regular and Chaotic Quantum Dynamics

We identify a border between regular and chaotic quantum dynamics. The border is characterized by a power-law decrease in the overlap between a state evolved under chaotic dynamics and the same state evolved under a slightly perturbed dynamics. For example, the overlap decay for the quantum kicked top is well fitted with $[1+(q-1)(t/\tau {)}^2{]}^{1/(1-q)}$ (with the nonextensive entropic index $q$ and $\tau$ depending on perturbation strength) in the region preceding the emergence of quantum interference effects. This region corresponds to the edge of chaos for the classical map from which the quantum chaotic dynamics is derived.

Figure 1

Overlap versus time for initial angular momentum coherent states located in the chaotic region and the regular region of the quantum kicked top. The system has a spin 120 and is evolved under the kicked top Hamiltonian with $\alpha =3$ and ${\alpha }^{\prime }=3.015$. The overlap of the state in the chaotic region decreases exponentially with the number of iterations of the map. The state in the regular region is practically an eigenstate of the system and therefore oscillates close to unity.

Figure 2

Overlap versus time for an angular momentum coherent state initially located at the border between regular and chaotic zones of the QKT of spin 120 and $\alpha =3$. This region is called the edge of quantum chaos and shows the expected power law decrease in overlap. The top figure is for a perturbation strength within the FGR regime, $\delta =0.015$ and the bottom figure is for a perturbation strength of $\delta =0.0003$, well below the FGR regime. On the log-log plot the power law decay region, from about 20–70 in the FGR regime and 2000–7000 in the Gaussian regime, is linear. We can fit the decrease in overlap with the expression $[1+(q_{\mathrm{r}\mathrm{e}\mathrm{l}}-1)(t/{\tau }_{q_{\mathrm{r}\mathrm{e}\mathrm{l}}}{)}^2{]}^{1/(1-q_{\mathrm{r}\mathrm{e}\mathrm{l}})}$ where, in the FGR regime, the entropic index $q_{\mathrm{r}\mathrm{e}\mathrm{l}}=5.1$ and ${\tau }_{q_{\mathrm{r}\mathrm{e}\mathrm{l}}}=40$ and in the Gaussian regime $q_{\mathrm{r}\mathrm{e}\mathrm{l}}=3.8$ and ${\tau }_{q_{\mathrm{r}\mathrm{e}\mathrm{l}}}=2500$. The insets of both figures show ${ln}_{q_{\mathrm{r}\mathrm{e}\mathrm{l}}}O\equiv (O^{1-q_{\mathrm{r}\mathrm{e}\mathrm{l}}}-1)/(1-q_{\mathrm{r}\mathrm{e}\mathrm{l}})$ versus $t^2$; since ${ln}_{q}x$ is the inverse function of $e_q^x\equiv [1+(1-q)x{]}^{1/(1-q)}$, this produces a straight line with a slope $-1/{\tau }^2$ (also plotted).

Figure 3

$q_{\mathrm{r}\mathrm{e}\mathrm{l}}$ versus perturbation strength. The value of $q_{\mathrm{r}\mathrm{e}\mathrm{l}}$ remains constant at 3.8 until the critical perturbation, after which it increases. The relationship of ${\tau }_{q_{\mathrm{r}\mathrm{e}\mathrm{l}}}$ versus perturbation strength is shown on a log-log plot in the inset and is well fit by a line with slope $-1.06$.

Figure 4

Classical phase space of the kicked top with angular momentum coherent wave functions. 10 000 iterations of a chaotic orbit starting from the point $x=0.6294$, $y=0.7424$, $z=0.2294$. The spherical phase space and the ellipsoidal coherent states are projected onto the $x\mathrm{\text{−}}z$ plane (only $y>0$ shown) by multiplying the $x$ and $z$ coordinates of each point by $R/r$ where $R=\sqrt{2(1-|y|)}$ and $r=\sqrt{(1-y^2)}$ [4]. The regular regions of the kicked top are clearly visible. Shown is a $j=120$ wave function (stars) and a $j=240$ (circles) wave function both at the edge of quantum chaos. Note that the variance of the $j=120$ wave function is much larger than the variance of the $j=240$ wave function. Hence, the behavior characteristic of the edge of quantum chaos appears further from the fixed point of the classical map.