Quantum double pendulum: Study of an autonomous classically chaotic quantum system

A numerical study of the quantum double pendulum is conducted. A suitable quantum scaling is found which allows us to have as the only parameters the ratios of the lengths and masses of the two pendula and a (quantum) gravity parameter containing Planck’s constant. Comparison with classical and semiclassical results is used to understand the behavior of the energy curves of the levels, to define regimes in terms of the gravity parameter, and to classify the (resonant) interactions among levels by connecting them to various classical phase space structures (resonance islands).

Figure 1

An ideal double pendulum.

Figure 2

Comparison of theoretical (dash) and numerical (full line) adimensional quantum scaled energy vs level number curves.

Figure 3

Energy surfaces in action variable representation. The plots show $I_2$ vs $I_1$ at constant energy $h=1$. The radii indicate the areas to be calculated to evaluate the density of states.

Figure 4

Quantum scaled energy curves.

Figure 5

(Color online) An avoided crossing of two levels, one of which is a principal resonance one. The points at which the Husimi functions are calculated are marked as bigger dots on the quantum scaled energy curves shown on the right of the figure. The quantum gravity parameter increases from top to bottom. For comparison, the classical Poincaré section for $\gamma =0.04888$ is superimposed on the corresponding Husimi function.

Figure 6

(Color online) An avoided crossing of three levels, one of which is a principal resonance one. The points at which the Husimi functions are calculated are marked as bigger dots on the quantum scaled energy curves shown on the right of the figure. The quantum gravity parameter increases from top to bottom. For comparison, the classical Poincaré section for $\gamma =0.0597877$ is superimposed on the corresponding Husimi function.

Figure 7

(Color online) An avoided crossing of two levels, one of which is a principal resonance one. The points at which the Husimi functions are calculated are marked as bigger dots on the quantum scaled energy curves shown on top of the figure. The quantum gravity parameter increases from top to bottom. For comparison, the classical Poincaré section for $\gamma =0.04888$ is superimposed on the corresponding Husimi function.

Figure 8

(Color online) Examples of Husimi functions in the global chaos region. (a) $n=33,\tilde{\gamma }=5.4,\gamma =0.22445$. (b) $n=36,\tilde{\gamma }=5.4,\gamma =0.21950$. (c) $n=127,\tilde{\gamma }=9.8,\gamma =0.15692$. (d) $n=118,\tilde{\gamma }=8.1,\gamma =0.14632$. (e) $n=117,\tilde{\gamma }=8.1,\gamma =0.14815$. (f) $n=74,\tilde{\gamma }=8.6,\gamma =0.19053$.