Shannon entropy at avoided crossings in the quantum transition from order to chaos

Shannon entropy is studied for the series of avoided crossings that characterize the transition from order to chaos in quantum mechanics. In order to be able to study jointly this entropy for discrete and continuous probability, calculations have been performed on a quantized map, the kicked Harper map, resulting in a different behavior, as order-chaos transition takes place, for the discrete (position representation) and continuous (coherent state representation) cases. This different behavior is analyzed in terms of the distribution of zeros of the Husimi function.

Figure 1

Classical kicked Harper map given, with random initial conditions, obtained for increasing control parameter values $\gamma =0.1$ (a), $\gamma =0.2$ (b), $\gamma =0.3$ (c), and $\gamma =0.4$ (d).

Figure 2

Eigenphase spectrum ${\omega }_n$ versus parameter $\gamma$ for $N=30$. The two series of avoided crossings studied are marked with open circles. Blue (smaller) circles correspond to hetero-resonances between states quantized on different torus families, and red (bigger) circles correspond to homo-resonances between states quantized on the same torus family.

Figure 3

Magnification of the avoided crossings (top panels) in the hetero-resonance series, along with the corresponding Shannon entropy of the involved states for Husimi probability density (middle panels) and for position probability (bottom panels). Columns (a)–(h) correspond to the first through eighth avoided crossing, respectively, in the series. Blue (dark gray) and magenta (light gray) lines correspond to upper and lower states, respectively, in the avoided crossings. Coordinates $\tilde{\gamma }=\gamma -{\gamma }_0$ and $\tilde{\omega }=\omega -{\omega }_0$ shifted to the center $({\gamma }_0,{\omega }_0)$ of each avoided crossing have been used. Parameter ${\tilde{\omega }}_{\text{b}}$ defines the different boundaries for the $\tilde{\omega }$ coordinate used in each magnification. All parameters determining each avoided crossing are listed in Table 1.

Figure 4

Upper state [upper panels: (a)–(f)] and lower state [lower panels: (g)–(l)] involved in the first avoided crossing in the hetero-resonance series, represented before [left panels: (a), (d), (g), and (j)], during [central panels: (b), (e), (h), and (k)], and after [right panels: (c), (f), (i), and (l)] the avoided crossing. Histograms [(a)–(c) and (j)–(l)] represent the probability in position representation, and grayscale depictions [(d)–(i)] represent the probability density in the coherent state representation (Husimi function). Zeros of the Husimi function have been marked with cyan (grayish-white) dots.

Figure 5

Same as described in the caption of Fig. 4 for the eighth avoided crossing in the hetero-resonance series.

Figure 6

Same as described in the caption of Fig. 3 for the homo-resonance series. Columns (a)–(f) correspond to the first through sixth avoided crossing, respectively, in the series. Horizontal gray line in the fourth, fifth, and sixth avoided crossings corresponds to symmetry line $\omega =\pm \pi$ rad. All parameters determining each avoided crossing are listed in Table 2.

Figure 7

Same as described in the caption of Fig. 4 for the first avoided crossing in the homo-resonance series.

Figure 8

Same as described in the caption of Fig. 4 for the sixth avoided crossing in the homo-resonance series.

Figure 9

Upper state [left panels: (a) and (c)] and lower state [right panels: (b) and (d)] involved in the second avoided crossing in the homo-resonance series, represented at the center of the avoided crossing. Histograms [(a) and (b)] represent the probability in position representation, and grayscale depictions [(c) and (d)] represent the probability density in the coherent state representation (Husimi function). Zeros of the Husimi function have been marked with cyan (grayish-white) dots. Vertical lines superimposed on the continuous Husimi function correspond to the histogram bin width of the discrete position representation. The index $k=0,1,...,30$ labeling the discrete positions $q_k$ has been included. Note that $k=0$ and $k=30$ label the same position $q_0=0$.