Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 67
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for K only from 0 to 2. Transition happens about at K= 0.6.
For further study of the the dips in Lyapunov exponent in areas surrounding K= 1 ,1= 1,2,3,...,
the resolution AK= 0.002 is selected for further K scanning in Fig. 6.3. Actually, all regions for ] up
to 15 are investigated and they show results similar to Fig. 6.3. They are shown in App. D. Three
conclusions can be summarized from those figures:
1) All those regions have points of Lyapunov exponents down to negative, which is the signature
of non-chaos in the chaotic sea.
2) There are two groups of K odd integer times of n and even integer times of n. For the
odd, nc (n odd integer) is located at the center of the region. While for the even, minn (m even
integer) is located at the left edge of the region, the same as the classical fundamental accelerator
islands. However these regions are much narrower than the correspondence classical accelerator
and oscillator mode islands.
3) The width of the regions decreases as the order increases.
Stability regions occur periodically in the chaotic sea with period of c is the conclusion drawn
from the numerical experiments. The width is very narrow and decreases as the K value increases.
The mapping of the quantum trajectories of the kicked rotor at quantum resonance is presented.
Lyapunov exponents from the Benettin et al. approach are obtained for the first time from Bohmian
trajectories. Quantum chaos is found for K > 0.6 for the quantum resonance case.
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Zheng, Yindong. Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor, dissertation, August 2005; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc4824/m1/84/: accessed October 15, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .