Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 64
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Since the kicked rotor model is 8- kick followed by a free evolution, the mapping of the Bohmian
trajectories at quantum resonance is
p(tN1+) = po+ (N+ 1)Ksin(x(tN)), x(tN+1) = x(tN) + p(tN+1) (6.14)
The momentum diffusion at quantum resonance is
<[p(tN)- po]2 -2 (6.15)
which is faster than that of the classical.
6.4 Numerical Experiment of Bohmian Trajectories
For quantum kicked rotor, the angle has 21r period. The system can be considered as space
limited from 0 to 21r so that Benettin et al. approach should be used for the Lyapunov exponent
Figure 6.1 shows convergence of LN on N going to infinity. For numerical computation, it is
sufficient for N to reach 2000 from Fig. 6.1. Four curves are provided for K= 12.5, 6.49, 27c and
0.55. For K= 12.5 and 6.49, 4N converges to a positive value, which is the signature of a chaotic
regime. For K= 21, it converges to zero. And for K= 0.55, it converges to a negative value. These
latter two are non-chaotic regimes.
For giving a general picture and showing the transition of the non-chaotic to chaotic of the
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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/m1/81/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .