Chaos and Momentum Diffusion of the Classical and Quantum Kicked Rotor Page: 75
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Figure 7.1 (b) shows the momentum diffusion for nonresonance with k = 2. The dash-dot line
is the classical linear momentum diffusion (K2/2)N, which serves as a reference. The dots are
the numerical experimental data which is linear between 0 and about 28 kicks, and then begins to
saturate. Figure 7.1 (d) shows the momentum diffusion for resonance with k = 4. The dash-dot
line is the momentum diffusion for an initial plane wave (K2/2)N2 in Eq. (7.3). The dots are the
numerical simulation data that shows quadratic growth with a quadratic diffusion rate close to K2/2.
A least squares fit to the numerical data gives a quadratic diffusion rate of D= 76.2, which differs
from the analytical diffusion rate of K2/2 by 2.4% as shown in Table 7.1.
Figure 7.2 gives the results of momentum distribution and momentum diffusion using the ini-
tial Gaussian momentum distribution in Eq. (7.4) with a control parameter K= 12.5 for both the
resonant and nonresonant cases. The initial Gaussian momentum distribution describes the atom
optics kicked rotor more closely than the initial zero momentum state .
Figure 7.2 (a) shows the momentum distribution as a function of time from 0 to 100 kicks for
the nonresonant case with k = 2. Figure 7.2 (c) shows the momentum distribution as a function
of time from 0 to 100 kicks for the resonant case with k = 4. Both distributions have an expected
symmetry around zero momentum. The probability distributions are similar to Fig. 7.1 (a) and (c).
Figure 7.2 (b) for the nonresonant case with k = 2 shows saturation after about 10 kicks, and
is similar to the experimental data for the atom optics kicked rotor . Figure 7.2 (d) shows the
momentum diffusion for the resonant case with k = 4, which is quadratic in kicks (time). For
K= 12.5 the theoretical momentum diffusion in Eq. (7.5) gives a rate D= 67.5, while the numerical
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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/m1/92/: accessed January 23, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .