Quasiperiodic kicked rotor

The quasi-periodic kicked rotor is obtained from the standard kicked rotor by an additional quasi-periodic temporal modulation of the kick strength. The Hamiltonian is written - for 2 additional frequencies - in the upper left equation.

As discussed in the lecture, it can be mapped on a 3d kicked pseudo-rotor, which iteslf can be mapped on a Anderson-like model. A big advantage of the quasi-periodic kicked rotor is that the dynamics can be numerically generated very easily, because it requires only one spatial dimension, but displays the richness of 3d systems.

The classical equations of motion for the pseudo-rotor are very simple. The map giving positions and momenta at kick n+1 are simply related to positions and momenta at kick n, see the second set of equations on the left side.

qpkr_p2_vs_time_classical.py K epsilon final_time number_of_trajectories number_of_bins

K: kick strength
epsilon: modulation depth
final_time: <p**2(t)> is computed for each (integer) time up to final_time
number_of_trajectories: number of initial conditions
number_of_bins: number of bins for the histogram of the momentum distribution

This script computes the average value of <p**2(t)> (along the 3 directions, i.e. p1**2, p2**2, p3**2) as a function of time for the CLASSICAL quasi-periodically kicked rotor, in the file "qpkr_p2_vs_time_classical.dat" (time in the first column, data in columns 2 to 4). Initial conditions are generated randomly. For a single initial condition, it displays large fluctuations. When averaged over several initial conditions, it displays an average linear increase in time, characteristic of a (chaotic) diffusion process, for K sufficiently large, say K>4. In addition, it computes the momentum distribution (along the 3 directions) at the final time and plots histograms, together with a fit by a Gaussian.

You should observe that, as soon as epsilon is significantly larger than 0., the classical dynamics is diffusive in the three directions. The diffusion coefficients should be close to K**2(1+epsilon**2/4)/4 along the "1" axis and (K*epsilon)**2/16 along the "2" and "3" axes. Use e.g. K=8, epsilon=0.1,0.2,0.5 and final_time=100.

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qpkr_quantum_dynamics.py K epsilon hbar final_time basis_size number_of_configurations

K: kick strength
epsilon: modulation depth
hbar: effective Planck constant
final_time: <p**2(t)> is computed for each (integer) time up to final_time
basis_size: size of the basis used. In momentum space, because of spatial periodicity, the momentum is an integer multiple of hbar. This integer runs from -basis_size/2 to basis_size/2
number_of_configurations: number of initial states used for averaging.

This script computes the quantum dynamics of the periodically kicked rotor by iteration of the evolution operator over one period. The latter is computed using a "split" technique where the kick operator is computed in configuration space (where it is diagonal) and the free propagation is computed in momentum space (where it is diagonal). Passing from configuration to momentum representation is done with a Fast Fourier Transform. For this reason, it is more efficient to use a basis size which is a power of 2. The initial state is chosen randomly near zero momentum, in a band defined by the half_width_initial_state parameter. Averaging is performed over the initial state, the quasi-momentum and the phases of the modulations. The expectation value <p**2> is printed at each kick in file "qpkr_p2_vs_time_quantum.dat". Also the (average) density in momentum space is printed at the final time in "qpkr_quantum_momentum_distribution.dat".

The dynamics is expected to be localized at small K,epsilon with <p**2(t)> constant at long time. It is completely diffusive at large K,epsilon with <p**2(t)> proportional to t. The two regimes are separated by a mobility edge (a line in the (K,epsilon) plane). At the mobility edge, <p**2(t)> is expected to scale like t**(2/3).

I suggest the following numerical experiments:

1. K=4.0, epsilon=0.1, hbar=2.0, final_time=100, basis_size=64 with 500 configurations. Observe localization and exponential momentum distribution.
2. K=8.0, epsilon=0.5, hbar=2.0, final_time=100, basis_size=256 with 500 configurations. Observe diffusion and Gaussian momentum distribution.
3. K=5.81, epsilon=0.281, hbar=2.0, final_time=100, basis_size=128 with 500 configurations. Observe the critical behavior. What is the shape of the momentum distribution?
4. Stick to the critical behavior, but with a much longer time. For example, K=5.85, epsilon=0.285 (note a small shift compared to the previous run, to avoid finite size effects), hbar=2.0, final_time=10000, basis_size=768 with 100 configurations.

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