Incorporation of Levy and Colored Noise in the Kick Model

This study investigates the impact of stochastic noise on the kick model. Levy noise is incorporated as timing noise, while colored noise serves as amplitude noise, as outlined in the work of Klappauf et al. (1998). The kick strength is represented as K_n = K + k_n, where K denotes the fixed value from the unperturbed model, and k_n represents a random deviation arising from the noise.

To manage the amplitude noise, a Levy renewal process is employed to control the timing noise. Specifically, k_n is set to zero for kicks occurring between Levy renewal events. This modification results in an updated Floquet operator for the noisy kick model, as shown in Equation \ref{fkn}. The noise contribution is incorporated into the operator via an exponential term.

\begin{equation} \hat{F}^{k_n}=e^{-i\frac{p^{2}}{2}T} e^{-iKcos^2\theta}e^{-ik_ncos^2\theta} = \hat{F}e^{-ik_ncos^2\theta} \label{fkn} \end{equation}