Effective time-independent analysis for quantum kicked systems

We present a mapping of potentially chaotic time-dependent quantum kicked systems to an equivalent approximate effective time-independent scenario, whereby the system is rendered integrable. The time evolution is factorized into an initial kick, followed by an evolution dictated by a time-independent Hamiltonian and a final kick. This method is applied to the kicked top model. The effective time-independent Hamiltonian thus obtained does not suffer from spurious divergences encountered if the traditional Baker-Cambell-Hausdorff treatment is used. The quasienergy spectrum of the Floquet operator is found to be in excellent agreement with the energy levels of the effective Hamiltonian for a wide range of system parameters. The density of states for the effective system exhibits sharp peaklike features, pointing towards quantum criticality. The dynamics in the classical limit of the integrable effective Hamiltonian shows remarkable agreement with the nonintegrable map corresponding to the actual time-dependent system in the nonchaotic regime. This suggests that the effective Hamiltonian serves as a substitute for the actual system in the nonchaotic regime at both the quantum and classical level.

Figure 1

Quasienergy spectrum (solid line) of the Floquet operator $\widehat{\mathcal{F}}$ and the energy eigenvalues (solid square) of the effective Hamiltonian ${\widehat{H}}_{\mathrm{eff}}$ are compared as a function of $\alpha$. Upper window shows the result for $\beta =0.1$, and the lower window shows the same for $\beta =0.5$. Whereas the integrable ${\widehat{H}}_{\mathrm{eff}}$ shows level crossing, the quasienergy spectrum indicates level repulsion (see inset).

Figure 2

Density of states of the exact quasienergy spectrum of $\widehat{\mathcal{F}}$ and of the energy spectrum of ${\widehat{H}}_{\mathrm{eff}}$ are compared. Here the parameter $\beta =0.1$.

Figure 3

The upper panel shows the phase space corresponding to the nonintegrable evolution governed by the map given in Eq. (14) for parameter $\beta =0.1$. The lower panel shows the corresponding phase for the effective time-independent integrable Hamiltonian in Eq. (18). The parameter $\alpha$ is increasing from left to right as $\alpha =0.2$ (left column), $1.0$ (middle column), and $6.0$ (right column). From left to right, the upper panel shows the system evolving towards chaotic regime. There is no such indication for the integrable system depicted in the lower panel.