We investigate precursors of critical behavior in the quasienergy spectrum due to the dynamical instability in the kicked top. Using a semiclassical approach, we analytically obtain a logarithmic divergence in the density of states, which is analogou
s to a continuous excited state quantum phase transition in undriven systems. We propose a protocol to observe the cusp behavior of the magnetization close to the critical quasienergy.
The decoherence induced on a single qubit by its interaction with the environment is studied. The environment is modelled as a scalar two-level boson system that can go through either first order or continuous excited state quantum phase transitions,
depending on the values of the control parameters. A mean field method based on the Tamm-Damkoff approximation is worked out in order to understand the observed behaviour of the decoherence. Only the continuous excited state phase transition produces a noticeable effect in the decoherence of the qubit. This is maximal when the system-environment coupling brings the environment to the critical point for the continuous phase transition. In this situation, the decoherence factor (or the fidelity) goes to zero with a finite size scaling power law.
