  We calculate the dynamical decoherence rate and susceptibility of a
nonequilibrium quantum dot close to the delocalized-to-localized quantum phase
transitions. The setup concerns a resonance-level coupled to two spinless
fermionic baths with a finite bias voltage and an Ohmic bosonic bath
representing the dissipative environment. The system is equivalent to an
anisotropic Kondo model.
  As the dissipation strength increases, the system at zero temperature and
zero bias show quantum phase transition between a conducting delocalized phase
to an insulating localized phase. Within the nonequilibrium functional
Renormalization Group (FRG) approach, we address the finite bias crossover in
dynamical decoherence rate and charge susceptibility close to the phase
transition. We find the dynamical decoherence rate increases with increasing
frequency. In the delocalized phase, it shows a singularity at frequencies
equal to positive or negative bias voltage. As the system crossovers to the
localized phase, the decoherence rate at low frequencies get progressively
smaller and this sharp feature is gradually smeared out, leading to a single
linear frequency dependence. The dynamical charge susceptibility shows a
dip-to-peak crossover across the delocalized-to-localized transition. Relevance
of our results to the experiments is discussed.
