  Anomalous decoherence in the Jaynes-Cummings model emerges for a certain
class of bosonic reservoirs, described by spectral densities with a band edge
frequency coinciding with the qubit transition frequency. The special
reservoirs are piecewise similar to those usually adopted in Quantum Optics,
i.e., sub-ohmic at low frequencies and inverse power laws at high frequencies.
The exact dynamics of the qubit is described analytically through Fox
$H$-functions. Over estimated long time scales, decoherence results in inverse
power laws with powers decreasing continuously to unity, according to the
particular choice of the special reservoir. The engineering reservoir approach
is a new way of strongly delaying the decoherence process with possible
applications to Quantum Technologies, due to the simple form of the designed
reservoirs.
