We discuss how the bath's memory affects the dynamics of a swap gate. We present an exactly solvable model that shows various dynamical transitions when treated beyond the Fermi Golden Rule. By moving continuously a single parameter, the unperturbed Rabi frequency, we sweep through different analytic properties of the density of states: (I) collapsed resonances that split at an exceptional point in (II) two resolved resonances ; (III) out-of-band resonances; (IV) virtual states; and (V) pure point spectrum. We associate them with distinctive dynamical regimes: overdamped, damped oscillations, environment controlled quantum diffusion, anomalous diffusion and localized dynamics respectively. The frequency of the swap gate depends differently on the unperturbed Rabi frequency. In region I there is no oscillation at all, while in the regions III and IV the oscillation frequency is particularly stable because it is determined by the environment's band width. The anomalous diffusion could be used as a signature for the presence of the elusive virtual states.

Numerically, we study the time fluctuations of few-body observables after relaxation in isolated dynamical quantum systems of interacting particles. Our results suggest that they decay exponentially with system size in both regimes, integrable and chaotic. The integrable systems considered are solvable with the Bethe ansatz and have a highly nondegenerate spectrum. This is in contrast with integrable Hamiltonians mappable to noninteracting ones. We show that the coefficient of the exponential decay depends on the level of delocalization of the initial state with respect to the energy shell.

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