Quantum speed-up transition in open system dynamics

The rate of the trace distance is used to evaluate quantum speed-up for arbitrary mixed states. Compared with some present methods, the approach based on trace distance can provide an optimal bound to the speed of the evolution. The dynamical transition from no speed-up region to speed-up region takes on in the spontaneous decay of an two-level atom with detuning. The evolution is characteristic of the alternating behavior between quantum speed-up and speed-down in the strong system-reservoir coupling regime. Under the off-resonance condition, the dynamical evolution can be accelerated for short previous times and then decelerated to a normal process either in the weak or strong coupling regime. From the time-energy uncertainty relation, we demonstrate that the potential capacity for quantum speed-up evolution is closely related to the energy flow-back from the reservoir to the system. The negative decay rate for short time intervals leads to the speed-up process where the photons previously emitted by the atom are reabsorbed at a later time. The values of the spontaneous decay rate becomes positive after a long enough time, which results in the normal evolution with no speed-up potential.