Dependence of the recoherence times and recoherence increments on the state of phonon bath in a single-qubit dephasing model

The recoherence times <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mi>t</a:mi> <a:mo>*</a:mo> </a:msup> </a:math> and the maximum values of the recoherence increments <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:msub> <b:mi>γ</b:mi> <b:mi>extr</b:mi> </b:msub> </b:math> are studied as functions of the bath parameters for a single qubit dephasing model, prepared initially by a special kind of the nonselective measurements. The recoherence-decoherence events (RDE), occurring at the initial stage of the system evolution, are found to be both similar and different from the system dynamics at large times. For instance, in contrast to the RDE observed on large timescales, the sub-Ohmic and Ohmic coupling regimes are more favorable for the short-time recoherence than the super-Ohmic one. On the other hand, the short-time behavior of the recoherence and the long-time dynamics of the decoherence are closely related: The domain of the ohmicity indexes, where the decoherence changes its type (from the complete to incomplete one), is, simultaneously, that of the weakest recoherence. The obtained results give us some hints about the basic characteristics of the environment, which might provide the most optimal values of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msup> <c:mi>t</c:mi> <c:mo>*</c:mo> </c:msup> </c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>γ</d:mi> <d:mi>extr</d:mi> </d:msub> </d:math> in some sense.