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We study non-Markovianity and information flow for qubits experiencing local dephasing with an Ohmic class spectrum. We demonstrate the existence of a temperature-dependent critical value of the Ohmicity parameter s for the onset of non-Markovianity and give a physical interpretation of this phenomenon by linking it to the form of the reservoir spectrum. We demonstrate that this link holds also for more general spectra. We unveil a class of initial states for which discord is forever frozen at a positive value. We connect time invariant discord to non-Markovianity and propose a physical system in which it could be observed. PACS numbers: 03.65.Ta, 03.65.Yz, 03.67.Mn Introduction.-Qubits subjected to local purely dephasing noise are ubiquitous models of open quantum systems, and they have been studied extensively in the literature. Examples include dephasing in quantum registers [1-3], ultracold gases [4, 5], quantum metrology protocols [6], quantum biological systems [7], and dynamical decoupling theory [8]. From a theoretical point of view the dephasing model is exactly solvable [1-3], and hence it is an ideal testbed to investigate one of the most thrilling fields of the theory of open quantum systems, that of non-Markovian quantum processes [9].Recently, a great deal of attention has been devoted to the study of systems whose reduced dynamics are characterised by memory effects and recoherence phenomena, emerging from non-trivial correlations with an environment. Such dynamics are typically called non-Markovian. Memory effects and non-Markovianity have been shown to be a resource for quantum technologies [6, 10-13] and consequently measures of non-Markovianity have become important as quantifiers of this resource [14][15][16]. Moreover, it has been shown that non-Markovianity of a quantum probe can indicate a quantum phase transition occurring in a complex environment, with which the probe is interacting [17].Non-Markovian features play an important role in systems where the frequency spectrum of the environment is structured. However, a connection between the general form of the spectrum and the memory effects in the reduced system dynamics has not been elucidated until now. In this Letter we establish this connection by unveiling a necessary condition on the form of the spectrum to induce non-Markovian dynamics for a dephasing qubit. We then focus on the widely used Ohmic class of reservoir spectra and show that the condition is both necessary and sufficient for this type of spectra. Moreover, we demonstrate that only super-Ohmic environments can induce non-Markovian dynamics. This means that even if the reduced dynamics is exact, and hence no Markovian approximation has been performed, the time evolution of the qubit does not present any memory effects or reco-

We study quantum information flow in a model comprised of a trapped impurity qubit immersed in a BoseEinstein-condensed reservoir. We demonstrate how information flux between the qubit and the condensate can be manipulated by engineering the ultracold reservoir within experimentally realistic limits. We show that this system undergoes a transition from Markovian to non-Markovian dynamics, which can be controlled by changing key parameters such as the condensate scattering length. In this way, one can realize a quantum simulator of both Markovian and non-Markovian open quantum systems, the latter ones being characterized by a reverse flow of information from the background gas (reservoir) to the impurity (system). More recently, hybrid systems composed of quantum dots, single trapped ions, and optical lattices coupled to Bose-Einstein condensates (BECs) have been studied both theoretically and experimentally [5]. These systems are studied in the framework of open quantum systems [6], effectively described as one or more two-level systems (qubits) interacting with a reservoir consisting of the ultracold gas. The possibility of manipulating crucial parameters of the reservoir, such as the scattering length [7], combined with the continuous improvements in quantum control of qubits, highlights the enormous potential of hybrid systems as quantum simulators of both condensed-matter models and open quantum systems.In this Rapid Communication, we study a qubit system composed of an impurity atom trapped in a double-well potential, interacting with a BEC environment. This model has been shown to describe an effective pure-dephasing model [8]. Our focus is on the dynamics of quantum information between the qubit system and the ultracold reservoir. We show how information flux can be manipulated by experimentally achievable means, such as changing the scattering length, the effective dimension of the background gas, or the trapping geometry of the qubit.Recently, dynamics of information flow has been an active area of research in the open quantum systems community due to several proposals to link it to the division of quantum processes into Markovian and non-Markovian ones [9][10][11][12]. The latter ones have been defined as processes where an

A simple relationship between recently proposed measures of non-Markovianity and the Loschmidt echo is established, holding for a two-level system (qubit) undergoing pure dephasing due to a coupling with a many-body environment. We show that the Loschmidt echo is intimately related to the information flowing out from and occasionally back into the system. This, in turn, determines the non-Markovianity of the reduced dynamics. In particular, we consider a central qubit coupled to a quantum Ising ring in the transverse field. In this context, the information flux between system and environment is strongly affected by the environmental criticality; the qubit dynamics is shown to be Markovian exactly and only at the critical point. Therefore non-Markovianity is an indicator of criticality in the model considered here.

We consider two recently proposed measures of non-Markovianity applied to a particular quantum process describing the dynamics of a driven qubit in a structured reservoir. The motivation of this study is twofold: on one hand, we study the differences and analogies of the non-Markovianity measures and on the other hand, we investigate the effect of the driving force on the dissipative dynamics of the qubit. In particular we ask if the drive introduces new channels for energy and/or information transfer between the system and the environment, or amplifies existing ones. We show under which conditions the presence of the drive slows down the inevitable loss of quantum properties of the qubit.