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We propose a scheme to characterize the non-Markovian dynamics and quantifythe non-Markovianity via the non-classicality measured by the negativity ofquantumness. By considering a qubit in contact with a critical Ising spin bathand introducing an ancilla, we show that revivals of negativity of quantumnessindicate the non-Markovian dynamics. Furthermore, a normalized measure ofnon-Markovianity based on the negativity of quantumness is introduced and theinfluences of bath criticality, bath temperature and bath size on thenon-Markovianity are discussed. It is shown that, at the critical point, thedecay of non-Markovianity versus the size of spin bath is fastest and thenon-Markovianity is exactly zero only in the thermodynamic limit. Besides,non-trivial behaviours of negativity of quantumness such as sudden change,double sudden changes and keeping constant are found for different relationsbetween parameters of the initial state. Finally, how the non-classicality ofthe system is affected by a series of bang-bang pulses is also examined.

Negativity of quantumness and non-Markovianity in a qubit coupled to a thermal Ising spin bath system