Temporal steering of a two-level system interacting with a coherent superposition of two environments

Quantumness of a two-level system which interacts with one of two environments in an indefinite way is studied in terms of violation of the temporal steering inequalities, where the indefiniteness is created by a coherent superposition of two environments. Two models of two-level system and environment are applied. One is a dephasing spin-boson model with the Ohmic-like spectral density and the other is a zero-temperature dissipative spin-boson model with the Lorentzian spectral density. It is shown that a coherent superposition of the two environments can enhance the quantumness of the two-level system. The parameter regions in which the indefiniteness is effective for enhancing the quantumness are clarified.