How nonequlibriumness influences macroscopic realism through Leggett-Garg inequality?

We study the macroscopic realism through Leggett-Garg inequality (LGI) for a two-qubit quantum system coupled with two environments characterized by either the bosonic (thermal and photonic) baths or fermionic (electronic) baths with different temperatures or chemical potentials respectively. Analytical form of LGI and the maximal values of LGI based on the quantum master equation beyond secular approximation are derived. We found that the nonequilibriumness quantified by the temperature difference or chemical potential difference can lead to the LGI violation or the increase of the maximal value of LGI, restoring the quantum nature from certain equilibrium cases where LGI is preserved giving classical realism. Our results shed light on the nature of the macroscopic realism and the relationship between the nonequilibriumness and quantum temporal correlation. Our finding of nonequilibrium promoted LGI violation suggests a new strategy for the design of quantum information processing and quantum computational devices to maintain the quantum nature and quantum correlations for long.