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Influence of equilibrium and nonequilibrium environments on macroscopic realism through the Leggett-Garg inequalities

We study the macroscopic realism (macrorealism) through the two- and three-time Leggett-Garg inequalities (LGIs) in a two interacting qubits system. The two qubits are coupled either with two bosonic (thermal or photonic) baths or fermionic (electronic) baths. We study both how the equilibrium and nonequilibrium environments influence the LGIs. One way to characterize the nonequilibrium condition is by the temperature difference (for the bosonic bath) or the chemical potential difference (for the fermionic bath). We also study the heat or particle current and the entropy production rate generated by the nonequilibrium environments. Analytical forms of LGIs and the maximal value of LGIs based on the quantum master equation beyond the secular approximation are derived. The LGI functions and the corresponding maximal value have separated contributions, the part describing the coherent evolution and the part describing the coupling between the system and environments. The environment-coupling part can be from the equilibrium environment or the nonequilibrium environment. The nonequilibrium dynamics is quantified by the Bloch-Redfield equation which is beyond the Lindblad form. We found that the nonequilibriumness quantified by the temperature difference or the chemical potential difference can lead to the LGIs violations or the increase of the maximal value of LGIs, restoring the quantum nature from certain equilibrium cases where LGIs are preserved. The corresponding nonequilibrium thermodynamic cost is quantified by the nonzero entropy production rate. Our finding of the nonequilibrium promoted LGIs violations 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.