  We explore the quantum metrology in an optical molecular system coupled to
two environments with different temperatures, using a quantum master equation
beyond secular approximation. We discover that the steady-state coherence
originating from and sustained by the nonequilibrium condition can enhance
quantum metrology. We also study the quantitative measures of the
nonequilibrium condition in terms of the curl flux, heat current and entropy
production at the steady state. They are found to grow with temperature
difference. However, an apparent paradox arises considering the contrary
behaviors of the steady-state coherence and the nonequilibrium measures in
relation to the inter-cavity coupling strength. This paradox is resolved by
decomposing the heat current into a population part and a coherence part. Only
the latter, coherence heat current, is tightly connected to the steady-state
coherence and behaves similarly with respect to the inter-cavity coupling
strength. Interestingly, the coherence heat current flows from the
low-temperature reservoir to the high-temperature reservoir, opposite to the
direction of the population heat current. Our work offers a viable way to
enhance quantum metrology for open quantum systems through steady-state
coherence sustained by the nonequilibrium condition, which can be controlled
and manipulated to maximize its utility. The potential applications go beyond
quantum metrology and extend to areas such as device designing, quantum
computation and quantum technology in general.
