Exact Relation between Internal Dissipation Power and Energy Relaxation Rate——A Rigorous Derivation in Markovian Quantum Systems

This paper rigorously proves the exact relation between the internal dissipation power Rinternal (the net power flowing from the system to the dissipative channel) and the energy relaxation rate Γ1 for a broad class of Markovian quantum systems. Starting from the standard Lindblad master equation incorporating an amplitude damping term, we provide a purely algebraic proof that, in the absence of driving and at zero temperature, the identity Rinternal = ℏωqΓ1⟨σ+σ−⟩ holds universally. This proof does not rely on any additional assumptions regarding the microscopic origin of Γ1 or the strength of pure dephasing. For systems subject to external driv ing, we precisely define the internal dissipation power according to the first law of quantum thermodynamics, and we provide a completely self-contained demon stration that, under the original driving form without invoking the rotating-wave approximation, the exact result is Rinternal = ℏωqΓ1⟨σ+σ−⟩ + ℏ 2 ΩΓ1 cos(ωdt)⟨σx⟩; this correction term originates from counter-rotating components and, under the rotating-wave approximation, averages to zero over time due to its fast oscilla tion, thereby recovering the same relation as in the undriven case. For finite temperature environments, we derive the modified relation Rinternal = ℏωqΓ1 [ (1+ 2nth)⟨σ+σ−⟩ − nth] and verify that it vanishes in thermal equilibrium. This arti cle elaborates on the minimal assumptions, conditions of independence, and limitations of this relation, and also presents its generalization to multi-level sys tems.