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

Published February 13, 2026 | Version v1

Preprint Open

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.

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Is continued by: Preprint: 10.5281/zenodo.18544462 (DOI); Preprint: 10.5281/zenodo.18363034 (DOI); Preprint: 10.5281/zenodo.18400122 (DOI)

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