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

Abstract

This paper rigorously proves the exact relation between the internal dissipation power $R_{\mathrm{internal}}$ (the net power flowing from the system to the dissipative channel) and the energy relaxation rate $\Gamma_1$ 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 $R_{\mathrm{internal}} = \hbar\omega_q \Gamma_1 \langle \sigma_+\sigma_-\rangle$ holds universally. This proof does not rely on any additional assumptions regarding the microscopic origin of $\Gamma_1$ or the strength of pure dephasing. For systems subject to external driving, we precisely define the internal dissipation power according to the first law of quantum thermodynamics, and we provide a \textbf{completely self-contained} demonstration that, under the original driving form without invoking the rotating-wave approximation, the exact result is $R_{\mathrm{internal}} = \hbar\omega_q \Gamma_1 \langle \sigma_+\sigma_-\rangle + \frac{\hbar\Omega}{2}\Gamma_1\cos(\omega_d t)\langle \sigma_x\rangle$; this correction term originates from counter-rotating components and, under the rotating-wave approximation, averages to zero over time due to its fast oscillation, thereby recovering the same relation as in the undriven case. For \textbf{finite-temperature} environments, we derive the modified relation $R_{\mathrm{internal}} = \hbar\omega_q \Gamma_1 \bigl[(1+2n_{\mathrm{th}})\langle \sigma_+\sigma_-\rangle - n_{\mathrm{th}}\bigr]$ and verify that it vanishes in thermal equilibrium. This article elaborates on the \textbf{minimal assumptions}, \textbf{conditions of independence}, and \textbf{limitations} of this relation, and also presents its generalization to multi-level systems. This work was first published as a preprint on Zenodo (DOI: https://doi.org/10.5281/zenodo.18624328) before being submitted to aiXiv. The content of this submission is consistent with the Zenodo version.

Keywords