Thermalization-Induced Entropy-Rate Representation of Internal Energy in a Damped Quantum Oscillator
We study a harmonic oscillator weakly coupled to a thermal bath and evolving under the standard quantum-optical master equation. For zero-mean Gaussian states, entropy alone does not determine the internal energy because the latter also depends on nonequilibrium Gaussian structure. We show, however, that thermal relaxation yields the exact representation $ E=E(S,\dot S) $ for every mixed Gaussian state evolving under this thermal quantum-optical master equation. Explicitly, \[ E(S,\dot S)= \frac{\omega\nu(S)}{\nu_{\rm th}} \left[ \nu(S)+ \frac{\dot S}{\gamma S'(\nu(S))} \right]. \] Once the thermal GKLS equation is assumed, no further expansion in the damping rate, entropy rate, or distance from equilibrium is made. The entropy rate is not an alternative instantaneous state coordinate: it depends on the open-system dynamics and therefore carries information about the coupling to the bath. This is exposed by the isolated limit, where $\dot S=0$ for all Gaussian states and the representation no longer determines the energy. The result thus identifies an exact thermalization-induced dynamical energetic relation, distinct from rate dependence generated by an external driving protocol. We also describe an operational test based on time-resolved purity and energy measurements. At fixed frequency the bath only relaxes pre-existing squeezing. We therefore also consider frequency modulation, which can drive an initially thermal state away from the instantaneous Gibbs family by generating off-Gibbs Gaussian structure. Within a local Gaussian continuation, the bath then relaxes this driven deformation, providing a dynamical realization of the entropy-rate dependence.