Zeroth Law in Quantum Thermodynamics at Strong Coupling: `in Equilibrium, not `Equal Temperature

The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of `equal temperature (ET), or `in equilibrium (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result cite{NEqFE} for Gaussian quantum systems shows that an effective temperature $T^{*}$ can be defined at all times during a systems nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the systems $T^*$ is slightly higher than the bath temperature $T_{textsc{b}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature $T_{textsc{b}}$ will not have the same equilibrated temperature as the first object. Thus $ET eq EQ $ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that `in equilibrium is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that `temperature appearing in thermodynamic relations becomes universally defined and makes better physical sense.