From the Bloch equation to a thermodynamically consistent master equation

Abstract: The Bloch equation that set the foundation for open quantum systems, was conceived by pure physical reasoning. Since then, the Lindblad (GKLS) form of a quantum master equation, its most general mathematical representation, became an established staple in the open quantum systems toolbox. It allows to describe a multitude of quantum phenomena, however its universality comes at a cost - without additional constraints, the resultant dynamics are not necessarily thermodynamically consistent, and the equation itself lacks an intuitive interpretation. We present a mathematically equivalent form of the Lindblad master equation under a single constraint of strict energy conservation. The "elemental Bloch" equation separates the system dynamics into its elemental parts, making an explicit distinction between thermal mixing, dephasing, and energy relaxation, and thus reinstating the physical intuition in the equation. We derive the equation for a many-level system by accounting for all relevant transitions between pairs of levels. Finally, the formalism is illustrated by calculating the fixed point of the dynamics and exploring the conditions for canonical invariance in quantum systems.