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Limitations of the Markovian approximation in the harmonic oscillator

The quantum fluctuation-dissipation theorem is a central ingredient in the construction of quantum dynamics of Brownian motion which necessarily is non-Markovian. Yet, often Markovian approximations to quantum dynamics are studied in the literature. In this work, we investigate the limitations of the Markovian approximation within two paradigmatic models describing a single damped harmonic oscillator. These models are governed by distinct quantum Langevin equations, although both are constructed to satisfy the same set of phenomenological criteria: the canonical commutation relations between position and momentum, the Kubo response relation, the virial theorem, and the equilibrium quantum variance. The limitations of the Markovian approximation are underscored by the classical limit, violations of the Ehrenfest theorem, the breakdown of complete and simple positivity in the reconstructed master equations, and anomalies in thermalisation behaviour. Further phenomenological differences between the two models are illustrated through their quantum relaxation dynamics and phase diagrams, derived from their reinterpretation as mean-field approximations of a many-body interacting magnet. Our analysis explicitly reveals intrinsic inconsistencies introduced by the Markovian approximation, emphasising the need for non-Markovian frameworks for a consistent description of open quantum dynamics.