Strong-coupling energetics of a Brownian quantum oscillator: the mean energy, its interpretation, and low-temperature behavior

In this talk, I will discuss the notion of thermally averaged mean energy of a quantum harmonic oscillator strongly coupled to a heat bath, defined as the expectation value of the bare-system Hamiltonian in the canonical Gibbs state of the composite system, i.e., the system and bath taken together with their interactions. This quantity differs fundamentally from the thermodynamic internal energy obtained from the reduced partition function and provides an alternative perspective on the strong-coupling energetics. Using the Brownian quantum oscillator as a paradigmatic example, I will show how this mean energy can be consistently interpreted within the frameworks of quantum thermodynamics and stochastic energetics.

Based on the fluctuation–dissipation theorem, I will first show how the Lehmann–Kubo representation of the generalized susceptibility allows one to interpret the mean energy as the bare oscillator's energy contained within the dressed eigenmodes of the composite system. I will then show, using the quantum Langevin equation, that the mean energy obeys an exact energy-balance relation consistent with the framework of stochastic energetics, even in the presence of non-Markovian dissipation.

Finally, I will discuss analytical results for Ohmic dissipation with a Drude cutoff and show that, at low temperatures, the thermal part of the mean energy exhibits the same universal power-law behavior as the thermodynamic internal energy. The remaining difference between these two notions of energy is a temperature-independent contribution originating from system-bath correlations, highlighting the persistent role of the environment deep into the quantum regime.