Exponentially Accurate Work Extraction by Swapping Detuned Bosonic Modes

A temperature imbalance between two thermal oscillators can yield exponentially little work, even under arbitrary unitary control. We determine when frequency detuning makes a simple mode swap nearly optimal, by bounding the entire extra work available beyond Gaussian operations. With detuning proportional to the temperature bias near equilibrium, this absolute improvement is exponentially small in the inverse bias. The bound separates exponentially weak extraction at resonance from a detuning window with quadratic work, where the swap captures the leading resource. In this limit, within that window, the accessible fraction of the energy excess above a Gibbs state of equal entropy follows a temperature-independent parabola in detuning normalized by the bias. Inside the window, any fixed fraction below 100% of the leading work can be extracted with an occupation cutoff independent of the bias. The finite bound determines when more general control offers negligible additional work.