Spectral thermodynamics of a soliton heat engine

We demonstrate a thermodynamic engine whose working substance is a sine-Gordon soliton in a heterogeneous current-driven Josephson junction. We show that solitons can act as thermodynamic working substances whose internal spectral structure enables energy conversion beyond conventional few-level engines. By dynamically deforming the soliton using a controllable dipole current, the internal bound-state spectrum of the soliton can be engineered in time, enabling a finite-time Carnot-like cycle based on spectral control, in close analogy with quantum heat engines. Mapping the instantaneous nonlinear field configuration to an effective Schrödinger operator, we reveal how bound states appear, approach the continuum threshold, and disappear during the cycle. Comparing three thermodynamic descriptions (full nonlinear field dynamics, a coarse-grained mesoscopic model, and a two-level spectral model), we show that few-level descriptions systematically underestimate the engine performance. The enhanced efficiency arises from the extended nature of the soliton, whose internal spectral degrees of freedom provide additional energy storage and transfer channels. Our results reveal a general thermodynamic principle: extended nonlinear excitations with particle-like behavior can serve as tunable working media, whose internal spectral degrees of freedom provide additional reversible channels for energy storage and transfer beyond those of few-level systems.