Spectral engineering of a soliton heat engine

We demonstrate a heat engine whose working substance is a sine–Gordon soliton in a heterogeneous current-driven Josephson junction. We show that solitons can serve as working substances whose internal spectral structure provides energy-storage and redistribution channels beyond those captured by conventional few-level descriptions. 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 the deformation of the trapped fluxon generates spectral configurations in which bound states appear, approach the continuum threshold, and disappear during the cycle. We compare the fluxon dynamics using three complementary descriptions: the complete nonlinear field, a coarse-grained mesoscopic approach, and a two-level spectral model. The full-field energy balance reveals nearly reversible mechanical energy recovery under slow driving, while the reduced descriptions show how progressively coarse-graining the extended soliton changes the amount of energy storage and redistribution that is explicitly resolved. Our results establish dynamically controllable internal soliton spectra as a resource for constructing finite-time spectral cycles in extended nonlinear working media.