Ergotropy from Geometric Phases in a Dephasing Qubit

Abstract

We analyze the geometric phase and dynamic phase acquired by a qubit coupled to anenvironment through pure dephasing, establishing a direct connection between phase accu-mulation and ergotropy. We show that the dynamic phase depends solely on the incoherentergotropy, reflecting its purely energetic origin. In contrast, the geometric phase exhibitsa nontrivial dependence on both the coherent and incoherent contributions to the totalergotropy, encoding the interplay between coherence, dissipation, and energy extraction.By performing a perturbative expansion in the qubit–environment coupling strength, wedemonstrate that, in the weak-coupling and long-time regime, the geometric phase becomesdetermined exclusively by the incoherent ergotropy, which coincides with the asymptoticvalue of the total ergotropy reached under decoherence. These results provide a clearphysical distinction between dynamic and geometric phases in open quantum systemsand establish geometric phases as sensitive probes of energetic resources. Furthermore, insuperconducting circuit implementations, our findings suggest that the ergotropy of atwo-level system could be inferred indirectly from geometric-phase measurements usingstandard techniques such as quantum state tomography.