Ergotropy from Geometric Phases in a Dephasing Qubit

Abstract

We analyze the geometric phase and dynamic phase acquired by a qubit coupled to an environment through pure dephasing, establishing a direct connection between phase accumulation and ergotropy. We show that the dynamic phase depends solely on the incoherent ergotropy, reflecting its purely energetic origin. In contrast, the geometric phase exhibits a nontrivial dependence on both the coherent and incoherent contributions to the total ergotropy, encoding the interplay between coherence, dissipation, and energy extraction. By performing a perturbative expansion in the qubit-environment coupling strength, we demonstrate that, in the weak-coupling and long-time regime, the geometric phase becomes determined exclusively by the incoherent ergotropy, which coincides with the asymptotic value of the total ergotropy reached under decoherence. These results provide a clear physical distinction between dynamic and geometric phases in open quantum systems and establish geometric phases as sensitive probes of energetic resources. Furthermore,~in superconducting circuit implementations, our findings suggest that the ergotropy of a two-level system could be inferred indirectly from geometric-phase measurements using standard techniques such as quantum state tomography.

Figures (6)

• Figure 1: Dephasing factor $F(t)$ of the dephasing model. On the left, we have varied the rate $\gamma_0/\Omega$ for a fixed initial quantum state ($\theta=\pi/3$). On the right: we also show the coherent ergotropy ${\cal E}_c(t)$ compared to the dephasing factor for an initial angle and a fixed environment coupling ($\gamma_0/\Omega=0.05$).
• Figure 2: Geometric phase for different initial states under different dephasing environments. On the left, we show the GP acquired for several times cycles under a dephasing model for different values of $\gamma_0$, considering an initial state superposition state with $\theta=\pi/3$. It is easy to see that the GP remains robust since all curves are superposed. On the right panel, we show the GP acquired by the system under a dephasing model with $\gamma_0/\Omega=0.05$ for different initial angles.
• Figure 3: Geometric phase acquired in a cycle ($T=2\pi/\Omega$) showing different dependencies : on the left and in the middle panels, we show the relation among the GP and the coherent ergotropy and the incoherent ergotropy, correspondingly, for different angles (lateral bar shows colors for $\theta$). On the right is the GP as a function of the angles for one cycle evolution. The parameters used are as follows: $\gamma_0/\Omega=0.05$, $\theta_0=\pi/3$.
• Figure 4: On the left: we present ${\cal E}_c(t)$ as function of the number of cycles elapsed for $\gamma_0/\Omega=0.01$ and different initial states. On the right: we show the coherent ergotropy (brown bars) ${\cal E}(T)$ (on the left y-axis) and the GP acquired (on the right y-axis) after 20 cycles ($T=20\tau$). Both quantities are plotted as function of the initial state ($\theta$). ${\cal E}_c(T)=1$ for all T, since there are no coherences to destroy. The green bars represent the incoherent ergotropy (that is not modified by the environment) and gray bars are the coherent ergotropy in a unitary evolution. As a benchmark, we present the unitary GP with gray small dots. $\gamma_0/\Omega=0.01$.
• Figure 5: $\Delta \Phi_g(T)=\Phi_{\rm exact}- \Phi_{\rm approx}$. On the left: $\Delta \Phi_g(T)$ for a zero-T dephasing model as a function of the different rate $\gamma_0/\Omega$ for a different initial angles. On the right: $\Delta \Phi_g(T)$ under a zero-T dephasing model as function of the initial angles for different $\gamma_0/\Omega$.