Local ergotropy and its fluctuations across a dissipative quantum phase transition

We investigate a two-qubit open Rabi model, focusing on local ergotropy--the maximum extractable work by acting solely on the two qubits--within a parameter regime where a Berezinskii-Kosterlitz-Thouless dissipative phase transition occurs. First, we aim to define a protocol for charging, storing, and discharging the two-qubit system, interpreted as the working principle of an open quantum battery. Second, we examine the impact of the phase transition on ergotropy and identify potential markers. To achieve these goals, we construct an ad-hoc charging unitary operator, leveraging our knowledge of the ground state near the transition to bring it into a decoherence-free state during storage. Using state-of-the-art numerics based on matrix product state representation, we reveal that high couplings to an external bath approximately double the local ergotropy immediately post-charging. Over time we observe oscillatory behaviors in ergotropy and its fluctuations, which undergo significant changes near the transition, signaling its occurrence. Furthermore, we optimize local ergotropy over time using a physically inspired ansatz, enabling work extraction at a generic time (local ergotropy never reaches zero). Our work proposes a tunable, experimentally realizable protocol for work extraction, leveraging decoherence-free states and phase transitions. Additionally, it sheds light on the complex interaction between local ergotropy and quantum phase transitions.