Qubit dynamics of ergotropy and environment-induced work

We investigate the dynamics of ergotropy in open systems under Markovian and non-Markovian evolutions. In this scenario, we begin by formulating the ergotropy of an arbitrary qubit state in terms of energy and coherence. Thus we determine the conditions for ergotropy freezing and ergotropy sudden death as a consequence of the system -bath interaction. In order to use ergotropy as a resource for energy extraction in the form of work in an open -system scenario, we adopt the entropy -based formulation of quantum thermodynamics. In this approach, the work gains an additional environment -induced component, which may be present even for constant Hamiltonians. We then establish an analytical relationship between the environment -induced work and ergotropy, providing an interpretation of environment -induced work in terms of variation of ergotropy. In particular, energy transfer by environment -induced work can be performed up to a limit, which is governed by the energy cost to transit between the initial and final passive states of the quantum dynamics. We illustrate these results for qubit states evolving under nondissipative and dissipative quantum processes.