**1. Introduction**

Besides its importance for nanoscience, nanotechnology and quantum engineering in general, quantum thermodynamics has a particular status in contemporary physics. It sheds fresh light on both classical and quantum theories. For instance, the quantum-mechanical violation of the law of equipartition of energy seems to require careful reexaminations of even the isothermal process of the ideal gas. It is also the case for the connection between the adiabatic and isoentropic processes. Classical mechanics is described by a set of dynamical variables, whereas quantum mechanics contains not only dynamical variables but also the Hilbert space. Accordingly, quantum mechanics may lead to novel concepts that have no counterparts in classical thermodynamics. The dephasing bath [1] and energy bath [2,3] can be thought of as typical such examples. Therefore, it is of significance to elucidate the implications of such baths to thermodynamics.

In the present paper, we develop a discussion about the isoenergetic process associated with the energy bath within the framework of finite-time quantum thermodynamics. We employ a single Pauli spin in a time-dependent magnetic field and describe its nonequilibrium subdynamics based on the Lindblad equation [4,5], which is known to be unique in the Markovian approximation. It is shown how the Lindbladian operators can be determined without detailed information about the interaction between the subsystem and the energy bath. The resulting subdynamics is seen to be unital [6] and accordingly has a non-negative entropy production rate. The power output and work done are evaluated in the leading order.

The paper is organized as follows. In Section 2, a succinct explanation is presented about the concept of weak invariants, which is directly connected to the isoenergetic process. In Section 3, the Lindbladian operators are determined for the Pauli spin in a time-dependent magnetic field, and the entropy production rate under the Lindblad equation thus obtained is discussed. Then, based on the result in Section 3, finite-time thermodynamics of the power output and work in the leading order is studied in Section 4. Section 5 is devoted to concluding remarks.
