  Heat and work for quantum systems governed by dissipative master equations
with a time-dependent driving field were introduced in the pioneering work of
Alicki [J. Phys. A 12, L103 (1979)]. Alicki's work was in the Schroedinger
picture; here we extend these definitions to the Heisenberg and interaction
pictures. We show that in order to avoid consistency problems, the full time
derivatives in the definitions for heat flux and power (work flux) should be
replaced by partial time derivatives. We also present an alternative approach
to the partitioning of the energy flux which differs from that of Alicki in
that the instantaneous interaction energy with the external field is not
included directly. We then proceed to generalize Alicki's definition of power
by replacing the original system and its external driving field with a larger,
bipartite system, governed by a time-independent Hamiltonian. Using the
definition of heat flux and the generalized definition of power, we derive the
first law of thermodynamics in differential form, both for the full bipartite
system and the partially traced subsystems. Although the second law (Clausius
formulation) is satisfied for the full bipartite system, we find that in
general there is no rigorous formulation of the second law for the partially
traced subsystem unless certain additional requirements are met. Once these
requirements are satisfied, however, both the Carnot and the Clausius
formulations of the second law are satisfied. We illustrate this thermodynamic
analysis on both the simple Jaynes-Cummings model (JCM) and an extended
dissipative Jaynes-Cummings model (ED-JCM), which is a model for a quantum
amplifier.
