Extended system-bath entanglement theorem with multiple baths in the presence of external fields

The system-bath entanglement theorem (SBET) was established in terms of linear response functions [Du et al., J. Chem. Phys. 152, 034102 (2020)] and generalized to correlation functions [Su et al., J. Chem. Phys. 160, 084104 (2024)] in our previous studies. This theorem connects the entangled system-bath properties to the local system and bare-bath ones. In this work, we extend the SBET to field-dressed conditions with multiple baths at different temperatures. As in reality, the external fields may interact with not only the system but also environments. The extended SBET facilitates, for example, photo-acoustic, photo-thermal, pump-probe related studies. The theorem under the field-free condition (multiple baths) and its counterpart in the classical limit is also presented.