  The system-bath entanglement theorem (SBET) was established in terms of
linear response functions [J. Chem. Phys. 152, 034102 (2020)] and generalized
to correlation functions [arXiv: 2312.13618 (2023)] in our previous works. This
theorem connects the entangled system-bath properties to the local system and
bare bath ones. In this work, firstly we extend the SBET to field-dressed
conditions with multiple bosonic Gaussian environments at different
temperatures. Not only the system but also environments are considered to be of
optical polarizability, as in reality. With the aid of the extended SBET
developed here, for the evaluation of the nonlinear spectroscopy such as the
pump-probe, the entangled system-bath contributions can be obtained upon
reduced system evolutions via certain quantum dissipative methods. The extended
SBET in the field-free condition and its counterpart in the classical limit is
also presented. The SBET for fermionic environments is elaborated within the
transport scenarios for completeness.
