  In traditional open quantum systems, the baths are usually traced out so that
only the system information is left in the equations of motion. However, recent
studies reveal that using only the system degrees of freedom can be
insufficient. In this work, we develop a stochastic c-number Langevin equation
method which can conveniently access the bath information. In our method, the
studied quantities are the expectation values of operators which can contain
both system operators and bath operators. The dynamics of the operators of
interest is formally divided into separate system and bath parts, with
auxiliary stochastic fields. After solving the independent stochastic dynamics
of the system part and the bath part, we can recombine them by taking the
average over these stochastic fields to obtain the desired quantities. Several
applications of the theory are highlighted, including the pure dephasing model,
the spin-boson model, and an optically excited quantum dot coupled to a bath of
phonons.
