  The Generalized Langevin Equation (GLE) can be derived from a particle-bath
Hamiltonian, in both classical and quantum dynamics, and provides a route to
the (both Markovian and non-Markovian) fluctuation-dissipation theorem (FDT).
All previous studies have focused either on particle-bath systems with
time-independent external forces only, or on the simplified case where only the
tagged particle is subject to the external time-dependent oscillatory field.
Here we extend the GLE and the corresponding FDT for the more general case
where both the tagged particle and the bath oscillators respond to an external
oscillatory field. This is the example of a charged or polarisable particle
immersed in a bath of other particles that are also charged or polarizable,
under an external AC electric field. For this Hamiltonian, we find that the
ensemble average of the stochastic force is not zero, but proportional to the
AC field. The associated FDT reads as $\langle
F_P(t)F_P(t')\rangle=mk_BT\nu(t-t')+(\gamma e)^2E(t)E(t')$, where $F_{p}$ is
the random force, $\nu(t-t')$ is the friction memory function, and $\gamma$ is
a numerical prefactor.
