  We consider a particle trapped by a generic external potential and under the
influence of a quantum-thermal Ohmic bath. Starting from the Langevin equation,
we derive the corresponding Schwinger-Keldysh action. Then, within the
path-integral formalism, we obtain both the semiclassical Fokker-Planck
equation and the quantum Fokker-Planck equation for this out-of-equilibrium
system. In the case of an external harmonic potential and in the underdamped
regime, we find that our Fokker-Planck equations contain an effective
temperature $T_{\rm eff}$, which crucially depends on the interplay between
quantum and thermal fluctuations in contrast to the classical Fokker-Planck
equation. In the regime of high temperatures, one recovers the classical
Fokker-Planck equation. As an application of our result, we also provide the
stationary solution of the semiclassical Fokker-Planck equations for a
superconducting Josephson circuit and for a Bose Josephson junction, which are
experimentally accessible.
