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We consider a particle trapped by a generic external potential and under theinfluence of a quantum-thermal Ohmic bath. Starting from the Langevin equation,we derive the corresponding Schwinger-Keldysh action. Then, within thepath-integral formalism, we obtain both the semiclassical Fokker-Planckequation and the quantum Fokker-Planck equation for this out-of-equilibriumsystem. In the case of an external harmonic potential and in the underdampedregime, we find that our Fokker-Planck equations contain an effectivetemperature $T_{\rm eff}$, which crucially depends on the interplay betweenquantum and thermal fluctuations in contrast to the classical Fokker-Planckequation. In the regime of high temperatures, one recovers the classicalFokker-Planck equation. As an application of our result, we also provide thestationary solution of the semiclassical Fokker-Planck equations for asuperconducting Josephson circuit and for a Bose Josephson junction, which areexperimentally accessible.

Fokker-Planck equations for a trapped particle in a quantum-thermal Ohmic bath: general theory and applications to Josephson junctions