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

Abstract

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 $T_{\mathrm{eff}}$, 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.

17 pages, 2 figures; invited review article published in AAPPS Bulletin

References in corpus (4)

• Symmetries of generating functionals of Langevin processes with colored multiplicative noise
• The quantum Fokker-Planck equation of stochastic inflation
• A path integral approach to the Langevin equation
• Noise in resistively shunted Josephson junctions