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

TL;DR

This paper derives quantum and semiclassical Fokker-Planck equations for a particle in a potential coupled to a quantum-thermal Ohmic bath, revealing an effective temperature that bridges quantum and classical regimes, with applications to Josephson circuits.

Contribution

It introduces a unified path-integral framework to derive Fokker-Planck equations incorporating quantum effects and applies it to superconducting and Bose Josephson junctions.

Findings

Effective temperature $T_{\rm eff}$ depends on quantum and thermal fluctuations.

Recovers classical Fokker-Planck at high temperatures.

Provides stationary solutions for Josephson systems.

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 $T_{\rm eff}$, 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.