Quantum effective action for dissipative semiclassical dynamics

Using the quantum effective action in the Schwinger-Keldysh formalism, we derive quantum corrections to the semiclassical Langevin dynamics of a dissipative system governed by a macroscopic degree of freedom. We discuss the connection with the Ehrenfest theorem and show that, in the low-temperature and weak-damping regime, quantum corrections are determined by the zero-point energy of fluctuations evaluated at the classical underdamped frequency, closely paralleling the conservative case. We apply these general results to the resistively and capacitively shunted superconducting Josephson junction and to an elongated bosonic junction, where quantum corrections can reach the percent level under realistic conditions.