Does canonical quantization lead to GKSL dynamics?

TL;DR

The paper addresses whether Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) dynamics with detailed balance can be derived from a generalized classical Brownian motion via canonical quantization. It introduces a generalized Brownian model with friction in both position and momentum, coupled to multiple heat baths, and derives a generalized Kramers equation whose stationary state is the thermal state $\hat{\rho}^* \propto e^{-\beta \hat{H}}$ under appropriate conditions. Canonical quantization then yields a quantum master equation for the density operator; for the harmonic oscillator, the dynamics can be cast in a GKSL form with detailed balance when parameters satisfy specific constraints, notably $\Gamma_{(i),1}/\Gamma_{(i),2}=e^{\beta_i \hbar \omega_i/2}$. This work highlights a quantum-classical correspondence in thermal relaxation and offers a framework to study decoherence and nonlocal dissipation in open quantum systems, while leaving open how to extend the GKSL derivation to more general Hamiltonians.

Abstract

We introduce a generalized classical model of Brownian motion for describing thermal relaxation processes which is thermodynamically consistent. Applying the canonical quantization to this model, a quantum equation for the density operator is obtained. This equation has a thermal equilibrium state as its stationary solution, but the time evolution is not necessarily a Completely Positive and Trace-Preserving (CPTP) map. In the application to the harmonic oscillator potential, however, the requirement of the CPTP map is shown to be satisfied by choosing parameters appropriately and then our equation reproduces a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation satisfying the detailed balance condition. This result suggests a quantum-classical correspondence in thermal relaxation processes and will provide a new insight to the study of decoherence.