Adiabatic elimination and Wigner function approach in microscopic derivation of Open Quantum Brownian Motion

Open Quantum Brownian Motion (OQBM) is a new class of quantum Brownian motion in which the dynamics of the Brownian particle depend not only on interactions with a thermal environment but also on the state of its internal degrees of freedom. For an Ohmic bath spectral density with a Lorentz-Drude cutoff frequency at a high-temperature limit, we derive the Born-Markov master equation for the reduced density matrix of an open Brownian particle in a harmonic potential. The resulting master equation is written in phase-space representation using the Wigner function, and due to the separation of associated timescales in the high-damping limit, we perform adiabatic elimination of the momentum variable to obtain OQBM. We numerically solve the derived master equation for the reduced density matrix of the OQBM for Gaussian and non-Gaussian initial distributions. In each case, the OQBM dynamics converge to several Gaussian distributions. To gain physical insight into the studied system, we also plotted the dynamics of the off-diagonal element of the open quantum Brownian particle and found damped coherent oscillations. Finally, we investigated the time-dependent variance in the position of the OQBM walker and observed a transition between ballistic and diffusive behavior.

💡 Research Summary

This paper presents a microscopic derivation of Open Quantum Brownian Motion (OQBM), a recently introduced class of quantum Brownian motion in which the particle’s dynamics are influenced not only by a thermal environment but also by an internal quantum degree of freedom. The authors consider a single particle of mass m confined in a one‑dimensional harmonic potential with frequency ω, and coupled to a two‑level system (TLS) with transition frequency Ω. The total Hamiltonian consists of three parts: the system Hamiltonian (\hat H_S = \hat p^2/(2m) + (m\omega^2/2)\hat x^2 + (\hbar\Omega/2)\hat\sigma_z); a bath of independent harmonic oscillators (\hat H_B = \sum_n