Open Quantum Brownian Motion (OQBM) was introduced as a scaling limit of discrete-time open quantum walks. This limit defines a new class of quantum Brownian motion, which incorporates both the external and internal degrees of freedom of the Brownian particle. We consider a weakly driven Brownian particle confined in a harmonic potential and dissipatively coupled to a thermal bath. Applying the rotating wave approximation (RWA) to the system-bath interaction Hamiltonian, we derive a completely positive Born-Markov master equation for the reduced dynamics. We express the resulting master equation in the coordinate representation and, utilizing the adiabatic elimination of fast variables, derive a completely positive hybrid quantum-classical master equation that defines OQBM. We illustrate the resulting dynamics using examples of initial Gaussian and non-Gaussian distributions of the OQBM walker. Both examples reveal the emergence of Gaussian distributions in the limiting behavior of the OQBM dynamics, which closely matches that of the standard OQBM. With the help of the obtained OQBM master equation, we derive the equations for the
n-th moments and the cumulants of the position distribution of the open Brownian walker. We subsequently solve these equations numerically for Gaussian initial distributions across various parameter regimes. Notably, we find that the third-order cumulant is nonzero, indicating that the dynamics' intrinsic generator is non-Gaussian.
We consider Quantum Brownian motion of a toy model, which is a harmonic oscillator that interacts with environment consisting of harmonic oscillators through both qQ and pP types of coupling. At first, we formally derive exact master equation following Unruh and Zurek's method. Subsequently we estimate the behavior of the coefficients in the master equation at initial stage and at late time stage under weak coupling approximation. We find that, as the model with qQ coupling only, the system in supraohmic environment is less dissipative and less diffusive beyond the cutoff time scale than the system in ohmic environment. The cause is due to the weakness of spectral densities of environment in infrared sector. We also find that the remarkable feature of the coexistence of qQ and pP coupling is steep dependence of dissipative coefficients on time at initial stage compared with the case of qQ coupling only or pP coupling only. Further we find that the sign of the dissipation coefficient at initial stage can be negative, that is, the system can be anti-dissipative.