c-number Quantum Generalised Langevin Equation for an open system

We derive a $c-$number Generalised Langevin Equation (GLE) describing the evolution of the expectation values $\left\langle x_{i}\right\rangle_{t}$ of the atomic position operators $x_{i}$ of an open system. The latter is coupled linearly to a harmonic bath kept at a fixed temperature. The equations of motion contain a non-Markovian friction term with the classical kernel {[}L. Kantorovich - PRB 78, 094304 (2008){]} and a zero mean \emph{non-Gaussian} random force with correlation functions that depend on the initial preparation of the open system. We used a density operator formalism without assuming that initially the combined system was decoupled. The only approximation made in deriving quantum GLE consists in assuming that the Hamiltonian of the open system at time $t$ can be expanded up to the second order with respect to operators of atomic displacements $u_{i}=x_{i}-\left\langle x_{i}\right\rangle_{t}$ in the open system around their exact atomic positions $\left\langle x_{i}\right\rangle_{t}$ (the "harmonisation" approximation). The noise is introduced to ensure that sampling many quantum GLE trajectories yields exactly the average one. An explicit expression for the pair correlation function of the noise, consistent with the classical limit, is also proposed. Unlike the usually considered quantum operator GLE, the proposed $c-$number quantum GLE can be used in direct molecular dynamic simulations of open systems under general equilibrium or non-equilibrium conditions.